Abstract
In this paper, we provide the ML (Maximum Likelihood) and the REML (REstricted ML) criteria for consistently estimating multivariate linear mixed-effects models with arbitrary correlation structure between the random effects across dimensions, but independent (and possibly heteroscedastic) residuals. By factorizing the random effects covariance matrix, we provide an explicit expression of the profiled deviance through a reparameterization of the model. This strategy can be viewed as the generalization of the estimation procedure used by Douglas Bates and his co-authors in the context of the fitting of one-dimensional linear mixed-effects models. Beside its robustness regarding the starting points, the approach enables a numerically consistent estimate of the random effects covariance matrix while classical alternatives such as the EM algorithm are usually non-consistent. In a simulation study, we compare the estimates obtained from the present method with the EM algorithm-based estimates. We finally apply the method to a study of an immune response to Malaria in Benin.
1 Introduction
The linear mixed-effects model [1, 1, 2, 3, 4, 5] has become a popular tool for analyzing univariate multilevel data which arise in many areas (biology, medicine, economy, etc.) thanks to its flexibility to model the correlation contained in these data, and the availability of reliable and efficient software packages for fitting it [6, 7, 8, 9]. Univariate multilevel data are referred to observations (or measurements) of a single variable of interest on several levels. For instance, schools grouped by village, and villages grouped at the city-level. In this context, multiple variables of interest measured at multiple levels characterize multivariate multilevel data. Examples include exam or test scores recorded for students across time and multi-dimensional subjects (e.g. maths, biology, French, etc.). In some situations where attention is focused on the analysis of multivariate multilevel data, some research questions can only be answered in a joint analysis of two or more variables, not separately [10]. A multivariate extension of a single response variable-based linear mixed-effects model is indeed having increasing popularity as a flexible tool for the analysis of multivariate multilevel data [11, 12, 13, 14].
For example, in the context of the malaria immune study, fourteen antibodies against malaria have been measured on children exposed to malaria-infected mosquitoes in certain Benin republic villages from West Africa. One of the objectives of the malaria study was to evaluate the effect of the malaria infection on the child’s immune against the disease. The antigens which characterize the child’s immune status interact together, and it makes more sense to evaluate the effect of the malaria infection on all these antigens, simultaneously. The only valid way of doing this evaluation is to analyze the characteristics of the joint distribution of these antigens, conditional on the malaria infection. In [15] the authors pointed out the drawback of performing a separate analysis instead. The malaria study is further described below and used as an illustration of the method at the end of the paper.
For the one-dimensional linear mixed-effects model, many methods for obtaining the estimates of the fixed and the random effects have been proposed in the literature. These methods include Henderson’s mixed model equations [16], approaches proposed by [17] as well as techniques based on two-stage regression. Searle et al. [18] and [19] gave more details of the latter. Concerning the variance parameters estimation in linear mixed-effects model, the discussed methods in the literature include the ANOVA method for balanced data which uses the expected mean squares approach [20, 21]. For unbalanced data, [22] proposed the minimum norm quadratic estimation (MINQUE) method, where the resulting estimates are translation invariant under unbiased quadratic forms of the observations. Lee and Nelder [23] suggested another method of estimating variance parameters using extended quasi-likelihood, meaning gamma-log generalized linear models. Gumedze and Dunne [24] presented more details on the parameters’ estimation methods in the linear mixed-effects model. The Maximum Likelihood (ML) and the Restricted Maximum Likelihood (REML) methods are the most popular estimation methods in the linear mixed-effects model [25]. The main attraction of these methods is the ability to handle a much wider class of variance models than simple variance components [24].
In the multivariate linear mixed-effects model, ML and REML estimates are frequently obtained through iterative schemes such as the EM algorithm [14, 26, 27, 28, 29]. This avoids the difficulties related to the direct calculation of the parameters’ likelihood, without increasing the implementation burden. Despite the existence of valid theorems proving the asymptotic convergence of the iterative sequences of these algorithms [27], the practical reality is often more complex. Verbeke et al. [10] explained that the main problem in fitting multivariate mixed-effects models is the loss of numerical stability due to the increasing of the dimension of the response variable. Indeed, the increase of the dimension of the response variable obviously leads to that of the dimension of the parameters vector of the model. Adjakossa et al. [15] illustrate through simulations studies that the random effects covariance matrix obtained by the EM algorithm has the worst Mean Square Error (MSE) among the parameters. In the same paper, the authors also gave a direct estimate of the random effects covariance matrix.
In this paper, we focus on the multivariate linear mixed-effects model with arbitrary correlation structure between the Gaussian random effects across dimensions, but independent (and possibly heteroscedastic) Gaussian residuals. In order to gain in numerical stability along with a better MSE for the variance parameter estimates, inspiring from [6] approach in the one-dimensional case, we optimize the profiled deviance which is obtained by partially maximizing the likelihood in the fixed effect parameters. This choice is justified by the fact that the fixed-effects vector is generally greater in dimension than the vector of variance parameters. Furthermore, avoiding a direct optimization on this vector of fixed-effects may decrease the complexity of the resulting optimization problem. Finally, the profiled deviance is a function of a small number of parameters that is easily optimized by any box-constrained optimizer, since some of the variance parameters are constrained. Our approach consists in directly calculating the likelihood of the model parameters after a reparametrization. This likelihood is used to obtain ML or REML estimates through the provided profiled deviance or the REML criterion, respectively. This strategy may explain the high quality of the estimates for the fixed effects parameters and the random effects variance parameters as well as the residual variance parameters. This approach may be viewed as the multi-dimensional generalization of the lme4 R package [6].
2 Multivariate linear mixed-effects model
Throughout the paper we adopt the following classical notations:
For the sake of simplicity, we focus on the bivariate case (d = 2) in most of the paper, but the generalization to higher dimensions (d > 2) is straightforward. Thus, in dimension 2, the model is the following:
where
For k ∈ {1,2},
The variance-covariance matrices of
In eq. (3),
where
and
where
with
where
2.1 ML criterion
The objective of this section is to express the log-likelihood of the model defined in eqs. (5) and (6) given an observation
If we denote by
Lemma 2.1.
Denoting
By using Lemma 2.1 the log-likelihood of Θ given
Theorem 2.1.
Assume that
The ML criterion expressed in eq. (8) is a differentiable convex function of
Corollary 2.1.
Consider the log-likelihood defined in eq. (8) and let
The procedure that we use to derive the ML estimate of Θ components is summarized in Corollary 2.1 through eq. (9). First, the ML estimate of the fixed-effects vector is analytically computed while supposing that all the variance parameters are known. This leads to
2.2 REML criterion
The REML criterion [4] is obtained by integrating the likelihood of the model (given
In eq. (8), we recognize the expression of a multivariate Gaussian distribution density in
Theorem 2.2.
Consider the log-likelihood defined in eq. (8). By using the notations of Theorem 2.1, the log-REML profiled deviance defined by
The ML and the REML deviance optimization tasks can be solved by using any box-constrained optimizer. In this paper, we use for example the optimization function nlminb available in the R software.
3 Simulation studies
In this section, the computational stability of the procedure is illustrated through simulation studies, and the present estimation procedure is compared with the estimation procedure based on the EM algorithm. For the sake of simplicity, the simulation studies are performed using simulated bivariate longitudinal data sets. In the following paragraph, we explain how we chose the parameters that have been used to simulate the “working" data sets.
The working data sets. We suppose that we are following up a sample of subjects where the goal is to evaluate how the increase of the weight and the height of the individuals of this population are jointly explained by the sex, the score of nutrition (Nscore), and the age. Each subject has been seen
where
In eq. (12),
The true values of
In order to have an almost strong correlation between the marginal random effects, we set ρ = 0.8 and randomly chose all other parameters involved in the expression of
3.1 Estimates’ performances
One practical way to show the numerical consistency of an estimator is by computing its Mean Square Error (MSE). If the MSE of an estimator is asymptotically zero, this estimator converges in probability, and is then consistent. We simulated datasets with gradually increasing sample sizes where n ∈ {50, 60, 100, 300} and N ∈ {600, 1000, 3000}. In order to account for the unbalanced nature of the longitudinal data we observe in practice, we used a multinomial distribution of the N observations among the n individuals under the constraint that
One should note that the chosen simulation design is challenging since it usually requires
Furthermore, it requires not only
Parameter | n | N = 600 | N = 1000 | N = 3000 |
---|---|---|---|---|
50 | 2.43 (0.11 | 1.89 (0.22 | 1.02 (0.07 | |
60 | 2.57 (0.14 | 2.13 (0.26 | 0.77 (0.10 | |
100 | 2.27 (0.16 | 1.55 (0.14 | 0.71 (0.04 | |
300 | 3.16 (0.14 | 1.70 (0.10 | 0.51 (0.04 | |
50 | 5.50(0.09 | 3.26 (0.02 | 2.06 (0.09 | |
60 | 5.06 (0.12 | 3.22 (0.02 | 1.78 (0.10 | |
100 | 4.33 (0.03 | 2.37 (0.02 | 1.06 (0.02 | |
300 | 4.58 (0.18 | 2.43 (0.05 | 0.90 (0.04 | |
50 | 0.03 (0.00 | 0.02 (0.00 | 0.00 (0.00 | |
60 | 0.04 (0.00 | 0.02 (0.00 | 0.00 (0.00 | |
100 | 0.03 (0.00 | 0.01 (0.00 | 0.00 (0.00 | |
300 | 0.05 (0.00 | 0.03 (0.00 | 0.00 (0.00 | |
50 | 0.04 (0.00 | 0.03 (0.00 | 0.00 (0.00 | |
60 | 0.04 (0.00 | 0.03 (0.00 | 0.01 (0.00 | |
100 | 0.06 (0.00 | 0.03 (0.00 | 0.01 (0.00 | |
300 | 0.13 (0.00 | 0.04 (0.00 | 0.01 (0.00 | |
50 | 0.90 (0.12 | 0.62 (0.06 | 0.45 (0.04 | |
60 | 1.07 (0.06 | 0.68 (0.08 | 0.25 (0.03 | |
100 | 0.90 (0.05 | 0.47 (0.03 | 0.21 (0.02 | |
300 | 1.45 (0.19 | 0.69 (0.06 | 0.17 (0.02 |
3.2 Comparison with EM-based estimates
In this section, we compare the estimation procedure based on the EM algorithm with NCEmlme. For both algorithms, we consider two initialization schemes: naive or advised. For the naive initialization the starting parameter is randomly chosen (without specific control), and for the advised one the starting values are those obtained by fitting separately each dimension of the bivariate model. For each replication, the starting values are the same for both the NCEmlme and EM-based algorithms. For each algorithm, the number of iterations required for convergence are reported too. Our methodology consists in simulating one hundred longitudinal datasets of size (N = 3000, n = 300) and fitting the model to each of these datasets using the EM-based algorithm and the NCEmlme, respectively. This allows to compute both the 95 % CI and the empirical mean of the one hundred estimates in each case (naive and advised starting values). The results are presented in Table 2 and Table 3.
Table 2 contains the empirical means of the estimates with their 95 % CI, and the minimum, the maximum and the average numbers of iterations. Table 3 contains the empirical relative error of the estimators with their 95 % CI. The results show that in the case of naive initialization, the NCEmlme estimators outperform the EM estimators. For example, the component of
Naive initialization | Advised initialization | ||||||||
---|---|---|---|---|---|---|---|---|---|
NCEmlme | EM | NCEmlme | EM | ||||||
Parameter | Value | Emp. Mean | 95 % CI | Emp. Mean | 95 % CI | Emp. Mean | 95 % CI | Emp. Mean | 95 % CI |
50.67 | 50.79 | 13.47 | 50.80 | 50.78 | |||||
–4.80 | –4.79 | –5.02 | –4.98 | ||||||
14.00 | 14.02 | –2.05 | 14.02 | 14.02 | |||||
2.70 | 2.70 | 2.69 | 2.70 | 2.70 | |||||
13.20 | 13.65 | –84.47 | 13.65 | 13.68 | |||||
–2.80 | –2.80 | –2.75 | –2.81 | –2.85 | |||||
27.00 | 27.00 | 0.90 | 27.00 | 27.00 | |||||
1.70 | 1.68 | 1.68 | 1.68 | 1.68 | |||||
5.80 | 5.79 | 5.78 | 5.78 | 5.79 | |||||
7.60 | 7.61 | 7.59 | 7.61 | 7.63 | |||||
Nbr. of iteration | Min | 56 | – | 63 | – | 48 | – | 14 | – |
Mean | 71 | – | 109 | – | 64 | – | 169 | – | |
Max | 103 | – | 157 | – | 89 | – | 645 | – |
Naive initialization | Advised initialization | ||||||||
---|---|---|---|---|---|---|---|---|---|
NCEmlme | EM | NCEmlme | EM | ||||||
Parameter | Value | R. Error | 95 % CI | R. Error | 95 % CI | R. Error | 95 % CI | R. Error | 95 % CI |
50.67 | 0.01 | 0.73 | 0.01 | 0.01 | |||||
0.21 | 0.21 | 0.21 | 0.21 | ||||||
14.00 | 0.02 | 1.14 | 0.02 | 0.02 | |||||
2.70 | 0.00 | 0.00 | 0.00 | 0.00 | |||||
13.20 | 0.07 | 7.39 | 0.07 | 0.07 | |||||
0.43 | 0.43 | 0.43 | 0.43 | ||||||
27.00 | 0.00 | 0.96 | 0.00 | 0.00 | |||||
1.70 | 0.01 | 0.01 | 0.01 | 0.01 | |||||
5.80 | 0.01 | 0.01 | 0.01 | 0.01 | |||||
7.60 | 0.01 | 0.01 | 0.01 | 0.01 |
The empirical mean of the random effects covariance matrix,
and
The matrices
For all the simulation studies, we use the ML deviance criterion and have minimized it using the nlminb function under the R software. Thus, the estimates obtained are from the ML estimators. In this paper, we do not provide an application of REML estimators.
4 Application on malaria dataset
4.1 Data description
The data that we analyze here come from a study which was conducted in nine villages (Avamé centre, Gbédjougo, Houngo, Anavié, Dohinoko, Gbétaga, Tori Cada Centre, Zébè and Zoungoudo) of Tori Bossito area (Southern Benin), where P.falciparum is the commonest species in the study area (95 %) [31] from June 2007 to January 2010. The aim of this study was to evaluate the determinants of malaria incidence in the first months of life of children in Benin.
Mothers (n = 620) were enrolled at delivery and their newborns were actively followed up during the first year of life. One questionnaire was conducted to gather information on women’s characteristics (age, parity, use of Intermittent Preventive Treatment during pregnancy (IPTp) and bed net possession) and during their pregnancy. After delivery, thick and thin placental blood smears were examined to detect placental infection defined by the presence of asexual forms of P.falciparum. Maternal peripheral blood as well as cord blood were collected. At birth, newborn’s weight and length were measured and gestational age was estimated.
During the follow-up of newborns, axillary temperature was measured weekly. In case of temperature higher than
Concerning the antibody quantification, two recombinant P.falciparum antigens where used to perform IgG subclass (IgG1 and IgG3) antibody. Recombinants antigens MSP2 (3D7 and FC27) were from La Trobe University [32, 33]. GLURP-R0 (amino acids 25-514, F32 strain) and GLURP-R2 (amino acids 706-1178, 140 F32 strain) were also expressed. The antibodies were quantified in plasma at different times and ADAMSEL FLPb039 software (http://www.malariaresearch.eu/content/software) was used to analyze automatically the ELISA optical density (OD) leading to antibody concentrations in (µg/mL).
In this paper, we use some of the data and we rename the proteins used in the study, for reasons of the protection of these data. We thus name A1, A2, B and C the proteins that we consider, which are related to the antigens IgG1 and IgG3 as mentioned above. The information contained in the multivariate longitudinal dataset of malaria is described in Table 4, where Y denotes an antigen which is one of the following:
N | Variable | Description |
---|---|---|
1 | id | Child ID |
2 | conc.Y | concentration of Y |
3 | conc | Measured concentration of Y in the umbilical cord blood |
4 | conc | Predicted concentration of Y in the child’s peripheral blood at 3 months |
5 | ap | Placental apposition |
6 | hb | Hemoglobin level |
7 | inf | Number of malaria infections in the previous 3 months |
8 | pred | Quarterly average number of mosquitoes child is exposed to |
9 | nutri | Quarterly average nutrition scores |
4.2 Data analysis
The aim of the data analysis is to evaluate the effect of the malaria infection on the child’s immune acquisition (against malaria). Since the antigens which characterize the child’s immune status interact together in the human body, we analyze the characteristics of the joint distribution of these antigens, conditionally on the malaria infection and other factors of interest. The dependent variables are then provided by conc.Y (Table 4) which describes the level of the antigen Y in the children at 3, 6, 9, 12, 15 and 18 months. All other variables in Table 4 are covariates. We then have d = 8 dependent variables which describe the longitudinal profile (in the child) of the proteins listed in eq. (15).
To illustrate the stability of our approach, we are fitting here a bivariate model (see eqs. (16) and (17)) to the data, with
with
Our strategy is to (1) fit the model to the data by running the NCEmlme algorithm using 25 different naive starting points and (2) retain the estimates related to the best likelihood (the minimum of the 25 deviances) as the true parameters and compute the estimators’ MSE using the 24 others estimates. This may allow to evaluate how much the NCEmlme algorithm is sensitive to the starting points. The results are contained in Table 5. Based on these results, the influence of the starting points on the NCEmlme algorithm is very low (see the MSE in Table 5). The estimated random effects covariance matrix is
with an MSE of 0.0095.
Response variables | ||||||
---|---|---|---|---|---|---|
Covariates | Joint estim. | MSE | Indep. estim. | Joint estim. | MSE | Indep. estim. |
Intercept | 0.648 | 0.451 | 0.032 | |||
ap | ||||||
0.127 | 0.174 | |||||
0.151 | 0.130 | |||||
0.092 | 0.044 | |||||
0.117 | 0.137 | |||||
hb | ||||||
inf_trim | 0.398 | 0.373 | 0.785 | 0.709 | ||
pred_trim | 0.001 | 0.017 | 0.019 | |||
nutri_trim | 0.072 | 0.046 | 0.271 | 0.106 | ||
1.345 | 1.426 | 1.495 | 1.672 |
In this kind of study, it is often useful to characterize the subjects (here, the children) by constructing their profiles. This is done through the computation of the expected random-effects vector (i.e.
5 Conclusion
In the context of multivariate linear mixed-effects models having homoscedastic dimensional residuals, we have suggested ML and REML estimation strategies by profiling the model’s deviance and Cholesky factorizing of the random effect covariance matrix. This approach can be considered as the generalization of the approach used by [6] in the R software lme4 package. Through extensive simulation studies, we have illustrated that the present approach outperforms the traditional EM-based estimates and provides estimates that are numerically consistent for both fixed effects and variance components. Another interesting characteristic is its robustness relative to the initial value of the optimization procedure which can be randomly chosen without affecting the estimation results. These characteristics of our procedure show their improved computational stability in comparison with the traditional approaches. Furthermore, the optimization of the profiled ML or REML criterion can be easily and rapidly performed using any existing box-constrained optimizer in the R software. Further considerations of this approach may include heteroscedastic residuals as well as residuals correlated with the random effects, where the theoretical consistency of the resulting estimators should be demonstrated.
Appendix
A Proof of Theorem 2.1 and Lemma 2.1
Proof.
Denoting by
where
Let us denote by
where
and
And in eq. (20),
for which the solution satisfy the normal equation:
where
This concludes the proof of Lemma 2.1. In eq. (21), we behave as if
By setting
And
where
Thereafter,
By setting
and returning to the calculation of
By setting
The log-likelihood to be maximized can therefore be expressed as,
B Proof of Theorem 2.2
Proof.
Using the expression of
where
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