Psychology 202a Advanced Psychological Statistics October 6, 2015

The plan for today Assumptions of the t test. Quantile-Quantile plots. The t test for repeated measures. The two-sample, independent groups t test.

Assumptions of the t test Independent observations. Distribution is normal. The idea of robustness. (not an assumption)

Assessing the assumptions Independence –Look at procedure, not at data Normality –Graphical methods Stem-and-leaf plots, histograms The normal quantile-quantile plot

Understanding the Q-Q plot Manual Q-Q plots Using R's “qqnorm” function The 'plot' subcommand in SAS's proc univariate

t tests for differences between means Why differences? A step in the right direction: the repeated measures t test. Classroom exercise

t tests for differences between means The two-sample Z test: The two-sample t test: can’t just substitute estimated standard deviation.

The pooled variance estimate Weighted average of the two individual variance estimates: df = n 1 +n 2 - 2

The two-sample independent-groups t test where

What’s the null hypothesis?

What if it doesn’t make sense to pool the variances? Satterthwaite’s approximation for degrees of freedom: Use unpooled variances for the standard error with adjusted degrees of freedom. The t test in SAS and R.

Assumptions of the t test Independence within each population. Independence between populations. Equal variances in the two populations. –Also known as “homoscedasticity.” Both populations normally distributed.

Evaluating the assumptions Independence within populations: examine the data collection procedure. Independence between populations: examine the process that created the groups. Random assignment guarantees independence between populations.

Evaluating the assumptions Homoscedasticity: –Graphical comparisons of the two groups –Comparison of the two sample standard deviations Normality: –Graphical examination of each group –Q-Q plots

The illogic of auxiliary hypothesis tests Auxiliary hypothesis tests: –Involve confirmation of the null hypothesis. –Are least likely to detect a problem under precisely the circumstances where the problem matters the most. –Involve assumptions of their own (implying infinite recursion).