Efficient regression analyses with zero-augmented models based on ranking

Abstract

Several zero-augmented models exist for estimation involving outcomes with large numbers of zero. Two of such models for handling count endpoints are zero-inflated and hurdle regression models. In this article, we apply the extreme ranked set sampling (ERSS) scheme in estimation using zero-inflated and hurdle regression models. We provide theoretical derivations showing superiority of ERSS compared to simple random sampling (SRS) using these zero-augmented models. A simulation study is also conducted to compare the efficiency of ERSS to SRS and lastly, we illustrate applications with real data sets.

Appendices

Appendix A

Assume Y = $(y_{1},..., y_{N})'$ as the vector of responses, the probability distribution function of a zero-inflated Poisson may be written as:

$$\begin{aligned} f(y; \lambda , p)= {\left\{ \begin{array}{ll} p_i + (1-p_i) e^{-\lambda _i} &{}\text {if }y_i=0\\ (1-p_i) \frac{e^{-\lambda _i}\lambda ^y_i}{y_i!}&{}\text {if }y_i=1,2,3...,\\ \end{array}\right. } \end{aligned}$$

the parameters ${\varvec{p}} = (p_1,..., p_N)$ and $\varvec{\lambda } = (\lambda _i,..., \lambda _N)$ are modeled via canonical link GLMs as $logit({\varvec{p}}) = {\varvec{G}}\gamma$ and $log(\varvec{\lambda }) = {\varvec{B}}\beta$, where ${\varvec{G}}$ and ${\varvec{B}}$ are design matrices.

As described in Lambert (1992), the ZIP model can be fit using maximum likelihood via the EM algorithm. The log likelihood for regression parameters $\gamma$ and $\beta$ based on all of the data is given by

$$\begin{aligned} lnL(\gamma , \beta ; {\varvec{y}})&= \sum \limits _{i=1}^{n} \Bigg \{u_i ln\bigg [e^{\varvec{G_i}\gamma } + exp(-e^{\varvec{B_i}\beta })\bigg ] \\ {}&\quad + (1 - u_i) \big (y_i\varvec{B_i}\beta - e^{\varvec{B_i}\beta } \big ) \\ {}&\quad - ln \big (1 + e^{\varvec{G_i}\gamma }\big ) - (1 - u_i) ln(y_i!) \Bigg \}, \end{aligned}$$

The EM algorithm is based on a latent variable $Z_i$, where we observe $Z_i$ as 1, when $Y_i$ is from the perfect, zero state and $Z_i$ as 0, when $Y_i$ is from the Poisson state. To formulate the log-likelihood for the complete data $({\varvec{y}}, {\varvec{z}})$, we have:

$$\begin{aligned} lnL_c(\gamma , \beta ; {\varvec{y}}, {\varvec{z}})&= \sum \limits _{i=1}^{n} log(f(z_i|\gamma )) + \sum \limits _{i=1}^{n} log(f(y_i|z_i, \beta )) \\ {}&= \sum \limits _{i=1}^{n} \Big (z_iG_i\gamma - log(1 + e^{G_i\gamma }) \Big ) + \sum \limits _{i=1}^{n} (1 - Z_i)(y_i{\varvec{B}}_i\beta - e^{{\varvec{B}}_i\beta })\\&\quad - \sum \limits _{i=1}^{n} (1 - z_i) log(y_i!) \\ {}&= lnL_c(\gamma ; y, z) +lnL_c(\beta ; y, z) - \sum \limits _{i=1}^{n} (1 - z_i) log(y_i!). \end{aligned}$$

To implement EM algorithm, the log-likelihood above is maximized iteratively by alternating between estimating $Z_i$ by its expectation under the current estimates of $(\gamma , \beta )$ (E step) and then maximizing $L_c(\gamma , \beta ; y, z)$ (M step). In detail, the EM algorithm begins with starting values $(\gamma ^{(0)}, \beta ^{(0)})$ and proceeds iteratively. At iteration $(k + 1)$ of the EM algorithm requires the following steps.

E Step. Estimate $Z_i$ by its posterior mean $Z_i^{(k)}$ under current estimates $\gamma ^{(k)}$ and $\beta ^{(k)}$. This posterior mean is calculated as

$$\begin{aligned} z_i^{(k)}= {\left\{ \begin{array}{ll} [1 + exp(-G_i\gamma ^{(k)} - e^{B_i\beta ^(k)})]^{-1} &{}\text {if }y_i=0\\ 0 &{}\text {if }y_i=1,2,3....\\ \end{array}\right. } \end{aligned}$$

M Step for $\gamma$. Find $\gamma ^{(k+1)}$ by maximizing $L_c(\gamma ;y,z^{(k)})$. This can be accomplished by performing an an unweighted binomial logistic regression of $z^{(k)}$ on design matrix ${\varvec{G}}$ using a binomial denominator of one for each observation.

M Step for $\beta$. Find $\beta ^{(k+1)}$ by maximizing $L_c(\beta ;y,z^{(k)})$. This can be accomplished from a weighted Poisson log-linear regression of y on ${\varvec{B}}$, with weights $1 - Z^{(k)}$.

The EM algorithm converges in this problem and the estimates from the final iteration are the maximum likelihood estimates (MLEs) for the log-likelihood. The MLEs $({\hat{\gamma }}, {\hat{\beta }})$ are asymptotically Gaussian with variances equal to the inverse of the observed Fisher information matrix. Other issues including the choice of starting values, convergence of the EM algorithm, asymptotic distribution of $({\hat{\gamma }}, {\hat{beta}})$ in zero-inflated regression models are discussed in Lambert (1992).

Appendix B: Tables

See Tables 5, 6 , 7, 8, 9, 10, and 11.

Table 5 Estimation of power of testing $H_0$:$\beta _h=0$ vs $H_a$:$\beta _h\ne 0$, bias, MSE, coverage probability, and efficiency for hurdle Poisson model of set size 2

Table 6 Estimation of power of testing $H_0$:$\beta _h=0$ vs $H_a$:$\beta _h\ne 0$, bias, MSE, coverage probability, and efficiency for hurdle Poisson model of set size 2 including 10 additional predictors in the model

Table 7 Estimation of power of testing $H_0$:$\beta _h=0$ vs $H_a$:$\beta _h\ne 0$, bias, MSE, coverage probability, and efficiency for hurdle negative binomial model of set size 3 including 5 additional predictors in the model

Table 8 Estimation of power of testing $H_0$:$\beta _h=0$ vs $H_a$:$\beta _h\ne 0$, bias, MSE, coverage probability, and efficiency for zero-inflated Poisson model of set size 3

Table 9 Estimation of power of testing $H_0$:$\beta _h=0$ vs $H_a$:$\beta _h\ne 0$, bias, MSE, coverage probability, and efficiency for zero-inflated Poisson model of set size 3 under the null setting $\beta _h=0$

Table 10 Estimation of power of testing $H_0$:$\beta _h=0$ vs $H_a$:$\beta _h\ne 0$, bias, MSE, coverage probability, and efficiency for hurdle Poisson model of set size 5

Table 11 Estimation of power of testing $H_0$:$\beta _h=0$ vs $H_a$:$\beta _h\ne 0$, bias, MSE, coverage probability, and efficiency for hurdle Poisson model of set size 10 including 5 additional model predictors

See Figs. 8 and 9 and Table 12.

Fig. 8

Distribution of the number of surfaces with tooth wear

Fig. 9

Frequency distribution of the number of retweet counts

Table 12 Comparison of the computation time in seconds