As an example, we consider the HELP data used extensively for examples in our book. Does homelessness affect physical health, as measured by the PCS score from the SF-36?
First, we consider modeling this relationship directly. This analysis only answers the question of whether homelessness is associated with poorer physical health.
Then we create a propensity score by estimating a logistic regression to predict homelessness using age, gender, number of drinks, and mental health composite score. Finally, we include the propensity score in the model predicting PCS from homelessness. If we accept that these propensity predictors fully account for the probability of homelessness, and there is an association between homelessness and PCS in the model adjusting for propensity, and the directionality of the association flows from homelessness to PCS, then we can conclude that homelessness causes differences in PCS.
We note here that this conclusion relies on other untestable assumptions, including linearity in the relationship between the propensity and PCS. Many users of propensity scores prefer to fit models within strata of the propensity score, or to match on propensity score, rather than use the regression adjustment we present in this entry. In a future entry we'll demonstrate the use of matching.
In a departure from our usual practice, we show only pieces of the output below.
SAS
We being by reading in the data and fitting the model. This is effectively a t-test (section 2.4.1), but we use proc glm to more easily compare with the adjusted results.
proc glm data="c:\book\help";
model pcs = homeless/solution;
run;
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept 49.00082904 0.68801845 71.22 <.0001
HOMELESS -2.06404896 1.01292210 -2.04 0.0422
It would appear that homeless patients are in worse health than the others.
We next use proc logistic to estimate the propensity to homelessness, using the output statement to save the predicted probabilities. We omit the output here; it could be excluded in practice using the ODS exclude all statement.
proc logistic data="c:\book\help" desc;
model homeless = age female i1 mcs;
output out=propen pred=propensity;
run;
It's important to make sure that there is a reasonable amount of overlap in the propensity scores between the two groups. Otherwise, we'd be extrapolating outside the range of the data when we adjust.
proc means data=propen;
class homeless;
var propensity;
run;
N
HOMELESS Obs Mean Minimum Maximum
---------------------------------------------------------------
0 244 0.4296704 0.2136791 0.7876000
1 209 0.4983750 0.2635031 0.9642827
---------------------------------------------------------------
The mean propensity to homelessness is larger in the homeless group. If this were not the case, we might be concerned the the logistic model is too poor a predictor of homelessness to generate an effective propensity score. However, the maximum propensity among the homeless is 20% larger than the largest propensity in the non-homeless group. This suggests that a further review of the propensities would be wise. To check them, we'll generate histograms for each group using the proc univariate (section 5.1.4).
proc univariate data=propen;
class homeless;
var propensity;
histogram propensity;
run;
The resulting histograms suggest a some risk of extrapolation. In our model, we'll remove subjects with propensities greater than 0.8.
proc glm data=propen;
where propensity lt .8;
model pcs = homeless propensity/solution;
run;
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept 54.19944991 1.98264608 27.34 <.0001
HOMELESS -1.19612942 1.03892589 -1.15 0.2502
propensity -12.09909082 4.33385405 -2.79 0.0055
After the adjustment, we see a much smaller difference in the physical health of homeless and non-homeless subjects, and find no significant evidence of an association.
R
ds = read.csv("http://www.math.smith.edu/r/data/help.csv")
attach(ds)
summary(lm(pcs ~ homeless))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 49.001 0.688 71.220 <2e-16 ***
homeless -2.064 1.013 -2.038 0.0422 *
We use the glm() function to fit the logistic
regression model (section 4.1.1). A formula object is used to specify the model. The predicted probabilities that we'll use as propensity scores are in the fitted element of the output object.
form = formula(homeless ~ age + female + i1 + mcs)
glm1 = glm(form, family=binomial)
X = glm1$fitted
As in the SAS development, we check the resulting values. Here we use the fivenum() function (section 2.1.4) with the tapply() function (section 2.1.2) to get the results for each level of homelessness.
> tapply(X,homeless, FUN=fivenum)
$`0`
398 97 378 69 438
0.2136787 0.3464170 0.4040223 0.4984242 0.7876015
$`1`
16 18 262 293 286
0.2635026 0.4002759 0.4739152 0.5768015 0.9642833
Finding the same troubling evidence of non-overlap, we fit a histogram for each group. We do this manually, setting up two output areas with the par() function (section 5.3.6) and conditioning to use data from each homeless value in two calls to the hist() function (section 5.1.4).
par(mfrow=c(2,1))
hist(X[homeless==0], xlim=c(0.2,1))
hist(X[homeless==1], xlim=c(0.2,1))
As noted above, we'll exclude subjects with propensity greater than 0.8. This is done with the subset option to the lm() function (as in section 3.7.4).
summary(lm(pcs ~ homeless + X,subset=(X < .8)))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 54.199 1.983 27.337 < 2e-16 ***
homeless -1.196 1.039 -1.151 0.25023
X -12.099 4.334 -2.792 0.00547 **
