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I'm currently playing around with linear regression in R, and I've come up with a regression that fits data quite well. I'm just having some problems with interpreting the coefficients of my model. I know how to interpret log-log models in a simpler form, but when I have interactions I'm not quite sure how to interpret them.

Here's my output from R:

Call:
lm(formula = log(y) ~ log(x1) + x2 * log(x1) + x3 * log(x1) + 
I(x3^2), data = Data)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.56943 -0.12082  0.00012  0.11123  0.54579 

Coefficients:
                Estimate   Std. Error t value          Pr(>|t|)    
(Intercept) -2.393889950  0.545879641  -4.385 0.000025149470154 ***
log(x1)      0.497477722  0.056113496   8.866 0.000000000000009 ***
x2          -0.000264760  0.000055476  -4.773 0.000005220020368 ***
x3           0.041126987  0.017930934   2.294           0.02357 *  
I(x3^2)     -0.000688879  0.000231778  -2.972           0.00358 ** 
log(x1):x2   0.000031580  0.000006691   4.720 0.000006494076511 ***
log(x1):x3   0.003145219  0.001277909   2.461           0.01528 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 0.1932 on 119 degrees of freedom
Multiple R-squared: 0.9865,     Adjusted R-squared: 0.9859 
F-statistic:  1454 on 6 and 119 DF,  p-value: < 2.2e-16 

I've been Googling for the past hour, but I can only find answers to some simpler models like the answer given here: Interpretation of log transformed predictor or http://www.ats.ucla.edu/stat/sas/faq/sas_interpret_log.htm

I hope someone out there can help me with interpreting the interaction terms and the polynomial term in my model.

marcopah
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1 Answers1

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Interpretation of complex interactions is always tricky. The best way, I think, is to graph them. One way to start is with 3 graphs, one for each IV. For each of these graphs, one IV will be on the x-axis, and the the DV on the y-axis. Then make lines for the predicted value of the DV for each of several combinations of the other two IVs (e.g. quartiles).

So, plot 1 would have x1 on the x-axis, and 6 lines a) 1st quartile of both x2 and x3 b) Median x2, 1st quartile x3 c) 3rd quartile x2, 1st quartile x3 etc.

You can also make a single table with various values of x1, x2 and x3 and the predicted y.

Then you can use these graphs and tables to try to come up with a verbal description, but you will probably wind up referring to the figures.

Peter Flom
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  • Does it make sense to only have the interaction with the the linear part of the x3, given that x3 is a quadratic in the model? – user20650 Dec 05 '13 at 12:15
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    Another option is contour plots or surface plots e.g. [here](http://stats.stackexchange.com/questions/74275/problem-with-response-optimization-with-three-variables-using-response-surface-i/74461#74461). It's really worth spending some time on plotting the model different ways; it can show where its predictions are *surprising*. – Scortchi - Reinstate Monica Dec 05 '13 at 12:20
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    @user20650 It makes sense; but whether it is the best choice is another matter. – Peter Flom Dec 05 '13 at 12:33
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    @user20650: It makes mathematical sense, but whether it makes substantive sense depends what $x_3$ measures. Does it make sense for the model to force the relation of the response to $x_3$, when $x_1=\mathrm{e}$ & $x_2$ is held constant, to be a parabola with a maximum/minimum at $x_3=0$? Does $x_3=0$ even have any special interpretation, or is it an arbitrary point on an interval measurement scale? The answers [here](http://stats.stackexchange.com/questions/11009/including-the-interaction-but-not-the-main-effects-in-a-model) may help. – Scortchi - Reinstate Monica Dec 05 '13 at 12:41
  • @Scortchi: thanks for the comment & link. I think my difficulty with this is that it is almost stripping part of a function out with which to have an interaction - i know its unattractively phrased - and still offering a clear interpretation – user20650 Dec 06 '13 at 00:19
  • Thanks for the answer Peter. I've just played around with different plots, and your suggestion actually made it a bit easier to interpret. :) – marcopah Dec 06 '13 at 08:43
  • @user20650: [Venables(2000), "Exegeses on Linear Models"](http://www.stats.ox.ac.uk/pub/MASS3/Exegeses.pdf) discusses the marginality principle. It can hardly be an absolute law (one man's $x$, say area, is another man's $x^2$, say diameter), & the final court of appeal is the plausibility of the restrictions imposed on the proposed model when it's violated, but it's a very handy guide for building empirical models. – Scortchi - Reinstate Monica Dec 06 '13 at 09:47
  • @MarcoDalFarra I think the `effects` package would come in handy to plot complex regression coefficients... – landroni Feb 05 '15 at 16:57