Interaction Effect Interpretation in Multiple Regression

[Molly Illingworth]

Hi!

I was wondering whether someone could help me with the following,

I am trying to interpret an interaction effect in stata (see below)

eth_other is a dummy variable, in comparison to those who identify as white

working is a dummy variable in comparison to those who are ILO unemployed or economically inactive.

MCZ_2 dependent variable = self esteem scores

How would i go about interpreting the interaction effect created by the newly generated variable work_eth_o? I have noted how when confounded with work, eth_other self esteem scores shift from a negative relationship upon self-esteem to a positive one. I am tempted to say that when confounded with being in work, those who identify with an ethnicity that is not white, have .81 higher self esteem scores in comparison to those who identify as white (comparison category for eth_other) and not in work (comparison category for not working?)

I hope that made sense - any help would be much appreciated!

[Carlo Lazzaro]

Molly:
you should avoid to create categorical variables and (even worse) interactions by hand: use the wonderful capabailities of -fvvarlist- notation, instead.
Besides, please do not post screenshots but share what you typed and what Stata gave you back via CODE delimiters (see the FAQ). Thanks.
That said, one of the best way to practising with interactions is to ask Stata for the fitted values of your regression (see -predict- under -regression postestimation-) and then trying to re-calculate them using the coefficients of your regression:

. use "C:\Program Files (x86)\Stata15\ado\base\a\auto.dta"
(1978 Automobile Data)
. regress price i.foreign##i.rep78 if rep78>=3

      Source |       SS           df       MS      Number of obs   =        59
-------------+----------------------------------   F(5, 53)        =      0.44
       Model |  19070228.2         5  3814045.63   Prob > F        =    0.8204
    Residual |   462156727        53  8719938.25   R-squared       =    0.0396
-------------+----------------------------------   Adj R-squared   =   -0.0510
       Total |   481226956        58  8297016.48   Root MSE        =      2953

-------------------------------------------------------------------------------
        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
      foreign |
     Foreign  |  -1778.407   1797.111    -0.99   0.327    -5382.955     1826.14
              |
        rep78 |
           4  |  -725.5185   1136.593    -0.64   0.526    -3005.235    1554.198
           5  |  -2402.574   2164.008    -1.11   0.272    -6743.024    1937.876
              |
foreign#rep78 |
   Foreign#4  |   2158.296   2273.185     0.95   0.347    -2401.136    6717.728
   Foreign#5  |   3866.574   2925.484     1.32   0.192    -2001.204    9734.352
              |
        _cons |   6607.074   568.2963    11.63   0.000     5467.216    7746.932
-------------------------------------------------------------------------------

. predict fitted, xb
(5 missing values generated)

*when foreign==0 and rep78 <4 (ie, no interaction), the fitted values equals _cons; this is the main conditional effect of -foreign-*
. list price rep78 foreign fitted if foreign==0 & rep78<4

     +-------------------------------------+
     | price   rep78    foreign     fitted |
     |-------------------------------------|
  1. | 4,099       3   Domestic   6607.074 |
      -------------------------------------
*when foreign==0 and rep78==4 (ie,interaction), the fitted values equals _cons + 4.rep78 coefficient* 

. list price rep78 foreign fitted if foreign==0 & rep78==4

     +-------------------------------------+
     | price   rep78    foreign     fitted |
     |-------------------------------------|
  5. | 7,827       4   Domestic   5881.556 |
     +-------------------------------------+

. di  6607.074 + (-725.5185)
5881.5555

*when foreign==0 and rep78==5 (ie,interaction), the fitted values equals _cons + 5.rep78 coefficient* 

. list price rep78 foreign fitted if foreign==0 & rep78==5

     +-----------------------------------+
     | price   rep78    foreign   fitted |
     |-----------------------------------|
 20. | 3,984       5   Domestic   4204.5 |
     +-----------------------------------+

. di  6607.074 + (-2402.574)
4204.5

.
*when foreign==1 and rep78 <4 (ie, no interaction), the fitted values equals _cons + 1.Foreign coefficient*

. list price rep78 foreign fitted if foreign==1 & rep78<4

     +------------------------------------+
     | price   rep78   foreign     fitted |
     |------------------------------------|
 54. | 6,295       3   Foreign   4828.667 |
     +------------------------------------+

. di  6607.074 + (-1778.407)
4828.667


*when foreign==1 and rep78==4 (ie,interaction), the fitted values equals _cons + 4.rep78 coefficient + the interaction between the two* 

. list price rep78 foreign fitted if foreign==1 & rep78==4

     +------------------------------------+
     | price   rep78   foreign     fitted |
     |------------------------------------|
 55. | 9,735       4   Foreign   6261.444 |
  +------------------------------------+


. di  6607.074 + (-1778.407) + (-725.5185) + 2158.296
6261.4445

.
*when foreign==1 and rep78==5 (ie,interaction), the fitted values equals _cons + 5.rep78 coefficient + the interaction between the two* 

. list price rep78 foreign fitted if foreign==1 & rep78==5

     +-------------------------------------+
     |  price   rep78   foreign     fitted |
     |-------------------------------------|
 53. |  9,690       5   Foreign   6292.667 |
     +------------------------------------+

. di  6607.074 + (-1778.407) + (-2402.574) + 3866.574
6292.667

.

[Phil Bromiley]

To add to Carlo's helpful comments, you should also look at the margins and marginsplot commands. These do predicted values for different values of the iv's (holding all other iv's at their means usually), and marginsplot does nice graphs of the predicted values.

Conceptually, I find Robert J. Friedrich's In Defense of Multiplicative Terms in Multiple Regression Equations, American Journal of Political Science, 1982 particularly helpful. You should also look at Richard William's course notes titled "Interpreting Interaction Effects" https://www3.nd.edu/~rwilliam/