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<p>I am running a linear mixed model and want to check its assumptions. The model I run is comparing if males and females behave differently over time (timeclass=1,2,3,4):</p>
<pre><code> x <- lme(response ~ timeclass*sex, random = ~ 1|subject, method="ML", data=dat)
</code></pre>
<p>The code to create the plots: </p>
<pre><code>plot(x)
plot(x, response ~ fitted(.) | sex, abline = c(0,1))
qqnorm(x,~resid(.)|timeclass)
qqnorm(x,~resid(.)|sex)
hist((resid(x) - mean(resid(x), na.rm=T)) / sd(resid(x), na.rm=T), freq=F); curve(dnorm, add = TRUE)
</code></pre>
<p>This is the output I get:</p>
<p><img src="http://i.stack.imgur.com/4gYnC.png" alt="residual plot">
<img src="http://i.stack.imgur.com/Pykhd.png" alt="linear relationship males and females">
<img src="http://i.stack.imgur.com/nm6aJ.png" alt="qqplot for sex">
<img src="http://i.stack.imgur.com/IyXNj.png" alt="qqplot for timeclass">
<img src="http://i.stack.imgur.com/66naf.png" alt="histogram"></p>
<p>The plots except the residual plot seem to show that there seems to be a linear relationship between the fitted values per sex and the response and that the errors are relatively normally distributed in both sex and across the four time classes. However, the first plot has a very clear diamond shape, almost so clear that it seems values are cut off in the four triangular angles. </p>
<p>Can someone help me understand what is going on and what this means? Is the model I run valid based on the plots for assumptions?</p> | 30,166 | [
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<p>Please help!
I have recently been criticized for using pairwise comparisons to explain all three levels of a factor within a negative binomial GLM rather than all levels at once. I was told that it is "long-winded" and "uneccessary". I was under the impression that in GLMs one cannot bulk all levels of a factor together to obtain a test statistic and corresponding p-value.</p>
<p>Obviously if a factor is "insignificant" at any level then carrying out a post-hoc analysis is pointless. My levels all have there own p-values therefore I discussed these values from the below global model. I was told to do an ANOVA instead which I don't believe is suitable for overdispersed, zero-inflated data.</p>
<p>p-value for all levels of a factor anyone?</p>
<p>(Below, lower field layer 0, upper field layer 1 and change1 is in intercept)</p>
<pre><code> Deviance Residuals:
Min 1Q Median 3Q Max
-2.4284 -0.7956 -0.3862 0.4045 2.4233
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 4.3410884 1.8219786 2.383 0.01719 *
Height 0.0373584 0.0119929 3.115 0.00184 **
Width -0.0007891 0.0008246 -0.957 0.33859
MeanMin -0.1731877 0.1404434 -1.233 0.21752
as.factor(Site_Treat)2 -0.4080256 0.2480438 -1.645 0.09998 .
as.factor(Change)2 -0.4940398 0.1755487 -2.814 0.00489 **
as.factor(Change)3 -0.1613766 0.1763677 -0.915 0.36019
as.factor(Lower_Field_Layer)1 0.4873488 0.2931585 1.662 0.09643 .
as.factor(Lower_Field_Layer)2 -0.3292409 0.3717863 -0.886 0.37585
as.factor(Upper_Field_Layer)2 -0.0081040 0.3257734 -0.025 0.98015
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for Negative Binomial(4.7795) family taken to be 1)
Null deviance: 96.392 on 46 degrees of freedom
Residual deviance: 47.968 on 37 degrees of freedom
AIC: 403.94
</code></pre>
<p>Best wishes,
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<p>I have a time series that follow an MA(1) process , I do non-overlapping aggregation with aggregation level $m=2,3,\dots$, and then I calculate autocorrelation of basic and aggregated series, for negative value of moving average parameter, aggregated autocorrelation becomes close to zero, means that the aggregated series want to be like white noise, I was wondering if anybody can explain the reasons behind this behavior?</p>
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<p>Suppose I have two sets $X$ and $Y$ and a joint probability distribution over these sets $p(x,y)$. Let $p(x)$ and $p(y)$ denote the marginal distributions over $X$ and $Y$ respectively. </p>
<p>The mutual information between $X$ and $Y$ is defined to be:
$$I(X; Y) = \sum_{x,y}p(x,y)\cdot\log\left(\frac{p(x,y)}{p(x)p(y)}\right)$$</p>
<p>i.e. it is the average value of the pointwise mutual information pmi$(x,y) \equiv \log\left(\frac{p(x,y)}{p(x)p(y)}\right)$.</p>
<p>Suppose I know upper and lower bounds on pmi$(x,y)$: i.e. I know that for all $x,y$ the following holds:
$$-k \leq \log\left(\frac{p(x,y)}{p(x)p(y)}\right) \leq k$$</p>
<p>What upper bound does this imply on $I(X; Y)$. Of course it implies $I(X; Y) \leq k$, but I would like a tighter bound if possible. This seems plausible to me because p defines a probability distribution, and pmi$(x,y)$ cannot take its maximum value (or even be non-negative) for every value of $x$ and $y$. </p> | 36,199 | [
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<p>Suppose I have a $95 \%$ confidence interval for $\mu$ with population variance $\sigma^{2}$ known. It will be of the form $$[\bar{X}-1.96 \sigma/\sqrt{n}, \bar{X}+1.96 \sigma/\sqrt{n}]$$ </p>
<p>The statistical interpretation of this is that $95\%$ of the confidence intervals will contain the true mean? I know individually, there is a $50\%$ chance for a confidence interval to contain the mean. But collectively, my above sentence is valid?</p> | 36,200 | [
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<p>Suppose we have two estimators $\alpha_1$ and $\alpha_2$ for some parameter $x$. To determine which estimator is "better" do we look at the MSE (mean squared error)? In other words we look at $$MSE = \beta^2+ \sigma^2$$ where $\beta$ is the bias of the estimator and $\sigma^2$ is the variance of the estimator? Whichever has a greater MSE is a worse estimator? </p> | 36,201 | [
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<p>Curious whether anyone knows a package, or has written an implementation themselves, for conducting instrumental variables regressions using LIML in R. All of the R packages I have seen for IV regressions seem to resort to 2SLS/GMM as opposed to LIML, which may have more desirable finite sample qualities compared to 2SLS <a href="http://press.princeton.edu/titles/6946.html" rel="nofollow">(Hayashi 2000)</a></p>
<p>For additional context, stata's ivregress command includes options to use LIML estimation, and hoping someone has already implemented something similar in R so I don't have to write it myself.</p> | 40,440 | [
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<p>Are there any statistical concepts or theories on how to effectively measure recency to where recent events are given more weight than older ones. I'm creating a logistic regression model and would like to apply an adjustment to various factors based on the events recency.</p>
<p>...or, is it purely up to me to come up with an arbitrary formula?</p>
<p>For example: One of my projects is predicting via logistic regression the performance of pro golfers in upcoming tournaments. Their recent form (how they played last week) is generally more important than how they played six months ago. Are there any specific techniques/approaches that tap into this concept?</p> | 73,739 | [
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<p>For my data, I need to run numerous linear regression analyses, separately for each of my two groups of subjects as well as different conditions. Together, I would have to run approximately 40 regression analyses across the 3 conditions for each group. I'm aware that we should correct for multiple comparisons for the regression coefficients within a model. But in my case, should I also do a similar correction for the p-value associated with the regression model itself (ie: the F test)? </p>
<p>For example, if in one condition, I have to run 25 regression analyses for each group, would it make sense to divide alpha=.05 by 25?</p>
<p>I've been going through different websites and textbooks, but can't seem to find an answer. Any suggestions/help would be appreciated!</p> | 36,205 | [
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<p>I have a surgical database in which there are approximately 100,000 observations and 200 features. Each observation corresponds to a separate patient while the features correspond to either preoperative, perioperative, or postoperative variables (e.g. preoperative labs, length of operation, and days until death). As such some of the features are factors, while others are continuous data types. The database is not "clean" in that many of the features contain NAs because that data was not collected. Each patient has some feature with NAs in it, so when I try to select only observations without NAs that results in 0 observations.</p>
<p>That being all said my end goal is to see whether any of these preoperative variables might be predictive of mortality. Initially I wanted to use L1 regularization (lasso) via glmnet to perform feature selection, however because of the NAs I can't run it. Are there any alternatives or techniques to bypass this problem? I assume that it's more common than not to have a database with NAs, so I wanted to see if you guys could fill me in. Thanks!</p> | 43,183 | [
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<p>We have an events based tracking system for our website, with split testing built-in and we are using <a href="http://www.thumbtack.com/labs/abba/" rel="nofollow">ABBA</a> for the calculations.</p>
<p>The problem comes up when we are doing consecutive split tests. For example, we test X and Y, Y wins, so the next test is Y and Z. We are seeing, most of the time, that the conversion rate of Y differs in the two tests... Sometimes by as much as 50%.</p>
<p>We have a tool to track conversion rate over time, and I've included two graphs of conversion rate between two different tests of new and baseline. You can see that between the two tests, the previous new and the subsequent baseline differ by a significant amount even though they should theoretically be the same.</p>
<p>Both experiments have about 200,000 runs for new and baseline - 400k total - 800k over two experiments, and the P values on both are .0001. I'm just unable to explain why the conversion rate of "new" in test1 was 1.28% but the conversion rate of "baseline" in test2 was .85% - when they are exactly the same content.</p>
<p>I can certainly think of some things that would contribute to something like this: only catching part of a user cycle/time of day variations/seasonal variations/holidays, but my gut tells me that it shouldn't really make a 50% difference.</p>
<p>We have, however, eliminated the possibility of Googlebot and other bots affecting the results, as we don't even record events for bots.</p>
<p>Unfortunately, we don't have anything we can compare with. We do use Google analytics, but we only record our key metrics there, so there's no apples – apples.</p>
<p>The experiments were run immediately after each other, and I'm not aware of any special marketing campaigns (We don't do very much marketing, and no CPC). We are seeing some pretty good changes in traffic from certain countries, and I can definitely look into that to see if it could contribute... But, again, 50%?</p>
<p>Test 1: "New" wins
<img src="http://i.stack.imgur.com/1WSDF.png" alt="Test 1 conversion"></p>
<p>Test2: The old "New" is now "Baseline"...note the mismatch in conversion
<img src="http://i.stack.imgur.com/l1LFv.png" alt="Test 2 conversion"></p> | 36,206 | [
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<p>I sometimes have difficulty understanding the intuition behind some tests. Is it important to know why some test works in a particular situation, or is it enough to learn things at the level "OK, now I can check homoscedasticity" or "I have no idea why we divide by 12 in Mann–Whitney U-test, but trust me, it works"? Or are there books where the authors have explained the ideas behind every test and provided some reasons to "prove" that the tests work? </p> | 73,740 | [
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<p>Principal component analysis <em>can use</em> matrix decomposition, but that is just a tool to get there.</p>
<p>How would you find the principal components without the use of matrix algebra?</p>
<p>What is your objective function (goal), and what are your constraints?</p>
<p>Thanks!</p> | 73,741 | [
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<p>What is the easiest way / method to compute the correlation between two time series that are exactly the same size? I thought of multiplying $(x[t]-\mu_x)$ and $(y[t] - \mu_y)$, and adding up the multiplication. So if this single number was positive, can we say these two series are correlated? I can think of some examples however where a linearly another exponentially growing time series would have no relation to eachother, but the computation above would report they were correlated. </p>
<p>Any thoughts?</p> | 42,713 | [
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<p>Can someone tell me why this is wrong: </p>
<p>I want to estimate the goodness-of-fit for a linear mixed effects model. I regress the fitted values against the raw values and calculate an $R^2$. </p>
<p>I suppose an accepted alternative would be to run the model with just the random effect and no fixed effects and compare AIC scores, but I'm specifically interested in the scenario above.</p> | 36,209 | [
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<p>I have just been reading a paper which uses the following method to calculate parameter uncertainties from a likelihood surface of two parameters.</p>
<p>Step 1) Calculate the likelihood for finding a given pair of parameters given the data at each point on the grid.</p>
<p>Step 2) Normalise the surface to 1. The Likelihood value now give the probability to the parameter given the as L(a|x) is proportional to p(x|a). </p>
<p>I'm not totally convinced by this - not least as the extent of the grid is arbitrary. I'm probably going to ticck the 'forward it on to a statician' box as it's a review and I'm not familiar with this method but I was curious to whether this works under any circumstances. I always assumed Wilkes' theorem was about as well as you could do this.</p> | 73,742 | [
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<p>This is an artificial example. I want to compare the Income of the <em>same</em> group of people (i.e. "pairing") while they were working in US, Spain, and Nigeria. In other words, I use the "US" income as a reference and simply conduct 2 Two-sample Paired T-tests:</p>
<pre><code>t.test(dat[Country=="US",]$Income, dat[Country=="Spain",]$Income, paired=TRUE)
t.test(dat[Country=="US",]$Income, dat[Country=="Nigeria",]$Income, paired=TRUE)
</code></pre>
<p>But how is this different from the following?</p>
<pre><code>pairwise.t.test(dat$Income, dat$Country, paired=TRUE)
</code></pre>
<p>The help for pairwise.t.test() states, "...with corrections for multiple testing". Is that the only difference?</p> | 73,743 | [
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<p>I'm trying to generate many draws (i.e., realizations) of
a Gaussian process $e_i(t)$, $1\leq t \leq T$ with mean 0 and covariance
function $\gamma(s,t)=\exp(-|t-s|)$. </p>
<p>Is there an efficient way to do this that wouldn't involve computing the square
root of a $T \times T$ covariance matrix? Alternatively can
anyone recommend an <code>R</code> package to do this?</p> | 73,744 | [
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<p>We have the following points:</p>
<p>$$ (0,0)(1,51.8)(1.9,101.3)(2.8,148.4)(3.7,201.5)(4.7,251.1)(5.6,302.3)(6.6,350.9)(7.5,397.1)(8.5,452.5)(9.3,496.3)$$</p>
<p>How can we find the best fitting line $y=ax$ through the points? My calculator has the option to find the best fitting line $y=ax+b$ through these points, which is:</p>
<p>$$y = 53.28x + 0.37$$</p>
<p>How can I find the best fitting $y=ax$? It seems to me we can't just remove the $0.37$ without compensating in the $a$?</p> | 36,214 | [
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<p>I am currently running some analyses on a linguistic data set with a mixed effect model. The problem is, I think that one random factor should be excluded while my colleague thinks it should be included. The two options are:</p>
<pre><code>lmer(intdiff ~ stress * vowel_group + (1|speaker) + (1|word), data)
lmer(intdiff ~ stress * vowel_group + (1|speaker), data)
</code></pre>
<p>How do we check which model best fits our data set? It was suggested that I use a likelihood ratio test, but as far as I can tell, there isn't a function in R that can be applied to 2 linear mixed effects models. Is there another way to tell which model is more predictive?</p>
<p>Thanks</p> | 73,745 | [
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<p>I have samples from two groups. Samples from each group follow a Dirichlet distribution. The two Dirichlet distributions have the same mean but different precision. How to find the MLE of mean and two precisions?</p> | 30,246 | [
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<p>Suppose you have 2 distinct features x1 and x2 with discrete values between 1 and 10.</p>
<p>When building a neural network, I have 2 options:</p>
<ol>
<li>Create a new feature x3 = 10*x1 + x2 and use this as the only input.</li>
<li>Use both inputs x1 and x2 for the neural network.</li>
</ol>
<p>How can I decide which is the best option? </p>
<p>Here is a practical example inspired from poker:
You have 7 cards. The first 2 cards are from player 1. The next 2 cards are from player 2 and the last 3 cards are the flop. You want to determine who has the strongest hand.
The two options are:</p>
<ol>
<li>Treat each card as a number from 1 to 52. </li>
<li>Make two inputs: One for the rank (2,3,4,5,6...J,Q,K,A) and one for the suit (club,diamond,heart,spade).</li>
</ol>
<p>More precisely, I want to understand how choosing each option will affect how much training data is needed, how many neurons in the hidden layer I should use and how much time will be needed for training.</p> | 73,746 | [
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<p>I'm reading through this paper on <a href="http://uai.sis.pitt.edu/papers/11/p736-wilson.pdf" rel="nofollow">Generalised Wishart Processes</a> (GWP). The paper calculates the covariances between different random variables (following <a href="http://en.wikipedia.org/wiki/Gaussian_process" rel="nofollow">Gaussian Process</a>) using squared exponential covariance function, i.e., $K(x,x') = \exp\left(-\frac{|(x-x')|^2}{2l^2}\right)$. It then says that this covariance matrix follows GWP.</p>
<p>I used to think that a covariance matrix computed from <strong>linear covariance function ($K(x,x') = x^Tx'$)</strong>, follows Wishart Distribution with appropriate parameters. </p>
<p>My question is, how can we still assume the covariance to follow a Wishart distribution with squared exponential covariance function? Also, in general, what is the necessary condition for a covariance function to produce a Wishart distributed covariance matrix?</p> | 47,211 | [
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<p>I'm trying to analyse a dataset to detect fraudulent insurance claims. Unfortunately, other than basic demographics the rest of the claim is a free format OCR scanned text file made from documents submitted by the patient. These documents include lab tests, hospital bills etc., patient summaries. </p>
<p>Right now, manual coding is one option but expensive. </p>
<p>Is there any chance that applying textual analysis / text-mining might yield some predictive strength? It need not be perfect but even if it can reliably identify high fraud potential claims, that's a plus. </p>
<p>Any ideas how I could model this?</p> | 73,747 | [
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<p>I have two matrices with multiple variables. The two matrices come from same sample.I would call them matrix A and Matrix B.Each of them have many correlated variables .</p>
<p>I would want to evaluate the relationship between matrix A and matrix B taking matrix B as explanatory and matrix A as dependent .
When i looked at the structure of Matrix A .I was able to describe it into 4 factors and matrix B into 3 factors.</p>
<p>I fitted a mancova model using factor scores of matrix A and factor scores of matrix B.</p>
<p>Is this a valid way to do the analysis.</p> | 36,219 | [
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<p>I'm working on a random forest classifier with 10-folds CV to aestimate the hyperparameter 'mtry' (chosen by maximizing AUROC). I decided to pre-split the training set in 8 samples equals in size (about 100 observations each).</p>
<p>I iterated the train process 8 times. At each iteration I added one sample to my actual training set. So, the first train model was fitted using a training set of 100 observations, the second had 200...the last one has the entire original training set (about 800 observations).</p>
<p>At each iteration I measured the missclassification error (FP + FN) for both training and test set in %.</p>
<p>I finally plot this learning curve. On the x axis I have the training set size, on the y axis the missclassification error in % (black for the training set, red for the test set).</p>
<p>The gap between train and test performance suggests me I have high variance in the data (overfitting) but I really cannot understand...why does the training error start so high, then suddenly drop, then start to rise again as training set size increases?</p>
<p>If I have a overfitting problem, how can I address it (close the gap) if I cannot obtain more training examples?</p>
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<p>I have two datasets, $D1$ and $D2$, each containing many samples $S_i$ of 100 numbers, each sample from a different (unknown) population. For example:</p>
<p>$S_1=\{0.125, 0.122, 0.053, \dots\} \\
S_2=\{0.001, 0.013, 0.034, \dots\} \\
\dots $</p>
<p>The are both quite large. $D1$ contains almost 6M samples, and $D2$ contains almost 500K.</p>
<p>I randomly split each sample $S_i$ in two disjoint subsamples of size 50 each, let them be $S_i'$ and $S_i''$. For each subsample I run a t-test for the null hypothesis of mean equal zero, so I have two p-values: $p_i'$ and $p_i''$.</p>
<p>My question of interest is: how often do I observe significance in $S_i''$ if I observed significance in $S_i'$. So, for different significance levels, I plot this proportion:</p>
<p>$\displaystyle\frac
{\#\{p_i' \leq \alpha~and~p_i''\leq\alpha\}}
{\#\{p_i' \leq \alpha\}}$</p>
<p>With dataset $D1$ there is nothing rare: the lower the significance level the smaller the proportion because it is harder to observe significance.
But when I plot it with dataset $D2$, I have this weird behavior in the plot: it kind of goes up with alpha, but then at some point it goes down with large alphas.</p>
<p>I can't figure out why this is happening. The same happens with other tests such as Wilcoxon and Sign. Any ideas?</p>
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<p>I have some data that is bounded between 0 and 1. I have used the <code>betareg</code> package in R to fit a regression model with the bounded data as the dependent variable. My question is: how do I interpret the coefficients from the regression?</p> | 36,223 | [
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<p>I'm using the <code>igraph</code> package in <code>R</code> to analyze network data. I'm currently trying to calculate some centrality measures for the vertices of my graph and the corresponding centralization scores. My network is both directed and weighted.</p>
<pre><code>require(igraph)
set.seed(12152)
m <- expand.grid(from = 1:4, to = 1:4)
m <- m[m$from != m$to, ]
m$weight <- sample(1:7, 12, replace = T)
g <- graph.data.frame(m)
</code></pre>
<p>I have no trouble using the <code>closeness</code> function to obtain the closeness centrality for each vertex:</p>
<pre><code>closeness(g, mode = "in")
closeness(g, mode = "out")
closeness(g, mode = "total")
</code></pre>
<p>However, it appears that the <code>centralization.closeness</code> function from <code>igraph</code> does not work for directed graphs. <code>igraph</code> does include a way to calculate a custom centralization from the individual centrality scores in a graph (the <code>centralize.scores</code> function), but that function requires the user to specify the theoretical maximum of the centrality measure, and it's not obvious to me what that would be in this weighted example (I believe the built-in <code>centralization.closeness.tmax</code> function in <code>igraph</code> assumes an unweighted graph).</p>
<p>Does anyone know how to calculate a centralization score in a weighted graph? Is there a good way to accomplish this in R with <code>igraph</code> or some other package?</p> | 32,214 | [
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<p>Consider a model with a continuous response variable and a categorical explanatory variable. I appreciate that in R, a summary.lm output of an anova on this data gives you rows that represent the mean value of each factor level. The significance stars represent the significance of the difference between the mean of each level and the "intercept", which represents the mean of the first level of the factor.</p>
<p>What I am wondering is what do significance stars on this intercept term represent? Simply that the mean of this particular factor level is different from zero?</p> | 36,225 | [
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<p>I recently learned about Fisher's method to combine p-values. This is based on the fact that p-value under the null follows a uniform distribution, and that $$-2\sum_{i=1}^n{\log X_i} \sim \chi^2(2n), \text{ given } X \sim \text{Unif}(0,1)$$
which I think is genius. But my question is why going this convoluted way? and why not (what is wrong with) just using mean of p-values and use central limit theorem? or median? I am trying to understand the genius of RA Fisher behind this grand scheme.</p> | 73,750 | [
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<p>What is the best method to determine the minimum number of training samples required for a classifier?</p>
<p>I am only comparing one classifier (four class problem), discriminant function analysis (DFA) with different training sample sizes from different data sets. However, I could be considered to be comparing more than one classifier as I am also comparing DFA using the within matrix and separate matrix methods. The data is accelerometer data used to classify animal behaviours. The different data sets are all from the same original data but have had treatments applied, eg smoothed, outlier removed. </p>
<p>I have run 100 interations of randomly selected increasing subsample sizes (10 - 2550) from each data set through each classifier. Averaging the classification accuracy or error rate for each subsample size separately for each data set gives me a learning curve for the classifier on each of the four classes.</p>
<p>Using the area at which the learning curve levels out, I would like to suggest a possible minimum training sample size for each class required for a future study. Rather than visually looking at the learning curve and making a recommendation, is there an appropriate statistical test. Am I talking about data-driven feature selection? I dont need to extrapolate as it looks like I have enough training samples. I dont think ROC curves are applicable as I only have the classification rates from each run of each sub sample sizes rather than predicted membership probabilities which is what I think i need for ROC curves. </p>
<p>I have looked over several posts here which have been useful in advancing my thinking.
This post (<a href="http://stats.stackexchange.com/questions/51490/how-large-a-training-set-is-needed">How large a training set is needed?</a>) relates to my requirement but I dont see it suggesting an appropriate test.</p>
<p>This paper (<a href="http://jmlr.org/papers/volume7/demsar06a/demsar06a.pdf" rel="nofollow">http://jmlr.org/papers/volume7/demsar06a/demsar06a.pdf</a>) suggests the Friedman test with the corresponding post-hoc tests for comparison of more classifiers over multiple data sets. However, it doesn't appear to take into account the increasing sub sample size from the same training data set, rather it uses completely independent data sets.</p> | 73,751 | [
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] |
<p>I would like to know what is the theoretical distribution of the vector of (signed) deviances $d_i$ from a logistic regression. Failing that, is there a way to to transform them unto approximately normally distributed r.v.? </p>
<p>My model for the data generating process is the simplest one, e.g.</p>
<p>$$y_i\sim\mathcal{B}(p_i)$$</p>
<p>where </p>
<p>$$p_i=\pmb x_i'\pmb\theta$$</p>
<p>and $\mathcal{B}$ is the Bernoulli distribution.</p>
<p>so in R code:</p>
<pre><code>n<-1000
p<-5
x<-matrix(rnorm(n*(p-1)),nc=p-1);
y<-cbind(1,x)%*%rep(1,p)
y0<-1/(exp(-y)+1);
y0<-rbinom(n,1,y0)
fit<-glm(y0~x,family="binomial")
y<-residuals(fit,type="deviance")
qqnorm(y)
qqline(y)
</code></pre> | 73,752 | [
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<p>I have a set of data points $\{x_i\}$. These data points are grouped so that (say) $i\in\{1,2,3\}$ is group $A$, $i\in\{4,5,6,7\}$ is group $B$, etc.</p>
<p>I would like to test the null hypothesis of no linear relationship between group means and group variances.</p>
<p>A naive approach would be to calculate, for each group, the sample mean $\overline{x}_g$ and sample variance $S^2_g$, then regress
$$
\overline{x}_g = \beta S^2_g + \varepsilon_g
$$
This doesn't work, since in small samples, $\hat{\beta}$ will be a biased estimate of $\beta$. For example, if $\beta=0$ but the $\{x_i\}$ are very positively skewed, a spuriously high $\hat{\beta}$ will be estimated. This is because</p>
<ul>
<li>Groups that do not contain positive outliers will have low $S^2_g$ and low $\overline{x}_g$</li>
<li>Groups that do contain these outliers will have high $S^2_g$ and high $\overline{x}_g$. </li>
</ul>
<p>Is there a general technique for solving this problem?</p>
<p>(Cross-posted from Math Stack Exchange.)</p> | 73,753 | [
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<p>I have a lot of financial instruments and I need to rank them based on price movement "stability", i.e. instruments with with big outliers should have the lowest ranks and instruments with stable and consistent moves should have the highest ranks. To solve this issue I use the ratio of the <code>Arithmetic Mean</code> to the <code>Standard Deviation</code> for every instrument and then sort them by this ratio. But there is a problem: this method works only roughly and some "bad" instruments have higher ranks than "good" instruments.</p>
<p>Are there are different approaches for ranking?</p>
<p>For data points I use $\ln(\text{High} / \text{Low})$ and period is 750 data points for every instrument.</p> | 36,229 | [
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<p>A co-worker has asked me to help setup a spreadsheet that is used in the analysis of customer care emails. In particular, we are looking at three categories - the assignment of meta-data, content and communication skills.</p>
<p>A typical example question may be:</p>
<p>"How well is the email tailored towards the customers questions?"
"How well did the employee answer the customers questions?"</p>
<p>I'm not a statistician but have conducted a few surveys in the past. After seeing the initial survey, which was basically an ad-hoc collection of questions combined with varying scaling mechanisms for each question. I figured the first step would be to iterate the goals and work back from there; otherwise I'd be basically devising my own methods from scratch, which feels a little flakey. All this led me to re-design the survey using the Likert scale due to it's simplicity in implementation and analysis. </p>
<p>However, without a deep statistical background I'm finding it hard to validate why it is better to use a proven methodology than a collection of different scales and measurements aggregated into a single number.</p>
<p>How can I convince them of the benefits if using a 'methodology'/scale such as Likert rather than various choice of measurements?</p> | 73,754 | [
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<p>Suppose I am building a binary logistic regression model with around 400 variables then how weight of evidence(WOE) can be used in here?
Also, I have known that WOE is helpful in dimension reduction in a model then why not go for factor analysis rather going for WOE?
What should be the size of the number of variables in order to use WOE?
Is there any other use of WOE in developing a model?
What is the most appropriate scenario where WOE would be best suited to use?</p> | 73,755 | [
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] |
<p>Question: Assume the following classical linear normal regression model:
\begin{gather*}
y_{i} = \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \beta_K x_{Ki} + e_i \\
\underbrace{\boldsymbol{y}}_{n \times 1} = \underbrace{\boldsymbol{X}}_{n \times K}\underbrace{\boldsymbol{\beta}}_{K \times 1} + \underbrace{\boldsymbol{e}}_{n \times 1}
\end{gather*}</p>
<p>where $e_i \sim iid \ N(0, \sigma^2)$ for $i = 1, 2, \cdots, n$ and $x_{1i} = 1$ for $i = 1, 2, \cdots, n$</p>
<p>Define $\boldsymbol{\beta} = (\beta_1, \beta_2, \cdots, \beta_K)'$ and assume a noninformative prior of the form $p(\boldsymbol{\beta}, \sigma) \propto \frac{1}{\sigma}$, derive the conditional posterior pdfs for $\boldsymbol{\beta}$ and $\sigma$, that is, $p(\boldsymbol{\beta}|\sigma, \mathbf{y})$ and $p(\sigma|\boldsymbol{\beta}, \mathbf{y})$.</p>
<hr>
<p>I can derive the conditional posterior pdf for $p(\boldsymbol{\beta}|\sigma, \mathbf{y})$ but I am not sure how to derive it for $p(\sigma|\boldsymbol{\beta}, \mathbf{y})$. The answer to the latter is:</p>
<p>$p(\sigma|\boldsymbol{\beta}, \mathbf{y})=\frac{2}{\Gamma\left(\frac{n}{2}\right)}\left(\frac{na}{2}\right)^{\frac{n}{2}} \frac{1}{\sigma^{n+1}}\exp\left[-\frac{na}{2\sigma^2}\right] $ </p>
<p>where $a = \frac{1}{n}\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)'\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)$</p>
<hr>
<p>The following is my working to derive the posterior pdf for $p(\boldsymbol{\beta}|\sigma, \mathbf{y})$</p>
<p>We have:
\begin{align*}
p(\boldsymbol{e}|\sigma) & = \prod_{i=1}^n p(e_i) \\
& = \prod_{i=1}^n (2\pi \sigma^2)^{-\frac{1}{2}}\exp\left[-\frac{1}{2\sigma^2}e_i^2\right] \\
& = (2\pi \sigma^2)^{-\frac{n}{2}}\exp\left[-\frac{1}{2\sigma^2}\sum_{i=1}^n e_i^2\right] \\
& = (2\pi \sigma^2)^{-\frac{n}{2}} \exp\left[-\frac{1}{2\sigma^2} \boldsymbol{e'}\boldsymbol{e}\right]
\end{align*}
Since $p(\boldsymbol{y}) = p(\boldsymbol{e})\left|\frac{\partial \boldsymbol{e'}}{\partial \boldsymbol{y}} \right|$ and $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{e} \Longleftrightarrow \boldsymbol{e} = \boldsymbol{y} - \boldsymbol{X}\boldsymbol{\beta}$, then $\frac{\partial \boldsymbol{e'}}{\partial \boldsymbol{y}} = \frac{\partial \boldsymbol{y'}}{\partial \boldsymbol{y}} = \boldsymbol{I}_{n}$. Hence, $\left|\frac{\partial \boldsymbol{e'}}{\partial \boldsymbol{y}} \right| = \left| \boldsymbol{I}_{n}\right| = 1$. So:
\begin{align*}
p(\boldsymbol{y}|\boldsymbol{\beta}, \sigma) & = p(\boldsymbol{e}) \\
& = (2\pi \sigma^2)^{-\frac{n}{2}} \exp\left[-\frac{1}{2\sigma^2}\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)'\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right] \ \ \text{since} \ \ \boldsymbol{e} = \boldsymbol{y} - \boldsymbol{X}\boldsymbol{\beta}
\end{align*}
Thus we have:
$$L(\boldsymbol{\beta}, \sigma|\boldsymbol{y}) \propto (\sigma^2)^{-\frac{n}{2}} \exp\left[-\frac{1}{2\sigma^2}\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)'\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right]$$
Given $p(\boldsymbol{\beta}, \sigma) \propto \frac{1}{\sigma}$, we have:
\begin{align*}
p(\boldsymbol{\beta}, \sigma|\boldsymbol{y}) & \propto L(\boldsymbol{\beta}, \sigma|\boldsymbol{y}) \times p(\boldsymbol{\beta}, \sigma) \\
& \propto \frac{1}{\sigma^n}\exp\left[-\frac{1}{2\sigma^2}\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)'\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right] \times \frac{1}{\sigma} \\
& = \frac{1}{\sigma^{n+1}} \exp\left\{-\frac{1}{2\sigma^2}\left[(n-K)\widehat{\sigma}^2 + (\boldsymbol{\beta} - \boldsymbol{b})'\boldsymbol{X}'\boldsymbol{X}(\boldsymbol{\beta}-\boldsymbol{b})\right] \right\} \\
& \text{where} \ \ \widehat{\sigma}^2 = \frac{1}{n-K} (\boldsymbol{y} - \boldsymbol{X}\boldsymbol{b})'(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{b}) \ \ \text{and} \ \ \boldsymbol{b} = (\boldsymbol{X}'\boldsymbol{X})^{-1}\boldsymbol{X}'\boldsymbol{y} \\
& = \underbrace{\frac{1}{\sigma^K} \exp\left[-\frac{1}{2\sigma^2}(\boldsymbol{\beta} - \boldsymbol{b})'\boldsymbol{X}'\boldsymbol{X}(\boldsymbol{\beta}-\boldsymbol{b})\right]}_{f^n(\boldsymbol{\beta}|\sigma, \boldsymbol{y})} \underbrace{\frac{1}{\sigma^{n-K+1}} \exp\left[-\frac{(n-K)\widehat{\sigma}^2}{2\sigma^2}\right]}_{f^n(\sigma|\mathbf{y})}
\end{align*}
Note that $f^n(\boldsymbol{\beta}|\sigma, \boldsymbol{y})$ is a kernel for $\boldsymbol{\beta}$, where $\boldsymbol{\beta} \sim MVN(\boldsymbol{b}, \sigma^2(\boldsymbol{X}'\boldsymbol{X})^{-1})$ and $f^n(\sigma|\mathbf{y})$ is a kernel for $\sigma$ where $\sigma \sim IG(v, \widehat{\sigma})$ and $v = n-K$, $\widehat{\sigma}^2 = \frac{1}{v} \left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)'\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)$. Thus we have:
$$p(\boldsymbol{\beta}, \sigma|\mathbf{y}) = p(\boldsymbol{\beta}|\sigma, \boldsymbol{y})p(\sigma|\mathbf{y})$$
where:
\begin{gather*}
p(\boldsymbol{\beta}|\sigma, \boldsymbol{y}) = (2\pi)^{-\frac{K}{2}}|\sigma^2(\boldsymbol{X}'\boldsymbol{X})^{-1}|^{-\frac{1}{2}}\exp\left[-\frac{1}{2\sigma^2}(\boldsymbol{\beta} - \boldsymbol{b})'\boldsymbol{X}'\boldsymbol{X}(\boldsymbol{\beta}-\boldsymbol{b})\right] \\
p(\sigma|\mathbf{y}) = \frac{2}{\Gamma\left(\frac{v}{2}\right)} \left(\frac{v\widehat{\sigma}^2}{2}\right)^{\frac{v}{2}}\frac{1}{\sigma^{v+1}}\exp\left[-\frac{v\widehat{\sigma}^2}{2\sigma^2}\right]
\end{gather*}</p> | 73,756 | [
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<p>I'm running the a ridge regression broadly analogous to the one at <a href="http://www.mathworks.com.au/help/stats/ridge.html" rel="nofollow">http://www.mathworks.com.au/help/stats/ridge.html</a>. </p>
<p>However, I have a lot more predictors (12) than the example (which only has 3). Thus when I include the interactions the number of parameters blows out. </p>
<p>What should I do? I don't have any particular reason for thinking there would/would not be interactions.</p> | 36,233 | [
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<p>Consider an i.i.d. sample $(X_{1},\ldots, X_{n})$ where the $X_{i}$ have density
$f(x) = k \cdot \exp(−(x − θ)^4)$ with $x$ and $\theta$ real, obtain the sufficient statistic and its dimension. What does this example show? Is the statistic that you have found also complete?</p> | 36,235 | [
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<p>I am looking for a toy similarity matrix of any kind of entity class to test my random walk code. Is there any set like this or how can I create my own set? I need to interpret the results of the random walk that I run to find the dominant instances of the data.</p> | 36,237 | [
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<p><strong>1. The problem</strong></p>
<p>I have some measurements of a variable $y_t$, where $t=1,2,..,n$, for which I have a distribution $f_{y_t}(y_t)$ obtained via MCMC, which for simplicity I'll assume is a gaussian of mean $\mu_t$ and variance $\sigma_t^2$. </p>
<p>I have a physical model for those observations, say $g(t)$, but the residuals $r_t = \mu_t-g(t)$ appear to be correlated; in particular, I have physical reasons to think that an $AR(1)$ process will suffice to take into account the correlation, and I plan on obtaining the coefficients of the fit via MCMC, for which I need <strong>the likelihood</strong>. I think the solution is rather simple, but I'm not pretty sure (it seems so simple, that I think I'm missing something).</p>
<p><strong>2. Deriving the likelihood</strong></p>
<p>A zero-mean $AR(1)$ process can be written as:
$$X_t = \phi X_{t-1}+\varepsilon_t,\ \ \ (1)$$
where I'll assume $\varepsilon_t\sim N(0,\sigma_w^2)$. The parameters to be estimated are, therefore, $\theta = \{\phi,\sigma_w^2\}$ (in my case, I also have to add the parameters of the model $g(t)$, but that's not the problem). What I observe, however, is the variable
$$R_t = X_t+\eta_t,\ \ \ (2)$$
where I'm assuming $\eta_t\sim N(0,\sigma_t^2)$, and the $\sigma_t^2$ are known (the measurement errors). Because $X_t$ is a gaussian process, $R_t$ is also. In particular, I know that
$$X_1 \sim N(0,\sigma_w^2/[1-\phi^2]),$$
therefore,
$$R_1 \sim N(0,\sigma_w^2/[1-\phi^2]+\sigma_t^2).$$
The next challenge is to obtain $R_t|R_{t-1}$ for $t\neq 1$. To derive the distribution of this random variable, note that, using eq. $(2)$ I can write
$$X_{t-1} = R_{t-1}-\eta_{t-1}.\ \ \ (3)$$
Using eq. $(2)$, and using the definition of eq. $(1)$, I can write,
$$R_{t} = X_t+\eta_t = \phi X_{t-1}+\varepsilon_{t}+\eta_t.$$
Using eq. $(3)$ in this last expression, then, I obtain,
$$R_{t} = \phi (R_{t-1}-\eta_{t-1})+\varepsilon_{t}+\eta_t,$$
thus,
$$R_t|R_{t-1} = \phi (r_{t-1}-\eta_{t-1})+\varepsilon_{t}+\eta_t,$$
and, therefore,
$$R_t|R_{t-1} \sim N(\phi r_{t-1},\sigma_w^2+\sigma_t^2-\phi^2\sigma^2_{t-1}).$$ Finally, I can write the likelihood function as
$$L(\theta) = f_{R_1}(R_1=r_1) \prod_{t=2}^{n} f_{R_{t}|R_{t-1}}(R_t=r_t|R_{t-1}=r_{t-1}),$$
where the $f(\cdot)$ are the distributions of the variables that I just defined, .i.e., defining $\sigma'^2 = \sigma_w^2/[1-\phi^2]+\sigma_t^2,$
$$f_{R_1}(R_1=r_1) = \frac{1}{\sqrt{2\pi \sigma'^2}}\text{exp}\left(-\frac{r_1^2}{2\sigma'^2}\right),$$
and defining $\sigma^2(t) = \sigma_w^2+\sigma_t^2-\phi^2\sigma^2_{t-1}$,
$$f_{R_{t}|R_{t-1}}(R_t=r_t|R_{t-1}=r_{t-1})=\frac{1}{\sqrt{2\pi \sigma^2(t)}}\text{exp}\left(-\frac{(r_t-\phi r_{t-1})^2}{2\sigma^2(t)}\right)$$</p>
<p><strong>3. Questions</strong></p>
<ol>
<li><strong>Is my derivation ok?</strong> I don't have any resources to compare other than simulations (which seem to agree), and I'm not a statistician!</li>
<li><strong>Are there any derivation of this kind of things in the literature for $MA(1)$ proccesses or $ARMA(1,1)$ proccesses?</strong> A study for $ARMA(p,q)$ proccesses in general that could be particularized to this case would be nice.</li>
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<p>(1) I preprocessed data with a matching method (CEM)</p>
<p>(2) On this data (binary dependent variable, explanatory variables) , I ran a logit model and estimate predicted probabilities or odds ratios.</p>
<p>I would like to integrate my estimates of the logit model into the potential outcome framework (I would like get an average treatment effect for the treated (ATT) of my explanatory variable)</p>
<p>I know it's possible for OLS (I found it in Gelman & Hill 2007, Chapter 9), but I did not find anything on logit.</p> | 73,758 | [
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<p>Is it possible to calculate the correlation between two proportions, while accounting for the individual sample size that is used to calculate the proportion per individual?</p>
<p>For example:
How do I calculate the correlation between the proportion of heads from one coin, and the proportion of heads from another coin, when tossed by the same individual. Not every individual tosses every coin as often (some individuals even toss one of the two coins once). </p>
<p>Is this even possible?</p> | 34,909 | [
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<p>I have two factors: <strong>Brainregion (4 levels)</strong> and <strong>Group (5 levels)</strong>.
Glutamate is my dependent variable.</p>
<p>I am running General Linear Models in SPSS (two factor ANOVA).</p>
<p>Can I get the output in such a way that:
I can compare means (post-hoc test) of control with other levels such as alcohol, EW etc. (Group) across CC, CB etc. (Brainregion)?</p> | 73,759 | [
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<p>I am looking for papers, preferably in economics, where simple OLS is used. The reason is that to convey the workings of linear regression, I want to have real data from real papers. Most research uses simultaneous models or other bells and whistles. Also few researchers release their data. I could generate my own data, but as I said, I want to have the whole story of the data and research behind it. Wooldridge (Introductory Econometrics) does a lot of simple analyses from past papers, but as far as I know, these are just simplified version of the real models.</p> | 73,760 | [
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<p>Let $X_1$, $X_2$, ..., $X_n$ be iid RV's with range $[0,1]$ but
unknown distribution. (I'm OK with assuming that the distribution
is continuous, etc., if necessary.)</p>
<p>Define $S_n = X_1 + \cdots + X_n$.</p>
<p>I am given $S_k$, and ask: What can I infer, in a Bayesian manner, about
$S_n$?</p>
<p>That is, I am given the sum of a sample of size $k$ of the RV's, and I would
like to know what I can infer about the distribution of the sum of all the RV's,
using a Bayesian approach (and assuming reasonable priors about the distribution).</p>
<p>If the support were $\{0,1\}$ instead of $[0,1]$, then this problem is well-studied, and (with uniform priors) you get beta-binomial compound distributions for the inferred distribution on $S_n$. But I'm not sure how to approach it with $[0,1]$ as the range...</p>
<p><strong>Full disclosure</strong>: I already <a href="http://mathoverflow.net/questions/90580/bayesian-inference-on-sum-of-random-variables" rel="nofollow">posted this on MathOverflow</a>, but was told it would be
better posted here, so this is a re-post.</p> | 47,785 | [
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<p>I am performing multiple t-tests to compare conditions of data. I have had to resort to permutation based t-tests (based on difference of means) due to high variance between samples.</p>
<p>My question is if I am using permutation based t-tests is FWE correction still required? </p>
<p>My understanding is that because is it non-parametric the probability of a false negative is not contingent on the number of tests applied to the same set of data. So FWE correction should not be required?</p>
<hr>
<p>I will expand on my specific issue. </p>
<p>I am comparing scan-path data for subjects looking at two different types of faces (inverted and upright). I need to compare a number of measures for different face areas (eyes, nose , mouth, ears and 'other') between the areas. For this I have set up a t-test for each area, for each measure, and then correct for multiple comparisons. For example, I am testing the number of fixations, and fixation duration for each face area. I am constrained these tests in order to be able to compare to previous work. I have not been able to rely on any omnibus test thus far because of the non normality of my data greatly reduced the power of any omnibus test I have tried.</p> | 73,761 | [
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<p>Gelman & Hill (2006) explain how a multilevel regression model can compensate for data sparsity by pooling parameter estimates. But what happens if the data are not sparse at random?</p>
<p>For example, suppose we are evaluating a new blood pressure medication. On every visit we measure a patient's blood pressure, administer the treatment, and measure blood pressure again. We may have fewer opportunities to observe the effect of a treatment in some patients than in others, so we may want to use pooling. But the number of observations per patient could be related to the fact that the treatment is ineffective for some patients (so they do not want to come in), or causes side effects two hours later (so they do not want to come in), or the medication puts patients in a coma and they drop out of the study.</p>
<p>Is pooling no longer appropriate in such a study?</p> | 73,762 | [
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] |
<p>I know the value for the 16% quartile, so I know the additive deviation for the given distribution. How do I find the deviation of the log of the given distribution on a multiplicative scale?</p> | 73,763 | [
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<p>I am trying to figure out good model to analyze social dynamics using a model. I have commenting data from a website. Such as 500 articles, 10000 users, and their 75000 comments. Every user does not comment on every article. </p>
<p>The opinions posed by each user is linked to the opinion of the previous users, as well as the opinion of the article itself. The opinions of previous users are also linked to the previous users and the article. </p>
<p>I want to estimate the impact of opinion of previous user, average user opinion, article opinion and variation of opinions on the current opinion. Can you suggest a methodology for such a problem? Everything is correlated. </p>
<p>I also would like to ask, what software and code would you use to model a problem like this? Codes are very very appreciated. </p> | 47,786 | [
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<p>I previously asked this question: <a href="http://stats.stackexchange.com/questions/24194/winning-percentage-logistic-regression-or-linear-regression">link</a></p>
<p>I've generated the two logistic regression models described in the link using all available data (8 seasons for all 30 teams), and now I want to decide which is a better predictor of win percentage in the coming season, winning percentage in the previous season or Pythagorean expectation from the previous season. </p>
<p>I know that if you want to evaluate the predictive abilities of a model, it's often good to evaluate on data different from that which generated the model. I wonder how important that is here, given that the models are of the same type, created from the same data. It seems as though we could say that the model which was a better fit with respect to the data which created both models is better in this case. </p>
<p>So, the question is: 1) do I need to see how the models compare on a different data set, and 2) what's the standard measure for model comparison in a situation like that I've described. </p>
<p>Basically, how do I know which is better? </p> | 73,764 | [
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] |
<p>I'm the author of the <a href="http://cran.r-project.org/package=ez" rel="nofollow">ez package</a> for R, and I'm working on an update to include automatic computation of likelihood ratios (LRs) in the output of ANOVAs. The idea is to provide a LR for each effect that is analogous to the test of that effect that the ANOVA achieves. For example, the LR for main effect represents the comparison of a null model to a model that includes the main effect, the LR for an interaction represents the comparison of a model that includes both component main effects versus a model that includes both main effects <em>and</em> their interaction, etc.</p>
<p>Now, my understanding of LR computation comes from <a href="http://www.ncbi.nlm.nih.gov/pubmed/15732688" rel="nofollow">Glover & Dixon</a> (<a href="http://www.thinkingaboutthinking.org/wp-content/uploads/2010/09/Glover-Dixon-2004-Likelihood-ratios-a-simple-and-flexible-statistic-for-empirical-psychologists.pdf" rel="nofollow">PDF</a>), which covers basic computations as well as corrections for complexity, and the appendix to <a href="http://rads.stackoverflow.com/amzn/click/0521009138" rel="nofollow">Bortolussi & Dixon</a> (<a href="http://www.thinkingaboutthinking.org/wp-content/uploads/2010/09/Dixon-Unknown-Evaluating-Evidence-Appendix-to-Psychonarratology.pdf" rel="nofollow">appendix PDF</a>), which covers computations involving repeated-measures variables. To test my understanding, I developed <a href="https://spreadsheets.google.com/ccc?key=0Ap2N_aeyRMGHdHJxUnVNeEl5VGtvY1RVLVc5UjU4Vmc&hl=en" rel="nofollow">this spreadsheet</a>, which takes the dfs & SSs from an example ANOVA (generated from a 2*2*3*4 design using fake data) and steps through the computation of the LR for each effect.</p>
<p>I would really appreciate it if someone with a little more confidence with such computation could take a look and make sure I did everything correctly. For those that prefer abstract code, <a href="http://gist.github.com/590789" rel="nofollow">here is the R code</a> implementing the update to ezANOVA() (see esp. lines 15-95).</p> | 73,765 | [
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<p>The Gauss-Markov theorem tells us that the OLS estimator is the best linear unbiased estimator for the linear regression model. </p>
<p>But suppose I don't care about linearity and unbiasedness. Then is there some other (possible nonlinear/biased) estimator for the linear regression model which is the most efficient under the Gauss-Markov assumptions or some other general set of assumptions?</p>
<p>There is of course one standard result: OLS itself is the best unbiased estimator if in addition to the Gauss-Markov assumptions we also assume that the errors are normally distributed. For some other particular distribution of errors I could compute the corresponding maximum-likelihood estimator.</p>
<p>But I was wondering if there is some estimator which is better-than-OLS in some relatively general set of circumstances?</p> | 73,766 | [
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<p>I want to conduct a statistical test that will allow me to say that two populations are similar rather than different. As statisticians know, I can't do this with tests like the T test because the null hypothesis of the test is that the populations' means are equal. Hence my question.</p> | 73,767 | [
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<p>E.g. from the <a href="http://en.wikipedia.org/wiki/Netflix_Prize" rel="nofollow">Netflix prize</a> I know that the best RMSE = 0.8563 where the test dataset has a size of n=1,408,789.</p>
<p>Can I calculate a boundary for the MAE. If not, why can't I calculate a boundary?</p>
<p>I started like that:</p>
<p>RMSE = $\sqrt{\frac{\sum_{i=1}^{n}(y_{i}-\hat y_{i})^2}{n}}=0.8563$</p>
<p>Now I want to calculate the MAE or a boundary for the MSE:</p>
<p>MAE = $\frac{1}{n}\sum_{i=1}^{n}{|y_{i}-\hat y_{i}|}$</p>
<p>I know that:</p>
<p>$\sum_{i=1}^{n}(y_{i}-\hat y_{i})^2=RMSE^2*n$</p>
<p>from the first formula.</p>
<p>Is it possible to determine a boundary for the MAE e.g. MAE < 1?</p> | 73,768 | [
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<p>To apply Bayes inference for data analysis or machine learning, we have to construct <em>prior</em> and <em>likelihood</em>, right? But if I fail to come up with a reasonable <em>prior</em> and <em>likelihood</em>, then the Bayes model will not be meaningful, right?</p>
<p>I wonder is there any technique could be utilized to construct reasonable Bayes models? </p>
<p>Let me make it concrete by an example, given a dataest with features $X$ and target $y$, where $y=\{0,1\}$ and $X$ is composed of $p$ variables. This is an ordinary binary classification problem, to do Bayes inference, what and how should I specify for the prior and likelihood?</p>
<p>One more question, if we could not specify a reasonable prior, then what's the point of using Bayes modelling?</p> | 73,769 | [
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<p>This should be a fairly straight forward, but I can't seem to figure out the answer. I'm doing some A/B(C/D/E...) testing on a website and measuring impressions and clicks. What method should I be using to determine the statistically significant winner?</p>
<p>This is sample data that I would have. There could be any number of tests, all being displayed in relatively-equal distribution and measuring clicks as a successful result. Such as:</p>
<pre><code>Test # | Impressions | Clicks
1 | 50 | 12
2 | 55 | 15
3 | 53 | 30
4 | 49 | 22
</code></pre>
<p>What algorithm should I be using to determine the winner in a statistically significant way?</p> | 39,353 | [
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<p>I am working on Markov transition matrices. I would like to find a statistical test to compare them.</p>
<p>The first matrix is considered the population transition matrix and the second one is obtained by estimation from data (counting the transitions).</p>
<p>My first idea was to use a Chi square test on each line of the two matrices. But I have an issue: some of the transitions might be null. When this happens the Chi square test does not work any more.</p>
<p>If someone has an idea, it would be great. In addition, if you have some academic paper on the subject I would like to have the references.</p> | 73,770 | [
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<p>EDIT 2: I originally thought I needed to run a two-factor ANOVA with repeated measures on one factor, but I now think a linear mixed-effect model will work better for my data. I think I nearly know what needs to happen, but am still confused by few points.</p>
<p>The experiments I need to analyze look like this: </p>
<ul>
<li>Subjects were assigned to one of several treatment groups</li>
<li>Measurements of each subject were taken on multiple days</li>
<li>So:
<ul>
<li>Subject is nested within treatment</li>
<li>Treatment is crossed with day</li>
</ul></li>
</ul>
<p>(each subject is assigned to only one treatment, and measurements are taken on each subject on each day)</p>
<p>My dataset contains the following information:</p>
<ul>
<li>Subject = blocking factor (random factor)</li>
<li>Day = within subject or repeated measures factor (fixed factor)</li>
<li>Treatment = between subject factor (fixed factor)</li>
<li>Obs = measured (dependent) variable</li>
</ul>
<p><strong>UPDATE</strong>
OK, so I went and talked to a statistician, but he's an SAS user. He thinks that the model should be:</p>
<p><strong>Treatment + Day + Subject(Treatment) + Day*Subject(Treatment)</strong></p>
<p>Obviously his notation is different from the R syntax, but this model is supposed to account for:</p>
<ul>
<li>Treatment (fixed)</li>
<li>Day (fixed)</li>
<li>the Treatment*Day interaction</li>
<li>Subject nested within Treatment (random)</li>
<li>Day crossed with "Subject within Treatment" (random)</li>
</ul>
<p>So, is this the correct syntax to use? </p>
<pre><code>m4 <- lmer(Obs~Treatment*Day + (1+Treatment/Subject) + (1+Day*Treatment/Subject), mydata)
</code></pre>
<p>I'm particularly concerned about whether the Day crossed with "Subject within Treatment" part is right. Is anyone familiar with SAS, or confident that they understand what's going on in his model, able to comment on whether my sad attempt at R syntax matches?</p>
<p>Here are my previous attempts at building a model and writing syntax (discussed in answers & comments):</p>
<pre><code>m1 <- lmer(Obs ~ Treatment * Day + (1 | Subject), mydata)
</code></pre>
<p>How do I deal with the fact that subject is nested within treatment? How does <code>m1</code> differ from: </p>
<pre><code>m2 <- lmer(Obs ~ Treatment * Day + (Treatment|Subject), mydata)
m3 <- lmer(Obs ~ Treatment * Day + (Treatment:Subject), mydata)
</code></pre>
<p>and are <code>m2</code> and <code>m3</code> equivalent (and if not, why)?</p>
<p>Also, do I need to be using nlme instead of lme4 if I want to specify the correlation structure (like <code>correlation = corAR1</code>)? According to <a href="http://circ.ahajournals.org/content/117/9/1238.full" rel="nofollow">Repeated Measures</a>, for a repeated-measures analysis with repeated measures on one factor, the covariance structure (the nature of the correlations between measurements of the same subject) is important. </p>
<p>When I was trying to do a repeated-measures ANOVA, I'd decided to use a Type II SS; is this still relevant, and if so, how do I go about specifying that?</p>
<p>Here's an example of what the data look like:</p>
<pre><code>mydata <- data.frame(
Subject = c(13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65),
Day = c(rep(c("Day1", "Day3", "Day6"), each=28)),
Treatment = c(rep(c("B", "A", "C", "B", "C", "A", "A", "B", "A", "C", "B", "C",
"A", "A", "B", "A", "C", "B", "C", "A", "A"), each = 4)),
Obs = c(6.472687, 7.017110, 6.200715, 6.613928, 6.829968, 7.387583, 7.367293,
8.018853, 7.527408, 6.746739, 7.296910, 6.983360, 6.816621, 6.571689,
5.911261, 6.954988, 7.624122, 7.669865, 7.676225, 7.263593, 7.704737,
7.328716, 7.295610, 5.964180, 6.880814, 6.926342, 6.926342, 7.562293,
6.677607, 7.023526, 6.441864, 7.020875, 7.478931, 7.495336, 7.427709,
7.633020, 7.382091, 7.359731, 7.285889, 7.496863, 6.632403, 6.171196,
6.306012, 7.253833, 7.594852, 6.915225, 7.220147, 7.298227, 7.573612,
7.366550, 7.560513, 7.289078, 7.287802, 7.155336, 7.394452, 7.465383,
6.976048, 7.222966, 6.584153, 7.013223, 7.569905, 7.459185, 7.504068,
7.801867, 7.598728, 7.475841, 7.511873, 7.518384, 6.618589, 5.854754,
6.125749, 6.962720, 7.540600, 7.379861, 7.344189, 7.362815, 7.805802,
7.764172, 7.789844, 7.616437, NA, NA, NA, NA))
</code></pre> | 47,796 | [
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<p>I am looking to generate 450 data points in R. There are three distinct sets 150 of each distributed in a circular band with different radii (at 1, 2.8 and 5).</p>
<p>In particular, I'm looking to reproduce the first graph on p546 of The Elements of Statistical Learning.
<img src="http://i.stack.imgur.com/rMILV.png" alt="enter image description here"></p>
<p>I would be very grateful for some assistance in the R code that could generate this data.</p>
<p>Thanks!</p> | 73,771 | [
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] |
<p>I would like to build a message classification system which classifies a given message into either of 2 class - Relevant/Not.</p>
<p>I don't have any labelled dataset. I only have certain keywords which defines relevance. I can include more keywords (synonyms, nearly same words, etc).</p>
<p>For example: If I want to define relevance for Automobile the relevant keywords I'll be having are - Cars, bikes, automobiles, gas, tires, engine, oil, ... Now, the classifier should check a new message (email/SMS) and classify them whether the message belongs to this category or not.</p>
<pre><code>Message1: Always use abc car oil.
Message2: My dish had too much of oil.
</code></pre>
<p>In this case, <em>Message1</em> is relevant to automobile domain rather <em>Message2</em>.</p>
<p>Please, advice me how to build a training dataset and how do I generalize this model for any domain?</p> | 36,243 | [
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<p>I am working with different transition diagrams and want to calculate the likelihood ratio statistic for testing time-homogeneous. I saw that there are already some comparable questions, but I still don't understand the method. For example, if I have the following transition diagram (probabilities) for 1 year:</p>
<pre><code>From state 1 to state 1,2,3: 0,3 0,4 0,3 <- (Based on 80 companies)
From state 2 to state 1,2,3: 0,2 0,7 0,1 <-(Based on 500 companies)
From state 3 to state 1,2,3: 0,5 0,2 0,3 <- (Based on 300 companies).
</code></pre>
<p>And the following diagram for 2 years:</p>
<pre><code>From state 1 to state 1,2,3: 0,2 0,5 0,3 <-(Based on 100 companies)
From state 2 to state 1,2,3: 0,1 0,2 0,7 <-(Based on 1000 companies)
From state 3 to state 1,2,3: 0,6 0,2 0,2 <-(Based on 500 companies)
</code></pre>
<p>How can I calculate the likelihood ratio statistic with the same method as done in the article Testing simple Markov structures for Credit Rating transitions (<a href="http://www.occ.gov/publications/publications-by-type/occ-working-papers/2008-2000/wp2004-3.pdf" rel="nofollow">http://www.occ.gov/publications/publications-by-type/occ-working-papers/2008-2000/wp2004-3.pdf</a>). I hope that someone can help me with this question based on this example!</p> | 73,772 | [
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<p>Here are my data:</p>
<pre><code> Substrate Species Human.Impact Count
1 rubbles E. quadricolor 0 0
2 corals and rubbles E. quadricolor 0 0
3 corals E. quadricolor 0 4
4 corals and sand E. quadricolor 0 0
5 sand E. quadricolor 0 0
6 rubbles S. mertensii 0 0
7 corals and rubbles S. mertensii 0 2
8 corals S. mertensii 0 10
9 corals and sand S. mertensii 0 9
10 sand S. mertensii 0 0
11 rubbles H. crispa 0 0
12 corals and rubbles H. crispa 0 6
13 corals H. crispa 0 6
14 corals and sand H. crispa 0 5
15 sand H. crispa 0 0
16 rubbles M. doreensis 0 0
17 corals and rubbles M. doreensis 0 0
18 corals M. doreensis 0 0
19 corals and sand M. doreensis 0 0
20 sand M. doreensis 0 0
21 rubbles H. magnifica 0 0
22 corals and rubbles H. magnifica 0 0
23 corals H. magnifica 0 1
24 corals and sand H. magnifica 0 0
25 sand H. magnifica 0 0
26 rubbles S. gigantea 0 0
27 corals and rubbles S. gigantea 0 0
28 corals S. gigantea 0 0
29 corals and sand S. gigantea 0 0
30 sand S. gigantea 0 0
31 rubbles H. aurora 0 0
32 corals and rubbles H. aurora 0 0
33 corals H. aurora 0 2
34 corals and sand H. aurora 0 4
35 sand H. aurora 0 0
36 rubbles E. quadricolor 1 0
37 corals and rubbles E. quadricolor 1 0
38 corals E. quadricolor 1 8
39 corals and sand E. quadricolor 1 0
40 sand E. quadricolor 1 0
41 rubbles S. mertensii 1 0
42 corals and rubbles S. mertensii 1 0
43 corals S. mertensii 1 0
44 corals and sand S. mertensii 1 0
45 sand S. mertensii 1 0
46 rubbles H. crispa 1 0
47 corals and rubbles H. crispa 1 0
48 corals H. crispa 1 18
49 corals and sand H. crispa 1 0
50 sand H. crispa 1 0
51 rubbles M. doreensis 1 0
52 corals and rubbles M. doreensis 1 0
53 corals M. doreensis 1 0
54 corals and sand M. doreensis 1 0
55 sand M. doreensis 1 0
56 rubbles H. magnifica 1 0
57 corals and rubbles H. magnifica 1 0
58 corals H. magnifica 1 3
59 corals and sand H. magnifica 1 0
60 sand H. magnifica 1 0
61 rubbles S. gigantea 1 0
62 corals and rubbles S. gigantea 1 0
63 corals S. gigantea 1 1
64 corals and sand S. gigantea 1 0
65 sand S. gigantea 1 0
66 rubbles H. aurora 1 0
67 corals and rubbles H. aurora 1 0
68 corals H. aurora 1 1
69 corals and sand H. aurora 1 0
70 sand H. aurora 1 0
71 rubbles E. quadricolor 2 0
72 corals and rubbles E. quadricolor 2 0
73 corals E. quadricolor 2 0
74 corals and sand E. quadricolor 2 1
75 sand E. quadricolor 2 0
76 rubbles S. mertensii 2 0
77 corals and rubbles S. mertensii 2 0
78 corals S. mertensii 2 2
79 corals and sand S. mertensii 2 1
80 sand S. mertensii 2 0
81 rubbles H. crispa 2 0
82 corals and rubbles H. crispa 2 0
83 corals H. crispa 2 0
84 corals and sand H. crispa 2 3
85 sand H. crispa 2 0
86 rubbles M. doreensis 2 0
87 corals and rubbles M. doreensis 2 0
88 corals M. doreensis 2 0
89 corals and sand M. doreensis 2 0
90 sand M. doreensis 2 0
91 rubbles H. magnifica 2 0
92 corals and rubbles H. magnifica 2 0
93 corals H. magnifica 2 0
94 corals and sand H. magnifica 2 0
95 sand H. magnifica 2 0
96 rubbles S. gigantea 2 0
97 corals and rubbles S. gigantea 2 0
98 corals S. gigantea 2 0
99 corals and sand S. gigantea 2 0
100 sand S. gigantea 2 0
101 rubbles H. aurora 2 0
102 corals and rubbles H. aurora 2 0
103 corals H. aurora 2 4
104 corals and sand H. aurora 2 5
105 sand H. aurora 2 0
106 rubbles E. quadricolor 3 0
107 corals and rubbles E. quadricolor 3 0
108 corals E. quadricolor 3 0
109 corals and sand E. quadricolor 3 0
110 sand E. quadricolor 3 0
111 rubbles S. mertensii 3 2
112 corals and rubbles S. mertensii 3 0
113 corals S. mertensii 3 1
114 corals and sand S. mertensii 3 0
115 sand S. mertensii 3 0
116 rubbles H. crispa 3 1
117 corals and rubbles H. crispa 3 0
118 corals H. crispa 3 6
119 corals and sand H. crispa 3 3
120 sand H. crispa 3 0
121 rubbles M. doreensis 3 0
122 corals and rubbles M. doreensis 3 0
123 corals M. doreensis 3 3
124 corals and sand M. doreensis 3 0
125 sand M. doreensis 3 0
126 rubbles H. magnifica 3 0
127 corals and rubbles H. magnifica 3 0
128 corals H. magnifica 3 0
129 corals and sand H. magnifica 3 0
130 sand H. magnifica 3 0
131 rubbles S. gigantea 3 1
132 corals and rubbles S. gigantea 3 0
133 corals S. gigantea 3 5
134 corals and sand S. gigantea 3 0
135 sand S. gigantea 3 2
136 rubbles H. aurora 3 0
137 corals and rubbles H. aurora 3 0
138 corals H. aurora 3 8
139 corals and sand H. aurora 3 3
140 sand H. aurora 3 0
</code></pre>
<p>I'd like to know if the species are found in random substrate or if each species seem to have its own personal substrate.</p>
<p>Here is I think might be good (coded in R):</p>
<pre><code>summary(glm(Count ~ substrate * species + Human.impact, data=myData, family='Poisson'))
</code></pre>
<p>Does it seem good to you? All p.values are extremely close to 1 (except for <code>Human.impact</code>). Am I missing something or is there really no difference in habitat choice per species?</p>
<p>Note: <code>Human.impact</code> is added in order to remove some unwilling variance of the model.</p> | 36,245 | [
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<p>I would like to show that $\sum \frac{X^m}{a_n k^2}$ from $k=1$ to $k=n$ converges in distribution to the degenerate random variable 0. We have that some constant $m > 0$, $X$ ~ $Cauchy$, or the ratio of two independent standard Normal random variables, and that $a_n$ goes to $\infty$ as $n$ goes to $\infty$. I am not sure how to show this - it seems intuitive if the denominator itself goes to infinity, but the Cauchy term itself could take on values of infinity too, and would be much larger than the denominator with a power $m > 0$. Given this, how would this possibly converge to the degenerate case?</p> | 36,247 | [
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<p>Does anybody have an idea about why conditional correlation between 2 variables is called "partial" correlation and the simple correlation between them (so, when not conditioned on any other variable) is called "marginal" correlation? What is the intuition behind words "partial" and "marginal"? What do they do with "parts" or "margins"?</p>
<p>It would be good to learn the answer in order to understand those concepts better.</p> | 36,248 | [
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<p>I have a population of interest ($N = 5000$) for which I know some demographic information. I have a sample 1500 members of that population. </p>
<p>So I have a good sized sample, and I know exactly how it is biased (at least based on the demographic information I have). </p>
<p>Is it possible for me to make generalizations about the entire population based on the responses of my sample? </p>
<p>Can someone point me to a good resource on where to read about how to handle drawing inferences and calculating stats in this situation? Do I treat this as a stratified sample, but not all strata are equal (or equally representative of the whole). </p> | 73,773 | [
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<p>Looking at the <a href="http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/geobugs12manual.pdf" rel="nofollow">GeoBUGS manual version 12</a>, there is a confusion how exactly the <code>car.normal</code> model is specified:</p>
<p>1) at page 12, I read</p>
<blockquote>
<h3>car.normal and car.l1 [top]</h3>
<p>The intrinsic Gaussian CAR prior distribution is specified using the
distribution car.normal for the vector of random varables ....</p>
</blockquote>
<p>2) looking into Appendix 1, how exactly is this <strong>intrinsic</strong> CAR model specified, I read (page 41):</p>
<blockquote>
<h3>Intrinsic CAR model [top]</h3>
<p>Besag, York and Mollie (1991) propose an intrinsic version of this CAR
model in which [...] Their model corresponds to choosing C<sub>ij</sub> = 1/n<sub>i</sub> if
areas i and j are adjacent and C<sub>ij</sub> = 0 otherwise (with C<sub>ii</sub> also set to
0) [...] Note that an equivalent specification is take unnormalised
weights W<sub>ij</sub> = 1 if areas i and j are adjacent and W<sub>ij</sub> = 0 otherwise,
and set C<sub>ij</sub> = W<sub>ij</sub> / W<sub>i+</sub> where W<sub>i+</sub> = ∑<sub>j</sub> W<sub>ij</sub>. The car.normal and
(car.l1) distributions in GeoBUGS requires the user to specify
unnormalised weights.</p>
</blockquote>
<p>Let alone that they use Both C<sub>ij</sub> and W<sub>ij</sub> for weights, which is confusing, and that setting C<sub>ij</sub> = W<sub>ij</sub> / W<sub>i+</sub> with W<sub>ij</sub> = 1 for neighbours is trivially the same as setting C<sub>ij</sub> = 1/n<sub>i</sub>, so why they say it's equivalent - they probably wanted to say that an equivalent specification is to set C<sub>ij</sub> = 1 for neighbours and 0 otherwise... There seems to be a lot confusion in the manual (e.g. what is v in the model at pg. 41, what is Φ in the Conditional specification at pg. 41 and how is it connected to tau parameter at pg. 12?) but anyway... </p>
<p>The major confusion is that <code>car.normal</code> allows specification of <em>arbitrary (?) weights</em>, yet they say that weights are restricted to 1 and 0 and in the model specification at the end of page 41 (and beginning of page 42) there are no weights in the expression. So it is not clear what the actual model of <code>car.normal</code> actually does. The description of <code>weights</code> parameter at pg 12 doesn't make it clearer:</p>
<blockquote>
<p><strong>weights[]</strong> : A vector the same length as adj[] giving unnormalised weights associated with each pair of areas. For the CAR model
described above, taking Cij = 1 (equivalently Wij = 1/ ni ) if areas i
and j are neighbours and 0 otherwise, gives a vector of 1's for
weights[].</p>
</blockquote>
<p>I don't know what they mean by "CAR model described above", there is only a general call to <code>car.normal</code> - so this is kind of circular definition. So is it necessary to have such weights or not? This comment adds a lot of confusion. (and btw they keep using both C and W for weights, here at pg 12 they use them in the opposite context than at pg 41 - sometimes W is normalized, sometimes C is normalized..)</p>
<p>I would like to see the real expression behind this model. I can only speculate if the model is like:</p>
<p>$$ S_i|S_j \sim \mbox{Normal}(\frac{1}{n_i}\sum_{j}{S_j}, \frac{1}{\tau \space n_i}) $$</p>
<p>as suggested (unclearly) by the documentation in the Appendix 1, or if it is like ($w_{i+} = \sum_{k}{w_{ik}}$) </p>
<p>$$ S_i|S_j \sim \mbox{Normal}(\frac{1}{w_{i+}}\sum_{j}{w_{ij} \space S_j}, \frac{1}{\tau \space n_i}) $$
</p>
<p>as you would expect from the function <code>car.normal</code> having weights as an argument (why would it be there if it must be all 1s?)</p>
<p>Is there any other authoritative/trustworthy source to answer that question?</p> | 73,774 | [
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<p>I'd like to match the outputs of lmer (really glmer) with a toy binomial example. I've read the vignettes and believe I understand what's going on.</p>
<p>But apparently I do not. After getting stuck, I fixed the "truth" in terms of the random effects and went after estimation of the fixed effects alone. I'm including this code below. To see that it's legit, you can comment out <code>+ Z %*% b.k</code> and it will match the results of a regular glm. I'm hoping to borrow some brainpower to figure out why I'm not able to match lmer's output when the random effects are included.</p>
<pre><code># Setup - hard coding simple data set
df <- data.frame(x1 = rep(c(1:5), 3), subject = sort(rep(c(1:3), 5)))
df$subject <- factor(df$subject)
# True coefficient values
beta <- matrix(c(-3.3, 1), ncol = 1) # Intercept and slope, respectively
u <- matrix(c(-.5, .6, .9), ncol = 1) # random effects for the 3 subjects
# Design matrices Z (random effects) and X (fixed effects)
Z <- model.matrix(~ 0 + factor(subject), data = df)
X <- model.matrix(~ 1 + x1, data = df)
# Response
df$y <- c(1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1)
y <- df$y
### Goal: match estimates from the following lmer output!
library(lme4)
my.lmer <- lmer( y ~ x1 + (1 | subject), data = df, family = binomial)
summary(my.lmer)
ranef(my.lmer)
### Matching effort STARTS HERE
beta.k <- matrix(c(-3, 1.5), ncol = 1) # Initial values (close to truth)
b.k <- matrix(c(1.82478, -1.53618, -.5139356), ncol = 1) # lmer's random effects
# Iterative Gauss-Newton algorithm
for (iter in 1:6) {
lin.pred <- as.numeric(X %*% beta.k + Z %*% b.k)
mu.k <- plogis(lin.pred)
variances <- mu.k * (1 - mu.k)
W.k <- diag(1/variances)
y.star <- W.k^(.5) %*% (y - mu.k)
X.star <- W.k^(.5) %*% (variances * X)
delta.k <- solve(t(X.star) %*% X.star) %*% t(X.star) %*% y.star
# Gauss-Newton Update
beta.k <- beta.k + delta.k
cat(iter, "Fixed Effects: ", beta.k, "\n")
}
</code></pre> | 30,414 | [
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<p>I am using a Neural Networks based classifier to run a classification for my data in n-dimensional.</p>
<p>Then I thought it may be a good idea to run dimension reduction like PCA for my data at first, and then put the PCA results into the classifier (I keep 3 PCs). However, the classification on the dimension reduced features are not as good as using the original high-dimensional features directly.</p>
<p>Then I came across this post
<a href="http://stats.stackexchange.com/questions/31462/how-exactly-do-neural-network-and-support-vector-machine-methods-reduce-dimensio">NN as a DR1</a>
that discussed Neural Networks as a dimension reduction method. Also some information can be found in this paper
<a href="http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=897072" rel="nofollow">NN as a DR2</a>
I am confusing now:</p>
<ol>
<li>If I use Neural Networks based classification (in Matlab), does it automatically do the dimension reduction for me?</li>
<li>Should I run dimension reduction like PCA before running Neural Networks classification?</li>
<li>Are there any other reasons the classification on PCA results is not as good as using the original high dimensional features?</li>
</ol>
<p>Thanks very much.
A.</p> | 1,022 | [
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<p>I'm trying to train a classifier on 10 classes, using 249 samples and a (currently) 16-dimensional feature vector. I'm using an SVM with RBF kernel, through Python's scikit-learn module. The distribution of classes in those 249 samples is as follows:</p>
<p>[6, 45, 51, 38, 34, 23, 12, 15, 19, 6]</p>
<p>When building my 200-sample training set, I'm careful to maintain this same relationship. I'm not currently applying any class weighting when training. I've used grid search to optimize my C and gamma values, and the feature vectors are normalized.</p>
<p>My problem is this: When I randomly create different test/training sets from the samples (again, always maintaining the class frequency relationship above), the learned SVMs, when cross-validated against the test set, give values for precision/recall/f-score that will be anywhere from ~0.20 to ~0.50. This is making it hard for me to evaluate relative performance changes when trying different feature vectors. </p>
<p>I should mention that the data is somewhat subjective; it consists of what I'll call 'squiggly lines' (to avoid the wrath of my employer) that have been graded by an expert we call the 'Golden Eyeball', who allegedly knows how and why some squiggles are better than others. So, we're all quite happy to be within +/- 1 grade of his 'true' grade.</p>
<p>My question: I have some learning curve data suggesting that more training examples will help, but can I glean anything else from these differing performances? Or does this just tell me my model is flawed?</p>
<p>And a side question: Am I fooling myself in thinking I can choose the SVM with the best performance and say 'Done!'?</p>
<p>For the visual: I like making animations to quickly scan through the different SVMs. The one below shows the confusion matrix for 20 different SVMs taught per above. Probably not overly helpful, but I think I can sometimes see 'flashes of brilliance' in some of them that keeps me plugging away.</p>
<p><img src="http://i.stack.imgur.com/S3NWu.gif" alt="enter image description here"></p>
<p>(The size of the blocks above is relative to the proportion of samples in that class; predicted class is on the horizontal axis)</p> | 73,775 | [
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<p>This is nearly the same headline as my previous <a href="http://stats.stackexchange.com/questions/67762/interpretation-of-logistic-regression-intercept-with-one-dummy-coded-categorical">question</a>, but it turned out I was asking about multiple issues. I still have not understood the more basic issues, which have to do with the interpretation of the intercept and the role of the reference category.</p>
<p>Let's say I do a regression (in this case, a logistic regression) with a single predictor, "color", a dummy coded categorical variable having three categories ("red", "blue", and "green"). The regression and results have two predictor variables, let's say "red" and "blue" with "green" as the omitted reference category.</p>
<p>In the results, the intercept is the log odds when "red" and "blue" are zero. But in that case, it's also the log odds when "green"=1. </p>
<p>The results have an intercept and a test of its significance. If the intercept is significant but the coefficients for "red" and "blue" are not, what does this say about:</p>
<ul>
<li><p>the value "green" as a predictor of the outcome?</p></li>
<li><p>the categorical variable "color" as a predictor of the outcome?</p></li>
</ul>
<p>(I have an inkling that the answers may depend on how perfectly "color" predicts the outcome, but I don't know how to talk about this.)</p>
<p>Thanks for your help with these elementary questions.</p> | 73,776 | [
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<p>I am rather new to Matlab and never had a lot to do with statistics, so I apologize already for possibly being ignorant of quite a bit of important knowledge. It also would be nice if you could answer as simple as possible. Thanks ☺</p>
<p>My question is about finding the parameters of a univariate distribution. I get two sets of data, which I want to analyze automatically. The first set is bimodal, with the two distributions generally being separated completely. The second set has at least three modes.
My data is about the fiber directions in a carbon composite. So if I have 2 general directions I get the bimodal distribution, but if I have 3, I get two high peaks and usually one or two smaller ones (the peak at 0° is wrong data I'm working on eliminating).</p>
<p>I already played around a bit with mle, and got it to work for the bimodal case, but it takes ages, even when given good starting values. Right now I am splitting the bimodal in two and fit two separate normal distributions, but I would like to find a more elegant way. As for the 3-fiber case, I’m still without any working idea. </p>
<p>I really appreciate any help with this :)</p>
<p>The distributions</p>
<p><img src="http://i.stack.imgur.com/zBdVG.png" alt="enter image description here"></p>
<p><img src="http://i.stack.imgur.com/PHoJP.png" alt="enter image description here"></p> | 73,777 | [
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<p>I try to model quasi-poisson family in bugs language, to handle overdispersion. According to <a href="http://www.mbr-pwrc.usgs.gov/software/kerybook/" rel="nofollow">Introduction to WinBUGS for ecologists</a>, this is done by:</p>
<p>$log(\lambda_i) = f(x_i) + \epsilon_i$</p>
<p>$N_i \sim Poiss(\lambda_i)$</p>
<p>where $\epsilon_i$ does the overdispersion. This is equivalent to scenario 8 in <a href="http://www.esajournals.org/doi/abs/10.1890/10-1831.1" rel="nofollow">Lindén & Mäntyniemi 2011</a>:</p>
<p>$log(\lambda_i) = f(x_i)$</p>
<p>$N_i \sim Poiss(\lambda_i \theta_i)$</p>
<p>However, <strong>how do I get the "dispersion parameter for quasipoisson family"</strong>, as reported by GLM when using quasipoisson family?</p>
<pre><code>> summary(glm.fit.with.OD)
Call:
glm(formula = C.OD ~ x, family = quasipoisson)
[...]
(Dispersion parameter for quasipoisson family taken to be 1.470301)
</code></pre>
<p>Here I include full reproducible code, taken from <a href="http://www.mbr-pwrc.usgs.gov/software/kerybook/R_WB_code.txt" rel="nofollow">Introduction to WinBUGS for ecologists, see chapter 14.1</a> (slightly modified):</p>
<pre><code>set.seed(123)
### 14.1. Overdispersion
### 14.1.2. Data generation
n.site <- 10
x <- gl(n = 2, k = n.site, labels = c("grassland", "arable"))
eps <- rnorm(2*n.site, mean = 0, sd = 0.5)# Normal random effect
lambda.OD <- exp(0.69 +(0.92*(as.numeric(x)-1) + eps) )
lambda.Poisson <- exp(0.69 +(0.92*(as.numeric(x)-1)) ) # For comparison
C.OD <- rpois(n = 2*n.site, lambda = lambda.OD)
C.Poisson <- rpois(n = 2*n.site, lambda = lambda.Poisson)
par(mfrow = c(1,2))
boxplot(C.OD ~ x, col = "grey", xlab = "Land-use", main = "With OD",
ylab = "Hare count", las = 1, ylim = c(0, max(C.OD)))
boxplot(C.Poisson ~ x, col = "grey", xlab = "Land-use", main = "Without OD",
ylab = "Hare count", las = 1, ylim = c(0, max(C.OD)) )
### 14.1.3. Analysis using R
glm.fit.no.OD <- glm(C.OD ~ x, family = poisson)
glm.fit.with.OD <- glm(C.OD ~ x, family = quasipoisson)
summary(glm.fit.no.OD)
summary(glm.fit.with.OD)
anova(glm.fit.no.OD, test = "Chisq")
anova(glm.fit.with.OD, test = "F")
### 14.1.4. Analysis using WinBUGS
# Define model
sink("Poisson.OD.t.test.txt")
cat("
model {
# Priors
alpha ~ dnorm(0,0.001)
beta ~ dnorm(0,0.001)
sigma ~ dunif(0, 10)
tau <- 1 / (sigma * sigma)
maybe_overdisp <- mean(exp_eps[])
# Likelihood
for (i in 1:n) {
C.OD[i] ~ dpois(lambda[i])
log(lambda[i]) <- alpha + beta *x[i] + eps[i]
eps[i] ~ dnorm(0, tau)
exp_eps[i] <- exp(eps[i])
}
}
",fill=TRUE)
sink()
# Bundle data
win.data <- list(C.OD = C.OD, x = as.numeric(x)-1, n = length(x))
# Inits function
inits <- function(){ list(alpha=rlnorm(1), beta=rlnorm(1), sigma = rlnorm(1))}
# Parameters to estimate
params <- c("lambda","alpha", "beta", "sigma", "maybe_overdisp")
# MCMC settings
nc <- 3 # Number of chains
ni <- 30000 # Number of draws from posterior per chain
nb <- 10000 # Number of draws to discard as burn-in
nt <- 5 # Thinning rate
# Start Gibbs sampling
out <- bugs(data=win.data, inits=inits, parameters.to.save=params,
model.file="Poisson.OD.t.test.txt", n.thin=nt, n.chains=nc,
n.burnin=nb, n.iter=ni, debug = TRUE)
print(out, dig = 3)
</code></pre>
<p>Here is the output of the <code>glm()</code> in R:</p>
<pre><code>> summary(glm.fit.with.OD)
Call:
glm(formula = C.OD ~ x, family = quasipoisson)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.2945 -0.5905 -0.4348 1.0798 1.8159
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.4055 0.3131 1.295 0.21166
xarable 1.2622 0.3546 3.559 0.00224 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for quasipoisson family taken to be 1.470301)
Null deviance: 51.032 on 19 degrees of freedom
Residual deviance: 28.524 on 18 degrees of freedom
AIC: NA
Number of Fisher Scoring iterations: 5
</code></pre>
<p>And here is the output of the jags model (using more iterations than in the above code):</p>
<pre><code>> print(out, dig = 3)
Inference for Bugs model at "Poisson.OD.t.test.txt", fit using jags,
3 chains, each with 300000 iterations (first 100000 discarded), n.thin = 5
n.sims = 120000 iterations saved
mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
alpha 0.275 0.324 -0.415 0.073 0.291 0.498 0.861 1.001 23000
beta 1.289 0.385 0.547 1.035 1.281 1.534 2.071 1.001 41000
lambda[1] 1.163 0.586 0.263 0.740 1.092 1.496 2.512 1.001 24000
lambda[2] 1.368 0.654 0.398 0.914 1.273 1.700 2.934 1.001 27000
lambda[3] 2.173 1.081 0.818 1.429 1.919 2.638 4.976 1.001 120000
lambda[4] 1.367 0.654 0.397 0.912 1.273 1.701 2.927 1.001 26000
lambda[5] 1.370 0.655 0.401 0.914 1.275 1.702 2.929 1.001 20000
lambda[6] 1.602 0.759 0.545 1.086 1.464 1.954 3.503 1.001 24000
lambda[7] 1.365 0.656 0.399 0.909 1.268 1.697 2.941 1.001 34000
lambda[8] 1.366 0.653 0.394 0.912 1.272 1.701 2.919 1.001 43000
lambda[9] 1.368 0.654 0.400 0.916 1.270 1.700 2.934 1.001 16000
lambda[10] 1.872 0.904 0.690 1.260 1.679 2.261 4.198 1.001 84000
lambda[11] 4.608 1.548 1.978 3.525 4.494 5.521 8.032 1.001 54000
lambda[12] 5.058 1.628 2.340 3.937 4.903 5.969 8.818 1.001 52000
lambda[13] 6.544 2.064 3.426 5.091 6.189 7.649 11.521 1.001 120000
lambda[14] 4.170 1.492 1.610 3.095 4.078 5.116 7.355 1.001 67000
lambda[15] 4.613 1.554 1.974 3.524 4.499 5.528 8.073 1.001 120000
lambda[16] 7.066 2.258 3.759 5.439 6.675 8.293 12.477 1.001 49000
lambda[17] 7.615 2.462 4.054 5.802 7.190 9.003 13.472 1.001 42000
lambda[18] 3.385 1.458 0.956 2.281 3.264 4.385 6.403 1.001 61000
lambda[19] 6.538 2.066 3.431 5.089 6.185 7.643 11.505 1.001 120000
lambda[20] 3.385 1.459 0.957 2.279 3.260 4.384 6.399 1.001 95000
maybe_overdisp 1.121 0.179 0.898 1.006 1.076 1.189 1.577 1.001 120000
sigma 0.465 0.236 0.051 0.301 0.449 0.606 0.983 1.001 27000
deviance 74.767 5.985 64.292 70.311 74.343 79.042 86.528 1.001 120000
For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
DIC info (using the rule, pD = var(deviance)/2)
pD = 17.9 and DIC = 92.7
DIC is an estimate of expected predictive error (lower deviance is better).
</code></pre>
<h3>Edit 2</h3>
<p>Modified model:</p>
<pre><code>model {
# Priors
alpha ~ dnorm(0,0.001)
beta ~ dnorm(0,0.001)
sigma ~ dunif(0, 10)
tau <- 1 / (sigma * sigma)
maybe_overdisp <- mean(exp_eps[])
# co mi poradili na stats.SE
sigma2 <- sigma*sigma
kappa <- exp(alpha + beta);
mean.x <- exp(alpha + 0.5*sigma2);
vari <- kappa*exp(sigma2/2)+kappa*kappa*exp(2*sigma2) - kappa*kappa*exp(sigma2)
DI <- vari/mean.x
# Likelihood
for (i in 1:n) {
C.OD[i] ~ dpois(lambda[i])
log(lambda[i]) <- alpha + beta *x[i] + eps[i]
eps[i] ~ dnorm(0, tau)
exp_eps[i] <- exp(eps[i])
}
}
</code></pre>
<p>output:</p>
<pre><code>> print(out, dig = 6)
Inference for Bugs model at "Poisson.OD.t.test.txt", fit using jags,
3 chains, each with 30000 iterations (first 10000 discarded), n.thin = 5
n.sims = 12000 iterations saved
mu.vect sd.vect 2.5% 25% 50% 75% 97.5% Rhat n.eff
DI 20.691012 574.969428 2.748532 4.908448 7.479553 13.014255 64.410436 1.001834 2100
alpha 0.276339 0.319067 -0.410506 0.081158 0.293487 0.491261 0.859875 1.001180 6500
beta 1.285987 0.378753 0.556607 1.042379 1.276188 1.523705 2.054021 1.001084 9500
lambda[1] 1.173864 0.584403 0.268459 0.749624 1.106648 1.508932 2.499185 1.001344 4300
lambda[2] 1.364399 0.653852 0.395182 0.913774 1.272952 1.693450 2.944849 1.001782 2200
lambda[3] 2.148461 1.056564 0.814731 1.422458 1.892689 2.620223 4.912488 1.001393 3900
lambda[4] 1.373973 0.651289 0.404761 0.922161 1.283126 1.707821 2.930640 1.001074 10000
lambda[5] 1.364438 0.646131 0.397013 0.915541 1.271812 1.700321 2.908050 1.000922 12000
lambda[6] 1.596684 0.737480 0.562846 1.098588 1.463223 1.931152 3.472518 1.001139 7500
lambda[7] 1.372055 0.667968 0.394750 0.912321 1.270109 1.702314 3.030247 1.001004 12000
lambda[8] 1.368805 0.641905 0.393292 0.926187 1.281149 1.694738 2.933341 1.001159 7000
lambda[9] 1.367516 0.644055 0.410916 0.914202 1.279055 1.693786 2.892084 1.001237 5500
lambda[10] 1.874077 0.903156 0.706316 1.261625 1.683053 2.256230 4.224203 1.001380 4000
lambda[11] 4.604620 1.551257 1.983965 3.515142 4.484644 5.530132 8.040010 1.000890 12000
lambda[12] 5.052838 1.631220 2.325490 3.949769 4.904707 5.942355 8.784829 1.000883 12000
lambda[13] 6.506786 2.050675 3.424231 5.043272 6.159473 7.621714 11.341993 1.000896 12000
lambda[14] 4.168782 1.468719 1.617240 3.104178 4.099795 5.114893 7.324785 1.002270 1400
lambda[15] 4.588128 1.529048 1.986595 3.537436 4.479550 5.476045 7.997842 1.000895 12000
lambda[16] 7.035670 2.235785 3.776619 5.428774 6.617842 8.235651 12.389102 1.001082 9600
lambda[17] 7.586794 2.444835 4.078003 5.780102 7.145878 8.986080 13.364148 1.000973 12000
lambda[18] 3.413945 1.459583 0.955955 2.306502 3.321237 4.407808 6.427175 1.002377 1300
lambda[19] 6.559620 2.077231 3.418885 5.098825 6.196538 7.664223 11.574183 1.001324 4400
lambda[20] 3.391437 1.459840 0.949251 2.262276 3.281995 4.407545 6.371617 1.001482 3300
maybe_overdisp 1.119677 0.178304 0.900895 1.006214 1.073253 1.185895 1.565368 1.002841 1000
sigma 0.458185 0.235224 0.048157 0.294464 0.441186 0.597933 0.972298 1.009139 960
deviance 74.818941 5.985537 64.100886 70.435067 74.417487 79.115472 86.443865 1.001312 4600
For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
DIC info (using the rule, pD = var(deviance)/2)
pD = 17.9 and DIC = 92.7
DIC is an estimate of expected predictive error (lower deviance is better).
</code></pre> | 36,254 | [
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<p>I've tried to create three models (using R): an intercept only linear regression, a simple mixed effects regression and a by-subject effects mixed effects regression.</p>
<p>An intercept only regression models the grand mean of a response variable plus error. In <code>mtcars</code>, the variable <code>drat</code> may be considered a response variable. In the model below, have I correctly modelled the grand mean of <code>drat</code>, plus error?</p>
<pre><code>interceptOnly <- lm(drat ~ 1, data=mtcars)
</code></pre>
<p>A simple mixed effects regression models the grand mean of a response variable, plus subject deviation, plus error. In <code>mtcars</code>, <code>drat</code> may be considered a response variable and <code>cyl</code> a subject deviation. In the model below, have I correctly modelled the grand mean of <code>drat</code>, plus the deviation of <code>cyl</code> from <code>drat</code>, plus error?</p>
<pre><code>library(lme4)
simpleMixedEffects <- lmer(drat ~ (1|cyl), data=mtcars)
</code></pre>
<p>A by-subject effects mixed effects regression models the grand mean of a response variable, plus subject deviation, plus condition effect, plus error. In <code>mtcars</code>, <code>drat</code> may be considered a response variable, <code>cyl</code> a subject deviation and <code>wt</code> a condition effect. In the model below, have I correctly modelled the grand mean of <code>drat</code>, plus the deviation of <code>cyl</code> from <code>drat</code>, plus the effect of <code>wt</code>, plus error?</p>
<pre><code>bySubjectMixedEffects <- lmer(drat ~ (1|cyl) + wt, data=mtcars)
</code></pre>
<p>I have one further question: </p>
<p>How can I model a by-subject varying condition effect model. This is a mixed effects model which models the grand mean of a response variable, plus group deviation from grand mean (random effect), plus condition effect (fixed effect), plus group deviation from condition effect (random effect), plus error. Could someone provide R code that outputs a "by-subject varying condition effect model"?</p> | 36,255 | [
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<p>I have a database of time series data generated by giroscope sensor. A small part of the data is already labeled and used to fit the model. My strong assumption is that most of the samples left in this database are of the same class of movement. Can I use this data to replenish the model without manual labeling? What are the steps required?</p> | 36,256 | [
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] |
<p>Also, are the standardized regression weights equivalent to the correlation between a factor and a manifest variable?</p>
<p>I write this question with reference to an example on p138-142 of the following document: <a href="ftp://ftp.software.ibm.com/software/analytics/spss/documentation/amos/20.0/en/Manuals/IBM_SPSS_Amos_User_Guide.pdf" rel="nofollow">ftp://ftp.software.ibm.com/software/analytics/spss/documentation/amos/20.0/en/Manuals/IBM_SPSS_Amos_User_Guide.pdf</a>. </p>
<p>Here are illustrative figures and a table:
<img src="http://i.imgur.com/Rmayocw.png" alt="CFA example"></p> | 36,257 | [
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<p>Please consider the following panel dataset:</p>
<pre><code>comp obs industry weekDay ind10 ind15 day3 day4 day5 marketRet tweets stockRet
-----------------------------------------------------------------------------------------
1 1 15 3 0 1 1 0 0 0.10 5321 -0.90
1 2 15 4 0 1 0 1 0 1.30 4244 -0.30
1 3 15 5 0 1 0 0 1 0.90 5543 1.32
2 1 10 3 1 0 1 0 0 0.10 789 0.10
2 2 10 4 1 0 0 1 0 1.30 842 0.16
2 3 10 5 1 0 0 0 1 0.90 734 0.00
</code></pre>
<ul>
<li>For a list of companies (<code>comp</code>) it describes the number of <code>tweets</code> and the stockreturn (<code>stockRet</code>) for a series of days (<code>obs</code>)</li>
<li><code>weekDay</code> gives the day of the week (<code>1</code> = monday, <code>2</code> = tuesday, ...); this has been extracted in dummies <code>day3</code> to <code>day5</code></li>
<li><code>industry</code> gives the company industry (<code>15</code> = IT, <code>10</code> = banking, ...); this has been extracted in dummies <code>ind10</code> and <code>ind15</code></li>
<li>The final variable (<code>marketRet</code>) gives the average return of the stockmarket that day. Notice that for each day (<code>obs</code>), the market return is the same.</li>
</ul>
<p><strong>Question 1:</strong> Say I am running an OLS regression with <code>tweets</code> as independent variable and <code>stockRet</code> as dependent. I'm also adding the dummies <code>day3</code> to <code>day5</code> and <code>ind10</code> & <code>ind15</code> to the model as independent variables. Does the model now include, as they say, "fixed effects for industry and day of week"?</p>
<p><strong>Question 2:</strong> I have read articles with similar research to mine, and they say they have "added the market return as a control". In SPSS preferably, how do I enter the variable <code>marketRet</code> as a control to the model? Just by adding it as an independent variable?</p>
<p><strong>Question 3:</strong> What is the difference between fixed effects and control variables?</p>
<p>These questions are probably very basic, but I have not been able to find a clear answer to them. For instance, articles mention they "control for market return" but make no mention of <em>how</em> and <em>why</em> they do so. Thus, any help is greatly appreciated :-)</p> | 36,258 | [
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<p>Given min and max values, how can I estimate shape parameter (tail index) of data generated by truncated pareto distribution ? I see a package tpareto but find no information on how to estimate tail index from given data.</p>
<p>kindly help.</p> | 36,260 | [
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<p>Suppose I have a dataset $x_i$ (with $x_i >= 0$) of N samples. I would like to determine the average of the data when the data is limited to some theshold c: </p>
<p>$limitedaverage(x,c) = \frac{1}{N} \sum_{i=1}^N max(x_i,c)$</p>
<p>Problem is I don't have access to the actual samples, I only know some aggregate statistics of the data: the sum, sum of squares, etc, up to a certain order k:</p>
<p>$\sum x$, $\sum x^2$, $\sum x^3$ ... $\sum x^k$</p>
<p>I'm looking for a way to estimate the above limited average for a given c and given said aggregates.</p>
<p>Intuitively I've been able to get a rough idea for k=2: given $\sum x$ and $\sum x^2$ one can calculate the average and the standard deviation and this tells me whether there are any samples that are significantly below c and would become clipped. But also given $\sum x^3$ and $\sum x^4$ I'm sure it's possible to get a more accurate estimate.</p>
<p>My idea is that it should be possible to exploit $max(x,c) = \lim_{k \to \infty} (x^k+c^k)^\frac{1}{k}$ but from here I'm stuck.</p>
<p>Any hints?</p> | 73,778 | [
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] |
<p>How can a stationary, invertible ARMA(1,1) process be represented as
either an infinite order AR or infinite order MA process?</p> | 73,779 | [
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<p>I am investigating the achieved mark in a math test in grade 8 in two different cities. The plot shows the two <em>Empirical Cumulative Distribution Functions</em> (<em>ECDF</em>).</p>
<p><img src="http://i.stack.imgur.com/6vLlw.png" alt="Plot of two ECDFs"></p>
<p>How can I evaluate whether these two <em>ECDFs</em> follow the same distribution or not?</p> | 73,780 | [
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<p>I'd be really grateful for recommendations of a robust package for fitting discrete choice models to a large amount ($n$ in the millions and $p$ in 2000 range) of data. I want a smoothed model that can deal with multi-colinear dependent variables and matrix inversion issues sensibly - like <a href="http://cran.r-project.org/web/packages/glmnet/index.html" rel="nofollow">glmnet</a>. I'm happy to bootstrap samples, which may be the only way to deal with big data in R.</p>
<p>I've tried using the <a href="http://cran.r-project.org/web/packages/mlogit/index.html" rel="nofollow">mlogit</a> package and it falls apart with more than a few hundred predictors, producing errors to do with matrix inversion.</p>
<p>My alternative is to use the <a href="http://cran.r-project.org/web/packages/glmnet/index.html" rel="nofollow">glmnet</a> package for binary regression and then use transforms to approximate the discrete choice model using something called Begg and Gray's approximation. </p>
<p>This data is not multinomial, in the traditional sense. It is discrete-choice, that is the classes themselves change from observation to observation -- possibly also the number of classes. Each of the classes has a set of predictors which are measured on the same scale and are class specific -- cf. <a href="http://en.wikipedia.org/wiki/Discrete_choice" rel="nofollow">Discrete Choice Models</a>. I wrote to the maintainer of <code>glmnet</code>, Trevor Hastie, who says there is no mapping to discrete choice models in their package.</p>
<p>Another name for discrete-choice is conditional logit, with the correct parameterization. I found the package <a href="http://cran.r-project.org/web/packages/pglm/index.html" rel="nofollow">pglm</a>, but it is also lacking in robustness. There's reference, <a href="http://stats.stackexchange.com/questions/10141/">Discrete choice panel models in R</a> to <a href="http://cran.r-project.org/web/packages/lme4/index.html" rel="nofollow">lme4</a> also, but I have found no examples of the conditional logit with it.</p> | 73,781 | [
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<p>I am trying to use <code>randomForest</code> in R to find interaction terms to add to a model. My plan was to fit trees with <code>maxnodes=4</code> (two deep stubs), then compute how often <code>var A</code> is a child of <code>var B</code> and its improved accuracy to measure the importance of the <code>A*B</code> interaction. Essentially this is leveraging <code>randomForest</code> for the sampling.</p>
<p>To make this work I also wanted <code>mtry=#</code> variables, so all variables would have a chance at being the child of the root variable.</p>
<p>This strategy failed in my data because all trees after the first few were identical. I did decrease <code>cp</code>, but that did not help. I removed <code>mtry=#vars</code>, and still got all 95% of trees the same (the last 95%, the first few are different).</p>
<p>When <code>maxnodes=32</code>, I got a good set of trees, but that's not what I needed for determining interactions. I need to do this because I can't deploy a <code>randomForest</code> in production.</p>
<p>Any ideas why this wouldn't work? I am coding this myself now using <code>rpart</code> on 2 variable models iterating over the potential good pairs.</p> | 36,265 | [
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<p>I have a general question about interpreting interaction effects in a non-linear model. I understand the reasons Ai and Norton (2004) suggest using the stata inteff command to help interpret interaction effects in a non-linear model.</p>
<p><a href="http://www.unc.edu/~enorton/NortonWangAi.pdf" rel="nofollow">http://www.unc.edu/~enorton/NortonWangAi.pdf</a></p>
<p>However, Buis (2010) seems to suggest that interpreting a logit model through odds ratios overcomes the problem in a much simpler way. </p>
<p><a href="http://www.stata-journal.com/sjpdf.html?articlenum=st0194" rel="nofollow">http://www.stata-journal.com/sjpdf.html?articlenum=st0194</a></p>
<p>It is not clear to me how Buis's suggestion helps with the issues of Ai and Norton (2004) namely that the sign and significance of interaction effects in non-linear models vary across observations. Any help in understanding this would be much appreciated?</p> | 73,782 | [
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<p>This is bordering on a philosophical question, but I am interested in how others with more experience think about distribution selection. In some cases it seems clear that theory might work best (mice tail lengths are probably normally distributed). In a lot of cases there is probably no theory to describe a set of data, so you just use something that fits what you have fairly well regardless of what it was originally developed to describe? I can imagine some of the pitfalls of going with one or the other of these, and then of course there seems to be the problem that maybe you should just use an empirical distribution if you really have no idea. </p>
<p>So I guess what I'm really asking: does someone have a coherent way of approaching/thinking about this problem? And are there any resources you can suggest that give a good treatment of this? </p> | 73,783 | [
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<p>I want to compare the distributions of $K$ contingency tables, where each table is $N \times M$.
To make things clearer, let's work on $K=M=N=2$.
Suppose I have a classical cases and controls test, where I have $2$ variables, treated or not, and cured or not.
Suppose that besides testing whether the treatment is effective or not, I want to test whether the effect is different among groups, say males and females. Then I would have $2$ $2 \times 2$ tables, one for males, one for females. I want to test whether the distribution of the tables is independent of sex.</p>
<p>Is there a standard test for this?</p> | 40,857 | [
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<p>I am just learning R. I have developed a regression model with six predictor variables. While developing it, I found the relationships are not very linear. So, maybe because of this the predictions of my model are not exact.</p>
<p>Here is Headers of my data set:</p>
<pre><code>1.bouncerate(To be predicted)
2.avgServerResponseTime
3.avgServerConnectionTime
4.avgRedirectionTime
5.avgPageDownloadTime
6.avgDomainLookupTime
7.avgPageLoadTime
</code></pre>
<p>Sample datasets:</p>
<pre><code>28.57142857,4.132,0.234,0,0.505,0,14.168
42.85714286,3.356777778,0.090777778,0.077333333,0.459,0.105444444,14.78644444
0,3.372,0.1105,0.0015,0.425,0.1305,34.3425
33.33333333,3.583,0.218,0,0.385,0.649,11.816
66.66666667,2.438,0.234,0,0.3405,0,8.645
100,2.805,0.179666667,3.203666667,0.000333333,0.11,13.47066667
66.66666667,0.977,0,0.003,0,0,12.847
0,2.776,0,7.888,0,0,14.393
100,2.59,0.261,0,0.517,0,6.216
</code></pre>
<p>Here is the summary of my model:</p>
<pre><code>Call:
lm(formula = y ~ x_1 + x_2 + x_3 + x_4 + x_5 + x_6)
Residuals:
Min 1Q Median 3Q Max
-125.302 -26.210 0.702 26.261 111.511
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 48.62944 0.27999 173.684 < 2e-16 ***
x_1 -0.67831 0.08053 -8.423 < 2e-16 ***
x_2 0.07476 0.49578 0.151 0.880143
x_3 -0.22981 0.06489 -3.541 0.000399 ***
x_4 0.01845 0.09070 0.203 0.838814
x_5 3.76952 0.67006 5.626 1.87e-08 ***
x_6 0.07698 0.01565 4.919 8.75e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 33.76 on 19710 degrees of freedom
Multiple R-squared: 0.006298, Adjusted R-squared: 0.005995
F-statistic: 20.82 on 6 and 19710 DF, p-value: < 2.2e-16
</code></pre>
<p>plot with all single variable are below:
<img src="http://i.stack.imgur.com/jsW0j.png" alt="bouncerate vs avgServerConnectionTime">
<img src="http://i.stack.imgur.com/uhrya.png" alt="bouncerate vs ServerResponseTime">
<img src="http://i.stack.imgur.com/iuROe.png" alt="bouncerate vs avgRedirectionTime">
<img src="http://i.stack.imgur.com/wbfsP.png" alt="bouncerate vs avgDomainLookupTime">
<img src="http://i.stack.imgur.com/arTC6.png" alt="bouncerate vs avgPageLoadTime"></p>
<p>I have certain questions about this model: </p>
<ol>
<li>Is there any way to improve the accuracy of this model? </li>
<li>Which of the values is most useful: residual standard error, degrees of freedom, multiple R-squared, adjusted R-squared, F-statistics, or p-values for choosing best model? </li>
<li>Is it appropriate to use polynomial transformations with these data? </li>
<li>In case I do use polynomial terms in my model, which degree is most appropriate? </li>
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<p>I'm doing statistical analysis on a psychology study I ran a short while ago, and I'm starting to think that I've designed myself into a corner.</p>
<p>In the study, we measured participants' pointing responses after traveling in a virtual environment. There were three factors, each nested within another:</p>
<p>Physical rotation: 2 levels
Visuals: 3 levels
Turning angle: 10 levels</p>
<p>Each participant was subjected to all variations of these treatments, so they ran 60 trials each. However, due to this large number of factors, we restricted the randomization of the order of treatments for each participant. In picture form, our design looks like this:</p>
<p><img src="http://i.stack.imgur.com/zEITj.png" alt="Split-split-split plot design"></p>
<p>M represents the physical rotation condition, V is the visual condition, and A is the turning angle condition. Each table represents one full session of 60 trials for a participant assigned to that group. The numbers in the cells represent the order in which participants assigned to that group perform trials. We ran 36 participants in total, and assigned each randomly to a group so that each of the four groups had 9 participants.</p>
<p>This is a bit complex, so please let me know if I can explain something better. The groups represent the four different permutations of the 60 conditions we decided on, to make the best out of our limited number of participants.</p>
<p>Group 1:</p>
<ol>
<li>M1|V1</li>
<li>M2|V1</li>
<li>M1|V2</li>
<li>M2|V2</li>
<li>M1|V3</li>
<li>M2|V3</li>
</ol>
<p>Group 2:</p>
<ol>
<li>M1|V3</li>
<li>M2|V3</li>
<li>M1|V2</li>
<li>M2|V2</li>
<li>M1|V1</li>
<li>M2|V1</li>
</ol>
<p>Group 3:</p>
<ol>
<li>M2|V1</li>
<li>M1|V1</li>
<li>M2|V2</li>
<li>M1|V2</li>
<li>M2|V3</li>
<li>M1|V3</li>
</ol>
<p>Group 4:</p>
<ol>
<li>M2|V3</li>
<li>M1|V3</li>
<li>M2|V2</li>
<li>M1|V2</li>
<li>M2|V1</li>
<li>M1|V1</li>
</ol>
<p>Within each of the six blocks, participants completed all ten turning angle conditions, in random order.</p>
<p>So, we have one factor that is randomized (turning angle), one that is alternated (movement condition) and one that is ordered (visual condition). Each of these is nested within another, making for a split-split-split plot design.</p>
<p>Now, what we are analyzing is how the pointing responses differ between conditions, and between participants. I have read everything I could find on split-plots, and even found some references to split-split-split plots, but I am still very unsure of how to actually crunch the numbers in R or SPSS.</p>
<p>I would be very grateful for any advice, hints or references to useful reading material that could help me in this. Please ask if you have any questions, or if I can explain something more clearly.</p> | 73,784 | [
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<p>Yesterday i read this blog post <a href="http://www.petrkeil.com/?p=393" rel="nofollow">Predictors, responses and residuals: What really needs to be normally distributed?
</a>
and I wonder what do you think about it. Personally I totally agree with the author about the fact that sometimes people even discard to build or evaluate models without log-transformations or things like that, just because of they have variables not normal distributed (incomes or other measures that are usually not gaussian), sometimes getting wrong about the predictor or outcome hypothesis for the models.. </p>
<p>I think this is an relevant point nowadays when the "machine-learning trend" seems to be way less restrictive than "classic statistics" and therefore leaves classic models a little bit aside while many times they produce equal or even better results..</p> | 47,833 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://stats.stackexchange.com/questions/35719/improvement-of-regression-model">Improvement of regression model</a> </p>
</blockquote>
<p>I have recently started learning regression. I have a dataset containing a response and a predictor variable. When I plot the response and predictor variables there is no linear or curvilinear relationship between them. I used polynomial regression with degree 6 to fit the data, but the prediction does not seem good (the difference between the actual and predicted values is large). So I want to know that how can I improve my regression model? </p> | 49,923 | [
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] |
<p>Is there a Box-Cox like transformation for independent variables? That is, a transformation that optimizes the $x$ variable so that the <code>y~f(x)</code> will make a more reasonable fit for a linear model?</p>
<p>If so, is there a function to perform this with <code>R</code>?</p> | 49,507 | [
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<p>I am not entirely sure if I should ask this question here, but does EViews has a function of converting <em>J</em> statistics to <em>p</em>-value of Jstats, or other way around. I am running several GMM estimations and I would like show both of them in my paper. Or is their a way to convert the <em>J</em> statistics to <em>p</em>-value of <em>J</em> statistics.</p> | 12,399 | [
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<p>I have a sample design that can be resumed in:</p>
<ol>
<li>Split the city in X areas and, for each area, sample 5% of the residences and get some informations about it;</li>
<li>For each residence, interview ALL residents;</li>
<li>In the end, weights to homes and residents are calculated.</li>
</ol>
<p>So, actually, I have 2 samples: one for residences and another for residents. My question is: what are the sampling designs and how declare it on survey()?</p>
<p>Each dataset has this variables:
CD_APONDE - A code to identify the splited areas
V0300 - A code to identify the residences
V0010 - The weight</p>
<p>The first one (for the residences), I believe that is a stratified sampling and should declare like <code>svydesign(strata = ~ CD_APONDE, weights = ~ V0010, data = dados)</code>.</p>
<p>The second (for residents, and my true question) I guess that is a stratified 1 - level Cluster Sampling design, but have not sure, and should use <code>svydesign(ids = ~ V0300, strata = ~ CD_APONDE, weights = ~ V0010, data = dados)</code>.</p>
<p>Am I right? Wrong? 50/50?</p> | 73,785 | [
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<p>I am thinking about how best to construct a data set containing one record for every individual in the entire United States population starting from something like the <a href="http://www.asdfree.com/search/label/american%20community%20survey%20%28acs%29" rel="nofollow">American community survey</a> or the <a href="http://www.asdfree.com/search/label/united%20states%20decennial%20census%20public%20use%20microdata%20sample%20%28pums%29" rel="nofollow">United States decennial census public use microdata files</a>.</p>
<p>Both of these "starting points" would be very large, they have between 1 and 5% of the entire United States population already.</p>
<p>So long as this concept is not majorly flawed from the start, having this synthetic but complete data set would make it much easier to merge on (cold-deck impute) information from other, smaller data sets.</p>
<p>One could obviously take a simplistic approach and - for every record - make the same number of records as that record's weights. This wouldn't be much different than just analyzing the data set with <a href="http://www.stata.com/help.cgi?weights" rel="nofollow">Stata's frequency weights</a> but that will obviously create problems in smaller geographic areas.</p>
<p>Say you've got a small county that you know for a fact has 10,000 residents, but in your sample only ten records. Obviously you cannot simply expand those ten records to the 10,000 residents on their own. Since you'll have a very lumpy (and wrong) age, race, income distribution within the county. However, if you took a bit of information from nearby areas and projected the probabilities of age, race, income, etc. of each of those 10,000 records you're creating from scratch, you could semi-randomly create 10,000 records that would look more reasonable for that specific county.</p>
<p>I am unsure whether this sort of thing has not been done because it's a terrible idea for some reason or if statisticians and demographers simply haven't had the computing power to quickly deal with 300 million records of data <a href="http://www.asdfree.com/2013/03/column-store-r-or-how-i-learned-to-stop.html" rel="nofollow">until very recently</a>.</p> | 73,786 | [
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<p>The regret in a multi-arm bandit model is given by </p>
<p>$$\underset{j}{\max}\sum_{t=1}^{T}x_j(t) -G_{A}$$</p>
<p>where $$G_A=\sum_{t=1}^{T}x_{it}(t)$$ is the total reward achieved by the learner, based on an action $i$, taken at each time interval $t$, i.e $i_{t}\in {1,2,...,K}$ and $x_j(t)$ is the reward assosciated with action $j$, at time $t$ with $K$, being the total number of actions available.</p>
<p>In a stochastic multi-arm bandit setting where each arm is modeled by a population distribution: </p>
<p>a) What is the connection between the sequential decisions taken by a chosen learner that guarantees a minimization of the expectation of the above regret?</p>
<p>I ask this, as I do not see the expectation of the best arm being accounted for- directly in the given form of the regret. Instead, is there a bound that connects this regret to the population mean parameters?</p>
<p>I looked at the expected regret being $$\mathbb{E}\left[\underset{j}{\max}\sum_{t=1}^{T}x_j(t)\right]-\mathbb{E}G_A $$ and want to have a connection between the steps in any chosen multi-arm-bandit algorithm and the minimization of the expected regret. Is it done directly- or is it based on the minimization of say, an indirect upper bound? </p> | 36,273 | [
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<p>How many lags should one use a Ljung-Box test for the returns on 1320 daily price quotes? Is there a rule of thumb? What is exactly the impact in the end result of misuse of lag number?</p> | 36,274 | [
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<p>I have four sets of data grouped by stages (Stage 1-4) with each having their own correlation coefficient. The 4 sets of data has the same dependent and independent variables with 1st n=106, 2nd n=23, 3rd n=89, 4th n=132. My hypothesis is that at stage 4 the correlation would get weaker.</p>
<p>From the numbers alone it supports the idea since the correlation is smaller on the 4th stage than on the 1st, 2nd and 3rd but is there a way to prove it statistically? Or is it just okay to say that there is a weakening in the correlation without having to resort to statistical significance testing? I have read about the Fisher z transform but can it be used to test 4 coefficients? And since I am only comparing the 4th stage to the rest can I just test 1-4 2-4 and 3-4 using Fisher z? Oh and is there a way to do the test using SPSS?</p> | 73,787 | [
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<p>I would like to try to make some sense of a neural network. The neural network has a single hidden layer and is used on 30-40 attributes, which are used to classify the probability that the observation belongs to 1 of 3 different classes. My goal is to the find 5-10 main interactions in order to understand the model better.</p>
<p>To do this I've been recommended by my adviser to look into the 'Optimal Brain Damage'-algorithm (<a href="http://yann.lecun.com/exdb/publis/pdf/lecun-90b.pdf" rel="nofollow">http://yann.lecun.com/exdb/publis/pdf/lecun-90b.pdf</a>), which removes connections between nodes one by one. However, I cannot seem to find any R packages that supports this algorithm.</p>
<p>I would therefore like to know what other algorithms exist to simplify neural networks and make them more easily interpreted. My efforts searching has so far been pretty useless, which is likely due to the fact that I don't have any clue of which key words to use.</p>
<p>Thanks!</p> | 73,788 | [
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<p>I'm creating Poisson GLMs in R. To check for overdispersion I'm looking at the ratio of residual deviance to degrees of freedom provided by <code>summary(model.name)</code>. </p>
<p>Is there a cutoff value or test for this ratio to be considered "significant?" I know that if it's >1 then the data are overdispersed, but if I have ratios relatively close to 1 [for example, one ratio of 1.7 (residual deviance = 25.48, df=15) and another of 1.3 (rd = 324, df = 253)], should I still switch to quasipoisson/negative binomial? I found <a href="http://data.princeton.edu/R/glms.html">here</a> this test for significance: 1-pchisq(residual deviance,df), but I've only seen that once, which makes me nervous. I also read (I can't find the source) that a ratio < 1.5 is generally safe. Opinions?</p>
<p>Thanks!</p> | 73,789 | [
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