Mantel–Haenszel estimators of a common odds ratio for multiple response data

Abstract

For a two-way contingency table, odds ratios are commonly used to describe the relationships between the row and column variables. In the ordinary case cells are mutually exclusive, that is each subject must fit into one and only one cell. However, in many surveys respondents may select more than one outcome category, commonly referred to as multiple responses. We discuss model-based and Mantel–Haenszel estimators of an assumed common odds ratio for several $2\times c$ tables, where the two rows refer to independent groups and the c columns to multiple responses, treating the multiple responses as an extension of the multinomial sampling model. We derive new dually consistent (co)variance estimators for the Mantel–Haenszel odds ratio estimators and show their performance in a simulation study and illustrate the estimators on a linguistic data set.

Appendices

Dual consistency of ${{\varPsi }}^*_{jh}$

The estimator

$$\begin{aligned} {\hat{\theta }}_h =\frac{ \sum _k X_{h|ak} \frac{n_{bk}}{N_k}}{ \sum _k X_{h|bk} \frac{n_{ak}}{N_k} } \end{aligned}$$

which converges to

$$\begin{aligned} \frac{ \sum _k \pi _{h|ak} \frac{n_{bk}n_{ak}}{N_k}}{ \sum _k \pi _{h|bk} \frac{n_{ak}n_{bk}}{N_k} }&= \frac{ \sum _k \exp (\gamma _{ak}+\alpha _{hk}+\beta _{ah} ) \frac{n_{bk}n_{ak}}{N_k}}{ \sum _k \exp (\gamma _{bk}+\alpha _{hk}+\beta _{bh} ) \frac{n_{ak}n_{bk}}{N_k} }\\&=\frac{ \exp (\beta _{ah}) \sum _k \exp (\gamma _{ak}+\alpha _{hk} ) \frac{n_{bk}n_{ak}}{N_k}}{ \exp (\beta _{bh})\sum _k \exp (\gamma _{bk}+\alpha _{hk} ) \frac{n_{ak}n_{bk}}{N_k} }\\&= \exp (\beta _{ah}-\beta _{bh}) \frac{ \sum _k \exp (\gamma _{ak}+\alpha _{hk} ) \frac{n_{bk}n_{ak}}{N_k} }{\sum _k \exp (\gamma _{bk}+\alpha _{hk} ) \frac{n_{ak}n_{bk}}{N_k} } \end{aligned}$$

Under model (1$^*$), the term on the right hand side becomes 1, and dual consistency applies because $\theta _h= \exp (\beta _{ah}-\beta _{bh})$. However under model (1), this is not the case because of the $\gamma _{ik}$ terms. This means that the estimator ${\hat{\theta }}_h$ converges to $\theta _h \times c$, where c is a constant.

Now ${\hat{\theta }}_j$ converges under model (1) to $\theta _j \times c$, therefore ${{\varPsi }}^*_{jh}={\hat{\theta }}_j/{\hat{\theta }}_h$ converges to $\frac{\theta _j \times c}{\theta _h \times c}={\varPsi }_{jh}$, and dual consistency even applies under model (1).

Dual consistency of covariance estimator of two MH relative risk estimators

Showing that $\text {Cov}(L_j,L_h)$ is consistent is equivalent to showing that $\text {Cov}({\hat{\theta }}_j,{\hat{\theta }}_h)$ is consistent by application of delta method to log-function. Hence we need to show that

$$\begin{aligned} \widehat{\text {Cov}({\hat{\theta }}_j,{\hat{\theta }}_h)}=\frac{\frac{n_{bk}^2}{N_k^2}(X_{jh|ak}-d_{jh|ak})+{\hat{\theta }}_j{\hat{\theta }}_h\frac{n_{ak}^2}{N_k^2}(X_{jh|bk}-d_{jh|bk})}{C_{j|ba} C_{h|ba} } \end{aligned}$$

is consistent for $\text {Cov}({\hat{\theta }}_j,{\hat{\theta }}_h)$.

We can show that

$$\begin{aligned} \lim \text {Cov}({\hat{\theta }}_j-\theta _j,{\hat{\theta }}_h-\theta _h)&=\lim \text {Cov}\left( \frac{C_{j|ab}-\theta _j C_{j|ba}}{C_{j|ba}},\frac{C_{h|ab}-\theta _h C_{h|ba}}{C_{h|ba}}\right) \\&=\frac{\sum _k \text {Cov}(c_{j|ab}-\theta _j c_{j|ba},(c_{h|ab}-\theta _h c_{h|ba}))}{\lim C_{j|ba}\lim C_{h|ba} } \end{aligned}$$

and

$$\begin{aligned} \text {Cov}(c_{j|ab}-\theta _j c_{j|ba},c_{h|ab}-\theta _h c_{h|ba})= & {} \frac{n_{ak}n_{bk}}{N_k^2}\left( n_{bk} (\pi _{jh|ak}-\pi _{j|ak}\pi _{h|ak})\right. \\&\left. +\, n_{ak}\theta _j \theta _h (\pi _{jh|bk}-\pi _{j|bk}\pi _{h|bk})\right) \end{aligned}$$

which can be estimated under both limiting models by

$$\begin{aligned} \frac{n_{bk}^2}{N_k^2}(X_{jh|ak}-d_{jh|ak})+{\hat{\theta }}_j{\hat{\theta }}_h\frac{n_{ak}^2}{N_k^2}(X_{jh|bk}-d_{jh|bk}) \end{aligned}$$

with $d_{jh|ak}=(X_{j|ak}X_{h|ak}-X_{jh|ak})/n_{ak}'$ and $n_{ak}'=n_{ak}-1$.

Dual consistency of ordinary MH estimator

1.1 Sparse data limiting model

For the sparse data limiting model, the number of observations per stratum is bounded ($O(N_k)=1$) and K approaches infinity.

From $\pi _{j|1k}\pi _{h|2k}= {\varPsi }_{jh}\pi _{h|1k}\pi _{j|2k}$, which follows from the assumption of a common odds ratio, and Eq. (11), we derive

$$\begin{aligned} \mathbb {E}\omega _{jh|k} = \mathbb {E}(c_{jh|k}-{\varPsi }_{jh}c_{hj|k})&= \mathbb {E}c_{jh|k}-{\varPsi }_{jh} \mathbb {E}c_{hj|k} \\&= \{ \mathbb {E}X_{j|1k} \mathbb {E}X_{h|2k}-{\varPsi }_{jh} \mathbb {E}X_{h|1k} \mathbb {E}X_{j|2k} \}/N_k \\&= \{ n_{1k}n_{2k} \pi _{j|1k}\pi _{h|2k} - {\varPsi }_{jh} n_{1k}n_{2k}\pi _{h|1k}\pi _{j|2k} \}/ N_k \\&= \{n_{1k}n_{2k} ( \pi _{j|1k}\pi _{h|2k} - \pi _{j|1k}\pi _{h|2k} )\}/N_k=0 \end{aligned}$$

We can write

$$\begin{aligned} {\hat{{\varPsi }}}_{jh}-{\varPsi }_{jh}&= \frac{\sum _{k=1}^K c_{jh|k}-{\varPsi }_{jh}c_{hj|k}}{\sum _{k=1}^K c_{hj|k}} = \frac{\sum _{k=1}^K (c_{jh|k}-{\varPsi }_{jh}c_{hj|k})/K}{\sum _{k=1}^K c_{hj|k}/K}&\end{aligned}$$

(13)

$$\begin{aligned}&= \frac{\sum _{k=1}^K \omega _{jh|k} /K}{\sum _{k=1}^K c_{hj|k}/K} = \frac{{\varOmega }_{jh} /K}{ C_{hj}/K}&. \end{aligned}$$

(14)

with with $\omega _{jh|k}:=c_{jh|k}-{\varPsi }_{jh}c_{hj|k}$ and ${\varOmega }_{jh}:=\sum _k\omega _{jh|k}$.

The term $c_{jh|k}$ is a bounded random variable under model II, hence, the variance of $C_{jh}$ is $o(K^2)$ and Chebyshev’s weak law of large numbers states $({\varOmega }_{jh}-\mathbb {E}{\varOmega }_{jh})/K$${\rightarrow }_p0$. Since $\mathbb {E}\omega _{jh|k}=0$, the expression $({\varOmega }_{jh}-\mathbb {E}{\varOmega }_{jh})/K$${\rightarrow }_p0$ reduces to ${\varOmega }_{jh}/K {\rightarrow }_p0$, that is, the numerator of ${\hat{{\varPsi }}}_{jh}-{\varPsi }_{jh}$ converges to zero in probability. Applying the Chebyshev weak law of large numbers again to the denominator yields

$$\begin{aligned} \sum _{k=1}^K c_{jh|k}/K \overset{K \rightarrow \infty }{\longrightarrow }_p \lim _{K \rightarrow \infty } \sum _{k=1}^K \mathbb {E}(c_{jh|k})/K<\infty . \end{aligned}$$

This limit is finite and nonzero. Thus, we conclude ${\hat{{\varPsi }}}_{jh}-{\varPsi }_{jh}{\rightarrow }_p0$ by Slutsky’s theorem.

1.2 Large stratum limiting model

Let us consider the case $N\rightarrow \infty$ with $N\alpha _{ik}=n_{ik}$ and $0<\alpha _{ik}<1$, that is, as N approaches infinity the number of subjects $n_{ik}$, for all rows i and strata k, also approaches infinity. Note $N_k=n_{1k}+n_{2k}=N\sum _{i}\alpha _{ik}$.

We have

$$\begin{aligned} C_{jh} /N&=\sum _{k=1}^K c_{jh|k} /N =\sum _{k=1}^K X_{j|1k} X_{h|2k} /(N_k N) \nonumber \\&=\sum _{k=1}^K \frac{n_{1k}n_{2k}}{N_kN} \frac{X_{j|1k}}{n_{1k}} \frac{X_{h|2k}}{n_{2k}}=\sum _{k=1}^K \frac{n_{1k}n_{2k}}{NN}\frac{N}{N_k} \frac{X_{j|1k}}{n_{1k}} \frac{X_{h|2k}}{n_{2k}} \\ \overset{ N \rightarrow \infty }{\longrightarrow }_p&\sum _{k=1}^K \alpha _{1k}\alpha _{2k}\left( \sum _i \alpha _{ik}\right) ^{-1} \pi _{j|1k} \pi _{h|2k} =\sum _{k=1}^K \left( \sum _i \alpha _{ik}^{-1}\right) ^{-1} \pi _{j|1k} \pi _{h|2k}. \end{aligned}$$

Therefore

$$\begin{aligned} {\hat{{\varPsi }}}_{jh}&=\frac{C_{jh}}{C_{hj}}=\frac{C_{jh}/N}{C_{hj}/N} \overset{N \rightarrow \infty }{\longrightarrow }_p \frac{\sum _{k=1}^K \left( \sum _i \alpha _{ik}^{-1}\right) ^{-1} \pi _{j|1k} \pi _{h|2k}}{\sum _{k=1}^K \left( \sum _i \alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k} \pi _{j|2k}}\\&={\varPsi }_{jh}\frac{\sum _{k=1}^K \left( \sum _i \alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k} \pi _{j|2k}}{\sum _{k=1}^K \left( \sum _i \alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k} \pi _{j|2k}} ={\varPsi }_{jh}. \end{aligned}$$

Asymptotic covariances

1.1 Sparse-data limiting model

Let $\text {Var}^a(\cdot )$ and $\text {Cov}^a(\cdot )$ refer to the asymptotic variance and covariance. From above ${\hat{{\varPsi }}}_{jh}-{\varPsi }_{jh}=\frac{{\varOmega }_{jh} /K}{ C_{hj}/K}=\frac{\sum _k \omega _{jh|k} /K}{ C_{hj}/K}$.

First by independence of rows $\text {Cov}({\varOmega }_{jh},{\varOmega }_{ts})=\sum _{k=1}^K \text {Cov}(\omega _{jh|k},\omega _{ts|k})$. Note that $\mathbb {E}|\omega _{jh|k}-\mathbb {E}\omega _{jh|k}|^3$$=\mathbb {E}|\omega _{jh|k}|^3$$=O(1)$ , because $c_{jh|k}$ is a bounded random variable under the sparse-data limiting model. By setting $\delta =1$, we conclude from the Multivariate Central Limit Theorem (Sen and Singer 1993, p.123) that ${K}^{-1/2}$$({\varOmega }_{jh},$${\varOmega }_{ts})=$$\sqrt{K}({\varOmega }_{jh}/K,$${\varOmega }_{ts}$  / K) converges to a zero mean multivariate normal distribution with covariance $\lim _{K\rightarrow \infty }$$\frac{1}{K}$$\sum _{k=1}^K \text {Cov}(\omega _{jh|k},\omega _{ts|k})$, by noting that $\mathbb {E}\omega _{jh|k}=0$ and $\text {Cov}(\omega _{jh},$$\omega _{ts})$ exists. We conclude the asymptotic covariance between ${\varOmega }_{jh}$ and ${\varOmega }_{ts}$ is $\lim _{K\rightarrow \infty } K \cdot \text {Cov}^a({\varOmega }_{jh} ,{\varOmega }_{ts})=\lim _{K\rightarrow \infty }$$\frac{1}{K}$$\sum _{k=1}^K \text {Cov}(\omega _{jh|k},\omega _{ts|k})$.

Therefore by the delta method, Slutsky’s theorem, Eq. (13), and using that the denominator terms $\lim _{K}\mathbb {E}C_{hj}/K$ are finite we obtain

$$\begin{aligned}&\lim _{K\rightarrow \infty }K\cdot \text {Cov}^a(\log {\hat{{\varPsi }}}_{jh} , \log {\hat{{\varPsi }}}_{ts})\\&=1/({\varPsi }_{jh}{\varPsi }_{ts}) \lim _{K\rightarrow \infty } K \cdot \text {Cov}^a({\hat{{\varPsi }}}_{jh} ,{\hat{{\varPsi }}}_{ts}) \\&=1/({\varPsi }_{jh}{\varPsi }_{ts}) \frac{\lim _{K\rightarrow \infty } K \cdot \text {Cov}^a({\varOmega }_{jh} ,{\varOmega }_{ts})}{(\lim _{K}\mathbb {E}C_{hj}/K)( \lim _{K}\mathbb {E}C_{st} /K )}\\&=1/({\varPsi }_{jh}{\varPsi }_{ts}) \frac{\lim _{K\rightarrow \infty } 1/K \cdot \sum _k \text {Cov}(\omega _{jh|k},\omega _{ts|k}) }{(\lim _{K}\mathbb {E}C_{hj}/K)( \lim _{K}\mathbb {E}C_{st} /K )}\\ \end{aligned}$$

for arbitrary indices $j,h,s,t \in \{1,\dots ,c \}$ with $j\ne h$ and $s\ne t$.

Now we obtain the following variance

$$\begin{aligned} \text {Var}(\omega _{jh|k})&= v_{jh|k}^1-2{\varPsi }_{jh} v_{jh|k}^2+{\varPsi }_{jh}^2 v_{jh|k}^3 \end{aligned}$$

and covariances

$$\begin{aligned} \text {Cov}(\omega _{jh|k},\omega _{js|k})&= v_{jhs|12,k} - {\varPsi }_{jh} v_{jh,js|k} - {\varPsi }_{js} v_{js,jh|k} + {\varPsi }_{jh}{\varPsi }_{js}v_{jhs|21,k}\\ \text {Cov}(\omega _{jh|k},\omega _{ts|k})&= v_{jt,hs|k}-{\varPsi }_{jh} v_{ht,js|k} - {\varPsi }_{ts}v_{js,ht|k} + {\varPsi }_{jh}{\varPsi }_{ts}v_{hs,st|k} \end{aligned}$$

with

$$\begin{aligned} v_{jh|k}^1&=\frac{n_{1}n_{2}}{N^2} \left( \pi _{j|1}\pi _{h|2} + n_1'\pi _{j|1}^2\pi _{h|2} +n_2'\pi _{j|1}\pi _{h|2}^2\right) \\ v_{jh|k}^2&=\frac{n_{1}n_{2}}{N^2}\left( n_1'\pi _{j|1}\pi _{h|1} \pi _{jh|2}+ n_2'\pi _{j|2}\pi _{h|2}\pi _{jh|1}+\pi _{jh|1}\pi _{jh|2}\right) \\ v_{jh|k}^3&=\frac{n_{1}n_{2}}{N^2} \left( \pi _{h|1}\pi _{j|2} + n_1'\pi _{h|1}^2\pi _{j|2} +n_2'\pi _{h|1}\pi _{j|2}^2\right) \\ v_{jh,ts|k}&=\frac{n_{1}n_{2}}{N^2} \left( \pi _{jh|1}\pi _{ts|2}+n_1'\pi _{j|1}\pi _{h|1}\pi _{ts|2}+n_2' \pi _{jh|1}\pi _{t|2}\pi _{s|2}\right) \\ v_{jhs|abk}&=\frac{n_{1}n_{2}}{N^2} v_{jhs|abk}^A+v_{jhs|abk}^B \; (a\ne b) \\ v_{jhs|abk}^A&=\frac{n_{1}n_{2}}{N^2} \pi _{hs|bk}\left( \pi _{j|ak}+n_a'\pi _{j|ak}^2 \right) \; (a\ne b)\\ v_{jhs|abk}^B&=\frac{n_{1}n_{2}}{N^2} n_b'\pi _{j|ak}\pi _{h|bk}\pi _{s|bk} \; (a\ne b). \end{aligned}$$

The subscript k is often suppressed for convenience only.

The (co)variance estimators were constructed in such a way that they converge exactly to the asymptotic (co)variance(s). We can also express $U_{jhs}$ as $U_{jhs}=U_{jhs}^{add}$ omitting $U_{jhs}^{old}$ but only if ${\hat{v}}_{jhs|abk}^B$ is amended to ${\hat{v}}_{jhs|abk}^B = \frac{1}{N_k^2}X_{j|ak}\{ X_{h|bk}X_{s|bk}-X_{hs|bk} \}$. Then for the covariance estimators we have $\sum _k {\hat{v}}_k/K \overset{K\rightarrow \infty }{\longrightarrow } \sum _k \mathbb {E}{\hat{v}}_k/K = \lim _K \sum _k v_k/K$ and $\sum _k c_{jh|k}/K \overset{K\rightarrow \infty }{\longrightarrow }\sum _k \mathbb {E}c_{jh|k}/K$ by Chebyshev’s weak law of large numbers.

1.2 Large-stratum limiting model

By the delta method, the large stratum limiting variance is

$$\begin{aligned}&\lim _{N\rightarrow \infty } N \cdot \text {Var}^a(\log {\hat{{\varPsi }}}_{jh})\\&=\dfrac{\sum _k \frac{\alpha _1^2\alpha _2}{(\sum _i \alpha _{ik})^2} \left\{ \pi _{j|1}^2\pi _{h|2}+{\varPsi }_{jh}^2 \pi _{h|1}^2\pi _{j|2}- 2{\varPsi }_{jh} \pi _{j|1} \pi _{h|1} \pi _{jh|2} \right\} }{\left( \sum _k \left( \sum _i\alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k}\pi _{j|2k}\right) ^2 } \nonumber \\&\quad +\, \dfrac{\sum _k \frac{\alpha _1\alpha _2^2}{(\sum _i \alpha _{ik})^2} \left\{ \pi _{j|1}\pi _{h|2}^2 +{\varPsi }_{jh}^2 \pi _{h|1}\pi _{j|2}^2 - 2{\varPsi }_{jh} \pi _{jh|1}\pi _{j|2}\pi _{h|2} \right\} }{\left( \sum _k \left( \sum _i\alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k}\pi _{j|2k}\right) ^2 } \end{aligned}$$

and the limiting covariances are

$$\begin{aligned}&\lim _{N\rightarrow \infty } N\cdot \text {Cov}^a(\log {\hat{{\varPsi }}}_{jh},\log {\hat{{\varPsi }}}_{js})\\&=\dfrac{\sum _k \frac{\alpha _1^2\alpha _2}{(\sum _i \alpha _{ik})^2} \left\{ \pi _{j|1}^2\pi _{hs|2}- {\varPsi }_{jh} \pi _{j|1}\pi _{h|1}\pi _{js|2} -{\varPsi }_{js}\pi _{j|1}\pi _{s|1}\pi _{jh|2} +{\varPsi }_{jh}{\varPsi }_{js}\pi _{h|1}\pi _{s|1}\pi _{j|2} \right\} }{\left( \sum _k \left( \sum _i\alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k}\pi _{j|2k}\right) ^2 }\\&\quad +\,\dfrac{\sum _k \frac{\alpha _1\alpha _2^2}{(\sum _i \alpha _{ik})^2} \{ \pi _{j|1}\pi _{h|2}\pi _{s|2}-{\varPsi }_{jh}\pi _{jh|1}\pi _{j|2}\pi _{s|2} -{\varPsi }_{js}\pi _{js|1}\pi _{j|2}\pi _{h|2} +{\varPsi }_{jh}{\varPsi }_{js}\pi _{hs|1}\pi _{j|2}^2\} }{\left( \sum _k \left( \sum _i\alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k}\pi _{j|2k}\right) ^2 } \end{aligned}$$

$$\begin{aligned}&\lim _{N\rightarrow \infty } N\cdot \text {Cov}^a(\log {\hat{{\varPsi }}}_{jh},\log {\hat{{\varPsi }}}_{ts})\\&=\dfrac{\sum _k \frac{\alpha _1^2\alpha _2}{(\sum _i \alpha _{ik})^2} \left\{ \pi _{j|1}\pi _{t|1}\pi _{hs|2} - {\varPsi }_{jh} \pi _{h|1}\pi _{t|1}\pi _{js|2} - {\varPsi }_{ts} \pi _{j|1}\pi _{s|1}\pi _{ht|2} + {\varPsi }_{jh}{\varPsi }_{ts} \pi _{h|1}\pi _{s|1}\pi _{jt|2}\right\} }{\left( \sum _k \left( \sum _i\alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k}\pi _{j|2k}\right) ^2 }\\&\quad +\,\dfrac{\sum _k \frac{\alpha _1\alpha _2^2}{(\sum _i \alpha _{ik})^2} \{\pi _{jt|1}\pi _{h|2}\pi _{s|2} - {\varPsi }_{jh} \pi _{ht|1}\pi _{j|2}\pi _{s|2} - {\varPsi }_{ts} \pi _{js|1}\pi _{h|2}\pi _{t|2} + {\varPsi }_{jh}{\varPsi }_{ts} \pi _{hs|1}\pi _{j|2}\pi _{t|2} \} }{\left( \sum _k \left( \sum _i\alpha _{ik}^{-1}\right) ^{-1} \pi _{h|1k}\pi _{j|2k}\right) ^2 }. \end{aligned}$$

The estimators were constructed such that

$$\begin{aligned} \lim _{N\rightarrow \infty } N \cdot \text {Var}^a(\log {\hat{{\varPsi }}}_{jh})&=\lim _N N \cdot U_{jhh}\\ \lim _{N\rightarrow \infty } N \cdot \text {Cov}^a(\log {\hat{{\varPsi }}}_{jh},\log {\hat{{\varPsi }}}_{js})&=\lim _{N\rightarrow \infty } N \cdot U_{jhs}\\ \lim _{N\rightarrow \infty } N \cdot \text {Cov}^a(\log {\hat{{\varPsi }}}_{jh},\log {\hat{{\varPsi }}}_{ts})&=\lim _{N\rightarrow \infty } N \cdot U_{jhts}. \end{aligned}$$