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WESTERN MICHIGAN UNIVERSITY
COMPUTER CENTER
LIBRARY PROGRAM #1.8.1
CALLING NAME: PATHAN
PROGRAMMED BY: SAM ANEMA*
PREPARED BY: BILL GRANET
APPROVED BY: JACK R. MEAGHER
DATE: AUGUST 22, 1975
PATH ANALYSIS
1.0 PURPOSE
GIVEN A CORRELATION MATRIX, AND AN ASSUMED CAUSAL RELATIONSHIP BETWEEN THE
VARIABLES WHICH GENERATE THE CORRELATION MATRIX, THIS PROGRAM PRODUCES PATH
COEFFICIENT, STANDARD ERROR, RESIDUAL PATH, AND R*R FOR EACH DEPENDENT VARIABLE
CONSIDERED.
THE NUMERICAL VALUES OF THE OUTPUT ARE INTENDED TO GIVE INFORMATION TO HELP
DETERMINE THE DEGREE OF CONFIDENCE THE USER SHOULD HAVE IN THE ASSUMED CAUSAL
RELATIONSHIP BETWEEN THE VARIABLES. FOR THEORY AND DETAILS SEE THE REFERENCES.
2.0 LIMITATIONS
(1) NO MORE THAN 50 VARIABLES
(2) NO MORE THAN 20 COLUMNS AS SHOWN IN FIG. 1
(3) NUMBER OF NODES (VARIABLES) PER LEVEL MUST BE NOT GREATER THAN 20
(4) THREE FORMAT LINES
(5) NO MORE THAN ONE LINE CAN BE USED FOR ENTERING THE NODES OF ONE LEVEL
(COLUMN) SEE FIG. 1
(6) SAMPLE SIZE MINUS NUMBER OF INDEPENDENT VARIABLES MINUS 1 MUST BE
GREATER THAN OR EQUAL TO 1 IN ORDER TO OBTAIN STANDARD ERRORS.
3.0 SAMPLE PROBLEM
LET THE FOLLOWING CORRELATION MATRIX ON FIVE VARIABLES X(1),X(2),X(3),X(4)
AND X(5) BE GIVEN
----------------------------------------------------------
X(1) X(2) X(3) X(4) X(5)
----------------------------------------------------------
X(1) 1 .516 .453 .332 .322
X(2) .516 1 .438 .417 .405
X(3) .453 .438 1 .538 .596
X(4) .332 .417 .538 1 .541
X(5) .322 .405 .596 .541 1
THE SAMPLE SIZE IS 5.
---------------
*SUBSTANTIAL MODIFICATIONS BY RUSS BARR
THE ASSUMED CAUSAL RELATIONSHIPS BETWEEN THE VARIABLES IS GIVEN BY THE FOLLOWING
DIAGRAM:
THE DIRECTION OF AN ARROW INDICATES THE CAUSAL RELATIONSHIP ASSUMED. FOR
EXAMPLE, X(1) CAUSES X(2) AND X(3) CAUSES X(4), ETC. THE ARROWS POINTING FROM
THE OUTSIDE THE MODEL TO X(2), X(3), X(4), X(5) INDICATE THAT THERE ARE FACTORS,
EXTERNAL TO THE MODEL, WHICH AFFECT THESE VARIABLES. THIS PROGRAM WILL PRODUCE
NUMBERS (PATH COEFFICIENTS AND RESIDUAL PATHS) WHICH ARE ASSOCIATED WITH ALL THE
ARROWS SHOWN IN FIG. 1 ABOVE.
4.0 PROGRAM PROMPTINGS AND USER RESPONSES
ALL USER RESPONSES MUST BE FOLLOWED BY THE ENTERING OF RETURN <CR>.
THE FIRST TWO PROMPTINGS WILL ALWAYS BE
OUTPUT?(TYPE HELP IF NEEDED)--
INPUT?(TYPE HELP IF NEEDED)--
IF THE RESPONSE TO INPUT? IS OTHER THAN <CR> OR TTY:<CR>, THEN A FILE
CONTAINING THE INPUT CORRELATION MATRIX MUST HAVE BEEN CREATED PREVIOUS TO THE
RUNNING OF THIS PROGRAM.
THE RESPONSES TO THESE TWO LINES DEFINE WHERE THE USER INTENDS TO WRITE HIS
OUTPUT FILE (OUTPUT?) AND FROM WHERE THE USER EXPECTS TO READ HIS INPUT DATA
(INPUT?).
THE PROPER RESPONSE TO EACH OF THESE QUESTIONS CONSISTS OF THREE BASIC PARTS:
A DEVICE, A FILENAME, AND A PROJECT-PROGRAMMER NUMBER.
THE GENERAL FORMAT FOR THESE THREE PARTS IS AS FOLLOWS:
DEV:FILE.EXT[PROJ,PROG]
1) DEV: ANY OF THE FOLLOWING DEVICES ARE APPROPRIATE WHERE INDICATED:
DEVICE LIST DEFINITION STATEMENT USE
TTY: TERMINAL INPUT OR OUTPUT
DSK: DISK INPUT OR OUTPUT
CDR: CARD READER INPUT ONLY
LPT: LINE PRINTER OUTPUT ONLY
DTA0: DECTAPE 0 INPUT OR OUTPUT
DTA1: DECTAPE 1 INPUT OR OUTPUT
DTA2: DECTAPE 2 INPUT OR OUTPUT
DTA3: DECTAPE 3 INPUT OR OUTPUT
DTA4: DECTAPE 4 INPUT OR OUTPUT
DTA5: DECTAPE 5 INPUT OR OUTPUT
DTA6: DECTAPE 6 INPUT OR OUTPUT
DTA7: DECTAPE 7 INPUT OR OUTPUT
MTA0: MAGNETIC TAPE 0 INPUT OR OUTPUT
MTA1: MAGNETIC TAPE 1 INPUT OR OUTPUT
INPUT MAY NOT BE DONE FROM THE LINE PRINTER NOR MAY OUTPUT GO TO THE CARD
READER.
2) FILE.EXT IS THE NAME AND EXTENSION OF THE FILE TO BE USED. THIS PART OF
THE SPECIFICATION IS USED ONLY IF DISK OR DECTAPE IS USED.
3) [PROJ,PROG] IF A DISK IS USED AND THE USER WISHES TO READ A FILE IN
ANOTHER PERSON'S DIRECTORY, HE MAY DO SO BY SPECIFYING THE PROJECT-PROGRAMMER
NUMBER OF THE DIRECTORY FROM WHICH HE WISHES TO READ. THE PROJECT NUMBER AND
THE PROGRAMMER NUMBER MUST BE SEPARATED BY A COMMA AND ENCLOSED IN BRACKETS.
OUTPUT MUST GO TO YOUR OWN AREA.
EXAMPLE:
OUTPUT?(TYPE HELP IF NEEDED)--LPT:/2
INPUT?(TYPE HELP IF NEEDED)--DSK:DATA.DAT[71171,71026]
IN THE EXAMPLE, TWO COPIES OF THE OUTPUT ARE TO BE PRINTED BY THE HIGH SPEED
LINE PRINTER. THE INPUT DATA IS A DISK FILE OF NAME DATA.DAT IN USER DIRECTORY
[71171,71026].
DEFAULTS:
1) IF NO DEVICE IS SPECIFIED BUT A FILENAME IS SPECIFIED THE DEFAULT DEVICE
WILL BE DSK:
2) IF NO FILENAME IS SPECIFIED AND A DISK OR DECTAPE IS USED THE DEFAULT ON
INPUT WILL BE FROM INPUT.DAT, ON OUTPUT IT WILL BE OUTPT.DAT.
3) IF THE PROGRAM IS RUN FROM THE TERMINAL AND NO SPECIFICATION IS GIVEN (JUST
A CARRIAGE RETURN) BOTH INPUT AND OUTPUT DEVICES WILL BE THE TERMINAL.
4) IF THE PROGRAM IS RUN THROUGH BATCH AND NO SPECIFICATION IS GIVEN, (A BLANK
CARD) THE INPUT DEVICE WILL BE CDR: AND THE OUTPUT DEVICE WILL BE LPT:
5) IF NO PROJECT-PROGRAMMER NUMBER IS GIVEN, THE USER'S OWN NUMBER WILL BE
ASSUMED.
NOTE: (1) IF LPT: IS USED AS AN OUTPUT DEVICE, MULTIPLE COPIES MAY BE OBTAINED
BY SPECIFYING LPT:/N WHERE N REFERS TO THE NUMBER OF COPIES DESIRED.
(2) THE FOLLOWING TWO OPTIONS ARE NOT APPLICABLE FOR THE FIRST DATA SET,
I.E., IT IS APPLICABLE ONLY WHEN THE PROGRAM BRANCHES BACK TO
INPUT?.
(A) SAME OPTION
UPON RETURNING TO INPUT?, IF THE SAME DATA FILE IS TO BE USED
AGAIN, SIMPLY ENTER SAME; OTHERWISE, EITHER USE THE FINISH
OPTION OR ENTER ANOTHER FILENAME, ETC.
(B) FINISH OPTION
THE USER MUST ENTER FINISH TO BRANCH OUT OF THE PROGRAM.
FAILURE TO DO SO MIGHT RESULT IN LOSING THE ENTIRE OUTPUT FILE.
THE OTHER PROMPTINGS ARE AS FOLLOWS IN THE ORDER IN WHICH THEY APPEAR:
NO. OF VARIABLES?
SAMPLE SIZE?
FORMAT: (F-TYPE ONLY)
ENTER IDENTIFICATION
ENTER ORDERING OF VARIABLES
MANUAL OR AUTOMATIC
THE PROMPT(S) THAT FOLLOW MANUAL OR AUTOMATIC ARE DETERMINED BY THE RESPONSE
GIVEN TO MANUAL OR AUTOMATIC.
NO. OF VARIABLES.
ENTER ON THE SAME LINE AN INTEGER EQUAL TO THE NUMBER OF COLUMNS OR ROWS IN
THE INPUT CORRELATION MATRIX. THIS INTEGER MUST NOT BE GREATER THAN 50.
SAMPLE SIZE?
ENTER ON THE SAME LINE AN INTEGER EQUAL TO THE NUMBER OF OBSERVATIONS USED IN
THE GENERATION OF THE CORRELATION MATRIX.
FORMAT: (F-TYPE ONLY)
ENTER ON THE NEXT LINE ONE OF THE 3 OPTIONS AVAILABLE FOR THE FORMAT, NAMELY:
(A) STANDARD FORMAT OPTION
UNLESS OTHERWISE SPECIFIED, THE PROGRAM ASSUMES THE STANDARD
OPTION. IN THIS OPTION, THE DATA ARE ARRANGED IN GROUPS OF 10
PER LINE, TWO VALUES BEING SEPARATED BY A COMMA.
TO USE THIS OPTION, SIMPLY TYPE IN "<CR>" ON TELETYPE JOBS OR
USE A BLANK CARD FOR BATCH JOBS OR ENTER "STD<CR>".
(B) OBJECT TIME FORMAT OPTION
IF THE DATA IS SUCH THAT A USER'S OWN FORMAT IS REQUIRED,
SIMPLY ENTER A LEFT PARENTHESIS FOLLOWED BY THE FIRST FORMAT
SPECIFICATION, A COMMA AND THE SECOND SPECIFICATION, ETC.
WHEN YOU FINISH, ENTER A RIGHT PARENTHESIS, AND THEN A CARRIAGE
RETURN. THERE CAN BE A MAXIMUM OF 3 LINES FOR THE FORMAT, EACH
LINE BEING 80 COLUMNS LONG.
NOTE THAT THE FORMAT SPECIFICATION LIST MUST USE THE FLOATING
POINT (F-TYPE) NOTATION AND MUST CONTAIN SPECIFICATION FOR EACH
OF THE VARIABLES. THE SPECIFICATIONS FOR THE FORMAT ITSELF
ARE THE SAME AS FOR THE FORTRAN IV FORMAT STATEMENT.
(C) SAME OPTION
THE SAME OPTION IS APPLICABLE ONLY TO JOBS THAT USE MORE THAN
ONE DATA FILE. IF AN OBJECT TIME FORMAT WAS USED ON A DATA
SET AND THE SUCCEEDING DATA SET UTILIZES SAME FORMAT, SIMPLY
ENTER "SAME<CR>".
ENTER IDENTIFICATION
ENTER ON THE NEXT LINE EITHER A RETURN, WHICH SIGNIFIES THAT NO IDENTIFICATION
IS DESIRED, OR ANY STRING OF SYMBOLS YOU MAY WANT FOR IDENTIFICATION OF THE
OUTPUT FROM THIS PROGRAM.
ENTERING ORDERING OF VARIABLES
ENTER ON THE FOLLOWING LINES THE LABELS FOR EACH NODE IN FIG. 1. THE LABELS
ARE SEPARATED BY COMMAS IF THERE IS MORE THAN ONE LABEL IN A LEVEL (COLUMN).
THE LABELS ARE THE INTEGERS FROM 1 TO N WHERE N IS THE NUMBER OF VARIABLES.
ENTER THE LABELS A COLUMN AT A TIME (SEE FIG. 1.) STARTING FROM THE LEFT.
USING THE LABELS SHOWN IN FIG. 1, WE WOULD ENTER IN RESPONSE TO THIS PROMPTING:
1
2,3
4,5
<CR>
WHERE "<CR>" SIGNIFIES THE ENTERING OF RETURN.
NOTES: (A) THIS DISCUSSION ASSUMES THAT THE FIRST COLUMN OF THE CORRELATION
MATRIX CORRESPONDS TO VARIABLE 1, THE SECOND COLUMN CORRESPONDS TO VARIABLE 2,
ETC. (B) ONE COULD HAVE ENTERED
4
1,2
3,5
PROVIDED THE LABELING IN FIG. 1 IS CHANGED ACCORDINGLY.
ONLY ONE LINE CAN BE USED FOR THE ENTERING OF A LEVEL. THE MAXIMUM NUMBER OF
ALLOWED NODES PER LEVEL IS 20. NO MORE THAN 20 LEVELS ARE ALLOWED.
MANUAL OR AUTOMATIC
IN BOTH METHODS SEE THE 6TH LIMITATION IN SECTION 2.
THERE ARE TWO WAYS ONE CAN PROCEED:
FIRST WAY
ON THE NEXT LINE ENTER M TO OBTAIN AN INTERACTIVE METHOD OF PATH ANALYSIS
WHERE THE USER SPECIFIES THE ARROWS IN FIG. 1 AS RESPONSES TO VARIOUS
QUESTIONS FROM THIS PROGRAM. THIS IS CALLED THE MANUAL METHOD.
FOR EXAMPLE, IN SECTION 3 THE FIRST QUESTION THAT WILL PRINT IS:
WHICH ARE THE VARIABLES CAUSING VARIABLE 1
THE QUESTIONING STARTS WITH THE FIRST LEVEL AS SHOWN IN FIG. 1. THE
RESPONSE IS ENTERED ON THE NEXT LINE. IN THIS CASE SINCE NO VARIABLE IS
ASSUMED TO CAUSE VARIABLE 1 A RETURN IS ENTERED.
HOWEVER, IT IS ALLOWABLE TO ASSUME A MODEL WHERE ANY COMBINATION OF
VARIABLES 2,3,4, AND 5 CAUSE VARIABLE 1. FOR EXAMPLE, IF 2,4, AND 5 ARE
ASSUMED TO CAUSE VARIABLE 1, THEN YOU ENTER 2,4,5 ON THE NEXT LINE.
TO AVOID REPETITION AND TO SHOW THE MORE GENERAL SITUATION WE WILL DISCUSS
IN DETAIL ONLY THE 3RD QUESTION OF THE REMAINING QUESTIONS IN THE MANUAL
METHOD.
THE 3RD QUESTION THAT WILL PRINT IS WHICH ARE THE VARIABLES CAUSING
VARIABLE 3
ACCORDING TO FIG. 1 WE SHOULD ENTER 1,2 ON THE NEXT LINE. AFTER THIS THE
PROGRAM WILL PRINT:
DEP = 3
RSQ = 0.02621
RESIDUAL PATH = 0.8590
IND BETA S.E.
1 0.3094 0.7091
2 0.2784 0.7091
OK?
RSQ IS THE COEFFICIENT OF MULTIPLE DETERMINATION (R*R)
RSQ = 0.2621=(.3094)(.453)+(.2784)(.438)
WHERE .453 AND .438 ARE IN THE 3RD COLUMN OF THE CORRELATION MATRIX GIVEN
IN SECTION 3. 0.3094 AND 0.2784 ARE IN THE BETA COLUMN ABOVE.
RESIDUAL PATH IS A NUMBER ASSOCIATED WITH THE ARROW POINTING FROM OUTSIDE
THE MODEL TO NODE 3 IN FIG. 1. BETA IS THE PATH COEFFICIENT ASSOCIATED
WITH THE ARROW IN FIG. 1 POINTING FROM THE NODE INDICATED AT BEGINNING OF
THE LINE TO THE NODE SHOWN AFTER DEP=.
RESIDUAL PATH = 0.8590 =SQRT(1-.2621)
"S.E." IS AN ABBREVIATION FOR STANDARD ERROR OF THE PATH COEFFICIENT. IT
IS GIVEN BY THE FORMULA:
S.E. = SQRT(((1-R*R)*A(I,I))/(N-K-1))
WHERE R*R IS THE COEFFICIENT OF MULTIPLE DETERMINATION, N IS THE SAMPLE
SIZE, AND K IS THE NUMBER OF INDEPENDENT VARIABLES. A(I,I) IS THE NUMBER
IN THE ITH ROW AND ITH COLUMN OF A CERTAIN INVERSE MATRIX.
FOR R*R AND RESIDUAL PATH SEE REFERENCE 1.
NOTE THAT THE TWO S.E.'S ARE EQUAL. THIS WILL ALWAYS BE TRUE FOR 2
INDEPENDENT VARIABLES AND IN GENERAL IS NOT TRUE FOR MORE THAN TWO
INDEPENDENT VARIABLES.
OK?
THE ENTERING OF NO ON THE SAME LINE WILL ALLOW YOU TO ENTER A DIFFERENT
SET OF VARIABLES TO CAUSE VARIABLE 3 AND TO OBTAIN THE CORRESPONDING R*R,
RESIDUAL PATH, PATH COEFFICIENTS, AND S.E.'S. THE SUMMARY TABLE THAT WILL
PRINT AT THE CONCLUSION OF THIS PATH ANALYSIS WILL INCLUDE ANSWERS ONLY
FOR THE LAST SET OF VARIABLES ENTERED AS CAUSING VARIABLE 3.
AFTER YOU HAVE RESPONDED TO:
WHICH ARE THE VARIABLES CAUSING VARIABLE 5
AND RESPONDED TO THE FOLLOWING OK? WITH YES, THEN THIS PROGRAM WILL PRINT:
INPUT?(TYPE HELP IF NEEDED)--
IF YOUR RESPONSE IS FINISH, THEN PROGRAM TERMINATION INFORMATION WILL
PRINT AND YOU WILL OBTAIN A PERIOD. IF YOU WANT TO ANALYZE ANOTHER MODEL,
SEE THE BEGINNING OF THIS SECTION FOR OTHER POSSIBLE RESPONSES.
SECOND WAY
ENTER A ON THE NEXT LINE IF YOU WANT ANSWERS PRINTED FOR ALL POSSIBLE
COMBINATIONS OF ARROWS IN FIG. 1 SUBJECT TO THE RESTRICTION THAT ARROWS
IN FIG. 1 CANNOT POINT FROM A LEVEL (COLUMN) TO A LEVEL ON THE LEFT (E.G.
FROM NODE 2 TO NODE 1) AND SUBJECT TO THE CRITERION THAT NO NUMBERS PRINT
FOR THOSE INDEPENDENT VARIABLES WHOSE PATH COEFFICIENTS ARE IN ABSOLUTE
VALUE LESS THAN A SPECIFIED NUMBER. THIS MEANS THAT WITH METHOD OF THE
SECOND WAY ONE CANNOT GET AS MUCH INFORMATION AS WITH THE FIRST WAY
METHOD. THIS SPECIFIED NUMBER WOULD BE ENTERED ON THE SAME LINE AS:
CRITICAL BETA VALUE?
AFTER YOU HAVE RESPONDED TO THIS QUESTION, THIS PROGRAM WILL PRINT ALL
THE ANSWERS AND THEN PRINT:
INPUT?(TYPE HELP IF NEEDED)--
THE EXPLANATION HERE WAS GIVEN UNDER THE FIRST WAY.
5.0 SAMPLE TERMINAL RUN
.R PATHAN
---WMU-CSR PATH ANALYSIS PROGRAM
OUTPUT? (TYPE HELP IF NEEDED)--TTY:
INPUT? (TYPE HELP IF NEEDED)--PATHAN.INP
ENTER # OF VARIABLES OR VARIABLE NAMES
5
SAMPLE SIZE? 5
FORMAT: (F-TYPE ONLY)
(5F)
ENTER IDENTIFICATION
PATHAN
ENTER ORDERING OF VARIABLES
1
2,3
4,5
MANUAL OR AUTOMATIC? M
WHICH ARE THE VARIABLES CAUSING VARIABLE 1
WHICH ARE THE VARIABLES CAUSING VARIABLE 2
1
DEP = 2
RSQ = 0.2663
RESIDUAL PATH = 0.8566
IND BETA S.E.
1 0.5160 0.4946
OK? YES
WHICH ARE THE VARIABLES CAUSING VARIABLE 3
1,2
DEP = 3
RSQ = 0.2621
RESIDUAL PATH = 0.8590
IND BETA S.E.
1 0.3094 0.7091
2 0.2784 0.7091
OK? YES
WHICH ARE THE VARIABLES CAUSING VARIABLE 4
1,3
DEP = 4
RSQ = 0.2993
RESIDUAL PATH = 0.8371
IND BETA S.E.
1 0.1111 0.6640
3 0.4877 0.6640
OK? NO
WHICH ARE THE VARIABLES CAUSING VARIABLE 4
1,2,3
DEP = 4
RSQ = 0.3306
RESIDUAL PATH = 0.8182
IND BETA S.E.
1 0.0254 0.9996
2 0.2144 0.9913
3 0.4326 0.9525
OK? YES
WHICH ARE THE VARIABLES CAUSING VARIABLE 5
1,2,3,4
DEP = 5
RSQ = 0.4338
RESIDUAL PATH = 0.7525
IND BETA S.E.
1 -0.0139 0.0000
2 0.1205 0.0000
3 0.3983 0.0000
4 0.2811 0.0000
**S.E. NOT CALCULATED BECAUSE NUMBER OF SAMPLES MINUS
NUMBER OF INDEPENDENT VARIABLES IS LESS THAN 2.
OK? YES
***SUMMARY MATRIX***
PATHAN
IND: 1 2 3
RSQ RESID BETA S.E. BETA S.E. BETA S.E.
DEP
1 0.0000 0.0000
2 0.2663 0.8566 0.5160 0.4946
3 0.2621 0.8590 0.3094 0.7091 0.2784 0.7091
4 0.3306 0.8182 0.0254 0.9996 0.2144 0.9913 0.4326 0.9525
5 0.4338 0.7525 -0.0139 0.0000 0.1205 0.0000 0.3983 0.0000
IND: 4 5
RSQ RESID BETA S.E. BETA S.E.
DEP
1 0.0000 0.0000
2 0.2663 0.8566
3 0.2621 0.8590
4 0.3306 0.8182
5 0.4338 0.7525 0.2811 0.0000
INPUT? (TYPE HELP IF NEEDED)--FINISH
END OF EXECUTION
CPU TIME: 0.47 ELAPSED TIME: 5.73
EXIT
6.0 BATCH OPERATION
THE FOLLOWING IS A BATCH JOB SET UP: (EACH LINE REPRESENTS ONE CARD. EACH CARD
STARTING IN COLUMN 1. DO NOT INCLUDE THE COMMENTS AT THE RIGHT.)
SEE COMPUTER CENTER USERS GUIDE #7 FOR OTHER BATCH SYSTEM COMMANDS.
--------------------------------------------------------------------------------
$JOB [#,#] ;JOB CARD; INSERT USERS' PROJECT-
PROGRAMMER NUMBER WITHIN THE BRACKETS.
$PASSWORD ###### ;IN PLACE OF THE 6 #'S, PUT IN THE
PASSOWRD.
$DATA ;SIGNIFIES BEGINNING OF THE OPTIONAL
DATA DECK.
(DATA CARDS) ;INSERT THE DATA CARD DECK TO BE
ANALYZED.
$EOD ;SIGNIFIES THE END OF DATA CARD DECK.
.R PATHAN
(RESPONSES AS EXPLAINED IN ;USERS' RESPONSE
SECTION 4 AND SHOWN
IN SECTION 5.)
(EOF) ;AN END-OF-FILE CARD.
--------------------------------------------------------------------------------
7.0 REFERENCES
1) NYGREEN, C.T. (1971), "INTERACTIVE PATH ANALYSIS", THE AMERICAN SOCIOLOGIST,
VOL. 6 (FEBRUARY): 37-43.
2) VANDEGEER, J.P. (1971), "INTRODUCTION TO MULTIVARIATE ANALYSIS FOR THE
SOCIAL SCIENCES", W.H. FREEMAN, SAN FRANCISCO.