Improving Linearity in Health Science Investigations

By definition, correlation between X and Y is the ratio of Cov(X,Y) and product of SD(X)and SD(Y). [...]average of k-number of correlations Г =(I_(j=1)Ak raja )/k is meaningless for correlations with mixed signs and of same sample size (Field, 2003). Correlation and Linearity Observations: If X is increasing and Y is decreasing, r_XY is negative viz. r_(X,1/X) , r_(X,CosX), etc. r_(X,f(X)) |>0.92 for non-linear f(X) = ХЛ2,ХЛ3, [log]_10AX, Cos X and Sin X despite non-linear relationship between X and f(X) Maximum improvement in correlation was observed for X and f(X) = Sin X where r_(X,SinX)=0.99982 followed by r_(X,CosX)=(-) 0.97156. [...]trigonometric transformations like f(X) = Sin X or f(X) = Cos X tend to improve absolute magnitude of correlation coefficient. [...]absolute value of r_XY could be high i.e. |r_XY |=1 even if X and Y are related by a non-linear fashion. S_E and results of AD-test for normality of error scores of the chosen non-linear functions of X are shown in (Table 3). [...]the assumption of normality of error of prediction needs to be verified for fitting regression line and not merely the magnitude of correlation.