Interactions with Categorical Predictors

Linear Statistical Models: Regression

Interactions With Categorical Predictors

Updated for Stata 11

So Far...

We have considered the interaction of continuous variables, called by some product variables.

We have also considered the interaction of dummy variables with continuous variables.

Now is is time to consider the interaction of two categorical variables.

With fixed levels of the categorical variable this model would be considered to be an analysis of variance type model

Example Using hsbdemo

We will look at a model that uses write as the response variable and female and prog as predictors.

use http://www.philender.com/courses/data/hsbdemo, clear
 
tab1 female prog
 
-> tabulation of female  

     female |      Freq.     Percent        Cum.
------------+-----------------------------------
       male |         91       45.50       45.50
     female |        109       54.50      100.00
------------+-----------------------------------
      Total |        200      100.00
 
-> tabulation of prog  

    type of |
    program |      Freq.     Percent        Cum.
------------+-----------------------------------
    general |         45       22.50       22.50
   academic |        105       52.50       75.00
   vocation |         50       25.00      100.00
------------+-----------------------------------
      Total |        200      100.00
  
table prog female, cont(mean write sd write freq)
 
------------------------------
type of   |       female      
program   |     male    female
----------+-------------------
  general | 49.14286     53.25
          | 10.36478  8.205248
          |       21        24
          | 
 academic | 54.61702  57.58621
          | 8.656622  7.115672
          |       47        58
          | 
 vocation | 41.82609  50.96296
          | 8.003705  8.341193
          |       23        27
------------------------------
 
/* model 2 -- no interaction */
 
anova write female prog

                           Number of obs =     200     R-squared     =  0.2408
                           Root MSE      = 8.32211     Adj R-squared =  0.2291

                  Source |  Partial SS    df       MS           F     Prob > F
              -----------+----------------------------------------------------
                   Model |  4304.40272     3  1434.80091      20.72     0.0000
                         |
                  female |  1128.70487     1  1128.70487      16.30     0.0001
                    prog |  3128.18888     2  1564.09444      22.58     0.0000
                         |
                Residual |  13574.4723   196  69.2575116   
              -----------+----------------------------------------------------
                   Total |   17878.875   199   89.843593   
 
regress write i.female i.prog


      Source |       SS       df       MS              Number of obs =     200
-------------+------------------------------           F(  3,   196) =   20.72
       Model |  4304.40272     3  1434.80091           Prob > F      =  0.0000
    Residual |  13574.4723   196  69.2575116           R-squared     =  0.2408
-------------+------------------------------           Adj R-squared =  0.2291
       Total |   17878.875   199   89.843593           Root MSE      =  8.3221

------------------------------------------------------------------------------
       write |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    1.female |   4.771211   1.181876     4.04   0.000     2.440385    7.102037
             |
        prog |
          2  |   4.832929   1.482956     3.26   0.001     1.908331    7.757528
          3  |  -4.605141   1.710049    -2.69   0.008      -7.9776   -1.232683
             |
       _cons |   48.78869   1.391537    35.06   0.000     46.04438      51.533
------------------------------------------------------------------------------
 
test 1.female

 ( 1)  1.female = 0

       F(  1,   196) =   16.30
            Prob > F =    0.0001
 
testparm i.prog

 ( 1)  2.prog = 0
 ( 2)  3.prog = 0

       F(  2,   196) =   22.58
            Prob > F =    0.0000
 
/* model 2 -- interaction */
 
anova write female prog female#prog

                           Number of obs =     200     R-squared     =  0.2590
                           Root MSE      = 8.26386     Adj R-squared =  0.2399

                  Source |  Partial SS    df       MS           F     Prob > F
             ------------+----------------------------------------------------
                   Model |  4630.36091     5  926.072182      13.56     0.0000
                         |
                  female |  1261.85329     1  1261.85329      18.48     0.0000
                    prog |  3274.35082     2  1637.17541      23.97     0.0000
             female#prog |  325.958189     2  162.979094       2.39     0.0946
                         |
                Residual |  13248.5141   194  68.2913097   
             ------------+----------------------------------------------------
                   Total |   17878.875   199   89.843593    
 
regress write i.female##i.prog

      Source |       SS       df       MS              Number of obs =     200
-------------+------------------------------           F(  5,   194) =   13.56
       Model |  4630.36091     5  926.072182           Prob > F      =  0.0000
    Residual |  13248.5141   194  68.2913097           R-squared     =  0.2590
-------------+------------------------------           Adj R-squared =  0.2399
       Total |   17878.875   199   89.843593           Root MSE      =  8.2639

------------------------------------------------------------------------------
       write |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    1.female |   4.107143   2.469299     1.66   0.098    -.7629757    8.977261
             |
        prog |
          2  |   5.474164   2.169095     2.52   0.012     1.196128      9.7522
          3  |   -7.31677   2.494224    -2.93   0.004    -12.23605   -2.397493
             |
 female#prog |
        1 2  |  -1.137957   2.954299    -0.39   0.701    -6.964625     4.68871
        1 3  |   5.029733    3.40528     1.48   0.141    -1.686391    11.74586
             |
       _cons |   49.14286   1.803321    27.25   0.000     45.58623    52.69949
------------------------------------------------------------------------------
 
test _IfemXpro_1_2 _IfemXpro_1_3

 ( 1)  1.female#2.prog = 0
 ( 2)  1.female#3.prog = 0

       F(  2,   194) =    2.39
            Prob > F =    0.0946
 
test 1.female

 ( 1)  1.female = 0

       F(  1,   194) =    2.77
            Prob > F =    0.0979
 
testparm i.prog

 ( 1)  2.prog = 0
 ( 2)  3.prog = 0

       F(  2,   194) =   18.69
            Prob > F =    0.0000

Please note: With dummy coding the tests of the highest order interaction is the same as that using anova. However, the tests of the main effects will not be the same as anova. We will need to use a different approach, such as, the anovalator program (findit anovalator).

anovalator female prog, main fratio

anovalator main-effect for female  
chi2(1) = 18.477509   p-value = .00001719
scaled as F-ratio = 18.477509

anovalator main-effect for prog  
chi2(2) = 47.946815   p-value = 3.877e-11
scaled as F-ratio = 23.973408

Some examples of 2x2 interactions

Consider the following 2x2 table of cell means and the regression results. The categorical predictors are A and B, each with two levels. The first example will not have a significant interaction effect.

	A0	A1
B0	50.02	55.09
B1	54.61	60.09

regress y1 a##b

      Source |       SS       df       MS              Number of obs =      40
-------------+------------------------------           F(  3,    36) =  185.27
       Model |  507.825013     3  169.275004           Prob > F      =  0.0000
    Residual |  32.8925799    36  .913682774           R-squared     =  0.9392
-------------+------------------------------           Adj R-squared =  0.9341
       Total |  540.717593    39  13.8645537           Root MSE      =  .95587

------------------------------------------------------------------------------
          y1 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         1.a |   5.068859    .427477    11.86   0.000     4.201896    5.935823
         1.b |   4.582972    .427477    10.72   0.000     3.716009    5.449936
             |
         a#b |
        1 1  |    .410215   .6045437     0.68   0.502    -.8158565    1.636286
             |
       _cons |   50.02439   .3022719   165.49   0.000     49.41135    50.63743
------------------------------------------------------------------------------

Let's interpret this regression table.

_cons = 50.02439: The expected value when both A and B equal 0. It is cell 1,1 in the table above.
1.a = 5.068859: This is the amount added to _cons when A equals 1 and B equals 0. The value 50.02439 + 5.068859 = 55.09 is found in cell 1,2.
1.b = 4.582972: This is the amount added to _cons when B equals 1 and A equals 0. The value 50.02439 + 4.582972 = 54.61 is found in cell 2,1.
a#b = .410215: This amount is added to 1.a plus 1.b when both A and B equal 1. The value 50.02439 + 5.068859+ 4.582972 + .410215 = 60.09 is found in cell 2,2.

Next, let's try an example in which the interaction term is significant and positive.

	A0	A1
B0	50.25	54.73
B1	55.10	65.08

regress y2 a##b

      Source |       SS       df       MS              Number of obs =      40
-------------+------------------------------           F(  3,    36) =  753.03
       Model |  1175.55141     3  391.850469           Prob > F      =  0.0000
    Residual |  18.7332456    36  .520367934           R-squared     =  0.9843
-------------+------------------------------           Adj R-squared =  0.9830
       Total |  1194.28465    39  30.6226834           Root MSE      =  .72137

------------------------------------------------------------------------------
          y2 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         1.a |   4.485349   .3226044    13.90   0.000     3.831077    5.139621
         1.b |   4.848378   .3226044    15.03   0.000     4.194106     5.50265
             |
         a#b |
        1 1  |   5.494711   .4562315    12.04   0.000     4.569431    6.419991
             |
       _cons |   50.24988   .2281157   220.28   0.000     49.78724    50.71252
------------------------------------------------------------------------------

Let's interpret this regression table for y2.

_cons = 50.24988: The expected value when both A and B equal 0. It is cell 1,1 in the table above.
1.a = 4.485349: This is the amount added to _cons when A equals 1 and B equals 0. The value 50.24988 + 4.485349 = 54.73 is found in cell 1,2.
1.b = 4.582972: This is the amount added to _cons when B equals 1 and A equals 0. The value 50.24988 + 4.848378 = 55.10 is found in cell 2,1.
a#b = .410215: This amount is added to 1.a plus 1.b when both A and B equal 1. The value 50.24988 + 4.485349 + 4.848378 + 5.494711 = 65.08 is found in cell 2,2.

Finally, we will run a model in which the interaction coefficient is negative and statistically significant.

	A0	A1
B0	50.33	55.21
B1	54.76	55.43

regress y3 a##b

      Source |       SS       df       MS              Number of obs =      40
-------------+------------------------------           F(  3,    36) =   33.39
       Model |  175.137941     3  58.3793136           Prob > F      =  0.0000
    Residual |   62.938775    36  1.74829931           R-squared     =  0.7356
-------------+------------------------------           Adj R-squared =  0.7136
       Total |  238.076716    39  6.10453118           Root MSE      =  1.3222

------------------------------------------------------------------------------
          y3 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         1.a |   4.879437   .5913204     8.25   0.000     3.680183     6.07869
         1.b |   4.427636   .5913204     7.49   0.000     3.228383    5.626889
             |
         a#b |
        1 1  |  -4.213134   .8362534    -5.04   0.000    -5.909134   -2.517133
             |
       _cons |   50.33252   .4181267   120.38   0.000     49.48452    51.18052
------------------------------------------------------------------------------

Let's interpret this regression table for y3.

_cons = 50.33252: The expected value when both A and B equal 0. It is cell 1,1 in the table above.
1.a = 4.879437: This is the amount added to _cons when A equals 1 and B equals 0. The value 50.33252 + 4.879437 = 55.21 is found in cell 1,2.
1.b = 4.4276362: This is the amount added to _cons when B equals 1 and A equals 0. The value 50.33252 + 4.427636 = 54.76 is found in cell 2,1.
a#b = -4.213134: This amount is added to 1.a plus 1.b when both A and B equal 1. The value 50.33252 + 4.879437 + 4.427636 - 4.213134 = 55.43 is found in cell 2,2.

We can see from the tables and regression results that the coefficient for the interaction term specifies the amount that is added to or subtracted from the sum of the coefficients for A and B plus the constant.

Linear Statistical Models Course

Phil Ender, 24sep10, 18dec99