Interaction Term Interpretation - Continuous*Categorical in Log-Linear OLS

[Sally Dee]

Hi! I have a continuous log-transformed health expenditure variable, and an interaction term of weight (continous variable) * diabetes (binary variable). I have a coefficient of 0.03 and I was wondering how to interpret it? My model specification is;

ln(expenditure) = intercept + age + sex + weight + diabetes + weight*diabetes

Any help would be greatly appreciated.

[Maarten Buis]

You are better of using a log link function rather than log transforming the dependend variable. For a detailed explanation see: https://blog.stata.com/2011/08/22/us...tell-a-friend/

. sysuse nlsw88, clear
(NLSW, 1988 extract)

. poisson wage i.union##c.ttl_exp i.race grade, irr vce(robust) nolog
note: you are responsible for interpretation of noncount dep. variable

Poisson regression                              Number of obs     =      1,876
                                                Wald chi2(6)      =    1047.04
                                                Prob > chi2       =     0.0000
Log pseudolikelihood = -4774.6602               Pseudo R2         =     0.1188

---------------------------------------------------------------------------------
                |               Robust
           wage |        IRR   Std. Err.      z    P>|z|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
          union |
         union  |    1.35593   .0968508     4.26   0.000     1.178794    1.559684
        ttl_exp |    1.04289    .002878    15.22   0.000     1.037264    1.048546
                |
union#c.ttl_exp |
         union  |   .9872035   .0048121    -2.64   0.008     .9778169    .9966803
                |
           race |
         black  |   .9010583   .0218959    -4.29   0.000     .8591492    .9450118
         other  |   1.088323   .1002268     0.92   0.358     .9085906    1.303609
                |
          grade |   1.081436   .0048295    17.53   0.000     1.072012    1.090943
          _cons |   1.494749   .0899274     6.68   0.000     1.328489    1.681816
---------------------------------------------------------------------------------
Note: _cons estimates baseline incidence rate.

In this case we see that union membership when one has no previous work-experience results in a 36% (\( (1.35593-1)\times100\% \approx 36\% \) ) wage increase, but that for every year experience the effect of union membership decreases by 1% (\( ( .9872035 -1)\times 100\% \approx -1\% \) )

. lincom 1.union + 10*1.union#c.ttl_exp, eform

 ( 1)  [wage]1.union + 10*[wage]1.union#c.ttl_exp = 0

------------------------------------------------------------------------------
        wage |     exp(b)   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   1.192077   .0363323     5.76   0.000     1.122952    1.265457
------------------------------------------------------------------------------

So the effect of union membership is "only" 19% after 10 years of work-experience.

[Sally Dee]

Hi Maarten - thank you so much for this response. I'm a junior researcher and I will keep this in mind for my next project, but at the current moment I'm not in a position to be able to change the model. Does the general outline of your interpretation cross over, such that for every kilogram of weight, a person with diabetes is spending approximately 3% more on health expenditure than someone without diabetes?

[Phil Bromiley]

The simple minded way to interpret such things is to use the margins statement. You can exponentiate the predicted value in margins (see detailed discussion of predict options for margins in pdf documentation), and get predicted values for differing values of the rhs variables.