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# Exploratory Factor Analysis of an Incomplete Correlation Matrix

Exploratory factor analysis (EFA) is a widely used multivariate statistical method in social and behavioral sciences. Researchers sometimes need to factor-analyze correlation matrices with unavailable elements. One situation to encounter such incomplete correlation matrices is when researchers synthesize information from multiple studies that involves a number of questionnaires designed for measuring a complex construct. For example, Daly (2014) investigated narcissism with a meta-analysis, which required to factor-analyze a 41 by 41 correlation matrix with 63$\%$ of elements unavailable. Another situation is to use a planned missingness design to reduce participant burden. There are currently no methods of factor analyzing incomplete correlation matrices.

I propose a one-step estimation method to conduct EFA with an incomplete correlation matrix. I adapt the iterated principal factor method to estimate unrotated factor loadings while accommodating missing correlation. The usual EFA identification conditions do not carry over to the current situation because the correlation matrix contains unavailable elements. I thus numerically check whether an EFA model is identified with an incomplete correlation matrix. In particular, I compute the rank of the Hessian matrix of the discrepancy function. If the rank is smaller than the effective number of EFA parameters, the EFA model is not identified.

I also use the parametric bootstrap to determine the number of factors and estimate standard errors. I generate bootstrap samples according to the EFA model with parameter values being the sample estimates. I reproduce the missing correlation pattern by mimicking the scale pairing process of empirical studies. I obtain bootstrap versions of discrepancy function values, rotated factor loadings, and factor correlations. These bootstrap replicates form reference distributions for the test statistic and standard errors of EFA parameters.

I illustrate the method using two studies. The first study entails the analyses at the population level to explore identifiability of EFA models with different different missing patterns in the incomplete correlation matrix. Incomplete correlation matrices whose observed elements spread across rows and columns allowed identifying larger models than those matrices whose observed elements are clustered. The second study entails simulated samples and explores the statistical properties. In addition to identifiability of EFA models, factor congruence coefficients, standard errors, and bootstrap tests are explored. Studies showed that (1) one may reduce the participant burden by half while still adequately recovering factors; (2) parametric bootstrap can provide adequate standard error estimates; and (3) bootstrap tests may underestimate the number of factors.

## History

## Date Created

2021-12-29## Date Modified

2022-01-27## Defense Date

2021-07-29## CIP Code

- 42.2799

## Research Director(s)

Guangjian Zhang## Committee Members

Ying Cheng Zhiyong Zhang Lijuan Wang## Degree

- Doctor of Philosophy

## Degree Level

- Doctoral Dissertation

## Alternate Identifier

1291625175## Library Record

6157793## OCLC Number

1291625175## Program Name

- Psychology, Research and Experimental