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Something to Remember About Factorial ANOVA using SPSS GLM
Descriptive Statistics
Dependent Variable: DEP
GENDER
male
female
Total
Mean Std. Deviation N
Tests of Between-Subjects Effects Dependent Variable: DEP
563.148a^1 563.148 12.964. 15697.727 1 15697.727 361.359. 563.148 1 563.148 12.964. 15682.141 361 43. 32432.000 363 16245.289 362
Source Corrected Model Intercept GENDER Error Total Corrected Total
Type III Sum of Squares df Mean Square F Sig.
a.R Squared = .035 (Adjusted R Squared = .032)
Parameter Estimates
Dependent Variable: DEP
0 a^...
Parameter
Intercept
[GENDER=1]
[GENDER=2]
B Std. Error t Sig.
a.This parameter is set to zero because it is redundant.
For the regression parameter estimates SPSS computes a dummy
code for the fixed factor, using the largest coded group (here
female) as the comparison group.
Notice that, as expected, the b weight matches the mean
difference – male mean is 2.497 larger than the female mean.
Also, the F the effect of gender is t² for the gender parameter.
Model Summary
.186a^ .035 .032 6.
Model 1
R R Square
Adjusted R Square
Std. Error of the Estimate
a.Predictors: (Constant), GENC
Coefficientsa
(Constant) GENC
Model 1
B Std. Error
Unstandardized Coefficients Beta
Standardized Coefficients t Sig.
a.Dependent Variable: DEP
ANOVAb
563.148 1 563.148 12.964 .000a 15682.141 361 43. 16245.289 362
Regression Residual Total
Model 1
Sum of Squares df Mean Square F Sig.
a.Predictors: (Constant), GENC b.Dependent Variable: DEP
We get the same thing all 3 ways …
Same df, SS & F
Same effect size
Same b
Same F = t²
GLM provides both “traditional ANOVA output” and “regression output” for the same analysis.
Here’s an example, using the relationship between Gender and depression.
Also, the results are the same as if we computed a dummy code with
females as the comparison group ourselves and then used
regression to perform the analysis.
if (gender = 1) genc = 1.
if (gender = 2) genc = 0.
Descriptive Statistics Dependent Variable: DEP
MARITAL
single married Total single married Total single married Total
GENDER
male
female
Total
Mean Std. Deviation N
Parameter Estimates Dependent Variable: DEP
0 a^... 3.150 .961 3.276. 0 a^... -4.207 1.474 -2.855.
0
a
...
0
a
...
0
a
...
Parameter Intercept [GENDER=1] [GENDER=2] [MARITAL=1] [MARITAL=2] [GENDER=1] * [MARITAL=1] [GENDER=1] * [MARITAL=2] [GENDER=2] * [MARITAL=1] [GENDER=2] * [MARITAL=2]
B Std. Error t Sig.
a.This parameter is set to zero because it is redundant.
Tests of Between-Subjects Effects Dependent Variable: DEP
1055.189a^3 351.730 8.313. 13262.668 1 13262.668 313.447. 278.122 1 278.122 6.573. 85.324 1 85.324 2.017. 344.868 1 344.868 8.151. 15190.100 359 42. 32432.000 363 16245.289 362
Source Corrected Model Intercept GENDER MARITAL GENDER * MARITAL Error Total Corrected Total
Type III Sum of Squares df Mean Square F Sig.
a.R Squared = .065 (Adjusted R Squared = .057)
YIKES!
Neither the gender main effect, nor the marital main effect match for the ANOVA and parameter estimates -- F? t² and the b values don’t reflect the marginal mean differences!
However, the interaction does match -- 2.855² = 8.151, and the interaction b (-4.207) reflects the difference between the simple effect of marital for males (-1.0562) and for females (3.1498) ‡ 4.207.
However, watch what happens when we switch to a factorial design – here with the fixed effects of gender and marital status and the outcome variable depression.
Why does this happen? The confluence of two things…
The ANOVA summary table reflects the use of “effects coding” (the highest coded group for each IV is the comparison group and is weighted -1 and the interaction term is the product of the two main effect codes), whereas the parameter estimates reflect the use of dummy coding (the highest coded group for each IV is the comparison group and is
weighted 0 and the interaction term is the product of the two main effect codes),
Effect codes and dummy codes are not linear transformations of each other, and so using them leads to different
patterns of colinearity between the main effects terms of the model, and so, different expressions of the main effects.
Two things to notice:
- The interaction term is the same for both effect coded and dummy coded versions. This may be why some folks suggest ignoring main effects if you have an interaction – because the interaction term is the same for various coding schemes. However, even if you delete a non-contributing interaction term, the main effects will still appear different for a dummy and effect coded analysis.
- This effect won’t show up if there is =n for the factorial design. If so, the main effects are orthogonal and so there’s
no colinearity to be partitioned differently by the different coding schemes. Some folks base upon this the suggestion that factorial analyses should be reserved for =n designs. Others counter that this is an artificial simplification of the patterns of “real colinearity” among variables. These folks suggest that differences between the main effects results using dummy vs. effect coding is no more “troublesome” than the differences between the main effects results when using alternative operational differences (manipulation or measurement) of the IVs. In either case you select your “preferred” operationalization and judiciously interpret the results.
