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I'm running a SEM in which I have several very positively skewed endogenous variables. Unfortunately, even when I log transform these variables they are still quite non-normal. Kline (2011) p64 writes that "Some distributions can be so severely non-normal that basically no transformation will work", and so I gave up trying to transform them more.

Kline (2011) provides another option on p177-178:

[An] option for analyzing continuous but severely non-normal endogenous variables is to use a normal theory method (i.e., ML estimation) but with nonparametric bootstrapping, which assumes only that the population and sample distributions have the same shape. In a bootstrap approach, parameters, standard errors, and model test statistics are estimated with empirical sampling distributions from large numbers of generated samples. Results of a computer simulation study by Nevitt and Hancock (2001) indicate that bootstrap estimates for a measurement model were generally less biased compared with those from standard ML estimation under conditions of non-normality and for sample sizes of N ≥ 200. For N = 100, however, bootstrapped estimates had relatively large standard errors, and many generated samples were unusable due to problems such as nonpositive definite covariance matrices. These problems are consistent with the caution by Yung and Bentler (1996) that a small sample size will not typically render accurate bootstrapped results.

In this answer @michael-chernick writes that

The theory of the bootstrap involves showing consistency of the estimate. So it can be shown in theory that it works for large samples. But it can also work in small samples. I have seen it work for classification error rate estimation particularly well in small sample sizes such as 20 for bivariate data.

Some questions:
1. Those two quotes seem at face value to be in conflict with each other, but I note that @michael-chernick was answering a question that did not involve SEM. Does SEM require a larger original sample size for successful bootstrapping? If so, why?
2. If Kline is right that bootstrapping will work poorly with a low N, what should I do if I have a low N? Say for the sake of argument I have a sample size of 100 and no way to collect more data. Should I go ahead with bootstrapping, and if so should I bootstrap using the transformed variable (remembering that the transformation didn't do a great job of resolving the non-normality) or the original variable?

Kline, R. B. (2011). Principles and practice of structural equation modeling. Guilford publications. Chicago

Yung, Y. F., & Bentler, P. M. (1996). Bootstrapping techniques in analysis of mean and covariance structures. Advanced structural equation modeling: Issues and techniques, 195-226.

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  • $\begingroup$ I might note that Michael's answer does not claim that the bootstrap works unconditionally in small samples. The more pertinent question may be: in what cases does the bootstrap work well in small samples? $\endgroup$ – AdamO Jan 17 '18 at 21:57
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Yes, SEM requires a larger size. The reason being that SEM is doing two things: First, it's trying to find a model, and then it finds the standard errors of that model.

There are two problems. The first is that you will have trouble estimating the model(s).

If you have problems with your standard errors (because, say, of non-normality) then bootstrapping might help you. But if you try to run a SEM model with a small sample size, you'll find that you don't get a sensible model to interpret - the model will frequently not converge, or converge with out of bounds estimates (variances < 0; correlations > 1 [perhaps MUCH greater than one - one sometimes sees correlations that are in the three digit range]).

So when you try to bootstrap a model with a small sample size you might find that 25% of the bootstrap samples are clearly wacky and should be discarded. And some proportion of the rest are also wacky, but you don't have a good way to decide which ones. If you did, you could go ahead and use the standard errors.

The second problem is that ML tends to be biased in small samples.

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The sandwich standard error estimator is a first order approximation to the bootstrap because it considers the empirical influence function of the data. You may know it by other names such as the Heteroscedasticity Consistent estimator, the Huber White standard error, or even just robust standard erorrs. Unlike the bootstrap, the sandwich is solved analytically, and while it performs poorly in small samples like the bootstrap, it does not perform quite as badly and generally converges faster to the solution than the bootstrap. Most SEM implementations have options to estimate the sandwich standard error as a 3rd alternative to model based or bootstrap standard errors.

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