tttjjj/elt_bench huggingface dataset:
1) Not sure what you mean; do you mean that your two parameters are on a line in the unit square (and are therefore really only one parameter?)  2) What 2 refers to is transforming the parameters to other parameters, e.g., y = x/(1-x) has range $(0,\\inf)$ when $x \\in (0,1)$ and sampling from y instead of x.  3) The $\\beta$ proposal works; you just have to incorporate the proposal probabilities in the calculation of the acceptance ratio (standard MCMC.)  You don't need to have a symmetric proposal (which is what I think you mean by "invertible") for ergodicity to hold.
@user4959, I don't know R. [Here](http://www.statmethods.net/management/operators.html) !X is recommended; however the result is boolean TRUE/FALSE, not numeric 1/0. Note that I updated my answer where I say that there is also another way to arrive at D matrix.
Thanks for doing the timings, Gavin - that's really helpful for questions like these.
+1. It's almost as if Nik were describing SVM's, not having heard of them. In R, you could use the (mysteriously-named) `e1071` package's `svm` with `kernel="linear"` and look at the prediction versus actual.
This related question might be useful, [Statistical test for Positive and Negative Predictive Value](http://stats.stackexchange.com/questions/6309/statistical-test-for-positive-and-negative-predictive-value).
@KjetilLoland That may be a reason to use doubly robust estimation - generally, it's something to look at whenever you're concerned that one method for controlling for variables is suffering from misspecificication. I'd also check to make sure your PS model isn't acting up and is giving you nice, overlapping propensity scores between the two groups.
+1: Very intuitive answer!
And the 5 samples differ substantially (the measure is reliable?) If the measures are decently correlated then maybe you could just use one. Alternatively, you could perhaps use the mean of the 5. Is that a possibility? What is the correlation between the 5 measurements?
You may be confusing the assumption of "homogeneity of variances" (explanatory variable has distinct groups as in ANOVA) with the assumption of "homoskedasticity" (explanatory variable is continuous as in regression). Levene's test cannot test the latter assumption. See [this answer](http://stats.stackexchange.com/questions/15427/kruskal-wallis-or-fligner-test-to-check-homogeneity-of-variances/15471#15471) for details.
AIC can also be used as (approximate) Bayesian model probabilities with non uniform model prior.  By using BIC to approximate the likelihood $P(D|M_i I)\\propto\\exp\\left(-\\frac{1}{2}BIC_i\\right)$, and then with the prior for the ith model is given by $P(M_i|I)=N^{\\frac{d_i}{2}}\\exp(-d_i)$, where $d_i$ is the dimension of model $i$.  We get posterior probability of $P(M_{i}|DI)\\propto\\exp\\left(-\\frac{1}{2}AIC_i\\right)$
True. I was worried whether people might take it that literally as "uniformly better than Bessel". I've updated the conclusion to reflect that.
@Michael Thanks for the clarification. If he wishes to relate two time series , one at a different level of aggregation/bucket size then he has to convert one of the series. In the case of y being monthly and x being quarterly, I have delivered code that linearly interpolates the quarterly numbers to obtain "monthly estimates". One could use a quadratic in time to interpolate as an alternative.
@MichaelChernick: I am not sure I agree with the first part of your reply. The distribution of $Z$ is the convolution of the distribution of $Y$ by an exponential distribution. For a given $\\lambda$, there should be a single correspondence between $f_Z$ and $f_X$.
@cyan - Note that while his resolution was flawed in some areas his underlying principles solved it.  The general rule of proper priors and their convergent limits means the mp cannot arise. The flaw is most likely due to the book being unfinished over most of part two.  I like the resolution [here]( http://arxiv.org/abs/math/0310006) better than ksvh version.  shorter and more general.
Close, but you're forgetting about the test statistic, which partitions the sample space & orders those partitions:  "...under the null model you would observe a test statistic at least as extreme as that calculated for your particular sample" would do it.
I believe I have tracked down the right person's work now; I just need to make sure I have the right paper.
@JakeWestfall You're right, when I first wrote this it was kind of a stream of consciousness question as the problem developed. I'll do some editing and see if it can be more helpful. Unfortunately, the code has wandered off somewhere...
Yes, I know psych principal wraps this up.  My purpose was to show what SPSS "factor analysis" was doing when using the principal components extraction method.  I agree that the eigenvalue rule is a poor way to select the number of factors.  But, that is exactly what SPSS does by default and this was what I was demonstrating.
I'm interested but could you explain a little more?
OK so I will use "binary hypothesis testing" if I want to be esoteric. It seems marketers have been doing "split testing" long before "A/B testing" was coined by google. The first such test I can find however was when William Sealy Gosset invented the t-test in 1908 for the Guinness brewery to compare beer. Would be interested if anybody can find something that predates this but it is rather poetic if it all started with beer.
Hi.Thanks to all for the suggestions. I ended up looking at my residuals as suggested and using the syntax above with my variables. My data was not as disasterously non-normal as I'd thought so I've used my parametric linear regressions with a lot more confidence and a clear conscience! Thanks again.
`auto.arima` requires time series to proceed correctly, so just make sure you supply the function with `ts` at needed seasonal frequency.
I have a meta-comment: Please consider which of your previous questions might have long-term value for others and delete any that don't qualify, so long as there were no up-voted answers. Some questions seem to clarify misunderstandings, which is fine, but if the route taken from start is end is too specific in terms of what you did, it's unlikely that they should remain on the site.
Thank you. I adjusted the question to "Should alice make wet weather plans for tomorrow ?". I copied this question directly from my CAs with nothing changed in the main body. The question is made up by me.
right, sorry, the weights are independent from $x$. The restrictions on the weights characterize, evidently, the class of 'coherent' risk measures. I have not seen any suggestions about how to set them, though, and this appears to be a black art. However, perhaps one would choose them to get the best discriminatory power (i.e. smallest standard error)...
I'll second the suggestions that usage can make a concept clearer, even though its original definition may not yet make sense.  A very good example of this is from Euclid's Elements.  It looks trivial when you first begin to read it, if you've already studied number theory, geometry, algebra, etc.  However, as you progress, it is a marvel of simplicity and clarity.  I frequently tried to reduce the number of steps needed for a proof, and I don't recall that I ever succeeded.  The process of reading this book readies the mind for reading and writing other definitions.
Aren't the two squared correlation values equal? The predicted values being left by both models are linear combinations of the same terms except for a constant term which does not affect a correlation; the correlation between predicted values of the two models is thus 1.
I would be wary of applying the test to many different scenarios as you may "unwittingly happen onto a solution by chance".  What I mean is that if you repeat the test many times, you are not using the original test anymore (most tests are not designed to handle being used repeatedly on the same data).  See this paper by [Salzberg](http://www.google.com/url?sa=t&rct=j&q=salzberg%20pitfalls&source=web&cd=1&ved=0CCAQFjAA&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.183.534%26rep%3Drep1%26type%3Dpdf&ei=0w-TT6LPIcOLswaBic2VBA&usg=AFQjCNFIsQZqExas4JrucNd69vhOBelP0w).
@Macro, could you expand on the idea?
I think it would be better for your to formulate this as a stand-alone question and post it separately rather than as an answer here.
I am not familiar with the software but if Std. Error ist the standard error of the estimate of the differences in means as I expect, the t-value is just the difference divided through the std. error (about 3.08 in the first row).
@GaetanLion Sorry, didn't have time to read all replies - only the question. I just shared my particular experience with building these stress scenarios forecasting models. Consider it as an extra evidence for your point of view.
Sounds like you need some sort of resampling or permutation test or something like that
@whuber: As you say, the cf, mgf, and pgf are more-or-less the same and easily transformable into one another, however Mathematica has a cf builtin that works with all the probability distributions it knows about, whereas it doesn't have a pgf builtin. This makes the Mathematica code for working with sums (and differences) of dice using cfs particularly elegant to construct, regardless of the complexity of dice expression as I hope I demonstrated above. Plus, it doesn't hurt to know how cfs, FTs, convolutions, and cross-correlations can help solve problems like this.
@whuber, I recommend to make "cart" tag a junior synonym of "decision-tree" or "classification-tree". Because, often CART acronym is regarded to be just a particular _method or algorithm_ of the tree, along with other methods such as CHAID or QUEST.
Thanks.  I thought something like this might be the problem, but I wasn't sure.  I was a little bit confused about the the finer points of what constitutes a nested model for the purpose of Wilks' Theorem.  Good point about identifiability under the null.
Gotchya... I *was* assuming that @SamPatch had a specific alternate distribution in mind.
Some more thought has convinced me that I cannot get a (natural-looking) smoother that satisfies my requirements above and also passes through the data points. (Let two series coincide on an upward trend then diverge with one continuing upwards and the other going back down. Think about the derivatives at the point where they part company.) Looks like this is the best I'm gonna get.
Have you plotted the population over time? Maybe the growth pattern will help in postulating a model.
Elvis has a good answer, but I would add (1) have you tried the classification model that predicts C for all cases? This seems like the better baseline prediction model. (2) Using the c-statistic to compare models can be suboptimal. There are many threads on this site discussing this issue.
Confidence intervals for what?
No problem, @Will, if you have any questions about GEE vs GLiMM's or any confusions, you can comment there & ping me as well. GL
1) Causality has nothing to do with it. 2) Be careful about "substantial" vs. "significant". With a large enough N, very small differences can be significant; with a small N, even large differences may not be.
Take that predictor with a large coefficient and change its units of measurement. For instance, if it is a length in kilometers, change it to meters.  Re-running the regression will create an inverse change in the coefficient (in this instance, it will divide it by 1000).  Suddenly, *due to an arbitrary and irrelevant change in how the data are recorded,* the coefficient undergoes an enormous change.  For more about these issues, please consult the "Related" links at the side of your screen.
@EMS, a Monte Carlo calculation is an *approximation*.  It therefore behooves one to know how accurate that approximation is, which is why it is reasonable for your advisor to ask for some assessment of the quality of that approximation.  More to the point, once you know how to quantify the connection between the number of MC iterations and precision of the p-value, you can determine (in advance) how many iterations may be necessary.
In the bivariate case, yes, but I'm assuming that $X_{1}$ and $X_{2}$ are both multi-dimensional.
"*Due to fact that there are only two sites, than random effect model or repeated measures ANOVA also can't be used.*" ... really? Why not? (This is not sarcasm; I may be ignorant of something obvious.)
I am also assuming that unemployment is a percentage.
In Stata `vce(robust)` rather than `robust` is what is now documented, but `robust` should still work.
Have you looked at the very first figure in the Wikipedia article on [consistent estimators](http://en.wikipedia.org/wiki/Consistent_estimator), which specifically explains this distinction?
Are you familiar with [tag:var] (vector autoregression)? That sounds a lot like what you are talking about.
Your description is good answer for the general scenario. Just to make the discussion a bit clearer in context of EM algorithm (as OP's original question), I would also add: because the EM algorithm replaces the missing observations in the complete data by their conditional expectations, in the context of exponential family, this is equivalent to replacing the missing observations in the sufficient statistics by their conditional expectations.
+1 for being gentle :-).  A nit, but maybe a revealing one: a suitable variance estimator in this situation is the number of successes themselves.  Thus one can indeed bootstrap from a single observation.  But of course that would be self-fulfilling: a Poisson bootstrapped dataset is going to look Poisson!
Thanks Mike. Are you saying in these situations, examination of the confidence intervals will show that they are so wide that we should not really trust the exact value of our estimates?
@B_Miner: you are right, if called with "standardize=TRUE" glmnet returns coefficients on the original scale. One way to get around that is to standardize the explanatory variables outside (e.g. using "scale()" function) and call glmnet with "standardize=FALSE". The resulting coefficients could then be ranked by magnitude to judge their importance.
MAybe this will help http://stats.stackexchange.com/questions/33053/incorporating-a-response-variable-in-principal-component-analysis
It's a good question, but in the interest of pursuing what might be an important issue, I would like to point out that the much simpler approximation $F(\\mathbb{E}(Y_n^{(n)}))\\approx 1/2$ already works pretty well: it is never off by more than $1/2$! The point, of course, is that it would behoove us to *quantify* the accuracy of your approximation and to establish a threshold to distinguish good from poor accuracy for your application.  Do you have any thoughts about that? (BTW, the right hand side has a Beta distribution for any order statistic provided $F$ is a continuous distribution.)
@Dail, look at the assumptions of CUSUM test. IrishStat summarized the violations of these assumptions.
Does `R` know wealth quintile is a factor?
Can you amplify on what DNA methylation data look like? It will help your audience to answer your question.
Can you say more about what your data are? How are they gathered / measured? What is the "background noise"? How is it assessed? etc.
All this begs the questions of why we need formal hypotheses vs. estimating a quantity of interest and reporting confidence intervals.  There is no bifurcation implied by estimation.
I agree you need to find a specification for the X partof the ARMAX Model.Classical ARIMA modelling died in about 1979 when work was done to empirically identify via Intervention Detection Pulses,Level Shifts and Local Time Trends (see I.Chang or G.C. Tiao or Ruey Tsay's literature on this). Their work and in some rare cases my developments of their work have lead to much more usable ARIMA structures incorporating Deterministic Structure such as multiple time trends and/or multiple Level Shifts.Level Shifts are simply first differences of a time trend as Pulses are to Level Shifts.
You have no placebo, no control group, the number of sessions attended may be correlated with significant individual traits (dropout NOT at random)... You are going to have some hard time persuading folks that your anxiety/depression treatment works, and if it does this is not just because of Hawthorne effect...
You could eliminate the last 80% of this work by starting with the observation that $\\rho$ is invariant under location and scale changes, thereby reducing the problem to expressing $\\sum x_iy_i$ in terms of $\\sum(x_i-y_i)^2$ when $\\sum x_i^2=\\sum y_i^2=1$; the formula obviously is $\\frac{1}{2}\\sum d_i^2=\\frac{1}{2}\\sum(x_i-y_i)^2=1-\\sum x_iy_i$. Then the only real work to be done is accomplished by your calculation of the denominator.
(+1) By the way, I believe the OP uses "multiple comparisons" in the last sentence to refer to the simultaneous comparison of the 4 groups, i.e. the omnibus test. But I could be wrong... I'm a little bit confused by the phrasing of the question.
I will be running a two-step cluster analysis, if that answers your question...
So, if your equality constraints are $A x = b$, at least a *work-around* for `constrOptim` would be to include both $A x \\leq b$ and $A x \\geq b$. No?
@Glen_b you were right. In another step the variable s is stochastic. Because of a bad initialization it couldnt find stationary distribution so the number added was extremely big.
Asserting $E[R(n)]$ goes to zero with large $n$ means, *for certain,* that the gun sights are absolutely dead-on accurate. I do not see how you can possibly know that; in fact, it seems to me that is the principal question you are addressing when you conduct this kind of study. Declaring $(\\mu_X,\\mu_Y)$ to be the origin doesn't work, *because you do not know these coordinates*! You have to use an origin you do know, such as the target center. The two big questions are (1) accuracy: what is $\\sqrt{\\mu_X^2+\\mu_Y^2}$ and (2) precision: what is $\\sigma$? *These* are what you need CIs for.
I'm not familiar w/ Rminc. Does it not report p-values? You could use the standard R function [?wilcox.test](http://stat.ethz.ch/R-manual/R-patched/library/stats/html/wilcox.test.html), which does report p-values.
@ Momo - thanks! That worked.
Doesn't make too much sense, I guess; did it anyway for curiosity.
Had an error in 4., see update.
I think my use of the word "state" probably does imply that -- although I did want to think about a "continuous" situation, where the "birth rate" of the process declines gradually (perhaps precipitousness) as the population grows.
@jsk Thank you for taking a look at this question and your thorough answer. I wish to determine if there is a difference in how many positive responders from the early time period (pos.early) shift to positive, negative, and neutral in the late time period. It looks like the chi-square goodness of fit might work for me - seeing my research question, do you still agree that this is the case?
Does the training set have patches with very tall or very short trees?  What does the model predict for those patches?
Look up "Conformal prediction".
A agree that that would be a nice resource. I don't know of any, but it'd be a worthwhile thing to set up (maybe in wiki format, so it can be kept up to date).
Especially for small datasets where maybe leaving out data from CV is just not feasible but the risks of overfitting your model are also high! References are needed to clarify this issue.
I don't think it is accurate. First, can someone write down the Likelihood of a T-Test? Then we can start talking about analogies. Well, and if you can't... we'll that picture does not make sense.
I'd recommend Lumley's book, particularly if you use R.  He has a web page with lots of resources here http://faculty.washington.edu/tlumley/survey/.  Yes, counts for each combination of age, sex and education is a classic list of combinations to do post-stratification weighting on.
Have you noticed your estimator is not unimodal, but that the recommendation you are following explicitly applies only to "unimodal" distributions?  That doesn't mean you're doing something wrong, but it should generate some hard thinking about what the answer might mean.
My guess is that you might want to use a Poisson distribution instead of a normal distribution because all of your data are counts. Then maybe try comparing your data with the expected values arising from a Poisson distribution with a mean of 5 using a Pearson's chi-square test.
Unfortunately, all of my books I could use to double-check all of that are in the mail.
See [this answer] of mine for ways of constructing random variables whose characteristic functions are the same as their pdfs.
There is no reason why you can't have a very large chi squared test statistic if the expectations vary a lot between groups and you have a large sample for each group. In such a case you are nearly certain that the differences between the groups didn't happen by chance.
Statistical projections into the future are often based on simple-minded assumptions, such as: the trend seen in the past will continue in the future.  Such assumptions are often, even typically, wrong.  For example, the rate of inflation may change, or we may have an unexpected (or expected) financial crisis.  So, think long and hard about how you make projections into the future.
Apart from the effect of discreteness, under the null, both sets of p-values should be uniform, shouldn't they? As for comparing their distribution, if you want a hypothesis test, there are a number of goodness of fit tests, but at sufficient sample size they'll reject any difference, even if it's not practically important. It may be better to identify the effect size (the size of the difference in cdf, say, near commonly used significance levels) in order to demonstrate their closeness.
Thank you. This is what I suspected, but my statistical knowledge is limited (thus far :) In my case I do have reason to suspect the "squareness" to affect the dependent variable. But I wasn't sure if the width and height variables already took things into account. Thanks for your info regarding ratios and nonlinearity. I'll need to study up some more there.
I cannot understand your last sentence. It may help you to read: [What is the difference between linear regression on y with x and x with y?](http://stats.stackexchange.com/q/22718/7290)
If you boil it back down to linear you should see the reason. The 95%CI of the predicted slope is the narrowest at the means of x and y. The behaviors you are expecting is perhaps of the prediction interval, not confidence interval. Why the narrower range in 4th month, I am not sure... probably the combined effects of the different independent variables.
@PeterFlom Also, are there any other descriptive statistics I can show on x that would show a significant relationship, maybe p-value or r-squared?
Michael, sarcasm? People *do* insist on a certain degree of precision, for a reason: many, if not most, of our questioners come to get advice to solve their problems and are not, themselves, experts in statistics. As you gain more experience here, you'll notice that overgeneralizing is one of the most likely mistakes a questioner will make. Often to seek improvement in both questions and answers, a form of Socratic questioning is used. Cheers.
Well, yes, "related" is better. CCA takes account for both inter-covariances and cross-covariances.
Sounds like you might be able to do some sort of bootstrap. @Whuber might be able to point you in the right direction.
Also, you probably don't want to use the logistic regression here since the restriction that results be between zero and 1 is most likely an unreasonable assumption on your part. As an aside, your data will be more readable if you follow the convention of using **x** to signify your predictors and **y** to be the response.
No problem, @Dostre, welcome to the site.