tttjjj/elt_bench huggingface dataset:
@Behacad, to put it simply, convergent validity is how well each variable loads onto its respective factor, whereas discriminant validity is how poorly each variable loads onto other factors other than the factor it loads most highly on. For example, if we had a questionairre with 4 items (Do you feel blue today? Do you ever have thoughts of suicide? Are you depressed? Are you happier than you have been your entire life?), we would expect the first 3 items to have convergent validity with one another and discriminant validity with the "happy" item, and vice versa.
A Google search on "Matlab variography" picks up ["Matlab Recipes for Earth Sciences"](http://books.google.com/books?id=mIzb7IaEtLAC&pg=PA238&lpg=PA238&dq=matlab+variography&source=bl&ots=ymBJLFNI9P&sig=32G-KeLHt4r7v1nlZwpTTXwOLeY&hl=en&sa=X&ei=ifjQT-GWHMje0gHbiIHQDQ&ved=0CE4Q6AEwAQ#v=onepage&q=matlab%20variography&f=false). The code at pp 244-45 looks like a good start. If you're not intimately familiar with variography, no amount of code will do you any good: read a good book on geostatistics first.
Even in *infinitely many dimensions,* any two vectors determine a subspace of dimension at most $2$: therefore the (Euclidean) relationships that hold in two dimensions among pairs of vectors hold *entirely without any change at all* in any number of higher dimensions, too.
My co-worker looked this over for me and thinks that this should work. Just in case someone else finds this. I'm going to try to post results using R's boot.ci function and some results I got with real data.
If you report high power and you didn't reject the null then people will tend to think that the null is true.  That's unlikely.  What's likely is that the effect is small.  However, consider the converse, a lower powered experiment where you do reject the null.  Your effect size must be quite large and it suggests you have very strong evidence.  In this case, what's likely true, is that you have an over estimate of the true effect and it's actually lower, perhaps substantially.  I think the former is more problematic.  It's better to quantify effects than just think of them as there or not.
This question appears to be off-topic because it is about how to use R without a [reproducible example](http://stackoverflow.com/q/5963269/1217536).
+1 GPs are a good alternative to KLR (although KLR often gives better performance because evidence based model selection can go wrong quite easily if there is model mis-specification) and cross-validation is often preferable.
I guess you misquoted the criteria numbers!
Related: http://stats.stackexchange.com/questions/421/what-book-would-you-recommend-for-non-statistician
You should speak of **excess** kurtosis being $0$ or kurtosis being $3$. Just to be sure.
Both might be important predictors. There's no requirement for predictors to be perfectly orthogonal in generalized linear models, any more than there is in linear regression or ANOVA. If the predictors are *too* collinear the model fit will still be valid, but useless because the variances for individual model coefficients become enormous. When that doesn't happen you have no cause to worry.
TOST stands for two one-sided tests - it is a generally applicable procedure.
I guessed you wouldn't want to homogenise your soil columns. Nevertheless, if would help if you can take more soil columns and use baseline measures to select fairly homogeneous ones to use in the experiment, or find group of 2 or more soil columns that are similar on your three outcome measures and randomly allocate equal numbers within each group to each treatment, i.e. a form of blocking.
+1 I think really of the answer so far, this probably best addresses the original post.  This question was almost certainly motivated by understanding the instrument rather than calibrating a proxy.  If you have just one X measurement for each Y, then I'm pretty sure (whuber's comment aside) that Y-on-X is a correct approach.  Multiple X's destroy that though, but X-on-Y is still correct (but not usable for predicting Y).
This is not a coding scheme because it is unable to distinguish clearly among the colors. For instance, it would equate the average of `Red` and `Yellow` with `Green`. It amounts to a kind of (arbitrary) *valuation* of the colors.  Therefore it does not have any standard name.
But what goes into the cells? Imagine that I asked people whether they prefer apples or pears 24 times, just phrasing it slightly different every time. I'm not interested in the individual answers on each of the questions (as I consider them equivalent) but whether people prefer apples or pears "overall". However, if I just sum up all the answers and put them in a Chi square table (by summing up I mean add all A's and B's, so if I have 10 participants I could find 130 times A and 110 times B), but am I then not throwing away valueable (per-subject) info?
I asked the question about [cultural differences in use of response scales on cogsci.stackexchange.com](http://cogsci.stackexchange.com/q/1033/52)
+1 very nice links
If at all possible, I would recommend simply putting the relevant text from your image directly in your question, and cite the source.
Ben, thank you very much for that answer! I wouldn't suppose this to be a reason, so your suggestion is of a great help for me!
Thank you very much. I did end up using a Gaussian Process. Thanks once again for the detailed answer.
1. *What is the exact meaning of the t-value in the student t test?* It's the number of standard errors of the mean difference that the means differ by. $ $ 2. What's a "BI"? Do you mean the confidence interval for the difference in population means? What's "B" then?
Great answer! I was really stunned how much information you packed in a single post
Peter, I think that's good advice. Plus, if the results aren't different then I think that would give evidence for the errors being more "at random" than systematically biased. If you get a chance, making that more explicit in your answer would probably help future readers. Cheers
Thanks for your quick reply! Although it didn't contain the solution, it did provide the needed insight.
You can model the counts of type $A$ crashes out of $A+B$ crashes in total as a binomial variate, on the assumption that type $A$ & type $B$ crashes occur independently. Locations with no crashes at all give no information on the proportion of type A crashes that they might have had. The next step is to consider the type of model to fit: is location the only predictor, & do you want to model it as a fixed or random effect (are you interested in making predictions for those locations with no crashes, or others)? Would you consider a Baysian multi-level model?
Covariance convey the information about the authentic variability while correlation does not. On the other hand, Pearson correlation captures only the strength of linear relation, whereas covariance isn't devoted just to linearity (unless the variabilities are taken under control). You may see my opinion [here](http://stats.stackexchange.com/a/62699/3277). Sometimes we will prefer covariances, sometimes correlations.
If the experiments are exploratory in nature then there is probably no point in doing any 'correction' for multiple comparisons.
Ok, therefore my test should be H0: mu >= 52 and H1: mu < 52, for the picture to be correct in the first case?
Do you believe this test is suitable for this amount of time poitns?
(+1) Nice to see you back answering questions!
I gave a derivation for 2 Poisson variables [here](http://stats.stackexchange.com/questions/88588/regression-model-for-proportion-or-count-when-counts-of-outcome-and-total-events/88690#88690). Note that the binomial distribution is conditional on the realized value of the random variable $N$, usually written as $n$.
That decreasing almost straight part marking the lower boundary of points on the first plot is almost certainly where $y=0$. That kind of thing is unavoidable, and that's also responsible for the apparent set of points lying on a sideways parabola in the Scale-location plot. Other than that, there's some suggestion of pattern but it's pretty mild; I'd probably want to see plots against the predictors.
Yes, take a look at the R penalizedSVM package. Other packages of interest are : penalized, elasticnet, ppls, lars, or more generally: http://cran.r-project.org/web/views/MachineLearning.html
"Most likely" has to be the mode, since it has the highest probability mass.
About the factors, I was conscious something bas was lurking in there — but I am not sure yet I fully understand.

In what I wrote, I have relations like $A_i = \\alpha_i A + \\epsilon$ — except that this $A_i$ is in fact a polytomy of this. Even if it where not ordinal, but quantitative, as I only give correlations (not even covariance) I don’t get how the scale of the $A_i$'s can be retrieved. I mean not only the variance of $\\epsilon$, but also the value of the $\\alpha_i$. However fixing it makes the fit worse! I understand things better when we fit raw data, but unfortunately this fails...
Wow, when I asked you to make it into an answer you didn't mess around! Thanks so much! Sorry for being whiny.
There's some discussion on this elsewhere on CV - http://stats.stackexchange.com/questions/2628/statistical-inference-when-the-sample-is-the-population.  While you need to be careful what inferential approaches you use, I cannot accept there is no point in many of the standard tests which test a null hypothesis that the population can be adequately modelled as produced by a specified data generating process.  There are many times we have the whole population but we want to model the data generating process - often much more interesting than the actual individuals who were generated.
@EMS, also Looking forward to your additions, thanks
+1 for mentioning the book. (and for having read a 440-pg book cover to cover!)
thank you! It seems that this could be applicable for my problem. I will play around a bit.
This (why that test uses two-tailed pvalues, rather than one, as in the F-test in ANOVA) is discussed in a number of posts on this site, including [this one](http://stats.stackexchange.com/questions/67543/why-do-we-use-a-one-tailed-test-f-test-in-analysis-of-variance-anova/73993#73993), and also in the extensive comments under [this one](http://stats.stackexchange.com/questions/55550/how-do-i-interpret-the-results-from-the-f-test-in-excel/55553#55553)
@MichaelBishop Why not simply resort on the [large-data](http://stats.stackexchange.com/questions/tagged/large-data) tag?
Just to clarify, are you talking about single items with 4 and 7 response options respective or scales that are the result of taking the sum or mean of a set of items where each item happens to have 4 or 7 response options?
Asking for a "hard-limit" is misguided when you haven't specified the purpose of the statistics: the replies are going to push back on this.  Even when the research objective is placed in evidence, there will always be mitigating factors modifying any rule of thumb.  Using statistical procedures effectively and correctly always requires *judgment.*  It's not blind application of algorithms.
I am reading up on dynamic linear regression and it seems to fit what I am looking for. Do you have any suggested reading materials?
If your residual of a point $i$ is zero, removing that point and re-running the regression will not change the line (it will still go through that point).
So if I change my code from "t(i)=1;" to "t(i)=0;" then it shifts the plot to the left, but it does not help the nonsmooth size change become intuitive.
I think we need a bit more information to answer your question.  For example, what are the main of the "various reasons" it doesn't follow the original plan.  Also, exactly what do you mean by calibrate - this has a narrow technical term in survey analysis, but your use suggests that perhaps you might mean post-stratification weighting to population.
Just to add that I tried the same thing in R and got the same behavior.  Namely, I opened two R windows, ran the command 
x <- rnorm(50)
and got the same vector of 50 random numbers in both windows.
We tend to (visually) expect that the line would go approximately through where the blue and green slices cross. But the expected value of the response at a specific value of $x$ is where the red line is. That shift from what we tend to expect by looking at the relationship toward a flatter line is literally regression toward the mean. It's one aspect of the tendency to see the table as suggesting a bad fit.
Can a discrete time series be projected/forecasted with continuous smooth cycles and moving averages models? Is Arima model more for series with trending & seasonal components?
Re: *"Ridge regression adds a small bias factor to the variables in order to alleviate this problem"* - can you say more about this, as it relates to the OP's query *"In what case should you use ridge regression as opposed to some other kind of regression"*? There are various forms of penalized regression that add a small bias factor, e.g. LASSO (which is known to have some bad properties with regard to collinearity). I think some intuition about why ridge regression, specifically, is a commonly used method when you have collinear predictors would improve this answer.
@AntonRoth: the standard textbook methods often have assumptions that are valid only for ordinary least squares (and possibly also only for the univariate case). For methods like PLS, they would seriously underestimate the real uncertainty. I added some rough explanations, a paper with an example and a theory paper to the answer.
Do they have an R-specific offering?  I'm not looking for a general cloud-computing provider, I'm looking for one that provides pre-configured R (preferably RStudio) servers.
Thanks for the appreciation DWin. I just graduated, and this is my first job as an analyst. Coincidentally, this kind of analysis is new to the company too. So, I am just trying to come up an analysis that is not completely nonsense. I will take your suggestion and try transformations on both $Y$ and $X$ variables. I will also go through the reference. I just found a pdf version online. Thanks for your input.
You may want to take a look at [this answer of mine](http://stats.stackexchange.com/a/22532/6633) which says pretty much the same things.
@pteetor That's interesting, I hadn't run across distance correlation before. Computationally, it seems more expensive than the entropy estimation approach for large sample sizes because you need the full distance matrices (where for the entropy estimators you can use indices to get only the first `k` neighbors). No idea how it compares in terms of statistical power / etc.
So are your data counts then?
Almost any continuous distribution on (0,1) might be argued to be suitable as a model for continuous proportions in some cases, as long as it has the properties we seek (we might look for smoothness, or unimodality for example - or we might not). Beta is certainly popular, and has some nice things about it. There's even some models of the situation that might lead us to the beta (such as independent unscaled components of the mixture we're looking at being gamma). Certainly a reasonable choice, but not the only one. It's not quite as obvious a 1st choice as the binomial is in count proportions
Because the negatives of left-skewed data are right-skewed, and negation is a linear operation, you can just negate all the coefficients (and residuals and predicted values) in a model of the negatives. However, it is doubtful that any such general-purpose mathematical manipulation will be appropriate for your data: you should select a model based on many more considerations beyond skewness of the responses (or even of the residuals).
+1 I hope that the OP can edit the question to match the high standard of the answer.
Good point Whuber. My apologies and I will try to provide better answers in future.
@dacongy Correlation will tell you how *consistently* the values change over time.  The slope/coefficient/"parameter" for time tells you how *much* they change per unit of time.  The *t*-statistic and *p*-value associated with that slope tells you something about how statistically significant it is--how rare such a slope would be if it were the result of chance alone (and if the slope in the larger population were zero).
5th D in color hue, 6th D in color intensity, 7th D in point size and we can go ;-)
Thank you for this superb answer :)
Welcome to this site! Could you clarify the following: "Interpretable" in what sense?
In Scala this would just return `Infinity`, but if your not using such an awesome language you'll have to change your cross-entropy function to be an tenary expression to handle the 0
Are the sample sizes fixed in advance? If not, you could use inverse binomial samplig, which basically reverses the problem: you set a desired confidence interval and confidence leve, and there's a stopping rule which tells you how long to continue sampling in order to guarantee that confidence interval and confidence level.
You must first figure out that barplots are made by barplot function... this is not that easy when you don't know that.
Your question is really the same as http://stats.stackexchange.com/questions/4220 (asking about values of a PDF) because the likelihood you are using is a product of PDFs. When you standardize them you change the unit of measure along the base of the graph, and so the height (as given by the PDF) has to compensate with an inverse change.  Have you noticed that the *differences* in the log likelihoods are nearly the same in both tables? (The differences should be identical, but considerable floating point roundoff error is involved in the calculations.)
This made laugh pretty hard: "I'm not really sure what MCMC is best suited to solve for, I just know that it does magic by smashing a bunch of random things together to get a good real answer."  Too perfectly stated.
I would add that the debate is not just philosophical - there are definitely times where it makes a difference which method you choose to adopt - particularly when it comes to quantifying the "error"/"uncertainty" in your estimate/conclusion.
This is good stuff (+1). However, I think you misuse conventional terminology and have co-opted "outlier" to refer to "influential observation." The concepts are both valuable, and you treat the latter well here, but they are not as interchangeable as you seem to indicate. For instance, an influential observation that *is* consistent with the majority of the data would fit your characterization of "observations that have an abnormal leverage (or 'pull') over the coefficients obtained from an LS fit" but would not be considered by most writers to be an "outlier" *per se.*
The formula given by @Mimshot is only for 2-class situation. See complete set of formulas to get the discriminants for k-class situation [here](http://stats.stackexchange.com/a/48859/3277)
Is your overall goal prediction of future cases, or trying to draw causal inferences that may guide future efforts at prevention? If your goal is the later, then you should be much more concerned with theoretical justifications of variable inclusion rather than statistical ones - i.e. accounting for bias is much more important than p-values for determining what should be in the model.
I think you've answered your own question @Baz. If we could do what you suggest, then why would we run the regression at all?
Something appears to have gone missing from the end of your first sentence.
Would R code that does a fully Bayesian analysis for this be useful for you? Likely it would be based on direct simulation and not use any packages or involve MCMC.
How many observations do you have? Sixteen?
Are you asking about differences in *rates* (# per unit time), differences in *counts* (total # of visits), or differences in the *durations* (amount of time passed until they were done w/ therapy)? Note also, that you probably didn't randomly assign the patients to differing diagnoses, so you can't infer the "effect" of the diagnosis, only that it is *associated* w/ differing responses.
See my response to User28.
Rotation of axes (factors) change nothing in the juxtaposition of the variables vis-a-vis each other in the space of common factors. Rotation only changes their coordinates on those axes (the loadings), which help to interpret the factors; the ideal here is some form of the so called "simple structure". Rotation is only for interpretation. You may rotate orthogonally, obliquely, rotate only this or that axis, or not rotate at all. That has nothing to do with the mathematical quality of your factor analysis. That's why they usually don't discuss `whether or not to rotate in the first place`.
+1 from me, excellent. So > 1.5 X inter-quartile range is the consensus definition of an outlier for time-dependent series? That would be nice to have a scale-independent reference.
@whuber: Thank you!
In 1 I think you have "fixed" and "random" just backward. Subject is pretty much always a random factor.
@MSalters: thank you for the clarification.  I'll leave my answer up for amusement value ;-).
Just added more info
Wow, thanks! This is a lot more efficient than recursively counting minima/maxima within a set interval... I'll have to go and check it out.
Thanks.  I try, as much as humanly possible, to translate statistical laments into corrective action, though an intensive education process with the client.
P.S. You can spot check their technique if you can find the median by state.  If their reported median national house selling price matches the median of reported state medians - then they are doing what you fear.  If not, then you can probably assume that they are using the simple median of all houses sold.
KL is not a random variable. Are you interested in the distribution of KL when the parameters are replaced with their estimates based on a random sample ? Your question should be precised.
The overal variance is the trace (sum of elements on the diagonal) of the matrix, so with correlation PxP matrix this "variance"=P. PCA (by eigendecomposition or SVD) aims to capture the possible greatest share of the trace - it is how it works, it is its objective function.
Yes, sorry about that. I got these values from the Discrete module in BayesTraits. Each value in the matrix is the difference in log-likelihood values between a 4-parameter model of independent evolution and an 8-parameter model of dependent evolution. I apologize if I used the terms "correlation" or "correlation matrix" incorrectly.

The documentation is at http://www.evolution.rdg.ac.uk/BayesTraitsV2Beta.html . The data here is the result of an analysis with two inputs: A phylogeny, or evolutionary tree, and a matrix of presence/absence data (1/0) for 2 traits, for each species in the tree.
THIS IS FANTASTIC THANK YOU
What is your explanation of the constancy of the F-statistic?
+1. Michael, given your many expert contributions on this site, it would help users if you would learn and use mathjax so that your answers are easier to read and follow. You can infer a lot about how it works by looking at the edits people have made on your answers. You can also see how any expression on CV was produced by right-clicking on it and then selecting 'Show Math As' -> 'TeX Commands'. I have also found [this](http://www.onemathematicalcat.org/MathJaxDocumentation/TeXSyntax.htm) to be a very comprehensive reference (albeit slow to load). Thanks again.
+1 But Types I and III do not apply to the question.
@TomRitter, yes those calculations are correct.  So there is a very high probability of randomly guessing more than 8 out of 126 bits.
Thanks for your answer Marc. So can we distinguish the suitable method based on the dataset we have? If so, can you give me some pointers? I have 40000 samples with 200 distinct outputs. So it can be thought of as 200 sets of 200 unique samples. The input for all 40000 are different though, just the output is unique for 200 samples.
I'm sorry, I should have added simulation results right from the start. See my edit. I don't expect the combination (2,2) to be generated at all. The situation in the original data is very similar.