Discussion Questions:
1. Broadly speaking, what kinds of questions can we test with a factorial ANOVA?
Factorial ANOVA is capable of testing questions that involve comparing two or more
independent variables. With this statistical tool, groups of four or more can be formed by
splitting samples based on the independent variables. Additionally, the use of a factorial ANOVA
can assume a cause-and-effect relationship, as it suggests that one or more of the independent
variables may cause significant differences in one or more characteristics.
2. In this model, are there significant main effects of gender and race? If so, how do the
levels of each factor differ from one another?
Significant main effects of gender and race are observed in this model. This is confirmed by
examining the p-values for each variable in the ANOVA and conducting post hoc comparisons
using the Holm's method. The p-value for gender was found to be less than 0.001, while the pvalue for race was 0.003. Both values being less than 0.05, we reject the null hypothesis,
indicating that the main effects are significant for the different levels of gender and race. The
post hoc comparisons reveal that there are no significant differences between Black males and
White females, Black males and Black females, and White females and Black females. However,
the largest mean difference (12919 and 13205) in income is observed between White males and
White females and between White males and Black females. Comparatively, the difference in
income between men and women (8373) is larger than between Black and White individuals
(4832).
3. Is there an interaction between race and gender on income? If so, which
combination(s) of race and gender are significantly different than the others?
When examining the n^2p values, we observe an interaction between race and gender on income
as all values are below 0.05. By analyzing the p-values, we can determine which combinations
are statistically significant and different. Our analysis indicates that there are significant
differences between White males and Black males, White males and White females, and White
males and Black females.
4. What do the effect sizes of each significant effect tell us?
The effect sizes for each significant effect reveal that only a small portion of the variance is
accounted for by the eta-values, and the rest of the variance is explained by other variables. In
this particular scenario, the eta-value for gender is 0.018, 0.006 for race, and 0.005 for the
interaction of race and gender. Although the effect sizes for gender, race, and the interaction of
race and gender are all significant, they only account for a small amount of variance.
ANOVA
ANOVA - income
Sum of Squares
df
Mean Square
F
p
η²
η²p
gender (2) (2)
1.08e+10
1
1.08e+10
26.11
< .001
0.018
0.018
race
3.59e0+9
1
3.59e0+9
8.70
0.003
0.006
0.006
gender (2) (2) ✻ race
3.18e0+9
1
3.18e0+9
7.70
0.006
0.005
0.005
Residuals
5.83e+11
1411
4.13e0+8
Post Hoc Tests
Post Hoc Comparisons - gender (2) (2)
Comparison
gender (2) (2)
gender (2) (2)
M
-
Mean Difference
W
8373
SE
df
t
ptukey
1639
1411
5.11
< .001
Note. Comparisons are based on estimated marginal means
Post Hoc Comparisons - race
Comparison
race
White
race
-
Black
Mean Difference
4832
SE
df
t
ptukey
1639
1411
2.95
0.003
Post Hoc Comparisons - race
Comparison
race
race
Mean Difference
SE
df
t
ptukey
Note. Comparisons are based on estimated marginal means
Post Hoc Comparisons - gender (2) (2) ✻ race
Comparison
gender (2)
(2)
M
White
Black
W
gender (2)
(2)
race
White
race
Mean
Difference
df
t
ptukey
-
M
Black
9378
2564
1411
3.657
0.002
-
W
White
12919
1167
1411
11.066
< .001
-
W
Black
13205
2078
1411
6.354
< .001
-
W
White
3541
2534
1411
1.397
0.501
-
W
Black
3827
3062
1411
1.250
0.595
-
W
Black
286
2041
1411
0.140
0.999
Note. Comparisons are based on estimated marginal means
Estimated Marginal Means
gender (2) (2)
SE
Estimated Marginal Means - gender (2) (2)
95% Confidence Interval
gender (2) (2)
Mean
SE
M
21449
W
13076
race
Lower
Upper
1282
18934
23965
1020
11075
15078
Estimated Marginal Means - race
95% Confidence Interval
race
Mean
SE
Lower
Upper
White
19679
584
18534
20824
Black
14847
1531
11843
17850
gender (2) (2) ✻ race
Estimated Marginal Means - gender (2) (2) ✻ race
95% Confidence Interval
race
White
Black
gender (2) (2)
Mean
SE
Lower
Upper
M
26138
871
24429
27847
W
13220
777
11695
14744
M
16761
2412
12030
21492
W
12933
1887
9232
16634