1. A simulation analysis

This simulation experiment provides a parsimonious test of the efficacy of the use of nonresponse rates to politically sensitive questions to measure preference falsification, as well as the viability of inferences drawn from analyses using such measures. Ultimately, this simulation will demonstrate that utilizing politically sensitive questions where the regime's preferred response is likely known, such as when questions relate to the government or regime itself, is not a reliable proxy for levels of preference falsification within a society. While increasing repression when levels of repression are relatively low (under certain reasonable conditions) may indeed increase nonresponse rates to politically sensitive questions, as repression increases beyond this level, the nonresponse rate to politically sensitive questions and SCI score should decrease. Thus, the nonresponse rates of relatively repressive states and nonrepressive states may be indistinguishable from one another, yet the repressive state may produce patterns of responses that are highly unrepresentative of the true preferences of people within the state.

The most direct and robust attempt to proxy for preference falsification using nonresponse rates was set forth by Shen and Truex (Reference Shen and Truex2021), where they produced an index of self-censorship across democratic and autocratic states using nonresponse rates to politically sensitive questions adjusted for the nonresponse rates to questions that are not politically charged. In their view, “item nonresponse rates can be used to measure self-censorship, which will indirectly allow us to assess the broader idea of preference falsification, and to measure its likely incidence across time and space” (Shen and Truex, Reference Shen and Truex2021: 1675). This view reflects the logic underlying the use of nonresponse rates to proxy for preference falsification in many other studies (Jiang and Yang, Reference Jiang and Yang2016; Robinson and Tannenberg, Reference Robinson and Tannenberg2019; Shamaileh, Reference Shamaileh2019). The goal is to capture to some reasonable extent the sensitivity bias associated with preference falsification through a measure of self-censorship since they ostensibly move in the same direction. The article and analysis are well crafted, and Shen and Truex are careful to note many of the potential limitations of their exploratory analysis. Moreover, their contention that there is heterogeneity among autocratic regimes with regard to the truthfulness of responses to politically sensitive questions is certainly likely and supported by the literature, as well as the findings of this article. Given that their measure represents the most thorough attempt to capture preference falsification using nonresponse rates, this analysis explores the viability of using nonresponse rates to proxy for preference falsification while focusing on the SCI that they developed (Shen and Truex, Reference Shen and Truex2021). Thus, this article should not be construed as narrowly focused on evaluating the efficacy of the SCI, but on evaluating the use of nonresponse rates to proxy for preference falsification more broadly by using the most developed measure of this nature.Footnote ¹

I simulated a set of true preferences of 1000 respondents within a model state (i ∈ {1,  …,  n} : n = 1000). For simplicity, it is assumed that the true preferences of individuals are evenly distributed across all contexts.Footnote ² The answers to three positively correlated political questions and three uncorrelated nonsensitive questions were simulated for the respondents. The distribution of these responses was intentionally made approximately equal for ease of interpretation. The number of questions simulated was chosen in order to reproduce the SCI in Shen and Truex (Reference Shen and Truex2021). Let x represent an individual's response to a question and $\hat {x}$ represent an individual's true preference. To match the survey items generally used by scholars, I assumed that each x is an integer ranging from 1 (least preferred by the regime) to 5 (most preferred by the regime). Where $x \neq \hat {x}$, an individual is engaging in some degree of preference falsification. True preferences were distributed across the population in the manner described in Table 1.Footnote ³ Individuals were also assigned a parameter associated with the desire to express their true preferences that are distributed uniformly across the population; 0 < c < 1. This was used to construct a cost function associated with expressing an opinion different from one's true preference. Thus, this represents the internal costs of preference falsification noted by Kuran (Reference Kuran1991).

Table 1. Simulated true preferences (proportions)

Utilizing the same underlying preferences, 1000 democratic and autocratic states were simulated (s ∈ {1,  …,  r} : r = 2000). Each of these states allocates some cost associated with expressing an opinion that differs from the regime's preferred stance (0 < d < 1). While repression and the costs of repression should generally be thought of as endogenous (Kuran, Reference Kuran1989; Davenport, Reference Davenport2007b; Ritter et al., Reference Ritter and Conrad2016), for the purposes of analyzing responses to a survey where the responses of others are not known, it is reasonable to treat repression as an exogenous parameter. This parameter should be thought of as generally capturing the overall perceived probability of being punished and the perceived expected punishment that might be allocated if authorities are informed of an individual's response. Such punishments should not be thought of strictly in terms of raids on dissident homes, imprisonment or torture by regime officials. The costs of deviating from the regime's preferred position may also involve less severe punishments such as the refusal of selective benefits or access to certain institutions by local officials or regime-aligned elites.

For the simulation analysis, I assumed that democratic institutions constrain the costs that can be allocated to punish deviations from the regime's preferred position, and, for simplicity, democratic states range from 0 to 0.1 for the repression parameter. The 0.1 limit on repression was arbitrarily chosen to allow some heterogeneity among democratic regimes, but lower or higher thresholds would produce qualitatively similar results. Authoritarian states are distributed uniformly between 0 and 1, producing greater heterogeneity in repression among them. It is further assumed that all governments prefer the maximum response to the survey question (e.g., individuals answer that they have a great deal of confidence in the government), and that their second preference is for moderate support, third is nonresponse, fourth is a moderately critical position and fifth is the most critical response.

Adopting Kuran's model would not allow for shifts toward nonresponse unless nonresponse is indeed the regime's preferred response since Kuran's model does not allow individuals to partially mask their opinions. Nonresponse would reveal that an individual is not a supporter. Therefore, Kuran's basic formulation is utilized, where the costs of repression and the psychological costs of lying factor into an individual's expression of preferences, while allowing for moderate positions to be adopted. For simplicity, quadratic loss functions are used to model the internal and external costs associated with deviating from the regime's preferred position. It is assumed that individuals weigh the internal costs of not expressing their true opinion against the external costs of expressing their opinion. The further an individual deviates from either the regime's preferred position or their own preferred position, the more costly such deviations are (decreasing at a decreasing rate). This produces an internal bargain between the regime's stance and an individual's true preference. The simple utility function used to capture this tradeoff is:

(1)$$U_{i}( x) = -d_s( x - x_{s}) ^{2} - c_i( x - \hat{x}_i) ^{2}$$

Thus, an individual's response, if the answer were measured continuously, would be a weighted average residing somewhere between their preferred response and the regime's:

(2)$$x^\ast = {d_s\over d_s + c_i}x_{s} + {c_i\over d_s + c_i}\hat{x}_i$$

No stochastic component was added to the model since adding such a component would not substantively contribute to this analysis. The discussion that follows will focus on the straightforward prediction that repression increases the degree to which an individual agrees with the regime:

(3)$${\partial x^\ast \over \partial d} = {c_i( x_{s}-\hat{x}_i) \over ( d_s + c_i) ^2}$$

The expressed preference for the regime's position is increasing when an individual's position is not perfectly aligned with the regime's; $x_{s} > \hat {x}_i$. Thus, while Shen and Truex (Reference Shen and Truex2021) observe no relationship between repression and their SCI scores, this simulation will demonstrate why the SCI as currently formulated is not constructed to provide a reliable measure of preference falsification.Footnote ⁴

There are several plausible alternative strategies for modeling the behavior of survey respondents answering regime assessment questions. One strategy would be to treat nonresponses as opting out of responding; however, since the question has already been asked, a nonresponse offers a response. Moreover, since the answers to regime assessment questions preferred by officials in authoritarian contexts, particularly direct questions related to support for the government, are known to virtually all respondents capable of participating in the survey, there is little room for masking an individual's preferences by blending in with a subset of the population that lacks such knowledge. A second strategy could use the same basic formulation provided within this article, but assume that only active dissent is punished. This would be plausible in certain contexts, and in such contexts measures of preference falsification that rely on nonresponse rates may be theoretically justified. Yet, the seemingly arbitrary shifts in what is permitted, when, by whom and which punishments are allocated create some uncertainty as to how authorities may respond to the mere withdrawal of active support in most authoritarian states (Wedeen, Reference Wedeen1999). Moreover, in many states, autocratic patronage networks and local distributive channels also create the possibility of the withdrawal of benefits from nonsupporters and decentralized punishment that is less devastating in nature than those used against dissidents by the security apparatus (Kuran, Reference Kuran1991; Carlson, Reference Carlson2018; Mazur, Reference Mazur2021). It should be noted that the parsimonious model presented above accounts for the reduced probability/cost of punishment for nonresponse by modeling the costs of deviating from the regime's preferred answer as increasing at an increasing rate. Finally, it should be noted that a high proportion of survey respondents in many contexts appear to believe that the surveys they are responding to are sponsored by the government and that there is evidence that fear of the state biases results on surveys (Zimbalist, Reference Zimbalist2018; Tannenberg, Reference Tannenberg2021). Of course, we cannot assume fear plays an important role in shaping response patterns to regime assessment questions in all authoritarian states [see Lei and Lu (Reference Lei and Lu2017); Stockmann et al. (Reference Stockmann, Esarey and Zhang2018)]. It is precisely the goal of tools such as the SCI to inform us as to when preference falsification is biasing responses.

I constructed the data-generating process for the simulations using the model presented above. Given that the questions used to construct the SCI require ordinal rather than continuous responses, for the simulated true preferences, internal individual psychological costs, and external country-level costs, each individual chooses the answer that maximizes her utility within the context of this model.Footnote ⁵ Figure 1 provides a simple representation of how an individual's response varies as the external costs of dissenting increase. Each of the three represented individuals possesses a preference for providing the answer most critical of the regime. As repression increases, the regime critic's response moves closer to the regime's. Theoretically, an increase in repression may push the regime critic to not respond, but it also may push the regime critic to move from nonresponse to an expression of support. It is for this reason that we cannot consistently expect nonresponse to provide a reliable indication of increased preference falsification.

Figure 1. Change in expressed preference as repression increases.

Note that each of the simulated countries has identical populations with regard to their underlying preferences, and that all that varies across these regimes in this analysis is the external cost imposed on deviating from the regime's preferred position. This produces different sets of expressed preferences, and it is the differences between these states in this regard that are of primary interest in this analysis. An SCI score was created for each simulated state using the equation developed by Shen and Truex (Reference Shen and Truex2021)Footnote ⁶ :

(4)$$SCI_{s} ={\mathop{{\sum_{\,j = 1}^{m}} \sum_{i = 1}^{n} nonresponse_{i, j}}\limits^{\rm Regime \ Assessment}\over m \cdot n}-{\mathop{{\sum_{\,k = 1}^{m}} \sum_{i = 1}^{n} nonresponse_{i, k}}\limits^{\rm Non \hyphen Sensitive}\over m \cdot n} $$

The distribution of the simulations across SCI scores is presented in Figure 2. The pattern produced is similar to the pattern of SCI scores exhibited in Shen and Truex (Reference Shen and Truex2021). While the distribution of SCI scores across authoritarian regimes is larger than across democratic regimes (sd _autocracy = 0.044, sd _democracy = 0.017), the average SCI score across autocracies and democracies are substantively similar (μ _autocracy = 0.029, μ _democracy = 0.025).Footnote ⁷ A low SCI score, however, does not necessarily or even generally indicate lower levels of preference falsification.

Figure 2. SCI scores and regime type.

Following Blair et al. (Reference Blair, Coppock and Moor2020), sensitivity bias induced by preference falsification is conceptualized as the absolute distance of an individual's response from their true preference. Aggregating for all questions across all individuals in each state, the mean deviation from true preferences isFootnote ⁸

(5)$$Bias_{s} = {\sum_{\,j = 1}^{m} \sum_{i = 1}^{n} \vert x_{i, j}-\hat{x}_{i, j}\vert \over m \cdot n}$$

Figure 3 presents the LOESS (locally weighted regression) curves for the relationship between the SCI scores, sensitivity bias, and the external costs of speaking out across authoritarian and democratic regimes. Sensitivity bias is consistently increasing in the external costs of criticism at a decreasing rate. At low levels of repression, SCI scores are indeed increasing in these external costs. As the external costs of repression increase, however, the SCI score begins to decrease, and at very high levels of repression, SCI scores should be lower than when repression is relatively low. More importantly, the SCI shares a similar relationship with the measure of sensitivity bias. Thus, as Shen and Truex (Reference Shen and Truex2021) found, there appears to be no consistent relationship between SCI scores and repression, but that does not imply that there is no consistent relationship between preference falsification and repression.

Figure 3. Change in SCI scores with repression.

To further examine how the use of nonresponse rates, and SCI scores in particular, may lead to invalid inferences, Figure 4 presents the responses for three simulated states with the same SCI score, 0.042, indicating relatively low (although nonnegligible) levels of self-censorship. The states include a democracy with low levels of repression, an autocracy with low levels of repression and an autocracy with high levels of repression. All three states have the same exact distribution of true preferences among the population, yet the expressed preferences for the repressive autocratic state differ dramatically from those of the other states. While this example may be relatively extreme in that the repressive state has no individuals who explicitly take an anti-government stance, this highlights the problematic nature of attempting to infer levels of preference falsification from the use of nonresponse rates. While approximately 42 percent of respondents’ true simulated preferences are critical of the regime, 0 percent provide critical responses. Preference falsification is far more prevalent in the repressive state, yet the repressive state produces the same SCI score as states with far lower levels of repression. Autocratic regimes bunched near democratic states in SCI scores may be producing relatively low or high levels of preference falsification, and the most repressive regimes should produce SCI scores lower than environments where there is little repression of speech. It should also be noted that even the minor costs associated with voicing dissent in states with low levels of repression may produce severe selection bias in the responses.

Figure 4. Change in item responses for simulated states with the same SCI score.

It could be posited that SCI scores are useful proxies only when repression is sufficiently low. Indeed, Figure 1 does show that at low levels of repression, the SCI is capturing sensitivity bias. However, this is the product of fixed distributional assumptions as to the true preferences of citizens within a state. This causes two related problems associated with the interpretation of such scores. First, the score itself is highly sensitive to the distribution of preferences within the state, even when holding true preferences for nonresponse constant. Second, we cannot gauge at what level of repression the SCI score loses its ability to act as a proxy for preference falsification without understanding the underlying distribution of true preferences. Figure 5 presents the LOESS curves for the relationship between SCI scores across simulations run on populations with preferences that are high and low relative to the analysis in the initial simulation, and demonstrates the sensitivity of SCI scores to the underlying true preferences of individuals within a state.Footnote ⁹

Figure 5. Distribution of true preferences and the SCI.

2. Some empirical evidence

Does the empirical evidence suggest that, despite its limitations, the SCI—as well as other indices based on nonresponse rates—can serve as a viable cross-country measure of preference falsification? If an index of nonresponse rates can serve as an indirect measure of preference falsification, it should be associated with an inflation in the assessment of regimes. While the results presented in this section are merely suggestive, there is little evidence to suggest that nonresponse rates to regime assessment questions can act as a suitable measure of preference falsification. Using six waves of the World Values Survey, V-Dem's Freedom of Expression Index as a proxy for the repression of speech (Pemstein et al., Reference Pemstein, Marquardt, Tzelgov, Wang, Krusell and Miri2018),Footnote ¹⁰ and the replication data provided by Shen and Truex (Reference Shen and Truex2021), this brief analysis demonstrates that the theorized effect of repression on responses to regime assessment questions that was developed in this article is plausible and that the SCI does not appear to capture sensitivity bias in favor of the regime.Footnote ¹¹ All estimated correlations in the analysis are conditional on a country being included in the World Values Survey, which is skewed toward less restrictive and more democratic settings. In addition, it is important to acknowledge the potential bias observed due to selection into the survey.

Figure 6 presents the response distributions from four countries for the question related to confidence in the government and highlights the inability of the SCI or nonresponse rates to proxy for preference falsification. The UK, Iraq, and Uzbekistan all have relatively low SCI scores and low nonresponse rates for this particular question, yet they differ in their Freedom of Expression scores.Footnote ¹² More importantly, there is substantial variation in the response patterns. In Iraq and the UK, criticism of the government is fairly common, indicating that levels of preference falsification are likely low. In Uzbekistan, however, less than 3 percent expressed a lack of confidence in the government, and less than 2 percent did not respond to the question. While the underlying true preferences of these three societies are likely substantially different, it is highly unlikely that we would observe such low levels of criticism of Uzbekistan's government if coercion was not affecting responses. Neither the broader SCI nor the nonresponse rate for this particular question points to preference falsification. Furthermore, a high SCI score does not imply higher levels of preference falsification, as the patterns from Pakistan in 2001 indicate. Although the nonresponse rate is high, criticism of the regime is pervasive with over 50 percent responding that they have no confidence or not very much confidence in the government. These countries’ scores on the Freedom of Expression index appear to provide a more reliable proxy for the prevalence of preference falsification. While this is just a small subset of cases, it should be noted that the pool of country-years in the analysis is small and that there were several combinations of countries chosen that would have produced the same result. For example, we could replace the UK with France, Iraq with Iran, Uzbekistan with Azerbaijan (2011), and Pakistan with Morocco (2007 or 2011) and reach the same conclusions. In fact, of the states with relatively high SCI scores, only China appears to exhibit signs of mass reticence to express dissent.

Figure 6. Response patterns for four states.

What is the relationship between freedom of expression, confidence in government, and the SCI for the full sample of country-years? Figure 7(a) presents the bivariate relationship between V-Dem's Freedom of Expression Index and the mean of Government Confidence for each country-year. There is a strong and consistent negative correlation between freedom of expression and confidence in the government which is decreasing at a decreasing rate. This result is consistent with similar recent analyses that have been conducted (Nathan, Reference Nathan2020; Tannenberg, Reference Tannenberg2021), as well as the theoretical prediction that preference falsification is consistently increasing in the costs of expressing criticism of the regime. Of course, the correlation could be due to confounding variables associated with the set of highly repressive regimes where the survey was conducted or the potential for authoritarian propaganda to alter preferences and beliefs. Nevertheless, the relationship between the Freedom of Expression Index and Government Confidence seems likely to be at least partially due to preference falsification. The relationship between the Freedom of Expression Index and the SCI, however, is weak and inconsistent [Figure 7(b)]. Moreover, the bivariate relationship between the SCI/nonresponse rates and Government Support does not appear to be substantively meaningful or consistently positive.

Figure 7. Government support, freedom of expression, and the SCI.

Ostensibly, if the SCI were capable of capturing preference falsification, we would see some inflation in the assessment of regimes as the SCI increases when holding other variables constant. Thus, the most relevant tests of the SCI's relationship to regime assessment require the inclusion of contextual variables that may shape nonresponse rates. For ease of exposition, the results of simple ordinary least squares (OLS) models examining the weighted mean values of regime assessments will be the centerpiece of the empirical examination. The unit of analysis in this set of regression models is the country-year. All country-years that were available were utilized. For the simple aggregate group-level OLS analyses, the dependent variables were the weighted means of the regime assessment questions utilized to construct the SCI: Government Confidence, Human Rights in the country, and how democratic the country is according to the respondents (Democracy). For each of the analyses in this section, I coded nonresponses as missing. An alternative coding scheme where nonresponses were coded as the middle category can be found in the online Appendix and does not alter the predictions of the analyses in the paper. In addition, I utilized listwise deletion due to sample size considerations, the lack of compelling alternatives for the analysis of this particular data, and its simplicity. Given the robustness of the results that follow, it is unlikely that the handling of the missing data impacted the results.

There is little evidence of a consistent relationship between the SCI and regime assessment. For the range of the SCI where the majority of states are found, the relationship between the SCI and Government Confidence is negative or flat, and many of the states with seemingly inflated levels of Government Support are at the lower end of the SCI [Figure 7(c)]. However, some of the most repressive states where World Values surveys were conducted were relatively wealthy. Figure 8 presents a simple country-year level OLS analysis that accounts for the country's polity score, freedom of expression, GDP per capita, the weighted average of the subjective evaluations of the financial conditions of the household, enrollment in secondary education, and a dummy variable capturing whether a country's oil rents exceeded 5 percent for all three regime assessment questions. Even when excluding the most repressive regimes from the analysis, there is little evidence that the SCI offers a reliable proxy for the sensitivity bias associated with preference falsification. While none of the analyses seemed to indicate that the SCI was associated with higher regime assessments, there is evidence that the Freedom of Expression Index may offer a reliable proxy when evaluating the potential for sensitivity bias. Intuitively, this result would imply that simply examining regime-imposed limitations on speech may offer the most appropriate method for evaluating whether the regime assessments provided by survey respondents are potentially reliable.

Figure 8. OLS regression results.

Finally, as a robustness check, I also utilized per-cluster regression to account for individual-level and group-level effects when exploring the relationship between the SCI and the expressed preferences of survey respondents across contexts. In particular, it is plausible that when accounting for the variables that might influence political preferences, the SCI is associated with inflated expressions of regime support. If this were the case, an argument could be made that the SCI or some other measure rooted in nonresponse rates could be adjusted in a manner that allows for the measurement of preference falsification. Per-cluster regression was chosen due to the potential bias associated with group-level estimates in random effects models, including bias-corrected multilevel models, and the inability of the standard fixed effects framework to include group-level covariates (Bates et al., Reference Bates, Castellano, Rabe-Hesketh and Skrondal2014; Hazlett and Wainstein, Reference Hazlett and Wainstein2022).

The dependent variable for these analyses is the individual's response to the relevant regime assessment question. In the first step, I ran OLS analyses that included country-year fixed effects and the set of individual-level covariates noted below. I account for fixed effects due to the possibility that contextual factors that are not measured influence the underlying preferences of respondents in ways that are not captured by the group-level variables in the model. At the individual level, the models included the respondent's age, education, gender, whether they reside in an urban or rural community, and their subjective evaluations of the financial condition of their household. The marginal effects of the individual-level covariates were then subtracted from the responses of the individuals for the dependent variable and averaged by group. OLS regression was then used to examine the relationship between the dependent variable and the country's SCI score, as well as other country-year variables measuring democracy (using Polity IV scores) and economic development (log of GDP per capita).

The results of the per-cluster regression analyses are presented in Figure 9 and are in line with those of the simple OLS analyses of country-year aggregates. The SCI does not appear to be associated with the inflation of regime assessment scores. If anything, higher SCI scores are generally associated with lower evaluations of the regime when holding other variables constant. The SCI is negatively associated with evaluations for each of the regime assessment questions, albeit at levels that do not cross traditional levels of statistical significance. While the results are not statistically significant or substantively large for the three analyses, and the number of country-year observations is relatively low, there are intuitively appealing explanations for a potential negative correlation. The same freedom that provides individuals with the leeway to criticize the government allows individuals to comfortably claim that they do not know. In addition, democratic contexts produce competing political groups, which may produce greater levels of uncertainty as to how people feel about the regime based on who is in power. Such uncertainty may be the product of not knowing which political faction to support, or it may be associated with uncertainty as to how to evaluate the overall performance of a regime when the side you do not support is in power.

Figure 9. Predicted probabilities for per-cluster regression analyses.

Furthermore, alternative specifications replacing the SCI with Freedom of Expression demonstrate a substantively meaningful and significant decrease in confidence in the government as Freedom of Expression increases, corroborating the theoretical model presented in this article. Although the results of the per-cluster regression analyses do not provide evidence of a decrease in the evaluations of democracy and human rights within the country, the questions themselves are conceptually closely related to freedom of expression. Thus, freedom of expression is likely to be positively correlated with evaluations of how democratic a country is and how much it respects human rights conditional on respondents answering the questions truthfully. In addition, one of the core findings of Shen and Truex (Reference Shen and Truex2021) is a statistically significant negative correlation between electoral competition for the executive and the SCI. As such, we would expect a negative correlation between executive competition and regime assessment when holding other variables constant.Footnote ¹³ None of the analyses point to evidence of a substantively meaningful inflation of scores among states without executive competition when holding other variables constant.Footnote ¹⁴ Nevertheless, these results should be interpreted particularly cautiously given the low number of groups in the analysis of executive competition. Overall, the results presented above are consistent across several robustness checks, including OLS regression analyses that include group-level variables and cluster-robust standard errors, linear mixed-effects regression analyses, and limiting the analysis to authoritarian regimes (online Appendix).

The evidence presented above is not conclusive. Nevertheless, while the primary basis for rejecting the use of missing responses to proxy for the sensitivity bias associated with preference falsification is theoretical, the empirical evidence presented in this section further suggests that measuring preference falsification via nonresponse rates is not a viable solution. Moreover, it provides some support for the underlying theoretical mechanisms driving preference falsification presented in this article.

Although the SCI as currently constructed cannot effectively serve as a proxy for preference falsification, it introduces an interesting setup for the continued exploration of how preference falsification can be measured. If those less supportive of a regime mimic the preferences of regime supporters, any attempt to measure preference falsification via nonresponse rates should try to choose questions where mimicry is difficult. One possible extension of the strategy outlined by (Shen and Truex, Reference Shen and Truex2021) would be to attempt to measure preference falsification using sensitive political questions where there is no clear answer that government supporters would provide (Shamaileh, Reference Shamaileh2019: 962). Such an index would generally provide a weak, but potentially meaningful, proxy for preference falsification. A more robust examination of this is provided in the online Appendix, yet this solution is also problematic.