The maximum likelihood (ML) estimate of a parameter is the value of that parameter under which your actual observed data are most likely, relative to any other possible values of the parameter. The ...

Post hoc power analysis That is, using power analysis after a study has been completed rather than before, and in particular plugging in the observed effect size estimate, sample size, etc. Some ...

P(HHHHH) There are 32 possible outcomes from flipping a coin 5 times. Here they are listed: HHHHH THHHH HTHHH TTHHH HHTHH THTHH HTTHH TTTHH HHHTH THHTH HTHTH TTHTH HHTTH THTTH HTTTH TTTTH HHHHT THHHT ...

There are several things going on here. These are interesting issues, but it will take a fair amount of time/space to explain it all. First of all, this all becomes a lot easier to understand if we ...

Consider the table below (copied from this website) representing joint probabilities of outcomes from rolling two dice: In this common and natural way of showing the distribution, the marginal ...

The short answer is that your conjecture is true when and only when there is a positive intra-class correlation in the data. Empirically speaking, most clustered datasets most of the time show a ...

Treatment coding doesn't always or necessarily result in intercept/slope correlation, but it tends to more often than not. It's easiest to see why this is the case using pictures, and considering the ...

1. A famous example in psychology and linguistics is described by Herb Clark (1973; following Coleman, 1964): "The language-as-fixed-effect fallacy: A critique of language statistics in psychological ...

This is not technically an error in statsmodels, rather it is because statsmodels.OLS does not add the intercept/constant term to the right-hand-side of the regression equation by default -- you have ...

Here is another geometric view of suppression, but rather than being in the observation space as @ttnphns's example is, this one is in the variable space, the space where everyday scatterplots live. ...

The direct answer to your question is that the last model you wrote, anova(lmer(y ~ a*b*c +(1|subject) + (1|a:subject) + (1|b:subject) + (1|c:subject) + (1|a:b:subject) + (1|a:c:subject) +...

Couldn't we have normal residuals at each predicted value of y, while having overall residuals that were quite non-normal? No -- at least, not under the standard assumption that the variance of the ...

I don't think there's a problem. The lesson from the model output is that although there is "obviously" variation in subject performance, the extent of this subject variation can be fully or virtually-...

Assuming $p$ is the probability of an event, $1 - p$ is the probability of its complement. If $p$ is not the probability of an event then I doubt that $1 - p$ has any special meaning or name.

The unregularized model is suffering from complete separation because you are trying to predict the dichotomized variable price_c from the continuous variable price from which it is derived. The ...

As Ben Bolker already mentioned in the comments, the problem is as you suspect: The lmer() model gets tripped up because it attempts to estimate a variance component model, with the variance component ...

In addition to the numerous (correct) comments by other users pointing out that the $p$-value for $r^2$ is identical to the $p$-value for the global $F$ test, note that you can also get the $p$-value ...

Yes, there is a consensus: you should use the variances, not the standard deviations, in computing the intra-class correlation (ICC). The two-level random-intercept-only model is  y_{ij} = \beta_0 +...

Put simply, the pooled variance is an (unbiased) estimate of the variance within each sample, under the assumption/constraint that those variances are equal. This is explained, motivated, and ...

Basically there's no single number or estimate that can summarize the degree of clustering in a random slopes model. The intra-class correlation (ICC) can only be written as a simple proportion of ...

There's more than one way to test this hypothesis. For example, the procedure outlined by @amoeba should work. But it seems to me that the simplest, most expedient way to test it is using a good old ...

The fourth point is an example of Berkson's paradox, also known as conditioning on a collider, also known as the explaining-away phenomenon. As an example, consider a young woman who is frequently ...

"I have always been taught that random effects only influence the variance (error), and that fixed effects only influence the mean." As you have discovered, this is only true for balanced, complete (...

Having random slopes at level 1 is not a necessary condition for examining cross-level interactions. All that is necessary is that you have 2 predictors that vary at different levels, and their ...

A simple two-sample t-test would actually be a conservative way of testing for time differences here. Let's say that $t_{paired}$ is the t-statistic that you would get from a paired t-test on your ...

Sort of. There is the so-called "one standard error rule," which does use the standard deviation of the prediction error estimates, although not in quite the way you mentioned: instead you divide the ...

Technically, z-scoring does not depend on any distributional assumptions, such as normality. It's just a way of describing how far observations are from the mean, no matter what the distribution ...

Your last 2 contrasts are right, but the first 3 are wrong. We can verify this by figuring out the linear combinations of coefficients that give each group mean, and then constructing the desired ...