Alpha Factor Analysis
/****************************************************************/
/* S A S S A M P L E L I B R A R Y */
/* */
/* NAME: ALPHA */
/* TITLE: Alpha Factor Analysis */
/* PRODUCT: IML */
/* SYSTEM: ALL */
/* KEYS: MATRIX STATAPP SUGI6 */
/* PROCS: IML */
/* DATA: */
/* */
/* SUPPORT: Rick Wicklin UPDATE: Sep 2013 */
/* REF: */
/* MISC: */
/* */
/****************************************************************/
proc iml;
/* Alpha Factor Analysis */
/* Ref: Kaiser et al., 1965 Psychometrika, pp. 12-13 */
/* Input: r = correlation matrix */
/* Output: m = eigenvalues */
/* h = communalities */
/* f = factor pattern */
start alpha(m, h, f, r);
p = ncol(r);
q = 0;
h = 0; /* initialize */
h2 = I(p) - diag(1/vecdiag(inv(r)));/* smc=sqrd mult corr */
do while(max(abs(h-h2))>.001); /* iterate until converges */
h = h2;
hi = diag(sqrt(1/vecdiag(h)));
g = hi*(r-I(p))*hi + I(p);
call eigen(m,e,g); /* get eigenvalues and vecs */
if q=0 then do;
q = sum(m>1); /* number of factors */
iq = 1:q;
end; /* index vector */
mm = diag(sqrt(m[iq,])); /* collapse eigvals */
e = e[,iq] ; /* collapse eigvecs */
h2 = h*diag((e*mm) [,##]); /* new communalities */
end;
hi = sqrt(h);
h = vecdiag(h2); /* communalities as vector */
f = hi*e*mm; /* resulting pattern */
finish;
/* Correlation Matrix from Harmon, Modern Factor Analysis, */
/* Second edition, page 124, "Eight Physical Variables" */
nm = {Var1 Var2 Var3 Var4 Var5 Var6 Var7 Var8};
r ={ 1.00 .846 .805 .859 .473 .398 .301 .382 ,
.846 1.00 .881 .826 .376 .326 .277 .415 ,
.805 .881 1.00 .801 .380 .319 .237 .345 ,
.859 .826 .801 1.00 .436 .329 .327 .365 ,
.473 .376 .380 .436 1.00 .762 .730 .629 ,
.398 .326 .319 .329 .762 1.00 .583 .577 ,
.301 .277 .237 .327 .730 .583 1.00 .539 ,
.382 .415 .345 .365 .629 .577 .539 1.00};
run alpha(Eigenvalues, Communalities, Factors, r);
print Eigenvalues,
Communalities[rowname=nm],
Factors[label="Factor Pattern" rowname=nm];