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How would you interpret this interaction? The structure of the data is all integer variables. Inc.fix= income, age.fix=age, profit99= profit

 Call:
  lm(formula = Profit99 ~ Age.fix + Inc.fix + Age.fix:Inc.fix, 
       data = pilg)

      Residuals:
           Min      1Q  Median      3Q     Max 
          -421.86 -148.76  -84.45   55.67 1938.70 

              Coefficients:
                            Estimate Std. Error t value Pr(>|t|)    
                 (Intercept)     -139.9706    11.1273 -12.579  < 2e-16 ***
                 Age.fix           37.8453     2.4430  15.491  < 2e-16 ***
                 Inc.fix           26.5252     1.9790  13.403  < 2e-16 ***
                 Age.fix:Inc.fix   -2.2217     0.4475  -4.965 6.92e-07 ***
                  ---
                 Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

                Residual standard error: 268.1 on 31630 degrees of freedom
                Multiple R-squared:  0.03443,   Adjusted R-squared:  0.03434 
                 F-statistic: 375.9 on 3 and 31630 DF,  p-value: < 2.2e-16
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migrated from stackoverflow.com Apr 24 '15 at 18:02

This question came from our site for professional and enthusiast programmers.

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    $\begingroup$ This is more of a statistics question than an R programming question so have flagged for transfer to Cross-Validated. $\endgroup$ – r.bot Apr 24 '15 at 17:47
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    $\begingroup$ It might be easier to pick a handful of ages, a couple incomes, plug those numbers into the equation with the betas, and see how profit changes as you vary one parameter, the other, or both. then see if you can generalize like the answers below do $\endgroup$ – rawr Apr 24 '15 at 18:16
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    $\begingroup$ Another possibility would be to try visualizing the results. car::scatter3d() might be useful. $\endgroup$ – Ben Bolker Apr 24 '15 at 18:18
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  • The expected profit when Age and Inc (income?) are both 0 equals -139
  • when Inc is zero, for every additional unit of Age, the expected profit increases by 38
  • when Age is zero, for every additional unit of Inc, the expected profit increases by 27
  • for every additional unit of Inc, the slope with respect to Age (expected increase in profit per unit of Age) decreases by 2.2 (i.e. changes by -2.2), and vice versa (for additional unit of Age, the slope with respect to Inc also decreases by 2.2).
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Interpreting regression models/linear models can be tricky. In my head, I always "read" the formula: Profit is a function of (or can be predicted by) of age and income, but not necessarily additive (hence the interaction term).

All of your P-values are signification, therefore you need to consider all of the term in the regression. Looking at the individual coefficients:

  • As age increase profits increase (because the coefficient is positive)
  • As income increases, profits increase (because the coefficient is positive as well)
  • As both age and income increase, profit decreases increase as much (because the coefficient is negative)

The last coefficient is hardest to interpret. The strength of the interaction depends upon the coefficient values. In your case,

Also, you might want to checkout the Wikipeida article on the topic for more background.

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    $\begingroup$ The only part of this I would disagree with is "All of your P-values are [significant], therefore you need to consider all of the term[s] ...". IMO you need to consider all of the terms that you put into the model, whether they're significant or not. $\endgroup$ – Ben Bolker Apr 24 '15 at 18:17
  • $\begingroup$ @BenBolker That's a good nuanced catch. Personally, in my own data analysis, I would either use confidence or credible intervals and avoid P-values, but that's a different topic for a different post... $\endgroup$ – Richard Erickson Apr 24 '15 at 18:19
  • $\begingroup$ @whuber, Thank you! I was answering the question on Stack Overflow when it got migrated over. Also, I browsed your CV. Your research looks interesting. $\endgroup$ – Richard Erickson Apr 24 '15 at 18:23
  • $\begingroup$ I hope you will consider answering more questions over here :-). $\endgroup$ – whuber Apr 24 '15 at 18:26

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