Tests for evaluating nonresponse bias in surveys Section 3. Propensity weightingTests for evaluating nonresponse bias in surveys Section 3. Propensity weighting

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An alternative to poststratification is to use inverse propensity weighting of the respondents (see, for example, Folsom 1991; Kim and Kim 2007).

In this framework, the true response propensity of unit $\left(hik\right)$ is $R_{hik}$ and a model is used to predict the propensity from characteristics known for everyone in the selected sample. Logistic regression is often used to estimate propensities. Suppose that the $p‑$vector $x_{hik}$ is known for each unit in $S.$ The modeled response propensity, if $x_{hik}$ and $R_{hik}$ were known for each unit in the population, is

$$R_{hik}^M\mathrm{<mo> =</mo>}{\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\beta \right)\right]}^{-1}\mathrm{,}$$

where $\beta$ is the solution to the expected population score equations

$${\displaystyle \sum _{hik\in U}}\left[R_{hik}-R_{hik}^M\right]x_{hik}\mathrm{<mo>=</mo>\; 0.}$$

The model removes the bias for the estimated population total of $y$ if

$$\theta \mathrm{<mo> =</mo>}{\displaystyle \sum _{hik\in U}}\left[R_{hik}\frac{y_{hik}}{R_{hik}^M}-y_{hik}\right](3.1)$$

equals 0. If $R_{hik}\mathrm{<mo>=</mo>}{R}_{hik}^M,$ that is, the response propensity model is correctly specified, then the weighting adjustments remove the bias for every possible response variable $y.$ The population parameter $\theta$ is estimated by

$$\widehat{\theta }\mathrm{<mo> =</mo>}{\displaystyle \sum _{hik\in S}}{w}_{hik}\left[r_{hik}{y}_{hik}\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\widehat{\beta }\right)\right]-y_{hik}\right]\mathrm{,}$$

where $\widehat{\beta }$ is the solution to the pseudolikelihood score equations

$${\displaystyle \sum _{hik\in S}}{w}_{hik}\left[r_{hik}-{\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\widehat{\beta }\right)\right]}^{-1}\right]x_{hik}\mathrm{<mo>=</mo>\; 0.}$$

Unlike the poststratification situation, the population parameter $\theta$ in (3.1) is not an explicit function of population totals. Similarly to Kim and Kim (2007), we can obtain the linearization variance and a linearization variance estimator of $\widehat{\theta }$ by using the estimating equation for $\left(\beta \mathrm{,}\mathrm{ \theta }\right),$ as derived in Binder (1983): $\left(\widehat{\beta },\widehat{\theta }\right)$ is the solution to

$$\widehat{A}\left(\beta \mathrm{,}\theta \mathrm{,}r\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{hik\in S}{w}_{hik}}u\left(y_{hik}\mathrm{,}{x}_{hik}\mathrm{,}{r}_{hik}\mathrm{,}\beta \right)-{\left[\mathrm{0,0,}\dots \mathrm{,0,}\theta \right]}^{\prime }\mathrm{<mo>=</mo>\; 0,}(3.2)$$

where

$$u\left(y_{hik}\mathrm{,}{x}_{hik}\mathrm{,}{r}_{hik}\mathrm{,}\beta \right)\mathrm{<mo>=</mo>}\left[\begin{array}{c}{u}_1\left(y_{hik}\mathrm{,}{x}_{hik}\mathrm{,}{r}_{hik}\mathrm{,}\beta \right)\\ u_2\left(y_{hik}\mathrm{,}{x}_{hik}\mathrm{,}{r}_{hik}\mathrm{,}\beta \right)\end{array}\right]\mathrm{<mo>=</mo>}\left[\begin{array}{c}\left[r_{hik}-{\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\beta \right)\right]}^{-1}\right]x_{hik}\\ r_{hik}{y}_{hik}\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\beta \right)\right]-y_{hik}\end{array}\right]\mathrm{.}$$

The population parameter $\theta$ solves the population estimating equation

$$A\left(\beta \mathrm{,}\theta \mathrm{,}R\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{hik\in U}u\left(y_{hik}\mathrm{,}{x}_{hik}\mathrm{,}{R}_{hik}\mathrm{,}\beta \right)}-{\left[\mathrm{0,0,}\dots \mathrm{,0,}\theta \right]}^{\prime }\mathrm{<mo>=</mo>\; 0.}$$

Theorem 4. Let $\widehat{U}\left(\beta \mathrm{,}\theta \right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{hik\in S}{w}_{hik}}u\left(y_{hik}\mathrm{,}{x}_{hik}\mathrm{,}{r}_{hik}\mathrm{,}\beta \right)\mathrm{<mo>=</mo>}{\left[{\widehat{U}}_1{\left(\beta \right)}^{\prime }\mathrm{,}{\widehat{U}}_2\left(\beta \right)\right]}^{\prime }.$  Suppose conditions (A2) $–$ (A5) are met and there exists a value $B$  such that $\left|x_{hik\mathrm{,}j}\right|\mathrm{<mo><</mo>}B$  for all units $\left(hik\right)$  and components $j.$  Then $V\left(\widehat{\theta }\right)\mathrm{<mo>=</mo>}{V}_L\left(\widehat{\theta }\right)+o\left(M^2/n\right),$  where

$$V_L\left(\widehat{\theta }\right)\mathrm{<mo>=</mo>}{T}^{\prime }QXCV\left[{\widehat{U}}_1\left(\beta \right)\right]C{X}^{\prime }QT-2T^{\prime }QXCCov\left[{\widehat{U}}_1\left(\beta \right)\mathrm{,}{\widehat{U}}_2\left(\beta \right)\right]+V\left[{\widehat{U}}_2\left(\beta \right)\right]\mathrm{,}(3.3)$$

$X$  is the $M\times p$  matrix with rows $x_{hik}^{{}^{\prime }},$   $T$  is the $M‑$vector with elements $R_{hik}{y}_{hik},$   $Q$  is the $M\times M$  diagonal matrix with entries $\exp \left(-x_{hik}^{{}^{\prime }}\beta \right),$  and $C\mathrm{<mo>=</mo>}{\left(X^{\prime }{\left[I+Q\right]}^{-2}QX\right)}^{-1}.$

A linearization variance estimator for $\widehat{\theta }$ may be obtained by substituting estimators for the population quantities in (3.3) to obtain

$$\begin{array}{ll}{\widehat{V}}_L\left(\widehat{\beta }\mathrm{,}\widehat{\theta }\right)\hfill & \mathrm{<mo>=</mo>}{t}_S^{{}^{\prime }}{W}_S{Q}_S{X}_S\widehat{C}\widehat{V}\left[{\widehat{U}}_1\left(\widehat{\beta }\right)\right]\widehat{C}{X}_S^{{}^{\prime }}{Q}_S{W}_S{t}_S\hfill \\ \hfill & -2t_S^{{}^{\prime }}{W}_S{Q}_S{X}_S\widehat{C}\widehat{\text{Cov}}\left[{\widehat{U}}_1\left(\widehat{\beta }\right)\mathrm{,}{\widehat{U}}_2\left(\widehat{\beta }\right)\right]+\widehat{V}\left[{\widehat{U}}_2\left(\widehat{\beta }\right)\right]\mathrm{,}\hfill \end{array}$$

where $X_S$ is the $m\times p$ matrix with rows $x_{hik}^{{}^{\prime }}$ for the sampled units with $m$ the size of the selected sample, $W_S$ is the $m\times m$ diagonal matrix of weights $w_{hik}$ for sampled units, $t_S$ is the $m‑$vector with elements $r_{hik}{y}_{hik}$ for sampled units, $Q_S$ is the $m\times m$ diagonal matrix with entries $\exp \left(-x_{hik}^{{}^{\prime }}\widehat{\beta }\right)$ for values of $x_{hik}$ in the sample, and $\widehat{C}\mathrm{<mo>=</mo>}{\left(X_S^{{}^{\prime }}{W}_S{\left[I+Q_S\right]}^{-2}{Q}_S{X}_S\right)}^{-1}.$

The jackknife variance estimator for inverse propensity weighting is defined using the formula in (2.8) with jackknife weights in (2.9). For the propensity setting,

$${\widehat{\theta }}^{\left(gj\right)}\mathrm{<mo>=</mo>}{\displaystyle \sum _{hik\in S}}{w}_{hik}^{\left(gj\right)}\left[r_{hik}{y}_{hik}\left[1+\exp \left(-x_{hik}^{{}^{\prime }}{\widehat{\beta }}^{\left(gj\right)}\right)\right]-y_{hik}\right]\mathrm{,}$$

where ${\widehat{\beta }}^{\left(gj\right)}$ solves

$${\displaystyle \sum _{hik\in S}}{w}_{hik}^{\left(gj\right)}\left[r_{hik}-{\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\beta \right)\right]}^{-1}\right]x_{hik}\mathrm{<mo>=</mo>\; 0.}$$

Theorem 5. Assume that the conditions of Theorem 4 hold. If $\left(n/M^2\right)V_L\left(\widehat{\theta }\right)$  converges to a positive constant, then $\left(n/M^2\right)\left[{\widehat{V}}_L\left(\widehat{\theta }\right)-V_L\left(\widehat{\theta }\right)\right]$  and $\left(n/M^2\right)\left[{\widehat{V}}_J\left(\widehat{\theta }\right)-V_L\left(\widehat{\theta }\right)\right]$  both converge in probability to 0.  

The proof of Theorem 5 follows by standard arguments in Fuller (2009) and Shao and Tu (1995) and is hence omitted.

The parameter $\theta$ for examining bias with inverse propensity weighting was defined for population totals. As with poststratification, it may be desired to compare means instead of totals, particularly if weight trimming is used to truncate large and influential values of the propensity weight $\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\widehat{\beta }\right)\right].$ In this case, the parameter to be evaluated is

$${\theta }_M\mathrm{<mo>=</mo>}\frac{{\displaystyle \sum _{hik\in U}}\left[R_{hik}{y}_{hik}/R_{hik}^M\right]}{{\displaystyle \sum _{hik\in U}}{R}_{hik}/R_{hik}^M}-\frac{{\displaystyle \sum _{hik\in U}{y}_{hik}}}{M}$$

with estimator

$${\widehat{\theta }}_M\mathrm{<mo>=</mo>}\frac{{\displaystyle \sum _{hik\in S}{r}_{hik}{w}_{hik}{y}_{hik}}\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\widehat{\beta }\right)\right]}{{\displaystyle \sum _{hik\in S}{r}_{hik}{w}_{hik}}\left[1+\exp \left(-x_{hik}^{{}^{\prime }}\widehat{\beta }\right)\right]}-\frac{{\displaystyle \sum _{hik\in S}{w}_{hik}{y}_{hik}}}{{\displaystyle \sum _{hik\in S}{w}_{hik}}}\mathrm{.}$$

Special adjustments are needed to account for weight trimming with the linearization variance estimator; in general, we recommend using the jackknife or another replication method for finding the variance of $\widehat{\theta }$ or  ${\widehat{\theta }}_M.$

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