Principio IV: Adquisición de conocimientos

The first step in Maximum Likelihood (ML) estimation is always to build the likelihood func- tion. For model (4.2.1), this function is:

Lj = nj Y

i=1

fi(Yi,j|Xi,j, Zj, β, uj)

Now, imagine that patient i in cluster j has a missing value in covariate X1_{, which is a binary} predictor at level 1. In order to be able to use the information provided by this patient when calculating the likelihood function, we need to sum the likelihood term over all the possible values for the missing covariate. Normally, when calculating the maximum likelihood estimates for a generalized linear mixed model, we do not consider any distribution for the covariates, that are just considered as fixed values we condition on. However in this case, we will have to introduce a distribution for the unobserved covariate. In the above case of a binary variable:

Lij = 1 X k=0 f (Yi,j|Xi,j1 , X 2,...,q i,j , Zj, β, uj)p(Xi,j1 = k, γ).

Where γ is the parameter defining the distribution of the covariate X1 and is distinct from (β, uJ). In this case, X1 being a binary variable, the calculation is straightforward since we just need to sum up two terms. This approach generalises naturally to the case where X1 _{is a categorical variables; in this situation we sum up over all the categories. This also} readily extends to the case of more than one categorical predictor missing, so for example for observation i in cluster j, if we had a missing binary covariate X1 _{and a missing 4-category} covariate X2_{, the likelihood term from this observation is:}

Lij = 1 X k=0 4 X t=1 f (Yi,j|Xi,j1 , X 2 i,j, X 3,...,q

i,j , Zj, β, uj)p(Xi,j1 = k, γ1)p(Xi,j2 = t, γ2)

Things are analytically more complex if we have missing values in continuous covariates. In this case, we must integrate over the distribution of the missing covariate instead of summing. For example, considering again patient i in cluster j, with a missing normally distributed covariate X_i,j1 , its contribution to the likelihood is:

Lij = Z +∞

−∞

f (Yi,j|X2,...,qi,j , Zj, β, uj)f (X1, µ, σ2)dX1

with µ and σ2 _{parameters of the distribution of X}1_{. The problem is that a closed form} solution for this integral is rarely available. Instead, we have to approximate the solution through numerical integration, for example using quadrature or adaptive quadrature, or to use Laplace approximations. The more continuous covariates with missing data we have, the more difficult it becomes to approximate this integral precisely. Furthermore, the integral needs to be recalculated as we search the parameter space for the maximum likelihood estimates.

Depending on the number and different kinds of partially observed covariates, finding a good joint model for the covariates and calculating the values of the parameters that maximize its likelihood becomes very challenging. Some situations can be solved with ad hoc methods, e.g. imagine that we have a model with covariate X1 _{continuous and X}2 _{and X}3 _{binary, subject} to missingness. In this case the contribution to the likelihood for subject (i, j) is:

Lij = Z +∞ −∞ 1 X k=0 1 X t=0 f (YI,J|Xi,j1 , X 2 i,j, X 3 i,j, X 4,...,q I,J , ZJ, β, uJ)f (Xi,j1 , µ, σ 2_)p(X2 i,j = k)p(X 3 i,j = t)dX 1 (4.4.1)

This problem can be greatly simplified in some particular situations, for example if the sub- stantive model is the general location model, introduced by (Olkin and Tate, 1961). Broadly, the idea behind this model is to define a contingency table according to the values of the binary variables. In this case:

X2 _{= 0 X}2 _{= 1 total} X3 _{= 0 n} 1 n2 n1+ n2 X3 _{= 1 n} 3 n4 n3+ n4 total n1+ n3 n2+ n4 n

We then give to each cell c the corresponding probability pc and, given that unit (i, j) belongs to cell c, the continuous variable follows a normal model with cell-specific mean and variance.

Therefore, this decomposition of the problem makes the integral (4.4.1) much simpler to cal- culate; however, the general location model is rarely (if ever) our substantive model. In a more general situation solving (4.4.1) is not so straightforward. Finding a closed form solution for the integral is even more difficult with increasing number of partially observed variables and possibly missing data patterns.

Missing data could also occur in Zs, i.e. in covariates that are included in the multilevel model with a random effect, further complicating the situation.

Furthermore, once we have our likelihood, we still need to maximize it in order to get the desired estimates. This process of maximizing the Full Likelihood directly is called Full Information Maximum Likelihood (FIML), also known as direct maximum likelihood. In order to calculate the estimates for the standard errors, we need to calculate the second derivative of the log- likelihood at its maximum, and this is often numerically complex. There are some particular cases of multilevel models where a closed form solution can be obtained without relying on numerical approximations, and these are mainly linear mixed models and the LISREL model, essentially a latent normal variable model, which is very common in Structural Equation Modeling (Skrondal and Rabe-Hesketh, 2004, Chap. 6).

FIML is implemented for example in SAS PROC CALIS and in MPlus. The problem with PROC CALIS, is that it can only handle normal data. The only commercial software able to do FIML in presence of missing data in predictors in generalized linear models is MPlus, which can deal with missing data in continuous, categorical or counting variables, in responses and in

covariates at level 1, but still not at higher levels. In case of missing data in level 2 covariates, the software authors suggest putting these as responses in the model, turning the substantive model into a multivariate model.

Muth´en and Brown (2001) and successively Little and Rubin (2002), proved that under the MAR mechanism FIML estimates and standard errors are unbiased. However, everything we said so far about maximum likelihood methods rely on the assumption that data are MAR. This is not always true, and if data are MNAR we need to model also the missing data mechanism.

One last issue, but practically very important, with maximum likelihood approaches is the difficulty in including auxiliary variables. These are variables that are not to be included in the substantive model, but which can be important to both recover information and improve the plausibility of the MAR assumption (Spratt et al., 2010). If present, they need to be properly included in the model likelihood and integrated out, presenting additional difficulties.

As explained in Section 3.2.3 for the simple case of single level data, another way of maximizing the likelihood is using the EM algorithm; however the same issues arise, i.e. possibly slow convergence of the algorithm and difficulty to calculate the SEs, which make the EM algorithm hardly widely applicable.

All the discussion so far focussed on calculating maximum likelihood estimates for multilevel models in presence of sporadically missing data only. The additional problems, that are very common in IPD-MA, of handling systematically missing variables and allowing for heterogene- ity between different studies, would even complicate things further.