Daniel J. Mundfrom

Minimum Sample Size Recommendations for Conducting Factor Analyses

There is no shortage of recommendations regarding the appropriate sample size to use when conducting a factor analysis. Suggested minimums for sample size include from 3 to 20 times the number of variables and absolute ranges from 100 to over 1,000. For the most part, there is little empirical evidence to support these recommendations. This simulation study addressed minimum sample size requirements for 180 different population conditions that varied in the number of factors, the number of variables per factor, and the level of communality. Congruence coefficients were calculated to assess the agreement between population solutions and sample solutions generated from the various population conditions. Although absolute minimums are not presented, it was found that, in general, minimum sample sizes appear to be smaller for higher levels of communality; minimum sample sizes appear to be smaller for higher ratios of the number of variables to the number of factors; and when the variables-to-factors ratio exceeds 6, the minimum sample size begins to stabilize regardless of the number of factors or the level of communality.

Sample Sizes When Using Multiple Linear Regression for Prediction

When using multiple regression for prediction purposes, the issue of minimum required sample size often needs to be addressed. Using a Monte Carlo simulation, models with varying numbers of independent variables were examined and minimum sample sizes were determined for multiple scenarios at each number of independent variables. The scenarios arrive from varying the levels of correlations between the criterion variable and predictor variables as well as among predictor variables. Two minimum sample sizes were determined for each scenario, a good and an excellent prediction level. The relationship between the squared multiple correlation coefficients and minimum necessary sample sizes were examined. A definite relationship, similar to a negative exponential relationship, was found between the squared multiple correlation coefficient and the minimum sample size. As the squared multiple correlation coefficient decreased, the sample size increased at an increasing rate. This study provides guidelines for sample size needed for accurate predictions.

A Monte Carlo comparison of the Type I and Type II error rates of tests of multivariate normality

Many multivariate statistical methods call upon the assumption of multivariate normality (MVN). However, many researchers fail to test this assumption. This omission could be due to either ignorance of the existence of tests of MVN or confusion about which test to use. Although at least 50 tests of MVN exist, relatively little is known about the power of these procedures. The purpose of this study was to examine the power of 13 promising tests of MVN with a Monte Carlo study. Ten thousand data sets were generated from several multivariate distributions. The test statistic for each procedure was calculated and compared with the appropriate critical value. The number of rejections of the null hypothesis of MVN was tabled for each situation. No single test was found to be the most powerful in all situations. The use of the Henze–Zirkler test is recommended as a formal test of MVN. Supplementary procedures such as Mardias skewness and kurtosis measures and the chi-square plot are also recommended for diagnosing possible deviations from normality.

Power Ratings for NCAA Division II Football

There is an abundance of entities that disseminate power ratings for NCAA Division I-A and I-AA football teams. This is not true for the NCAA Division II level. Statistical data on all NCAA Divisions II football games for two years were examined to develop a rating system. Several statistical techniques were performed on the data. Ultimately, the ratings were based on won-lost percentage, margin of victory, strength of schedule, and quality wins. Results are given for the 2001 and 2002 seasons. Ratings for the 2002 season for Divisions I-A and I-AA are also given using the techniques developed.