  We developed a single factor model with measure-specific sample weights for
multivariate data with multiple observed indicators clustered within a higher
level subject. The factor is therefore a latent variable shared by multiple
indicators within a same subject and the sample weights are different across
different indicators and different subjects. Even after integrating out the
latent variable, the likelihood of the data cannot be written as the sum of
weighted likelihood of each subject because a subject has different sample
weights respectively for its multiple indicators. In addition, the number of
available indicators varies across subjects. We derive a pseudo likelihood for
the latent variable model with measure-specific weights. We investigate various
statistical properties of the latent variable model with measure-specific
sample weights and its connection to the traditional factor analysis. We found
that the latent variable model provides consistent estimates for its variances
when the measure-specific sample weights are properly re-scaled. Two estimation
procedures are developed - EM algorithm for the pseudo likelihood and
marginalization of the pseudo likelihood by directly integrating out the latent
variable to obtain the parameter estimates. This approach is illustrated by the
analysis of publicly reported hospitals with indicators and sample weights.
Numerical studies are conducted to investigate the influence of weights and
their sample distribution.
