Introduction
One of the most subjective decisions made when using ordination techniques
is the number of components to retain for interpretation. Generally, the
researcher subjectively decides, but more objective methods are described
below. Based on a literature review of the use of factor analysis and principal
components analysis (Franklin et al. 1995, J. Veg. Sci. 6:99-106), more
objective methods are necessary.
The parallel analysis (PA) must match the number of samples (n) and number
of variables (p) in your real data set (i.e. that data collected from the
field). Further, the PA must match the type of analysis (e.g. factor analysis
or principle component analysis), and must correspond to the type of matrix
being decomposed (i.e. correlation of covariance).
By setting criteria for the interpretation of ordination results, a
researcher can objectively decrease the number of axes for interpretation and
number of variables of interest , thus facilitating interpretation.
Decomposing a correlation matrix:
The program LONGMAN.SAS given below is the easiest program for interpreting
the number of components to retain when decomposing a correlation matrix. Simply
change the N and P values (on the first line after explanation text only) to
match your field data set and run the program. The values derived from the
analysis are the 95th percentile eigenvalues for each component based on the
regression equations of Longman et al. 1989 (Multi. Behav. Res. 24:59-69). The
regression equations are based on a Monte Carlo analysis. Eigenvalues from the
analysis of the field data that are greater than the PA eigenvalues (comparing
each axis separately - PA eigenvalue for axis one versus real eigenvalue for
axis one, PA eigenvalue for axis two versus real eigenvalue for axis two, etc.)
are significant at the 0.05 level, and should be retained for interpretation.
The analysis should be rerun using the appropriate number of components.
Decomposing of covariance matrix:
The program COVRPARA.SAS given below will produce a pseudorandom matrix
based on the means and standard deviations of the real data. Here, each variable
in the analysis contributes to the distributions and variance of the pseudorandom
matrix. Insert the standard deviations and means where appropriate and add or
subtract variables as necessary. In this analysis, the program performs the
Monte Carlo analysis, the number of which can be set as K (99 or 999 are
recommended). Change appropriately the X1-X?? for number of variables, and the J
size for the number of samples. Make sure the analysis used (e.g. factor analysis
with rotations) on the collected data is the same as on the randomized data set
created in this program. The results derived are the maximum eigenvalues from
all the K analyses. Eigenvalues retained for interpretation should be greater
than these eigenvalues. The analysis should be rerun using the appropriate number
of components.
Determining Significant loadings:
When decomposing a correlation matrix, use the program LOADINGS.SAS. When
decomposing a covariance matrix, use the program COVRPARA.SAS (see above). This
program creates a random data set with the same n and p size as the collected
data set and should be subjected to the same analysis (e.g. factor analysis
with rotations). This step follows the determination of the number of components
to retain, and this number (?) should be set in the PROC FACTOR statement as
N=?. Set the number of samples (yyy), variables (www), and randomizations (zzz)
in all areas and run. The results give you the maximum loadings from all the
data sets analyzed. To determine the 95th percentile, multiply the number of
the components by the number of variables (3 * 16 = 48), then multiply that
number by 0.05 (48 * 0.05 = 3 with rounding). Thus, the third highest loading
from the maximums shown gives the 95th percentile cutoff, and only loadings
greater than this value should be interpreted. This program is written for
a factor analysis. The univariate procedures need to be modified for any other
types of analyses, so that the univariate analysis is performed on the appropriate
variable.
TITLE 'PROGRAM TO DECOMPOSE A CORRELATION MATRIX - THE LONGMAN METHOD'; OPTIONS LS=73 NOCENTER; DATA LONGMAN; * THIS PROGRAM PRODUCES ESTIMATES OF THE 95TH PERCENTILE EIGENVALUES FROM A PARALLEL ANALYSIS, USING THE WORK OF LONGMAN ET AL. (1989). CHANGE THE VALUES OF N AND P TO THOSE OF YOUR DATA MATRIX. THIS PROGRAM SHOULD ONLY BE USED WHEN DECOMPOSING A CORRELATION MATRIX; N=36; P=13; * N = SAMPLE SIZE, P = NO. OF VARIABLES. ; LN = LOG (N); LP = LOG(P); LEIG1 = 0.0316*LN +0.7611*LP -0.0979 *(LN*LP) -0.3138; LAM1 =EXP (LEIG1); LEIG2 = 0.1162*LN +0.8613*LP -0.1122 *(LN*LP) -0.9281; LAM2 =EXP (LEIG2); LEIG3 = 0.1835*LN +0.9436*LP -0.1237 *(LN*LP) -1.4173; LAM3 =EXP (LEIG3); LEIG4 = 0.2578*LN +1.0636*LP -0.1388 *(LN*LP) -1.9976; LAM4 =EXP (LEIG4); LEIG5 = 0.3171*LN +1.1370*LP -0.1494 *(LN*LP) -2.4200; LAM5 =EXP (LEIG5); LEIG6 = 0.3809*LN +1.2213*LP -0.1619 *(LN*LP) -2.8644; LAM6 =EXP (LEIG6); LEIG7 = 0.4492*LN +1.3111*LP -0.1751 *(LN*LP) -3.3392; LAM7 =EXP (LEIG7); LEIG8 = 0.5309*LN +1.4265*LP -0.1925 *(LN*LP) -3.8950; LAM8 =EXP (LEIG8); LEIG9 = 0.5734*LN +1.4818*LP -0.1986 *(LN*LP) -4.2420; LAM9 =EXP (LEIG9); LEIG10= 0.6460*LN +1.5802*LP -0.2134 *(LN*LP) -4.7384; LAM10=EXP(LEIG10); PROC PRINT; VAR N P LAM1-LAM10; RUN;
TITLE ' PARALLEL ANALYSIS DECOMPOSING A COVARIANCE MATRIX';
*NOTE: THIS USES MEANS AND STANDARD DEVIATIONS FROM FIELD DATA;
DATA ONE;
OPTIONS LS=73;
DO K = 1 TO 100;*SET NUMBER OF PERMUTATIONS;
DO J = 1 TO 30; * GIVE NUMBER OF SAMPLES IN DATA SET;
*** THE FOLLOWING FUNCTIONS CREATE PSEUDORANDOM DATA SETS USING
THE MEANS AND STANDARD DEVIATIONS FROM THE REAL DATA;
x1 = normal(0)*24 + 21.43;
x2 = normal(0)*0.5 + 0.345;
x3 = normal(0)*1.8 + 0.3;
x4 = normal(0)*18 + 7.93;
x5 = normal(0)*2.5 + 1.16;
x6 = normal(0)*10 + 6.1;
x7 = normal(0)*0.3 + 0.03;
x8 = normal(0)*0.6+ 0.287;
x9 = normal(0)*40 + 40.97;
x10 = normal(0)*0.2 + 0.043;
output;
end;
end;
RUN;
DATA TWO; SET ONE;
PROC FACTOR COV METHOD=PRINCIPAL N=6 OUTSTAT=RESULTS;VAR X1-X28;BY
K;
DATA FINAL; SET RESULTS; IF _TYPE_ = 'EIGENVAL';
OPTIONS LS=73;
PROC UNIVARIATE NOPRINT; VAR X1-X10; OUTPUT OUT=EIGEN
MAX=MAXE1-MAXE10;
PROC PRINT;
RUN;
DATA FINAL2; SET RESULTS; IF _TYPE_ = 'PATTERN' AND _NAME_ = 'FACTOR1';
OPTIONS LS=73;
PROC UNIVARIATE NOPRINT; VAR X1-X10; OUTPUT OUT=PATTERN
MAX=MAXP1-MAXP10;
PROC PRINT;
RUN;
TITLE 'THIS PROGRAM WILL GENERATE SIGNIFICANT LOADINGS';
* THIS PROGRAM SHOULD ONLY BE USED WHEN DECOMPOSING A CORRELATION MATRIX;
OPTIONS LS=73;
DATA ONE;
* GENERALIZED PARALLEL FACTOR ANALYSIS PROCEDURE:
1. NUMBER OF VARIABLES IS SET WITH THE WWW
INDEX VALUE,
2. NUMBER OF OBSERVATIONS IS SET WITH THE
YYY INDEX FOR J IN THE FIRST DO STATEMENT
AND IN THE VAR STATEMENT,
3. NUMBER OF ANALYSES IS SET WITH THE ZZZ INDEX,
4. LASTLY, PERFORM THE SAME FACTOR ANALYSIS ON THE
SIMULATED DATQA MATRIX THAT YOU PERFORMED ON
THE ACTUAL DATA MATRIX.;
******WARNING - THIS PROGRAM GENERATES A BIG LISTING****** ;
ARRAY X (I) X1-X16; * SET THE NUMBER OF VARIABLES (WWW);
DO K = 1 TO 50; * SET THE NUMBER OF ANALYSES (ZZZ);
DO J = 1 TO 133; * SET THE SAMPLE SIZE (YYY);
DO OVER X;
X = NORMAL (0);
END;
OUTPUT;
END;
END;
RUN;
DATA TWO; SET ONE;
* SET THE NUMBER OF FACTORS WITH THE N = PARAMETER;
PROC FACTOR COV METHOD=PRINCIPAL N=3 OUTSTAT=RESULTS; VAR X1-X16; BY
K;
RUN;
DATA FINAL; SET RESULTS; IF _TYPE_ = 'EIGENVAL';
OPTIONS LS=73;
PROC UNIVARIATE NOPRINT; VAR X1-X16; OUTPUT OUT=EIGEN
MAX=MAXE1-MAXE10;
PROC PRINT;
RUN;
DATA FINAL2; SET RESULTS; IF _TYPE_ = 'PATTERN' AND _NAME_ = 'FACTOR1';
OPTIONS LS=73;
PROC UNIVARIATE NOPRINT; VAR X1-X16; OUTPUT OUT=PATTERN
MAX=MAXP1-MAXP16;
PROC PRINT;
RUN;
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©1996, Southern Illinois University at Carbondale.
Last revised: 10 Sept, 2003 by DJG.