Bayesian estimation for an item response tree model for nonresponse modeling

Abstract

Nonresponse data are common in achievement tests or questionnaires. Chang et al. (Br J Math Stat Psychol 74:487–512, 2021) proposed an Item Response tree model, namely TR4, for modeling some potential mechanisms underlying nonresponses so that the estimates of parameters of interest would not be biased due to missing not at random (Rubin in Biometrika 63:581–592, 1976). TR4 has two notable degenerate cases, both with insightful practical meanings. When TR4 is fitted to data originated from some degenerate cases, there exist model identifiability issues so that the existing frequentist inference for the TR4 model is not suitable. In the current study, we propose a Bayesian estimation procedure that incorporates the Markov chain Monte Carlo technique for estimating the TR4 model. We conducted simulation studies to demonstrate the effectiveness of the Bayesian estimation procedure in solving the model unidentifiability issue. In addition, the TR4 model is further extended in the present study to effectively accommodate the complexity underlying some real data. The advantage of the extended models over TR4 is demonstrated in the real data analysis where we apply our method to the data of a geography test for college admission in Taiwan.

Appendix: Details of the Markov chain Monte Carlo procedure

In this appendix, we present details regarding the MCMC procedure for estimating the TR4 model. The estimation procedures for the TR4-I and TR4-SI models in Sect. 4 follow directly.

At each iteration, we update three categories of parameters or variables sequentially: (i) $V_{pj}$; (ii) parameters in $\varvec{\zeta }_1$; (iii) hyperparameters in $\varvec{\xi }_1$. In each full conditional, “$|\cdots$” denotes conditioning on all other parameters at their current values and on observed data. At the $(t+1)$-th iteration, we update parameters and missing variables as follows.

(i) Update $V_{pj}$:

The partially latent binary variable $V_{pj}$ has a full conditional

$$\begin{aligned}&p(v_{pj}|\cdots ) \;\; \nonumber \\&\quad \displaystyle \propto \left[ \left\{ \pi _{pj}^{y_{pj} \cdot d_{pj}} \left[ (1-\pi _{pj}) \cdot t_0 \right] ^ {(1-y_{pj}) \cdot d_{pj}} \left[ (1-\pi _{pj})(1-t_0) \right] ^ {1-d_{pj}} \right\} \cdot \phi _{pj} \right] ^ {v_{pj}} \nonumber \\&\quad \cdot \displaystyle \left[ (1-\phi _{pj}) \cdot I\{D_{pj}=0\} \right] ^ {1-v_{pj}}. \end{aligned}$$

(13)

According to (13), $V_{pj}$ must be 1 when $d_{pj}=1$. Therefore, we do not need to update $V_{pj}$ whenever $d_{pj}=1$. When $d_{pj}=0$, the full conditional $V_{pj}|\cdots$ is a Bernoulli random variable with success probability $q_1/(q_0+q_1)$, where

$$\begin{aligned} q_1= & {} \left\{ \pi _{pj}^{y_{pj} \cdot d_{pj}} \left[ (1-\pi _{pj}) \cdot t_0 \right] ^ {(1-y_{pj}) \cdot d_{pj}} \left[ (1-\pi _{pj})(1-t_0) \right] ^ {1-d_{pj}} \right\} \cdot \phi _{pj} \;\; , \\ q_0= & {} 1 - \phi _{pj}. \end{aligned}$$

(ii) Update parameters in $\varvec{\zeta }_1$:

For the full conditional of every parameter in $\varvec{\zeta }_1$, the posterior distribution is not a standard distribution. We develop a Metropolis–Hastings algorithm for each parameter in $\varvec{\zeta }_1$. We first introduce the sampling scheme for $\rho$.

The full conditional for $\rho \in [-1,1]$ is

$$\begin{aligned}&f( \rho | \cdots ) \propto \; e^{ \frac{ -\left( \log { \frac{1 + \rho }{1 - \rho } } - \mu _\eta \right) ^ 2 }{2 \sigma _\eta ^ 2}} \cdot \\&\frac{2}{ (1 + \rho )(1 - \rho ) } \cdot \text{ I } \left\{ -1 \le \rho \le 1 \right\} \cdot \prod _{p=1}^{N} \left( \frac{1}{ \sqrt{1 - \rho ^ 2} } \, e^{ \frac{ -(\theta _p^2 - 2 \rho \theta _p \tau _p + \tau _p^2) }{2 (1 - \rho ^2) } } \right) . \end{aligned}$$

For a normal symmetric random walk, we first transform the $\rho$ to the real line scale, called $\eta$, which is the same as (7). A proposal distribution for the Metropolis–Hastings algorithm is imposed on the real line scale $\eta$: we generate $\eta ^*$ from the proposal $N(\eta ^{(t)}, c_\eta )$, where $\eta ^{(t)}=\log ((1+\rho ^{(t)})/(1-\rho ^{(t)}))$ and $c_\eta$ is a constant to be determined empirically so that the acceptance rate in the following acceptance–rejection rule is about $30\%{\sim }40\%$. Subsequently, $\rho ^*$ can be obtained through $\rho ^* = \frac{2 \exp ( \eta ^*) }{1 + \exp ( \eta ^*)}-1$. The following acceptance–rejection rule is adopted:

$$\begin{aligned} \rho ^{(t+1)} = \left\{ \begin{array}{ll} \rho ^* &{} \text{ w.p. } \;\; r_\rho \equiv \text{ min }\left\{ \frac{ f\left( \rho ^*|\cdots \right) \cdot (1+\rho ^*)(1-\rho ^*) }{ f\left( \rho ^{(t)}|\cdots \right) \cdot (1+\rho ^{(t)})(1-\rho ^{(t)}) } \; , 1 \right\} \;, \\ \rho ^{(t)} &{} \text{ w.p. } \;\; 1-r_\rho \;. \end{array} \right. \end{aligned}$$

According to the Metropolis–Hastings algorithm, $\rho ^{(t+1)}$ follows $f( \rho | \cdots )$.

The Metropolis–Hastings algorithms for other parameters in $\varvec{\zeta }_1$ are similar to the above procedure. Basically, for a parameter whose range is $\varvec{\mathcal {R}}$, we directly adopt a normal symmetric random walk proposal. For a parameter whose range is not the whole real line, we transform the parameter to a new scale on $\varvec{\mathcal {R}}$ and then apply a normal random walk to the new scale. The transformations and the ratios in the acceptance–rejection rules for Metropolis–Hastings algorithms of other parameters are summarized in Table 4. The Metropolis–Hastings algorithms for $\tau _p$, $b_j$, and $\gamma _{0,j}$ are the same as that for $\theta _p$.

Table 4 The transformations and the ratios in the acceptance–rejection rules for Metropolis–Hastings algorithms of parameters $t_0$, $a_j$, $\gamma _1$, and $\theta _p$

(iii) Update hyperparameters in $\varvec{\xi }_1$:

The full conditional of every parameter in $\varvec{\xi }_1$ is a commonly used distribution and can be sampled directly by using available functions in computing software such as R (R Core Team 2020). The distributions of such full conditionals are as follows:

$$\begin{aligned} \mu _b^{(t+1)} | \cdots&\sim \mathrm {N} \left( \frac{ \sigma _1^2 ( \sum _{j=1}^{J} b_j ) + \sigma _b^2 \, \mu _1 }{ \sigma _b^2 + J \cdot \sigma _1^2 } , \;\; \frac{ \sigma _b^2 \, \sigma _1^2 }{ \sigma _b^2 + J \cdot \sigma _1^2 } \right) , \\ \mu _{\eta }^{(t+1)} | \cdots&\sim \mathrm {N} \left( \frac{ \sigma _2^2 \cdot \text{ log }\left( \frac{1 + \rho }{1 - \rho } \right) + \sigma _{\eta }^2 \, \mu _2 }{ \sigma _{\eta }^2 + \sigma _2^2 } , \;\; \frac{ \sigma _{\eta }^2 \, \sigma _2^2 }{ \sigma _{\eta }^2 + \sigma _2^2 } \right) ,\\ \mu _\gamma ^{(t+1)} | \cdots&\sim \mathrm {N} \left( \frac{ \sigma _3^2 ( \sum _{j=1}^{J} \gamma _{0,j} ) + \sigma _\gamma ^2 \, \mu _3 }{ \sigma _\gamma ^2 + J \cdot \sigma _3^2 } , \;\; \frac{ \sigma _\gamma ^2 \, \sigma _3^2 }{ \sigma _\gamma ^2 + J \cdot \sigma _3^2 } \right) , \\ \mu _{\kappa }^{(t+1)} | \cdots&\sim \mathrm {N} \left( \frac{ \sigma _4^2 \cdot \log ( \frac{ t_0 }{1 - t_0} ) + \sigma _{\kappa }^2 \, \mu _4 }{ \sigma _{\kappa }^2 + \sigma _4^2 } , \;\; \frac{ \sigma _{\kappa }^2 \, \sigma _4^2 }{ \sigma _{\kappa }^2 + \sigma _4^2 } \right) , \\ { \sigma _b^2 } ^ {(t+1)} | \cdots&\sim \mathrm {IG} \, \left( \alpha _1 + \frac{J}{2} , \; \beta _1 + \frac{1}{2} \sum _{j=1}^{J} \left( {b_j} - { \mu _b} \right) ^ 2 \right) , \\ { \sigma _{\eta }^2 } ^ {(t+1)} | \cdots&\sim \mathrm {IG} \, \left( \alpha _4 + \frac{1}{2} , \; \beta _4 + \frac{1}{2} \left[ \log \left( \frac{1 + \rho }{1 - \rho } \right) - \mu _\eta \right] ^ 2 \right) , \\ { \sigma _{\gamma }^2 } ^ {(t+1)} | \cdots&\sim \mathrm {IG} \, \left( \alpha _5 + \frac{J}{2} , \; \beta _5 + \frac{1}{2} \sum _{j=1}^{J} ( \gamma _{0,j} - \mu _\gamma ) ^ 2 \right) , \\ { \sigma _\kappa ^2 } ^ {(t+1)} | \cdots&\sim \mathrm {IG} \, \left( \alpha _7 + \frac{1}{2} , \; \beta _7 + \frac{1}{2} \left[ \log \left( \frac{ t_0 }{1 - t_0 } \right) - \mu _\kappa \right] ^ 2 \right) , \\ { \beta _a } ^ {(t+1)} | \cdots&\sim \mathrm {IG} \, \left( \alpha _2 + 2J , \; \beta _2 + \sum _{j = 1}^{J} a_j \right) , \\ { \beta _\gamma } ^ {(t+1)} | \cdots&\sim \mathrm {IG} \, \left( \alpha _6 + 1 , \; \beta _6 + \gamma _1 \right) . \end{aligned}$$