Interaction Between Continuous Variable and Indicator

First off, let's start with what a significant categorical by continuous interaction means. It means that the slope of the continuous variable is different for one or more levels of the categorical variable.

We will use an example from the hsbdemo dataset that has a statistically significant categorical by continuous interaction to illustrate one possible explanatory approach.

The categorical variable is female, a zero/one variable with females coded as one (therefore, male is the reference group). The continuous predictor variable, socst, is a standardized test score for social studies. We will begin by running the regression model and graphing the interaction. Please note that we use c.socst to indicate that socst is a continuous variable.

                use https://stats.idre.ucla.edu/stat/data/hsbdemo, clear  regress write female##c.socst                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] -------------+----------------------------------------------------------------     1.female |   15.00001    5.09795     2.94   0.004     4.946132    25.05389        socst |   .6247968   .0670709     9.32   0.000     .4925236    .7570701              |       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 ------------------------------------------------------------------------------                twoway (scatter write socst, msym(oh) jitter(3)) ///        (lfit write socst if ~female)(lfit write socst if female), ///        legend(order(2 "male" 3 "female"))                Image catcon12_0              

Looking at the graph, we can see that the two regression lines are not parallel and that the line for females falls above the line for males. How could we tell that females are higher than males? The coefficient for female is positive (15.00) which tells us that the level for females is higher than for males.

Let's interpret the coefficients for this model starting with the constant (17.76). This is the value of the intercept for socst regressed on write for males. i.e., the expected value for write when both socst and female equal zero.

The coefficient for socst is .6247 which is the slope of the regression line for the male group (i.e., female=0). The value for the female by socst interaction is -.2047 which is the difference in slope between the male and female group, i.e., the slope for the female group would be about .6248 – .2047 = .4201.

We can also get the slopes for the two groups using the margins command.

                margins female, dydx(socst)                Average marginal effects                          Number of obs   =        200 Model VCE    : OLS  Expression   : Linear prediction, predict() dy/dx w.r.t. : socst  ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- socst        |       female |           0  |   .6247968   .0670709     9.32   0.000     .4933403    .7562533           1  |    .420068   .0678044     6.20   0.000     .2871739    .5529622 ------------------------------------------------------------------------------

The difference between males and females may or may not be significantly different for different values of socst. What we will do is look at the male-female difference for various values of socst using the margins command. We will let socst vary between 25 and 70 in increments of 5.

                margins, at(female=(0 1) socst=(25(5)70)) vsquish                                Adjusted predictions                              Number of obs   =        200 Model VCE    : OLS  Expression   : Linear prediction, predict() 1._at        : female          =           0                socst           =          25 2._at        : female          =           0                socst           =          30 3._at        : female          =           0                socst           =          35 4._at        : female          =           0                socst           =          40 5._at        : female          =           0                socst           =          45 6._at        : female          =           0                socst           =          50 7._at        : female          =           0                socst           =          55 8._at        : female          =           0                socst           =          60 9._at        : female          =           0                socst           =          65 10._at       : female          =           0                socst           =          70 11._at       : female          =           1                socst           =          25 12._at       : female          =           1                socst           =          30 13._at       : female          =           1                socst           =          35 14._at       : female          =           1                socst           =          40 15._at       : female          =           1                socst           =          45 16._at       : female          =           1                socst           =          50 17._at       : female          =           1                socst           =          55 18._at       : female          =           1                socst           =          60 19._at       : female          =           1                socst           =          65 20._at       : female          =           1                socst           =          70  ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------          _at |           1  |   33.38182    1.94946    17.12   0.000     29.56095    37.20269           2  |    36.5058   1.645495    22.19   0.000     33.28069    39.73091           3  |   39.62979   1.356406    29.22   0.000     36.97128    42.28829           4  |   42.75377   1.094051    39.08   0.000     40.60947    44.89807           5  |   45.87775   .8825999    51.98   0.000     44.14789    47.60762           6  |   49.00174   .7654688    64.02   0.000     47.50145    50.50203           7  |   52.12572     .78602    66.32   0.000     50.58515    53.66629           8  |   55.24971   .9352206    59.08   0.000     53.41671     57.0827           9  |   58.37369   1.164634    50.12   0.000     56.09105    60.65633          10  |   61.49767   1.436326    42.82   0.000     58.68253    64.31282          11  |   43.26361   2.015017    21.47   0.000     39.31425    47.21297          12  |   45.36395   1.700513    26.68   0.000       42.031    48.69689          13  |   47.46429   1.397521    33.96   0.000     44.72519    50.20338          14  |   49.56463   1.115464    44.43   0.000     47.37836    51.75089          15  |   51.66497   .8748286    59.06   0.000     49.95033     53.3796          16  |   53.76531   .7185139    74.83   0.000     52.35704    55.17357          17  |   55.86565   .7050327    79.24   0.000     54.48381    57.24748          18  |   57.96599   .8412798    68.90   0.000     56.31711    59.61486          19  |   60.06633   1.071589    56.05   0.000     57.96605     62.1666          20  |   62.16667   1.348602    46.10   0.000     59.52346    64.80988 ------------------------------------------------------------------------------

So, the write value for males at socst = 25 is 33.38182 as shown in row 1. The same value for females is 43.26361 as shown in row 11. Here is the differences in the two values, 43.26361 – 33.38182 = 9.88179. We can obtain this difference for all of the values of socst using the margins command with the dydx option.

                margins, dydx(female) at(socst=(25(5)70)) vsquish                                Conditional marginal effects                      Number of obs   =        200 Model VCE    : OLS  Expression   : Linear prediction, predict() dy/dx w.r.t. : 1.female 1._at        : socst           =          25 2._at        : socst           =          30 3._at        : socst           =          35 4._at        : socst           =          40 5._at        : socst           =          45 6._at        : socst           =          50 7._at        : socst           =          55 8._at        : socst           =          60 9._at        : socst           =          65 10._at       : socst           =          70  ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- 1.female     |          _at |           1  |   9.881789   2.803692     3.52   0.000     4.386654    15.37692           2  |   8.858145   2.366305     3.74   0.000     4.220273    13.49602           3  |   7.834501   1.947538     4.02   0.000     4.017396     11.6516           4  |   6.810857   1.562436     4.36   0.000     3.748538    9.873176           5  |   5.787213   1.242702     4.66   0.000     3.351562    8.222863           6  |   4.763569   1.049859     4.54   0.000     2.705882    6.821255           7  |   3.739925   1.055888     3.54   0.000     1.670423    5.809426           8  |   2.716281   1.257931     2.16   0.031      .250782    5.181779           9  |   1.692637   1.582617     1.07   0.285    -1.409236    4.794509          10  |   .6689926   1.970219     0.34   0.734    -3.192565     4.53055 ------------------------------------------------------------------------------ Note: dy/dx for factor levels is the discrete change from the base level.

Now, we can graph these differences using the marginsplot command.

                marginsplot, yline(0)                Image catcon12_1              

We can see that the differences between males and females is significant for values of socst below about 60. This graph is nice and tells the story we want to know but it is not the best looking graph we can draw. By recasting the lines and confidence intervals we get a much sharper looking graph.

                marginsplot, recast(line) recastci(rarea) yline(0)                Image catcon12_2              

Ah, that's much better.

The graph shows that the male/female differences decreases as the value of socst increases. Whenever the 95% confidence interval for the difference does not include zero, the difference can be considered to be statistically significant. This looks to be the case for all values of socst up to about 60. For socst values greater than 60 the males/female difference is not significant.

A three level categorical variable

What if your categorical variable has more than two levels? The dataset catcon3l has a categorical predictor, b, with three levels. The response variable is y, the categorical predictor is b and it is interacted with a continuous predictor x, specified in Stata as c.x.

                use https://stats.idre.ucla.edu/stat/data/catcon3l, clear  regress y b##c.x                Source |       SS       df       MS              Number of obs =     200 -------------+------------------------------           F(  5,   194) =   32.02        Model |  8083.95798     5   1616.7916           Prob > F      =  0.0000     Residual |  9794.91702   194   50.489263           R-squared     =  0.4522 -------------+------------------------------           Adj R-squared =  0.4380        Total |   17878.875   199   89.843593           Root MSE      =  7.1056  ------------------------------------------------------------------------------            y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------            b |           2  |  -14.56722   7.178201    -2.03   0.044    -28.72455   -.4098837           3  |  -20.16644   6.779483    -2.97   0.003    -33.53739   -6.795485              |            x |   .2728512    .086295     3.16   0.002     .1026544    .4430479              |        b#c.x |           2  |   .1349579   .1453856     0.93   0.354    -.1517813    .4216972           3  |   .3012428   .1203257     2.50   0.013     .0639282    .5385573              |        _cons |   42.03102   4.907897     8.56   0.000     32.35134    51.71071 ------------------------------------------------------------------------------                /* test of overall significant of the interaction */  testparm b#c.x                ( 1)  2.b#c.x = 0  ( 2)  3.b#c.x = 0         F(  2,   194) =    3.15             Prob > F =    0.0453              /* or */                contrast b#c.x                Contrasts of marginal linear predictions  Margins      : asbalanced  ------------------------------------------------              |         df           F        P>F -------------+----------------------------------        b#c.x |          2        3.15     0.0453              |     Residual |        194 ------------------------------------------------

The testparm and/or contrast commands show that the overall interaction is statistically significant.

Next, we will compute the simple slopes using the margins command.

                margins b, dydx(x)                Average marginal effects                          Number of obs   =        200 Model VCE    : OLS  Expression   : Linear prediction, predict() dy/dx w.r.t. : x  ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- x            |            b |           1  |   .2728512    .086295     3.16   0.002     .1037161    .4419862           2  |   .4078091   .1170049     3.49   0.000     .1784837    .6371345           3  |   .5740939   .0838538     6.85   0.000     .4097435    .7384444 ------------------------------------------------------------------------------

The slope for b = 1 seems to be different from the slopes of b equal 2 or 3.

Now, let's graph the slopes along with a scatterplot of the data. We will do this by quietly running margins (to suppress the large output) followed by a marginsplot command with added scatterplot.

                quietly margins b, at(x=(25(5)70))  marginsplot, recast(line) noci addplot(scatter y x, jitter(3) msym(oh))                Image catcon12_3              

Let's see if the slope for b = 3 is significantly different from each of the other two slopes. We will test this using reference contrasts with the margins command. We will indicate that slope 3 is the reference using b3 and reference group coding with r which combine to rb3.

                margins rb3.b, dydx(x)                margins rb3.b, dydx(x)  Contrasts of average marginal effects Model VCE    : OLS  Expression   : Linear prediction, predict() dy/dx w.r.t. : x  ------------------------------------------------              |         df        chi2     P>chi2 -------------+---------------------------------- x            |            b |    (1 vs 3)  |          1        6.27     0.0123    (2 vs 3)  |          1        1.33     0.2480       Joint  |          2        6.29     0.0430 ------------------------------------------------  --------------------------------------------------------------              |   Contrast Delta-method              |      dy/dx   Std. Err.     [95% Conf. Interval] -------------+------------------------------------------------ x            |            b |    (1 vs 3)  |  -.3012428   .1203257     -.5370769   -.0654086    (2 vs 3)  |  -.1662848     .14395     -.4484217     .115852 --------------------------------------------------------------

We see that slope 3 is significantly different from slope 1 but is not different from slope 2.

Looking at the three slopes one might wonder where the differences between groups are statistically significant. The most natural way to do this is to pick a reference group, this time b = 1 and see where the values for b = 2 are different and then the same for b1 versus b3. Again, the margins command with the dydx option comes to mind.

                margins, dydx(b) at(x=(25(5)70)) vsquish                Conditional marginal effects                      Number of obs   =        200 Model VCE    : OLS  Expression   : Linear prediction, predict() dy/dx w.r.t. : 2.b 3.b 1._at        : x               =          25 2._at        : x               =          30 3._at        : x               =          35 4._at        : x               =          40 5._at        : x               =          45 6._at        : x               =          50 7._at        : x               =          55 8._at        : x               =          60 9._at        : x               =          65 10._at       : x               =          70  ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- 2.b          |          _at |           1  |  -11.19327   3.704771    -3.02   0.003    -18.45448    -3.93205           2  |  -10.51848   3.055425    -3.44   0.001      -16.507   -4.529954           3  |  -9.843688   2.450055    -4.02   0.000    -14.64571   -5.041668           4  |  -9.168898   1.930483    -4.75   0.000    -12.95258   -5.385221           5  |  -8.494108   1.583543    -5.36   0.000    -11.59779   -5.390422           6  |  -7.819319   1.531437    -5.11   0.000    -10.82088   -4.817758           7  |  -7.144529   1.799955    -3.97   0.000    -10.67238   -3.616682           8  |  -6.469739   2.278427    -2.84   0.005    -10.93537   -2.004105           9  |   -5.79495   2.863471    -2.02   0.043    -11.40725   -.1826503          10  |   -5.12016   3.502078    -1.46   0.144    -11.98411    1.743787 -------------+---------------------------------------------------------------- 3.b          |          _at |           1  |  -12.63537   3.853374    -3.28   0.001    -20.18784   -5.082896           2  |  -11.12916   3.285978    -3.39   0.001    -17.56956   -4.688757           3  |  -9.622942   2.733264    -3.52   0.000    -14.98004   -4.265844           4  |  -8.116729   2.206291    -3.68   0.000    -12.44098   -3.792477           5  |  -6.610515   1.728765    -3.82   0.000    -9.998831   -3.222199           6  |  -5.104301   1.354048    -3.77   0.000    -7.758187   -2.450415           7  |  -3.598087   1.184137    -3.04   0.002    -5.918953   -1.277221           8  |  -2.091873   1.301856    -1.61   0.108    -4.643464    .4597175           9  |  -.5856595    1.64663    -0.36   0.722    -3.812996    2.641677          10  |   .9205543   2.109945     0.44   0.663    -3.214862    5.055971 ------------------------------------------------------------------------------ Note: dy/dx for factor levels is the discrete change from the base level.

The first block of results, 2.b compares b1 with b2 and are significant for x values less thatn 70. The second block, b3 compares b1 with b3 and are significant for x values less thatn 60. Let's graph these margins results.

                marginsplot, recast(line) recastci(rarea) yline(0)                Image catcon12_4              

The blue shaded area is b1 vs b2 while the red shaded area is b1 vs b3. The graph reaffirms our interpretation of the margins table.

croommeme1948.blogspot.com

Source: https://stats.oarc.ucla.edu/stata/faq/how-can-i-understand-a-categorical-by-continuous-interaction-stata-12/

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