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Background:

I’m exploring if hair cortisol levels (pg/mg) are related with executive functions like working memory (measured on WISC-scale), inhibition control (BRIEF-scale) and attention shifting (BRIEF-scale) with SPSS. My sample sizes are 11 (group 1) and 21 (group 2). I want to study the groups together (are cortisol and executive functions related) and separately (is the relationship different in different groups). The hair cortisol values are positively skewed.

Questions:

  1. Other studies in this field have used log transformation of the (hair) cortisol without bootstrapping, but I want to use bootstrapping because of the small sample size and I’m now wondering whether to log transform the data before bootstrapping or not?

  2. Is the regression analysis right method for estimating the relationship between cortisol and executive functions? If not, what alternatives could you recommend? My hypothesis is that the relationship is U-shaped (very low and very high cortisol levels are related with more problems in executive functions), so I would add cortisol to the model in the first step, and squared cortisol in the model to the second step.

  3. I want to study if the sex, age and grouping variable have an impact on the relationship of working memory and cortisol. Can I explore this with a general linear model?

I would really appreciate for any feedback!

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  • $\begingroup$ It doesn't matter if you log transform before or after standard bootstrap... $\endgroup$ – Tim Apr 29 '17 at 7:45
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Let $X_1$ be the indicator of group 1, and $X_2$ be hair cortisol levels. $Y$ be working memory. The model will be as following:

$Y=\beta_0 + \beta_1X_1 + \beta_2 X_2 +\beta_3X_2^2 +\beta_4X_1X_2+\beta_5X_1X_2^2 + \epsilon $

If $E(Y|X_2)$ and $\log(X_2)$ have better linear relation for given group, then transform it. Otherwise, do not.

You can delete the non-significant terms step by step.

Replace $Y$ with inhibition control and attention shifting, fit other two models.

Do not use bootstrapping because of small sample size. Bootstrapping is used for some difficult situations, but cannot increase the sample size.

Theoretically, you can add other factors such as gender into the model. But given the sample size is 32, I would not do it if I analyze these data.

Example of not using bootstrapping:

When you estimate the mean from a sample following normal distribution, then sample mean +/- t * SE is the best regardless the sample size. If you perform bootstrapping with 1,000,000 replica, you may get the same result. Example of using bootstrapping:

You find a way to estimate parameter u with statistic T, but you cannot derive the variance of T mathematically, and of course you cannot estimate the variance of T from sample. Under this situation, bootstrapping can be used to estimate the variance of T.

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  • $\begingroup$ Thank you for your answer. My intention is not to "increase the sample size", but to gain more reliable significance levels. I have understood that bootstrap samples the data several times and can be done across many analyses, which is why I thought I could use it also with my t-tests (when I compare the group's executive functions together). I have watched this [short educational video about bootstrapping(youtube.com/watch?v=9VjzPnoUBJQ). Could you elaborate why bootsrap won’t be suitable in my case? I was wondering why doesn't everybody use it to be surer about their results? $\endgroup$ – HappyZebra Apr 29 '17 at 11:21
  • $\begingroup$ I found this discussion from here link, is my small sample size reason for not to use bootstrapping at all? I'm very confused, because some people at the university recommended to use it precisely because of the small sample size.. $\endgroup$ – HappyZebra Apr 29 '17 at 11:32
  • $\begingroup$ Misusing bootstrapping is very common. If you really want to know bootstrapping, the paper by Efron is the best. I added two examples in Answer. Sample size is not a factor to determine use or not use bootstrapping. $\endgroup$ – user158565 Apr 29 '17 at 14:11

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