Fitting the multivariate meta-analysis

and i is a vector of random sampling errors associated with the study i which is independent of δiand it is assumed to that i ∼ M V N (0, S_i). It can easily be shown that marginally yi∼ M V N (X_iβ, V)where V = ∆+Si. It is noted that the structure of V may differ depending on the main interest of the model.

3.5.2 Fitting the multivariate meta-analysis

n this subsection, we are going to describe only the random effect model since the fixed effect is taken as a special case where the heterogeneity is equal to zero (∆ = 0) (Mavridis & Salanti, 2013). the main interest is on the effect sizes, the vector of summary effect for T outcomes ( for simplicity let us make Xiβ = µ) The focus of study interest is only on the random-effects model; the fixed effect model is taken as a special case where the heterogeneity is zero (Mavridis & Salanti, 2013). The main interest is on the effect sizes, the vector of the summary effects for the T outcomes (for this case lets us make µ = xitβ; and the uncertainty which can be estimated by the T × T variance-covariance matrix V and ∆.

Likelihood

If we assume that all studies are independent experiments, as suggested by Mavridis

& Salanti (2013), the log-likelihood to estimate the model parameter can be given as:

L ≈ −1

2Σⁿ_i=1log|∆+S_i| −1

2Σⁿ_i=1⁰_i(∆+S_i)⁻¹_i

Then according to Ishak et al. (2007); Trikalinos & Olkin (2012); Mavridis & Salanti (2013) and Steel & Kammeyer-Mueller (2008) stated that this likelihood can be max-imized by a numerical iteration methods using the expectation-maximization nu-merical, Newton-Raphson or Fisher scoring algorithms subject to restriction that ∆ is positive semi-definite. Although this method is good on estimating the estimates of interest, it can be computationally intensive and time demanding.

Maximum likelihood estimates

Assuming that all studies report the same outcomes and there are no missing values, then the summary estimates can be obtained by maximizing the likelihood:

µ =b the summary estimates are approximately normally distributed with the variance-covariance matrix:

V = (Σb ⁿ_i=1( b∆+S_i)⁻¹)⁻¹ (3.41)

The parameter µ and ∆ are estimated iteratively in expectation-maximization nu-merically algorithm with equation 3.40 being one of the 2 steps. An approximation (1 − α)100%confidence interval can be obtained forbµ_j asµb_j± z^α

2

q

Vb_{j j}, where bV_{j j}is the j-diagonal element of the bVmatrix. The alternative method that can be used is the quantile method based on the t-distribution (Mavridis & Salanti, 2013).

In case of missing outcomes 3.40 cannot be used directly to compute the estimates as the dimension of matrix ∆ and Siwill not be the same across studies (Mavridis

& Salanti, 2013). This issue can be resolved by imputing those entries in the covari-ance matrix by allocating very large within-study varicovari-ance to the missing outcomes and zero within-study correlation, ensuring that missing outcomes contribute neg-ligible weight and information (Mavridis & Salanti, 2013; Trikalinos & Olkin, 2012) in the estimation of the combined effect sizes. Most of the difficulties will lie in the

3.5. Dependence of effect sizes between time points

estimation of between-study covariance matrix parameter ∆, which will be used in summary effect sizes estimation in equation 3.40.

In maximum likelihood estimation theory, the restricted maximum likelihood (REML) play an important role in producing unbiased estimate of variance and covariance parameter matrix in mixed models (Mavridis & Salanti, 2013). In fact REML has the capacity to deal with unbalanced data by design or that due to missing observa-tions. In this case it can also be used to estimate the covariance matrix ∆ in the case of meta-analysis. The modified log-likelihood is given by:

RL ≈ L − 1

Using Cholesky decomposition, ∆ can be ensured to be positive definite when max-imizing the likelihood function, then to compute µ and V, b∆can be used (Mavridis

& Salanti, 2013).

Methods of moment

Method of moment estimation is another approach that can be used to estimate µ and V, for that reason theµband bVwill still be defined as in equation 3.40 and 3.41, but ∆ is estimated by employing a multivariate extension of the Q ∼ statistic and the method of moments was extended by Jackson et al. (2010) from DerSimonian &

Laird (1986) method of moments

where Yj is weighted mean of Yij across studies and the weights are W_ij = 1

σ²_ij

Then on estimating the expectation, the Qj that will include estimation of τj and µj

is estimated as a weighted average with weights W_ij^∗ = 1

σ²_ij+τ_j².

When the multivariate approach will be similar to that of univariate. We can have p outcomes and Q is a matrix given by:

Q =

where this element denotes Qjj=P

iNjj In this case Njjdenotes the set of studies where only outcome j is reported and N_jj⁰ is the set of studies where both outcomes j and j⁰ are reported. Yj is the weighted average of outcomes over the studies reporting only the outcome j (with the weights W_ij across studies) and Y_j⁰_j is the weighted average of outcomes over the studies that report the outcomes of j and j⁰ (with the weightspWijW_ij⁰ across studies). If there are missing values, they can be, handled by replacing them with arbitrary val-ues with large within-study variance and summing all studies (Mavridis & Salanti, 2013; Jackson et al., 2010).

Jackson et al. (2010) show how to estimate ∆, by showing that it can be estimated by equating the expected value of Q, E(Q) = E. It can be shown that the diagonal Ejjare function of σij and τj, given as: a linear function of unknown between study variance τ_j², hence by equating Ejj = E(Q_(jj)) and we can solve the linear equation to estimate τ_j² in a similar way as in univariate case. Off-diagonal elements E_jj⁰ are a function of σij, σ_ij⁰, τ_j, τ_j⁰, ρ_τ,