  This paper discusses estimation and limited information goodness-of-fit test
statistics in factor models for binary data using pairwise likelihood
estimation and sampling weights. The paper extends the applicability of
pairwise likelihood estimation for factor models with binary data to
accommodate complex sampling designs. Additionally, it introduces two key
limited information test statistics: the Pearson chi-squared test and the Wald
test. To enhance computational efficiency, the paper introduces modifications
to both test statistics. The performance of the estimation and the proposed
test statistics under simple random sampling and unequal probability sampling
is evaluated using simulated data.
