Nonresponse adjustments with misspecified models in stratified designs 2. SettingNonresponse adjustments with misspecified models in stratified designs 2. Setting

• Previous
• Next

Survey weights compensate for different types of missing data $–$  sampling or base weights adjust for those that are not sampled, noncoverage adjustment weights account for those that are not in the sampling frame, and nonresponse adjustment weights compensate for those that are sampled but do not respond. We focus on nonresponse adjustment weights and the effect of using the base weights in creating the nonresponse adjustments.

We begin with the unadjusted Horvitz-Thompson estimator of the total

$${\widehat{y}}_{un}={\displaystyle {\sum }_s{R}_i}{d}_i{y}_i,(2.1)$$

where $d_i$ is the inverse of the probability of selection of unit $i,R_i=1$ if unit $i$ responds and $=0$ otherwise, and the sum is over the units in sample $s.$ The ratio mean is ${\widehat{\overline{y}}}_{un}={\widehat{y}}_{un}/{\displaystyle {\sum }_s{R}_i{d}_i}.$ If all the sample data are observed and the frame is complete, then $E({\widehat{y}}_{un})=Y,$ and the ratio mean is consistent for $\overline{Y}.$

When there is unit nonresponse, we assume that response is a random variable and the probability or propensity of response $\left({\varphi }_i=\mathrm{Pr}\left(R_i=1\right)\right)$ is like the probability from an additional phase of sampling (Särndal, Swensson and Wretman 1992). If we assume ${\varphi }_i>0$ for all $i,$ then the nonresponse bias of an estimated ratio mean under the stochastic model is

$$\text{bias}\left({\widehat{\overline{y}}}_{un}\right)\approx {\overline{\varphi }}^{-1}{\sigma }_{\varphi }{\sigma }_y{\rho }_{\varphi ,y},(2.2)$$

where $\overline{\varphi }$ is the population mean of the response propensities, ${\sigma }_{\varphi }$ is the standard deviation of $\varphi ,$ ${\sigma }_y$ is the standard deviation of $y,$ ${\rho }_{\varphi ,y}$ is the correlation between $\varphi$ and $y$ (Bethlehem 1988). The estimated respondent mean is unbiased if $\varphi$ and $y$ are uncorrelated. Brick and Jones (2008) extend these results to other types of statistics and estimators.

To reduce nonresponse bias, auxiliary variables associated with the sample can be used to support nonresponse adjustments to the base weights. The adjustments can be implemented by modeling either the distribution of $\varphi$ or $y,$ or both using the auxiliaries. We are specifically interested in modeling the response mechanism.

The estimated response propensities are applied as if they were the actual probabilities of responding. In other words, the nonresponse adjustment factor is the inverse of the estimated propensity of responding for sampled unit $i\left({\widehat{\varphi }}_i\right).$ The response propensity can be estimated by a variety of methods such as logistic regression, but most surveys form mutually exclusive groups called weighting classes or response homogeneity groups which contain units with similar estimated propensities and adjust the weights in each group or class by a common factor, say ${\widehat{f}}_c={\widehat{\varphi }}_c^{-1}$ for all $i\in c$ (Särndal et al. 1992, and Little 1986). When this approach is used, the adjusted estimator is called a weighting class estimator and is

$${\widehat{y}}_{wc}={\displaystyle {\sum }_c{\displaystyle {\sum }_{i\in s_c}{R}_{ci}}{d}_{ci}{\widehat{f}}_c{y}_{ci},(2.3)}$$

where $c=1,2,\dots ,C$ are the nonresponse adjustment classes and $i\in s_c$ is a sampled unit in class $c.$

The specific issue we address is the effect of weighting the adjustment factor. The unweighted factor is

$${\widehat{f}}_c^u=\frac{{\displaystyle {\sum }_{i\in s_c}{\delta }_{ci}}}{{\displaystyle {\sum }_{i\in s_c}{R}_{ci}{\delta }_{ci}}}=\frac{n_{c+}}{r_{c+}}$$

where ${\delta }_{ci}=1$ if $i\in c$ and ${\delta }_{ci}=0$ if $i\notin c,$ and $n_{c+}$ and $r_{c+}$ are the number of sampled and responding units in class $c.$ The weighted adjustment factor is

$${\widehat{f}}_c^w=\frac{{\displaystyle {\sum }_{i\in s_c}{d}_{ci}}}{{\displaystyle {\sum }_{i\in s_c}{R}_{ci}{d}_{ci}}}=\frac{{\widehat{N}}_c}{{\widehat{N}}_c^{\prime }},$$

where ${\widehat{N}}_c={\displaystyle {\sum }_{i\in s_c}{d}_{ci}}$ and ${\widehat{N}}_c^{\prime }={\displaystyle {\sum }_{i\in s_c}{R}_{ci}{d}_{ci}}.$ The factors correspond to the unweighted and weighted response rates, respectively. Substituting the factors into the estimator (2.3) yields two alternative estimators (2.4) and (2.5) of the total population. These are both weighting class estimators but we have changed notation to emphasize whether the weighted or unweighted response rate is used.

$${\widehat{y}}_{urr}={\displaystyle {\sum }_c{\widehat{f}}_c^u}{\displaystyle {\sum }_{i\in r_c}{d}_{ci}{y}_{ci}}={\displaystyle {\sum }_c\frac{n_{c+}}{r_{c+}}}{\displaystyle {\sum }_{i\in r_c}{d}_{ci}{y}_{ci}},(2.4)$$

$${\widehat{y}}_{wrr}={\displaystyle {\sum }_c{\widehat{f}}_c^w}{\displaystyle {\sum }_{i\in r_c}{d}_{ci}{y}_{ci}}={\displaystyle {\sum }_c\frac{{\widehat{N}}_c}{{\widehat{N}}_c^{\prime }}}{\displaystyle {\sum }_{i\in r_c}{d}_{ci}{y}_{ci}}.(2.5)$$

These two estimators are the building blocks for all the types of statistics that we consider in the simulation study. For example, estimators of means, domain means, and ratios are simple functions of estimators (2.4) and (2.5).

To be consistent with the structure, notation, and simulations in L&V, we restrict our study to the same population with a stratified simple random sample where two strata are defined by the binary design variable, $Z,$ and two nonresponse adjustment classes are defined by a binary auxiliary variable, $C,$ that cross the strata as shown in Table 2.1. We replaced the $X$ used in L&V with $C$ for weighting cell as introduced above to easily identify the nonresponse adjustment cell. Consistent with L&V, the population size is set at $N=\text{10,000}.$

The variable of interest, $Y,$ is a binary variable with the probability that $Y=1$ defined by a logistic model with $\mathrm{logit}\left(Y=1|C,Z\right)=0.5+{\gamma }_C\left(C-\overline{C}\right)+{\gamma }_Z\left(Z-\overline{Z}\right)+{\gamma }_{CZ}\left(C-\overline{C}\right)\left(Z-\overline{Z}\right).$ The response variable $R$ is also binary with the probability of $R=1$ generated from a logistic model with $\mathrm{logit}\left(R|C,Z\right)=0.5+{\beta }_C\left(C-\overline{C}\right)+{\beta }_Z\left(Z-\overline{Z}\right)+{\beta }_{CZ}\left(C-\overline{C}\right)\left(Z-\overline{Z}\right).$ Different populations and response propensities are generated depending on the values of ${\gamma }_C, {\gamma }_Z, {\gamma }_{CZ}, {\beta }_C, {\beta }_Z$ and ${\beta }_{CZ}$ as shown in Table 2.2. We have adopted the generalized linear model notation L&V used to make comparison to their work easier. The tabled values are the same populations and response variables that L&V generated by assigning values to $\left({\gamma }_C,{\gamma }_Z,{\gamma }_{CZ},{\beta }_C,{\beta }_Z,{\beta }_{CZ}\right).$ In the notation ${\left[A\right]}^B$ in Table 2.2, the population $\left(Y\right)$ or the response propensity $\left(R\right)$ are indicated by the superscript $B$ while the parameters and interactions of the model for the distribution of the population or response are indicated by $A$ inside the brackets. For example, the additive logistic model that generates the distribution of $Y$ within the sampling stratum $Z$ and nonresponse cell $C$ is indicated by ${\left[C+Z\right]}^Y.$ Similarly, models where $R$ depends on $C$ only, $Z$ only or neither $C$ nor $Z$ are denoted by ${\left[C\right]}^R,{\left[Z\right]}^R,$ and ${\left[C+Z\right]}^R$ respectively. L&V give more details on their rationale for choosing these populations and response models.

L&V computed estimates of means that are, in our notation,

$${\widehat{\overline{y}}}_{urr}=\frac{{\widehat{y}}_{urr}}{{\displaystyle {\sum }_c{\widehat{f}}_c^u}{\displaystyle {\sum }_{i\in s_c}{R}_{ci}}{d}_{ci}}=\frac{{\widehat{y}}_{urr}}{{\displaystyle {\sum }_c{\widehat{f}}_c^u{\widehat{N}}_c^{\prime }}},(2.6)$$

and

$${\widehat{\overline{y}}}_{wrr}=\frac{{\widehat{y}}_{wrr}}{{\displaystyle {\sum }_c{\widehat{f}}_c^w}{\displaystyle {\sum }_{i\in s_c}{R}_{ci}}{d}_{ci}}=\frac{{\widehat{y}}_{wrr}}{{\displaystyle {\sum }_c{\widehat{N}}_c}}.(2.7)$$

The denominators of the means are estimates of the population size $N.$ In estimator (2.7), the denominator is a constant and equal $N,$ but in estimator (2.6) the denominator is a random variable. In the simulation setting with the stratified simple random sample design described below, or in any design where ${\displaystyle {\sum }_{i\in s}{d}_i}=N$ for every $s,$ the estimator (2.7) reduces to the linear estimator ${\widehat{\overline{y}}}_{wrr}=N^{-1}{\widehat{y}}_{wrr};$ whereas (2.6) is a ratio estimator. This is an important point we return to later.

Domain means may have properties that differ from overall means because the denominators of the weighted and unweighted domain means are both random variables. One exception is when the domains match the sampling strata and therefore both the domain sizes and stratum sizes are known. L&V did not discuss domains, so these estimates are not studied in their simulation. We create domains by randomly generating a random variable ${\nu }_i$ from a uniform (0, 1) distribution, and defining the membership function $\tau \left(a\right)=1$ if $a<0$ and $\tau \left(a\right)=0$ if $a\ge 0.$ Domain means of 50% were created by substituting $d_{ci}^{*}=\tau \left({\nu }_i-0.5\right)d_{ci}$ into expressions (2.6) and (2.7) to produce the estimators ${\widehat{\overline{y}}}_{urr,0.5}$ and ${\widehat{\overline{y}}}_{wrr,0.5},$ respectively. Weighted and unweighted estimators of domain totals ${\widehat{y}}_{urr,0.5}$ and ${\widehat{y}}_{wrr,0.5}$ were formed similarly. We used the same device to create 25 percent domain means and 25 percent domain totals. Since we are interested in the effect of the nonresponse adjustments in means computed as ratio estimators, other domains such as those defined close to 100 percent of the population were excluded from the analysis because the denominator of the domain means approaches the constant population total $N$ and the mean becomes a linear estimator. Domains closer to 0 percent were excluded because of small sample sizes.

• Previous
• Next