Regression procedures for relationships between random variables

Finding association and relationship among measured random variables is a common task in biological research and regression analysis plays a major role for this purpose. Which type of regression model to use depends on the nature of the predictor variable and on the purpose of the analysis. The most commonly used model ordinary least squares (Type I regression) applies when measurement errors affect only the response variable. The predictor is either without measurement errors or under the control of the investigator. However, if the predictor variable does have measurement errors then an 'errors in both variables' or Type II regression model is more appropriate and takes into account variation in both the response and the predictor variables simultaneously. When repeatability error variances in both variables are known then the maximum likelihood solution (or Deming regression) is appropriate. If the measurement error variance is unknown then a suitable Type II model should be selected. In this respect, Bartlett's three-group method, major axis regression, standard major axis regression (also called reduced major axis), ranged major axis and instrument variable methods are discussed in relation to energy balance studies with cattle.