Table 1 The pseudocode for the proposed COLA-GLMM algorithm

COLA-GLMM Algorithm

Input: Patient-level clinical data from K participating sites, including pre-specified set of covariates x, well-defined outcome of interest y.

Output: Fixed effects \(\hat{{\boldsymbol{\beta }}}\)

  for each site \(k=1,\ldots ,K\), do

     1. Calculate q-dimensional vector \({{\boldsymbol{C}}}_{k}=\{{c}_{k1},\ldots ,{c}_{{kq}}\}\), where \({c}_{{kj}}=\mathop{\sum }\limits_{i=1}^{{n}_{k}}I\left({{\boldsymbol{x}}}_{{ki}}={{\boldsymbol{x}}}^{(j)}\right)\)

     2. Calculate q-dimensional vector \({{\boldsymbol{S}}}_{k}=\{{s}_{k1},\ldots ,{s}_{{kq}}\}\), where \({s}_{{kj}}=\mathop{\sum }\limits_{i=1}^{{n}_{k}}{y}_{{ki}}I\left({{\boldsymbol{x}}}_{{ki}}={{\boldsymbol{x}}}^{(j)}\right)\)

     3. Calculate q × p-dimensional matrix \({{\boldsymbol{U}}}_{k}=\{{{\boldsymbol{u}}}_{k1}^{T},\ldots ,{{\boldsymbol{u}}}_{{kq}}^{T}\}\), where \({{\boldsymbol{u}}}_{{kj}}=\mathop{\sum }\limits_{i=1}^{{n}_{k}}{{\boldsymbol{x}}}_{{\boldsymbol{ki}}}^{T}{y}_{{ki}}I\left({{\boldsymbol{x}}}_{{ki}}={{\boldsymbol{x}}}^{(j)}\right)\)

     4. Transfer Items 1-3 to the coordinating center

  end

Within the coordinating center, reconstruct the log-likelihood function in Eq. (4) and estimate the parameter of interest, \(\hat{{\boldsymbol{\beta }}}\)