The effects of number of samples and random error on the factor analysis/multiple regression (FA/MR) receptor modeling technique

Abstract

The effects of normally distributed random errors and of sample size on the stability of FA/MR results were examined as applied to a data set from the ATEOS studies. Three levels of normally distributed random errors were added to the original data set to produce five perturbed data sets for FA testing and a total of 10 for testing the MR results. Reductions of sample size from 138 cases to 70 and 35 cases (24-h daily samples) were performed both randomly and systematically. The addition of random errors reduced squared multiple correlation coefficients to the same extent for different sample sizes. The effect of a reduction in sample had a more significant impact on the standard errors of regression coefficients than did added random errors of up to ± 30% of the measured independent variables. Reducing the data sets to 70 and 35 cases resulted in nonsignificant coefficients for a variable whose t-value was relatively small in the data set with 138 cases. This analysis demonstrated the usefulness of the degrees of freedom criteria suggested by Henry (1984, Atmospheric Environment 18, 1507–1515) for minimum sample size in multivariate analysis. The results also showed that data sets with smaller sample sizes tend to give larger R 2 s than the data sets with larger samples sizes, due to their fewer degrees of freedom. It is recommended that the relative significance of parameters that can contribute to the variance of the data set be investigated.