2.1 Multivariate Regression Models: A Review

Suppose that we have a simple random sample of $N$ respondents from a population. In standard list experiments, we have a total of $J$ binary control questions and one binary sensitive question. Let $T_{i}$ be the randomized treatment assignment indicator. That is, $T_{i}=1$ indicates that respondent $i$ is assigned to the treatment group and is asked to report the total number of affirmative responses to the $J+1$ items ( $J$ control items plus one sensitive item). In contrast, $T_{i}=0$ implies that the respondent is assigned to the control group and is asked to report the total number of affirmative answers to $J$ control questions. We use $X_{i}$ to represent the set of $K$ pretreatment covariates (including an intercept).

Let $Y_{i}$ denote the observed response. If respondent $i$ belongs to the treatment group, this variable can take any nonnegative integer less than or equal to $J+1$ , i.e., $Y_{i}\in \{0,1,\ldots ,J+1\}$ . On the other hand, if the respondent is assigned to the control group, the maximal value is $J$ , i.e., $Y_{i}\in \{0,1,\ldots ,J\}$ . Furthermore, let $Z_{i}$ represent the latent binary variable indicating the affirmative answer to the sensitive question. If we use $Y_{i}^{\ast }$ to represent the total number of affirmative answers to the $J$ control questions, the observed response can be written as,

(1) $$\begin{eqnarray}\displaystyle Y_{i}=T_{i}Z_{i}+Y_{i}^{\ast }. & & \displaystyle\end{eqnarray}$$

In the early literature on list experiments, researchers estimated the proportion of respondents with the affirmative answer to the sensitive item using DiM, but could not characterize the respondents most likely to have the affirmative response. To overcome this challenge, Imai (Reference Imai2011) considers the following multivariate regression model,

(2) $$\begin{eqnarray}\displaystyle \mathbb{E}(Y_{i}\mid T_{i},X_{i})=T_{i}\mathbb{E}(Z_{i}\mid X_{i})+\mathbb{E}(Y_{i}^{\ast }\mid X_{i}) & & \displaystyle\end{eqnarray}$$

where the randomization of treatment assignment guarantees the following statistical independence relationships: $T_{i}\bot \!\!\!\bot Z_{i}\mid X_{i}$ and $T_{i}\bot \!\!\!\bot Y_{i}^{\ast }\mid X_{i}$ .

Although this formulation can accommodate various regression models, one simple parametric model, considered in Imai (Reference Imai2011), is the following binomial logistic regression model,

(3) $$\begin{eqnarray}\displaystyle Z_{i}\mid X_{i} & \stackrel{\text{indep.}}{{\sim}} & \displaystyle \text{Binom}(1,g(X_{i};\unicode[STIX]{x1D6FD}))\end{eqnarray}$$

(4) $$\begin{eqnarray}\displaystyle Y_{i}^{\ast }\mid X_{i} & \stackrel{\text{indep.}}{{\sim}} & \displaystyle \text{Binom}(J,f(X_{i};\unicode[STIX]{x1D6FE}))\end{eqnarray}$$

where $f(X_{i};\unicode[STIX]{x1D6FE})=\text{logit}^{-1}(X_{i}^{\top }\unicode[STIX]{x1D6FE})$ and $g(X_{i};\unicode[STIX]{x1D6FD})=\text{logit}^{-1}(X_{i}^{\top }\unicode[STIX]{x1D6FD})$ , implying the following regression functions, $\mathbb{E}(Y_{i}^{\ast }\mid X_{i})=J\cdot f(X_{i};\unicode[STIX]{x1D6FE})$ and $\mathbb{E}(Z_{i}\mid X_{i})=g(X_{i};\unicode[STIX]{x1D6FD})$ . Note that $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6FE}$ represent vectors of regression coefficients.

Imai (Reference Imai2011) proposes two ways to estimate this multivariate regression model: nonlinear least squares (NLSreg) and maximum likelihood (MLreg) estimation. NLSreg is obtained by minimizing the sum of squared residuals based on equation (2).

(5) $$\begin{eqnarray}\displaystyle (\hat{\unicode[STIX]{x1D6FD}}_{\mathsf{NLS}},\hat{\unicode[STIX]{x1D6FE}}_{\mathsf{NLS}})=\underset{(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE})}{\operatorname{argmin}}\mathop{\sum }_{i=1}^{N}\{Y_{i}-T_{i}\cdot g(X_{i};\unicode[STIX]{x1D6FD})-f(X_{i};\unicode[STIX]{x1D6FE})\}^{2} & & \displaystyle\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D6FD}}_{\mathsf{NLS}}$ and $\hat{\unicode[STIX]{x1D6FE}}_{\mathsf{NLS}}$ are the nonlinear least squares (NLS) estimates of the coefficients. NLSreg is consistent so long as the regression functions are correctly specified and does not require the distributions to be binomial. One can obtain more efficient estimates by relying on distributional assumptions. In particular, MLreg is obtained by maximizing the following log-likelihood function,

(6) $$\begin{eqnarray}\displaystyle (\hat{\unicode[STIX]{x1D6FD}}_{\mathsf{ML}},\hat{\unicode[STIX]{x1D6FE}}_{\mathsf{ML}}) & = & \displaystyle \underset{(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE})}{\operatorname{argmax}}\mathop{\sum }_{i\in {\mathcal{J}}(1,0)}[\log \{1-g(X_{i};\unicode[STIX]{x1D6FD})\}+J\cdot \log \{1-f(X_{i};\unicode[STIX]{x1D6FE})\}]\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{y=0}^{J}\mathop{\sum }_{i\in {\mathcal{J}}(0,y)}y\log f(X_{i};\unicode[STIX]{x1D6FE})+(J-y)\log \{1-f(X_{i};\unicode[STIX]{x1D6FE})\}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{i\in {\mathcal{J}}(1,J+1)}\left\{\log g(X_{i};\unicode[STIX]{x1D6FD})+J\log f(X_{i};\unicode[STIX]{x1D6FE})\right\}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{y=1}^{J}\mathop{\sum }_{i\in {\mathcal{J}}(1,y)}\log \left[g(X_{i};\unicode[STIX]{x1D6FD})\binom{J}{y-1}f(X_{i};\unicode[STIX]{x1D6FE})^{y-1}\{1-f(X_{i};\unicode[STIX]{x1D6FE})\}^{J-y+1}\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\{1-g(X_{i};\unicode[STIX]{x1D6FD})\}\binom{J}{y}f(X_{i};\unicode[STIX]{x1D6FE})^{y}\{1-f(X_{i};\unicode[STIX]{x1D6FE})\}^{J-y}\right]\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D6FD}}_{\mathsf{ML}}$ and $\hat{\unicode[STIX]{x1D6FE}}_{\mathsf{ML}}$ are the maximum likelihood (ML) estimates of the coefficients and ${\mathcal{J}}(t,y)$ represents the set of respondents who have $T_{i}=t$ and $Y_{i}=y$ .

The choice between NLSreg and MLreg involves a fundamental tradeoff between bias and variance. MLreg is more efficient than NLSreg because the former makes additional distributional assumptions. In particular, MLreg models each cell of the observed response, including the $Y_{i}=J+1$ and $Y_{i}=0$ cells in the treatment group, which can greatly affect parameter estimates (Ahlquist Reference Ahlquist2018). In contrast, NLSreg only makes an assumption about the conditional mean functions and hence is more robust to measurement errors in specific cells. In line with these theoretical expectations, simulations in Ahlquist (Reference Ahlquist2018) report that DiM, which is a special case of NLSreg without covariates, is more robust for estimating the proportion of the sensitive trait than MLreg.

Two identification assumptions are required for DiM, NLSreg, and MLreg. First, respondents in the treatment group are assumed not to lie about the sensitive item, i.e., no liars. Any other behavior implies misreporting, and any estimator based on mismeasured responses is likely to be biased. The second assumption is that respondents’ answers to the control items are not affected by the treatment, i.e., no design effect. Because list experiments rely upon the comparison of responses between the treatment and control groups, responses to the control items must remain identical in expectation between the two groups. The violation of this assumption also leads to mismeasured responses, yielding biased estimates. We emphasize that DiM is a special case of NLSreg and is not exempt from these assumptions. In Appendix A, we prove that DiM is biased under top-biased and uniform error. The bias is large when the prevalence of the sensitive trait is small. NLSreg adds an assumption about the correctly specified regression function and MLreg imposes an additional distributional assumption.

The main difficulty of multivariate regression analysis for list experiments stems from the fact that the response to the sensitive item is not observed except for the respondents in the treatment group who choose the maximal or minimal response. Regression analysis under this circumstance is more challenging than when the outcome is directly observed. If the sensitive trait of interest is a rare event, then MLreg is likely to suffer from bias. Such bias is known to exist even when the outcome variable is observed (King and Zeng Reference King and Zeng2001), and is likely to be amplified for list experiments. In addition, dealing with measurement error will also be more difficult when the outcome variable is not directly observed. Below, we consider several methodological strategies for addressing this issue.

2.2 Strategic and Nonstrategic Measurement Errors

Researchers most often use indirect questioning techniques to study sensitive topics, which present respondents with strong incentives to disguise the truth. For this reason, much of the existing literature on list experiments has been rightly concerned with mitigating strategic measurement error, particularly floor and ceiling effects (e.g., Blair and Imai Reference Blair and Imai2012; Glynn Reference Glynn2013). These errors arise because the list experiment fails to mask the sensitive trait for respondents whose truthful response under treatment occupies the floor or ceiling cells.

Although the literature has provided many tools for ameliorating such bias, much less attention has been paid to nonstrategic measurement error, arising for example from poor survey implementation or respondent inattention. Because these behaviors run against the assumptions of the estimators described in Section 2.1, it is no surprise that they can induce similar forms of bias. Illustrating this point, Ahlquist (Reference Ahlquist2018) examines a specific nonstrategic error mechanism—called top-biased error, where a random fraction of respondents provide the maximal response value regardless of their truthful answer to the sensitive question—and demonstrates that MLreg can be severely biased under this error mechanism.

Although we provide the tools to diagnose and correct this and other nonstrategic measurement error mechanisms below, we are skeptical that top-biased error is common in practice, at least for truly sensitive questions. The reason is that under the treatment condition, giving the maximal value reveals that the respondent answers the sensitive question affirmatively. This implies, for example, that respondents are willing to admit engaging in such sensitive behaviors as drug use and tax evasion or having socially undesirable attitudes such as gender and racial prejudice. This scenario is unlikely so long as the behavior or attitudes researchers are trying to measure are actually sensitive.

In our experience, when answering truly sensitive questions, respondents typically avoid reporting the extreme values (rather than gravitating toward them as assumed under the top-biased error mechanism). As an example of this phenomenon, we present a list experiment conducted by Lyall, Blair, and Imai (Reference Lyall, Blair and Imai2013) in the violent heart of Taliban-controlled Afghanistan, which was designed for estimating the level of support for the Taliban. The control group was given the following script ( $J=3$ ):

For the treatment group, the sensitive actor, i.e., the Taliban, is added.

Table 1 presents descriptive information, which shows clear evidence of floor and ceiling effects. Indeed, no respondent gave an answer of 0 or 4. By avoiding the extreme responses of 0 and 4, respondents in the treatment group are able to remain ambiguous as to whether they support or oppose the Taliban. This strategic measurement error may have arisen in part because of the public nature of interview. As explained in Lyall, Blair, and Imai (Reference Lyall, Blair and Imai2013), interviewers are required to ask survey questions to respondents in public while village elders watch and listen. Under this circumstance, it is no surprise that respondents try to conceal their truthful answers. Because of this sensitivity, the authors of the original study used endorsement experiments (Bullock, Imai, and Shapiro Reference Bullock, Imai and Shapiro2011), which represent a more indirect questioning technique, in order to measure the level of support for the Taliban. On the other hand, Blair, Imai, and Lyall (Reference Blair, Imai and Lyall2014) find that in the same survey the list experiment works well for measuring the level of support for the coalition International Security Assistance Force, which is a less sensitive actor to admit support or lack thereof for than is the Taliban.

2.3 Detecting Measurement Error

Although researchers are unlikely to know the magnitude of measurement error, whether strategic or not, we can sometimes detect measurement error from data. In addition to the tests developed by Blair and Imai (Reference Blair and Imai2012) and Aronow et al. (Reference Aronow, Coppock, Crawford and Green2015), we extend and formalize the recommendation by Ahlquist (Reference Ahlquist2018) to compare the results of multiple models to assess their robustness to measurement error. We focus on comparisons between MLreg and NLSreg, in order to focus on comparisons of the quantity of interest most commonly used by applied researchers.

We employ a general specification test due to Hausman (Reference Hausman1978) as a formal means of comparison between MLreg and NLSreg, both of which are designed to examine the multivariate relationships between the sensitive trait and respondents’ characteristics. The idea is that if the regression modeling assumptions are correct, then NLSreg and MLreg should yield statistically indistinguishable results. If their differences are significant, we reject the null hypothesis of correct specification. Note that model misspecification can arise for various reasons, with measurement error being one possibility. Furthermore, the test assumes the linear regression specification shared across NLSreg and MLreg. Then the test statistic and its asymptotic distribution are given by,

(7) $$\begin{eqnarray}\displaystyle (\hat{\unicode[STIX]{x1D703}}_{\text{ML}}-\hat{\unicode[STIX]{x1D703}}_{\text{NLS}})^{\top }(\widehat{\mathbb{V}(\hat{\unicode[STIX]{x1D703}}_{\text{NLS}})}-\widehat{\mathbb{V}(\hat{\unicode[STIX]{x1D703}}_{\text{ML}})})^{-1}(\hat{\unicode[STIX]{x1D703}}_{\text{ML}}-\hat{\unicode[STIX]{x1D703}}_{\text{NLS}})^{\top }\stackrel{\text{approx.}}{{\sim}}\unicode[STIX]{x1D712}_{\text{dim}(\unicode[STIX]{x1D6FD})+\text{dim}(\unicode[STIX]{x1D6FE})}^{2} & & \displaystyle\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D703}}_{\text{NLS}}=(\hat{\unicode[STIX]{x1D6FD}}_{\text{NLS}},\hat{\unicode[STIX]{x1D6FE}}_{\text{NLS}})$ and $\hat{\unicode[STIX]{x1D703}}_{\text{ML}}=(\hat{\unicode[STIX]{x1D6FD}}_{\text{ML}},\hat{\unicode[STIX]{x1D6FE}}_{\text{ML}})$ are the NLS and ML estimators and $\widehat{\mathbb{V}(\hat{\unicode[STIX]{x1D703}}_{\text{NLS}})}$ and $\widehat{\mathbb{V}(\hat{\unicode[STIX]{x1D703}}_{\text{ML}})}$ are their estimated asymptotic variances. We view this test as a logical extension and formalization of the recommendation in Ahlquist (Reference Ahlquist2018) to compare the results from DiM and MLreg.

Table 1. An Example of Floor and Ceiling Effects from the List Experiment in Afghanistan Reported in Lyall, Blair, and Imai (Reference Lyall, Blair and Imai2013). No respondent in the treatment group gave an answer of 0 or 4, suggesting that the respondents were avoiding revealing whether they support the Taliban.

2.4 Modeling Measurement Error Mechanisms

One advantage of the multivariate regression framework proposed in Imai (Reference Imai2011) is its ability to directly model measurement error mechanisms, an approach which has demonstrated its value in a variety of contexts (see e.g., Carroll et al. Reference Carroll, Ruppert, Stefanski and Crainiceanu2006), including list experiments (Blair and Imai Reference Blair and Imai2012; Chou Reference Chou2018). Measurement error models strike a valuable middle ground between MLreg and NLSreg. First, these models include the model without measurement error as their limiting case, requiring fewer and weaker assumptions than standard models. As a result, we can apply the specification test as shown for NLSreg and MLreg above. Second, these models can be used to check the robustness of empirical results to measurement error. Third, researchers can use these models to test the mechanisms of survey misreporting in order to understand when list experiments do and do not work.

Although we believe that top-biased error is unlikely to obtain in applied settings, we show how to model this error process as an illustration of how our modeling framework can flexibly incorporate various measurement error mechanisms. We then show how to model uniform error in which “a respondent’s truthful response is replaced by a random uniform draw from the possible answers available to her, which in turn depends on her treatment status” (Ahlquist Reference Ahlquist2018, p. 5). We think that this uniform response error process is more realistic and hence the proposed uniform error model will be useful for applied researchers. As shown in Appendix A, DiM is biased under these error processes.

Top-biased error. Under top-biased error, for the NLS estimation, equation (2) becomes,

(8) $$\begin{eqnarray}\displaystyle \mathbb{E}(Y_{i}\mid T_{i},X_{i}) & = & \displaystyle pJ+T_{i}\{p+(1-p)\mathbb{E}(Z_{i}\mid X_{i})\}+(1-p)\mathbb{E}(Y_{i}^{\ast }\mid X_{i})\end{eqnarray}$$

where $p$ is the additional parameter representing the population proportion of respondents who give the maximal value as their answer. When $p=0$ the model reduces to the standard model given in equation (2). The NLS estimator is obtained by minimizing the sum of squared error,

(9) $$\begin{eqnarray}(\hat{\unicode[STIX]{x1D6FD}}_{\text{NLS}},\hat{\unicode[STIX]{x1D6FE}}_{\text{NLS}})=\underset{(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE},p)}{\operatorname{argmin}}\mathop{\sum }_{i=1}^{N}[Y_{i}-pJ-T_{i}\{p+(1-p)\mathbb{E}(Z_{i}\mid X_{i})\}-(1-p)\mathbb{E}(Y_{i}^{\ast }\mid X_{i})]^{2}.\end{eqnarray}$$

We can also model top-biased error using the following likelihood function,

(10) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\prod }_{i\in {\mathcal{J}}(1,J+1)}[g(X_{i};\unicode[STIX]{x1D6FD})f(X_{i};\unicode[STIX]{x1D6FE})^{J}+p\{1-g(X_{i};\unicode[STIX]{x1D6FD})f(X_{i};\unicode[STIX]{x1D6FE})^{J}\}]\mathop{\prod }_{i\in {\mathcal{J}}(0,J)}[f(X_{i};\unicode[STIX]{x1D6FE})^{J}+p\{1-f(X_{i};\unicode[STIX]{x1D6FE})^{J}\}]\nonumber\\ \displaystyle & & \displaystyle \mathop{\prod }_{i\in {\mathcal{J}}(1,0)}(1-p)\{1-g(X_{i};\unicode[STIX]{x1D6FD})\}\{1-f(X_{i};\unicode[STIX]{x1D6FE})\}^{J}\mathop{\prod }_{y=0}^{J-1}\mathop{\prod }_{i\in {\mathcal{J}}(0,y)}(1-p)\binom{J}{y}f(X_{i};\unicode[STIX]{x1D6FE})^{y}\{1-f(X_{i};\unicode[STIX]{x1D6FE})\}^{J-y}\nonumber\\ \displaystyle & & \displaystyle \mathop{\prod }_{y=1}^{J}\mathop{\prod }_{i\in {\mathcal{J}}(1,y)}(1-p)\left[g(X_{i};\unicode[STIX]{x1D6FD})\binom{J}{y-1}f(X_{i};\unicode[STIX]{x1D6FE})^{y-1}\{1-f(X_{i};\unicode[STIX]{x1D6FE})\}^{J-y+1}\right.\nonumber\\ \displaystyle & & \displaystyle \left.\qquad +\,\{1-g(X_{i};\unicode[STIX]{x1D6FD})\}\binom{J}{y}f(X_{i};\unicode[STIX]{x1D6FE})^{y}\{1-f(X_{i};\unicode[STIX]{x1D6FE})\}^{J-y}\right].\end{eqnarray}$$

Again, when $p=0$ , this likelihood function reduces to the likelihood function of the original model, which is given on the logarithmic scale in equation (6). While this likelihood function is too complex to optimize, we can use the expectation–maximization (EM) algorithm (Dempster, Laird, and Rubin Reference Dempster, Laird and Rubin1977) to maximize it. The details of this algorithm are given in Appendix B.1.

Uniform error. Under the uniform error mechanism, we modify the regression model given in equation (2) to the following,

(11) $$\begin{eqnarray}\displaystyle \mathbb{E}(Y_{i}\mid T_{i},X_{i}) & = & \displaystyle \frac{p_{0}(1-T_{i})J}{2}+T_{i}\left\{\frac{p_{1}(J+1)}{2}+(1-p_{1})\mathbb{E}(Z_{i}\mid X_{i})\right\}\nonumber\\ \displaystyle & & \displaystyle +\,\{(1-T_{i})(1-p_{0})+T_{i}(1-p_{1})\}\mathbb{E}(Y_{i}^{\ast }\mid X_{i})\end{eqnarray}$$

where $p_{t}=\Pr (S_{i}\mid T_{i}=t)$ represents the proportion of misreporting individuals under the treatment condition $T_{i}=t$ . Again, when $p_{0}=p_{1}=0$ , this model reduces to the original model without measurement error. As before, we can obtain the NLS estimator by minimizing the sum of squared error. We can also formulate the ML estimator using the following likelihood function,

(12) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\prod }_{i\in {\mathcal{J}}(1,J+1)}\left\{(1-p_{1})g(X_{i};\unicode[STIX]{x1D6FD})f(X_{i};\unicode[STIX]{x1D6FE})^{J}+\frac{p_{1}}{J+2}\right\}\nonumber\\ \displaystyle & & \displaystyle \mathop{\prod }_{i\in {\mathcal{J}}(1,0)}\left\{(1-p_{1})\{1-g(X_{i};\unicode[STIX]{x1D6FD})\}\{1-f(X_{i};\unicode[STIX]{x1D6FE})\}^{J}+\frac{p_{1}}{J+2}\right\}\nonumber\\ \displaystyle & & \displaystyle \mathop{\prod }_{y=0}^{J}\mathop{\prod }_{i\in {\mathcal{J}}(0,y)}\left\{(1-p_{0})\binom{J}{y}f(X_{i};\unicode[STIX]{x1D6FE})^{y}\left\{1-f(X_{i};\unicode[STIX]{x1D6FE})\right\}^{J-y}+\frac{p_{0}}{J+1}\right\}\nonumber\\ \displaystyle & & \displaystyle \mathop{\prod }_{y=1}^{J}\mathop{\prod }_{i\in {\mathcal{J}}(1,y)}\left[(1-p_{1})\left\{g(X_{i};\unicode[STIX]{x1D6FD})\binom{J}{y-1}f(X_{i};\unicode[STIX]{x1D6FE})^{y-1}\left\{1-f(X_{i};\unicode[STIX]{x1D6FE})\right\}^{J-y+1}\right.\right.\nonumber\\ \displaystyle & & \displaystyle \left.\left.\qquad +\,\{1-g(X_{i};\unicode[STIX]{x1D6FD})\}\binom{J}{y}f(X_{i};\unicode[STIX]{x1D6FE})^{y}\left\{1-f(X_{i};\unicode[STIX]{x1D6FE})\right\}^{J-y}\right\}+\frac{p_{1}}{J+2}\right].\end{eqnarray}$$

As shown in Appendix B.2, the EM algorithm can be used to obtain the ML estimator.

2.5 Robust Multivariate Regression Models

As another approach, we also show how to conduct multivariate regression analysis while ensuring that the estimated proportion of the sensitive trait is close to DiM. To do this, we follow Chou, Imai, and Rosenfeld (Reference Chou, Imai and Rosenfeld2017), who show how to incorporate available auxiliary information such as the aggregate prevalence of sensitive traits when fitting regression models. The authors find that supplying aggregate truths significantly improves the accuracy of list experiment regression models. Adopting this strategy, we fit the multivariate regression models such that they give an overall prediction of sensitive trait prevalence consistent with DiM. To the extent that DiM rests on weaker assumptions, this modeling strategy may improve the robustness of the multivariate regression models.

Specifically, we use the following additional moment condition,

(13) $$\begin{eqnarray}\displaystyle \mathbb{E}\{g(X_{i};\unicode[STIX]{x1D6FD})\}=\mathbb{E}\left\{\frac{\mathop{\sum }_{i=1}^{N}T_{i}Y_{i}}{\mathop{\sum }_{i=1}^{N}T_{i}}-\frac{\mathop{\sum }_{i=1}^{N}(1-T_{i})Y_{i}}{\mathop{\sum }_{i=1}^{N}(1-T_{i})}\right\}. & & \displaystyle\end{eqnarray}$$

For the NLS estimation, we combine this moment condition with the following first order conditions from the two-step NLS estimation.

(14) $$\begin{eqnarray}\displaystyle \mathbb{E}\left[T_{i}\{Y_{i}-f(X_{i};\unicode[STIX]{x1D6FE})-g(X_{i};\unicode[STIX]{x1D6FD})\}g^{\prime }(X_{i};\unicode[STIX]{x1D6FD})\right] & = & \displaystyle 0\end{eqnarray}$$

(15) $$\begin{eqnarray}\displaystyle \mathbb{E}\left[(1-T_{i})\{Y_{i}-f(X_{i};\unicode[STIX]{x1D6FE})\}f^{\prime }(X_{i};\unicode[STIX]{x1D6FE})\right] & = & \displaystyle 0\end{eqnarray}$$

where $f^{\prime }(X_{i};\unicode[STIX]{x1D6FE})$ and $g^{\prime }(X_{i};\unicode[STIX]{x1D6FD})$ are the gradient vectors with respect to $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FD}$ , respectively. Altogether, the moments form a generalized method of moments (GMM) estimator. In fact, we can use the same exact setup Chou, Imai, and Rosenfeld (Reference Chou, Imai and Rosenfeld2017) and use their code in the list package in R to obtain the NLS estimator with this additional constraint, although the standard errors must be adjusted as the auxiliary constraint does not provide additional information.

We can also incorporate the constraint in equation (13) in the ML framework. To do this, we combine the score conditions obtained from the log-likelihood function with this additional moment condition. Then, the GMM estimator can be constructed using all the moment conditions. The details of this approach are given in Appendix B.3.

3.1 Simulation Settings

We begin by replicating the “designed list” simulation scenario, which Ahlquist (Reference Ahlquist2018) found to be most problematic for MLreg.Footnote ¹ In addition to the introduction of top-biased error, this simulation scenario violates the assumptions of MLreg in two ways. First, Ahlquist follows the advice of Glynn (Reference Glynn2013) and generates a negative correlation among the control items. By contrast, MLreg assumes conditional independence of the control items. Second, control items are generated with different marginal probabilities, which is also inconsistent with the binomial distribution.Footnote ²

The data-generating process for these problematic “designed” lists is as follows. For the control outcome $Y_{i}^{\ast }$ , the marginal probabilities are fixed for each control item. For the simulations with $J=3$ control items, the probabilities of latent binary responses are specified to be $(0.5,0.5,0.15)$ , whereas in the simulations with $J=4$ , the study uses $(0.5,0.5,0.15,0.85)$ . The rmvbin() function in the R package bindata is used to generate the latent responses to the control items such that the correlation between the first two items is negative 0.6. To generate the sensitive trait, $Z_{i}$ , first a single covariate, $X_{i}$ , is sampled independently from the uniform distribution for each observation $i=1,2,\ldots ,N$ . Together with an intercept, we form the model matrix $\mathbf{X}_{i}=(1,X_{i})$ . The sensitive trait is then drawn according to the logistic regression given in equation (3). The coefficients are set to $\unicode[STIX]{x1D6FD}=(0,-4)$ corresponding to the prevalence of the sensitive trait approximately equal to $0.12$ . Finally, we assign the half of the sample to the treatment group ( $T_{i}=1$ ) and the other half to the control group ( $T_{i}=0$ ). The outcome variable then is generated according to equation (1). We conduct 1,000 simulations with these parameters.

To introduce top-biased error, Ahlquist (Reference Ahlquist2018) uses complete randomization to select 3% of the sample and changes the outcome variable $Y_{i}$ to $J+1$ ( $J$ ) if observation is assigned to the treatment (control) group, independently of the values of $\mathbf{X}_{i}$ , $Y_{i}^{\ast }$ , and $Z_{i}$ . To generate uniform error, 3% of the observations are similarly sampled, but are assigned their outcome variable with uniform probability to one of the $J+2$ ( $J+1$ ) possible values, depending on their treatment status. We follow these procedures in our simulations.

3.2 Detecting Model Misspecification

Given the discrepancies between this data-generating process and the process assumed by MLreg, MLreg is shown to be severely biased by these procedures (Ahlquist Reference Ahlquist2018). However, as explained in Section 2.1, NLSreg should be more robust than MLreg, though it is less efficient. This bias-variance tradeoff arises because NLSreg does not assume the binomial distribution for the control items. Under the assumptions of NLSreg, the control items can be arbitrarily correlated and have different marginal probabilities (although a specific functional form—here, the logistic function—is assumed for the conditional expectation). This implies that NLSreg should only be subject to the potential bias from response error.

This theoretical expectation suggests that the Hausman test proposed in Section 2.3 may be able to detect departures from the modeling assumptions. We find that this is indeed the case. Table 2 shows that our test diagnoses the inconsistency of MLreg in the presence of such severe model misspecification. The table presents the rejection rate for our simulations at different combinations of $J$ and $N$ with a $p$ -value of 0.1 as the threshold. As the “ $p$ -value” columns show, we find sufficiently large (and positive) test statistics to reject the null hypothesis of no misspecification in a large proportion of trials, especially when there is no response error. The finding is consistent with substantial model misspecification, in excess of response error, introduced by the designed list procedure in Ahlquist (Reference Ahlquist2018). Interestingly, the top-biased error appears to mask this misspecification.

Table 2. Results of the Model Misspecification Test for the Designed List Simulations. The proportions rejecting the null hypothesis of no model misspecification are shown. The “ $p$ -value” column is based on the standard Hausman test with a $p$ -value of 0.1 as the threshold, while the “ $p$ -value & negative” column is based on the combined rejection criteria where we reject the null hypothesis if the $p$ -value is less than 0.1 or the test statistic takes a negative value.

Importantly, we discovered that in all cases a large proportion of the trials yielded a negative value of the test statistic, which corresponds to extremely poor fit of MLreg relative to NLSreg. Such values are only consistent with model misspecification so long as the test is sufficiently powered (Schreiber Reference Schreiber2008). While the test statistic can by chance take a negative value in a finite sample, in our simulations such statistics are strikingly prevalent. As shown in the “ $p$ -value & negative” columns of the table, by using a negative or large positive test statistic as the criterion for rejection, we obtain a much more powerful test for misspecification even in the case of top-biased error. Although this test may be conservative, leading to overrejection of the null hypothesis, the fact that these rejection rates are large even when the sample size is moderate suggests that the test is well powered in this simulation study.

In sum, the proposed model misspecification test readily diagnoses the designed list misspecification studied in Ahlquist (Reference Ahlquist2018). Although the test may not work in all settings, we find here that it detects the significant problems in the designed lists simulations at a high rate.

3.3 Robustness of the Nonlinear Least Squares Regression Estimator

Our second main finding is that NLSreg is robust to these misspecifications. This fortifies our previous result that the model misspecification test, which is based on the divergence between NLSreg and MLreg, is effective for diagnosing model misspecification. To illustrate the robustness of NLSreg, in Figure 1, we present the bias, root-mean-squared error (RMSE), and the coverage of 90% confidence intervals of our estimates of the population prevalence of the sensitive trait. We also present results for MLreg, filtered using the $p$ -value plus negative value criterion for rejection described above.Footnote ³ Last, given the goals of multivariate regression, we also compute these statistics for the estimated coefficients.

Figure 1. Robustness of the Nonlinear Least Squares Regression Estimator in the Presence of Several Model Misspecifications Considered in Ahlquist (Reference Ahlquist2018). We consider the three estimators of the prevalence of the sensitive trait: the DiM estimator (open circle with dotted line), the ML regression estimator (solid square with solid line; filtered by the model misspecification test using the combined criteria), and the NLS estimator (solid triangle with dashed line). The result shows that the NLS regression estimator is as robust as the DiM estimator.

Figure 1 shows the results for $J=3$ control items for top-biased error (left three columns) and uniform error (right three columns). Given space limitations, the analogous figure for $J=4$ control items is shown in Figure 2 in the Supplementary Appendix. We include the two estimators considered in Ahlquist (Reference Ahlquist2018): DiM and MLreg (solid square with solid line), as well as the NLSreg estimator proposed in Imai (Reference Imai2011) (solid triangle with dashed line). Our main finding here is that NLSreg is robust to all of these model misspecifications, doing as well as DiM. This is consistent with our prior expectation: DiM is a special case of NLSreg.

Although filtering based on the model misspecification test addresses the overestimation of the sensitive trait under top-biased error observed in Ahlquist (Reference Ahlquist2018), we note that MLreg does not perform well for the estimation of the coefficients, nor does it improve inference for the prevalence of the sensitive trait under uniform error. However, as Table 2 showed, these results are based on the small fraction of trials that did not result in a negative or large positive test statistic. In such trials, the NLS estimates were also inaccurate due to sampling error. This suggests that, while our proposed statistical test will often be able to detect misspecification in practice, in the instances where it does not, NLSreg (and, by extension, DiM) will also be biased.

The results confirm our theoretical expectation that NLSreg is robust to various types of misspecification. As a final point, we note that our simulation results, based on the grossly misspecified data-generating process described above, do not necessarily imply that MLreg will perform badly for designed lists. The simulation settings we adapted from Ahlquist (Reference Ahlquist2018) are not realistic. They imply, for example, that the individual covariates have no predictive power for responses to the control items. Furthermore, our model misspecification test is able to detect the distributional misspecification in these simulations. Thus, in practice, such a misspecification will often be apparent from the divergence of the NLSreg and MLreg results.

3.4 Addressing Response Error

The simulation settings adopted above include several violations of the modeling assumptions, including correlation between control items, varying control item propensities, model misspecification, and measurement error. As such, it is difficult to disentangle which model misspecifications, or combination of them, are affecting the performance of different methods. In this section, we focus on assessing the impacts of top-biased and uniform error processes and examine how the multivariate models proposed in Section 2 can address them. To be sure, applied researchers will rarely know which (if any) of these mechanisms characterize their data. Nevertheless, we show here that these methods can eliminate these errors when they arise, and illustrate in Section 4 how they can be used to assess the robustness of empirical results.

To isolate the effects of measurement errors, we develop a data-generating process that assumes no other model misspecification. First, we draw the latent response to the sensitive item $Z_{i}$ and the control items $Y_{i}^{\ast }$ according to the model defined in equations (3) and (4). Following Ahlquist (Reference Ahlquist2018), we set the true values of the coefficients for the control items to $\unicode[STIX]{x1D6FE}$ to $(0,1)$ , corresponding to a conditional mean of observed response about $J\times 0.62$ , whereas the coefficients for the sensitive item are set to $\unicode[STIX]{x1D6FD}=(0,-4)$ , generating a low propensity of approximately $0.12$ . In the Supplementary Appendix, we present a high propensity scenario of about 0.38. Last, we introduce each response error using the same procedure described earlier. We conduct 5,000 simulations with these parameters.

Figure 2. Robustness of the Constrained and Measurement Error Maximum Likelihood Estimators in the Presence of Response Errors when the Propensity of Sensitive Trait is Low. We consider four estimators of the prevalence of the sensitive trait and slope and intercept regression coefficients: the standard ML estimator (solid square with solid line), the constrained ML estimator (solid triangle with dot-dash line), the ML estimators adjusting for top-biased response error (open circle with dashed lines) and uniform response errors (open diamond with dot-long-dash line). The result shows that both the constrained ML estimator and the models adjusting for response error are an improvement over the performance of the ML estimator.

Figure 2 presents the findings for $J=3$ , whereas Figure 3 of the Supplementary Appendix shows the results for $J=4$ . In the left-hand columns, we show the top-biased error simulation with the standard ML estimator (solid square with solid line), the constrained ML estimator (solid triangle with dot-dash line), the top-biased ML model (open circle with dash line), and the uniform ML model (open diamond with dot-long-dash line) introduced in Section 2. The right-hand columns present the uniform error simulation results. As before, we show the bias, RMSE, and coverage of 90% confidence intervals.

As the upper left-hand corner plot shows, we replicate the main finding in Ahlquist (Reference Ahlquist2018) that a small amount of top-biased error is enough to significantly bias the standard ML estimator. Looking at the regression coefficients, we find that this positive bias follows from the bias in the estimated slope. Our proposed methods appear to address this issue effectively. The constrained estimator slashes the bias of the overall prevalence by almost 75%. This is unsurprising, as it constrains the regression-based prediction to the DiM estimate. However, because the constrained ML does not model the error mechanism directly, it does not improve the bias of the estimated regression coefficients. Indeed, the dashed lines show, the constrained model reduces the bias by decreasing the intercept rather than the slope, which does not help in this particular simulation setting where the bias for the intercept is small to begin with. As a result, the coverage of the confidence interval for the slope is only slightly improved.

As expected, the top-biased error model most effectively addresses this measurement error mechanism, eliminating the bias of the three quantities of interest almost entirely. Likewise, coverage is at the nominal rate for all three quantities of interest. We find that the uniform error model, which models a different error process to the one assumed, nevertheless is no worse than the standard ML model. Indeed, it exhibits less bias, better coverage, and lower RMSE than MLreg. In both cases, there is a small finite-sample bias that reduces as sample size increases.

The right-hand columns of Figure 2 examine the performance of the same four estimators under uniform error. We find several interesting results. Most importantly, we find that MLreg is significantly less biased under this measurement error mechanism than under the top-biased error process. Given the greater plausibility of uniform error relative to top-biased error, this finding suggests that MLreg may be more robust to nonstrategic measurement error than the simulations of Ahlquist (Reference Ahlquist2018) suggest.

We find that the uniform error model leads to some underestimation of the sensitive trait prevalence. While no estimator is unbiased for estimating the intercept, the uniform error model yields a large finite-sample bias for the estimation of the slope coefficient. However, these biases are small relative to the standard error as shown by the proper coverage of the corresponding confidence intervals, and they go to zero as the sample size increases. In contrast, the constrained ML estimator appears to perform well with small bias and RMSE as well as proper coverage of confidence intervals. We also note that the top-biased error model, which assumes a different error process than the simulation DGP, performs well under uniform error, exhibiting low bias, RMSE, and nominal coverage.

4.1 Extremely Rare Sensitive Traits and Multivariate Regression Analysis

As a general rule of thumb, we caution against the use of multivariate regression models when studying extremely rare or even nonexistent sensitive traits. The reason is simple. The goal of multivariate regression analysis is to measure the association between sensitive traits and respondent characteristics. If almost all respondents do not possess such traits, then multivariate regression analysis is likely to be unreliable because no association exists in the first place (and any existing association is likely to be due to noise). In line with these expectations, Ahlquist (Reference Ahlquist2018) finds the ML regression estimator to be misleading for the list experiments on voter fraud and alien abduction but unproblematic for the list experiment on texting while driving, the more common phenomenon by far.

Table 3. Estimated Proportion of Respondent Types by the Number of Affirmative Answers to the Control and Sensitive Items. The table shows the estimated proportion of respondent type ${\mathcal{J}}(y,z)$ , where $y\in \{0,\ldots ,J\}$ denotes the number of affirmative answers to the control items and $z\in \{0,1\}$ denotes whether respondents would answer yes to the sensitive item. In the last row, we also present the DiM estimator for the estimated proportion of those who would affirmatively answer the sensitive item. The voter fraud and alien abduction list experiments have an extremely small proportion of those who would answer yes to the sensitive item, and for voter fraud some proportions are estimated negative, suggesting the problem of list experiment.

Indeed, we find that the voter fraud and alien abduction list experiments elicit extremely small proportions of the affirmative answer. As discussed in Section 2.3 and recommended in Blair and Imai (Reference Blair and Imai2012), Table 3 presents the estimated proportion of each respondent ${\mathcal{J}}(y,z)$ type defined by two latent variables, i.e., the total number of affirmative answers to the control items $Y_{i}^{\ast }=y$ and the answer to the sensitive item $Z_{i}=z$ . We also present the DiM for each list experiment. The list experiment on voter fraud is most problematic, yielding an overall negative estimate and exhibiting three negative estimates for respondent types who would answer the sensitive item affirmatively.

Although these negative estimates are not statistically significant, this simple diagnostic shows that the list experiment on voter fraud suffers from either the violation of assumptions or an exceedingly small number of respondents with the sensitive trait or both. The descriptive information clearly suggests that multivariate regression analysis cannot be informative for the list experiments on voter fraud and alien abduction. Virtually no respondent would truthfully answer yes to these questions; thus, there is no association to be studied.

The application of the multivariate ML regression model to the alien abduction and voter fraud lists compounds the weakness of indirect questioning methods, which are ill-suited to studying extremely rare sensitive traits. Although indirect questioning methods seek to reduce bias from social desirability and nonresponse by partially protecting the privacy of respondents, they are much less efficient than direct questioning. As a consequence, the estimates will lack the statistical precision required for measurement and analysis of extremely rare traits. Indirect methods further amplify the finite-sample bias associated with rare events (King and Zeng Reference King and Zeng2001).

Given these tradeoffs, we recommend that list experiments be used only when studying truly sensitive topics. The increased cognitive burden on respondents and the loss of statistical efficiency are too great for this survey methodology to be helpful for nonsensitive traits. Among the three list experiments, the one on alien abduction provides the least insight into the efficacy of list experiments. In fact, such questions may themselves increase measurement error if they prompt respondents to stop taking the survey seriously. Better designed validation studies are needed to evaluate the effectiveness of list experiments and their statistical methods (see e.g., Rosenfeld, Imai, and Shapiro Reference Rosenfeld, Imai and Shapiro2016).

Despite our reservations about the application of the multivariate regression models to two of the three list experiments, we now examine whether the methods proposed in Section 2 can detect and adjust for measurement error by applying them to these list experiments.

4.2 Comparison Between the ML and NLS Regression Estimators

We conduct two types of analyses. First, we fit the multivariate regression model with age, gender, and race as covariates using the ML and NLS estimation methods. We show that the NLS regression estimator does not overestimate the incidence of voter fraud and abduction. Next, we implement the test outlined in Section 2.3, and show that the difference between MLreg and NLSreg clearly indicates model misspecification.

Figure 3. The Estimated Prevalence of Voter Fraud, Alien Abduction, and Texting while Driving. Along with the DiM estimator, we present the estimates based on various multivariate regression analyses including the maximum likelihood and nonlinear least squares regression estimators and estimates are much more stable. The results based on the two measurement error models, i.e., top-biased and uniform errors, are also presented.

We begin by showing that NLSreg does not overestimate the prevalence of voter fraud and alien abduction. Figure 3 presents the estimated proportion of the sensitive trait based on DiM, NLSreg, and MLreg as well as two other estimators that will be described later. We find that the NLS regression estimates closely track the DiM estimates. Indeed, the NLS estimate is statistically indistinguishable from the DiM estimate in all three cases. In the case of voter fraud, the NLS estimate—around 1.4% and statistically indistinguishable from zero—exceeds the DiM estimate by $2.59$ percentage points, with a 90% bootstrap confidence interval equal to $[-3.74,9.30]$ . For the alien abduction list experiment, the NLS estimate exceeds the DiM estimate by $1.70$ percentage points, also an insignificant difference (90% CI: $[-0.60,5.90]$ ). Last, for the texting-while-driving list experiment, the difference is $1.10$ percentage points (90% CI: $[-1.40,2.60]$ .)

Table 4. Results of the Proposed Specification Tests. The $p$ -values are based on the absolute value of the test statistic. The results show that for the list experiments based on voter fraud and alien abduction we detect model misspecification. For the list experiment on texting while driving, we fail to reject the null hypothesis.

Having shown that NLSreg does not yield meaningfully larger estimates than DiM, we now apply the statistical test developed in Section 2.3 to these list experiments. Table 4 presents the results of this proposed specification test. For the list experiments on voter fraud and alien abduction, which our earlier analysis found most problematic, we obtain negative and large, positive values of the Hausman test statistic. For the negative test statistic, we compute $p$ -values based on the absolute value. The results of the statistical hypothesis tests strongly suggest that the model is misspecified for the voter fraud and alien abduction experiments. In contrast, for the list experiment on texting while driving, we fail to reject the null hypothesis of correct model specification.

In sum, the proposed model specification test reaches the same conclusion as the descriptive analysis presented above: multivariate regression models should not be applied to list experiments with extremely rare sensitive traits. For the list experiments on voter fraud and alien abduction, we find strong evidence for model misspecification, suggesting the unreliability of multivariate regression analysis for these list experiments. In contrast, we fail to find such evidence for the list experiment on texting while driving, for which the proportion of affirmative answers is relatively high.

4.3 Modeling Response Error

Next, we apply the nonstrategic measurement error models developed in Section 2.4 to these data and examine whether they yield different results. Earlier, we argued that the top-biased error process is often implausible, as it implies that respondents are willing to reveal sensitive traits. We would expect top-biased error to be most unlikely for the list experiment on voter fraud, as this is the most unambiguously stigmatized trait. On the other hand, uniform error may be the more plausible measurement error mechanism, for example due to satisficing.

As shown in Figure 3, the results based on these measurement error models are consistent with our arguments. For the list experiment on voter fraud, the top-biased error gives a relatively large estimate that is statistically indistinguishable from the ML estimate. In contrast, the uniform error model provides an estimate that is indistinguishable from zero with a 90% confidence interval that is narrower than any other estimator. The list experiment on alien abduction yields a similar result. Like all other models, the top-biased error model gives an estimate that is statistically distinguishable from zero, suggesting that 6 percent of respondents were abducted by aliens (even the DiM estimate is barely statistically insignificant). On the other hand, the prevalence estimate based on the uniform error process model has the narrowest 90% confidence interval that contains zero, suggesting a superior model fit. The results indicate that the uniform error model is more effective for mitigating nonstrategic respondent error in these data than the top-biased error model.

Finally, the results for the list experiment on texting while driving show that the top-biased and uniform measurement error models yield estimates that are much more consistent with the other models. Although the estimate based on the uniform error model is smaller, it has a wider confidence interval than other estimates, suggesting a possibly poor model fit. Together with the other results shown in this section, this finding implies that only the estimates based on the list experiment on texting while driving are robust to various model specification and measurement errors, whereas the estimates for voter fraud and alien abduction are quite sensitive.

Our statistical analysis suggests that the problems described in Ahlquist (Reference Ahlquist2018) arise mainly from the fact that voter fraud in the US and alien abduction are rare to nonexistent events. Given all three lists were implemented in the same survey on the same sample, and were consequently subject to the same nonstrategic measurement error, the robustness of the texting-while-driving lists indicates that researchers should be concerned more with the rarity of the traits under study than with nonstrategic measurement error per se. We provide additional evidence for this statement below by showing that accounting for measurement error does not alter any of the multivariate inferences that one would draw from the texting-while-driving list experiment.

4.4 Multivariate Regression Analysis of Texting While Driving

Recall that the goal of multivariate regression models for list experiments is to measure the association between the sensitive trait and respondent characteristics. We reiterate that voter fraud and alien abduction are so rare that multivariate analysis of these traits is likely to be unreliable. By contrast, the texting-while-driving list offers a unique opportunity to examine how accounting for measurement error affects the estimated regression parameters. Studies commonly assume that younger drivers are especially likely to text while driving. However, frequently used methods, such as analysis of traffic accidents, are unable to measure this association directly (e.g., Delgado, Wanner, and McDonald Reference Delgado, Wanner and McDonald2016). Clearly, DiM also fails to shed light on this relationship.

Figure 4 and Table 5 present the results from our multivariate analysis of the texting-while-driving list. In Figure 4, we present predicted values for the different subgroups defined by the three covariates. These values are calculated by changing the variable of interest while fixing the other variables to their observed values. The highlighted comparisons correspond to statistically significant coefficients at the $\unicode[STIX]{x1D6FC}=0.10$ level (see Table 5). As the bottom-left panel of the figure shows, we find a consistent association between age and texting while driving. In all models younger respondents are more likely to text while driving than older respondents. We find some evidence of gender differentiation as well; male respondents appear to be consistently more likely to text while driving than female respondents, although this difference is not generally statistically significant.

The overall conclusion is that accounting for uniform error does not have a substantial effect on the conclusions that one would draw from the standard ML or NLS models. Moreover, the results illustrate the primary advantage of multivariate regression modeling, which is to assess the relationship between the sensitive trait and covariates.

Table 5. Multivariate Regression Analysis of the Texting-While-Driving List. This table shows the estimated coefficients from the baseline NLS and ML multivariate regression models, as well as the proposed robust ML, top-biased, and uniform measurement error models. Younger respondents are more likely to text while driving. We also find suggestive evidence that male respondents are more likely to text while driving.