I.     One-Way ANOVA

A. Use when?

1.              DV –

2.              IV –

3.              IV –

a)               Why multiple groups?

b)                 Why not do multiple t-tests for 2 groups?

B. Hypotheses:

1.              H₀ = Group Means are Equal

2.              H₁ = Group Means not Equal

C.Variability

1.              Between-Group Variability – the difference between group means, 2 factors contribute

a)              Sampling error

b)             Effect of IV

2.              Within-Group Variability – variability of scores within each group

a)              Affected by sampling error

b)             Not affected by IV

II.  Partitioning Variability – how do we determine between and within groups variability

A. The Grand Mean = Mean of all scores for all Groups

B. Group Means = Means within each group/level of IV

C.So, the deviation of each score from the Grand Mean is made up of 2 parts

1.               

2.               

D.Therefore,

1.              SS_T = Sum of Squares Total = Sum of deviation of each score from the Grand mean squared

2.              SSw = Sum of Squares Within = Sum of deviation of each score from it’s own Group Mean

3.              SS_B = Sum of Squares Between = Sum of deviation of each Group Mean from the Grand Mean times sample size (if sample sizes are equal)

4.              SS_T = SS_W + SS_B

E. Variance Ratio

1.              Calculate Mean Squares = sum of squares/df

a)              Degrees of Freedom

(1)          SS_W = N – k

(2)          SS_B = k – 1

b)             MS_W = SS_W/df_w

c)              MS_B = SS_B/df_B

III.          The F ratio

A. F = MS_B/MS_W recall:

1.              MS_B – measures differences/variability between groups

2.              MS_W – measures differences/variability within groups

3.              So, If IV has an effect, then difs between groups should be _____ than within groups, and

4.              Since we can assume the differences within groups are the same (Homogeneity of variance) then MS_B also contains variability within groups so it will be larger than MS_W

5.              F will almost always be >1

B. The larger the F, the more likely we are to reject H₀

IV.         The F Distribution

A. Calculate Fobt = ratio of 2 independent estimates of the same population variance

B. Sampling Distribution of F

1.              Take all possible samples of size n

2.              Estimate the population variance from each sample

3.              Calculate Fobt for all possible combinations

4.              Calculate probability of F for each different possible Fobt

C.F varies with df

D.Properties

1.              Never negative

2.              Positively skewed

3.              Median = 1

4.              Family of curves

E. Compare Fobt to Table F to determine whether or not to accept or reject H₀

V. Computational Formulas

A. SS_T =

B. SS_W =

C.SS_B =

VI.         Example -

A. Calculate SS_B

B. Calculate SS_W

C.Calculate SS_T - check on steps A and B

D.Calculate df

1.              Df for SSB

2.              Df for SSw

E. Calculate MS_B

F. Calculate MS_W

G.           Calculate Fobt

H.Evaluate Fobt using Table F

I.      Standard Presentation of Results

VII.      Relationship Between ANOVA and t-Test

A. If analyzing 2 groups, then t² = F

VIII.  Assumptions of ANOVA

A. Populations are Normally Distributed

B. Homogeneity of Variance

C.It is robust

1.              Minimally affected by population abnormality

2.              Relatively insensitive to homogeneity of variance if ns are equal

IX.         Effect Size

A. Similar to omega squared in t-test

B. Compute an estimate of the percentage of total variability of the dv that is accounted for by the iv

C.Equation = SS_B/SS_T

X. Two-Way ANOVA

A. What if have multiple Ivs?

B. Called a factorial experiment

1.              Get effect of each IV

2.              Get effect of interaction

C.A by B designs - number of levels of each IV

D.Fixed-effects design

E. Effect Types

1.              Main Effects -

2.              Interaction effects -