  We address measurement error bias in propensity score (PS) analysis due to
covariates that are latent variables. In the setting where latent covariate $X$
is measured via multiple error-prone items $\mathbf{W}$, PS analysis using
several proxies for $X$ -- the $\mathbf{W}$ items themselves, a summary score
(mean/sum of the items), or the conventional factor score (cFS , i.e.,
predicted value of $X$ based on the measurement model) -- often results in
biased estimation of the causal effect, because balancing the proxy (between
exposure conditions) does not balance $X$. We propose an improved proxy: the
conditional mean of $X$ given the combination of $\mathbf{W}$, the observed
covariates $Z$, and exposure $A$, denoted $X_{WZA}$. The theoretical support,
which applies whether $X$ is latent or not (but is unobserved), is that
balancing $X_{WZA}$ (e.g., via weighting or matching) implies balancing the
mean of $X$. For a latent $X$, we estimate $X_{WZA}$ by the inclusive factor
score (iFS) -- predicted value of $X$ from a structural equation model that
captures the joint distribution of $(X,\mathbf{W},A)$ given $Z$. Simulation
shows that PS analysis using the iFS substantially improves balance on the
first five moments of $X$ and reduces bias in the estimated causal effect.
Hence, within the proxy variables approach, we recommend this proxy over
existing ones. We connect this proxy method to known results about
weighting/matching functions (Lockwood & McCaffrey, 2016; McCaffrey, Lockwood,
& Setodji, 2013). We illustrate the method in handling latent covariates when
estimating the effect of out-of-school suspension on risk of later police
arrests using Add Health data.
