Interpreting Interaction between Continuous Variables

[Odrija Karmakar]

Hello,

I have an unbalanced panel data set and I run regression with fixed effects. I am looking at the impact of interaction between two variables on Sales. These two variables are continuous in nature, IT intensity and IT investment. Individually, they are negative significant and their interaction is positive and significant (IT investment X IT intensity). How do I interpret the result. Is it not possible to interpret the results without graphically plotting them?

Should I interpret it as the impact of IT investment on sales is less for firms with more IT intensity or the impact of IT investment on sales is more for firms with more IT intensity.
My results are as follows:
The Standard error are in parentheses.
Please suggest

Dependent Variable Sales	MODEL 2
IT Investment	-13.378*** (0.231)
IT intensity	-0.351*** (0.026)
IT Investment X IT Intensity	0.009*** (0.000)
Size	-1.160*** (0.020)
Lev	0.104 (0.220)
MB	-0.004 (0.002)

[Clyde Schechter]

So the interpretation is that the effect of IT investment is -13.378 when IT intensity = 0. As IT intensity increases, the effect of IT investment increases (which means, initially, that it becomes less negative). If IT intensity increases sufficiently, it will be come zero and then positive and continue to increase. More generally, the effect of IT investment is given by:

Effect of IT investment = -13.378 + 0.009*IT intensity.

Is it not possible to interpret the results without graphically plotting them?

Well, is you can see, it is possible. But if you intend to try to present your findings to a non-technical audience, this explanation will just leave them bewildered, and their eyes will glaze over. So, yes, you should make a graph!

[Odrija Karmakar]

Thank you for the explaination

[Richard Williams]

I find interactions between continuous variables to be a bit confusing. Some suggestions for making them a little less confusing are at

https://www3.nd.edu/~rwilliam/stats2/l55.pdf

[Phil Bromiley]

You might also look at Friedrich, American J of Political Science, 1982.

[Odrija Karmakar]

Thank you so much for all your replies.

[Matthew Williams]

Dear Professor Clyde Schechter and Richard Williams,

I know that this thread is quite old, and there are several similar topics as this one in this forum, however, I would highly appreciate it if you could look at my interpretation of an interaction term between two continuous variables. The health outcome is measured as a continuous variable (its range is from 0-13) such that a higher value indicates poor health. The two interested independent variables are income and pension indicators measured as %GDP. Here I want to examine how pension modify the association between health and income, so I ran multilevel regressions with an interaction term between income and pension as follows:

mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country:, variance covariance(unstructured)

Results

------------------------------------------------------------------------------------------------------------
                                 health    | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------------------------------------+----------------------------------------------------------------
                                           |
                                    income |   .0003857   .0260837     0.01   0.988    -.0507374    .0515088
                                   pension |  -.2269578   .2170212    -1.05   0.296    -.6523115    .1983959
                                           |
                        c.income#c.pension |  -.0079466   .0041762    -1.90   0.057    -.0161318    .0002386
                                           |
                                     _cons |    14.2035   1.363194    10.42   0.000     11.53169    16.87531
------------------------------------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects parameters  |   Estimate   Std. err.     [95% conf. interval]
-----------------------------+------------------------------------------------
country: Identity            |
                  var(_cons) |   .5895703   .2103578      .2929733    1.186433
-----------------------------+------------------------------------------------
               var(Residual) |   9.082687   .0722386        8.9422    9.225382
------------------------------------------------------------------------------
LR test vs. linear model: chibar2(01) = 1764.81       Prob >= chibar2 = 0.0000

Marginsplot (I selected smallest to largest values of both income and pension)

    margins, at(income=(-15(5)15) pension=(5(1)7)) vsquish
    marginsplot, noci scheme(sj) ytitle("Predicted health")

My understanding is that the effect of the interaction term is the effect of income on health by one unit (1%? since pension is measured as %GDP) increase in pension, is that correct? As for coefficients, the effect of income on health = 0.0004 - 0.079*pension. Based on the marinsplot (please see the attached graph), we can say that higher pension benefits lower the association between income and health? or in the field of health inequality, it is equivalent to interpret that countries with higher pension benefits tend to have lower income-health inequalities. Are my interpretations correct?

Thank you.

[Clyde Schechter]

I was in agreement with all you say until we got to "Based on the marinsplot (please see the attached graph), we can say that higher pension benefits lower the association between income and health? or in the field of health inequality, it is equivalent to interpret that countries with higher pension benefits tend to have lower income-health inequalities. Are my interpretations correct?"

I think your interpretations are exactly backwards. This is a problem that arises because of the conflict between the mathematical use of the terms "less" or "lower" and their colloquial use.

Mathematically, with the interaction term being negative, higher pensions are associated with lower marginal effect of income on health. But through the range of pensions you have graphed, at least, all of those marginal effects are negative. And a lower negative number is a number that has a larger magnitude than a higher negative number. Consequently, in the colloquial sense of lower, the strength of association between income and health is actually greater in the high pension countries. In terms of income related health inequalities, this means that the higher pension countries show more income-associated health inequality than the lower pension countries.

You can see this directly in your graphs. Look at the top graph and the bottom graph. The health index plummets from about 12.1 all the way down to just under 10.5. This is a mujch greater magnitude of difference in health outcome associated with the same difference in income. So there is more income-related health inequality in your high pension countries in the sense that people usually mean the term.

While we are on the topic of terminology and mathematical vs colloquial language, with regard to the units of the income and GDP variables, there is the confusing matter of dealing with variables defined as percentages of something. The unit of meaurement for these variables is percent(age) of GDP. But a one unit difference or change in the variable is a difference (change) of 1 percentage point.

[Matthew Williams]

Thank you Prof. Clyde Schechter for the detailed explanations as always (I am a fan :D).

You are totally correct about this point and I now clearly understand the interpretation of the interaction term.

And a lower negative number is a number that has a larger magnitude than a higher negative number

[Weiwen Ng]

I have nothing to add to what Clyde said. However, while you didn't ask this, I see that your model includes a request for an unstructured covariance option.

mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country:, variance covariance(unstructured)

That option tells Stata about what to do about the random effects, plural. The default is to assume that the random effects aren't correlated, e.g. the value of the random intercept for country is not correlated with the random slope on time. Unstructured, with two random effects, would assume those two random effects are correlated (e.g. random slope increases with random intercept), and estimate the correlation.

With only one random effect, it doesn't do anything. It doesn't hurt, unless you hurt your finger typing. I mention this just in case you thought you were doing something else with that option.

[Matthew Williams]

Hi Weiwen Ng,

I am pretty new to mixed command, so I am grateful for your insightful inputs. Let me clarify a few things to make sure that I understand you correctly.

With only one random effect, it doesn't do anything. It doesn't hurt, unless you hurt your finger typing

I think what you meant here is that the two below commands do the same thing for only one random effect, and "one random effect" here would mean that the random intercept for country, is my understanding correct?

mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country:, variance covariance(unstructured)
mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country:,

Now, turning to two random effects, assuming that I let income varies across countries, so Stata command for two random effects would be:

* this case is the value of the random intercept for country is correlated with the random slope of income
mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country: income, variance covariance(unstructured)

* this case is the value of the random intercept for country is not correlated with the random slope of income
mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country: income,

Please feel free to correct me if I was wrong in any points. Also, would you suggest which models above would be more appropriate for two random effects?

Thank you.

[Weiwen Ng]

...
I think what you meant here is that the two below commands do the same thing for only one random effect, and "one random effect" here would mean that the random intercept for country, is my understanding correct?

mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country:, variance covariance(unstructured)
mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country:,

That's correct. You can go ahead and verify that by looking through the output tables.

Now, turning to two random effects, assuming that I let income varies across countries, so Stata command for two random effects would be:

* this case is the value of the random intercept for country is correlated with the random slope of income
mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country: income, variance covariance(unstructured)

* this case is the value of the random intercept for country is not correlated with the random slope of income
mixed health m5064 m6575 f5064 f6575 i.eduisced c.income##c.pension || country: income,

...

Correct also. You can think of it this way:

	Var(country intercept)	Var(slope of income)
Var(country intercept)	u_1	Covar(intercept, slope)
Var(slope of income)	Covar(intercept, slope)	u_2

I know that "variance-covariance matrix" sounds intimidating, but the above is the variance-covariance matrix of the two intercepts in the model. Let u_1 and u_2 be the variances of the country's intercept and the slope of income. Those are always going to be estimated. The key is what you assume about the covariance of the two - remember that covariance can be mathematically transformed to correlation. The default option without adding the covariance structure is to assume they're uncorrelated. There are a few other types of structures you can estimate, but ignore that for now since you only have two random effects. By adding the unstructured option, you're now telling Stata to estimate u_3.

As to selecting which model, that's a complex question. You could just fit the more complex model if you have substantive grounds to do so. That would effectively be saying that you think that countries with higher baseline health (higher value of the random intercept) have their health more strongly related to income. Maybe that doesn't make sense here, but there are situations where it might. Anyway, if you have substantive grounds to make that assumption, I think many people would accept you doing so even if the covariance is very low (i.e. you might as well have fitted the more restrictive model).

Alternatively, if you run the more complex model, and you see the covariance is ignorable, you could just report the more constrained model (without the covariance) and mention in discussion that the model with a covariance produced basically identical results, or the covariance is close to 0.

Often, to compare nested models, we would run likelihood ratio tests. I am actually not sure how it works for testing a model with a covariance between the random effects and the same model with no covariance. The test may be invalid. I would Google if you go this route.