Repeated responses in misclassification binary regression: A Bayesian approach

Abstract

Binary regression models generally assume that the response variable is measured perfectly. However, in some situations, the outcome is subject to misclassification: a success may be erroneously classified as a failure or vice versa. Many methods, described in existing literature, have been developed to deal with misclassification, but we demonstrate that these methods may lead to serious inferential problems when only a single evaluation of the individual is taken. Thus, this study proposes to incorporate repeated and independent responses in misclassification binary regression models, considering the total number of successes obtained or even the simple majority classification. We use subjective prior distributions, as our conditional means prior, to evaluate and compare models. A data augmentation approach, Gibbs sampling, and Adaptive Rejection Metropolis Sampling are used for posterior inferences. Simulation studies suggested that repeated measures significantly improve the posterior estimates, in that these estimates are closer to those obtained in a case with no misclassifications with a lower standard deviation. Finally, we illustrate the usefulness of the new methodology with the analysis about defects in eyeglass lenses.

Keywords

Bayesian analysis Data augmentation logistic regression misclassification repeated responses

1 Introduction

Binary data are currently subject to misclassification in a variety of applied fields, such as epidemiological and medical research, and econometric and social sciences. As first observed by Bross (1954), the parameter estimators are extremely biased when the classification system is subject to error, that is, when a success may be erroneously classified as a failure (known as a false negative in medical applications), or vice versa (a false positive). Examples of cases where misclassification can occur include smoking cessation (McInturff et al. 2004), cognitive severity in patients with Alzheimer's disease (Luo et al. 2016), human papillomavirus detection (Paulino et al. 2003), cancer death due to radiation (Roy et al. 2005), and serological tests (Fujisawa and Izumi 2000; Menten et al. 2012). Many methods for correcting the estimates have been proposed. Examples include a double sampling scheme in frequentist modelling (Tenenbein, 1970) and using an informative prior distribution in a Bayesian point of view (Gaba and Winkler, 1992).

Some approaches are also available for regression settings for data arising from longitudinal studies. For an unobservable alternating binary Markov process (a subject may alternate between two states over time), Cook et al. (2000), Nagelkerke et al. (1990), Rosychuk and Thompson (2003) and Rosychuk and Thompson (2003) discussed the impact of misclassification on transition probability estimates. García-Zattera et al. (2010) focused on misclassified longitudinal binary data where the true response follows a progressive or monotone process (subjects cannot alternate between the two stages once the severe stage is reached over time). In a Bayesian approach, they also extended the proposed model to describe the relationship between the covariates and the prevalence and incidence, allowing for different time intervals between examinations for each subject, and different classifiers. Garcia-Zattera et al. (2012) extended the obtained results and proposed a multivariate binary inhomogeneous Markov model for the case of unobserved, correlated response variables subject to an unconstrained misclassification process and a monotone behaviour.

Misclassification issues in survival analysis can also be found in the statistical literature. Some methods estimate time-to-event distributions for time-dependent sensitivity with no covariates Balasubramanian and Lagakos 2001, 2003) or with auxiliary covariates in the Cox proportional hazards model (Snapinn, 1998). Gu et al. (2015) proposed a semiparametric likelihood-based approach to time-to-event models to estimate the association of one or more covariates with an error-prone, self-reported outcome. For current status data, a method for nonparametric estimation of the distribution function for the one-sample problem was proposed by McKeown and Jewell (2010). They also extended this methodology to parametric regression models, and Sal Y Rosas and Hughes (2011) extended it to situations in which sensitivity and specificity may vary across groups of individuals, using nonparametric and semiparametric methods. In a discrete-time context, Meier et al. (2003) extended the discrete proportional hazards model to incorporate misclassified responses. Hunsberger et al. (2010) incorporated sensitivity and specificity quantities into a subject-specific and time-dependent covariates model, and Richardson and Hughes (2000) implemented an expectation-maximization (EM) algorithm to estimate the product limit of the survival function with no covariates and misclassification. Garca-Zattera et al. (2016) extended the accelerated failure time (AFT) frailty modelling approach to account for misclassification of the determination of the event in the context of interval-censored data.

In the context of binary regression models for cross-sectional studies, ignoring misclassification errors can produce biased covariate effect estimates (Neuhaus, 1999; McInturff et al., 2004; Meyer and Mittag, 2017; Savoca, 2011; Carroll et al., 2006). McInturff et al. (2004) also noted in their example that some variables that were significant if misclassification was accounted for were not significant if misclassification was ignored.

Many frequentist methods have been developed to handle the problem of logistic regression with outcome misclassification. These methods generally consider a validation subsample (Lyles et al., 2011; Carroll et al., 2006; Meyer and Mittag, 2017). Magder and Hughes (1997) derived maximum likelihood estimators (MLEs) using an EM algorithm. Roy et al. (2005) also considered maximum likelihood estimation, but allowed for measurement error in the covariates. Semiparametric MLEs were proposed by Cheng and Hsueh (2003) for handling differential (covariate-dependent) misclassification and by Cheng and Chen (2006) for case-control data. Edwards et al. (2013) presented an alternative approach, using missing data methods (multiple imputation with internal validation data).

Bayesian methods have also been developed to address the problem of binary regression with outcome misclassification. For case-control studies with differential misclassification, Prescott and Garthwaite (2005) proposed a Bayesian approach in which validation data were used to correct misclassification. Tu et al. (1999) proposed a method that accommodates error probabilities in a great family of link functions and uses a latent variable approach with an MCMC algorithm to perform posterior inferences. Paulino et al. (2003) also considered latent variables, but prior distributions for the regression coefficients were elicited using the so-called conditional means prior (CMP; Bedrick et al., 1996). Naranjo et al. (2013) used two types of latent variables in probit and t-link regressions. McInturff et al. (2004) followed the methods proposed by Paulino et al. (2003), but their likelihood function was based only on observed variables, as the software WinBUGS was used.

However, the mere consideration of misclassification does not eliminate the possibility of inferential problems, as in assessing the significance of the important variables in the study. This is particularly true when the misclassification probabilities are unknown or gold standard tests are unavailable or prohibitively expensive. So an alternative approach is using a repeated-measurement sampling procedure, in which repeated and independent measures of the subjects are performed by the same error-prone diagnostic test.

Accounting for multiple measures in a misclassification context has already been explored. Evans et al. (1996) considered a Bayesian estimation of the success probability of binary data subject to misclassification errors when repeated measures are available, using both Gauss–Jacobi quadrature and Gibbs sampling for posterior analyses. A similar approach was developed by Swartz et al. (2004) regarding misclassification in multinomial data. Fujisawa and Izumi (2000) proposed maximum likelihood estimation for prevalence, sensitivity and specificity based on repeated classifications. Quinino et al. (2013) and Morais et al. (2017) proposed an estimator based on simple majority results of repeated classifications as an alternative to the method of moments for obtaining parameter estimators of a mixture of two- or three-binomial distributions.

For case-control studies in which exposure assessments are error-prone, Lai et al. (2007) assumed non-differential (covariate-independent) misclassification, performed repeated imperfect measurements and obtained the maximum likelihood estimates by using an EM algorithm. In the same context, Zhang et al. (2013) accounted for both differential and dependent misclassification and, in a Bayesian approach, proposed a method to estimate the measurement-error corrected exposure-disease association.

The purpose of this study is to develop a Bayesian method that takes into account repeated and independent measures of misclassified responses in binary regression. Repeated measures are not considered a validation subsample because they are performed using the same imperfect test, that is, there is not a gold-standard test available. We evaluated two possible specific situations usually observed in practice: (a) the results of all repeated measures are available and (b) only the result of the majority of the repeated measures is known. In Section 2, we propose models that incorporate the results of repeated classifications. Bayesian procedures are described in Section 3. Simulation studies and a real-data application are presented in Sections 4 and 5, respectively. Discussion and comments are provided in Section 6.

2 Statistical methods

Let $D_{k}$ be the true (unobserved) status of the subject $k$ ( $k = 1, \dots, n)$ and assume that $D_{k}$ does not change, such that $D_{k} = 1$ when the true status is success and $D_{k} = 0$ otherwise. Let $θ_{k 1}$ ( $θ_{k 0}$ ) be the success (failure) probability of the subject $k$ , that is, $θ_{k 1} = P (D_{k} = 1 | x_{k})$ , and $i = 0, 1$ , where $x$ ’s are known $p \times 1$ vectors of covariates and $\sum_{i} θ_{ki} = 1$ .

Using generalized linear models (McCullagh and Nelder, 1989) we can express $θ_{ki}$ as a function of a linear predictor $θ_{K I} = g^{'} ({x^{'}}_{k} β)$ , where $β$ is an unknown $p$ × 1 vector of regression coefficients, and g⁻¹(·)=F(·), where F(·) is a continuous cumulative distribution function (cdf) with density f(·). The most common choices for F(·) are $F (x^{'} β) = e^{x^{'} β} / - [1 + e^{x^{'} β}]$ (logistic regression), $F (x^{'} β) = Φ (x^{'} β)$ (probit regression, where $Φ (\cdot)$ is the standard normal cdf) and $F (x^{'} β) = 1 - \exp [- e^{x^{'} β}]$ (complementary log-log regression).

Suppose that a non-destructive fallible mechanism measures each sample subject $m$ independent times. Let $Y_{kl}$ $(k = 1, \dots, n; l = 1, \dots, m)$ be the reported status in the $l$ -th classification of the $k$ th subject, such that $P (Y_{kl} = j | D_{k} = i, x_{k}) = λ_{kij}$ and $\sum_{j} λ_{kij} = 1$ , $j = 0, 1$ . In the following two subsections, we present two methods for addressing these repeated classifications of the response variable.

2.1 Frequency of success model (FSM)

Let $T_{k}$ be the number of successes (or positive tests) obtained by subject $k$ after $m$ independent classifications (in practice, $m$ is usually odd). Given $D_{k}$ , $T_{k}$ is distributed as follows:

P (T_{k} = t) \propto \sum_{i} θ_{ki} {(λ_{ki 1})}^{t} {(1 - λ_{ki 1})}^{m - t} .

Therefore, the likelihood function is expressed as

L (θ, λ | t) \propto \prod_{k = 1}^{n} [\sum_{i} θ_{ki} {(λ_{ki 1})}^{t_{k}} {(1 - λ_{ki 1})}^{m - t_{k}}] .

An augmented data approach can be used to obtain a more tractable likelihood function. Let $d_{kit}$ be a non-observed indicator variable, such that $d_{kit} = 1$ indicates that $T_{k} = t$ and $D_{k} = i$ . Therefore, $P (d_{kit} = 1) \propto θ_{ki} λ_{ki 1}^{t} {(1 - λ_{ki 1})}^{m - t}$ and $\sum_{i} d_{kit} = d_{k 0 t} + d_{k 1 t} = 1$ .

For each $k$ , after conditioning on the observed data $t$ , the augmented data $d$ are independently distributed according to a Bernoulli distribution:

d_{kit} | β, λ, t \sim Bernoulli (\frac{λ_{ki 1}^{t} {(1 - λ_{ki 1})}^{m - t} θ_{ki}}{\sum_{j = 0}^{1} λ_{kj 1}^{t} {(1 - λ_{kj 1})}^{m - t} θ_{kj}}) .

(2.1)

Based on the augmented data, $d = d_{kit}$ , the likelihood function can be written as:

L (θ, λ | d) \propto \prod_{k, i} {(θ_{ki})}^{d_{kit}} \prod_{k, i} {(λ_{ki 1})}^{t_{k} d_{kit}} {(1 - λ_{ki 1})}^{(m - t_{k}) d_{kit}} .

(2.2)

For the sake of simplicity and without a lack of generality, we can use the logit link function (or an alternative function, according to the nature of the data) and assume unknown and non-differential errors to rewrite the likelihood function (4) in terms of $β$ for the frequency of success model (FSM):

L (β, λ | d) \propto {\prod_{k =1}^{n} F ({x^{'}}_{k} β)}^{d_{k 1 t}} {[1 - F ({x^{'}}_{k} β)]}^{d_{k 0 r}} {\prod_{i} (λ_{i 1})}^{\sum_{k} t_{k} d_{k i t}} {(1 - λ_{i 1})}^{\sum_{k} (m - t_{k}) d_{k i t}} .

(2.3)

2.2 Simple majority model (SMM)

An alternative way to address repeated classifications is to consider the response variable as equal to the most frequent test result (i.e., the simple majority result) after $m$ repeated independent classifications, as in Quinino et al. (2013). This can occur when it is not permitted to reveal the results of all repeated measures or these results are not available owing to ethical reasons. To avoid ties, assume that $m$ is odd. Let $S_{k}$ be a Bernoulli variable such that $S_{k} = 1$ , only if $T_{k} = \sum_{l = 1}^{m} Y_{kl} > 0.5 m$ .

Then, we have

P (S_{k} = 1) = \sum_{i} θ_{ki} [1 - Bin (m, λ_{ki 1}; 0.5 m)],

where Bin $(m, λ_{ki 1}; 0.5 m) = \sum_{t = 0}^{0, 5 m} (\begin{array}{l} m \\ t \end{array}) λ_{ki 1}^{t} {(1 - λ_{ki 1})}^{m - t}$ .

It is reasonable to assume that if $λ_{k 10} < 0.5$ , $λ_{k 01} < 0.5$ and $m$ is sufficiently large, the probability that an individual $k$ is classified as a success approximates to the true success probability, $θ_{k 1}$ . This assumption is easily verified using a binomial approximation to the normal distribution, as demonstrated by Quinino et al. (2013).

The likelihood function can, therefore, be written as

\begin{matrix} L (θ, λ | s) \propto \prod_{k = 1}^{n} {\{\sum_{i} θ_{ki} [1 - Bin (m, λ_{ki 1}; 0.5 m)]\}}^{s_{k}} \\ \times {\{1 - \sum_{i} θ_{ki} [1 - Bin (m, λ_{ki 1}; 0.5 m)]\}}^{1 - s_{k}} . \end{matrix}

An augmented data approach can also be used. Let $d_{kis}$ be an indicator variable such that $D_{k} = i$ and $S_{k} = s$ . Consequently, $d_{k + 1} = \sum_{i} d_{ki 1} = s$ , $d_{k + 0} = 1 - s$ and $\sum_{i, j} d_{kij} = 1$ . Conditional on the observed data, the augmented data $s$ are distributed according to independent Bernoulli distributions for each $k$ :

d_{k 01}^{} | β, λ, s \sim Binomial (s_{k}, \frac{θ_{k 0}^{} [1 - Bin (m; λ_{k 01}^{}; 0.5 m)]}{\sum_{i} θ_{ki}^{} [1 - Bin (m; λ_{ki 1}^{}; 0.5 m)]})

d_{k 10}^{} | β, λ, s \sim Binomial (1 - s_{k}, \frac{θ_{k 1}^{} [1 - Bin (m; λ_{k 10}^{}; 0.5 m)]}{\sum_{i} θ_{ki}^{} [1 - Bin (m; λ_{ki 0}^{}; 0.5 m)]}) .

(2.4)

The likelihood function based on the augmented data $d = d_{kit}$ can be written as:

L (θ, λ | d) \propto \prod_{k, i} {(θ_{ki})}^{d_{ki +}} \times \prod_{k, i, j} {[1 - Bin (m, λ_{kij}; 0.5 m)]}^{d_{kij}} .

Using the logit link function and non-differential errors, we obtain the simple majority model (SMM):

L (β, λ | d) \propto {\prod_{k =1}^{n} F ({x^{'}}_{k} β)}^{d_{k 1+}} {[1 - F ({x^{'}}_{k} β)]}^{d_{k 0+}} {\prod_{i, j} [1 - B i n (m, λ_{i j}; 0.5 m)]}^{\sum_{k} d_{k i j}}

(2.5)

When $m = 1$ , it is observed that Equations (2.3) and (2.5) are equal to the likelihood function derived by Hausman et al. (1998), Paulino et al. (2003) and McInturff et al. (2004). This finding indicates that the FSM and SMM are expansions of these models, and were obtained by introducing repeated classifications.

The identifiability analysis of the proposed models is discussed in Appendix A. In the next section, we develop a Bayesian approach to the FSM and SMM, presented earlier.

3 Bayesian analysis

A Bayesian approach to binary regression with misclassification requires prior knowledge about the regression coefficients and error probabilities. As in Paulino et al. (2003), Gerlach and Stamey (2007), Naranjo et al. (2013) and others, it is appropriate to independently specify the error probabilities as $λ_{kij} \sim Beta (c_{kij 1}, c_{kij 2})$ . Considering non-differential errors (and therefore omitting index $k$ ), the joint prior distribution of $λ = (λ_{01}, λ_{10})$ is a product of beta distributions:

π (λ) \propto λ_{10}^{c_{101} - 1} {[1 - λ_{10}]}^{c_{102} - 1} \times λ_{01}^{c_{011} - 1} {[1 - λ_{01}]}^{c_{012} - 1} .

As reported by O'Hagan et al. (1990), it is extremely difficult to directly specify a prior distribution for the regression coefficients directly. Many methods have been proposed, but the standard approach is to assume a normal (e.g., West, 1985) or diffuse distribution $(π (β) = 1)$ (Sweeting, 1981) for these parameters.

Another approach to binomial regression is to evaluate the success probability for various covariate values instead of evaluating the regression coefficients, as in Tsutakawa (1975), Tsutakawa and Lin (1986) and Bedrick et al. (1996). As suggested by Bedrick et al. (1996), this procedure is less complicated, especially when considering competing models. When comparing competing models, such as a logistic versus probit regression, model coefficients may have different interpretations and require different information. However, this problem is circumvented if information on the probabilities is elicited, as the prior distributions for regression coefficients can be induced from this information.

Bedrick et al. (1996) proposed evaluating prior information at $p$ locations ${\tilde{x}}_{r}$ , $(r = 1, \dots, p)$ in the predictor variables and then specifying a prior distribution for the mean of the potential observations conditional at each location. In a binary regression context, $E (D_{k} | {\tilde{x}}_{k})$ is the probability of success for a potential observation with covariate vector ${\tilde{x}}_{k}$ , that is, $E (D_{k} | {\tilde{x}}_{k}) = {\tilde{θ}}_{k 1}$ . These distributions are known as conditional means priors (CMPs). It is assumed that the $p$ vectors are linearly independent covariates. It is convenient, although not theoretically necessary, to specify that independently ${\tilde{θ}}_{r 1} \sim Beta (a_{1 r}, a_{2 r})$ . Therefore, the prior distribution of $\tilde{Θ} = ({\tilde{θ}}_{1}, . . ., {\tilde{θ}}_{p})$ (CMP) is:

π (\tilde{Θ}) = \prod_{r = 1}^{p} π ({\tilde{θ}}_{r}) \propto \prod_{r = 1}^{p} {\tilde{θ}}_{r}^{a_{1 r} - 1} {(1 - {\tilde{θ}}_{r})}^{a_{2 r} - 1} .

(3.1)

\indent In practice, previous studies or expert knowledge about a given phenomenon indirectly supplemented the determination of the hyperparameters of prior distributions. By obtaining the prior mode and quantities associated with a percentile (e.g., 95%), mean or variance, one can numerically determine the beta distribution that satisfies these requirements.

The prior distribution $π (β)$ is induced from the CMP (Equation 3.1) (details in Bedrick et al., 1996):

π (β) \propto \prod_{r = 1}^{p} F {({\tilde{x}}^{'}_{r} β)}^{a 1 r - 1} {[1 - F ({\tilde{x}}^{'}_{r} β)]}^{a 2 r - 1} f ({\tilde{x}}^{'}_{r} β) .

For instance, under the logistic regression model, we have $f (\cdot) = F (\cdot) [1 - F (\cdot)]$ , and the prior distribution for the regression coefficients vector is:

π (β) \propto \prod_{r = 1}^{p} F {({\tilde{x}}^{'}_{r} β)}^{a 1 r - 1} {[1 - F ({\tilde{x}}^{'}_{r} β)]}^{a 2 r} .

The likelihood functions (2.3) and (2.5) can be factored as $L (β, λ | d) = L (β | d) \times L (λ | d)$ . This fact enables the derivation of the posterior distribution, and it can be written as:

π (β, λ | d) \propto L (β | d) \times L (λ | d) \times π (β) \times π (λ) \propto π (β | d) \times π (λ | d) .

For the FSM, the posterior distributions are

π (β | d) \propto \prod_{k = 1}^{n} F {({\tilde{x}}^{'}_{r} β)}^{d_{k} 1 t} {[1 - F ({\tilde{x}}^{'}_{r} β)]}^{d_{k}_{0 t}} \times \prod_{i = 1}^{p} F {({\tilde{x}}^{'}_{r} β)}^{a_{1 i}} {[1 - F ({\tilde{x}}^{'}_{r} β)]}^{a_{2 i}}

(3.2)

\begin{matrix} π (λ | d) \propto {(λ_{10})}^{c_{101} - 1 + \sum_{k} (m - t_{k}) d_{k 1 t}} {[1 - λ_{10}]}^{c_{102} - 1 + \sum_{k} t_{k} d_{k 1 t}} \\ \times {(λ_{01})}^{c_{011} - 1 + \sum_{k} t_{k} d_{k 0 t}} {[1 - λ_{01}]}^{c_{012} - 1 + \sum_{k} (m - t_{k}) d_{k 0 t}} . \end{matrix}

(3.3)

Note that (3.3) is a product of beta distributions.

For the SMM, we have

π (β | d) \propto \prod_{k = 1}^{n} F {({x^{'}}_{r} β)}^{d_{k} 1 +} {[1 - F ({x^{'}}_{r} β)]}^{d_{k}_{0 +}} \times \prod_{i = 1}^{p} F {({\tilde{x}}^{'}_{r} β)}^{a_{1 i}} {[1 - F ({\tilde{x}}^{'}_{r} β)]}^{a_{2 i}}

(3.4)

\begin{matrix} π (λ | d) \propto \prod_{i, j} {[1 - Bin (m; λ_{ij}; 0.5 m)]}^{\sum_{k} d_{kij}} \times λ_{10}^{c_{101} - 1} {[1 - λ_{10}]}^{c_{102} - 1} \\ \times λ_{01}^{c_{011} - 1} {[1 - λ_{01}]}^{c_{012} - 1} . \end{matrix}

(3.5)

MCMC methods are necessary to conduct Bayesian analysis. We will consider the use of the Gibbs sampler algorithm (Gelfand and Smith, 1990). The procedure, described in Appendix B, was implemented in the object-oriented matrix programming Language Ox, Version 5.1 (Doornik, 2007), and the code is available from the first author upon request.

4 Simulation studies

In this section, we consider the data from two studies that were previously presented in the existing literature: O-ring data (a small sample with only one independent variable) and Trauma data (a larger sample, considering five candidate independent variables). In both cases, the authors did not discuss the possibility of misclassification. Therefore, we simulated ‘misclassified’ responses and evaluated the impact of ignoring this misclassification. Then, we carried out simulation studies to compare the performance of the repeated classifications approach with that of a single classification approach in which some prior information about the errors probabilities was available. We conducted a sensitivity analysis that accounted for some different prior distribution specifications for the misclassification errors.

4.1 Single regressor: O-ring data

Challenger, the American space shuttle, exploded in 1986, 73 seconds after its take-off. The cause of the accident was attributed to the failure of two O-rings (circular rubber rings that sealed the joints of the shuttle's solid rocket boosters). It is suspected that the low temperature at launch time (31 $^{\circ}$ F) was a major factor contributing to the failure of the O-rings. Data on field O-ring failures in the 23 pre-Challenger space shuttle launches was first analysed in Dalal et al. (1989). Bedrick et al. (1996) and Christensen's (1997) analysed the binary version of the data, which views each flight as an independent trial with $1$ , if at least one field O-ring failure occurred on the flight, and a $0$ , if an O-ring failure did not occur.

The authors proposed a logistic regression model in which the probability that any O-ring failed was modelled as a function of temperature; the CMPs were Beta(1, 0.577) and Beta(0.577, 1) for 55 $^{\circ}$ F and 75 $^{\circ}$ F, respectively. Trace plots and autocorrelations were examined to determine the burn-in iterations and assess the MCMC convergence (Appendix C). Posterior sample sizes of 1 000 were obtained after considering a burn-in period of 50 000 iterations (to ensure convergence) and a lag of 10 to reduce correlations.

Table 1 presents the posterior means, standard deviations (SD) and percentiles of the regression coefficients (5% and 95%). Additionally, we made inferences about modelled probabilities in 53 $^{\circ}$ F ( ${\tilde{θ}}_{53, 1}$ ), 67 $^{\circ}$ F ( ${\tilde{θ}}_{67, 1}$ ) and 81 $^{\circ}$ F ( ${\tilde{θ}}_{81, 1}$ ) temperatures. The posterior estimates obtained using our Ox routine are similar to those obtained by Christensen's (1997).

The analysis of the original dataset did not consider the possibility of misclassification. However, to improve the understanding of the effect of misclassification, we generated misclassified responses that were simulated under the assumption that some non-differential error could occur in the response variable, perhaps due to the technical difficulty of detecting failure or a registration mistake. Thus, from the original data, we generated a sample with $λ_{01} = λ_{10} = 0.10$ , and a logistic regression model, ignoring the presence of such errors, was fitted (naive model). The results are presented in Table 1.

Table 1:
Posterior estimates for O-ring data

Coefficients and modelled probabilities Original data Naive model

Mean SD 5% 95% Mean SD 5% 95%

Intercept 12.946 5.672 4.477 23.027 5.519 4.553 $-$ 1.732 13.228

Temperature $-$ 0.202 0.084 $-$ 0.351 $-$ 0.076 $-$ 0.095 0.067 $-$ 0.212 0.011

${\tilde{θ}}_{53, 1}$ 0.854 0.136 0.571 0.990 0.598 0.213 0.217 0.913

${\tilde{θ}}_{67, 1}$ 0.372 0.114 0.195 0.573 0.312 0.097 0.161 0.485

${\tilde{θ}}_{81, 1}$ 0.056 0.060 0.004 0.179 0.137 0.106 0.019 0.343

Coefficients and modelled probabilities	Original data	Naive model
Intercept	12.946	5.672	4.477	23.027	5.519	4.553	$-$ 1.732	13.228
Temperature	$-$ 0.202	0.084	$-$ 0.351	$-$ 0.076	$-$ 0.095	0.067	$-$ 0.212	0.011
${\tilde{θ}}_{53, 1}$	0.854	0.136	0.571	0.990	0.598	0.213	0.217	0.913
${\tilde{θ}}_{67, 1}$	0.372	0.114	0.195	0.573	0.312	0.097	0.161	0.485
${\tilde{θ}}_{81, 1}$	0.056	0.060	0.004	0.179	0.137	0.106	0.019	0.343

Notably, ignoring misclassification leads to some inference problems: the intercept and the variable temperature became non-significant (90% central posterior intervals for the coefficients covered $0$ ). Furthermore, the probability that at least one field O-ring failure occurs in fixed temperatures was very different.

Suppose that three repeated evaluations of the O-rings are available and the logistic regression model accounting for misclassification was fitted. First, we considered just a single measure (FSM $m = 1$ ), and then the frequency of success after three evaluations (FSM $m = 3$ ). A Beta(2, 10) prior distribution for the misclassification probabilities was used. The posterior means, standard deviations, and percentiles are presented in Table 2.

Table 2:

Posterior estimates for O-ring data accounting for misclassification

Coefficients and modelled probabilities	FSM $m = 1$				FSM $m = 3$
Coefficients and modelled probabilities	Mean	SD	5%	95%	Mean	SD	5%	95%
Intercept	11.044	9.040	$-$ 1.142	27.840	12.610	7.897	3.187	28.796
Temperature	$-$ 0.185	0.139	$-$ 0.452	$-$ 0.003	$-$ 0.204	0.126	$-$ 0.466	$-$ 0.057
${\tilde{θ}}_{53, 1}$	0.677	0.259	0.196	0.990	0.791	0.171	0.450	0.988
${\tilde{θ}}_{67, 1}$	0.243	0.152	0.039	0.517	0.298	0.161	0.046	0.560
${\tilde{θ}}_{81, 1}$	0.074	0.105	0.000	0.298	0.069	0.084	0.000	0.241

When considering a single evaluation of the O-rings, the intercept was not significant and the posterior estimates for modelled probabilities in 53 $^{\circ}$ F and 67 $^{\circ}$ F temperatures were greatly underestimated. Specifically, the probability that any O-ring failed is approximately 0.68 in 53 $^{\circ}$ F, whereas in reality this probability is 0.85. In practice, this is a serious problem, since obtaining good estimates of the probability that at least one field O-ring failure occurs in low temperatures is extremely important. The model that considers repeated measures (FSM $m = 3$ ) provides a much better estimate, a probability of 0.79. Moreover, both coefficients were significant, the posterior SDs were lower, and the other modelled probabilities were closer to those obtained with the original data. These facts suggest that repeated measures improve the performance of inferential procedures in misclassification problems. It is also reasonable to believe that, by increasing the number of repeated measures $m$ , better posterior estimates would be obtained. We will confirm these hypotheses through a simulation study. The SMM proposed in Section 2.2 will also be evaluated.

Two prior beta distributions for the misclassification probabilities were used (Figure 1): Beta(2, 10) (for Case 1) and Beta(89.9, 809.1) (for Case 2). The hyperparameters for these two distributions were obtained from prior specified means (0.17 and 0.10, respectively) and variances (0.01 and 0.0001, respectively). The latter distribution is more informative. For the regression coefficients, we used the aforementioned CPMs produced by Christensen's (1997) and Bedrick et al. (1996).

The details of the simulation study are described below: $•$

Step I: Generate the $m = 9$ measures of $Y_{kl}$ from $D_{k}$ using $P (Y_{kl} = j | D_{k} = i, x_{k}) = λ_{kij}$ .

•

Step II: Using Case 1’s prior distribution, obtain posterior estimates under the FSM (or the SMM) using the first result of the test obtained in Step I.

•

Step III: Given the data $(Y, x)$ obtained in Step I and using a logit link and the Case 1’s prior distribution, fit the following for $m = 3$ , 5, 7, 9:

the FSM based on the number of successes obtained in the first $m$ classifications and

the SMM based on the simple majority classification obtained in the first $m$ classifications.

•

Step IV: Repeat Steps II and III using Case 2 prior distribution.

•

Step V: Repeat Steps I through IV for a large number ( $R$ ) of times and determine the posterior estimates ${\hat{θ}}_{(l)}$ , $l = 1, \dots, R$ .

•

Step VI: Find the average of ${\hat{θ}}_{(l)}$ , $l = 1, \dots, R$ .

We used $R = 100$ replications. Although this value of R results in some non-negligible simulation error, it was not possible to conduct many more replications given the computational complexity and computing time of the MCMC method. Nevertheless, some tests with $R = 1 000$ showed no relevant differences between the estimates.

The posterior mean, standard deviation (i.e., the square root of the average of the posterior variances, denoted by SD) and coverage probabilities (CP) of 90% credible (central posterior) intervals are shown in Tables 3 and 4. For $m = 1$ , the FSM and SMM are equivalent, and their results are, therefore, the same.

Table 3:

Estimates of the mean, standard deviation (SD) and coverage probabilities (CP) of 90% credible intervals for the intercept coefficient $^{a}$ –O-ring data

7* $m$	Case 1						Case 2
	FSM			SMM			FSM			SMM
	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP
1	13.741	8.686	1.00	13.741	8.686	1.00	13.770	7.947	0.99	13.770	7.947	0.99
3	13.429	6.582	1.00	14.040	7.544	1.00	13.510	6.340	1.00	12.999	6.503	1.00
5	12.986	5.961	1.00	14.162	7.061	1.00	12.837	5.874	1.00	13.082	6.033	1.00
7	12.954	5.818	1.00	13.780	6.678	1.00	12.866	5.757	1.00	12.873	5.818	1.00
9	12.894	5.775	1.00	13.711	6.506	1.00	12.914	5.765	1.00	12.947	5.772	1.00

Notes: ^aError free estimate = 12.946.

Table 4:

Estimates of the mean, standard deviation (SD), and coverage probabilities (CP) of 90% credible intervals for Temperature $^{a}$ –O-ring data

$m$	Case 1						Case 2
	FSM			SMM			FSM			SMM
	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP
1	$-$ 0.219	0.134	1.00	$-$ 0.219	0.134	1.00	$-$ 0.218	0.122	0.99	$-$ 0.218	0.122	0.99
3	$-$ 0.210	0.098	1.00	$-$ 0.221	0.115	1.00	$-$ 0.211	0.094	1.00	$-$ 0.203	0.097	1.00
5	$-$ 0.202	0.088	1.00	$-$ 0.222	0.107	1.00	$-$ 0.200	0.087	1.00	$-$ 0.204	0.089	1.00
7	$-$ 0.202	0.086	1.00	$-$ 0.216	0.100	1.00	$-$ 0.200	0.085	1.00	$-$ 0.201	0.086	1.00
9	$-$ 0.201	0.085	1.00	$-$ 0.214	0.097	1.00	$-$ 0.201	0.085	1.00	$-$ 0.202	0.085	1.00

Notes: ^aError free estimate $= - 0.202$ .

Biased parameter estimates were obtained under the standard model ( $m = 1$ ). After increasing the number of repeated classifications, both the bias and standard deviation decreased. This finding suggests that parameter estimates and standard deviations are closer to the values obtained when there is no misclassification (Table 1).

4.2 Multiple regressors: Trauma data

We also considered data from a random sample of 300 patients admitted to the University of New Mexico Trauma Center between 1991 and 1994. This dataset was first analysed in Bedrick et al. (1996) and Christensen's (1997) studies to explore CMPs. For each patient, they obtained data describing the injury severity score (ISS), revised trauma score (RTS), age (AGE), predominant type of injury (TI) and survival. ISS is the overall index of a patient's injuries and ranges from 0 (no injuries) to 75 (a patient with serious injuries at 3 or more body areas). RTS is an index of physiological injury, constructed as a weighted average of an incoming patient's respiratory rate, systolic blood pressure and a coma scale. RTS ranges from 0 (no vital signs) to 7.84 (normal vital signs). TI is a dichotomous variable, where TI $=$ 0 represents a blunt injury (e.g., the result of a car crash) and TI $=$ 1 represents a penetrating injury (e.g., a gunshot wound).

A trauma surgeon proposed a logistic regression model to estimate the probability of a patient's death. The model includes an intercept, the predictors ISS, RTS, AGE and TI, and the interaction of AGE and TI. This model also utilized CMPs, that is, the beta prior distributions for death probabilities on six sets of conditions ${\tilde{x}}_{i} = (1, IS S_{i}, RT S_{i}, AG E_{i}, T I_{i}, AG E_{i} \times T I_{i})$ . Bedrick et al. (1996) and Christensen's (1997) studies can be referred to for further details. The Beta prior distribution parameters for the six death probabilities are presented in Table 5.

Table 5:
Prior distribution parameters for death probabilities

i ${ISS}_{i}$ ${RTS}_{i}$ ${AGE}_{i}$ ${TI}_{i}$ ${AGE}_{i}$ * ${TI}_{i}$ Beta hyperparameters

$a_{1 i}$ $a_{2 i}$

1 25 7.84 60 0 0 1.1 8.5

2 25 3.00 10 0 0 3.0 11.0

3 41 3.00 60 1 60 5.9 1.7

4 41 7.84 10 1 10 1.3 12.0

5 33 6.00 35 0 0 1.1 4.9

6 33 6.00 35 1 35 1.5 5.5

i	${ISS}_{i}$	${RTS}_{i}$	${AGE}_{i}$	${TI}_{i}$	${AGE}_{i}$ * ${TI}_{i}$	Beta hyperparameters
1	25	7.84	60	0	0	1.1	8.5
2	25	3.00	10	0	0	3.0	11.0
3	41	3.00	60	1	60	5.9	1.7
4	41	7.84	10	1	10	1.3	12.0
5	33	6.00	35	0	0	1.1	4.9
6	33	6.00	35	1	35	1.5	5.5

The obtained prior and posterior distributions for the regression coefficients are displayed in Figure 2. The posterior estimates obtained using our Ox routine (Table 6) are similar to those obtained by Christensen's (1997).

The analysis of the original dataset did not consider the possibility of misclassification and, as in the previous example, we generated contaminated ‘reported’ responses with $λ_{01} = λ_{10} = 0.10$ to illustrate the effect of misclassification. The results for the naive model are presented in Table 6.

Table 6:

Posterior estimates for Trauma data

Coefficients	Original data				Naive model
Coefficients	Mean	SD	5%	95%	Mean	SD	5%	95%
Intercept	$-$ 1.739	1.129	$-$ 3.635	0.079	$-$ 0.405	1.002	$-$ 2.068	1.239
ISS	0.065	0.021	0.030	0.100	0.018	0.016	$-$ 0.008	0.045
RTS	$-$ 0.602	0.143	$-$ 0.838	$-$ 0.374	$-$ 0.331	0.118	$-$ 0.522	$-$ 0.135
AGE	0.047	0.014	0.025	0.071	0.008	0.011	-0.009	0.025
TI	1.098	1.129	$-$ 0.749	2.948	$-$ 0.401	0.819	$-$ 1.728	0.975
AGE*TI	$-$ 0.017	0.029	$-$ 0.065	0.028	0.029	0.021	$-$ 0.005	0.064

For original data (no misclassification), ISS, RTS and AGE were considered significant variables (the 90% central posterior interval for the regression coefficients associated with these variables did not cover $0$ ). However, for contaminated data, using the naive model, only RTS is significant.

We supposed that three repeated evaluations of the patients were available and the logistic regression models accounting for misclassification were fitted. The results of the FSM with $m = 1$ and $m = 3$ are presented in Table 7. A Beta(2, 10) prior distribution for the misclassification probabilities was used.

Table 7:

Posterior estimates for Trauma data accounting for misclassification

Coefficients	FSM $m = 1$				FSM $m = 3$
Coefficients	Mean	SD	5%	95%	Mean	SD	5%	95%
Intercept	$-$ 0.744	1.912	$-$ 3.764	2.665	$-$ 1.361	1.708	$-$ 3.760	1.652
ISS	0.086	0.058	0.003	0.188	0.098	0.055	0.030	0.201
RTS	$-$ 1.139	0.534	$-$ 2.082	$-$ 0.411	$-$ 1.004	0.503	$-$ 1.994	$-$ 0.401
AGE	0.078	0.052	0.000	0.168	0.077	0.044	0.021	0.161
TI	3.868	3.219	$-$ 0.614	9.518	3.500	2.802	$-$ 0.061	9.091
AGE*TI	$-$ 0.100	0.099	$-$ 0.274	0.043	$-$ 0.098	0.085	$-$ 0.275	0.009

Considering a single measure, ISS and RTS are significant, but the variable AGE is incorrectly considered to be a non-significant variable. Fitting the FSM with $m = 3$ , significant and non-significant variables are correctly identified, again suggesting that repeated measures improve the inferential procedure.

Adjusted ORs were similar between the models considering the direction of effect, with the exceptions of TI and AGE*TI in the naive model.

A simulation study was carried out again to demonstrate the good performance of repeated measures. The same prior beta distributions for the misclassification probabilities were used: Beta(2, 10) (for Case 1) and Beta(89.9, 809.1) (for Case 2). For the regression coefficients, we used the CPMs that were described in Table 5. Posterior estimates obtained after $R = 100$ replications are shown in Tables 8 through 13.

Table 8:

Estimates of the mean, standard deviation (SD) and coverage probabilities (CP) of 90% credible intervals for the intercept coefficient $^{a}$ –Trauma data

$m$	Case 1						Case 2
	FSM			SMM			FSM			SMM
	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP
1	$-$ 1.383	1.845	1.00	$-$ 1.383	1.845	1.00	$-$ 1.589	1.659	1.00	$-$ 1.589	1.659	1.00
3	$-$ 1.686	1.219	1.00	$-$ 1.692	1.476	1.00	$-$ 1.657	1.213	1.00	$-$ 1.672	1.291	1.00
5	$-$ 1.723	1.163	1.00	$-$ 1.761	1.358	1.00	$-$ 1.739	1.157	1.00	$-$ 1.690	1.177	1.00
7	$-$ 1.747	1.148	1.00	$-$ 1.777	1.296	1.00	$-$ 1.769	1.153	1.00	$-$ 1.715	1.148	1.00
9	$-$ 1.757	1.139	1.00	$-$ 1.775	1.263	1.00	$-$ 1.762	1.143	1.00	$-$ 1.737	1.145	1.00

Notes: ^aError free estimate $= - 1.739 .$

Table 9:

Estimates of the mean, standard deviation (SD) and coverage probabilities (CP) of 90% credible intervals for ISS $^{a}$ –Trauma data

$m$	Case 1						Case 2
	FSM			SMM			FSM			SMM
	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP
1	0.100	0.056	0.99	0.100	0.056	0.99	0.090	0.048	0.97	0.090	0.048	0.97
3	0.067	0.025	1.00	0.086	0.039	1.00	0.065	0.024	1.00	0.076	0.033	0.99
5	0.065	0.022	1.00	0.081	0.032	1.00	0.065	0.022	1.00	0.068	0.025	1.00
7	0.065	0.022	1.00	0.077	0.029	1.00	0.065	0.022	1.00	0.065	0.022	1.00
9	0.065	0.021	1.00	0.074	0.027	1.00	0.065	0.021	1.00	0.065	0.022	1.00

Notes: ^aError free estimate $= 0.065 .$

Table 10:

Estimates of the mean, standard deviation (SD) and coverage probabilities (CP) of 90% credible intervals for RTS $^{a}$ –Trauma data

$m$	Case 1						Case 2
	FSM			SMM			FSM			SMM
	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP
1	$-$ 1.063	0.393	0.73	$-$ 1.063	0.393	0.73	$-$ 0.909	0.352	0.90	$-$ 0.909	0.352	0.90
3	$-$ 0.633	0.170	1.00	$-$ 0.815	0.249	0.98	$-$ 0.629	0.165	1.00	$-$ 0.716	0.215	1.00
5	$-$ 0.610	0.150	1.00	$-$ 0.756	0.214	1.00	$-$ 0.606	0.149	1.00	$-$ 0.631	0.167	1.00
7	$-$ 0.603	0.146	1.00	$-$ 0.718	0.192	1.00	$-$ 0.603	0.146	1.00	$-$ 0.609	0.150	1.00
9	$-$ 0.602	0.144	1.00	$-$ 0.692	0.179	1.00	$-$ 0.599	0.144	1.00	$-$ 0.605	0.146	1.00

Notes: ^aError free estimate $= - 0.602 .$

Table 11:

Estimates of the mean, standard deviation (SD) and coverage probabilities (CP) of 90% credible intervals for AGE $^{a}$ –Trauma data

$m$	Case 1						Case 2
	FSM			SMM			FSM			SMM
	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP
1	0.080	0.033	0.87	0.080	0.033	0.87	0.067	0.030	0.97	0.067	0.030	0.97
3	0.049	0.016	1.00	0.059	0.023	1.00	0.049	0.016	1.00	0.054	0.019	1.00
5	0.048	0.014	1.00	0.056	0.020	1.00	0.047	0.014	1.00	0.048	0.016	1.00
7	0.048	0.014	1.00	0.054	0.018	1.00	0.048	0.014	1.00	0.048	0.014	1.00
9	0.048	0.014	1.00	0.053	0.017	1.00	0.047	0.014	1.00	0.048	0.014	1.00

Notes: ^aError free estimate $= 0.047 .$

Table 12:

Estimates of the mean, standard deviation (SD) and coverage probabilities (CP) of 90% credible intervals for TI $^{a}$ –Trauma data

$m$	Case 1						Case 2
	FSM			SMM			FSM			SMM
	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP
1	3.345	2.384	0.89	3.345	2.384	0.89	2.600	2.229	0.95	2.600	2.229	0.95
3	1.243	1.269	1.00	2.062	1.668	1.00	1.262	1.246	1.00	1.680	1.473	1.00
5	1.180	1.141	1.00	1.838	1.503	1.00	1.119	1.142	1.00	1.276	1.238	1.00
7	1.135	1.121	1.00	1.651	1.402	1.00	1.130	1.124	1.00	1.144	1.149	1.00
9	1.105	1.115	1.00	1.538	1.333	1.00	1.092	1.113	1.00	1.134	1.122	1.00

Notes: ^aError free estimate $= 1.098 .$

Table 13:

Estimates of the mean, standard deviation (SD) and coverage probabilities (CP) of 90% credible intervals for AGE*TI $^{a}$ –Trauma data

$m$	Case 1						Case 2
	FSM			SMM			FSM			SMM
	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP	Mean	SD	CP
1	$-$ 0.096	0.067	0.78	$-$ 0.096	0.067	0.78	0.071	0.062	0.90	$-$ 0.071	0.062	0.90
3	$-$ 0.024	0.033	1.00	$-$ 0.052	0.046	1.00	$-$ 0.024	0.032	1.00	$-$ 0.039	0.042	1.00
5	$-$ 0.020	0.029	1.00	$-$ 0.043	0.041	1.00	$-$ 0.018	0.029	1.00	$-$ 0.024	0.033	1.00
7	$-$ 0.019	0.029	1.00	$-$ 0.036	0.038	1.00	$-$ 0.018	0.029	1.00	$-$ 0.019	0.030	1.00
9	$-$ 0.018	0.028	1.00	$-$ 0.031	0.035	1.00	$-$ 0.017	0.028	1.00	$-$ 0.018	0.029	1.00

Notes: ^aError free estimate $= - 0.017 .$

The results obtained in this example are similar to those in the previous example: when increasing the number of repeated classifications, the bias and standard error decrease. Thus, these results indicate that the proposed models, which incorporate repeated test results, markedly improve parameter estimation in terms of bias and efficiency.

5 Real Data analysis

An industry that produces lenses for glasses contacted the Department of Statistics of the Federal University of Minas Gerais to evaluate the increase in the number of defective lenses after the diversification of the feedstock source (Region 0 or 1) and the use of an alternative laboratory built for lens production (Laboratory 0 or 1). The lenses were submitted to a visual inspection in order to detect defects such as scratches, cracks and excessive thickness or thinness, among others. If a lens showed one or more defects, it was classified as ‘defective’; otherwise, it was classified as ‘perfect’. Four inspectors took part in the evaluation study, and they performed five repeated independent classifications of 400 randomly selected lenses from the production. The selection was made in equal quantities for each combination of feedstock source ( $R$ ) and laboratory ( $L$ ) ( $R 0 L 0 = 100$ ; $R 0 L 1 = 100$ ; $R 1 L 0 = 100$ and $R 1 L 1 = 100$ ). The allocation of the lenses to the inspectors was randomized. It is quite reasonable to consider that the mechanism of misclassification is not related to the covariates (i.e., it is non-differential) and error probabilities are the same for all inspectors, since they are periodically and equally trained.

A summary for the data obtained after five repeated independent classifications is described in Table 14. It was observed that, for example, among five repeated classifications, 13 L0 $R$ 0 lenses had no ‘perfect’ classification, whereas 34 L1 $R$ 1 lenses had 4 ‘perfect’ classifications.

In this example, we will restrict the analysis to logistic regression. For the regression coefficients, we used the aforementioned CMPs (Bedrick et al., 1996). Since we have two covariates, we choose three covariate configurations to induce the prior distribution for the regression coefficients. We carried out the elicitation of the prior with the kind collaboration of the process and quality control manager of the industry, who was asked to choose the three configurations and to provide values for the 1%, 50% and 99% quantiles of his prior density for the probability of a perfect lens with those covariate characteristics. This is a purposeful over-specification of the beta distributions; therefore, to calculate the prior hyperparameters, we used the two quantiles about which the expert was more confident, namely 50% and 99%, and the third one to assess the consistency of the choice. This process was iterated until a consistent set of values was found and then the hyperparameters were obtained numerically.

Table 14:
Frequency of ‘perfect’ classifications among five repeated classifications

Number of perfect classifications $L$ 0 $R$ 0 $L$ 0 $R$ 1 $L$ 1 $R$ 0 $L$ 1 $R$ 1

0 13 8 5 1

1 11 9 2 1

2 5 4 2 2

3 3 9 5 11

4 20 27 31 34

5 48 43 55 51

Number of perfect classifications	$L$ 0 $R$ 0	$L$ 0 $R$ 1	$L$ 1 $R$ 0	$L$ 1 $R$ 1
0	13	8	5	1
1	11	9	2	1
2	5	4	2	2
3	3	9	5	11
4	20	27	31	34
5	48	43	55	51

A similar procedure was used to induce the beta prior distributions for the misclassification probabilities. The complete set of elicited quantiles and obtained hyperparameters are given in Table 15.

Table 15:

Expert elicitation of prior quantiles for $θ$ and $λ$ .

	q1%	q50%	q99%	Hyperparameters
$L 0 R 0$	0.42	0.70	0.90	12.44	5.52
$L 0 R 1$	0.55	0.80	0.95	15.42	4.10
$L 1 R 0$	0.78	0.90	0.97	45.46	5.35
$λ_{10}$	0.062	0.120	0.200	14.67	105.51
$λ_{01}$	0.106	0.170	0.250	25.07	121.11

To conduct a Bayesian analysis, posterior distributions for FSM and SMM were obtained by the MCMC method. Figures 3 displays the plots of the prior and posterior for perfect probabilities for each combination of Laboratory ( $L$ ) and Source ( $R$ ). Posterior distributions are drawn as smoothed histograms.

The posterior estimates for all parameters are given in Table 16. The estimated regression coefficients associated with Laboratory and Supplier and the misclassification probabilities were quite similar under SMM and FSM, but the extra amount of information available for FSM results in narrower credibility intervals for all the parameters.

Table 16:

Posterior estimates for lens data

	SMM					FSM
	Mean	Median	SD	HPD CI 95%		Mean	Median	SD	HPD CI 95% $^{a}$
Intercept	0,827	0,823	0,21	0,432	1,244	0,832	0,83	0,196	0,463	1,23
Laboratory	1,681	1,664	0,341	1,035	2,355	1,648	1,641	0,318	1,059	2,298
Region	0,618	0,615	0,292	0,061	1,221	0,593	0,591	0,281	0,052	1,146
$λ_{01}$	0,172	0,171	0,031	0,113	0,233	0,160	0,160	0,019	0,123	0,199
$λ_{10}$	0,122	0,120	0,030	0,068	0,183	0,105	0,105	0,008	0,090	0,120

Notes: ^aHighest posterior density 95% credible interval.

In practical terms, using the FSM, the feedstock originating in Region 1 and Laboratory 1 ( $L 1 R 1$ ) possesses the highest probability of producing a perfect lens (Figure 4). Therefore, procedures practiced in Laboratory 1 were standardized and utilized in Laboratory 0. As for the feedstock from Region 0, its supervision will be more rigorous, in order for its quality parameters to match those of Region 1. The conclusion is the same when using the SMM.

6 Conclusions

In the context of binomial regression with misclassification, we proposed to perform multiple assessments of the sample units. This additional information was incorporated using two different models. The first model uses the total number of successes obtained (FSM), while the second model uses simple majority classification (SMM). The new methodology we introduced is quite general and can be used to address a variety of practical problems.

The FSM results are closer to those obtained in a no-misclassification case with a lower standard deviation than the results of the SMM. However, both methods lead to similar conclusions. In this sense, the SMM may be a reasonable recommendation for cases in which we only have the final classification of the individuals.

A sensitivity analysis that took into account different prior specifications was performed. We demonstrated that the repetitive test procedure is a good alternative if there is some (or even little) information about classification errors, as the proposed models did not appear to be sensitive to different prior specifications.

We provided a discussion about the local identifiability of the proposed models. As far as we know, proving global identifiability under binary regression model subject to non-differential misclassification of the response variable is an open problem that would befit from future studies. Future studies should also perform Bayesian model selection (as in Gerlach and Stamey, 2007) and extend the proposed model to accommodate random effects (or frailties).

Appendix

Appendix A: Identifiability issue

Usually, consistent estimators of identifiable model parameters can be obtained in likelihood or Bayesian approaches, in the sense that estimates will converge to true parameter values as the sample size increases. However, if the true distribution of observable data in fact corresponds to more than one set of parameter values (non-identifiable situations), then no amount of data can lead to the true parameter values (Gustafson, 2004). Therefore, although Bayesian analysis can be performed under a non-identified model if proper prior distributions are used, the arising posterior distribution may not have certain appealing theoretical properties. For example, the joint posterior distribution of the parameters may not converge asymptotically (Dendukuri et al., 2010), the posterior distribution may not concentrate the mass around the true value of a parameter when the sample size increases, and there tend to be strong correlations between parameters in the posterior distribution, resulting in poor mixing of Markov chains used to explore the posterior, and thus rendering estimators that may fail to converge within reasonable computing times (Swartz et al., 2004; Gelfand and Sahu, 1999). Furthermore, even large sample sizes may be unable to overcome prior information (Johnson et al., 2001), or the posterior distributions arising from different priors will not match as the sample size increases.

In this sense, even when adopting a Bayesian approach, it is important to evaluate the situations that guarantee the identifiability of parameters in the proposed models, in order to minimize practical problems.

First, consider FSM (or SMM) with $m = 1$ . Thus,

P_{ϕ} (Y_{k} = 1) = \sum_{i} θ_{ki} λ_{i 1} = λ_{01} + (1 - λ_{01} - λ_{10}) F (x_{k}^{,} β),

(A.1)

where $ϕ = (λ_{01}, λ_{10}, β)$ , $Y_{k}$ , $θ_{ki}$ , $λ_{10}$ and $λ_{01}$ were previously defined. As mentioned in Section 2.2, in this case SMM (and FSM) are equivalent to the model evaluated by Hausman et al. (1998), Paulino et al. (2003) and McInturff et al. (2004).

Following Newey and McFadden (1994), Hausman et al. (1998) state, in Theorem 1 (p. 243) that, under the probit link function, $E [x^{'} x] < \infty$ non-singular, and the $λ_{01} + λ_{10} < 1$ assumption, the model parameters in (7.1) are identified by non-linear least squares (NLS) and maximum likelihood estimator (MLE). The existence and non-singularity of the second moment of the predictors are assumptions adopted for the traditional binary regression model (see, e.g., Example 1.2 of Newey and McFadden, 1994; Manski, 1988). In fact, under its assumptions, Theorem 1 states the local identifiability, that is, there exists some open neighbourhood $τ$ of $ϕ$ such that

P_{ϕ} (Y_{k} = 1) = P_{ϕ_{0}} (Y_{k} = 1) Ø ϕ = ϕ_{0}, \forall ϕ \in τ \subset Φ,

where $Φ$ denotes the set of all possible $ϕ$ restricted to $λ_{01} + λ_{10} < 1$ . In order to show the necessity of the $λ_{01} + λ_{10} < 1$ , Hausman et al. (1998), consider a symmetric F(where F(v)=1−F(−v)), such as the normal or logistic distribution. Setting ${\tilde{λ}}_{10} = 1 - λ_{01}, {\tilde{λ}}_{01} = 1 - λ_{10}$ and $\tilde{β} = - β$ , (7.1) is

P (Y_{k} = 1) = {\tilde{λ}}_{01} + (1 - {\tilde{λ}}_{01} - {\tilde{λ}}_{10}) F (x_{k}^{,} \tilde{β}) = λ_{01} + (1 - λ_{01} - λ_{10}) F (x_{k}^{,} β) .

Thus, estimators based on (7.1), such as NLS and MLE, cannot distinguish between the parameter values $(λ_{01}, λ_{10}, β)$ and $(1 - λ_{10}, 1 - λ_{01}, - β)$ . The assumption that $λ_{01} + λ_{10} < 1$ rules out such situations, and is usually satisfied in practice, because misclassification rates are expected to be small (generally less than 0.5). Otherwise, misclassification will not occur purely by chance.

Note that restriction $λ_{01} + λ_{10} < 1$ also implies that $λ_{01} + (1 - λ_{01} - λ_{10}) F (v)$ is strictly increasing in $v$ if $F$ is strictly increasing, like the normal or logistic distributions (Hausman et al., 1998).

FSM local identifiability

If $m > 1$ , then in addition to the assumptions used for the traditional binary regression, the condition that $λ_{01} + λ_{10} < 1$ is also required for FSM, because

\begin{matrix} P (T_{k} = t) = {\tilde{θ}}_{k} (\begin{matrix} m \\ t \end{matrix}) (1 - {\tilde{λ}}_{10})^{t} {\tilde{λ}}_{10}^{m - t} + (1 - {\tilde{θ}}_{k}) (\begin{matrix} m \\ t \end{matrix}) {\tilde{λ}}_{01}^{t} (1 - {\tilde{λ}}_{01})^{m - t} \\ = (1 - θ_{k}) (\begin{matrix} m \\ t \end{matrix}) λ_{01}^{t} (1 - λ_{01})^{m - t} + θ_{k} (\begin{matrix} m \\ t \end{matrix}) (1 - λ_{10})^{t} λ_{10}^{m - t}, \end{matrix}

(A.2)

where ${\tilde{θ}}_{κ} = F ({x^{'}}_{k} β)$ . For a better numerical understanding of the necessity of this restriction, Table A1 presents the probability distribution under FSM with $m = 3$ , $λ_{01} = 0.1$ , $λ_{10} = 0.2$ , $β = (0.5, 0.07, 0.08)$ , $x = (1, 1, 2)$ and the logit link function. Note that the same $P (T = t)$ is obtained for all $t$ , if $λ_{01} = 0.8$ , $λ_{10} = 0.9$ , $β = (- 0.5, - 0.07, - 0.08)$ .

Table A1:

Probability distribution under FSM in a non-identifiability case

1* $λ_{01}$	ensuremath \lambda_10	ensuremath \beta_0	ensuremath \beta_1	ensuremath \beta_2	ensuremath t	$P (T = t)$
0.1	0.2	0.5	0.07	0.08	0	0.2424
0.8	0.9	$-$ 0.5	$-$ 0.07	$-$ 0.08	0	0.2424
0.1	0.2	0.5	0.07	0.08	1	0.1438
0.8	0.9	$-$ 0.5	$-$ 0.07	$-$ 0.08	1	0.1438
0.1	0.2	0.5	0.07	0.08	2	0.2679
0.8	0.9	$-$ 0.5	$-$ 0.07	$-$ 0.08	2	0.2679
0.1	0.2	0.5	0.07	0.08	3	0.3458
0.8	0.9	$-$ 0.5	$-$ 0.07	$-$ 0.08	3	0.3458

It is also possible to prove FSM local identifiability through the results obtained by Teicher (1963) and Blischke (1962). To do so, consider the simple case with no covariates:

P (T_{k} = t) = θ (\begin{matrix} m \\ t \end{matrix}) {(1 - λ_{10})}^{t} {λ_{10}}^{m - t} + (1 - θ) (\begin{matrix} m \\ t \end{matrix}) {λ_{01}}^{t} {(1 - λ_{01})}^{m - t},

(A.3)

where $T$ , $m$ , $θ$ , $λ_{10}$ , and $λ_{01}$ were defined in Section 2. Because (7.2) is a (finite) mixture of two binomials, its identifiability is well established, and has been proved in the literature (Teicher, 1963; Blischke, 1962) under the necessary and sufficient conditions that $m \geq 3$ and $λ_{10} + λ_{01} < 1$ . This coincides with García-Zattera et al. (2010) and Garcia-Zattera et al. (2012) quotes, that at least three measurements are needed for the model's parameters to be identifiable (in Markov models) in the context of longitudinal data.

If there are $p$ covariates $X_{1}, . . ., X_{p}$ , then the number of parameters to be estimated increases ( $p + 3$ parameters: $λ_{10}$ , $λ_{01}$ , $β_{0}, β_{1}, . . ., β_{p}$ ). In this sense, let $θ_{κ} = F ({x^{'}}_{k} β)$ be the marginal success probability distribution of the $k$ th subject. It follows that if at least three classifications are performed ( $m \geq 3$ ) and $λ_{10} + λ_{01} < 1$ , then the local identifiability proved by Teicher (1963) and Blischke (1962) for the univariate case implies that the marginal parameters $(θ_{k}, λ_{01}, λ_{10})$ are locally identified. Therefore, if the rank of the design matrix $X$ is greater than or equal to $p$ , then the continuity of the logit function implies that the vector of regression coefficients $β$ is also locally identified, because a unique solution for $β$ is obtained through the system of equations defined by

\log i t (θ_{κ}) = {x^{'}}_{k} β, k = 1, ..., n .

Note that in this step the parameters of interest $β$ are expressed as functions of locally identified parameters $θ_{k}$ .

This completes the proof that $(β, λ_{01}, λ_{11})$ are locally identifiable under FSM.

SMM local identifiability

For SMM with $m \geq 1$ , the assumptions used for the traditional binary choice model are required as well as $(λ_{01} + λ_{10} < 1)$ , because $(λ_{01}, λ_{10}, β)$ and $(1 - λ_{10}, 1 - λ_{01}, - β)$ generate the same probability distribution:

\begin{matrix} P (S_{k} = 1) = {\tilde{θ}}_{k} [1 - \sum_{t = 0}^{0.5 m} (\begin{matrix} m \\ t \end{matrix}) {(1 - {\tilde{λ}}_{10})}^{t} {\tilde{λ}}_{10}^{m - t}] + (1 - {\tilde{θ}}_{k}) [1 - \sum_{t = 0}^{0.5 m} (\begin{matrix} m \\ t \end{matrix}) {\tilde{λ}}_{01}^{t} {(1 - {\tilde{λ}}_{01})}^{m - t}] \\ = (1 - θ_{k}) [1 - \sum_{t = 0}^{0.5 m} (\begin{matrix} m \\ t \end{matrix}) λ_{01}^{t} {(1 - λ_{01})}^{m - t}] + θ_{k} [1 - \sum_{t = 0}^{0.5 m} (\begin{matrix} m \\ t \end{matrix}) {(1 - λ_{10})}^{t} λ_{10}^{m - t}] . \end{matrix}

This fact is illustrated in Table A2, with $m = 3$ , $λ_{01} = 0.1$ , $λ_{10} = 0.2$ , $β = (0.5, 0.07, 0.08)$ , $x = (1, 1, 2)$ and the logit link function. Note that $P (S = s)$ is the same for all $s$ if $λ_{01} = 0.8$ , $λ_{10} = 0.9$ , $β = (- 0.5, - 0.07, - 0.08)$ .

Table A2:

Probability distribution under SMM in a non-identifiability case

$λ_{01}$	ensuremath \lambda_10	ensuremath \beta_0	ensuremath \beta_1	ensuremath \beta_2	ensuremath s	$P (S = s)$
0.1	0.2	0.5	0.07	0.08	0	0.3863
0.8	0.9	$-$ 0.5	$-$ 0.07	$-$ 0.08	0	0.3863
0.1	0.2	0.5	0.07	0.08	1	0.6137
0.8	0.9	$-$ 0.5	$-$ 0.07	$-$ 0.08	1	0.6137

SMM local identifiability can also be proved directly using Hausman's probability function (Hausman et al., 1998), as described in the following. Suppose that each sample subject is evaluated $m$ independent times, where $m$ is odd to avoid ties in determining the most frequent test result. Therefore, for $m \geq 3$ , it is reasonable to imagine that the parameters are also identifiable, because a greater amount of information is available. To prove this, let $λ_{i 1}^{*} = 1 - Bin (m, λ_{i 1}; 0.5 m)$ , $i = 0, 1$ . Therefore, P(S_k=1) (defined in Section 2.2) can be written as

P (S_{k} = 1) = \sum_{i} θ_{ki} λ_{i 1}^{*} .

Following the results obtained for (A.1) by Hausman et al. (1998), $(β, λ_{01}^{*}, λ_{11}^{*})$ will be locally identifiable if the assumptions used for the traditional binary regression model hold and $λ_{01}^{*} + λ_{10}^{*} < 1$ . The second condition holds, if $λ_{01} + λ_{10} < 1$ . Therefore, $λ_{i 1}$ , $i = 0, 1$ , are locally identified, because they can be expressed as one-to-one functions of the identified parameters $λ_{i 1}^{*}$ .

This completes the proof that $(β, λ_{01}, λ_{11})$ are locally identifiable under SSM.

8 Appendix: Implementation of the Gibbs sampler

After choosing adequate initial values for $β^{0}$ and $λ^{0}$ , the procedure is repeated $q$ times. It can be summarized in three steps ( $r = 1, \dots, q$ ):

Sample $d^{(r)}$

For FSM, given β^(r−1),λ^(r−1) and t, sample d^(r) in:

d_{k 1 t}^{(r)} | β^{(r - 1)}, λ^{(r - 1)}, t \sim Bernoulli (\frac{{(λ_{k 11}^{(r - 1)})}^{t} {(1 - λ_{k 11}^{(r - 1)})}^{m - t} θ_{k 1}^{(r - 1)}}{\sum_{j = 0}^{1} {(λ_{kj 1}^{(r - 1)})}^{t} {(1 - λ_{kj 1}^{(r - 1)})}^{m - t} θ_{kj}^{(r - 1)}}),

(B.1)

where $θ_{k (j)}^{(r -1)} = F ({x^{'}}_{k} β_{j}^{^{(r -1)}})$ and $θ_{k 0}^{(r - 1)} = 1 - θ_{k 1}^{(r - 1)}$ .

Do $d_{k 0 t}^{(r)} = 1 - d_{k 1 t}^{(r)}$ .

For SMM, given $β^{(r - 1)}, λ^{(r - 1)}$ and $s$ , sample $d^{(r)}$ in: $d_{k 01}^{(r)} | β^{(r - 1)}, λ^{(r - 1)}, s \sim Binomial (s_{k}, \frac{θ_{k 0}^{(r - 1)} [1 - Bin (m; λ_{k 01}^{(r - 1)}; 0.5 m)]}{\sum_{i} θ_{ki}^{(r - 1)} [1 - Bin (m; λ_{ki 1}^{(r - 1)}; 0.5 m)]})$

$d_{k 10}^{(r)} | β^{(r - 1)}, λ^{(r - 1)}, s \sim Binomial (1 - s_{k}, \frac{θ_{k 1}^{(r - 1)} [1 - Bin (m; λ_{k 10}^{(r - 1)}; 0.5 m)]}{\sum_{i} θ_{ki}^{(r - 1)} [1 - Bin (m; λ_{ki 0}^{(r - 1)}; 0.5 m)]})$

Do $d_{k 01}^{(r)} = 1 - d_{k 11}^{(r)}$ and $d_{k 00}^{(r)} = 1 - d_{k 10}^{(r)}$ .

Sample $β^{(r)} = [β_{0}^{(r)}, . . ., β_{p}^{(r)}]$ :

For FSM, given $d^{(r)}$ , sample $β_{j}^{(r)}$ , $j = 0, . . ., p$ in $π (β_{j}^{(r)} | d^{(r)}) \propto \prod_{k = 1}^{n} {(θ_{k (j)}^{(r - 1)})}^{d_{k 1 t}} {[1 - θ_{k (j)}^{(r - 1)}]}^{d_{k 0 t}} \times \prod_{i = 1}^{p} F {({\tilde{θ}}_{i (j)}^{(r - 1)})}^{a_{1 i}} {[1 - {\tilde{θ}}_{i (j)}^{(r - 1)}]}^{a_{2 i}},$

where $θ_{k (j)}^{(r -1)} = F ({x^{'}}_{k} β_{(j)}^{^{(r -1)}})$ , ${\bar{θ}}_{k (j)}^{(r -1)} = F ({\tilde{x^{'}}}_{k} β_{(j)}^{^{(r -1)}})$ and $β_{(j)}^{(r - 1)} = [β_{0}^{(r)}, . . .,$ $β_{j - 1}^{(r)}, β_{j}^{(r - 1)}, . . ., β_{p}^{(r - 1)}]$ .

For SMM, given $d^{(r)}$ , sample $β_{j}^{(r)}$ , $j = 0, . . ., p$ in

π (β_{j}^{(r)} | d^{(r)}) \propto \prod_{k = 1}^{n} {(θ_{k (j)}^{(r - 1)})}^{d_{k 1 +}} {[1 - θ_{k (j)}^{(r - 1)}]}^{d_{k 0 +}} \times \prod_{i = 1}^{p} {({\tilde{θ}}_{i (j)}^{(r - 1)})}^{a_{1 i}} {[1 - {\tilde{θ}}_{i (j)}^{(r - 1)}]}^{a_{2 i}} .

(B.2)

Sample $λ^{(r)} = [λ_{00}^{(r)}, λ_{01}^{(r)}, λ_{10}^{(r)}, λ_{11}^{(r)}]$

For FSM, given $d^{(r)}$ , sample $λ_{10}^{(r)}$ in

$π (λ_{10}^{(r)} | d^{(r)}) \sim Beta (c_{101} + \sum_{k} (m - t_{k}) d_{k 1 t}, c_{102} + \sum_{k} t_{k} d_{k 1 t})$ ,

and sample $λ_{01}^{(r)}$ in $π (λ_{01}^{(r)} | d^{(r)}) \sim Beta (c_{011} + \sum_{k} t_{k} d_{k 0 t}, c_{012} + \sum_{k} (m - t_{k}) d_{k 0 t}) .$

Do $λ_{11}^{(r)} = 1 - λ_{10}^{(r)}$ and $λ_{00}^{(r)} = 1 - λ_{01}^{(r)}$ .

For SMM, given $d^{(r)}$ , sample $λ_{01}^{(r)}$ in

\begin{matrix} π (λ_{01}^{(r)} | d^{(r)}) = \prod_{j} {[1 - Bin (m; λ_{0 j}^{(r - 1)}; 0.5 m)]}^{\sum_{k} d_{k 0 j}} \\ \times {[λ_{01}^{(r - 1)}]}^{b_{011} - 1} {[1 - λ_{01}^{(r - 1)}]}^{b_{012} - 1}, \end{matrix}

(B.3)

and sample $λ_{10}^{(r)}$ in

\begin{matrix} π (λ_{10}^{(r)} | d^{(r)}) = \prod_{j} {[1 - Bin (m; λ_{1 j}^{(r - 1)}; 0.5 m)]}^{\sum_{k} d_{k 1 j}} \\ \times {[λ_{10}^{(r - 1)}]}^{b_{101} - 1} {[1 - λ_{10}^{(r - 1)}]}^{b_{102} - 1} . \end{matrix}

(B.4)

Then, do $λ_{11}^{(r)} = 1 - λ_{10}^{(r)}$ and $λ_{00}^{(r)} = 1 - λ_{01}^{(r)}$ .

The Adaptive Rejection Metropolis Sampling (ARMS) algorithm (Gilks et al., 1995) can be used to sample from (B.1), (B.2), (B.3) and (B.4).

Appendix C: Convergence issue in FSM and SMM

A simple way to evaluate the convergence of a chain is to monitor the ergodic means. Convergence occurs when the ergodic means of chains that started from different initial values converge to the same values. Therefore, we use a single chain to produce inferences. Monitoring autocorrelation is also very useful: low values indicate fast convergence and high values indicate slow convergence.

Figure C1 and C2 show the monitoring charts of the ergodic means and autocorrelation of SSM parameters for a sample generated from the O-ring and Trauma simulation studies for one generated sample of O-ring and Trauma simulation study, respectively.

This analysis was performed for many generated samples and scenarios using the R package ‘MCMCpack’. Therefore, a burn-in period of 50 000 and a lag of 10 iterations were used. Figure C3 shows convergence results of FSM for the Lens data, for which we used a burn-in period of 50 000 and a lag of 15 iterations.

Acknowledgments

This work was supported by CAPES, CNPq and Pró-Reitoria de Pesquisa–UFMG [grant number 04/2016]. We would like to thank the editor, an associate editor and the two referees for their detailed and insightful comments, which led to a much improved manuscript.

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Repeated responses in misclassification binary regression: A Bayesian approach

Abstract

Keywords

1 Introduction

2 Statistical methods

2.1 Frequency of success model (FSM)

4.1 Single regressor: O-ring data

Table 5: Prior distribution parameters for death probabilities i ISS i RTS i AGE i TI i AGE i * TI i Beta hyperparameters a 1 i a 2 i 1 25 7.84 60 0 0 1.1 8.5 2 25 3.00 10 0 0 3.0 11.0 3 41 3.00 60 1 60 5.9 1.7 4 41 7.84 10 1 10 1.3 12.0 5 33 6.00 35 0 0 1.1 4.9 6 33 6.00 35 1 35 1.5 5.5

Posterior estimates for Trauma data

Posterior estimates for Trauma data accounting for misclassification

Table 14: Frequency of ‘perfect’ classifications among five repeated classifications Number of perfect classifications L 0 R 0 L 0 R 1 L 1 R 0 L 1 R 1 0 13 8 5 1 1 11 9 2 1 2 5 4 2 2 3 3 9 5 11 4 20 27 31 34 5 48 43 55 51

Expert elicitation of prior quantiles for θ and λ .

Posterior estimates for lens data

Appendix

FSM local identifiability

Probability distribution under FSM in a non-identifiability case

SMM local identifiability

Probability distribution under SMM in a non-identifiability case

Appendix C: Convergence issue in FSM and SMM

Acknowledgments

References

Table 14:
Frequency of ‘perfect’ classifications among five repeated classifications

Number of perfect classifications $L$ 0 $R$ 0 $L$ 0 $R$ 1 $L$ 1 $R$ 0 $L$ 1 $R$ 1

0 13 8 5 1

1 11 9 2 1

2 5 4 2 2

3 3 9 5 11

4 20 27 31 34

5 48 43 55 51

Expert elicitation of prior quantiles for $θ$ and $λ$ .