Ordinary Least Products: Let's welcome back an old friend

The inappropriate use of statistical methods within the scientific and medical literature is unfortunately not new. Although the increase in specialized statistical reviewers for most journals has gone some way to alleviating this problem, limited statistical training and a reliance on basic biostatistical methods by many scientific researchers translates into substandard approaches to data analysis still appearing in many published studies. A tendency for researchers to then reapply these methods to their own data without first checking their validity helps perpetuate the errors. In this issue of CEPP, researcher and statistician John Ludbrook discusses the use, or rather lack of use, of Model II regression. His review of the literature suggests that it is seldom used. Instead, researchers instinctively turn to the Model I approach, known more commonly as ordinary least squares (OLS) linear regression. This, despite the fact (or perhaps because of it! ) that most datasets in existence have independent variables measured with error to a greater or lesser extent. In these circumstances, the essential assumption of the Model I method, that the residuals of the model are exogenous and not associated with the predictors, is violated and results are therefore biased. Perhaps the most widespread misuse of OLS is seen in method comparison studies and/or the calibration of one method against another. For example, if we compare two methods for measuring blood pressure, measurements from both methods (represented by the xand y-variables) will contain error compared with the true underlying blood pressure of the subject at the moment of measurement; error that is due to either the inherent error of the instrument, error induced by the observer or both. It is simple to demonstrate that the slope of the resulting linear relationship is biased towards zero (i.e. underestimated). This effect is known as measurement error attenuation bias and increases when either systematic or proportional bias between the two methods is present. More generally, in any circumstance where both of the two variables in a simple linear regression equation are subject to error, or are random rather than fixed values (i.e. not controlled by the researcher vs controlled by the researcher), some form of ‘errors in variables’ or ‘measurement errors’ regression is required. Two approaches recommended by Ludbrook are Model IIA regression (also named ordinary least products (OLP) or major axis regression) and Model IIB regression (orthogonal or reduced major axis regression). So what is the difference between Model I and Model II regression, when should we use each, how do we perform Model II regression and which form (IIA or IIB) should we use? Model I (OLS) regression minimizes the distances between the observed y data and the estimated regression line, which passes through the centroid of the bivariate x and y distribution, although its slope still varies according to which of the two variables is chosen to be y. As referred to above, the method requires strict exogeneity of the residual errors, which also implies no correlation between x and the residual errors of the model. As such, the model effectively assumes no measurement error in x. That said, estimation using OLS is still considered to be a reasonable approach in circumstances where it is known that the error in x is small compared with that in y (say, approximately 1⁄3) and if the aim is simply to obtain predictions of y for given fixed values of x. However, for defining the form of the linear relationship (slope) between two error-prone variables, methods are required in which the error between the observed and predicted values is minimized for both the x and the y data. This approach provides a regression line that bisects the ellipsoid of data created by the two variables and is constant irrespective of which variable we choose as the regressor. Indeed, the estimated slope from OLP is the geometric mean of the slopes obtained from OLS. That is: