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DOCUMENT RESUME
. ED 073 122 TM 002 364
AUTHOR Joreskog, Karl'G..;,van Thillo, MarielleTITLE LISREL: A General Computer Program for Estimating a
Linear Structural Equation System Involving MultipleIndicators of Unmeasured Variables.
INSTITUTION Educatidnal Testing Service, Princeton, N.J.REPORT NO ETS-RB-72-56.PUB DATE Dec 72NOTE 73p.; Draft
EDRS PRICE MF-$0.65 HC-$3.29DESCRIPTORS Bulletins; *Computer Programs; *Data Analysis; *Inp,Ac
Output; *Linear Programing; *Mathematical Models
ABSTRACTA,,general computer program for estimating the 'unknown
coefficients in a set of linear structural equations is described. Inits most general form, the variables in the equation system may he.Unmeasured hypdthetical constructs or latent variables, and there may 4'be several measured variables or multiple indicators for each'unmeasured variable. Also, the method allows for both errors inequations (residuals, disturbances) and errors.in the observedvariables (errors of measurement, observational errors) and yieldsestimates of the disturbance variance-covariance matrix and themeasurement error variances, as well as estimates of the unknowncoefficients in the.structual equations, provided that all theseparameters are identified. The method is so general and flexible thatit is possible to handle a wide range of models. The model consideredhere 4s a generalization of the model considered by Joreskog (1973).'(Author/DB)
Ca
e
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bISREL
A GENERAL COMPUTER PROGRAM FOR ESTIMATING A LiN;LAE STRUCTAL
EQUATION SYTEM INVOLVING'MULTIPLEINDICATOPS OF
UNMEASURED. VARIABLES
Karl G. Jgreskog
University of Uppsala, Sweden,
and
Marlene van ThilloEducational Testing Service
.
This'BulletinZis a draft for interoffice circulation.
Corrections and suggestions for revision/ are solicited.
The Bulletin should not be cited as a reference without
the specific permission of the authqrs. It is automaLi-,
cally superseded upon formal publication of the material.
Educational Testing Service
Princeton, New J sey
December 297
tab
0
0.
Irk
A J:enral ComputE.,r Program for Istimating a Linear M.,l'uctural%
Eqation Syst(!m Involving Multiple Indicators'of
Unmeasured Variables
1. Introduction ,
We shall describe a general computer-program for estimating unkno4n
coefficients in a set of linear structural equations In its most general
.form the variables in the equa-tioncsystem may be unmeasured hypothetical
constructs or latent variables and there may be several measured variables
or -multiple indicators for each unmeasured variable. Also, the method allc,c
for both erorg in equations (residuals, disturbances) and errors in the
observed variables (errors of measurement, observational errors) and yields
estimates of the disturbance variance - covariance matrix and the measurement
error variances as well as estimates of the unknown coefficients in the
structural equationS, provided that all these paraMeters are idehtiried.
The method is so general and flexible that it is possible to ha-1.2J a wide
range of models, for example, path analysis models (Wright,
1960; Turner & Stevens, 1959; Duncan, 1966; Duncan, Haller.& Portes, I
Land, 1969; Heise, 1970; Blalock, 1969,- 1971; Costner, 1969; Hauser &
Goldbtrger, 1971), econometric models (Goldberger, 1964; Mal.invaud, 1,0:
Johnston, 1972), factor analysis and covariance si,ructure models Wreskoc
1969, 1970). The model. considered hen? is a generalization 0 th( m(k1
considered -by ,IBreskog (1975)
4
ti
a
1.1. Th( (1neral 11od(21.
nsider random vectors and
of true de-nend(nt., and imdependeA variables, r*.si,ectively, and the
.,:,.ster of linear structural rclaticTs
Sri = ( :.)
.). . .
wheve B(m x n) and P(m x A) are coefficient matrices an., P. .(('.:_.( ,
,.)i- 2
i
;..,( ) is a /:andom vector of residuals (r.:rr%Jrs in erluations, random.
disturbance ,terms). Without loss of generality it may be assumed that
t(11) = e(c) 0 and t( ) = 0 . It is furthe2rmore'assumed that C is
uncorrelated with -t; and that P, is nonsingular.
The vectors r and are not observed but instead vectors y'
(Y,, ) and -x' = (x ) are observed, such that
f A E
X = V 1
w-here,4
k = e(y) , v = (x) and c. and '6 are vectors of errors of.
measurement ih y and x , respectively. The matrices A.An xm) and
(A
.
A (q x n) are regression matrices of y on T1 and of x on ,
xI
restectively..
It is converient to rcP,r to y and x as the-observed
) , 1
variables and T1 and as the true variables. The errors of measure-
ment are assume
variates.
e uncorrelated with each other and with the tru'e
J4
,
-
.'
Let (1_.(n n) and 'f(m n) be the variance-cVarian, -Latrines ,:)1
and respectively, and _ the diagonal* matrices of error 4
S
variances for y and ..- , respectively. . Then it f,J1lows, f,rer, the abovery
* assumptions, tll'at the variance-covariance matri,: E((:
z =
The
and equal
A A:
is
(P-1
B 6 11,'B )A1
A or,B,-I
A'-x-- -y
elements of E are0
functions of the e_ e+nts of- A
q)3 of
AY0
- ,
86
,
'
and' 6e
'In applications some of these elements are fi-cd
to assigned values.
B and F , but we shall allow for fixed values in .the othel- ma-,
trices also. For the remaining*nonfixed elements of the six'parameter mairo!Lr:
In particular this js so for elements in
. ,
one or more subsets may have identiCal but unknown values'. rhas ,I.ements.
in A-Y
(i) fixed parameters1
B. 'n, and (L: are of three .inds':-o -e
that have been assigned given :values, (ii) constrained
parameters that are unknown but equal: to one or more other-parameters and
(iii) free
any other parameter.
parameters that are unknown and not constra(ined-to bc oT
Before an attempt is made to estimate a model of this kind,-the identi-.
fication.problem must be examined. Identifiability depends op the speafica-
tion of fiXed, constrained, and tree parameters. Under a given,
86 ; -(3 generate.:tion, a given structure B q, 5
one and only one E bul there may be several structures generatin.g.th(a
same . If two or more structures penerate the same Y:73 the,str'uctur.-
ti
are said t: be f' a iabal..tk_.r has th.. sa.,ne 1.rai
C-
- lent str.:cturfs LI_AL* barametr is said to f e id-ntlf1(.d. -1r an 7tarametf_..rs
of the gode._ are ideiltified, the hole model is said to identified.,
.When 'a Abdel is identified one can usually i tr.:on.; f./.
arameters. Identification problems ,L der somc th(
general model are considered Of-raci and goldberger (Lir]
Eotimation of the OenC:ral
.It is assurie(.% that
witlt:.mean vectzr8
6T',>1 has a multivariat( 'normal distribution
and variance-covarianc,_ E
Let b obsenfations (y1,-%)" . inc.-S no-±
on the meaniector ma:inun:constraints are imposed
hood estimate of this is the usual sample mean vector =
with
as
t
y's 1
a.17 1
1 , be the usl.ai sample variance-t:ovarianc( iticl.cd.
8[(b x (12 .4 -
-d
4 A :
The logarithm of the likelihy,,,d Nnecion,,omittinc
observaf,ions, is ir,:iven by
X _L N l El titr(gX
1)j .
'
16
4-
a
04.
.1'
a.
a
VIP
2
,
is i ; .t.e,;:ard-d la:: a ',f :aoat,irters
-12 r , ? -,; aria -,, arid .i.--, a-,_ : ,,:a:. imiz( ci c,-itii
v' ..k. . .
re:Tr_zot to thse, taking into p.c-...o.;_rit th.....1 :--,,-,- Hi: ,-:.t:- ,,.ay L, f'i:.&,d ar.d.* . ."
equal to ::ore other.,,. :"a:,..Imizil. .Lor- L..,..;one may be cuistrained to be
14
eqltivaleht to 1%ini,Mizing
(172)[ 1o;r, 12-:1 tr(SZ ) - IG
a minimization probl&rn ma:/- be formalized,
Let ?\' = ) be, a vf.r..tor a4i the elements.
cis and arranged in a r,rescribcdI) A
Nay be regarded as a f,Inctirin 17(n) of .)\1 2' ' '
r,and has continuous derivatives oF/t.1?\s and ',
2Fie
second *der; ,..;he-i.e J. -is sincular. 7Tie t'otillity of thesi:
which is cont
of first and
derivatives is represented It, y a cradient, vector 011,A and a symmetr
Matrix2F/0.6?: . ::o let some
the
fna.
of t?le 7's be fixed and denot(-*-
remaininc Vs by :1.1,g ...,a r ., The function. "
considered,as a f:_tnction (;(r) -of r.:1;:t2,.., -irs 7)-rivativs
s now4
2and t.,:.-G/c 7:5t0' are obtained from ,,FP ancr o
2110,,..;)%e by omitting
and columns c2rres-pondir.:: to e fi 9 Amoni:', ir, , -.12, ., let
Cher be some t distinct pa.ram--ters denoted K12 2" tthat 'each .11. is c:Tial to on, --,.?:d -n,1:" ..;n( .. b,tt sibly f.7CIP ra',
4 I l4/,T . ? v,qal r' Ca :710 I., t it. . (_ .) b,- a matri" of order s : t., with
7 . .
dements
(or -G nol,; a functon c'f.
and we have,
0.
if 7t.' t. and 1, 0 oth,,,reise. The unction
1.1
V
-6-
V, .
1
, .
2hus, thvilerivatives of F. are simple sums Of the Terivatives of i
The -linimization of li(K) now a straightforward application o2 the
i 7 s-7avidon-El(Acher-Powe7 l methrA (Fletcher & Powell 7,65) 'u.sin a cor-..zter
rrogram by ::-uvaeas and'reskog (19yo): This method makes ,z.re of a matri-
. 4, z2valualied in each iteration.. E any positive
.
definite matrix api-royinatinj, the inverse of .i/oKor . In.subsegaen-tIN N
iterations is improved, using the information built an a' the f.Inc--, . -
. \
~
.'
Lion so that ultimately converges to art appro.'imation of the 1 -erse c)f
t' ti /i'ui'ti t the minklum. If there d/te many taratheters, the number o
iterations may be excessive, but .::;an be considerably decreased by the
ifrovision of a good initial estimate of
tined b, 'inverting the information matri:.
e filltolri) is obtained fr-,C
e(S-Fic27.07..1)
Such an estimate may b ob -
A
by omitting rows and columns4eo rresponding to .the fixed Vs. When the. .
minimum o1 .; has been found, the inverse of the information matrix. may
be (2-miputed ;gain to obtain standard e ro s ofall the parameters. in -c
ge,erai method for obtaining the elements of .e(oF/oXF/W) has i)e(n
ivi=n °re'skog (1-9(5)-
.4k
describ,,d here th information is
the -. inf,r4lation !-n.atri> in thr- r.27ar.hwo%Ild require the writin:;
a fal,:17 subroutine. will 1-,Dsibly be don a
t it ,r,-,7nt the proFram :orks as foliws. The star in point -lay L'
a rb t ri 1 . Fron; thc startinp; -.oint a n.:Inber of zteq,est c3.,:sent
it,(A'rt: a _ :,(_ef,rred until the decrea.;e 1n furration:val,_zez is less thar
+.h<- n-w point, so obtaine(2... the Davidon-Fletcher-PowAl rcrocedur,I.
()st,,,1-,;s :.itn
I'-fn ).,?r. -::: ._.=n identi t:, mat,: Ly .
I
The application. of the vidon-Fletcher-Powell method resuires formula..
47
for th,_ d.:rilrtives of- F with rspect to the elements of A AA -Y
,.._ -7
..-
and . These may be obtained by rrtrix differenti-.. _,
,.. ' )
atbn. ';:ritini; A = 11-' D = B-1r
, = DDDI + ATAT , and
Ts
r
dcrivatives are
_ P. :j= A Cy7
b. D'
-FPA , A pp n J,
Z.)FP.1.1 = -A'A' (.2 A C ni A W)-y
A
-y7-y-
Att., (' A D 4 r1),' A ),T.-Y
/10, diac,"(Z)
4
::her
/
4
O
6/
T
7 = D' A' D i At;/ A D Di AT . .c2 /
4
1.
A'A'n A A - diag(A'A'!] A 0-Y-YY-Y-
5
,1E/b9 = diag(n )
uF/o) diag( )5 xx-6
Tests of Hynotheses
4
When the maximum likelihood estimates of the parameters nave.been ob-
tainea, tne g6odness of fit of'the model,may be tested, in 2;arce samples,
by the likelihood ratio technique. Let Ho be the null hypothesis of the
model,under the riven specifications of fixed, constrained, and free laram-
eters.1 First consider the'case when the alternative hypothesis HI is
that is aviinositive definite matrix. Then minus twice the logarithm
of the likelihood-ratio is TIF0
where F0 is the minimum value of F .
IT the model holds, this is distributed, in large samples, as X2
with
d -1 (m n)(m n - 1) -t
degrees of freedom, irhere, as before, t is the total number of independent
oarameters estimated under Hr,, ,_,
Let H be any specific hypothesis concerning the parametric structur(ir 0
of the general model and let1
be an alternative hypothesis. In large
samples one can then test Ho against Hi . Let F0 be the minimum
of, under0
and let F be the minimum of F under Hi Then
'71< F and minus twice the logarithm of the likelihood ratio becolles
-- 0
K(F - v1.
) . Under n this is distributed approximately as X' with' 0
degrees of freedoM, equal to the difference in number of independent
parameters estimated kInder H1
and 10 .
A H.y:thetical
2o illustrate the ideas of tht :receding sections consider the mdel
dieted in 1, where circles denote true variables and squa--...es
do note 'observed variables. The other variables in the fiEre are residuals
or error variables. A one-way arrow denotes a direct causal influtnce
whereas a double E.rrow denotes correlation or covariation -Atliout a.
causal interpretation., The two arrows between
vocal interaction (simultaneity or interthpendence).
The model in Figure 1 has P. 4 y -variables, qA
= 7 y -variables,. ,
and denote recip-
m --- 2 Ti -variables n -variables. The structural equations
are
or
-1'112 "
yL ' 722 ' 1
2 1-
1P 4 (
?
-1 0
The twe 7 -coefficients in the fh:st equation are assuMed to be equal
111w-trate the idea of a constrained parameter. The equations relatin,7
the observed and true variables are
is
k.
.-1
N. I (1)\ CL. ' ..-4
',. -4I i-i.,
N .,1
\N
0
CO
and
-1
:2
5
x71
v.0
VT!
11
0
TI 0 /p
O 1
0. T2-112/
0 0
?\_. 0 0
N. '2\5 0
O 1 0.
O 2\, 0
O 0 1
O 0 2\7_
El\
E2
E_
1E4)
52
1 \55
254
5 /55
56
571
c
(15-0
(17c)
(15b) and (15c) one T in each column of 7-/i---4nd A has been set
echial to one to fix the scales of measuiement in the true -Variables. When
a solution has been obtained, one can scale this so that all true variables
have unit variance if this is desired.
Data for this model were generated,by assigning the following values
1.0000.902
0.00.0
11.0001.500
A0.9000.00.00.00.0
BF1.000110.595
0.00.0
1.00011.095
0.0 0.00.0 0.01.201 0.01.000 0.0
1.098 0.0
0.0 1.0000.0 1.400
70.4951
1.000
ti
T _(*1.000
ep =
0.599
0.9990.700:0.601
(
(0.506
0.586
0.5990.0
.
1.199
0.300
0.705)
-12-
0.0
1.198) '
1.398 '
ae. diag(0.522, 0.432, 0.556, 0.452) ,
8 diag(6.613, 0.515, 0.418, 0.522, 0.614, 0.526, 0.414) I-b
to each of the parameter matrices. These generate a E according to (4),
where
=-TY
5.8153.195
5.8904.259
3.0685.5083.842
5.7756.182 6.975
0.76h 0.690 0.176 0.1930.994 0.896 0.229 0.251
11.690 1.525 0.347 0.3800.835 0.753 0.157 0.1720.917 0.827 0.173 0.1891259 1.136 1.824 1.9971: 63 1.591 2.554 2.797
1.5741.298 1.952
1.739 2..260 4.2240.700 0.909 2.069 1.4710.768 0,998 2.272 1.317 1.822
0.601 0.781 0.902 0.300 0.3300.841 1.094 1.262 0.421 0.462
1.6751.958
In the Appendix this model is analyzed using the above matrices as S ,
YY
S , and S_xy -xx
-15--
1.5. A Model of Duncan; Haller and Portes
In a study on peer influences on aspirations., Duncan, Haller, and
'Fortes (1968) gave severa*1 examples of path analysis models. Their model
ry is particularly interesting since it involveso unmeasured variables.
This model is reproduced here, in different notation, in Figure 2. In this
model, p = 4 , ,q = 6 , m = 2 , n = 6 . The six x -variables are
assumed to be measured without error, so we take = x v.
i = 1,2,...,6 ; i.e.; in terms of equation 0) we have A =. I and S = 0-x
The structural equations are
1
2.
(
1 111 yi y2 y3 y4 0 0..)
(32 1 n2 o o 75 76 77 78 3/4
and the equations relating the l's to the y's are
h\1
/12
)4 "
0 1\.. ri
2
(11)
11u4
?\
2
r2.
(1 2)
(16a)
(16b)
Duncan, Haller, and Port©s postulate that the u's ale correlated with each
other except that u1
and u2
are uncorrelated and also u_ and u,4 u
However, the four correlations p(u1.u2) , p(al,u4) , p(u2 5,u.) and.
p(u2'
u4) are not alliidentified. To make the model identified one of them
must be fixed and ire have chosen to set p(u2'u4) = 0 . Equation (16b) is
not in the form required by the general model. This is easily remedied, by
u as
4
ti
s
Flgure
)del D iiaa.ler, and 1),,rtes
I-
3
Jf
here
pi T5
u2
0
u_.
) 9\u1 / 10
T3, 14,
variances. It should be noted that there is a one-to-one correspondence
0
?\
70
0
0
T8
T10
0
-15-
0
T11
x'131
T14
5 O
(16e)
r15 , and r16 are mutu115 uncorrelated and of unit
between the nonzero variances and covariances of u and T5'T6' 11
.
Introducing t7 E 115 ', E Tht ,
915
, andX10 E T16
the whole
model may be specified as follows
X31X31 0 0
'321 0 0' 0
0 0 1 0 0
0 0 0 1, 0
0 0 0 0 1
0 0 0 0 0
x5
0
0
0
0
1
11
_ 2,
yi .72 73 7 0 0 0 0 0 0
TI20 0 75 76 77 78 0 0 0 0
4
'94= C 0 0 0 0 0 0 .. 1 0 0 8
6
-,- 01
1 3 0 0 0 0 0 0 0 1 0 0 05
115 0 0 0 0.0 0 0.6107
0
0 0 0 0 0 0 0 0 0 i...
80
i9 i
h.C1. (17a)
116
v1
v3
v4
V5
6
1 o o o o 0 o o o o100000000o o 1 o o o 0 o o o
0 0 0 1 0 0' 0 0 0 0
o o o 0 1 o o o o o
O o o 0 0 1 0 0 L0 0
(6 1
X21
0
0
0
40
50
7
C
Yl
"2
Y3
-16-
h 0 h_ 0yl-
0 0)2
0 ?\
3?\
90 'N10
11 , LO hit 0 0
Furthermore one specifies
and
:DOD x 10)
1/(6, x 6) .
0(4 x 6)
1
PRI.2)0
0
0
0
0b(6 x- 6) = 0 ,
e (4 x 4) . 0 .
1(4 x 4)
1
1
?\6' T12
011
30
2\1
96
:115
-1
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
This model is analyzed in the appendix using the following data
Ak.
from Table 1 of-Duncan, Haller, and Portes (1968).(N = 329 ):
s=
x2
x1
x3
-1.0000 .1839 .222o
1.0000 .0489
1.0000
.#
a
Yi 3,-2x5 x6 x4
...Y4 Y3
.4105 .4045 .3355 .1021 .1861 .2598 .290-A
.2137 .2742 .0782 .1147 -0186 .0839. .1124'- .3240 .4047 .2302 .0931 .2707 .2786 .3054:
1.0000 .6247 .2995 .07.60 .2930 .4216 .32691
1.0000 .2863 .0702 .2407 .3275 .3669!
1.00001$ .2087 .2950 .5007 .5191.
1.000o -,0438 .1988 .2734"
1.000o .3607 .4105,
1.0000 . -.64041
1.0000;.._. ,-. ,
.
1.6. A Restricted Factor Analysis Model
Consider the model shown in Figure 3. This is a restricted' factor
analysis model in the sense of.areskog (1969) and may be most conveniently
analyzed by the method describedzin that paper. However, it can also be
analyzed using the lel considered in this paper. To do so one specifies
E , i.e., one chooses B(3 x 3) = I r(3 x 3) . I , and = 0 . The
matrices A and A are as follows:-Y -x
A .
?1 0 0
?2
0 0
0 ?., 0
0* ?4
0
6 ot
a5
0 0 ?6..
7,7 0 0
Ay.
. 0 ?8
0-
79_0 0
The matrix (1) is specified to be a correlation matrix and 82
and -!-)
2
-e 0 6
contain the unique variances of the tests. An example is given in the
Appendix.
:r
6.3
e
/ 4.
_18--
T.,-;- -r .--,1 e )
; ?.. stricte4 pct r- Analysis Y,Jdel
7-i
A ,,-
/
CY
./
19-
2. The Program
.
In this section we describe briefly'what the program does., Details
'
about the input and output are given in sections 3 and 4 respectively.
2.1. .What the Program Does
.
The input ,data may 13e a dispersion matrix, a correlation matrix,A
a
correlation matrix With standard deviations, or raw data from which the
matrix to be analyzed can be computed. From the,inpul matrix, variables
may be selected to be excluded in the analysis, so that the matrix to be
analyzed may be of smaller order than the input matrix. This selection
procedure also"allows columns (rows) of the input tatrix to be interchanged.
The matrix to be, analyzed may bsums of squares and cross products,
deviation sums of squares and cross products, a dispersion matrix, or a
correla:(donmatrix.
Simplified versions;of the general model maybe estimated. That is,
the user can specify to estimate a model where there are no disturbances,
or a model when there is no x , or a model where B = I . In these cases
the equations are simplified and the minimization procedure is faster.
Also, the user can request an accurate or an approximate solution. If an
accurate solution is requested, the iterations of the minimization method are
continued until the minimum is found, the convergence criterion being, that
the magnitude of all derivatives be less than .00005. The solution is
then usually correct to three significant digits. If an approximate solutioh
is requested, the iterations terminate when the decrease in function values,/
is less than 5'.. The approximate solution may be substantially different
from the exact solution, but the residuals and the value of X2
will usually
;4*
.-20-
#
give an indication:of how reasonable the hypothesized model is. The option
of an approximate solution has been included in the program for the purpose-
4
of saving computer time in exploratory studies when-the primary purpose is.
to find a reasonable model. Once such a model has been ound, an accurate
solution may be computed.a 4
A variety of options for the output is available. 'Residuals may.
be printed. These are defined as the difference between reproduced (i),s
and observed (S) variances and covariances, which are useful for judging
'she godness of it of the model to the data. The standardized solution can
be computed and printed if requested. X2
-is`printed as an_overall goodness
of fit test statistic.. The final maximum likelihood, solution may be parched.
on' cards if requested.
2.2. z Fixed, Free and Constnined Parameters Are Specified.
The elements .".f the eight parameter m-rices are assumed to be in the
order Ay ,A , B, r, (1, 1 T_-xelements are ordered row-wise. The diagonal elements of the diagonal ma trice:.
e and 26 are treated as row vectors and only the lowerdiagonalparts
6Y symmetric '., and T are taken into account.vo. -
For each of the eight parameter matrices, a pattern matrix is defined,
with elements 0 , 1 , 2 and 3 depending on whether the corresponding
element in'the parameter matrix is fixed, free, constrained follower and
constrained leader, respective3. A constrained parameter is called-a con -
strained leaderthe first time it appears in the sequence. The parameters,
appearing later in the sequence and assumed to be equal to the constrained
leader are called constrained followers.
7 eE 7 28 and within each matrix, the
O
-21-
1
The :above technique defines uniauelY the positions of the f_ ed, free
and constrained leader arameters. It does not define, however, whichA
followers go with which leader, if there is more than one leader. To :do
so one must also specify all the fpllowers associated with a given,leqffejr-.
This is done by assigning to each leader and follower a 'our -digit number
NCCC, where M defines the matrix in which the constrained parameter appears
and CCC defines the position of the parameter in that matrix. Pius,
M = 1 for , 2 , 3 for B ,kfor 5for 0 , 6-for 'r
7 for 8E
, and 8 for ?5 For example,
4001 40,05. 50144
defines the first element in P 73.
, to be equal to fifth element in
P, as well as the fourteenth element in , where is the4)14
leaderandy-and are the followers.2 .
14,
Pattern mOrices have to be provided for each matrix containing both
fixed and free parameters and for each matrix containing constrained param-
eters. Patterns for matrices whose elements are all fixed or all free arc
sei up by the program.
We give a simpli;' example-to illustrate the above specifications..
Suppose B (5 x 3) and P (5 7. 5) = I , all elements in both B and
P fixed, (3 y, 5) with all diagonal elements lite, and
A =
p\3.
A2 b 0
o 2,
30
O 2\4- 0
o 0 A5
O p 2\ 6
)7 0 0
A = 0 2\8 0 i
0 "0 ;,9 I
. diag
1
, ,3
ti
-22-
with ?2 ?\8
5= ?\,
0and 0 =0 , 0
c3= e
e,,
El ,
O = 9 . The pattern matrices for A A (D and Jr
5c,
3 0 0
2 0 0 0 0 0
PA
0 1 0 PA = 0 2 0 P = 1 0
Y 0 1 0 X [o o 1 2 )
o 0 3
O 0 2_ C 3 2 3 2 3 2
C
and the specifications of leaders and followers are
1001 1004 2005 5004
1015 1018
7001 70027003 7004
7005 7006
0
In this model fourteen independent parameters willthe estimated.
In addition to the above specifications for fixed, free and constrainrd
parameters, start values have to be given for all parameter matrices, except
when a simplified iiiodel is to be estimated (see 2.1). That is, if there are
no disturbances, start values for T are not read in. If B = I , B is nott,-
read in. 'An'd if there is no x start values for A., r , (;) and 0
are not read in. The start values define the fixed parameters and the
initial values for the minimization procedure for the other parameters.
Constrained parameters assumed to be equal must be given the same values.
Otherwise, start values may be chosen arbitrarily but the closer they,
are to the final solution the less computer time it will take to reach
the solution..
a
13.
2.5. Limitations 0
-23-
4
The program is written in FORTRAN IV - G and has been tested out on
the IBM 360/65 at Educational Testing Service. Double precision is used
in floating point arithmetic tlroughout the program. With minor changes
the program should run ofPany computer with a FORTRAN IV compiler. Iz
computers with a single word length of 36 bits or more, single precision
is probably sufficient.
Limitations as to the,maximum number of variables, the maximum number
of independent and nonfixed parameters, and the maximum order of the .pasan-
"e matrices allowed, as well as the core requirements of the prbgram
on the IBM 360/6) are given in the following table.
max. no,. of variables (p0 + go) before selection = 80 -1
max. no. of variables + q) after seldc:tign == 30
max. p
max. 0
max. m
max. n .
ma ;. no. of independent parameters
max. ho. of nmfixed parameters
core requirements (K = 1024 bytes = 256 words)
2.4. Availability
= 15
= 15
= 15
= 1.5
= 80
= 80
. 140K
A copy of the program may be obtained by writing to Marielle van Thillo
at ETS. The user must provide a tarp on which the program will be loaded.
The program will be written on the take with 80 characters per record. The
tape will .be unlabeled. The user must specify whether he wants the tape
0a
-24-
blocked or unblocked, on,7 -track or c)-track, in EBCDIC Or BCD mode, as
well as the density and parity required. Test data will \e_ at the endft
,
of the program. The test data are .described in the Appendix. Anyone
using the program for the first time should make sure that the test data
run correctly.
2.5. Disclaimer
Although the program has been working satisfactorily for all data
analyzed so far, no claim is made thct it is free of error ano warranty
is given as to the accuracy and functioning of the program.
5. Input Data
Fcr each data to be analyzed, the input consists of the following:
1. Title card
P. Parameter card
3. Input matrix
4. Specifications for selection of variables froM the
input matrix
Pattern matrices for the parameter matrices
C. Fqualities
7. Start values for the parameter matrices
8. New data set or a STOP card
Sections 5.1 through 5.8 describe in general terms the functiOn and setup
of each of the above quantities. Illustrative examples are given in the
4.ppendix.
Whenever a matrix or vector is read in it is preceded by a format
card, containing at most.80 columns, beginning with a left parenthesis and'
ending with a right parenthesis. The format must specify floating point
numbers for the input matrix and parameter matrices, and fixed point numbers
for the pattern matrices, consistent with the way in which the elements of
the matrix are punched on the following cards. Users who are unfamiliar
with FORTRAN are referred to a FORTRAN where format rules are given.
Matrices are to be punched as one long vector, reading row-wise. For the
symmetric matrices only the love} half of the matrix including the diagonal
should be punched.
3.1. Title Card
Whatever appears on this card will appear on the first page of the
printed output. All 80 columns of the card are available to the user.
4
5.2. Parameter Card
All quantities' on this card, except for the logical indicators, must
be punched as integers right adjusted within the field.
cols. 1 - 5
cols. 6 - 10
number Of variables (p0 ) in y before seleCtion
number of variables ( qo ) in x before selection
(i.e., the input matrix S is of order
(p0 ± q0) x (p0 + qo) before: selection of
variables)
cols. 11 - 15 number of columns in A ( r )
cols. 16 - 20
cols. 21 - 25
cols. 26 - 5)
-y
number of columns in A ( n )x
number of observations (m )
total estimated execution time in seconds for all
stacked data ( SEC ). This should beta number
slightly less than the time requested on the
control cards sothat the program will have time
to print and/or punch results up to that point
(Note! SEC should be read in for each data set
and should be the same for all data sets in the
stack.)
col. 41 logical variable
= F if ,.here is no x , i.e, there are no A ,-.x
ei) . (In this case n = q = 0 )
T otherwise
col. 42 logical variable
F if there are no disturbances i.e., there
is no tif
= T otherwise
col. 4:5 logical variable
=F if B I
= T otherwise
col. 44 logical variable
col: 4,-)
col. 46
col. 47
col. 51
= F if the exact solution is to be computed
= T if the approximate solution is to be computed .
logical variable
= F if the solution is not to be punched on cards
= T if the solution is to be punched on cards.
This will automatically be done if IND / 0
(see 4.1 and 4.6)
logical variable
= F if variables are not to be selected frpm theY
ii
input matrix
= T if variables are to be selected from the
input matrix
logical variable.*
= F if the standardized solution is not to be printed
T if the standardized solution is to be
printed (see 4.4),4s
= 1: if raw data ( YIX ) is read in to compute
the matrix. S to be analyzed
= 2 if the input matrix is a.dispersion matrix
3 if the input matrix is a correlation matrix
= 4 if the matrix to be analyzed is the same as
in the previous data set, i.e., the input
matrix is not to be read in
-28-
col. 52 = 1 if the matrix to be analyzed is sums of
squares and cross products
= 2 if.the matrix to be analyzed is deviation
sums of squares and cross products
Note: Co152 can be 1 or 2 only if col. 51 =
3 if the matrix to be analyzed is a dispersion
matrix
= 4 if the matrix to be analyzed is a'correlation
matrix
col. 53 = 0 if only the standard output is to be printed
(see 4.1)
= 1 if the input matrix, the specification
matrices and the initial solution are to be
printed
= 2 if -theresiduals and matrices C and D are
to be printed (see 4.3)
3 if both 1 and 2 apply
. 4 if technical output from the minimization
procedure is to be printed (see 4.5)
. 5 if both 1 and 4 apply
= 6' if both 2 and 4 apply
. 7 if 1, 2 and 4 apply
3.3.1tMatrix
If col. 51 . 1 on the paramete) card an M x (p0 4 go) matrix ( yiz )
of raw data is read in, one root at a time, Starting a new card for each row.
-1RNote that this is the only input matrix not read in as.one continuous long
vector. The matrix is preceded by 6 format card.
-29-
If col. 51 = 2 or.3 on the parameter card the lower triangular part,
includng the diagonal, of the input matrix S is read in, reading row-wise.
By S we mean the partitioned matrix
s(P0 go x po q0)
.
xPO)
xy(qo x PO 2xx(q0 x q0)
The input matrix is preceded by a format card. If'a correlation matrix is
read in but a dispersion matrix is to be analyzed (i.e., col. 51 - 3 and
col. 52 = 5),the input matrix S is followed by a format card and a
vector of standard deviations on subsequent cards.
If col. 51 = 4 on the parameter card the input matrix is not to be
read in.
5.4. Specifications for Selection of Variables from S
These cards will be read in only if column 46 = T and column 51 4
on the parameter card. Omit otherwise.
The first card will have the integer values p and q punched in
columns 1-5 and 6-10 respectively, ri-ht adjusted within the field. These
integers will specify the order of S after selection (p p0 + go ).
The next card(s) will contain integers, right adjusted in five
column fields (i.e., sixteen such values will fit on one card) specifying
which columns (rows) are to be included. For example, if p0 qo = 3
and p = 3 q = 1 and columns (rows) 1, 2, 5, 8 and 9 of S are to be
exci_____.1._jided. then this card will have a 3 punched in column 5, and a 4
L30-
punched in column 10, a 6 punched in column 15 and a 7 punched in column '
20.
Note: If p0 + go = q there will be no reduction in the size of., !
S but columns (rows) can be interchanged.
3.5. Pattern Matrices for the Parameter Matrices
The pattern matrices are preceded by a data card with entries in
columns 1-8, the column defining the matrix in question, 1 for A ,
-Y
2 for A 3 for B , for F 5 for (1.) 6 for 7 for e 8 for. ,
cols. 1 - 80 CCCCCCCC where C = 0 if the matrix is fixed
C = 1 if the matrix is free
C = 2 if the matrix has mixed values
and/or constraints
A pattern matrix should be provided only when C = 2 . (See 2.2.)
For example, if columns 1 - 8 are punched 22002021 the matrices
AY
A ,> ) contain mixed values and/or 2onstraints, the matrices
B , -E , * are all fixed and matrix 76 is all free. In this case only
pattern matrices for Ay
, A , a) and e are to be read in.-s
The pattern matrix consists of a format card specifying an I-forMat
and subsequent cards with the integer entries of the pattern matrix.
3.6. Equalities
Omit if the pattern matrices do not contain any elements 2 or 3.
Otherwise starting in column 1 punch the four-digit, numbers MCCC as
described in section 2.2. For each new constrained leader start a new
card. The last entry on each "equality'. card should be a zero indicating
-31-
more "equality" cards are to follow or a nine indicating it is the last
one. In the example used in section 2.2 these cards would look as
ti
follows:
100110040100810110 _
101510180700170020700370040700570069
3.7. Start Values for the Parameter Matrices
Start values for each:of the parameter matrices are read in, each
preceded by a format card, and in the order previousiy'described (i.e.,
Ay , Ax , , P, (1) , , , e ). Only the lower half of (1)
E
and *- are read in.
1,4
If col. 41 = F on the parameter card do not read in start values for
A r (1) and 8x ' 5
If col. 42 = F on the parameter card do not read in a start value for
If col. 43 = F. on the parameter card do not read in a start value for
B .
3.8. Stacked Data
In sections 3.1 to 3.7 we have described how-each set of data should
be punched. Any uuMber of such sets of data may be stacked together and
analyzed in one run. After the. last set of data in the stack, there must
be a card with the word STOP punched in columns 1 - 4.
-32-
4-. Printed and Punched Output
-The output consists of a series o1 printed and punched tables as
described in sections 4.1- 4.5. Examples of printed outputs are given
in the Appendix,
4.1. Standard Output
The standard output is alvq,rs obtained, regardless of the value
punched in column 55 of the na!rameter card (see 5.2). t" The standard
output consists of the title with parameter listing, the final solution,
and the result of the test of goodness of fit.
The parameter listing gives the information supplied on the param-
eter card.
4
4,
The'final solution consists of the eight matrices A , A , B , 4
Y -x
(I) . ) 9 All numbers are printed with three decimals.
The test of goodness of fit giyes the value of X2
and the cor-
responding degrees of freedom. The probability level is also given. This
is defined as the probability of getting a X value larger than that
actually obtained, given that the hypothesized model is true.
Just above the table giving the final solution, the following message
is Printed'
"IND = X"
Usually X = 0 , but if, for some reason, it has not been possible to
determine the final, solution, X will be 1, 2, 3, 4 or 5. If IND is
1, 2 or 3 , "serious problems" have been encountered and the minimization
the function cannot continue. One reason for this may be erroneous
-55-
input data. Another reason may be that insuf icient arithmetic precision
is used. If IUD is 4, the number of iterati ns has exceeded 250. If
IND is 5, the time limit SEC has been exce ded (see3.2).. IND 0 ,
the solution obtained so far is automatically pu. ,ed on cards in such a
way as to be immediately available as initial estimates for a new run"
. with the same data Thus there is little loss of infor6.tion when
execution is terminated with IND / 0 .
).2. Input Matrix S and Parameter Specificat4ons
If column 55 of the parameter card is 1, 3, 5 or 7 (see 3.2), the
matrix to be analyzed,. S , as 6btained after exclusion or interchanging of
variables, if any, is printed. By S we mean-the partitioned matrix
S .
(S,
-YY
S-xy
S-xx
The matrix S71
will be Printed first, followed by Sxy
and Sxx- -
A table of parameter spec4cations, containing the information provided
By the pattern matrices and e,cluality cards (see 2.2) is also printed.
Integer taf.rices are priited corresponding to the parameter matrices. In
each matrix an element is an integer eqUal to the index of the corresponding
parameter in the sequence of independent parameters. Elements corre-
spondingto fixed parameters are 0 an_i elements corresponding to the same
constrained parameter have the same value. Examples are given in the
Appendix.
The initial solution or start values for the parameter matrices will
also be printed.
.t4
.04
5
-v
-34-
4.3. Matrices. Z , C , D and Residuals
If column 73 of the parameter card is 2, 3, o or 7 (see 3.2), the
matrices and residuals are printed.
The matrices E C , D are computed from the final solUtion. By
C , D we mean
where
E
Z
-xy
=-YY -Y- -Y -
E = DT'iT-xy x, -y
7. c^7Pl)
-x--x 0 .-xx
t-1,311-1,Movev ~ evM
If the fit is good Z should agree well with S. and the residual
matrix, Z - S , should be small. Elements of the residual matrices may
suggest how the hypothesized structure should be modified to obtain a
better fit. The matrices are p.'inted row-wise, each element with three
decimals.
4.4. Standaf-dized Solution
If column 47 or the parametet card is T (see 3.2), tlie standardized
,,
solution ( 11:m. '../. fi* F*., 6* and ** ) will be printed, as well,
x,
as the standardized matrices & and D* . That is:, ,
7
`
51.* = 11.7 A ,
-Y -Y-I
?* '
a,
-11
where
Tiv
A = (diag 6)1/2'
A = (diag $)1/2
-14
4.5. Technical Output
If column 55 of the parameter'card is 4, 5 or 7 (see 5.2),-the techni-,
cal output is printed. This consists of a series of tables which describe
the behavior of the iterative procedure and give yarious measures of the
accuracy of the final solution. Ordinary users will have little interest
-in these tables.
The tables show the behavior of the iterative procedure under the
steepest descent iterations and under the following iterations by the Davidon-
Fletcher.-Powell method., For interpretation of these tables the reader
-r-
-36-.
4
is referred to Gruiraeus and J8reskog (1970). If something goes wrong, so
that IND is 1, 2 or 3 (see 4.1), these tables may contain valuable
information.
.6. Punched Output .
If column 45 of the parameter card is T (see 3.2), the final solution
is punched on cards. The matrices are punched on cards in vector form,
reading row-wise, each preceded by a format card. Only the lower diagonal
parts of $ and * will be punched. column 41 of the parameter card
is F Ax
, 8, and r"-will not be punched. If column 42 of the-o
parameter card is F , If will not be punched. And if column 43 of the
parameter card is F , B will not be punched.
If IND / 0 (see-4.1), the final-solution will be automatically
punched, regardless of the value of column 45 on the parameter card.
-37-
References
Bla_,ock, H. M., Jr. Multiple indicators and the causal approach to measure-
ment error. American Journal of Sociology, 1969;22, 264-272.
Blalock, H.,M., Jr. Causal models in the social sciences. Chicago:
Aldine-Atherton; 1971.
Costner, 'H. L. Theory, deduction, and rules of correspondence. AmericanIL
Journal of Sociology, 1969, /5; 245-263.
Duncan, 0. D. Path analysis: Sociological examples. American Journalof
Sociology) 1966, 72 1-16.
Duncan, 0. D., Haller, A. 0., & Portes, A. Peer influences on aspirations:
A reinterpretation. American Journal of Sociology, 1968, 2141, 119-137.
Fletcher, R., & Powell, M. J. D. A rapidly convergent descent method for
minimization. Computer Journal, 1963, ..;) 163-168.
Geraci, V. J., & Goldberger, A. S. Simultaneity and measurement error
Social Systems Reseach Institute Workshop Paper ENE 7026. Madisdn,
Wisc.: University of Wisconsin, 1971.
Goldberger, A. S. Econometric theory. New York: Wiley, 1964.
Gruvaeus, G. T., & J8reskog, K. G. A computer progma for minimizing a
function of several variables. Research Bulletin 70-14. Princeton,
N.J.: Educational Testing Service, 1970.
Hauser, R. M., & Goldberger, A. S. The treatment of unobservable variables
in path analysis. In H. L. Costner (Ed.), Sociological methodology
1971... San Francisco: Jossey-Bass, 1971. Pp. 81-117.,
Heise, D. R. Problems in path analysis and causal inference. In F.
Borgatta (Ed.), Sociological methodology 1969. San-Francisco:
Jossey-Bass, 1970. Pp. 3-27..
-38-
Johnston, J. J. Econometric methods. (2nd ed.) New V rk: McGraw-Hill,
1972.
'gresKog, K. G. A general approach to confirmatory maximum likelihood
factor analysis. Psychometrika, 1969, 2., 183-202.
J8reskog, K. G. A general method for analysis of covariance structures.
Biometrika, 1970, 2], 239-251.
J3reskog, K. G. A general method for estimating a linear structural
equation system. In A. S. Goldberger& 0. D. Duncan (Eds.),
Structural equation models in the social sciences. New York:
Seminar Press, 1973, in press.
Land, K. C. Principles of uath analysis. In E. F,Borgatta (Ed.),
Sociological methodology 1969. San Francisco: Jossey-Bass, 1969.
Pp. 3-37.
Malinvaud, E. Statistical methods of econometrics. (2nd ed.) Amsterdam:
North-Holland, 1970.
Turner, M. E.I & Stevens, C. D. The regression analysis of causal paths.
Biometrics, 1959, 12,,236-278.,
Wright, S. The method of.path coefficients. Annals of Mathematical
Statistics,. 1934, 2, 161-215.
Wright, S. The interpretation of multivariate systems. In 0. Kempthorne,
et al. (Eds.), Statistics and mathematics in biology. Ames, Iowa:
Iowa State University, 1954.
Wr_ght, S. Tha treatment of reciprocal interaction, with'br withut lag,
in path analysis-. Biometrics, 1960, 16, 423-L45.
-Al-
Appendix
We shall illustrate how input data are set up anr, what the printout
"_ooks like by means of three small data sets. These data also serve as
test data to be run when the program has been compiled on a different
computer.
Pages A5 - Alf show card by card how the input data are punched. One
line corresponds to one card. Pages A5 -A54 show the corresponding printout'
obtained.
Tne first set of data is the artificial data discussed in'section 1.4.
N_ intermediate outputdis requested and the standardized solution is not
printed.
The second set of data is the model of Duncan, Haller and Fortes
discussed in section 1.5. :There is,no selection of variables from the
matrix, but columns (rows) are to be reordered. The standardized solution
is requested and printed. Note that the correct number of degrees of freedom
is twelve and not 35 as given by the program. Since we know the solution
S , we treated (1) as fixed at S when the program was run. But1 X -xx
should be considered as free, which accounts for the discrepancy in
degrees of freedom.
The third set of data-is the restricted factor analysis model dis-.
cussed in section 1.6. Note that this model n'ither W nor B are
be read Columns (rows) of the input matrix, are to be reordered.
Intermediate output is reauested.
For all three data sets both the input matrix and the matrix to be
ana1'zed are correlation maqices.
1 -711.11/
-P2a.
At various places in the output, time estimates are printed. The
time shown is the time taken to compute the solution that follows the time
estimate. This time includes only the iterations and not the time for
printing, except possibly the technical printout.
-45-
ART)
F).0
1AL
47
18F1"."
/
DA
TA 2
31.1
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343
3.815
3 15
3.,
413 85
'3.
5:4
5.773
4.295
3.84?
6.192
6.073
.764
.69
.176
.193
1.374
.594
.896
.229
.251
1.298
1.952
1.65.
.1.529
.347
.38
1.739
2. N,
'4.224
.835
.753
.197
.172
.7
.9...0
2..6c
1.471
.917
.827
.173
.189
.768
.998
2.272
1.317
1.822
1.259
1.136
1.824
1.997
.6( 1
.781
.9.2
.3
.33
1.675
1.763
1.591
2.554
2.797
.841
1.094
1.262
.421
.452
1.958
2.914
22221111
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PEER INFLUENCES ON
7 T7FF7T
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ASPIRATIONS
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