Relative risk estimation with clustered/longitudinal data: solving convergence issues in fitting the log binomial generalized estimating equations (GEE) Presenter: Zhu Chao Co-authors: David W Hosmer Jim Stankovich Karen Wills Leigh Blizzard

Introduction Log binomial generalized estimating equations (GEE) 2 Log binomial generalized estimating equations (GEE) It is possible to estimate risk and risk ratios in clustered/longitudinal data by fitting a log binomial GEE. However, the estimating equations may fail to converge if the iterations commence from inappropriate starting value, the fitted mean value of one or more observations is equal to unity. Starting values are inappropriate if the initial fitted mean value of one or more observations is greater than unity. This problem can be rectified by providing improved initial values. Solving the second problem requires a specialised approach.

Method Exact method The exact method (Petersen and Deddens, 2010; Zhu et al, 2019) solves convergence difficulties in estimation of the log binomial model for independent data by re- parameterizing the covariates to eliminate the boundary vectors (those covariate vectors with fitted probabilities equal to unity). Because of similarities in functional form and estimation, we postulated that the exact method could be used to solve convergence difficulties encountered in fitting the log binomial GEE for clustered/longitudinal data. Issues in applying the exact method to the log binomial GEE Identifying the boundary vector(s), Determining which optimisation criterion should be used to confirm the solutions.

Example The data are for a subset of 1000 subjects with a burn injury sampled from the National Burn Repository dataset by Hosmer, Lemeshow and Sturdivant (2013). The subjects were treated in 40 different burn facilities (FACILITY). Our goal was to estimate the probability of death (DEATH) taking account of the correlation within each of the 40 burn facilities. The covariates are: age (AGE), total burn surface area (TBSA), inhalation injury involved (INH_INJ, 0 = no, 1 = yes), race (RACE: 0 = non-white, 1 = white), flame involved in burn injury (FLAME: 0 = no, 1 = yes) and gender (GENDER: 0 = male, 1 = female). A log binomial GEE applied to these data failed to converge. There are four boundary vectors representing four subjects from different burn facilities with fitted mean values of unity when evaluated at the convergent solution. Table 1: Four boundary vectors involved in the model. DEATH AGE GENDER TBSA INH_INJ FLAME RACEC FACILITY 1 81.7 34 47 60.7 97 4 86.1 83 19 54.6 86 13

Coefficient errors (%)† Example Re-parameterizing the data to eliminate these boundary vectors in accordance with the exact method produced a convergent solution. Table 2: Convergent log binomial GEE solution by the exact method (with coefficient estimates from the Poisson GEE Shown for comparison). Log binomial GEE (95% CI) Poisson GEE Coefficient errors (%)† AGE 0.0263 (0.0244, 0.0283) 0.0393 (0.0330, 0.0456) 49.43 TBSA 0.0159 (0.0148, 0.0171) 0.0284 (0.0254, 0.0314) 78.62 INH_INJ 0.4461 (0.4131, 0.4791) 0.6207 (0.3704, 0.8710) 39.01 RACE -0.3359 (-0.3608, -0.3111) -0.2183 (-0.4844, 0.0478) 35.01 FLAME 0.7316 (0.2504, 1.2127) 0.5076 (0.0260, 0.9893) 30.62 GENDER -0.1144 (-0.1229, -0.1060) -0.1838 (-0.3567, -0.0108) 60.66 Constant -3.8715 (-4.2755, -3.4674) -4.7488 (-5.2865, -4.2112) 22.66 † Coefficient errors of the Poisson GEE calculated as the absolute percentage difference relative to the log binomial GEE estimates of the coefficient. Also shown are the Poisson GEE estimates of the coefficients, which differed by 23 – 79%.

Discussion and Conclusion Inadmissible starting values are one source of failure of standard fitting algorithms in fitting the log binomial GEE. Boundary solutions are responsible for the remaining failures of standard fitting algorithm. The exact method can be used to overcome convergence difficulties caused by boundary vectors in the log binomial GEE. The coefficients and standard errors of the commonly-used Poisson GEE method are biased and can be badly biased. We recommend estimation of the log binomial GEE using the exact method.