Bivariate relationships incorporating method comparison: A review of linear regression methods

In this review, we describe and illustrate the selection and use of some appropriate regression models for bivariate statistical relationships. The most commonly used method, ordinary least squares (OLS) or type I regression, may be inappropriate when the predictor variable is subject to measurement errors since this violates a fundamental assumption of OLS and as a result estimates of slope are likely to be biased or attenuated. The y-axis intercept will be biased too as it is a function of slope estimate and the means of y-and x-variables. This bias can have some undesirable consequences if OLS regression parameters and/or functions of them are used further with meaningful interpretations. For example, in animal energy balance studies, slope estimate represents efficiency of metabolizable energy utilization for body mass growth or milk production in dairy cows. The x-axis intercept, a function of y-intercept and slope, gives an estimate of the animal's body mass maintenance energy requirement. The choice of an alternative type II or functional regression model (e.g. maximum likelihood solution, major axis, reduced major axis and others) depends on the availability and ratio of measurement or precision variances of both y-and x-variables; otherwise non-parametric models (e.g. Theil-Sen non-parametric regression or Bartlett's three-group method) can be used. When the ratio of y-and x-variable error variances is not constant over the data range then the reiterated weighted functional model as described by Ripley and Thompson in 1987 may be necessary. Application of these models and other tests (e.g. mean-square prediction error, analysis of concordance) in analytical method comparisons is outlined. Data scrutiny and outlier diagnostics are included because outliers affect most of the non-robust statistics. © 2016 CAB Internationa