# Principles Of Linear Regression

In the past chapters we discussed different statistical methods to test statistically experimental data from clinical trials. We did not emphasize correlation and regression analysis. The point is that correlation and regression analysis test correlations, rather than causal relationships. Two samples may be strongly correlated e.g., two different diagnostic tests for assessment of the same phenomenon. This does, however, not mean that one diagnostic test causes the other. In testing the data from clinical trials we are mainly interested in causal relationships. When such assessments were statistically analyzed through correlation analyses mainly, we would probably be less convinced of a causal relationship than we are while using prospective hypothesis testing. So, this is the main reason we so far did not address correlation testing extensively. With epidemiological observational research things are essentially different: data are obtained from the observation of populations or the retrospective observation of patients selected because of a particular condition or illness. Conclusions are limited to the establishment of relationships, causal or not. We currently believe that relationships in medical research between a factor and an outcome can only be proven to be causal if the factor is introduced, and, subsequently, gives rise to the outcome. We are more convinced when such is tested in the form of a controlled clinical trial. A problem with multiple regression and logistic regression analysis as method for analyzing multiple samples in clinical trials is closely related to this point. There is always an air of uncertainty about such regression data. Interventional trials usually use hypothesis-testing and 95 % confidence intervals (CIs) of the data to describe and analyze data. They use multiple regression for secondary analyses, thus enhancing the substance of the research, and making the readership more willing to read the report, rather than proving the primary endpoints. Regression analysis may not be so important to randomized clinical trials, it is important to one particular study design, the crossover study, where every patient is given in random order test-treatment and standard treatment (or placebo). Figure 1 gives three hypothesized examples of crossover trials. It can be observed from the plots that in the left and right graph there seems to be a linear relationship between treatment one and two. The strength of relationship is expressed as r (=correlation coefficient) which varies between −1 and +1. The strongest association is given by either −1 or +1 (all data exactly on the line), the weakest association 0 (all data are parallel either to x-axis or to y-axis, or half one direction, half the other. A positive correlation in a crossover study is observed if two drugs from one class are compared. The patients responding well to the first drug are more likely to respond well to the second. In contrast, in crossover studies comparing drugs from different classes a negative correlation may be observed: patients not responding well to one class are more likely to respond well to the other.

## Keywords

Multiple Linear Regression Peripheral Vascular Disease Crossover Trial Independent Determinant Unstandardized Coefficient
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