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.
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.