1. Restriction. The analysis can be restricted to persons who are alike with regard to a confounding variable.⁵ For example, if sex is a confounder, the study could be limited to men.

2. Matching. Treated persons can be matched to controls on variables thought to be confounders. In a cohort study, confounding by the matching variables will be removed if the matching is sufficiently close, follow-up information is complete, and the matched sets have equal numbers of treated and control persons.^5-9 Matching can involve exact matches, nearest neighbor matches, or methods that use a measure constructed from several variables.^{6, 8-9}

The size of a treatment effect (such as a risk ratio or risk difference) may vary with levels of some participant characteristics; this variation is called effect modification, interaction, heterogeneity of effect, or variation between subgroups.⁵ If there are more controls than treated persons, matching each treated person to 1 control produces a treatment estimate for persons with the characteristics of the treated persons (sometimes called the average treatment effect among the treated).^{8, 10-11} If there is variation in treatment effect, the average effect among the treated may not be the same as the average effect in the entire study population. Likewise, if each control is matched to a treated person, the analysis will estimate the average effect of treatment among persons like the controls. If matching uses the entire cohort, the estimate applies to all cohort members.^{8, 10-11}

Matching treated persons to controls can be in ratios of 1:1, 1 to many, many to many, or many to 1. If matched sets do not have equal numbers of treated and control participants, the analysis should account for the matching to remove confounding by the matching variables,^{5, 7} for example, by using a t test for matched pairs, Mantel-Haenszel methods for matched pairs, conditional Poisson regression for matched sets, or a hazard ratio stratified on the matched sets.⁷

This long discussion of matching is not meant to suggest that matching is best, but is given because matching in cohort studies is often ignored in textbooks of epidemiology and biostatistics. Matching has received much attention in the literature about propensity scores,^{6, 9} but matching does not require the use of propensity scores or vice versa.

3. Stratified analysis. The data can be divided into strata within which treated and control participants are similar. Then treatment effects can be estimated within each stratum and results can be summarized across strata using Mantel-Haenszel weights or other weighting schemes.⁵ The distinction between stratification and matching is somewhat arbitrary; if we stratify participants into male and female groups, we have matched them with regard to sex.

4. Regression. Confounding can be addressed by adjustment in a regression model, with treatment group and confounding variables as the explanatory (independent) variables.^{5, 11}

These methods can be combined. For example, matched sets can be created and regression adjustment can be used to reduce confounding within each matched set.

PROPENSITY SCORES
Rozé et al¹ used regression to reduce confounding in their analysis. But instead of adjusting for a set of confounding variables, they used gestational age and a single variable called a propensity score, an estimate of the probability (risk or propensity) of treatment (exposure) for each study infant.^{9, 11-15} These have also been called exposure, exposure-balancing, or confounder scores.¹¹

The propensity score can be estimated using logistic regression with treatment as the outcome and independent variables selected from measured characteristics of study participants. Other estimation methods include probit regression,¹⁶ complementary log-log regression,¹⁷ and discriminant analysis.¹³

To remove confounding bias, variables thought to be confounders of the association between treatment and outcome should be used to create the propensity score, even if they are not strong predictors of treatment.^{11, 18} Important confounders can be identified using prior knowledge. If the outcomes are known, the study data can be used to find confounding variables.^{5, 19} Selection of variables for estimating the propensity score should not be based on statistical significance.¹¹ Continuous terms may need to be transformed using quadratic expressions or other polynomials.^{15, 20-22} Interaction terms should be considered.^{15, 21} There is less need for parsimony in variable selection for a propensity score, as we have no interest in the regression coefficients,¹⁵ compared with a regression model for an outcome.^23-24 Nevertheless, including many variables that have no association with the study outcome will lead to some loss of precision (loss of statistical power) when the score is used.^{11, 18}

For a randomized trial, we compare treated and control participants to check that randomization balanced characteristics such as age, sex, and blood pressure.²⁵ A propensity score is supposed to induce balance in characteristics of treated and control participants within propensity score strata; the score should be checked to see if this was achieved.¹⁵ For each variable used to generate the score, the association between exposure status (treated or not) and that variable can be examined after adjustment for the score.^{11, 13, 15, 26} For example, if age is in the score model, then the difference in mean age between the treated and control participants, adjusted for propensity score, could be estimated in linear regression. If the score works well, the adjusted age difference will be close to 0. Alternatively, participants can be stratified by propensity score values (quintiles are often used) and the analyst can check that, within each stratum, there is little difference in mean age by treatment group.

On average, participants who received treatment will have higher propensity score values compared with controls. We can compare treated and control propensity score distributions using density plots, box plots, histograms, or tabulations to see how well treated and control participants overlap in regard to the joint distribution of the variables used in the score.¹⁵ The score can be used to check this overlap, even if it is not used for the final analysis. If there are few treated participants with low scores or few controls with high scores, it is worth considering restriction of the study sample to a score range with greater overlap.²⁷ Otherwise the analysis may create estimates in regions of the data where nearly all the information comes from only treated or only control participants; this extrapolation may introduce bias.

Some authors report the area under the receiver operating characteristics curve to show how well their propensity score can distinguish treated persons from controls.¹⁸ The score is supposed to create balance between treated and control participants with regard to characteristics that may influence their outcome¹⁵; good discrimination between treated and control participants, as indicated by a larger area under the curve, is not the goal.¹¹ A large area under the curve may indicate an undesirable lack of overlap and poor performance for the score.¹⁸ In the study by Rozé et al,¹ area under the curve was 0.92, suggesting that overlap was less than optimal, with few untreated babies with higher scores. The article does not reveal how many treated infants had scores greater than any control or how many controls had scores less than any treated child.

A propensity score can be used to estimate treatment effects in several ways, including:

1. Restriction. The analysis can be restricted to participants who overlap sufficiently in their scores. This might be done even if the score is not otherwise used in the analysis.
   
2. Matching. Treated and control participants can be matched on values or categories of the propensity score. All the matching variations that I listed earlier can be considered. Matching on the score may be simpler than matching on a set of variables; several methods have been proposed.^{6, 8-9}
   
3. Stratified analysis. Participants are divided into score categories (quintiles are popular), estimates are created within each category, and these estimates are then summarized.^{6, 8-9}
   
4. Weighting. Each treated participant is given a weight equal to the inverse of their score. Each control is given a weight equal to the inverse of 1 minus their score: the inverse of their probability of being a control. The treatment effect can then be estimated using a method, such as regression, that can use these weights.^{8, 11, 28-30}
   
5. Regression. A regression model for the study outcome can use only the treatment and the score as explanatory variables. Flexible methods for the score variable should be considered, such as quadratic splines or fractional polynomials, as the association between the score and the outcome may not be linear.^{8, 11, 20}
   

Combinations of these methods can be used. For example, an analysis stratified on score could use regression adjustment within each stratum. Another variation is to use regression with the study outcome as the dependent variable, treatment as an explanatory variable, and to adjust both for the score and for important confounding variables; this method has been called doubly robust, as theory suggests that if either the propensity score adjustment or the confounding variable adjustment is correct, the final estimate of effect should be correct.^{8, 11} Rozé et al¹ used this method for part of their analysis, adjusting both for the score and other variables.

Propensity scores can be used in many ways to estimate treatment effects, but in most reports only 1 method need be presented. Redundant analyses are rarely of interest, as all methods should usually produce similar results. If there are different treatment estimates according to the method of analysis, the goal will usually be to understand why this occurred and either correct problems that produced differences or present the best analysis. Similarly, studies that model outcomes without the use of propensity scores should usually present 1 analysis, not several.

OTHER USES OF PROPENSITY SCORES
Propensity scores have been used for purposes other than to reduce confounding:

1. Missing data. Missing data can result in biased estimates; methods exist for reducing this bias.^31-34 If we have information for all participants at study entry but missing outcome data for some, we can create a propensity score for the probability that a participant will have a known outcome. This can be used, either through stratification or weighting, to create treatment estimates that are unbiased if outcome data are missing at random for participants with similar scores.^{33, 35-36}
   
2. Selection bias. If randomization in a clinical trial is subverted, persons selected for treatment may differ from the controls.³⁷ To check for selection bias, a reverse propensity score can be estimated for each participant in a randomization block.^{25, 38} If randomization is done in blocks of 2, the first participant has a reverse propensity score of 0.5. If that participant is assigned to treatment, the second person has a score of 0; if the first person is assigned to the control arm, the second has a score of 1. If allocation is truly random, there should be no association between the reverse score and the actual treatment assignment; this can be checked in the data.^{25, 38}
   
3. Missing or poorly measured confounding variables. Imagine we have data from a large cohort study and we estimate a propensity score to remove confounding, but this score is flawed because our data lack important variables or some variables are poorly measured. Assume there are data from a smaller group of similar participants with more variables and better measurements. Using this second group, we can create a better propensity score. Finally, assuming that the better score has all the predictive ability and more of the flawed score, we can use the better score to correct the flawed score in the large cohort, accounting for confounding information that was not actually measured for those participants.^{30, 39}
   

CAUTIONS
While propensity scores can often create balance in regard to measured factors, they cannot remove confounding bias due to unmeasured factors. Possible confounding from unmeasured factors is a limitation of all analytic methods, including propensity scores.¹⁵ Similarly, error in the measurement of variables used to create propensity scores can result in biased estimates, as in any analysis that relies on measured variables to remove confounding.^40-42

The propensity score is created by a model; errors in this model can produce an incorrect score that may result in bias. For example, failure to include important covariates, to use appropriate transformations for continuous terms, or to include interaction terms may result in bias. Inclusion of a variable that is intermediate in the pathway between the treatment and the outcome may bias the estimated association.⁸ Rozé et al¹ found an imbalance in days of mechanical ventilation (median of 27 days for treated newborns and 4 days for controls). If sedation or analgesia sometimes prolonged ventilator time, those extra ventilator days should not be treated as a confounder. Adjusting for the extra days or using them in a score may bias the estimate of treatment. The authors conducted an analysis in which ventilator days were omitted from the score and say that this produced similar results. Unfortunately, the estimated risk ratio was not reported.

CASE-CONTROL STUDIES
This discussion applies to cohort studies. Propensity scores have been used in a few case-control studies, but this involves special issues: (1) analysts must choose from among several score estimating methods; (2) some methods will produce estimates of treatment effect that erroneously vary by level of the propensity score; and (3) some residual confounding may remain.^{11, 43}

OUTCOME SCORES
Propensity scores were first described in 1983.¹² But the idea of using a single score to control confounding dates back at least to the 1970s; exposure scores, similar to what are now called propensity scores, and outcome scores were proposed.^{11, 44-45} An outcome score, also called a risk, disease, or prognostic score, is created by a regression model for the study outcome using all potential confounding variables and the study treatment. The score is estimated using the regression coefficients and each participant's covariate values, with the treatment variable set to 0.^{11, 45-46} Alternatively, the score is estimated using controls only.⁴⁷ The score is then used for a stratified analysis or as a continuous variable as if it were a propensity score.^{11, 45-47} This method has been used in several recent studies.^47-50

CONCLUSIONS
Propensity score methods are uncommon in the biomedical literature (about 32 studies before 2000 and only 71 in 2003).²⁷ To roughly assess current use, I searched PubMed on January 25, 2008, for English-language articles published in 2007 with the words propensity score; I found 237, compared with 27 036 articles for case-control study and 36 274 for cohort study.

Propensity scores are another tool that can be used to reduce confounding. They may be most useful when there are many potential confounders, many participants in both the treated and control groups, and few outcomes relative to the number of possible confounders.

But investigators need not jettison other methods. For many studies, regression modeling of the outcome will provide treatment estimates similar to those based on a propensity score that came from a regression model for the treatment.^{27, 51-52} Depending on the study topic, researchers may have more confidence in their ability to model the outcome, not the treatment (or other exposure), and may therefore prefer not to use a propensity score.¹¹ When the ratio of outcome events to confounding variables is 8 or more, a regression model for the outcome may offer more statistical power compared with propensity scores.⁵¹ In some situations, there may be an overriding interest in modeling several exposure variables simultaneously or assessing confounding effects in a regression outcome model. Researchers have many analytic options and the choice of method will vary with the study purpose, study design, and type of data.

AUTHOR INFORMATION
Correspondence: Dr Cummings, 250 Grandview Dr, Bishop, CA 93514 (peterc{at}u.washington.edu).

Financial Disclosure: None reported.