## 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];

```