Explanatory Factor Analysis: Alpha and Omega

Dominique Zephyr Applied Statistics Lab

University of Kenctucky

Factor Analysis

Factor analysis is a statistical method that allows us to understand large quantities of observable variables in terms of a smaller number of unobservable variables.

In factor analysis, latent variables represent unobserved

constructs that are referred as factors or dimensions.

Two types of factor Analysis Exploratory Factor Analysis (EFA) - It is exploratory when you do not

have a pre-defined idea of the structure or how many dimensions are in a set of variables.

Confirmatory Factor Analysis (CFA)- It is confirmatory when you want to test specific hypothesis about the structure or the number of dimensions underlying a set of variables. CFA is used to study how well a hypothesized factor model fits a new sample from the same population or a sample from a different population

The EFA Model

Factor analysis (FA) aims to describe a set of Q variables x1, x2 ,..., xQ in terms of a smaller number of m factors and residuals e ( unique factors)

The EFA model

where xi (i=1,…,Q) represents the original variables but standardized with zero mean and unit variance

F1, F2,...,Fm are m uncorrelated common factors, each with zero mean and unit variance;

αi1, αi2, ..., αim are the factor loadings related to the variable Xi

ei are the Q unique factors supposed independently and identically distributed with zero mean

EFA Extraction Methods

Methods of extraction available in Stata pf : principal factor; the default pcf: principal-component factor ipf: iterated principal factor ml: maximum-likelihood factor

We will use ml so that the results are comparable with CFA

EFA results

Factor GSE1-GSE6, ml

---------------------------------------------------------------------- Factor | Eigenvalue Difference Proportion Cumulative -------+------------------------------------------------------------ Factor1 | 1.29942 0.50316 0.5331 0.5331 Factor2 | 0.79625 0.45460 0.3267 0.8598 Factor3 | 0.34166 . 0.1402 1.0000 --------------------------------------------------------------------

----------------------------------------------------------- Variable | Factor1 Factor2 Factor3 | Uniqueness -------------+------------------------------+-------------- GSE1 | 0.1672 0.3052 0.3568 | 0.7516 GSE2 | 0.2981 0.3809 0.2999 | 0.6762 GSE3 | 1.0000 -0.0000 -0.0000 | 0.0000 GSE4 | 0.1956 0.5643 -0.3081 | 0.5484 GSE5 | 0.2681 0.3042 -0.1344 | 0.8175 GSE6 | 0.2691 0.3835 0.1068 | 0.7691 -----------------------------------------------------------

How many factors to retain? Kaiser criterion. Drop all factors with eigenvalues below 1.0. The

simplest justification for this is that it makes no sense to add a factor that explains less variance than is contained in one individual indicator.

  Scree plot. This method, proposed by Cattell, plots the successive

eigenvalues, which drop sharply and then level off. It suggests retaining all eigenvalues in the sharp descent before the first one on the line where they start to level off.

  Variance explained criteria. Some researchers simply use the rule

of keeping enough factors to account for 90% (sometimes 80%) of the variation.

Kaiser Criterion

------------------------------------------------------------------ Factor | Eigenvalue Difference Proportion Cumulative -------+------------------------------------------------------------ Factor1 | 1.29942 0.50316 0.5331 0.5331 Factor2 | 0.79625 0.45460 0.3267 0.8598 Factor3 | 0.34166 . 0.1402 1.0000 --------------------------------------------------------------------

factor GSE1-GSE6, ml

screeplot

Rotation

The goal of rotational strategies is to obtain a clear pattern of loadings.

The sum of eigenvalues is not affected by rotation, but changing the axes will alter the eigenvalues of particular factors and will change the factor loadings

The most common rotation method is the “varimax rotation”

Oblimin Rotation Resultsfactor GSE1-GSE6, factor(1) mlrotate, oblimin Factor analysis/correlation Number of obs = 700 Method: maximum likelihood Retained factors = 1 Rotation: orthogonal oblimin (Kaiser off) Number of params = 6 Schwarz's BIC = 74.2601 Log likelihood = -17.47681 (Akaike's) AIC = 46.9536

Rotated factor loadings (pattern matrix) and unique variances

--------------------------------------- Variable | Factor1 | Uniqueness -------------+----------+-------------- GSE1 | 0.3583 | 0.8716 GSE2 | 0.5186 | 0.7311 GSE3 | 0.5483 | 0.6994 GSE4 | 0.4109 | 0.8312 GSE5 | 0.4072 | 0.8342 GSE6 | 0.5043 | 0.7457 ---------------------------------------

Cronbach Coefficient Alpha

The Cronbach Coefficient Alpha is the most common estimate of internal consistency of items in a model or survey

How large the α must be:Nunnally (1978) suggests 0.7 as an acceptable reliability threshold.

Some authors use .75 or .80 as a cut-off value, while others are as lenient as to go to 0.6

Alpha results

alpha GSE1-GSE6, item std detail average item-test item-rest interitemItem | Obs Sign correlation correlation correlation alpha-------------+-----------------------------------------------------------------GSE1 | 700 + 0.5220 0.2671 0.2300 0.5990GSE2 | 700 + 0.6178 0.3892 0.1965 0.5501GSE3 | 700 + 0.6279 0.4026 0.1930 0.5445GSE4 | 700 + 0.5635 0.3188 0.2155 0.5787GSE5 | 700 + 0.5584 0.3123 0.2173 0.5813GSE6 | 700 + 0.6112 0.3804 0.1988 0.5537-------------+-----------------------------------------------------------------Test scale | 0.2085 0.6125-------------------------------------------------------------------------------

Bootstrapped CI for Alpha

bootstrap r(alpha), reps(1000): alpha GSE1-GSE6, item std detail

estat bootstrap, all------------------------------------------------------------------------------ | Observed Bootstrap | Coef. Bias Std. Err. [95% Conf. Interval]-------------+---------------------------------------------------------------- _bs_1 | .61251427 .0005385 .02538602 .5627586 .6622699 (N) | .5559364 .6589164 (P) | .5535116 .6563407 (BC)------------------------------------------------------------------------------(N) normal confidence interval(P) percentile confidence interval(BC) bias-corrected confidence interval

Omega after EFA

set more offscalar drop _allfactor GSE1-GSE6, factor(1) mlrotate, oblimin matrix c = e(L)matrix e = e(Psi)'

* sum(loadings)^2 matrix one = J(rowsof(c),1,1)matrix c = one'*c*c'*one

* sum(error variances)matrix e = J(1,rowsof(e),1)*e

matrix list cmatrix list escalar omega= c[1,1]/(c[1,1]+e[1,1])scalar list

matrix list c: sum(loadings)^2

c1 7.5489134

. matrix list e: sum(error variances)r1 4.7131344

. scalar omega= c[1,1]/(c[1,1]+e[1,1])

. scalar list omega = .61563236

Bootstrapped CI for Omegacapture program drop omegaefaprogram define omegaefa, rclassversion 12.0set more offpreservesyntax [varlist] [if] [in]factor `varlist', factor(1) mlrotate, obliminmatrix c = e(L)matrix e = e(Psi)'

* sum(loadings)^2 matrix one = J(rowsof(c),1,1)matrix c = one'*c*c'*one

* sum(error variances)matrix e = J(1,rowsof(e),1)*e

matrix list cmatrix list ereturn scalar omega=c[1,1]/(c[1,1]+e[1,1])restoreend

bootstrap omega=r(omega), reps(1000): omegaefa GSE1-GSE6estat bootstrap, all

Bootstrapped ResultsBootstrap results Number of obs = 700 Replications = 1000

command: omegaefa GSE1-GSE6 omega: r(omega)

------------------------------------------------------------------------------ | Observed Bootstrap | Coef. Bias Std. Err. [95% Conf. Interval]-------------+---------------------------------------------------------------- omega | .61563236 .0011574 .02429982 .5680056 .6632591 (N) | .5679913 .6597112 (P) | .5620664 .6560227 (BC)------------------------------------------------------------------------------(N) normal confidence interval(P) percentile confidence interval(BC) bias-corrected confidence interval