Iterated Reweighted Rank-Based Estimates for GEE Models

Abstract

Repeated measurement designs occur in many areas of statistical research. In 1986, Liang and Zeger offered an elegant analysis of these problems based on a set of generalized estimating equations (GEEs) for regression parameters, that specify only the relationship between the marginal mean of the response variable and covariates. Their solution is based on iterated reweighted least squares fitting. In this paper, we propose a rank-based fitting procedure that only involves substituting a norm based on a score function for the Euclidean norm used by Liang and Zeger. Our subsequent fitting, while also an iterated reweighted least squares solution to GEEs, is robust to outliers in response space and the weights can easily be adapted for robustness in factor space. As with the fitting of Liang and Zeger, our rank-based fitting utilizes a working covariance matrix. We prove that our estimators of the regression coefficients are asymptotically normal. The results of a simulation study show that the our proposed estimators are empirically efficient and valid. We illustrate our analysis on a real data set drawn from a hierarchical (three-way nested) design.

Appendix

Proof of Theorem 4.1.

Let $\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }) = \hat{\boldsymbol{\alpha }} (\boldsymbol{\beta },\hat{\phi }(\boldsymbol{\beta }))$. Let k ≥ 1 be arbitrary but fixed. For i = 1, …, K, let

$$\displaystyle\begin{array}{rcl} \mathbf{Z}_{i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }))& =& \mathbf{D}_{ i}^{T}\hat{\mathbf{V}}_{ i}^{-1/2}\mathbf{W}_{ i}\hat{\mathbf{V}}_{i}^{-1/2}\left [\mathbf{Y}_{ i} -\mathbf{a}^{{\prime}}(\boldsymbol{\theta }) -\mathbf{M}^{{\ast}}\left (\boldsymbol{\beta }\right )\right ] \\ & =& \mathbf{D}_{i}^{T}\hat{\mathbf{V}}_{ i}^{-1/2}\mathbf{W}_{ i}[\mathbf{Y}_{i}^{{\ast}}-\mathbf{G}_{ i}^{{\ast}}(\boldsymbol{\beta }) -\mathbf{M}\mathbf{1}]. {}\end{array}$$

(4.27)

We then write the GEERB estimating equations (4.17) in the compact form

$$\displaystyle{ \sum _{i=1}^{K}\mathbf{Z}_{ i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta })) = \mathbf{0}\;. }$$

(4.28)

The GEERB estimator $\hat{\boldsymbol{\beta }}_{R}^{(k)}$ solves this equation.

Similar to Liang and Zeger (1986), we first expand $K^{-1/2}\sum _{i=1}^{K}\mathbf{Z}_{i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }))$ in a Taylor series about the true parameter $\boldsymbol{\beta }$ and evaluated at $\hat{\boldsymbol{\beta }}_{R}^{(k)}$. By the chain rule, the gradient in this expansion is given by,

$$\displaystyle\begin{array}{rcl} \boldsymbol{\bigtriangledown }_{i}& =& \frac{\partial \mathbf{Z}_{i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }))} {\partial \boldsymbol{\beta }} + \frac{\partial \mathbf{Z}_{i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }))} {\partial \boldsymbol{\alpha }} \frac{\partial \boldsymbol{\alpha }} {\partial \boldsymbol{\beta }} \\ & =& \mathbf{A}_{i} + \mathbf{B}_{i}\mathbf{C}\;. {}\end{array}$$

(4.29)

Because $\hat{\boldsymbol{\beta }}_{R}^{(k)}$ solves Eq. (4.28), the Taylor expansion evaluated at $\hat{\boldsymbol{\beta }}_{R}^{(k)}$ is

$$\displaystyle{\mathbf{0} =\sum _{ i=1}^{K}\mathbf{Z}_{ i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta })) +\sum _{ i=1}^{K}\boldsymbol{ \bigtriangledown }_{ i}(\hat{\boldsymbol{\beta }}_{R}^{(k)} -\boldsymbol{\beta })\;.}$$

Solving for $\sqrt{K}(\hat{\boldsymbol{\beta }}_{R}^{(k)} -\boldsymbol{\beta })$, we obtain

$$\displaystyle{ \sqrt{K}(\hat{\boldsymbol{\beta }}_{R}^{(k)} -\boldsymbol{\beta }) = \left \{ \frac{1} {K}\sum _{i=1}^{K}\boldsymbol{ \bigtriangledown }_{ i}\right \}^{-1}\left [ \frac{1} {\sqrt{K}}\sum _{i=1}^{K}\mathbf{Z}_{ i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }))\right ]\;. }$$

(4.30)

Secondly, we fix $\boldsymbol{\beta }$ and expand $K^{-1/2}\sum _{i=1}^{K}\mathbf{Z}_{i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }))$ about the true parameter $\boldsymbol{\alpha }$ and evaluated at $\boldsymbol{\alpha }^{{\ast}}$ to get

$$\displaystyle\begin{array}{rcl} \frac{1} {\sqrt{K}}\sum _{i=1}^{K}\mathbf{Z}_{ i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }))& =& \frac{1} {\sqrt{K}}\sum _{i=1}^{K}\mathbf{Z}_{ i}(\boldsymbol{\beta },\boldsymbol{\alpha }) + \frac{1} {K}\sum _{i=1}^{K}\frac{\partial } {\boldsymbol{\alpha }} \mathbf{Z}_{i}(\boldsymbol{\beta },\boldsymbol{\alpha })\sqrt{K}(\boldsymbol{\alpha }^{{\ast}}-\boldsymbol{\alpha }) + o_{ p}(1) \\ & =& \frac{1} {\sqrt{K}}\sum _{i=1}^{K}Z_{ i}(\boldsymbol{\beta },\boldsymbol{\alpha }) + \mathbf{B}^{{\ast}}\mathbf{C}^{{\ast}} + o_{ p}(1)\;, {}\end{array}$$

(4.31)

where the o _p (1) term is due to regularity conditions which imply that the remainder term is $\frac{1} {K}O_{p}(1)$. Note that the weights are evaluated at the true parameters in this expansion too.

Because $\mathbf{Z}_{i}(\boldsymbol{\beta },\boldsymbol{\alpha })$ is evaluated at the true parameters, we use the notation given in (4.18). Letting h _it ^T be the tth row the product $\mathbf{D}_{i}^{T}\mathbf{V}_{i}^{-1/2}$, we then have

$$\displaystyle\begin{array}{rcl} \frac{1} {\sqrt{K}}\sum _{i=1}^{K}Z_{ i}(\boldsymbol{\beta },\boldsymbol{\alpha })& =& \frac{1} {\sqrt{K}}\sum _{i=1}^{K}\sum _{ t=1}^{n_{i} }\mathbf{h}_{it}^{T}w_{ it}[y_{it}^{\dag }- g_{ it}^{\dag }(\boldsymbol{\beta }) - m(\boldsymbol{\beta })] \\ & =& \frac{1} {\sqrt{K}}\sum _{i=1}^{K}\sum _{ t=1}^{n_{i} }\mathbf{h}_{it}^{T}a[R(y_{ it}^{\dag }- g_{ it}^{\dag }(\boldsymbol{\beta }))]. \\ & =& \frac{1} {\sqrt{K}}\sum _{i=1}^{K}\mathbf{D}_{ i}^{T}\mathbf{V}_{ i}^{-1/2}\mathbf{a}[R(\mathbf{Y}_{ i}^{\dag }-\mathbf{G}_{ i}^{\dag }(\boldsymbol{\beta }))].{}\end{array}$$

(4.32)

The second equality holds because the weights are evaluated at the true parameters.

By Assumptions [A.5] and [A.6], it follows from Theorem 3.1 of Brunner and Denker (1994) and the usual Cramer-Wold device that $\frac{1} {\sqrt{K}}\sum _{i=1}^{K}Z_{i}(\boldsymbol{\beta },\boldsymbol{\alpha })$ is asymptotically normal with mean 0 and variance-covariance matrix

$$\displaystyle{ \mathbf{M} = \frac{1} {K}\sum _{i=1}^{K}\mathbf{D}_{ i}^{T}\mathbf{V}_{ i}^{-1/2}\mbox{ Var}(\boldsymbol{\varphi }_{ i}^{\dag })\mathbf{V}_{ i}^{-1/2}\mathbf{D}_{ i}. }$$

(4.33)

Note that the form of the variance-covariance matrix follows from (4.32) and the independence between subjects.

Returning to expression (4.31), from the assumptions we have C ^∗ = O _p (1). Because the scores can change value at only a finite number of points [e.g., Sect. 3.2.1 of Hettmansperger and McKean (2011)], we can write B ^∗ as

$$\displaystyle{ \mathbf{B}^{{\ast}} = \frac{1} {K}\sum _{i=1}^{K}\left \{\frac{\partial } {\partial \boldsymbol{\alpha }}\mathbf{D}_{i}^{T}\mathbf{V}_{ i}^{-1/2}\right \}\mathbf{a}[R(\mathbf{Y}_{ i}^{\dag }-\mathbf{G}_{ i}^{\dag }(\boldsymbol{\beta }))]. }$$

(4.34)

Assuming Lindeberg-Feller conditions for the quantity in braces, as in (A.4), it follows, similar to (4.32) that $\sqrt{K}\mathbf{B}^{{\ast}} = O_{p}(1)$, and, hence, B ^∗ = o _p (1).

To finish the proof, we need to consider the terms in expression (4.29). A simple derivation shows that

$$\displaystyle{\mathbf{A}_{i} = -\mathbf{D}_{i}^{T}\mathbf{V}_{ i}^{-1/2}\mathbf{W}_{ i}\mathbf{V}_{i}^{-1/2}\mathbf{D}_{ i}^{T}.}$$

By assumption, C = O _p (1). Since,

$$\displaystyle{\mathbf{B}_{i} = \frac{\partial \mathbf{Z}_{i}(\boldsymbol{\beta },\boldsymbol{\alpha }^{{\ast}}(\boldsymbol{\beta }))} {\partial \boldsymbol{\alpha }} }$$

arguments similar to those above show that $K^{-1}\sum _{i=1}^{K}\mathbf{B}_{i} = o_{p}(1)$. Hence,

$$\displaystyle{ \frac{1} {K}\boldsymbol{ \bigtriangledown }_{i} = \frac{1} {K}\sum _{i=1}^{K}\mathbf{D}_{ i}^{T}\mathbf{V}_{ i}^{-1/2}\mathbf{V}_{ i}^{-1/2}\mathbf{D}_{ i} + o_{p}(1).}$$

This and the discussion around expressions (4.32) and (4.33) finish the proof of Theorem 4.1.

As a final note, the asymptotic representation of the estimator is

$$\displaystyle{ \sqrt{ K}(\hat{\boldsymbol{\beta }}_{R}^{(k)}-\boldsymbol{\beta }) = \left \{ \frac{1} {K}\sum _{i=1}^{K}\mathbf{A}_{ i}\right \}^{-1}\left [ \frac{1} {\sqrt{K}}\sum _{i=1}^{K}\mathbf{D}_{ i}^{T}\mathbf{V}_{ i}^{-1/2}\mathbf{a}[R(\mathbf{Y}_{ i}^{\dag }-\mathbf{G}_{ i}^{\dag }(\boldsymbol{\beta }))]\right ]+o_{ p}(1). }$$

(4.35)