A simple case of over-fitting. Consider that a researcher is studying the effect of TP53 expression level (x) on survival (y) of a group of breast cancer patients. (a) Simple regression: from knowing the expression level and survival (the variables) for each patient, the relationship between the two variables can be modelled with a simple univariable linear regression equation of the form y = a + bx, where a is the interception point with the y axis and b is the slope of the equation line. Applying this equation to a TP53 expression value will result in a new y value that corresponds to predicted survival. However, the equation seldom gives a perfect match between the real survival (triangles) and the predicted survival from the equation (circles) for any given x. In general, the closer the predicted values are to the real values, the better the equation (model) is in explaining the observations or the better the 'fit' of the model. The fit of the model is therefore used as a measure of its performance. (b) Over-fitting: an equation that is dependent on only two observations will always result in a line that passes between these two observations, giving an artificially perfect match between the predicted and the observed data. This represents meaningless good performance of a model or 'over-fitting'. This results from using too few observations (patients) per variable (gene) studied. To make a more complex 'multi-variable analysis' requires even more observations (patients) required to avoid over-fitting. In practice, a working ratio of 10 patients for every variable studied is recommended. However, in microarray studies few patients are evaluated for many thousands of genes.