On Systematic Errors in the Least Squares Regression Analysis, with Application to the Atmospheric Effects on the Cosmic Radiation

It is shown that if the variables employed in a least squares regression analysis are subject to random errors of measurement, the expectation values of the partial regression coefficients, of the partial correlation coefficients and of the multiple correlation coefficient may all differ from those which would have existed, had no errors been present. If there is no intercorrelation between the errors of different variables, random errors in a given variable always reduce the numerical expectation values of the corresponding partial regression and correlation coefficients. Coefficients corresponding to other variables may, however, be influenced in either direction depending on the intercorrelations between the variables. The expectation value of the multiple correlation coefficient is reduced by errors in any variable. The general case, in which the errors of different variables are intercorrelated, has also been briefly discussed.

The problem of determining the atmospheric effects on the cosmic radiation is then discussed. It is shown that some previously unexplained discrepancies between empirical and theoretical estimates, and also between empirical estimates obtained from the study of day-to-day variations, and such obtained from the seasonal variations of the cosmic-ray intensity, are probably due to systematic effects of random errors in the aerological data employed in the regression analysis.

Estimates of error variances and covariances of aerological data from the upper troposphere and the lower stratosphere have been obtained by analysing differences between data from two closely situated stations on Spitzbergen. They have then been used to obtain corrected estimates of the cosmic-ray atmospheric effects, which are now found to agree fairly well with the theoretical ones.