During this last week I focused on an implantation of the classical bootstrap, as well as the bootstrap-t technique (see previous post for a detailed description of the latter), to provide a robust estimate of significance for the results of the group-level linear regression analysis framework for neural time-series we've been working on during the last few weeks.

In particular, this week I was able to put together a set of functions in a tutorial that shows how the second-level (i.e., group-level) regression analysis can be extended to estimate the moderating effects of a continuous covariate on subject-level predictors. In other words, how variability in the strength of the effect of a primary predictot can be attributed to inter-subject variability on another, putative secondary variable (the subject’s age, for instance).

On a first step, the linear model is fitted each subject’s data (i.e., first level analysis) and the regression coefficients are extracted for the predictor in question. Then the approach consists in sampling with replacement an n number of second level design matrices, with n being the number of subjects in the original sample. Here, the link between subjects and covariate values is maintained, so for simplicity the subject indices (or IDs) are sampled. Thus, the linear model is fitted on the previously estimated subject-level regression coefficients of a given predictor variable, this time however, with the covariate values on the predicting side of the equation.

Next the second-level coefficients sorted in ascending order and the 95% confidence interval is computed. In the added tutorial (see here), we use 2000 bootstraps, although "as little as" 599 bootstraps have been previously shown to be enough to control for false positives in the inference process (see for instance here).

One challenge is however that no P-values can be computed with this technique. One was to derive a decision on the the statistical significance significance of this effect cane be achieved via the confidence interval of the regression coefficients: a regression coefficient is significant if the confidence interval does not contain zero.