Interpretation interaction between dummy and continuous variable in Poisson irr model with exposure

[Chiara Brouns]

Hello,

For my master thesis I'm estimating a Poisson model with exposure. The dependent variable is price difference, the exposure a maximum limit to these prices (so I get a relative diversion in percentage from the maximum price). To separate the effect of market power on prices between 2 different groups of health care providers, I created interaction terms with the market power (MP) measures and the dummy for group 1 (1) and group 2 (0) of providers. For easier interpretation I estimated the model in IRR. How do I interpret the interaction coefficients in the model? Are they group 1 relative to group 2? Or can they be interpreted as having separate effects? It's not count data, I know, but OLS or logit/probit is not feasible for several reasons.

Thanx a lot!
Chiara

[Clyde Schechter]

If you have run an interaction model, you are best off not trying to interpret the regression output. Rather, run -margins- to get the marginal effects of market power in each group. So if your model looks like this:

poisson price i.group##c.market_power other_variables // etc.

// AVERAGE MARGINAL EFFECT OF MARKET POWER IN EACH GROUP
margins group, dydx(market_power)

// MARGINAL EFFECT OF MARKET POWER IN EACH GROUP AT
// INTERESTING VALUES OF MARKET POWER
margins group, dydx(market_power) at(market_power = (mp1 mp2 mp3...)) 
marginsplot

// PREDICTED VALUES OF PRICE IN EACH GROUP AT
// INTERESTING VALUES OF MARKET POWER
margins group, at(market_power = (mp1 mp2 mp3...))

Note: Replace mp1 mp2 mp3... by actual numbers that are interesting values of market power.

Remember that this is a non-linear model, so even if you had no interaction with group, the effect of market power would depend on the level of market power you are starting from. Having an interaction makes it depend on group as well.

If you are not familiar with the i.group##c.market_power notation, read -help fvvarlist-. Factor variable notation and -margins- are among the most useful features of modern Stata releases. Moreover, they are nearly indispensable tools when modeling interactions. It is possible to do the calculations to interpret interaction models by hand, but it is tedious at best and error-prone. Letting -margins- do it for you, painlessly and accurately, is far the better way.

There is no problem using -poisson- for this kind of dependent variable. I do question, however, the rationale for using the maximum possible price as an exposure variable here. I'm having trouble understanding that--think carefully about whether it really makes sense.

[Chiara Brouns]

Hi Clyde,

Thank you for your answer! Very clear, except for 1 thing. What do you mean by that the effect of market power depends on the level of market power I'm starting from?

Actually in this setting makes sense to take the maximum price as exposure, because these prices are regulated and capped at this level. So the effect of competition in this case is measured by how far away prices are from the maximum price allowed by the government.

Thanx a lot!
Chiara

[Chiara Brouns]

Different question: how can I make a marginsplot with predicted values for the marginal effects?

[Clyde Schechter]

What do you mean by that the effect of market power depends on the level of market power I'm starting from?

This is a general property of nonlinear models like -poisson-. The change in outcome associated with a one unit change in market_power from, say, 1 to 2 will be different from the change in outcome associated with a one unit change in market_power from, say, 10 to 11. (I don't know what the range of actual values of this variable are and 1, 2, 10, and 11 are just chosen for illustrationi. But any such pair of numbers would similarly illustrate the point.)

Actually in this setting makes sense to take the maximum price as exposure, because these prices are regulated and capped at this level. So the effect of competition in this case is measured by how far away prices are from the maximum price allowed by the government.

Yes, but setting the maximum price as the exposure() option doesn't accomplish this. A better approach, I think, would be to make the outcome variable itself the difference between the observed price and the maximum price, or perhaps their ratio or something like that.

Regarding #4, you can use -marginsplot- after any -margins- command. The only constraint is that -marginsplot- must be run immediately after the -margins- command. No other commands can run between them.

[Chiara Brouns]

Ok, clear. Thanx!

The outcome variable is the price difference between the maximum price and the reimbursed price (max price - price paid), with exposure maximum price.

How do I take the exposure factor into account if I want to calculate marginal effects or does Stata do this automatically?

Thanx!

[Chiara Brouns]

One more question... How do I interpret the coefficients without the interaction terms in the interaction term model? Are they the total effects or the effects for group 2?

Price Difference	NBR	IRR
Inverse LOCI	-1.289** (0.540)	0.276** (0.149)
Market share insurer provider	0.0244 (0.0289)	1.025 (0.0296)
Market share largest insurer GHOR	1.146 (0.704)	3.146 (2.215)
Group
DTC dummies	0	1
Insurer dummies	0	1
GHOR region dummies	0	1
Number diagnosis categories	0.00513 (0.00754)	1.005 (0.00758)
Complexity	-0.506** (0.218)	0.603** (0.132)
Comorbidity	0.363 (0.224)	1.437 (0.322)
Budgeted	-1.771*** (0.556)	0.170*** (0.0947)
Inverse LOCI * group	0.823 (0.532)	2.278 (1.213)
Market share insurer provider * group	-0.0459 (0.187)	0.955 (0.178)
Market share largest insurer GHOR * group	0.812** (0.358)	2.253** (0.805)
Constant	-0.796 (0.648)	0.451 (0.293)
lnalpha	-0.0854 (0.0654)	0.918 (0.0600)
N	53601	53601
Log lik.	-330713.6	-330713.6
Chi-squared	14025.0	14025.0
alpha	0.918	0.918

Thanx a lot!!

[Clyde Schechter]

Effects that do not participate in interaction terms are average effects for the entire population.

Regarding #6, -margins- takes the exposure() into account, so you don't have to adjust the -margins- output in anyway. That said, I still cannot make any sense of what you are doing here. If what you are interested in is the difference between the maximum allowable price and the actual price as a fraction of the maximum price, then you should calculate that outcome and use a suitable regression model for it (not Poisson!) Putting maximum price as the -exposure()- does not accomplish this and I think the results you are getting are simply wrong.

[Chiara Brouns]

I was told the effect on the dependent variable can be interpreted as the effect on count/exposure (= rate). Of course I do not have a real count variable, but you could look at it that way.... How would you model a rate with a Poisson model?

[Clyde Schechter]

I was told the effect on the dependent variable can be interpreted as the effect on count/exposure (= rate).

Yes, but in an entirely different sense. The typical situation where this applies is if you are modeling counts of, say potholes in roads, and the exposure is the number of miles of road observed. You can interpret the effects as potholes per mile of road. If you are counting ticks of a photon detector and the exposure is the period of time observed, you can interpret it as ticks per second (or minute or whatever). If you are counting chocolate chips in bags of cookies, you can interpret it as chocolate chips per bag, or perhaps chocolate chips per pound of cookies. If you are counting incident cases of breast cancer in a population observed over a number of years, then you have a rate of cancer cases per 100,000 person-years. But notice that these rates have numerators that are not pieces of their denominators: they represent proportionality between things of a different nature. And in particular, in none of these examples does the exposure variable represent any kind of ceiling on the counted outcome.

If I have a rate where the rate I am interested in consists of a numerator that is a subset of the denominator, so that the denominator count is a ceiling on the numerator, I would not use a Poisson model. I would consider alternatives such as a -glm-, family(binomial)- model, or -fracreg-, or maybe -betareg-.