# hullEFA: Hull method for selecting the number of common factors In EFA.MRFA: Dimensionality Assessment Using Minimum Rank Factor Analysis

## Description

Performs the Hull method (Lorenzo-Seva, Timmerman, & Kiers, 2011), which aims to find a model with an optimal balance between model fit and number of parameters.

## Usage

 1 2 hullEFA(X, maxQ, extr = "ULS", index_hull = "CAF", display = TRUE, graph = TRUE, details = TRUE)

## Arguments

 X Raw sample scores. maxQ Maximum of dimensions to be tested. By default it will be determined by the Parallel Analysis advised dimensions, but the user can define it manually. extr Extraction method, the two options available being: "ULS" (Unweigthed Least Squares, by default) and "ML" (Maximum Likelihood). index_hull The index that will be used for determining the number of dimensions. The available options are the following: "CAF", "CFI""RMSEA", being "CAF" by default. display Determines if the output will be displayed in the console, TRUE by default. If it is TRUE, the output is returned silently and if it is FALSE, the output is returned in the console. graph Request a plot representing the Hull curve. details If detailed table will be displayed, containing the factors outside the convex Hull.

## Details

hullEFA is based on the procedure proposed by Lorenzo-Seva, Timmerman, & Kiers (2011) which is designed for assessing the dimensionality of a variable set. The hull heuristic was originally proposed by Ceulemans & Kiers (2006) in the context of model selection in multiway data analysis.

The hull analysis is performed in four main steps:

1. The range of factors to be considered is determined.

2. The goodness-of-fit of a series of factor solutions is assessed.

3. The degrees of freedom of the series of factor solutions is computed.

4. The elbow is located in the higher boundary of the convex hull of the hull plot.

The number of factors extracted in the solution associated with the elbow is considered the optimal number of common factors.

In the Lorenzo-Seva, Timmerman, & Kiers (2011) simulation study, the Hull method outperformed the other selected methods in recovering the corrrect number of major factors.

## Value

 Matrix Matrix containing the results of the Hull method using for the selected index. n_factors Number of advised dimensions by the selected index.

## Author(s)

David Navarro-Gonzalez

Urbano Lorenzo-Seva

## References

Lorenzo-Seva, U., Timmerman, M. E., & Kiers, H. A. (2011). The Hull Method for Selecting the Number of Common Factors. Multivariate Behavioral Research, 46(2), 340-364. doi: 10.1080/00273171.2011.564527

Ceulemans, E., & Kiers, H. A. L. (2006). Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method. British Journal of Mathematical and Statistical Psychology, 59: 133–150. doi: 10.1348/000711005X64817

## Examples

 1 2 3 4 ## Perform the Hull method defining the maximum number of dimensions to be tested by the ## Parallel Analysis + 1 rule, with Maximum Likelihood factor extraction method and CAF ## as Hull index. hullEFA(IDAQ, extr = "ML")

### Example output

HULL METHOD - CAF INDEX

q      f          g       st
0      0.2509   253       0.0000
1      0.4010   230       3.8330
2*     0.4371   208
3      0.4742   187       1.6631
4      0.4947   167       0.0000