Chapter 1 Introduction
5.2 The Generalized Linear Mixed Model
5.2.9 Remarks on the problem of Bias in Generalized Linear ModelsLinear Models
antilogit function
µ= g−1(η) = exp(η)/{1 + exp(η)}.
Genstat also issues a statement to emphasize that the “means are proba- bilities not expected values”. Thus inference should be carried out on the transformed scale, and the back-transformed means are only to give an in- tuitive guide in interpreting the results. Note in Genstat there is a choice to use either the Schall (PQL) method or the marginal method of Breslow and Clayton (MQL) method. As stated earlier the only difference in the two methods is in the way the parameter estimates are formed.
5.2.9 Remarks on the problem of Bias in Generalized
creased. Inference on variance parameters was less satisfactory under the PQL method, with a tendency for the procedure to converge to a non-positive definite variance matrix when the binomial denominator was 1 or 2. They point out that this is due, at least partially, to the fact that when the re- sponse probabilities are small and the data are highly discrete, only limited information is present for estimating random effects and their associated variances and covariances. They also note that a further limitation of the PQL method is the failure to account for the contribution of the estimated variance components when the link function is not the identity, but this is not the case with the MQL method.
Rodriguez and Goldman (1995) carried out a large simulation study to in- vestigate the performance of the MQL approach.The study was implemented using the statistical program ML3, which was the forerunner of MLn. The simulation study was based on a Guatemalan data set which consisted of a sub-sample of respondents in the 1987 National Survey of Maternal and Child Health. The survey was based on a national multistage clustered sam- ple of 5610 women aged 15-44 years living in 240 communities. The logistic model selected as the basis of their simulations examined the determinants of use of modern (versus traditional) prenatal care during pregnancy among women who reported having obtained some kind of prenatal care, for births during the 5 years before the survey. The sample size of interest here consists of the 2449 births that occurred to mothers who received any prenatal care, with 45% of this sample having received modern prenatal care. The data are clustered on two higher order levels; the 2449 births were to 1558 mothers who live in 161 communities. Among the 1558 families, 52.4% had one child (born during the 5 years before the survey), 38.2% had two children, 9.1%
had three children and 0.3% had four children. The sample sizes per com-
munity ranged from 1 to 50 with a mean of 15 children.
In the simulation Rodriguez and Goldman considered three covariates, one at each level. In the first four simulation sets they included random family and cluster effects in addition to the fixed components, with two magnitudes for the random effects, namely small (standard deviation 0.4) and large (stan- dard deviation 1). These random effects were generated from independent normal distributions with mean 0 and designated variance (0.16 or 1) while the individual random component was generated from a standard logistic distribution.
They found that the fixed effects exhibited a clear downward bias, as Bres- low and Clayton (1993) suggested would happen on theoretical grounds. Ro- driguez and Goldman (1995) found that the bias was moderate when the random effects were small (σ= 0.4) but was fairly substantial when at least one of the random effects was large (σ = 1). For both random effects large they obtained fixed effects estimates of the order of 0.75 of the true values.
They point out that the bias of 0.25 translates to an odds ratio of 0.78, indi- cating that the effects of the covariates on the odds of using modern prenatal care services would be underestimated by 22%. The estimates of the vari- ance components had an even more pronounced downward bias, with a large number of simulations leading to estimates of the family effect equal to zero.
Rodriguez and Goldman (1995) then carried out further simulations
1. To see whether the problems of bias were related to the use of a three level model. To achieve this, they used a two level model including only the family random effect (set to large or small) or the community random effect(set to large or small).
2. To see whether the problems of bias were related to the fact that in the the Guatemalan data structure (with an average of 1.5 children per
mother) they had very limited information on within family variation.
To do this they carried out two additional data sets of simulations using rectangular structure of 20 communities, each of which having 20 families with 20 children each, for a total sample size of 800.
For the fixed effects the downward bias remained, and was of a similar magni- tude with the corresponding previous simulation ( i.e. large or small variance) for the three level Guatemalan structure. For the two level model including only a family random effect, the estimates of the variance components still had a downward bias, and in the case of the small variance 19% of the sim- ulations led to estimates of the family effect equal to zero. The two level model including only a community random effect and the three level rect- angular structure model showed a slight downward bias, but none of the family effect equal to zero. This suggests that the cluster size is important in estimating the variance components. It is noted that in their estimation, Rodriguez and Goldman used Iterative Generalized Least Squares (IGLS) which is equivalent to using maximum likelihood (ML) and this would lead to a downward bias in the variance components anyway; for improved results the Restricted Iterated Generalized Least Squares (RIGLS) or REML should be used in each iteration. Goldstein (1995, Chapter 5) and Goldstein and Rasbash (1996) suggest an improved approximation which largely eliminates the downward biases in the estimates from GLMMs. Goldstein (1995, Chap- ter 5 and Chapter 7) proposed an alternative approach of the estimation for nonlinear models, including GLMMs. Rodriguez and Goldman (1995) point out that Goldstein’s procedure in general will produce the same results as those produce by the GLMM algorithm based on quasilikelihood. This follows since the two methods use exactly the same approximating linear model, based on a result due to Browne (1974) who proves that Generalized
Least Squares (GLS) and maximum likelihood are equivalent in the normal case, and that the Fisher’s scoring method and GLS coincide when the vari- ance matrix is linear in the unknown parameters, as in the case in variance component models if the parameterization is in terms of σu2 (the variance component corresponding to the random effect u.
Rodriguez and Goldman (1995) considered second order MQL estimation, and found that this improved the estimates, but only slightly. Goldstein and Rasbash (1996) carried out simulations based on those of Rodriguez and Goldman, and found that the PQL procedure considerably improved the model estimates with the only bias being in about 20% underestimation in their level-2 model. The greatest improvement occurred from a move from first to second order PQL. They also report the results of Ayis (1995) who showed that the second order PQL produced almost unbiased estimates for the fixed parameters and estimates that are no greater than 4% for the ran- dom parameters. Goldstein and Rasbash (1996) also reported on an iterated form of bootstrapping due to Kuk (1995) for producing asymptotically un- biased estimates.
Breslow and Lin (1995) and Lin and Breslow (1996) consider the bias in estimates of both fixed and random parameters, and give suggestions for bias correction procedures. Breslow and Lin (1995) studied the asymptotic bias of the variance component i.e. a single random component and the regres- sion parameter (fixed parameter) estimates in GLMMs with a canonical link function and a single source of extraneous variation. In addition to PQL, they also considered approximations based on Laplace approximations of the integrated likelihood (to be revisited in later sections); the Laplace approxi-
mation approach has been used by Liu and Pierce (1993), Solomon and Cox (1992) and Wolfinger (1993) among others. Breslow and Lin (1995) provided a correction factor for the variance component estimate derived from Laplace approximation and from PQL, and also a first order correction term for the regression coefficients estimated by PQL. They found that the proposed bias corrected PQL estimates significantly improve the asymptotic performance of the uncorrected quantities.
Lin and Breslow (1996) generalize these results of Breslow and Lin (1995) to GLMMs with multiple sets of random effects. They focus on correcting the bias in PQL estimates, since Breslow and Lin (1995) found that in some circumstances the Laplace approximation methods may be numerically un- stable. However they use a generalization of the asymptotic expressions de- rived by Solomon and Cox (1992) for the Laplace approximations to multiple components of dispersion to derive their bias correction procedure and de- rive a quadratic expansion of the integrated log-quasilikelihood. These issues then led Lin and Breslow to propose a 4-step algorithm to achieve the bias corrected PQL estimates of the regression coefficients and variance compo- nents. Lin and Breslow (1996) evaluated the performance of these correction procedures by reanalyzing the well known example of the salamander mating experiment reported by McCullagh and Nelder (1989, section 14.5) and car- rying out simulation studies. For the salamander data they found that the performance of the bias correction procedure was unsatisfactory and they attributed this to the large variability of the random effects in the actual salamander data. Their simulation studies showed more positive results; in particular the simple correction procedure for the variance components effec- tively reducing the bias in the PQL estimates of θ and the associated mean
square error when the sample size was reasonably large. Note that θ is the vector of variance component parameters.
They note that attempts to reduce bias are not always desirable, and that the effectiveness of the correction procedure for a particular problem will de- pend on both the sample size and the conditional form of responses. The corrections often inflate the variances of the parameter estimates, especially in problems involving very small variance components and small sample sizes.
The biases in first order and second order corrected regression coefficients are negligible for small amounts of dispersion. When the variance components are between 0.5 and 1 in problems involving binary outcomes, the second order correction perform better. However they point out that both correc- tions break down for larger variance components. They also note that, from the results of other simulations studies, caution is required when applying corrected PQL (CPQL) to the regression coefficients when the binomial de- nominators are small. Further as the binomial denominator increases, the PQL method itself yields satisfactory estimates and the corrections may not be necessary. They suggest that the best procedure for general use may be the correction of the variance components and recalculation of the PQL re- gression components β using the corrected PQL variance components.
Engel and a number of his co-workers have also investigated the problem of bias in the estimates from GLMM, and in particular the estimates of the variance components. Engel and Keen (1994) point out that the obvious estimates of the iterative weights, used in GLMMs are:
ˆ
w−1 =V(ˆµ)[g0(ˆµ)]2
where the estimate of µ is obtained in the first step of the estimation pro- cedure. They note that V(ˆµ)[g0(ˆµ)]2 may often be an accurate prediction for V(µ)[g0(µ)]2, particularly in the case of a single random effect, where the variance component σ2u is small and is not necessarily a consistent estimator.
They recommend the use of alternative weights which depend on both the link and the assumed variance function. In the case of the logit link with a binomial variance function, the alternative weights suggested by Engel and Keen (1994) are given by
w0 ={2 + 2expσ2u
2 cosh(x0β)}−1.
Engel, Buist and Visscher (1995) carried out a number of simulation studies in animal breeding and found out that both the magnitude and direction of the bias in the estimate of the variance component depend on the number of fixed effects and also on the underlying response probability with over estimation of the variance component σ2u when there are a large number of fixed effects and the overall incidence is above 0.9. They consider models in animal breeding which potentially can have over 100 fixed effects.
Engel and Buist(1998) further investigated bias in GLMMs, also looking at the correction method of Lin and Breslow (1996). They found out that while the correction of Lin and Breslow is useful for small or moderate num- bers of fixed effects, it is of little benefit in animal breeding studies which commonly have a large number of fixed effects. They also report that the alternative weights of Engel and Keen may alleviate the bias and reduce the MSE, but not for a large number of observations per random effect (in survey data analysis, say more than 40 observations per cluster). Engel (1998) con- siders a single example to illustrate the asymptotic bias in GLMM estimates of the variance component σu2 and finds that this can be underestimated by
almost half. He suggests that the best procedure for overcoming bias could be the use of a Markov chain Monte Carlo method such as the Gibbs sampler.
However such methods will not be the focus in the current work.