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