1 The Vital Role of Response Instruments in MNAR Models

There are two approaches to estimating population values in the face of missing data (Chen, Li, and Wu Reference Chen, Li and Wu2020):

1. 1. Weighting is the most widely used method to deal with nonresponse. “Inverse propensity weighting” (IPW) weights observations by the inverse of the probability of being observed conditional on a set of covariates X. Let $R_i = 1$ if individual i responds to the survey. In a MAR context, $\hat {\pi } = Pr(R=1|X)$ . An IPW estimate of the population mean is

   $$ \begin{align*} \hat{\overline{Y}}_{IPW} = \sum_i \frac{R_i Y_i}{\hat{\pi}_i}. \end{align*} $$

2. 2. Imputation estimates the expected value of Y for individuals who did not respond (Vermeulen and Vansteelandt Reference Vermeulen and Vansteelandt2016). There are many possible ways to impute missing outcome data, ranging from OLS to multilevel regression with post-stratification to nearest neighbor imputation (Chen et al. Reference Chen, Li and Wu2020). Assuming that Y is dichotomous, we can specify that value of Y for an individual who did not respond as $\hat {Y}^0_i = Pr(Y_i=1|X_i, R_i=0)$ . In a MAR context, $Pr(Y_i=1|X_i, R_i=0) = Pr(Y_i=1|X_i, R_i=1)$ . An imputation estimate of the population mean is

   $$ \begin{align*} \hat{\overline{Y}}_{Imp} = \frac{1}{N}\sum_i [R_iY_i + (1-R_i)\hat{Y}^0_i]. \end{align*} $$

These approaches are straightforward if the data are MAR. Weights are generated via a model predicting response as a function of X; imputed values of Y for missing data are generated as the fitted values of Y based on observed data.

There are two ways that the MAR assumption may be violated. First, an unmeasured factor may affect both response propensity and Y even after controlling for observed covariates. For example, unmeasured social trust may predict response and disapproval of Donald Trump. If true, conventional weighting will produce a sample less supportive of Trump than the population, even after weighting on standard demographics (Clinton et al. Reference Clinton2021). Second, Y may directly affect response propensity. It could be that supporting Trump directly predicts nonresponse even after controlling for the weighting variables. This paper focuses on models in which Y directly affects R; a model with common unobserved factors in response and outcome can be subsumed into this model.

If survey data are MNAR, bias can be large. Meng (Reference Meng2018) derives a general equation that characterizes the sampling error for any sample, be it random or nonrandom:

(1) $$ \begin{align} \overline{Y}_n-\overline{Y}_N =\underbrace{\rho_{R,Y}}_{\text{ data defect correlation }} \times \underbrace{\sqrt{\frac{N-n}{n}}}_{\text{ data quantity }} \times \underbrace{\sigma_Y}_{\text{ data difficulty }}. \end{align} $$

Equation (1) shows that sampling error is a function of the “data defect correlation” which is the correlation of R and Y, a data quantity term and a data difficulty term.Footnote ¹ The data defect correlation characterizes the degree of non-ignorability in the sample. If it is zero, then there is no error. (Since the correlation is unlikely to be literally zero even for a random sample, there will still be sampling error in random samples.) The key insight from Meng’s equation is that the non-ignorability term interacts with a data quantity term that is a function of population size. This comes as a bit of a shock for survey researchers steeped in random sampling theory, but for nonrandom sampling even a small amount of non-ignorability (via the $\rho _{R, Y}$ term) can induce large sampling errors if the population is large. We have known that large samples can go awry since the Literary Digest polling fiasco of 1936 (Lusinchi Reference Lusinchi2012).

Recent advances have expanded our toolkit for dealing with non-ignorable nonresponse. Hartman and Huang (Reference Hartman and Huang2023) use sensitivity analysis to assess when non-ignorable nonresponse is a threat. Isakov and Kuriwaki (Reference Isakov and Kuriwaki2020) and Bradley et al. (Reference Bradley, Kuriwaki, Isakov, Sejdinovic, Meng and Flaxman2021) combine the Meng equation with known benchmarks to present evidence that even seemingly small levels of non-ignorability can lead to large biases. Such sensitivity-based methods do not provide estimates of the influence of non-ignorability based on the data and models, however.

In this paper, I model nonresponse by allowing the response variable to directly affect the probability an individual responds. Suppose, for example, that

(2) $$ \begin{align} Pr(R=1|X, Y) = g(\gamma_0 + \gamma_{X}X + \gamma_YY). \end{align} $$

The data generating process is MNAR if $\gamma _Y \ne 0$ . We cannot use standard methods of estimating the probability of response because such a model needs observations for which R = 1 and R = 0 and we will have missing data for Y for all individuals with R = 0. We also cannot implement MAR-type imputation because $Pr(Y_i=1|X_i, R_i=0) \neq Pr(Y_i=1|X_i, R_i=1)$ when Y and R are jointly determined.

While some believe that MNAR data are inherently beyond the scope of modeling, there is a vast literature on how to estimate such models. One theme is that these models should incorporate a response instrument. A response instrument

• • influences response, conditional on covariates and

• • is not directly related to the outcome of interest in the population, conditional on covariates.

While it is not strictly necessary to have a response instrument (because, e.g., some parametric models can be estimated without one), the field recognizes the poor performance of MNAR models when there is no response instrument (see, e.g., Peress Reference Peress2010).

While a response instrument can in theory be observational, a response instrument is ideally based on a randomization treatment that affects response propensity and does not directly affect Y. Such an instrument can be created by randomly assigning individuals to a treatment protocol that increases the response probability. For example, New York Times/Sienna pollsters randomly assigned adults in Wisconsin in 2022 to a treatment in which they received $30 for responding (Cohn Reference Cohn2022). Randomization makes it easier to accept the condition that the instrument has no direct effect on Y; randomization also increases statistical power by reducing dependence of the instrument on covariates.

Sun et al. (Reference Sun, Liu, Miao, Wirth, Robins and Tchetgen-Tchetgen2018) explain how to use an instrument to estimate MNAR models. They formulate their models in terms of the following selection bias function that “quantifies the degree of association between Y and R given (X, Z) on the log odds scale”:

(3) $$ \begin{align} \eta(x, y, z) = log [\frac{Pr(R=1|x, y, z)}{Pr(R=0|x, y, z)}/ \frac{Pr(R=1|x, Y=0, z)}{Pr(R=0|x, Y=0, z)}]. \end{align} $$

The Supplementary Material shows that if we use a logit function for the response equation (and impose the condition described momentarily), then $\eta $ is simply the coefficient on Y in the response equation (Equation (2)). I use a logit model in what follows as doing so simplifies presentation and implementation; the model can be conceptualized and implemented with other specifications.

Sun et al. (Reference Sun, Liu, Miao, Wirth, Robins and Tchetgen-Tchetgen2018, 6) derive the conditions under which MNAR models are identified when one has a correct model for the selection bias and one assumes (as in MAR models) that every $(X, Y, Z)$ has a nonzero probability of responding.Footnote ² The identification conditions can be summarized with reference to a sufficient condition: the MNAR models discussed below are identified as long as Y and Z do not interact in the response equation.

To illustrate the identification assumption of an MNAR model as compared to a MAR model, consider the following model in which response depends on a response instrument Z, covariates X, the outcome Y, and all linear interactions.

$$ \begin{align*} Pr(R=1|Z, X, Y) = g(\gamma_0 + \gamma_{Z}Z + \gamma_{x}X + \gamma_YY + \gamma_{ZX}ZX + \gamma_{XY}XY + \gamma_{ZY}ZY). \end{align*} $$

• • Assumptions in MAR-model: $\gamma _Y = \gamma _{XY} = \gamma _{ZY}= 0$ .

• • Assumptions in MNAR-model: $\gamma _{ZY}= 0$ .

In a MAR model, Y does not affect response, whether or not this effect interacts with an X variable that also affects Y or an instrument that only affects R. In other words, MAR models assume that conditional on covariates, the people who respond do not have systematically different values of Y than the people who do not respond.

MNAR models allow Y to affect response. The effect of Y can vary depending on X; the effect of Z can also depend on X. In the data analysis below, I estimate MNAR models by subgroup, which allows us to measure the effects of Y and Z on response for selected values of X.

The MNAR models do have an important assumption, however: that $\gamma _{ZY}= 0$ . This assumption is violated if the treatment interacts with the Y in affecting response propensity as would happen, for example, in a survey about support for President Biden if the treatment had a different effect on those who approved of Biden. In the extreme, a response instrument that pulled in only Biden supporters could be taken by an MNAR model as indicating that the survey underestimated Biden support. This could lead to a biased estimate—and potentially a worse estimate than a MAR-based model.

A researcher using an MNAR model therefore needs to pay considerable attention to the design and defense of the instrument. In a political context, the instrument should be depoliticized and reflect the kinds of steps pollsters generically use to affect response propensity.

Suppose there are two survey protocols A and B, with B having a higher response rate. An MNAR model will perform well if the underlying response propensity function is the same for both protocols, with the only difference being the threshold above which people respond is different under treatment. If protocol B induces a differential response that depends on Y, then this would be problematic for MNAR models. Note, however, that a survey firm using only protocol B would have non-ignorable nonresponse and would also have bias. In short, for MAR models, Y cannot affect R; for MNAR models, Y can affect R but needs to do so in the same way under control and treatment.Footnote ³

MNAR models cannot be taken as comprehensive tests of non-ignorability. Finding a significant effect of Y on R in an MNAR model could be due to actual non-ignorability or due to an instrument that violates the identifying assumption of the model. MNAR models should instead be treated as an important part of the diagnostic toolkit. If one finds MNAR models diverging from MAR models—as we do below in turnout and other models, for example—it seems prudent to be transparent about assumptions underlying each set of estimates. If one believes the response in the overall survey is ignorable, the weighted estimates are best; if one believes that the effect of the treatment does not depend on Y, the MNAR estimates are best.

There are many ways to estimate MNAR models. In this paper, I focus on weighting and imputation. Other options include maximum likelihood (Heckman Reference Heckman1979; Tchetgen and Wirth Reference Tchetgen and Wirth2017) and copula methods (Marra et al. Reference Marra, Radice, Barnighausen, Wood and McGovern2017). Bailey (Reference Bailey2024) argues that all these methods benefit from randomized response instruments.

2 MNAR Weighting

MAR-based weighting uses weights based on results from a first-stage model that predicts whether or not someone responds. If Y directly affects response, however, a conventional MAR-based model would require information that is unobserved for non-respondents.

MNAR-based weighting models proceed differently: they leverage the orthogonality condition implied by randomized response instruments to generate weights in which Y does affect response.

Consider the following MNAR model for response:

(4) $$ \begin{align} \pi = Pr(R=1|Z, X, Y) = logit(\gamma_0 + \gamma_ZZ + \gamma_XX+ \gamma_YY), \end{align} $$

where Z is a randomized response instrument and X is a vector of covariates that affect response and outcome.

There are two key parameters in this model. The coefficient on Z ( $\gamma _Z$ ) reflects the effect of the response instrument on response. This parameter needs to be nonzero for the MNAR models discussed here to be identified. The coefficient on Y in the response equation ( $\gamma _Y$ ) reflects the effect of Y on response. If this parameter is nonzero, then Y directly affects response, which indicates a non-ignorable/MNAR data generating process. If $\gamma _Y=0$ , conventional weighting is better on efficiency grounds as I demonstrate in the Supplementary Material (see also Bailey Reference Bailey2024).

The theory of Sun et al. (Reference Sun, Liu, Miao, Wirth, Robins and Tchetgen-Tchetgen2018) allows for semi- and non-parametric estimation of these values. In this paper, I use a logit link function that is of the form above. In the simulations, there are no interactions; in the application, I allow for Y and Z to interact with X by looking at subgroups based on X.

An MNAR-IPW estimator is in the form

(5) $$ \begin{align} \hat{\overline{Y}} &= \sum_i \frac{R_iY_i}{\hat{\pi}} \nonumber\\ &= \sum_i \frac{R_iY_i}{logit(\gamma_0 + \gamma_ZZ_i + \gamma_XX_i+ \gamma_YY_i)}. \end{align} $$

As noted above, we cannot use the conventional weighting approach of estimating the response equation first because Y is missing for all $R=0$ observations. We therefore estimate the model with estimating equations. These are similar to methods of moment estimators in which “population parameters be estimated by sample statistics which have the same property in the sample as the parameters do in the population” (Cameron and Trivedi Reference Cameron and Trivedi2005, 135).

The estimating equations impose the following properties on the model: the weights sum to the sample size, the IPW weighted sums of the Z and X covariates are the same as the sums in the population and, critically, that Z conditional on X is orthogonal to the weighted Y which is the estimated Y in the full population. Given Equation (4), the estimating equations are

$$ \begin{align*} h[1]&:\ \ \sum_i \frac{R_i}{\hat{\pi}_i} =N\nonumber, \\ h[2]&:\ \ \sum_i \frac{R_i}{\hat{\pi}_i}Z_i =\sum Z_i\nonumber, \\ h[3]&:\ \ \sum_i \frac{R_i}{\hat{\pi}_i}X_i = \sum X_i\nonumber, \\ h[4]&:\ \ \sum_i \frac{R_iY_i}{\hat{\pi}_i}(Z_i- \hat{Z}_i)\nonumber =0, \end{align*} $$

where $\hat {Z}_i$ is the expected value of Z given X, which for randomized Z should be close to the proportion of people assigned to the treatment condition.

The estimation process finds the $\gamma $ values that satisfy the estimating equation conditions. The third equation (h[3]) corresponds to a sample weighting condition as the weighted distribution of X will reflect the (known) population distribution of X. The fourth equation imposes the orthogonality of weighted Y and Z minus $\hat {Z}$ . Standard errors are calculated based on a sandwich estimator as described in the Supplementary Material of Sun et al. (Reference Sun, Liu, Miao, Wirth, Robins and Tchetgen-Tchetgen2018).

Digging into the estimation helps us appreciate how estimating equations produce a coefficient on Y in the response equation even though Y is missing for $R=0$ individuals. Suppose that $\gamma _Y>0$ . This will produce more $R=1$ observations among the $Y=1,\ Z=1$ group than the $Y=0,\ Z=1$ group. In the unweighted data, Y and Z will be correlated. The orthogonality condition in the estimating equation requires that Y and Z are conditionally uncorrelated in the full sample, which is achieved with a $\hat {\gamma }_Y>0$ which produces a smaller weight on the $Y=1,\ Z=1$ observations compared to the $Y=0,\ Z=1$ observations. If $\gamma _Y$ were zero, we would not systematically observe more $R=1$ observations among the $Y=1,\ Z=1$ group than the $Y=0,\ Z=1$ group and Y and Z can be orthogonal in the population without down-weighting the $Y=1$ observations. Section 4 of the Supplementary Material provides more detail on the intuition behind estimating equations in this context by providing a numerical example.

3 MNAR Imputation

MNAR imputation models require a model that predicts $Y|X$ for $R=0$ individuals. In MAR models, $Y|X$ is the same by assumption among respondents and non-respondents. This is not the case for MNAR models. Hence, we need to formulate a model of $Y|X$ for unobserved individuals as a function of the $Y|X$ for the observed individuals. Tchetgen-Tchetgen, Robins, and Rotnitzky (Reference Tchetgen-Tchetgen, Robins and Rotnitzky2010) formulate the joint density of $(y, r)$ as a function of the selection bias function (see also Burger and McLaren Reference Burger and McLaren2017; Little and Rubin Reference Little and Rubin2020, 353):

$$ \begin{align*} f(y, r) \propto exp[(r-1)\eta]f(y|R=1, x, z)Pr(R=1|Y=0, x, z). \nonumber \end{align*} $$

The marginal distribution of y is

$$ \begin{align*} f(y|r, x, z) =& \frac{P(y, r|z, x)} {\int P(y, r|z, x) d\mu(y)}\nonumber\\ =& \frac{exp[(r-1)\eta] f(y|R=1, x, z)}{E[exp[(r-1)\eta]| R=1, x, z]}, \nonumber \end{align*} $$

which implies that the distribution of y when R = 0 can be written as a function of $y|R=1$ :

(6) $$ \begin{align} f(y|R=0, x, z) = \frac{exp[-\eta] f(y|R=1, x, z)}{E[exp[-\eta]| R=1, x, z)]}. \end{align} $$

To estimate $\gamma _Y$ for the MNAR imputation model, first estimate $f(y|R=1, x, z)$ . Note that the distribution of Y given that $R=1$ is a function of X and Z. Even though Z is assumed not to affect Y in the full population, it can affect the expected value of Y in the observed sample via the effects of non-random selection. This equation can be, for example, a saturated model such as $Y = \lambda _0 + \lambda _ZZ + \lambda _XX + \lambda _{ZX}ZX$ for a simple model with only Z and a single X. With this model in hand, one can calculate $\hat {Y}^0_i$ , the expected value of Y for $R=0$ observations given the density in Equation (6), which is a function of the selection bias function ( $\eta $ ) and the distribution of $Y|X, Z, R=1$ .

The expected value of $Y_i$ for each i is $R_iY_i + (1-R_i)\hat {Y}^0_i$ . To satisfy the estimating equation condition that Y and Z are independent in the full population, we find the value of $\gamma _Y$ that solves

(7) $$ \begin{align} \sum_i (Z_i - \hat{Z}_i)\{ R_iY_i + (1-R_i)\hat{Y}^0_i \}=0. \end{align} $$

The MNAR imputation estimate uses the estimates of $\eta $ and $\gamma _Y$ from the above process to estimate a population average

(8) $$ \begin{align} \hat{\overline{Y}} = \frac{1}{N}\sum_i [R_iY_i + (1-R_i)\hat{Y}^0_i]. \end{align} $$

4 Doubly Robust Estimation in MNAR Context

A doubly robust estimator is consistent if one or both of the weighting and imputation models is correct (Scharfstein, Rotnitzky, and Robins Reference Scharfstein, Rotnitzky and Robins1999; Vermeulen and Vansteelandt Reference Vermeulen and Vansteelandt2016).

A doubly robust estimator for the population average is

(9) $$ \begin{align} \hat{\overline{Y}} & = \frac{1}{N}\sum [\frac{R_i Y_i}{\hat{\pi}_i} - \frac{R_i - \hat{\pi}_i} {\hat{\pi}_i}\hat{Y}^0_i] \end{align} $$

(10) $$ \begin{align}\ \ \ \ \, & = \frac{1}{N}\sum [ \frac{R_i}{\hat{\pi}_i}(Y_i - \hat{Y}^0_i) + \hat{Y}^0_i]. \end{align} $$

If the propensity model is correct, Equation (9) converges to the (correctly) weighted observed data because the average of $R_i- \hat {\pi }$ converges to zero for a correctly specified propensity model. Equation (10) is mathematically identical. In a MAR context, if the imputation is correct, the observed difference of Y and $\hat {Y}^0_i$ will converge to zero and the doubly robust estimate will converge to the correctly specified (by assumption) value of Y. In an MNAR context, this convergence depends on knowing the $\eta $ function (Equation (3)); see the proof of Proposition 3 in the Supplementary Material of Sun et al. (Reference Sun, Liu, Miao, Wirth, Robins and Tchetgen-Tchetgen2018).

Figure 1 illustrates how doubly robust estimation works. These simulations highlight (1) the well-known result that MAR models (even doubly robust ones) are biased when nonresponse is non-ignorable and (2) that even when one is very wrong with one or the other propensity or imputation models, the doubly robust MNAR estimator can produce unbiased results. With better propensity or imputation models, the differences between those models and the doubly robust model would be less stark. Suppose that the true models are

$$ \begin{align*} Pr(R=1|X, Y, Z) &= logit(\gamma_0 + \gamma_{Z}Z + \gamma_{X1}X1 + \gamma_{X2}X2+ \gamma_{Y}Y),\\ Pr(Y=1|X, Y, Z) &= logit(\beta_0 + \beta_{X1}X1 + \beta_{X2}X2), \end{align*} $$

where $\gamma _Z>0$ and $\gamma _Y>0$ . I estimate the true specifications and also two misspecified models

$$ \begin{align*} Pr(R=1|X, Y, Z) &= logit(\gamma_0),\\ Pr(Y=1|X, Y, Z) &= logit(\beta_0). \end{align*} $$

The misspecified response equation assumes everyone (whatever their X values) has the same probability of responding. The misspecified outcome equation assumes everyone (whatever their X values) has the same probability of having $Y=1$ . The point is not that these are good models; rather, the point is that the doubly robust approach can overcome weaknesses in one (but not necessarily both) of the propensity and imputation models.

The upper left of Figure 1 shows the observed mean in the data and the population estimates for the IPW, imputation and doubly robust estimators assuming MAR and MNAR, respectively, given that the correct specifications are used. The bars indicate the 5th and 95th percentile results from 200 simulations. The MAR models do poorly because they do not capture the effect of Y on R. All the MNAR models do well, which is unsurprising because the weighting and imputation models are correctly specified.

Figure 1 Simulation results for MNAR data.

The upper right of Figure 1 shows estimates when the response equation is misspecified. The MAR models do poorly. The IPW weighted model here is not an MNAR model because it does not include a $\gamma _Y$ term; it too does poorly. The MNAR imputation model is fine and—importantly—the MNAR doubly robust model overcomes the poor properties of the misspecified propensity model.

The lower left panel shows what happens when the response model is correct and the imputation model is incorrect. The MAR models do poorly. MNAR weights produce good estimates, while the MNAR imputation model does poorly; critically, the MNAR doubly robust overcomes the weakness of the imputation model.

The lower right panel shows that the doubly robust estimator can be off if both specifications are incorrect. This is an illustrative example; it is possible to create simulations in which one or both of the MNAR IPW and MNAR imputation methods are less poorly specified and the MNAR doubly robust estimates do better than in Figure 1. In Section 3 of the Supplementary Material, I show simulations in which the MNAR estimators become less precise when the instrument has a weaker effect on response.

5 Survey Example

This section analyzes a March 2019 Ipsos Knowledge Panel survey that included a randomized treatment that discouraged response. This approach is only one of many possible instruments (see, e.g., Cohn Reference Cohn2022). Generally, what is needed is for a survey firm to implement two protocols: one standard protocol and one that increases or decreases response rates.

Figure 2 shows the structure of the data. To be concrete, the figure shows sample sizes for the turnout question; there are slight differences in nonresponse patterns for other dependent variables. The original pool of $3,573$ people is drawn from Ipsos’s online panel that was recruited using an address-based probability sample. These individuals are randomly assigned to control and treatment. For the contacted people who logged in, people in the control condition were asked political questions in a standard way. Of the 1,805 people assigned to control, 693 did not respond to Ipsos at all, 4 responded to Ipsos but did not answer the turnout question, and 1,108 responded to the turnout question. The response rate for those randomly assigned to control was 61%.

Figure 2 Survey design for turnout analysis.

People in the treatment group who logged in were given a chance to opt out of the survey by asking them if they would like to discuss politics, sports, movies, or health. Of the 1,768 people assigned to treatment, 662 did not respond to Ipsos at all (and therefore presumably did not see the treatment), 711 selected a topic other than politics, 393 chose politics and answered the turnout question, and 2 people chose politics but did not answer the turnout question. Those who chose politics are respondents, and the others are non-respondents. The response rate for those assigned treatment was 22%, indicating that the treatment reduced response rates by 39 percentage points. The resulting data have 1,501 respondents and 2,072 non-respondents. Bailey (Reference Bailey2024) analyzes these data with other MNAR protocols, all producing the same general conclusions as described below; I leave modeling differences across the types of nonresponse for future work. Table A16 in the Supplementary Material provides descriptive statistics for the survey.

Figure 3 Analysis of turnout question.

5.2 Trump Approval

Polling related to Donald Trump has challenged pollsters since 2016. In 2016, the national polls were roughly correct but polls in key states, especially in the Midwest, tended to understate support for Trump. In 2020, all polls tended to understate Trump support (Clinton et al. Reference Clinton2021).

This survey asked respondents whether they approved of how Trump was handling his job as president, with response categories ranging from strongly disapprove to strongly approve plus a not sure category. I coded strongly and somewhat approve answers as approving Trump and coded not sure answers as missing.

The left panel of Figure 4 displays the estimated MNAR doubly robust $\gamma _Y$ coefficient for Trump approval for two estimates. The lines indicate 95% confidence intervals. The right panel of the figure shows three different estimates of average support for Trump. The lightest shade for each population group displays the observed approval and the middle bar for each group displays the estimated approval for the group as estimated by a conventional MAR-type weighting. The darker bar for each group displays Trump approval as estimated by the doubly robust MNAR model. Full results are in Supplementary Tables A3 and A4.

Figure 4 Analysis of Trump approval.

Figure 5 Analysis of Trump approval by party.

The left panel of Figure 4 shows that $\gamma _Y$ is close to zero when we use all observations, suggesting that there was no non-ignorable nonresponse. The weighting and doubly robust population estimates in the right panel are therefore similar to each other.

As noted earlier, the nature of non-ignorability may vary by subgroups. A direct way to assess such heterogeneity is to subset the data. Figure 4 also displays results for white people in the Midwest.Footnote ⁴ Many assessments of the 2016 and 2020 polling errors noted that polling in the Midwest seemed particularly difficult with Trump on the ballot. In Wisconsin in 2020, for example, Biden only beat Trump by 0.6 % even though most polls suggested Biden would win easily. One late October Washington Post poll even showed Biden up 17% after weighting.

The estimate of $\gamma _Y$ for the Midwest is $-$ 1.34 with a confidence interval that does not include zero. A negative $\gamma _Y$ coefficient is consistent with non-ignorable nonresponse: people who disliked Trump were more eager to respond and therefore the observed sample was skewed against Trump, even when weighted with conventional weights.

The effect of non-ignorable nonresponse was substantial. The right panel of Figure 4 shows that for white Midwesterners, Trump support was 45% in the observed sample and 43% in the conventionally MAR-weighted sample. In the doubly robust MNAR estimate, Trump support among white Midwesterners was 58%. In other words, the analysis suggests that Trump approval among white Midwesterners was more than 10 percentage points higher when accounting for non-ignorable nonresponse. Differences of such magnitudes were not evident in other regions.

The effects of non-ignorable nonresponse went in different directions for the two parties. Figure 5 shows results for white Democrats and Republicans. The top line in the panel on the left shows that among Democrats there was little evidence of non-ignorable nonresponse with respect to Trump approval. Only 7% of Democrats in the raw data approved of Trump, however, meaning that the statistical power was quite low (as is typical for low probability outcomes). I also present a model labeled “not strong approve” which is 0 for Democrats who strongly disapprove of Trump and 1 for all other Democrats who answered the question (which includes those who disapproved, but not strongly). For this model, the $\gamma _Y$ is negative (with a confidence interval that does not include zero) and the estimated proportions are different. In the observed data (weighted or not), 13% of Democrats did not strongly disapprove of Trump, while this number rose 35% in the doubly robust MNAR estimate.

The opposite pattern occurred for Republicans. The estimate of $\gamma _Y$ was not statistically significant when distinguishing between approval and disapproval. However, the estimate of $\gamma _Y$ was positive and significant in a model in which Republicans who strongly approved of Trump were coded as 1 and all other Republicans who answered the question were coded as 0 (which includes those who approved, but not strongly in addition to those who disapproved). For strong approval, the observed and weighted estimates indicated that about 49% of Republicans strongly approved of Trump, while the MNAR doubly robust estimate was that 34% of Republicans strongly approved of Trump. Full results are available in Supplementary Tables A5–A8.

Figure 6 Analysis of tax cuts question.

This analysis prompts the question as to how best to identify subgroups. In principle, one could break the sample into cells (as in cell weighting) and estimate separate models for each cell, after which one could patch together a population estimate based on population proportions across the cells. There would likely be statistical power issues, however. My approach was to assess substantively important subgroups with a two-step process. In an earlier survey, I explored subgroups based on party and region in light of the evaluations of recent polling lapses. I found variation by region and party. The second step was to assess these subgroups in the subsequent survey reported here.

In general, the IPW, imputation and doubly robust estimates were similar. For Midwesterners, for example, the $\gamma _Y$ estimates for the IPW, imputation and doubly robust estimators were $-$ 1.46, $-$ 1.41, and $-$ 1.34, respectively.

5.4 Racial Conservatism

On questions related to race, some people may avoid expressing conservative views that they consider to be socially undesirable (Berinksy Reference Berinksy1999). The survey asked a battery of race-related questions and here we illustrate how the models described perform on race using a question asking whether the respondents agreed with the statement that “Black athletes should not take a knee during the national anthem.” Support for the conservative position is coded as 1 and support for the liberal position is coded as 0.

Figure 7 shows there is little sign of non-ignorable nonresponse and the full population. For Democrats, however, there is strong evidence of non-ignorable nonresponse. The estimated means for Democrats differ substantially: conventional weighting suggests 20% of Democrats opposed kneeling during the national anthem while doubly robust estimation suggests 36% of Democrats supported the racially conservative position. For Republicans, conventional weighting suggests 79% of Republicans supported the racially conservative position while doubly robust estimation suggests 71% of Republicans supported the racially conservative position. Full results are available in Supplementary Tables A13–A15.

Figure 7 Analysis of race question (high values are more conservative).