#### Date of Award

1980

#### Degree Type

Thesis

#### Degree Name

Master of Arts in Psychology

#### Department

Psychology

#### First Advisor

Wayne F. Velicer

#### Abstract

A common problem encountered in the applied use of principal components analysis (PCA) as a data reduction technique is the determination of the number of components to retain. This problem is particularly acute when some method of rotation is to be employed. This study examined the consequences of employing each of four decision methods upon PCA's of correlation matrices made up of systematically differing numbers of variables and subjects and based upon known systematically differing underlying component structures. Three levels of number of variables (36, 72, 144), sample size (75, 150, 450) and number of components (3, 6, 12) were examined at each of two (.50 and .80) levels of component saturation. Within each example each component was defined by the same number of variables. No unique components nor complex variables were included in the set of computer generated data

It was found the different rules lead to dramatically different results. In general, the scree test (SCREE) and the minimum average partial test (MAP) were the most accurate. The Bartlett test (BART) was somewhat less accurate and the eigenvalue greater than one rule (Kl) was quite inaccurate. Increases in sample size and component saturation improved the performance of the rules. Increases in the number of variables examined increased the accuracy of SCREE and MAP, did not effect BART markedly but drasticly decreased Kl's accuracy. Kl often greatly overestimated the number of components to retain.

Combinations of factors lead to the most inaccurate cases for each rule. Kl performed worst at low component saturation with high numbers of variables. MAP and SCREE performed worst at low component saturation with small samples, small numbers of variables and a large number of components. BART performed worst when the number of variables was almost equal to the sample size.

The single and interacting effects of the variables upon the rules is discussed. Guidelines for applied use are presented. Useful future research in this area is indicated.

#### Recommended Citation

Zwick, William R., "Factors Influencing Four Rules for Determining the Number of Principal Components to Retain" (1980). *Open Access Master's Theses.* Paper 1577.

https://digitalcommons.uri.edu/theses/1577