  Amid historically low response rates, survey researchers seek ways to reduce
respondent burden while measuring desired concepts with precision. We propose
to ask fewer questions of respondents and impute missing responses via
probabilistic matrix factorization. A variance-minimizing active learning
criterion chooses the most informative questions per respondent. In simulations
of our matrix sampling procedure on real-world surveys, as well as a Facebook
survey experiment, we find active question selection achieves efficiency gains
over baselines. The reduction in imputation error is heterogeneous across
questions, and depends on the latent concepts they capture. The imputation
procedure can benefit from incorporating respondent side information, modeling
responses as ordered logit rather than Gaussian, and accounting for order
effects. With our method, survey researchers obtain principled suggestions of
questions to retain and, if desired, can automate the design of shorter
instruments.
