### Stata FAQ Why don't my anova and regression results agree? (Stata 11)

We recently received a question asking why an anova and regression results do not agree. The model 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 http://www.ats.ucla.edu/stat/stata/data/hsbdemo, clear

anova write c.socst##i.female

Number 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   89.843593

regress write c.socst##i.female

Source |       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 introduced in Stata 11. 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 levles of female.

Method 2: using the margins command:
margins, dydx(socst) asbalanced post

Average 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 vaiable (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 (introduced in Stata 11). Stata use 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.fem

Source |       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.female

Number 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   89.843593

regress write c.socst i.female

Source |       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

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