doi:10.1016/S0167-9473(02)00263-3 
Copyright © 2002 Elsevier B.V. All rights reserved.
A pairwise likelihood approach to estimation in multilevel probit models
Didier Renard
, 
, Geert Molenberghs and Helena Geys
Center for Statistics, Department of Biostatistics, Limburgs Universitair Centrum, Universitaire Campus, building D 3590, Diepenbeek, Belgium
Received 28 March 2001;
revised 28 August 2002.
Available online 24 October 2002.
References and further reading may be available for this article. To view references and further reading you must
purchase this article.
Abstract
A pairwise likelihood (PL) estimation procedure is examined in multilevel models with binary responses and probit link. The PL is obtained as the product of bivariate likelihoods for within-cluster pairs of observations. The resulting estimator still enjoys desirable asymptotic properties such as consistency and asymptotic normality. Therefore, with this approach a compromise between computational burden and loss of efficiency is sought. A simulation study was conducted to compare PL with second-order penalized quasi-likelihood (PQL2) and maximum (marginal) likelihood (ML) estimation methods. The loss of efficiency of the PL estimator is found to be generally moderate. Also, PL tends to show more robustness against convergence problems than PQL2.
Author Keywords: Binary response data; Composite likelihood; Maximum marginal likelihood; Multilevel modeling; Penalized Quasi-Likelihood; Pairwise likelihood
Fig. 1. Boxplots of ML, PL and PQL2 simulated parameter estimates under Model (9) with random intercept 
N(0,σb02). Top panel: 20 clusters with σb02=0.5; Bottom panel: 20 clusters with σb02=1.
Fig. 2. Boxplots of ML, PL and PQL2 simulated parameter estimates under Model (
9) with random intercept
N(0,σ
b02). Top panel: 50 clusters with σ
b02=0.5; Bottom panel: 50 clusters with σ
b02=1.
Fig. 3. Boxplots of ML, PL and PQL2 simulated parameter estimates under Model (
9) with random intercept and random slope
N(0,Σ
b). Top panel: 20 clusters with σ
b002=0.5, σ
b01=0 and σ
b112=0.5; Bottom panel: 20 clusters with σ
b002=1, σ
b01=0 and σ
b112=1.
Fig. 4. Boxplots of ML, PL and PQL2 simulated parameter estimates under Model (
9) with random intercept and random slope
N(0,Σ
b). Top panel: 50 clusters with σ
b002=0.5, σ
b01=0 and σ
b112=0.5; Bottom panel: 50 clusters with σ
b002=1, σ
b01=0 and σ
b112=1.
Fig. 5. Boxplots of ML, PL and PQL2 simulated parameter estimates under Model (
10) with random intercept.
Fig. 6. Boxplots of ML, PL and PQL2 simulated parameter estimates under Model (
10) with random slope.
Table 1. Asymptotic efficiency of PL versus WPL in the random-intercept model. Cell entries are asymptotic relative efficiencies (percentages) for Not-found (first row) and Not-found (second row)
Table 2. Simulations results (1000 replicates) to compare PQL2 estimates in the random-intercept model with logit and probit link. Means are reported with Monte Carlo error given between parentheses
Table 3. Median computing times (seconds) for fitting random-effects model (9) to 100 simulated data sets using ML and PL
Simulation settings:
– 20 clusters of size between 10 and 30
– random-effects variance equal 1
Abstract
Purchase PDF (354 K)
E-mail Article
Add to my Quick Links
Cited By in Scopus (12)
Purchase PDF (2174 K)
