Psychology 202aAdvanced 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

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t tests for differences between means

• The two-sample Z test:

• The two-sample t test: can’t just substitute estimated standard deviation.

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The pooled variance estimate

• Weighted average of the two individual variance estimates:

• df = n1+n2 - 2

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The two-sample independent-groups t test

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What’s the null hypothesis?

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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.

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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).