1.8b - Chapter summary
In this chapter, we reviewed basic statistical inference methods in the context of a two-sample mean problem. You were introduced to using R to do permutation testing and generate bootstrap confidence intervals as well as obtaining parametric t-test and confidence intervals in this same situation. You should have learned how to use a for loop for doing the nonparametric inferences and the t.test function for generating parametric inferences. In the two examples considered, the parametric and nonparametric methods provided similar results, suggesting that the assumptions were at least close to being met for the parametric procedures. When parametric and nonparametric approaches disagree, the nonparametric methods are likely to be more trustworthy since they have less restrictive assumptions but can still have problems. When the noted conditions are not met in a hypothesis testing situation, the Type I error rates can be inflated, meaning that we reject the null hypothesis more often than we have allowed to occur by chance. Specifically, we could have a situation where our assumed 5% significance level test might actually reject the null when it is true 20% of the time. If this is occurring, we call a procedure liberal (it rejects too easily) and if the procedure is liberal, how could we trust a small p-value to be a "real" result and not just an artifact of violating the assumptions of the procedure? Likewise, for confidence intervals we hope that our 95% confidence level procedure, when repeated, will contain the true parameter 95% of the time. If our assumptions are violated, we might actually have an 80% confidence level procedure and it makes it hard to trust the reported results for our observed data set. Statistical inference relies on a belief in the methods underlying our inferences. If we don't trust our assumptions, we shouldn't trust the conclusions to perform the way we want them to. As sample sizes increase and violations of conditions lessen, then the procedures will perform better. In Chapter 2, we'll learn some new tools for doing diagnostics to help us assess how much those conditions are violated.