Hello All

On 9 February 2002 I sent the query quoted below to Allstat in the hope 
that somebody might know a reference justifying and/or demonstrating the 
use of propensity scores to adjust for confounder effects where the 
exposure of main interest is continuous, or, alternatively, a reason why I 
shouldn't do this. I have since received 2 emails expressing interest in 
any answer I might get and one from Merete Jørgensen of Novo Nordisk A/S 
giving the following 2 references.

Rubin DB, Thomas N. Combining Propensity Score Matching with Additional 
Adjustments for Prognostic Covariates. Journal of the American Statistical 
Association 2000; 95(450): 573-585.

D'Agostino RB. (1998) Propensity Score Methods for bias reduction in the 
comparison of teh treatment to a non-randomised control Group. Statistics 
in Medicine 1998; 17: 2265-2281.

These are about propensity scores for binary exposures, not continuous 
exposures, but are still useful in their own right. I myself managed to 
find another paper on propensity scores for non-binary outcomes:

Imbens GW. The role of the propensity score in estimating dose-response 
functions. Biometrika 2000; 87(3): 706-710.

This, however, is highly theoretical, and discusses mostly vector 
propensity scores for multinomial exposures (one dimension for each 
possible exposure value), rather than scalar propensity scores for 
continuous exposures (as I was proposing).

However, I also had another look at one of the references I originally 
quoted in my previous email to Allstat (Lu et al., 2001), and noticed that 
it mentioned, in passing, the possibility of using the same approach that I 
was proposing. They state that (as I expected) the use of scalar propensity 
scores for non-binary data is valid, if the conditional distribution of the 
exposure, given the confounders, is specified uniquely by that scalar 
propensity score. They go on to state that this possibility includes 
deriving the propensity score from an ordinary homoskedastic linear 
regression model predicting the exposure from the confounders. By 
implication, this possibility also includes deriving the propensity score 
using any other correctly specified generalized linear model, with a 
correctly specified link function and variance function, predicting the 
exposure from the confounders. (However, it does not include the 
possibility of certain types of heteroskedastic regression, in which the 
conditional mean exposure is predicted from the counfounders using one 
scalar function, and the conditional variance of the exposure is predicted 
from the confounders using another scalar function. For instance, if IQ 
score is the exposure, and the confounders include gender, then it might 
not be appropriate to define a propensity score for IQ using gender, 
because gender affects the variance of IQ but not the mean.)

Thanks to Merete Jørgensen for the 2 extra references and to John Hsu and 
to Xavier de Luna for their interest in the problem.

Best wishes

Roger Newson

On Sunday 9 February 2003 I wrote to Allstat:

>Fellow Allstatters:
>
>
>A query about the method of propensity scores, introduced by Rosenbaum and 
>Rubin (1983) and used increasingly in the last few years as a method to 
>measure exposure effects on an outcome, adjusted for large confounder 
>sets. (Two recent review articles on the method are Rubin (1997) and Joffe 
>and Rosenbaum (1999).) So far, they have been used mainly for binary 
>exposures (and computed using logistic regression), and sometimes for 
>ordinal categorical exposures (and computed using ordinal logistic 
>regression - see Lu et al. (2001) for an example). Does anybody out there 
>know a reason why they should not be used with continuous exposures? In 
>the continuous-exposure case, the propensity score would presumably be 
>calculated by fitting a linear regression model, with the continuous 
>exposure as the Y-variable and the confounders as the X-variables, and 
>defining the propensity score to be the fitted value of the continuous 
>exposure arising from this linear regression model. Once the propensity 
>score has been calculated in this way, we could then proceed in the usual 
>way, defining equal-sized propensity categories (in the case of a cohort 
>study) or propensity-matched pairs (in the case of a nested matched-pairs 
>study). By my reckoning, the mathematical arguments justifying propensity 
>scores for binary exposures can be generalised fairly readily to justify 
>propensity scores for continuous exposures, and are subject to the same 
>cautions. However, my literature searches so far have failed to uncover 
>any examples of propensity scores for continuous exposures, although they 
>seem to be well-established for binary exposures. Does anybody out there 
>know of any objections to using them for continuous exposures? And does 
>anybody know of examples where they have been used for continuous exposures?
>
>
>Best wishes (and thanks in advance)
>
>
>Roger Newson
>
>
>References
>
>
>Joffe MM, Rosenbaum PR. Propensity scores. American Journal of 
>Epidemiology 1999; 150(4): 327-333.
>
>
>Lu B, Zanutto E, Hornik R, Rosenbaum PR. Matching with doses in an 
>observational study of a media campaign against drug abuse. Journal of the 
>American Statistical Association 2001; 96(456): 1245-1253.
>
>
>Rosenbaum PR, Rubin DB. The central role of the propensity score in 
>observational studies for causal effects. Biometrika 1983; 70(1): 41-55.
>
>
>Rubin DB. Estimating causal effects from large data sets using propensity 
>scores. Annals of Internal Medicine 1997; 127: 757-763.


--
Roger Newson
Lecturer in Medical Statistics
Department of Public Health Sciences
King's College London
5th Floor, Capital House
42 Weston Street
London SE1 3QD
United Kingdom

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