I am working with a two part regression model for semi-continuous data slightly modified from Duan et al.

The Duan et al. model is used to predict medical expenses over the course of a year. There is a fraction of people who have zero expenses, and the rest have positive expenses that are log-normal distributed.

The Equations From Duan et al.

$I_i = x_i\delta_1+\eta_{1i}$,       $\eta_{1i} \sim N(0,1)$

Where $\mbox{MED} > 0$ if $I \ge 0$, and $\mbox{MED} = 0$ otherwise. The second equation is a linear model on the log scale for positive expenses:

$\mbox{log}(\mbox{MED}_i|I_i > 0) = x_i\delta_2 +\eta_{2i}$,               $\eta_{2i} \sim N(0,\sigma^2)$

I'm using a similar model but allowing for negative values, so use a normal distribution for the values rather than a log-normal. I'm fitting the model using the BayesGLM package in R with normal priors on the regression coefficients, which is equivalent to L2/Ridge Regression.

The model has two equations the first is probit for whether or not the event is zero, the other is a linear model.

I would like to be able to find the mean effect, and confidence intervals around the mean. I know that for each part of the model, I can calculate this but I'm not sure how to calculate it for the combined model.

For the linear model I can look at the estimate of the effect and the standard error to get the confidence intervals, but how do I layer in the probit model? For example if a variable has a positive effect in the linear componenent and a negative effect in the probit component, how can I figure out if the effect is overall positive or not?

Duan, N., Jr, W. M., & Morris, C. (1983). A comparison of alternative models for the demand for medical care. Journal of Business & Economic Statistics


3 Answers 3


I'm not familiar with Duan et al. model but by the way you wrote the equations, it's also a Tobit II model or Heckman's model. If you refer to the Tobit II notation in http://en.wikipedia.org/wiki/Tobit_model $y_1:=I_i$ and $y_2^*:= \log(MED|I>0)$ etc. you will find the equivalence.

In this case, the conditional expectation $E(y_2|I>0)$ will be determined by an additional correction term, the inverse mills ratio $G()$ that depends on the probit parameter $\delta_2$. In that way this expression takes into the account the direct and indirect effect of $x_i$ on the expectation:

$E(y_2|I>0) = x_i\delta_2 +\lambda G(x_i\delta_2) $

and the mean effect of $x_i$ will be non linear. (more details here http://www.scribd.com/doc/85806008/138/The-Tobit-II-Model )

If you want a quick computational tool that does everything, I can suggest STATA. The heckman command estimates the model while the margins post-estimation command calculates the estimated expectation, confidence intervals and marginal effects ($\partial \hat{E}()/\partial x_i)$. Sorry for not giving more details on the stata procedures, but will do if you want to give stata a try. good luck!

  • $\begingroup$ Thanks it looks like the Duan et al. is equivalent tobit II model with uncorrelated error terms. Could you provide more details on how to calculate the confidence intervals of the marginal effects? I'm using R and don't have access to STATA. $\endgroup$
    – aaronjg
    Jun 28, 2012 at 15:42
  • $\begingroup$ You wrote uncorrelated right? meaning that you impose $E(\eta_1 \eta_2) = 0$ to your model (data)? If true, then the inverse mills ratio term dissapears as it is premultiplied by $\lambda=\rho\sigma$ where $\rho$ is $E(\eta_1 \eta_2)=\rho$. $\endgroup$
    – JDav
    Jun 29, 2012 at 13:38

your problem is ordinairily faced by P&C actuaries (as me). Setting a rate for any coverages means building models (more or less complicated) for the frequency, E[N], and the severity, E[X] and then combining them. I would suggest you to analyze separately the frequency and the severity of the expenses, then you can combine them by the convolution formula. In order to adapt the framework to your specific problem I would do this:

  • Model the frequency, E[N] by a logistic equation (the event would be filing one or more medical expense claim). Then you could get estimates for E[N] and Var[N] for each subject.
  • Model the cost, fitting a regression model. Since you are assuming also negative value to have positive probability a OLS regression could be meaningful. I would otherwise suggest to to check the GAMLSS package for wider conditional distribution choices. However the regression model you build need to give you estimates of E[X] and Var[X].
  • The total cost random variable, S, i.e., your original variable, will have following moments according to convolution formula. E[S]=E[N]E[X], Var[S]=E[N]Var[X]+VarN^2
  • $\begingroup$ How do I obtain Var[N] with the logistic transformation? $\endgroup$
    – aaronjg
    Jul 2, 2012 at 2:46
  • $\begingroup$ if you get E[N], then Var[N]=E[N]*(1-E[N]) $\endgroup$ Jul 6, 2012 at 22:37

What do you mean by "mean effect"? I think it's important when doing an analyses like this to re-establish what question you are trying to answer by running a specific model.

In the probit model, if I understand you correctly, you are trying to determine the impact of an independent variable on the propensity for an individual to have nonzero medical expenses.

In the linear model, then, you are looking at the direct impact of an independent variable on personal medical expenses. You might want to then throw out the zero medical expense observations before running the linear model. That way, you can explain the coefficients in the linear model as:

"The impact of (explanatory variable X) on medical expenses, conditional on having nonzero medical expenses, is Z."

By virtue of you using a probit model, it doesn't appear you are interested in the "mean impact" on the entire population because you are implicitly dividing that population into two groups: nonzero and zero medical expense people.

So, explanatory variable X first acts to "push" a person into (or out of) nonzero medical expense, then once they have been pushed into one of the two groups, acts on them separately in the linear model.

  • $\begingroup$ Yes. That is the two part model described by Duan et al. Suppose I run the model and see that a coefficient has a positive effect in the probit model, but a negative effect in the linear model. How do I figure out the net effect of the covariate? Does it serve to increase medical expenses through increased propensity in the probit part, or is that completely offset by the decreased amount in the linear part of the model? $\endgroup$
    – aaronjg
    Jun 26, 2012 at 1:51
  • $\begingroup$ Take individual "i". His medical expenses, in expectation, and according to your model, would be Prob(ME>0)*E[ME|ME>0]. Increase your covariate by 1 incremental unit, then calculate the change in E[ME] for individual "i". The impact will necessarily be nonlinear due to the probit part. Repeat for all individuals, sum them up, and there is your expected net impact for a one unit change in your covariate. $\endgroup$
    – baha-kev
    Jun 26, 2012 at 5:09
  • $\begingroup$ That would give me the estimate for the effect, what about standard error? $\endgroup$
    – aaronjg
    Jun 26, 2012 at 15:10
  • $\begingroup$ ...if X and Y are independent random variables Var(XY) = E[(XY)^2] - E[XY]^2; Var(XY) = Var(X) Var(Y) + Var(X) E[Y]^2 + E[X]^2 Var(Y) $\endgroup$
    – baha-kev
    Jun 26, 2012 at 16:14
  • $\begingroup$ But the variance of the probit model returned by the GLM is in the probit space... $\endgroup$
    – aaronjg
    Jun 26, 2012 at 17:50

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