Smoothing combined estimating equations in quantile regression for longitudinal data

Abstract

Quantile regression has become a powerful complement to the usual mean regression. A simple approach to use quantile regression in marginal analysis of longitudinal data is to assume working independence. However, this may incur potential efficiency loss. On the other hand, correctly specifying a working correlation in quantile regression can be difficult. We propose a new quantile regression model by combining multiple sets of unbiased estimating equations. This approach can account for correlations between the repeated measurements and produce more efficient estimates. Because the objective function is discrete and non-convex, we propose induced smoothing for fast and accurate computation of the parameter estimates, as well as their asymptotic covariance, using Newton-Raphson iteration. We further develop a robust quantile rank score test for hypothesis testing. We show that the resulting estimate is asymptotically normal and more efficient than the simple estimate using working independence. Extensive simulations and a real data analysis show the usefulness of the method.

Appendix

To prove the theorems, we first give a set of regularity conditions. For any matrix A, ∥A∥ denotes the modulus of the largest singular values of A. We mainly follow Johnson and Strawderman (2009) for the proof and make the following standard assumptions (Koenker 2005).

Assumption A.1

The dimension p of covariates x _ij is fixed; m→∞ and max{n _i } is bounded. The distribution functions $F_{ij}(z)=P(y_{ij}-x_{ij}^{T}\beta_{\tau}\le z|x_{ij})$ are absolutely continuous, with continuous densities f _ij and its first derivative uniformly bounded away from 0 and ∞ at the points 0, i=1,…,m;j=1,…,n _i .

Assumption A.2

The true β _τ is in the interior of a compact set Θ∈ℝ^p.

Assumption A.3

There exist finite matrices A _lk (l,k=1,…,a) and nonsingular matrices G _l (β _τ ),l=1,…,a such that

• (1) $\lim_{m\rightarrow\infty} \frac{1}{m}\sum_{i=1}^{m}x_{i}^{T}\varGamma_{i}M_{li}M^{T}_{ki}\varGamma_{i}x_{i}=A_{lk},\ l,k= 1, \ldots,a$.

• (2) $\lim_{m\rightarrow\infty} \frac{1}{m}\sum_{i=1}^{m}x_{i}^{T}\varGamma_{i}M_{li}\varGamma_{i}x_{i}=G_{l}(\beta_{\tau}), l=1,\ldots,a$.

• (3) $\lim_{m\rightarrow\infty}\frac{1}{\sqrt{m}} \max\|x_{ij}\|=0$.

Proof of Theorem 1

Without loss of generality, we consider the lth component of S(β) and let $\beta=\beta_{\tau}+\delta/\sqrt{m}$,

(9)

where 0<δ<∞, ε _i =y _i −x _i β _τ and $Z_{i}=I(\varepsilon_{i}<x_{i}\delta/\sqrt{m})-I(\varepsilon_{i}<0)$. For the second term, write

By Assumption A.3 (1) and (2), we have

(10)

By Cauchy-Schwartz Inequality and Assumption A.3, for all ζ∈ℝ^p with ζ ^T ζ=1,

(11)

Therefore combining (9)–(11), we have

$$ m^{1/2}S(\beta)=m^{1/2}S(\beta_\tau)-G_m( \beta_\tau)\delta+o_p(1), $$

(12)

where $G_{m}(\beta_{\tau})=(G_{m,1}^{T}(\beta_{\tau}),\ldots ,G_{m,a}^{T}(\beta_{\tau}))^{T}$ with $G_{m,l}(\beta_{\tau})=\frac{1}{m}\sum_{i=1}^{m} x_{i}^{T}\varGamma_{i}M_{li}\varGamma_{i}x_{i}$, l=1,…,a.

Let S ^∗(β)=S(β _τ )−G _m (β _τ )(β−β _τ ) and $Q^{*}_{m}(\beta)=\{S^{*}(\beta)\}^{T}\{\varSigma^{*}_{m}(\beta)\}^{-1}S^{*}(\beta)$ where $\varSigma^{*}_{m}(\beta)=\frac{1}{m}\sum_{i=1}^{m}S_{i}^{*}\cdot\{S_{i}^{*}\}^{T}-S^{*}(\beta )\{S^{*}(\beta)\}^{T}$ with $S_{i}^{*}(\beta)=S_{i}(\beta_{\tau})-G_{m,i}(\beta-\beta_{\tau}), i=1,\ldots,a$. We then have that

$$\sup_{\|\beta-\beta_\tau\|<t/\sqrt{m}}\bigl \Vert Q_m(\beta)-Q_m^*(\beta) \bigr \Vert =o_p\bigl(m^{-1}\bigr) $$

for any fixed t>0. By (1) of Assumption A.3 and the boundedness of $\psi_{\tau}(y_{i}-x_{i}'\beta_{\tau})\psi^{T}_{\tau}(y_{i}-x_{i}'\beta_{\tau})$, we have

$$\sup_{\|\beta-\beta_\tau\|<t/\sqrt{m}}\bigl \Vert \varSigma^*_m(\beta)-\varSigma ( \beta_\tau)\bigr \Vert \rightarrow0, $$

for any fixed t>0, in probability, where $\varSigma(\beta_{\tau})= \mathit{cov}(\sqrt {m}S(\beta_{\tau}))$.

From (12) and the definition of S ^∗(β), we can see that S(β) is asymptotically equivalent to S ^∗(β). Thus in a neighborhood of β _τ , the objective function Q _m (β) is asymptotically equivalent to the smoothed objective function $Q_{m}^{*}(\beta)$ at the rate of 1/m. We then conclude that the minimizer of Q _m (β) in a neighborhood of β _τ is also minimizing the smoothed objective function $Q_{m}^{*}(\beta)$ asymptotically. Since $\hat{\beta}$ minimizes Q _m (β), and equivalently $Q_{m}^{*}(\beta )$, we obtain that

The second derivative matrix is asymptotically positive definite, which guarantees a unique minimum. Since $\hat{\beta}$ satisfies $\partial Q_{m}^{*}(\beta)/\partial\beta |_{\hat{\beta}}=0$ and $Q_{m}(\beta_{\tau})=Q_{m}^{*}(\beta_{\tau})$, and by the continuity of $\partial Q_{m}^{*}(\beta)/\partial\beta$ at β _τ , $\hat{\beta}$ convergences to β _τ in probability, as m→∞.

Since m ^1/2 S(β _τ ) converges to a zero-mean normal distribution with a variance-covariance matrix Σ(β _τ ), letting G(β _τ )=lim _m→∞ G _m (β _τ ) and by Slutsky’s theorem, we have

$$\sqrt{m}(\hat{\beta}-\beta_\tau)\rightarrow N\bigl(0, \bigl(G^T(\beta_\tau )\varSigma^{-1}( \beta_\tau)G(\beta_\tau)\bigr)^{-1}\bigr). $$

 □

Proof of Theorem 2

Without loss of generality, we consider the lth component of $\tilde{S}$, $\tilde{S}_{(l)}(\beta)=E_{\vartheta}S_{(l)}(\beta+m^{-1/2}\varOmega^{1/2}\vartheta)$, where ϑ∼N(0,I _p ). Then by the differentiability of $\tilde {S}_{(l)}(\beta)$ and Taylor expansion, for all ∥δ∥≤C for some finite constant C, we have

$$ \sqrt{m}\tilde{S}_{(l)}(\beta_\tau+\delta/ \sqrt{m})=\sqrt {m}\tilde {S}_{(l)}(\beta_\tau)-E \tilde{S}'_{(l)}(\beta_\tau)\delta+o(1), $$

(13)

where $E\tilde{S}'_{(l)}(\beta_{\tau})=\frac{1}{m}\sum_{i=1}^{m} x_{i}^{T}\varGamma_{i}M_{li}D_{i}x_{i}$ and D _i is a n _i ×n _i diagonal matrix with elements $E_{\varepsilon_{ij}}\phi(\sqrt{m}\frac{\varepsilon_{ij}}{r_{ij}})\frac{\sqrt{m}}{r_{ij}}$. Notice that

where ∫ϕ(x)f _ij (0)dx=f _ij (0) and $|{\frac{r_{ij}}{\sqrt{m}}}\int\!\phi(x)f_{ij}(w^{*})x dx| \le M{\frac{r_{ij}}{\sqrt{m}}}\int|x|\phi (x)dx\rightarrow0$. Thus $E\tilde{S}'_{(l)}(\beta_{\tau})=G_{ml}(\beta_{\tau})+o(1)$.

Following the proof of Theorem 1, if

$$ m^{1/2}\bigl \Vert \tilde{S}(\beta_\tau)-S( \beta_\tau)\bigr \Vert \rightarrow0 $$

(14)

holds in probability, then

$$\sup_{\|\beta-\beta_\tau\|<t/\sqrt{m}}\bigl \Vert \tilde{Q}_m(\beta )-Q_m^*(\beta )\bigr \Vert =o_p\bigl(m^{-1} \bigr), $$

and thus $\tilde{\beta}$ converges to β _τ in probability.

To see that (14) holds, write

$$\tilde{S}(\beta_\tau)-S(\beta_\tau)=\int _{\mathbb{R}^p}\bigl[S\bigl(\beta_\tau +m^{-1/2}u \bigr)-S(\beta_\tau)\bigr]\phi_\varOmega(u)du, $$

where ϕ _Ω (⋅) denotes the pdf of Ω ^1/2 ϑ. Let K _m (u,β _τ )=∥S(β _τ +m ^−1/2 u)−S(β _τ )−m ^−1/2 G(β _τ )u∥. Then, since $\int_{\mathbb{R}^{p}}u\phi_{\varOmega}(u)du=0$, the triangle inequality implies

(15)

for any ϵ _m >0. By Assumption A.3 and the proof of (12), it is easy to see that

$$ \sup_{\|b-\beta_\tau\|\le d_m}\frac{\|S(b)-S(\beta_\tau)+G(\beta_\tau )(b-\beta_\tau)\|}{1+m^{1/2}\|b-\beta_\tau\|}=o_p \bigl(m^{-1/2}\bigr) $$

(16)

for any positive sequences d _m →0. Suppose ϵ _m =o(m ^1/2), then taking b=β _τ +m ^1/2 u, d _m =m ^−1/2 ϵ _m , (16) implies

$$ \sup_{\|u\|\le\epsilon_m}\frac{K_m(u;\beta_\tau)}{1+\|u\|}=o_p \bigl(m^{-1/2}\bigr). $$

(17)

An easy calculation, in combination with (17), now shows that the first integral on the right-hand side of the inequality in (15) converges in probability to zero, even if ϵ _m →∞. With regard to the second term on the right-hand side of (15), we may use the definition of K _m (⋅;β _τ ) and the triangle inequality to write $\sqrt{m}\int_{\|u\|> \epsilon _{m}}K_{m}(u,\beta_{\tau})\phi_{\varOmega}(u)du\le A_{1}+A_{2}$, where

For all β∈Θ, ∥S(β)∥≤A for some positive constant A<∞ by Assumptions A.2 and A.3, hence A ₁≤2Am ^1/2⋅P(∥Ω ^1/2 ϑ∥>ϵ _m )→0 as m→∞. Similarly, $\int_{\|u\|>\epsilon_{m}}\|u\|\phi_{\varOmega}(u)du\rightarrow0$. Therefore, the second integral on the right-hand side of the inequality in (15), also converges in probability to zero. It follows that (14) converges in probability to zero as m→∞.

The asymptotic normality of $\tilde{\beta}$ is obtained directly following the proof of Theorem 1. The proof is completed. □

Proof of Theorem 3

Following the similar argument in the proof of Theorem 1, under Assumptions A.1–A.3 and the null hypotheses H ₀, we obtain

$$ \hat{\alpha}-\alpha_\tau=O_p\bigl(m^{-1/2} \bigr), $$

(18)

where $\hat{\alpha}=\arg\min_{\alpha}\tilde{Q}(\alpha)$ under the null hypothesis H ₀. Let

where $M_{i}=(M_{1i}^{T},\ldots,M_{ai}^{T})^{T}$. Then following a similar argument as in the proof of Lemma A.2 in Wang and He (2007), for some constant C,

$$ \sup_{\|t\|\le C}\bigl \Vert r_m(t)-E\bigl \{r_m(t)\bigr\}\bigr \Vert =o_p(1). $$

(19)

A Taylor expansion of E{r _m (t)} around 0 gives

(20)

where the last step is due to the fact that Z ^T Δ=0 by construction. Now, (20) together with (19) and (18) yields

$$ U_{(1)}-m^{-1/2}\sum_i \bigl[I_a\otimes\bigl(Z_i^T \varGamma_i\bigr)\bigr]M_i\psi_\tau ( \varepsilon_i)=o_p(1). $$

(21)

Note that U ₍₁₎ is a a(p−q)-dimensional vector. The asymptotic normality of U ₍₁₎ follows then from the Lindberg-Feller Central Limit Theorem, which together with similar argument as in the proof of Theorem 1 completes the proof. □