title: A review of statistical estimators for risk-adjusted length of stay: analysis of the Australian and new Zealand Intensive Care Adult Patient Data-Base, 2008-2009.
abstract: BACKGROUND
  For the analysis of length-of-stay (LOS) data, which is characteristically right-skewed, a number of statistical estimators have been proposed as alternatives to the traditional ordinary least squares (OLS) regression with log dependent variable.
  
  
  METHODS
  Using a cohort of patients identified in the Australian and New Zealand Intensive Care Society Adult Patient Database, 2008-2009, 12 different methods were used for estimation of intensive care (ICU) length of stay. These encompassed risk-adjusted regression analysis of firstly: log LOS using OLS, linear mixed model [LMM], treatment effects, skew-normal and skew-t models; and secondly: unmodified (raw) LOS via OLS, generalised linear models [GLMs] with log-link and 4 different distributions [Poisson, gamma, negative binomial and inverse-Gaussian], extended estimating equations [EEE] and a finite mixture model including a gamma distribution. A fixed covariate list and ICU-site clustering with robust variance were utilised for model fitting with split-sample determination (80%) and validation (20%) data sets, and model simulation was undertaken to establish over-fitting (Copas test). Indices of model specification using Bayesian information criterion [BIC: lower values preferred] and residual analysis as well as predictive performance (R2, concordance correlation coefficient (CCC), mean absolute error [MAE]) were established for each estimator.
  
  
  RESULTS
  The data-set consisted of 111663 patients from 131 ICUs; with mean(SD) age 60.6(18.8) years, 43.0% were female, 40.7% were mechanically ventilated and ICU mortality was 7.8%. ICU length-of-stay was 3.4(5.1) (median 1.8, range (0.17-60)) days and demonstrated marked kurtosis and right skew (29.4 and 4.4 respectively). BIC showed considerable spread, from a maximum of 509801 (OLS-raw scale) to a minimum of 210286 (LMM). R2 ranged from 0.22 (LMM) to 0.17 and the CCC from 0.334 (LMM) to 0.149, with MAE 2.2-2.4. Superior residual behaviour was established for the log-scale estimators. There was a general tendency for over-prediction (negative residuals) and for over-fitting, the exception being the GLM negative binomial estimator. The mean-variance function was best approximated by a quadratic function, consistent with log-scale estimation; the link function was estimated (EEE) as 0.152(0.019, 0.285), consistent with a fractional-root function.
  
  
  CONCLUSIONS
  For ICU length of stay, log-scale estimation, in particular the LMM, appeared to be the most consistently performing estimator(s). Neither the GLM variants nor the skew-regression estimators dominated.
authors.affiliation: Department of Intensive Care Medicine, The Queen Elizabeth Hospital, Woodville SA 5011, Australia. john.moran@adelaide.edu.au
authors.forename: John L
authors.initials: JL
authors.lastname: Moran
authors.forename: Patricia J
authors.initials: PJ
authors.lastname: Solomon
mesh_terms: D018806:APACHE; D001315:Australia; D003422:Critical Care / Q000706:statistics & numerical data*; D003627:Data Interpretation, Statistical; D016208:Databases, Factual; D006801:Humans; D007362:Intensive Care Units; D007902:Length of Stay / Q000706:statistics & numerical data*; D015233:Models, Statistical; D009520:New Zealand; D012044:Regression Analysis; D012720:Severity of Illness Index