Interpreting continuous by continous interaction in Poisson regression

[Monzur Alam]

Dear statalisters,

I am trying to estimate a model using Poisson regression in Stata 13. My dependent variable is a household food consumption score (a count variable that ranges from 0-12), and my key independent variable is an “empowerment” score, (emp_score2, continuous variable) of the primary female decisionmaker in the household. I have interacted this score with the amount of land the household owns (land, continuous variable), because I want to examine how the relationship between women’s empowerment and food consumption varies by the amount of land owned by the household . The interaction term is significant as well as the empowerment score, and land. If this were an OLS regression, I would have interpreted this as “food consumption score tends to decrease in larger landowner households where women have higher empowerment scores”. However, I am not quite sure how to interpret this in a Poisson model and was looking for some advice.

. poisson hhdietscore12 emp_score2 empxland age_hhhead agesq_hhhead eduy_hhhead hhsize depratio
land elec_conn owns_handtbwell san_latrine rice_price d1-d2 d4-d7 num_allcrops, vce(robust)
 
 
hhdietscore12 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
   emp_score2 |   .0623765   .0129364     4.82   0.000     .0370216    .0877314
     empxland |    -.08131   .0380617    -2.14   0.033    -.1559097   -.0067104
   age_hhhead |  -.0004532    .001327    -0.34   0.733     -.003054    .0021476
 agesq_hhhead |   2.99e-06   .0000137     0.22   0.828    -.0000239    .0000299
  eduy_hhhead |   .0075075   .0007078    10.61   0.000     .0061202    .0088949
       hhsize |   .0091887   .0017194     5.34   0.000     .0058187    .0125586
     depratio |  -.0170076   .0047506    -3.58   0.000    -.0263187   -.0076965
         land |   .0822558   .0258982     3.18   0.001     .0314963    .1330153
    elec_conn |   .0401311   .0055735     7.20   0.000     .0292072     .051055
owns_handtb~l |   .0261323   .0062202     4.20   0.000     .0139409    .0383238
  san_latrine |   .0355472   .0061283     5.80   0.000     .0235359    .0475584
   rice_price |   .0020904   .0009557     2.19   0.029     .0002172    .0039636
           d1 |  -.0292366   .0118613    -2.46   0.014    -.0524844   -.0059888
           d2 |   .0114597   .0099809     1.15   0.251    -.0081026     .031022
           d4 |  -.0096846   .0089651    -1.08   0.280     -.027256    .0078868
           d5 |  -.0055402   .0092899    -0.60   0.551     -.023748    .0126676
           d6 |  -.0448051   .0109061    -4.11   0.000    -.0661806   -.0234295
           d7 |   .0015101   .0090206     0.17   0.867    -.0161699    .0191901
 num_allcrops |   .0102112   .0018483     5.52   0.000     .0065886    .0138338
        _cons |   2.075979   .0426326    48.69   0.000     1.992421    2.159538
-------------------------------------------------------------------------------

[Maarten Buis]

I would add the irr option to your poisson model to make the interpretation easier.

If you do so than you will find that a unit increase in empowerment if the household holds no land is a associated with an increase in the foodconsumption by a factor exp(.0623765) = 1.06. In other words a (1.06-1)*100% = 6% increase in food consumption.

This effect of empowerment decreases by a factor of exp(-.08131) = 0.92 or (0.92-1)*100% = -8% for every unit (ha?) increase in landownership. So if the family owns 2 ha then the effect of empowerment is already negative: exp(.0623765 +2*-.08131) = 1.06*.92^2 = 0.90 or -10%

Also see: http://maartenbuis.nl/publications/interactions.html

[daniel klein]

Being super-cautious, you could think about testing for non-linear main effects as those may lead to spurious interactions (see this presentation by Patrick Royston). I have no clue about your research topic, but might it be reasonable to assume some kind of marginal return for empowerment and/or landownership? If your theory suggests such pattern, or it has been demonstrated by previous research on the topic, you might want to give it a shot.

Aside from this, I like to use margins and marginsplot to visually inspect continuous by continuous interactions.

Best
Daniel

[Jeff Wooldridge]

Because the mean function is exponential, you can write

log E(y|x) = b0 + b1*emp_score2 + b2*emp_score2*land + ...

So, subject to approximation error, the proportionate change in E(y|x) is

b1 + b2*land

At land = 0 (is this possible in the data?) the effect is b1. As Martin says, you can use the irr option to get a better estimate of the percentage effect, which is about 6% at land = 0. As land increases, the effect gets smaller, and eventually is negative. You need to say something about the units of measurement of land. The effect of emp2 is zero when

land = -b1/b2 = .768

Does this make sense? It depends on how land is measures. Then the proportionate effect of emp2 becomes negative for land > .768.

[Monzur Alam]

Thank you very much, Maarten, Daniel and Jeff. I should have mentioned this in my original post. The land distribution (originally in acres) was highly skewed, so the land variable that I generated is log (land+1) [I added the 1 because there were a large number of landless households, i.e. land=0). Thus I was thinking of interpreting changes in the land variable, not in terms of hectares or acres, but just as one unit, zero units, etc. I wonder if that makes sense.

[高佳]

I would add the irr option to your poisson model to make the interpretation easier.

If you do so, than you will find that a unit increase in empowerment if the household holds no land is a associated with an increase in the foodconsumption by a factor exp(.0623765) = 1.06. In other words a (1.06-1)*100% = 6% increase in food consumption.

This effect of empowerment decreases by a factor of exp(-.08131) = 0.92 or (0.92-1)*100% = -8% for every unit (ha?) increase in landownership. So if the family owns 2 ha then the effect of empowerment is already negative: exp(.0623765 +2*-.08131) = 1.06*.92^2 = 0.90 or -10%

Also see: http://maartenbuis.nl/publications/interactions.html

Dr. Buis, I'm confused with what has been offered here and on the text book. According to Cameron and Trivedi (2005, p.123-124), one can directly interpret the coefficient of a Poisson regression model as semi-elasticity. So, in OP's case, the marginal effect of emp_score2 is 6.23% percent increase of the dependent variable. This is very close to the interpretation based on irr, which is exp(0.0623) - 1. This is intuitive given that exp(b) - 1 approximate b when b is very small. But what if b is large, say the coefficient of emp_score2 is 1.67 (raw coefficient)? Then the irr would be exp(1.67), and the marginal effect based on the calculation your offered is exp(1.67) - 1 = 4.31: one unit increase of emp_score2 increasing conditional mean by 431%. But the marginal effect based on the semi-elasticity explanation from Cameron and Trivedi (2005, p/123-124) is that one unit increase in emp_score2 increases the conditional mean by 167%. I am really confused, and I may miss something very important here, but I do not know what it is. I would be more than appreciative if you can kindly help me with this.