Psychology 202aAdvanced Psychological
Statistics
November 17, 2015
The Plan for Today
• ANOVA: the traditional approach (continued)
• ANOVA in SAS• ANOVA assumptions• Visualizing ANOVA• ANOVA as a special case of regression
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
• Analog of the pooled variance estimate• Variance estimate based on the Central
Limit Theorem
Analog 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:
k
i i
k
i iiWE
n
snMSMS
1
1
2
1
1
Variance estimate based on the Central Limit Theorem
• The CLT says that
• If we substitute sample estimates and do a little algebra, this becomes
.2
2
nM
σσ =
.22Msns =
Variance estimate based on the Central Limit Theorem
• That idea leads to
.
11
2
k
i
iBM k
MMnMSMS
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• Assessing the assumptions in R• Visualizing ANOVA in R