Investigating health-related time use with partially observed data

Abstract

This paper suggests analytical strategies for obtaining informative parameter bounds when multivariate health-related time use data are partially observed in a particular yet common manner. One familiar context is where M>1 outcomes’ respective totals across N>1 time periods are observed but where questions of interest involve features—probabilities, moments, etc.—of their unobserved joint distribution at each of the N time periods. For instance, one might wish to understand the distribution of any type of unhealthy day experienced over a month but have access only to the separate monthly totals of physically unhealthy and mentally unhealthy days that are experienced. After demonstrating methods to partially identify such distributions and related parameters under several sampling assumptions, the paper proceeds to derive bounds on partial effects involving exogenous covariates. These results are applied in three empirical exercises. Whether the proposed bounds prove to be sufficiently tight to usefully inform decisionmakers can only be determined in context, although in this paper’s empirical analysis some of the estimated bounds turn out to be perhaps surprisingly tight. Moreover, it is suggested in the paper’s conclusion that the issues considered in this paper may become increasingly salient for analysts as data privacy policies increasingly constrain analyses.

Appendix A: General results on bounds

It proves useful for several of the derivations presented in the main text to describe some general results. For a set of generic parameters Θ = {θ_j} suppose lower and upper bounds, LB{θ_j} and UB{θ_j}, have been determined. Suppose h{θ_j} is a smooth, monotone function. Then:

$$\begin{array}{*{20}{c}} {LB\left( {h\left( {\theta _j} \right)} \right) = \left\{ {\begin{array}{*{20}{l}} {h\left( {LB\left( {\theta _j} \right)} \right),} \hfill & {h^\prime\, > \,0} \hfill \\ {h\left( {UB\left( {\theta _j} \right)} \right)} \hfill & {h^\prime\, < \,0} \hfill \end{array}} \right.} & {UB\left( {h\left( {\theta _j} \right)} \right) = \left\{ {\begin{array}{*{20}{l}} {h\left( {UB\left( {\theta _j} \right)} \right),} \hfill & {h^\prime\, > \,0} \hfill \\ {h\left( {LB\left( {\theta _j} \right)} \right)} \hfill & {h^\prime\, < \,0} \hfill \end{array}} \right.} \end{array}$$

(a)

$$\begin{array}{*{20}{c}} {LB\left( {\mathop {\sum}\limits_j {h_j\left( {\theta _j} \right)} } \right) = \mathop {\sum}\limits_j {LB\left( {h_j\left( {\theta _j} \right)} \right)} } & {UB\left( {\mathop {\sum}\limits_j {h_j\left( {\theta _j} \right)} } \right) = \mathop {\sum}\limits_j {UB\left( {h_j\left( {\theta _j} \right)} \right)} ,} \end{array}$$

(b)

where the summations in (b) run over some or all of the elements of Θ.

For example (and supposing θ_j > 0): $LB\left( {1/\theta _j} \right) = 1/UB\left( {\theta _j} \right)$; $UB\left( { - \theta _j} \right) = - LB\left( {\theta _j} \right)$; $UB\left( {\theta _j^2} \right) = \left( {UB\left( {\theta _j} \right)} \right)^2$; LB(θ_j−θ_k) = LB(θ_j)−UB(θ_k). Note that even when LB(h_j(θ_j)) and LB(h_j(θ_j)) are the tightest bounds on each h_j(θ_j) from (a) this does not imply that the bounds on their sums defined in (b) are the tightest possible bounds on Σ_jh_j(θ_j) even though in general the bounds in (b) will be valid.