We recently received a question asking why the results from the same model specified as anova versus a regression would not agree. The model in question had both categorical and continuous predictors. This question is really just a variation of questions concerning dummy (zero/one) coding versus effect coding. There are several FAQs that address this issue: How can get anova main-effects with dummy coding?, How can I get anova simple main effects with dummy coding?, How can I understand a three-way interaction in anova? and others.
Here is an example that is similar to the question asked by our client. It involves a model that has a categorical by continuous interaction.
use https://stats.idre.ucla.edu/stat/data/hsbdemo, clear anova write c.socst##i.femaleNumber of obs = 200 R-squared = 0.4299 Root MSE = 7.21161 Adj R-squared = 0.4211 Source | Partial SS df MS F Prob > F -------------+---------------------------------------------------- Model | 7685.43528 3 2561.81176 49.26 0.0000 | socst | 6242.19751 1 6242.19751 120.03 0.0000 female | 450.252986 1 450.252986 8.66 0.0036 female#socst | 239.648735 1 239.648735 4.61 0.0331 | Residual | 10193.4397 196 52.0073455 -------------+---------------------------------------------------- Total | 17878.875 199 89.843593regress write c.socst##i.femaleSource | SS df MS Number of obs = 200 -------------+------------------------------ F( 3, 196) = 49.26 Model | 7685.43528 3 2561.81176 Prob > F = 0.0000 Residual | 10193.4397 196 52.0073455 R-squared = 0.4299 -------------+------------------------------ Adj R-squared = 0.4211 Total | 17878.875 199 89.843593 Root MSE = 7.2116 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- socst | .6247968 .0670709 9.32 0.000 .4925236 .7570701 1.female | 15.00001 5.09795 2.94 0.004 4.946132 25.05389 | female#| c.socst | 1 | -.2047288 .0953726 -2.15 0.033 -.3928171 -.0166405 | _cons | 17.7619 3.554993 5.00 0.000 10.75095 24.77284 ------------------------------------------------------------------------------test socst( 1) socst = 0 F( 1, 196) = 86.78 Prob > F = 0.0000
As you can see the F-ratio for socst in anova is 120.03 and in regress
86.78. They are very different. What is going on here?
The answer is, of course, that the anova and the regression F-ratios are testing two different things.
The anova F-ratio is computed from the partial sum of squares for socst with all
of the other effects partialed out. The sum of squares is divided by its degrees of freedom (one) and
is in turn divided by the mean square residual (the pooled within cell variance). Although the
anova F-ratio is significant, you wouldn’t want to spend much effort trying to interpret it
since socst is also part of the significant socst#female interaction.
This particular regression model has a categorical variable, female, that is dummy coded (zero/one)
using the built_in factor variables notation.
The F-ratio in the regression is testing the slope of write on socst for the reference
group, in this case female = 0 (males). In fact, the regression coefficient (.6247968) is
the slope of write on socst for the males.
So, how can you get the anova F-ratio from the regress model. We will demonstrate
three ways of doing this.
Method 1: using the test command:
quietly regress write c.socst##i.female/* rerun regression model */test c.socst + 1.female#c.socst/2 = 0/* divide by 2 because there are two levels of female */ ( 1) socst + .5*1.female#c.socst = 0 F( 1, 196) = 120.03 Prob > F = 0.0000
This method shows that the “main” effect for socst is made of of the effect for socst
plus the average of the interaction effect over the two levels of female.
Method 2: using the margins command:
margins, dydx(socst) asbalanced postAverage marginal effects Number of obs = 200 Model VCE : OLS Expression : Linear prediction, predict() dy/dx w.r.t. : socst at : female (asbalanced) ------------------------------------------------------------------------------ | Delta-method | dy/dx Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- socst | .5224324 .0476863 10.96 0.000 .428969 .6158959 ------------------------------------------------------------------------------test socst( 1) socst = 0 chi2( 1) = 120.03 Prob > chi2 = 0.0000
For the margins command we need to use both the post and asbalanced options.
The post option allows us to use the test command after margins and the
asbalanced is needed both because the categorical variable (female) have unequal
cell size and also because we have a continuous predictor in the model.
Method 3: using a sum-to-zero coding:
You indicate categorical variables for regress using the i. prefix. This indicates
that Stata should use factor variables. Stata uses dummy (zero-one)
coding for its factor variables. The use of dummy coding is the reason that the anova
and regress results are different. If you were to use a sum-to-zero coding then the
results would be the same. We will demonstrate this using effect coding in which the reference
group is coded as minus one (-1). Technically, this coding scheme does not actually sum to zero in an
unbalanced design but it still works the way we want it to.
recode female (0 = -1), gen(fem)/* effect coding for female */regress write c.socst##c.femSource | SS df MS Number of obs = 200 -------------+------------------------------ F( 3, 196) = 49.26 Model | 7685.43528 3 2561.81176 Prob > F = 0.0000 Residual | 10193.4397 196 52.0073455 R-squared = 0.4299 -------------+------------------------------ Adj R-squared = 0.4211 Total | 17878.875 199 89.843593 Root MSE = 7.2116 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- socst | .5224324 .0476863 10.96 0.000 .4283883 .6164766 fem | 7.500004 2.548975 2.94 0.004 2.473066 12.52694 | c.socst#| c.fem | -.1023644 .0476863 -2.15 0.033 -.1964085 -.0083203 | _cons | 25.2619 2.548975 9.91 0.000 20.23496 30.28884 ------------------------------------------------------------------------------test c.socst( 1) socst = 0 F( 1, 196) = 120.03 Prob > F = 0.0000
For the sake of completeness, we need to mention that if there is no interaction then the anova
and regress results agree perfectly, as shown below.
anova write c.socst i.femaleNumber of obs = 200 R-squared = 0.4165 Root MSE = 7.27735 Adj R-squared = 0.4105 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 7445.78654 2 3722.89327 70.30 0.0000 | socst | 6269.5727 1 6269.5727 118.38 0.0000 female | 906.143844 1 906.143844 17.11 0.0001 | Residual | 10433.0885 197 52.9598399 -----------+---------------------------------------------------- Total | 17878.875 199 89.843593regress write c.socst i.femaleSource | SS df MS Number of obs = 200 -------------+------------------------------ F( 2, 197) = 70.30 Model | 7445.78654 2 3722.89327 Prob > F = 0.0000 Residual | 10433.0885 197 52.9598399 R-squared = 0.4165 -------------+------------------------------ Adj R-squared = 0.4105 Total | 17878.875 199 89.843593 Root MSE = 7.2774 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- socst | .5235458 .0481182 10.88 0.000 .428653 .6184386 1.female | 4.280318 1.034786 4.14 0.000 2.239637 6.320998 _cons | 23.00581 2.606248 8.83 0.000 17.86608 28.14554 ------------------------------------------------------------------------------test socst( 1) socst = 0 F( 1, 197) = 118.38 Prob > F = 0.0000