Two-Way ANOVA Flashcards
Two-Way ANOVA
An analysis of variance where there are 2 nominal IVs. The IV levels are unrelated, and the DV is interval/ratio. Analysis of variance for a two-way factorial research design.
Main Effect
Direct influence of IV on DV.
Interaction
Effect of IV1 in the presence of IV2. The effect of one IV on DV DEPENDS on the other IV.
Determining if there is a main effect
You tell whether there is a main effect by looking at the marginal means. There is a main effect for a particular IV if the marginal means for the different levels of one IV are DIFFERENT from each other.
Determining if there is an interaction (cells)
You tell whether there is an interaction effect by looking at the pattern of cell means. There is an interaction effect if the pattern of cell means in one row is DIFFERENT from the pattern of cell means across another row. IGNORE main effects (marginal means) when determining if there is an interaction.
Determining interaction by bar graph
Whenever there is an interaction, the pattern of bars on one section of the graph is DIFFERENT from the pattern of bars on the other section of the graph.
Marginal mean
In a factorial design in ANOVA, the mean score for all the participants at a particular level of one of the IVs.
Grand mean (GM)
Overall mean of all the scores, regardless of what group they are in. The average of the means within a particular group is the grand mean for that group, which is needed for conducting simple effects.
Between group variance
Consists of effect of IV1, effect of IV2, effect of IV1 x IV2, and random things (sampling error).
Within-group variance
Consists of error variance (sampling error).
Total variability
Consists of between group and within group variance.
Factor
Independent variable
Calculating F-ratio steps
- Calculate SS’s (may be provided)
- Calculate df’s (6)
- Calculate MS’s (4)
- Calculate F-ratios (3)
SS(b/w) formula
SS(b/w) = SS(b/w-IV1) + SS(b/w-IV2) + SS(b/w-IV1 x IV2).
SS(BW-IV1) concept
Group’s deviations from the Grand Mean
SS(BW-IV2) concept
Group’s deviations from the Grand Mean
SS(b/w-IV1 x IV2) formula
SS(b/w-IV1 x IV2) = SS(Total) - SS(b/w-IV1) - SS(b/w-IV2) - SS(w/in)
SS(w/in) conceptual
Score’s deviations from their group means
SS(Total) formula
SS(Total) = SS(b/w) + SS(w/in)
df(B/w)
df(B/w) = df(B/w-IV1) + df(B/w-IV2) + df(B/w-IV1 x IV2)
df(IV1)
df(IV1) = k(IV1) - 1
df(IV2)
df(IV2) = k(IV2) - 1
df(IV1 x IV2)
df(IV1 x IV2) = df(IV1) X df(IV2) (multiply!)
df(w/in)
df(Within) = df(Total) - df(b/w)
df(Total)
df(Total) = N - 1
Order of df calculations for Two-Way ANOVA
- Calculate df(IV1)
- Calculate df(IV2)
- Calculate df(IV1 x IV2)
- Calculate df(Between)
- Calculate df(Total)
- Calculate df(Within)
MS formula
MS(x) = SS(x)/df(x)
F-Ratio formula
F(x) = MS(x)/MS(w/in)
4 MS values
- MS(IV1)
- MS(IV2)
- MS(IV1 x IV2)
- MS(w/in)
3 F-ratio values
- F-ratio for IV1
- F-ratio for IV2
- F-ratio for IV1 x IV2
Two-Way ANOVA hypothesis testing steps
- Compute F-ratios: will have 3.
- Set criteria for decisions: add F[crit] column to source table.
a. Use df(b/w-IV1), df(w/in) and alpha level to find F[crit] for IV1.
b. Use df(b/w-IV2), df(w/in), and alpha level to find F[crit] for IV2.
c. Use df(b/w-IV1 x IV2), df(w/in), and alpha level to find F[crit] for IV1 x IV2. - Make decisions: See if you have any main effects and/or interaction.
Partial eta squared formula
ηp2(x) = SS(b/w-x)/[SS(b/w-total) - SS(b/w-y) - SS(b/w-z)]
Partial eta squared definition
Percentage of variance in the DV that can be attributed to the IV, when multiple IVs are present. ONLY calculate for significant results.
If the interaction is NOT significant, but main effect(s) are:
2 groups: compare means directly
3+ groups: Run a Tukey or a priori to compare means
If the interaction IS significant:
Ignore main effects, as the interaction becomes more interesting to look at statistically. Conduct simple effects.
Simple effects definition
Examine the effect of the IV(x) within one level of IV(y). Basically a series of ANOVAs.
Simple effects steps
- Choose direction (horizontal or vertical) in which you will conduct comparisons using the data table. Run comparisons in a way that you are comparing 2 levels to each other so that you don’t have to run a Tukey test if you do get significant results, you can just compare the 2 means to each other with a one-way ANOVA.
- Complete one-way ANOVA source tables for each level.
a. Make 3 new source tables, keeping the within values from the two-way ANOVA source table.
b. Calculate SS(b/w).
c. Calculate df(b/w).
d. Calculate MS(b/w).
e. Calculate F-ratio. - Make a decision featuring a Bonferroni correction of the original alpha level.
a. Divide original alpha level by m (the # of comparisons you are running for the one-way ANOVA)
b. Round the new alpha level down to use F-ratio table (.10, .05 or .01).
c. Find new F[crit] value and plot on an F- table graph to make a decision (based on significance) - Interpret the meaning of your decision. If it is significant, compare the group means from your IVs to interpret the meaning.
SS(b/w) formula for simple effects
SS(b/w) = n(M1 - GM)^2 + n(M2 - GM)^2, where n = # of participants, M1 = mean of IV1, M2 = mean of IV2, and GM = (M1 + M2)/2.
df(b/w) formula for simple effects
df(b/w) = k -1
MS(b/w) formula for simple effects
MS(b/w) = SS/df
F-ratio formula for simple effects
F = MS(b/w)/MS(w/in)
