nBartlett | R Documentation |
This function computes the Bartlett, Anderson and Lawley indices for determining the number of components/factors to retain.
nBartlett(x, N, alpha = 0.05, cor = TRUE, details = TRUE, correction = TRUE, ...)
x |
numeric: a |
N |
numeric: number of subjects |
alpha |
numeric: statistical significance level |
cor |
logical: if |
details |
logical: if |
correction |
logical: if |
... |
variable: additionnal parameters to give to the |
Note: the latex formulas are available only in the pdf version of this help file.
The hypothesis tested is:
(1) \qquad \qquad H_k: λ_{k+1} = … = λ_p
This hypothesis is verified by the application of different version of a
χ^2 test with different values for the degrees of freedom.
Each of these tests shares the compution of a V_k value:
(2) \qquad \qquad V_k = ∏\limits_{i = k + 1}^p ≤ft\{ \frac{\displaystyle λ_i}{\frac{1}{q}∑\limits_{i = k + 1}^p {λ _i } } \right\}
p is the number of eigenvalues, k the number of eigenvalues to test,
and q the p-k remaining eigenvalues. n is equal to the sample size
minus 1 (n = N-1).
The Anderson statistic is distributed as a χ^2 with (q + 2)(q - 1)/2 degrees
of freedom and is equal to:
(3) \qquad \qquad - n\log (V_k ) \sim χ _{(q + 2)(q - 1)/2}^2
An improvement of this statistic from Bartlett (Bentler, and Yuan, 1996, p. 300;
Horn and Engstrom, 1979, equation 8) is distributed as a χ^2
with (q)(q - 1)/2 degrees of freedom and is equal to:
(4) \qquad \qquad - ≤ft[ {n - k - {{2q^2 q + 2} \over {6q}}}
\right]\log (V_k ) \sim χ _{(q + 2)(q - 1)/2}^2
Finally, Anderson (1956) and James (1969) proposed another statistic.
(5) \qquad \qquad - ≤ft[ {n - k - {{2q^2 q + 2} \over {6q}}
+ ∑\limits_{i = 1}^k {{{\bar λ _q^2 } \over {≤ft( {λ _i
- \bar λ _q } \right)^2 }}} } \right]\log (V_k ) \sim χ _{(q + 2)(q - 1)/2}^2
Bartlett (1950, 1951) proposed a correction to the degrees of freedom of these χ^2 after the
first significant test: (q+2)(q - 1)/2.
nFactors |
numeric: vector of the number of factors retained by the Bartlett, Anderson and Lawley procedures. |
details |
numeric: matrix of the details for each index. |
Gilles Raiche
Centre sur les Applications des Modeles de
Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca
Anderson, T. W. (1963). Asymptotic theory for principal component analysis. Annals of Mathematical Statistics, 34, 122-148.
Bartlett, M. S. (1950). Tests of significance in factor analysis. British Journal of Psychology, 3, 77-85.
Bartlett, M. S. (1951). A further note on tests of significance. British Journal of Psychology, 4, 1-2.
Bentler, P. M. and Yuan, K.-H. (1996). Test of linear trend in eigenvalues of a covariance matrix with application to data analysis. British Journal of Mathematical and Statistical Psychology, 49, 299-312.
Bentler, P. M. and Yuan, K.-H. (1998). Test of linear trend in the smallest eigenvalues of the correlation matrix. Psychometrika, 63(2), 131-144.
Horn, J. L. and Engstrom, R. (1979). Cattell's scree test in relation to Bartlett's chi-square test and other observations on the number of factors problem. Multivariate Behavioral Reasearch, 14(3), 283-300.
James, A. T. (1969). Test of equality of the latent roots of the covariance matrix. In P. K. Krishna (Eds): Multivariate analysis, volume 2.New-York, NJ: Academic Press.
Lawley, D. N. (1956). Tests of significance for the latent roots of covarianceand correlation matrix. Biometrika, 43(1/2), 128-136.
plotuScree
, nScree
, plotnScree
, plotParallel
## ................................................ ## SIMPLE EXAMPLE OF A BARTLETT PROCEDURE data(dFactors) eig <- dFactors$Raiche$eigenvalues results <- nBartlett(x=eig, N= 100, alpha=0.05, details=TRUE) results plotuScree(eig, main=paste(results$nFactors[1], ", ", results$nFactors[2], " or ", results$nFactors[3], " factors retained by the LRT procedures", sep=""))
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