Intro to StatsANOVAs

Analysis of Variance (ANOVA)

Difference in two or more average scores in different groups

Simplest is one-way ANOVA (one variable as predictor); but can include multiple predictors

ANOVA

Differences between the groups are separated into two sources of variance◦ Variance from within the group◦ Variance from between the groups

The variance between groups is typically of interest

What it does

Use when:◦ you are examining differences between groups on

one or more variables, ◦ the participants in the study were tested only

once and◦ you are comparing more than two groups

When to use it

Factor: the variable that designates the groups to be compared

Levels: the individual comparable parts of the factor

Factorial designs have more than one variable as a predictor of an outcome

Terms

F is based on variance, not mean differences

Partial out the between condition variance from the within condition variance

Conceptual Calculation

F = MSbetween

MSwithin

MSbetween = SSbetween/dfbetween

MSwithin = SSwithin/ dfwithin

Calculation

Therapist wants to examine the effectiveness of 3 techniques for treating phobias. Subjects are randomly assigned to one of three treatment groups. Below are the rated fear of spiders after therapy.

X1: 5 2 5 4 2 X2: 3 3 0 2 2 X3: 1 0 1 2 1

Example 1

1. State hypotheses Null hypothesis: spider phobia does not differ

among the three treatment groups◦ μTreatement1 = μTreatment2 = μTreatment3

Research hypothesis: spider phobia differs in at least one treatment group compared to others OR there is an effect of at least one treatment on spider phobia ◦ XTreatment1≠ XTreatment2

◦ XTreatment1 ≠ XTreatment3

◦ XTreatment2 ≠ XTreatment3

◦ XTreatment1 ≠ XTreatment2≠ XTreatment3 (just write this one for ease, but all are made)

Example 1

F = MSbetween

MSwithin

MSbetween = SSbetween/dfbetween

MSwithin = SSwithin/ dfwithin

Calculation

SSbetween = Σ(ΣX)2/n – (ΣΣX)2/N

ΣX = sum of scores in each groupΣΣX = sum of all the scores across groupsn = number of participants in each groupN = number of participants (total)

SS between

SSwithin = ΣΣ(X2) – Σ(ΣX)2/n

ΣΣ(X2) = sum of all the sums of squared scores

Σ(ΣX)2 = sum of the sum of each group’s scores squared

n = number of participants in each group

SS within

Sstotal = ΣΣ(X2) – (ΣΣX)2/N

ΣΣ(X2) = sum of all the sums of squared scores

(ΣΣX)2 = sum of all the scores across groups squared

N = total number of participants (in all groups)

SS total

F = MSbetween

MSwithin

MSbetween = SSbetween/dfbetween

Dfbetween = k-1 (k=# of groups)

Example 1 - Dfs

6. Determine whether the statistic exceeds the critical value◦ 6.01 > 3.89◦ So it does exceed the critical value

7. If over the critical value, reject the null

& conclude that there is a significant difference in at least one of the groups

Example 1

For an ANOVA, the test statistic only tells you that there is a difference

It does not tell you which groups were different from other groups

There are numerous post-hoc tests that you can use to tell the difference

Here, we will use Bonferroni corrected post-hoc tests because they are already familiar (similar to t-tests, but with corrected critical value levels to reduce Type 1 error rates)

Example 1

In results◦ There was a significant effect of type of treatment on spider

phobia, F(2, 12) = 6.01, p < .05. With post-hoc tests

◦ There was a significant effect of type of treatment on spider phobia, F(2, 12) = 6.01, p < .05. Participants who received treatment X3 were less afraid of spiders (M = 1.00, SD = 0.71) than participants who received treatment X1 (M = 3.60, SD = 1.52), t(8) = 3.47, p = .008, but did not differ from participants who received treatment X2 (M = 2.00, SD = 1.22, t(8) = 1.58, n.s. Participants who received treatments X1 and X2 did not significantly differ, t(8) = 1.84, n.s.

If it had not been significant:◦ There was no significant effect of type of treatment on spider

phobia, F(2, 12) = 2.22, n.s.

Example 1

SPSS Check

SourceType III Sum of Squares df Mean Square F Sig.

Corrected Model17.200(a) 2 8.600 6.000 .016

Intercept72.600 1 72.600 50.651 .000

cond17.200 2 8.600 6.000 .016

Error17.200 12 1.433

Total107.000 15

Corrected Total34.400 14