# Adjusting variable for covariate to calculate rates or metric

In this report by Dartmouth Institute, rates of utilisation (per patient) were reported by year and regions. The input measure of utilisation were said to have been adjusted for differences in age, sex, race, primary chronic condition using ordinary linear regression and poisson regression (see page 3 and 38). I think it also means it is conditioned on these variables.

I am wondering how was it done. Was regression model fitted with a particular utilisation variable (e.g. hospital days) as dependent variable and the variables they want to adjust for as independent variable? Subsequently was predicted values used to get the rates?

I have an additional questions. Is this method valid? Appreciate if someone could show how conditional expectation is derived from log-linked generalized linear model and how to perform it in R. A working example would be the best.

## 1 Answer

The report goes into their adjustment strategy a little bit, and in general your instincts are correct: when people talk about adjustment, they usually mean they're including those variables in a regression equation to get a conditional estimate.

What likely happened, based on the report, is that they fit hospital days as a dependent variable in a Poisson model, which when you include an offset term can fit rates. They then included the other variables they were interested in as covariates.

In R, this would look something like glm(HospitalDays ~ Age + Sex + Illness + + offset(log(TotalDays)),family=quasipoisson,data=data)

Note I used "quasipoisson" here because the report states that they handled over-dispersion in their Poisson model, which is when the variance of a Poisson model exceeds the mean.

Once they have this model fitted, you can use the output to predict the rates under different circumstances.

• So if I use predict in R to get the fitted values, I will get the conditional expectation of each patients given their individual age, sex and illness? – tatami Jun 28 '18 at 4:24
• @tatami You can do this straight from the regression results as well. – Fomite Jun 28 '18 at 4:25
• so I could just get the conditional expectation from the fitted values? could you elaborate how is fitted values equivalent to conditional expectation? – tatami Jun 28 '18 at 14:48
• The conditional expectation for a Poisson model is: E[y|x1, . . . , xp] = exp(β0+β1x1+···+βpxp). Those β terms are the coefficients in the regression output. – Fomite Jun 28 '18 at 15:11