Psychology 202a Advanced Psychological Statistics November 7, 2017

The Plan for Today ANOVA: the traditional approach ANOVA in SAS

ANOVA: the Traditional Approach A motivating example Speed with which math problems are performed Three practice conditions: massed, spaced, none The multiple testing problem A way out: first, ask if any means differ then worry about which means differ

How ANOVA works Logic: develop two ways of estimating variance: one that always makes sense (given some assumptions) one that depends on the null hypothesis Analogue of the pooled variance estimate Variance estimate based on the Central Limit Theorem

Analogue of the pooled variance estimate When we dealt with the t test, we pooled variance using a weighted average of the variance estimate in each group. This is easily modified to accommodate more than two groups:

Variance estimate based on the Central Limit Theorem The CLT says that If we substitute sample estimates and do a little algebra, this becomes

Variance estimate based on the Central Limit Theorem That idea leads to

Illustration with example Massed practice: mean = 55.125, variance = 925.839286 Spaced practice: mean = 94.000, variance = 936.857143 No practice: Mean = 112.625, variance = 1668.26786 In each case, n = 8.

Organizing the information Source SS df MS F Between 13771.75 2 6885.875 5.85 Within 24716.75 21 1176.988 Total 38488.5 23

Assumptions of the ANOVA Independence between groups Independence within groups Homoscedastic populations Normal populations In other words, the assumptions are identical to those of the t test, generalized to more than two groups.

Practical ANOVA ANOVA in SAS ANOVA in R Assessing the assumptions in R Visualizing ANOVA in R

Next time ANOVA as a special case of regression