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Michigan State
University

 

This is to certify that the
dissertation entitled

Estimating the Covariance Components of an Unbalanced

Multivariate Latent Random Model Via the EM Algorithm
presented by

Leonard Joseph Bianchi

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Counseling,

Educational Psychology & Special
Education (Statistics & Research Design)

ém

Major professor

Date May 16, 15287

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ESTIMATING THE COVARIANCE COMPONENTS OF AN UNBALANCED

MULTIVARIATE LATENT RANDOM MODEL VIA THE EM ALGORITHM

by

Leonard Joseph Bianchi

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Counseling, Educational
Psychology and Special Education

1987

ABSTRACT

ESTIMATING THE COVARIANCE COMPONENTS OF AN UNBALANCED

MULTIVARIATE LATENT RANDOM MODEL VIA THE EM ALGORITHM

/ /I- By
I. /..
r'
x? I
f/ f
qu Leonard Joseph Bianchi
II 7 f [1" 4 ’7 q

Iélthgugh statistical procedures’gze aggilable_£gx £§£1QQELEE.
_E£§atment effects in: students taught in glggggpoms, these.prgcedures
are applicable only when every class has the same number of students.
The present study‘investigated a procedure that was originally
established to handle missing data (EM Algorithm) but which also
provides a solution to the problem of estimating parameters in
multivariate analysis when samples contain unequal group\sizes. The
focus of thepresent dissertation was on the estimation of latent group

nd individual level variances and covariances with measurement error
emoved when group sizes varied in a sample. Previous methods could
nly find maximum likelihood estimates for this problem if the dataset
contained groups of equal size. The EM Algorithm offers A method for
finding maximum likelihood estimates of parameters in situations where
classical maximum likelihood procedures faiI.

The estimate of balanced and unbalanced samples were both studied

while varying two factors, mainly the number of groups in the sample

Leonard Joseph Bianchi

(the size) and the particular model being estimated (that is to say,
the unrestricted.model, the correctly specified model and the
incorrectly specified model). Only I? replications ere used in this
demonstration of the algorithm under different circumstances.

Tests of the model based on the criteria of convergence showed
this estimation procedure to be a satisfactory and effective method in
theory. However, problems in the use of this algorithm appearred in

the form of large number of iterations needed for convergence and lack

of a universally accepted criterian for convergence.

This dissertation is dedicated to Robert Garden, Chris
Mooney and the small group of IEA workers in New Zealand

from 1981 to 1983 from whom I learned so much.

ii

ACKNOWLEDGMENTS

I can never fully express the appreciation and thanks I have for
Dr. William Schmidt who as both a friend and my chairman never gave up
on me. His support was invaluable.

To Dr. Richard Houang I owe much for both his time and effort.
He forced me to think, a pesky thing to do, and attempted to open my
eyes and mind and look beyond the obvious meaning of the research.

The other two members of my committee also have my thanks. Dr.
Stephen Raudenbush gave freely of both his time and knowledge while Dr.
Robert Floden contributed suggestions and ideas.

A very special thanks is due to JoAnn Green who, as a friend
and university employee, helped steer me through the sea of regulations
and red tape which, by necessity, exist in a university. Without her
kindness and help, I know all would have been lost.

Thanksis also given to Janice Benjamin for the use of her
equipment and to Laurence Bates who helped me with the logistics which must
be faced by every doctorial student. I

My gratitude goes to Dr. Arthur Tabachneck for his long hours
spent in an attempt to teach me the lost art of technical writing. His
kindness is more than appreciated.

My special appreciation goes to Peri who put up with my tirades,

depressions and periods of paranoia with patience and care.

{y-‘ iii

Lastly my appreciation goes to my parents who never faltered in
their belief in me, who helped me out during the tough times and who
never once asked "When are you going to graduate and get a job?". (Well,

hardly ever......)

iv

(yo :LKHS u »
‘I d": is» Mbfl’» .. I
'"y p, I
x}; ‘ '
” ‘\ I IV
TABLE OF CONTENTS _ . /3,_:-
. /.I '. . /o I 1 ’ ‘
LIST or TABLES .................................................. vii
LIST or FIGURES ................................................. viii

Chapter

/ / INTRODUCTION ............................................... 1 '

LATENT COVARIANCE STRUCTURAL MODELS ........................ 6

//II z/l/ Single Level Covariance Structural Model ............... <IGI7
I . Multilevel Covariance Structural Model ................. l7
3. Extension of G' and O to causal models ................. 22

/ T/I. REVIEW OF RELEVENT won; 2/ ................................... 23. o.
'l. Schmidt's Structural Model a ............................ 23
.2. Unbalanced Designs ...... a .............................. 32
"I y MISTATEMENT or PROBLEM 2" ...................................... ("3‘s
3 fIESTIMATION PROCEDURE ....................................... 37
I §l. Expectation-Maximization (EM) Algorithm ................ 37
I2. Theory for the Restricted Model ........................ 39
I3. The EM algorithm ....................................... 48

~‘1
x.

.r‘

F 4, v/ DESIGN OF STUDY 2 ........................................... 51.9.." ’

 

. Design ................................................. 49
. Generation of Data .9 ................................... 52
. Starting Values of Parameters . ......................... ( 55:3
CompuEer Program ..g .................................... 57
gfl RESULTS OF STUDY, ....};.;;2.. .................................... ~5&9,
I
. Design and Measures ..... N ............................... 59
2. Results of Estimation Procedure , ........................ 62
3 Results of the Process ........ 3 ......................... 86
I /"'>

/’6 /L7\/
VIII. DISCUSSION ................................................. 90

1. Summary and Conclusions .............................. éffi;
99

2. Future Explorations ..................................

APPENDICES v
A. Equations for the Estimation of the Covariance

Components of the Two-Parameter Model

Using the EM Algorithm ........................ 101
B. Equations for the Estimation of the Covariance

Components of the Four-Parameter Model

Using the EM Algorithm ........................ 103
C. Computer Program for the Estimation of the Three

Parameter Latent Model through the

SAS Package ................................... 106
D. Computer Program for the Estimation of the Four

Parameter Latent Model through the

SAS Package ................................... ll5‘

BIBLIOGRAPHY ..................................................... _123 -

vi

Table

0‘

10

11

12

LIST OF TABLES

Parameter Values Used in Data Generation ...................

Summary Statistics of the Latent Models for Balanced

and Unbalanced Data Sets with 100 Groups ...................

Summary Statistics of the Latent Models for Balanced

and Unbalanced Data Sets with 25 Groups ....................

Summary Statistics of the Unrestricted Model for Balanced

and Unbalanced Data Sets for both 25 and 100 Groups ........

Average B/E of the Latent Models for Balanced and

Unbalanced data Sets for both 25 and 100 Groups ............

Average Ratio(1) of the Ph, Om and Ps Matrices for the
Latent Models in Data Sets for both 25 and 100 Groups ......

Average Ratio(2) of the Ph, 0m and PS Matrices for Balanced

and Unbalanced Data Sets for both 25 and 100 Groups ........

Maximum Likelihood Ratio Test of the Fit of the Correct and
Incorrect Models ..........................................

Average B/E of the Unrestricted Model for Balanced and
Unbalanced Data Sets both 25 and 100 Groups ...............

Average Ratio(1) of the Phi and Psi Matrices of the
Unrestricted Model for Balanced and Unbalanced Data Sets
for both 25 and 100 groups ................................
Iterations Required by Algorithm for Convergence ..........

Summary Statistics of the Four Parameter Latent Model for
Balanced and Unbalanced Data Sets with 100 Groups .........

vii

Page

53

63

68

72

77

78

8O

82

84

85

87

94

LIST OF FIGURES

Figure Page

1 Design of Study ............................................... 51

viii

CHAPTER I: INTRODUCTION

Although statistical procedures are available for estimating
treatment effects for students taught in classroom groups, these
procedures are only applicable when every class has the same number of
students. Since equal class size is rare in schools, however, it is
important to develop practical methods that extend current procedures to
cover all patterns of class size. The present study investigates a
procedure that was originally established to handle missing data (the EM
algorithm), but which also provides a solution to the problem of unequal
sample sizes in multivariate analyses. After presenting a review of
V existing procedures, this dissertation will: (i) show how the EM
algorithm can be applied to this case; (ii) exhibit a computer program
that uses this procedure to analyze such data; and (iii) illustrate the
procedure and program with an analysis that estimates parameters from a
sample data set generated from a known distribution.

Many educational researchers have engaged in attempting to
identify the various factors which affect student achievement.
Laboratory studies have identified how individuals respond to different
educational treatments, but most formal education occurs in classroom
settings in which students receive treatments as a group. Two effects
are introduced in the latter situation, however, which cannot be
overlooked.

First, there may be some process which\affects the class as a
whole. A teacher with a class having a mean IQ of 120 may decide to

cover more material than one with a class having a mean of 100. Such an

.-'\ w " .fiw ‘_ _

\

_ NJ

3

2

effect, unless accounted for, could obscure the relatiggghgiLJufigfiggLJUL

"od-d .1.- ‘5“ ----- ~w“ I

and

Second, similarly, there may be an interactive effect between
individuals and the class. A person with an IQ_of 110 may do much

differently in a class with a mean of 100 than in W,

‘V I,”
{I
. ectibetween_
120 That is,\class level processe caQMaff ‘Sgass analyses,

>W -
4W gan influence__iih_in class anal ses and

undividuals can have an effect on both.\\The analysis of such data must

   

 

e interpreted carefully.

Estimates of relationships between variables at the class and
individual levels can either be low, or high, as two effects can either
combine to indicate a spuriously high relationship or work against each
other to reduce it. A number of models and strategies have been
recently developed to analyze this type of data. One line was the
development of regression models to study the individual and group
effects. Others\developed\models\to estimate underlying latent
variances at each level.

Keesling and Wiley (1974) used the relationship between two
student level variables to adjust class level scores. The aggregated
values of the variables were used with the estimated student level
regression coefficients to compute an expected group score. This score
was then subtracted from the aggregated class scores in order to obtain
residual scores which, in turn, were then used within regular linear
regression models with class level variables. Keesling and Wiley's
model was based on the assumption that the relationship between two

variables would be identical for all levels of the model.

3

Cronbach (1976) proposed an analytic approach that focussed on/
processes going on both between and within groups. He felt regression"L
effects were composed of two components, a between and a within effect.3
His model allowed for the relationship between two variables to varyv
between the individual and classroom levels. .1

Burstein, Linn and Capell (1980) developed a model allowing /
regression coefficients between two variables to change from class to ,5
class. These coefficients were then used as dependent variables in g;
regression analyses at the group level. Raudenbush (1986) applieda
empirical Bayes theory to develop a procedure for producing Maximum, .3.»
Likelihood (ML) estimates of the regression coefficients in Burstein's
2241 model. I)"

ffiNone of the univariate regression models, however, offered

/’{,;methods for estimating measurement error. Tests and i used in

If" education, virtually gy_definition, contain measurement errors which can

i
f L {Elate the analyses' errorWe model. _
I“? Schmidt (1971) addressed this problem by developing a

 

 

a." ,émultivariate structural model. By fitting an 5.91.1211 structure the

Nfivariance matrix of the student's test?, the variance and covariance of

Tathe latent dimensions and ’measurement errors {could be estimated for both
. , ‘ r

significant as both the variance and

1’ levels. This was especiall

   
 

‘1

alveovariance of latent dimensions are frequently thegitems/ of importance
“fixto researchers.

aann example of the applicability of this notion was evidenced in

“gifthewlnternational Associationfor the Study of Educational Achievement's
If:

’5£(IEA) ”Second International Mathematics Study", where items within].

W— m

4
7"“; academic tests were systematicallyvzonstructed, from a number 0f

i
4

/ ‘ "‘
I

dimensionsuf In one case, one dimension was word problems v§,numerical
examples.while another was arithmetic vs algebra. All items,containing
the two dimensions, in turn, could be combined to make four subtests.

The subtests, theoretically, were assumed to have served as con

  

tests (see Chapter 8, Lord and Novich, 1976).

The four subtests contained the following dimensions: IN7‘
PM”
Waffles Wales
Subscore A Word Problems Arithmetic
Subscore 8 Word Problems Algebra
Subscore CI Numerical Arithmetic
Subscore D Numerical Algebra
/ /

Of interest to the IEA study/was not the four subscores/but the

/

variance and covariance of the twolunderlying dimensioms. Schmidt's

—,—

.‘ A ‘ 3‘ "-_ .—-— ‘WY—m nun-o...
".

model/bould have been used/to provide estimates of theflatentiéariance

and covariance/é; both the class‘and individual leve .3

“.4-"

  

3: [covariance matricegycan then ’ sis
numerous models”.

Wisenbaker (1981), following the same logic, further developed the
estimation procedures necessary to estimate parameters of a causal model
for latent covariance structures.: The structural parameters of the
between and within levels, according to Wisenbaker's model, are
simultaneously estimated yielding ML estimators. 3/”.

’0 \ V
Schmidt's and Wisenbaker's models, however, both require groups C;~V¢V
\\;g

(classes) to be of equal size. The underlying multivariate normal

distribution upon which the ML equations are based is a vector of length
up where n is the number of students in each class and p is the number
of observed measures (tests) taken by each student. The ML procedure
requires that each group contains the same number of subjects - n. In
educational research, however, the number of subjects usually varies
from classroom to classroom.

The present dissertation concentrates on the problem of estimating
the latent between and within covariance matrices when the number of
subjects (students) varies between groups (classes). The early chapters
contain information on the background of the problem. Chapter Two,
Latent Structural Models, describes the development and background of
those models. Chapter Three describes the background of the specific
model used in this study and the development of different techniques for
estimating variance components in the unbalanced random model. Chapter
Four contains a statement of the problem.

The last chapters contain the derivation of the procedure and an
example of its use. Chapter Five contains the derivation of the
equations needed in the estimation procedure. Chapter Six details the
design of a monte carlo study for illustrating the use of the EM
algorithm under the current model. The results of the study are
described in Chapter Seven. Chapter Eight, finally, presents a

discussion of the results and conclusions.

CHAPTER II: LATENT COVARIANCE STRUCTURE MODELS
1. Single Level Covariance Structure Model

Latent Covariance structure analysis was developed along two

\ -’\ 7,” Hz“ “\J' " ~ # -2 ,—

different lines of inquiry. The,first approach, factor analysis, was

_f ‘__,,P’

derived explicitly for the purpose of finding latent structures
(Spearman, 1904). The second line of inquiry applied the  existing
random analysis of variance model toward solving the same  problem (e. g.
832k, 1960).

Factor analysis was developed by Spearman as a method for
confirming his theory on ability. Spearman sought to show that IQ tests
measured two components, a "general or G factor” common to all IQ tests
and a second factor specific only to the test. Through the application
of factor analysis, he was able to isolate the variance component of
each test attributable to the G factor, as well as the variance
component specific to the individual tests. As the mathematics for
factor analysis were expanded and refined, however, its use changed from
confirmatory to exploratory and became a method for reducing a set of
items or measures to a lesser number of underlying latent dimensions.
These dimensions, in turn, were used to form factor scores for
discriminating between subjects. These later exploratory methods of
factor analysis lacked a firm theoretical basis and were simply
algebraic manipulations of the data.

A confirmatory approach to factor analysis did not resurface
until the 1950's when such an approach was considered by Howe (1955),

Anderson and Rubin (1956), and Lawley (1958). The advantage of

6

7

confirmatory approaches lie in their use of statistical theory and
ability to test the fit of the latent models, but early efforts went
largely ignored because of computational difficulties. Interest
gradually rose after Joreskog (1966) developed more efficient estimation
procedures.

Joreskog (1969) developed a general approach to confirmatory
maximum likelihood factor analysis. Unlike the prior models, this model
had the flexibility of allowing/)esearchers the capability of selecting
different structures from possible solutions: orthogonal, oblique and
various mixtures of the two. The factor analysis model is based on the

‘\fundamental equation

r

(2.1a) x- 2+Azs+£

where y is a p x 1 vector of observed variables, a is a vector of grand
means, A is p x q matrix connecting the p observed values to the q
latent factors (with q 5 p), 3 is a q x 1 vector of the latent factors,
and z is a p x 1 vector of the error or unique parts of the test. It is
assumed that E(x)-E(z)eQ, E(xx')- O, E(zz')- i, and E(yy')- 2;. The

dispersion matrix for y is

Z, - AQA'+ W

Assuming 1 has a multivariate normal distribution, the maximum

likelihood equation is

(2.3a) L - 11:11am“ Izyl'“2 exm -§ (1, - 2y)’ 2’; (x, - a?) 1

The efficient part of the log(L) is

(2.4a) log(L) - - it n { logIEI + tr(SZ-1) )
Minimizing the following function

(2.5a) F( A, O, ‘1') - logIEI + tr(SE-1) - logISI - p

yields the likelihood ratio test statistic of goodness of fit.

/A second approach“ as developed through the use of the random

p; ——————‘

 

analysis of variance model. Burgߣl947) was the first to point out the
<:analogy between the analysis of variance (ANOVA) and factor analysis.>>
This was further elaborated EZ,§EE§§X;S}2§EZ; #Bock;(lgggl showed that a
formal relationship exists/between the two approaches. This relationship
only becomes clear if a distinction is made between factor analysis used
as a ”structural” versus "discriminal" analysis. According to Bock
(1960, p153): 3 N
”By Cstructural' analysis is meant a measure which attempts to make
l/éausal statements about test performances by assigning to definite

sourcesvthe covariation which arises between certain psychological

1‘

( .
tests: this was the original use of factor analysis. In its

subsequent application to the construction of3£esqibatteries, factor
analysis was also used to assess whether tests of known measurement
error yield reliable distinctions/between individuals, and, if so, in
how many dimensions: it seems appropriate to designate this

/ldiscriminal'fénalysis.' Factor analysis doesn't separate\fhese two

uses or give clear answers for either.

Bock showed that a Model II (Rendom) ANOVA,mgggl.can be applied
to tests in light of specific hypothesis about their composition and
suitably adjusting their psychometricvcharacteristics. The analysis
could be used to study st mctural and discriminal propertig§,bf the

1".—
. ‘x
' tests,(free of diffiéfilt statistical and interpretation problems} The

 

 

 

 

purposes of this dissertation deal only with the structural analysis and

 

ing an example to facilitate the discussion.

w
Consider the design of four testslfrom two dichotomgus
r"...- 3---.--,_ -__,,,,..... -»~~....__.-m-v-~Mm -......._,.,.—...... W..- 3 __3_ ~,”____,i...,-.. ,..- ,1...an

shall concentrate on it

dimensions, as referred to in Chapter 1, namely Type of Problem_((11,v

_.- ”Mr-1...”
._—

V“!

Word Problems vs (2) Numerical) and Type of Mathematics ((1) Arithmetic-
vs (2) Algebra).
Plnnour3example, the four tests may be _identified by the following

------------

ordered pairs:

 

 

A(J)
1 2
1 ll 12
B(k) '
2 21 22
TestUk) W W
/Test 113 Word Problems Arithmetic
f’Test 12 Word Problems Algébra
/ Test 2; Numerical Arithmetic
/ Test 22 Numerical Algbbra

A mpdelvfor the structural analysis of this design is

2.6a ' - -+ + + +
( ) xijkt ai fiiJ 71k 6 13k 6 13kt

10

where xnfit is the score of individual i on test jk on occasionat, a1 is a
component pf score specific \to individual i‘on all tests, flu and 1“ are
components of scoré\specific q? individual 1 en dimensions B and C respectively,

6am is a component of score specific\to individual i and the test jk

(with the dimensions effect exdiuded) and GL1: is a replication

k

error specific to individual, test, and occasion}: These‘components are

considered random effectsyhnd are assumed normal and independent,

N

a ~ N(O, 0‘);
p-Nw.{fl
1~Nw.{f
5 ~ N(O, 0:):
e - N(0, of)

f
Because the number of components/with a distribution over individuals is

equal/to the number of tests/in the dichotomous fac ,/the

covariance structure may be fully estimated. This will not be true when

 

 

there are more than 2 levels to a dimension._!

The/design‘jor our exampletcan be represented in matrix form as

v

the Hamadand design matrix

p B C
,1 1 l
1 l -l
(2.7a) P -1/2 1 -1 1
l -l -1

'The purpose of the structural analysis is to test whether the sample
covariances between tests fit the model. The covariance matrix of the

data from our example is

11

(2.8a) Sr - X.'X. - ( M /(n-1) )

where X is the N x 4 matrix of means of r replicate scores and M is the
matrix of corrections to the sample data. Pre and post multiplying the
population covariance matrix by P will reduce it to its canonical or
diagonal form (P ZIP'). If the sample covariance matrix is treated in
this way, the off diagonal elements will not necessarily be equal to
zero but if the model fits though, any non zero value will attributed

simply to sampling variance. Therefore a statistical test of the off

5..___ _‘

diagonals being equal to zero will be a test of the fit of the model.

 

W _ w—
wvv ‘|

A maximum likelihood ratio test given by Wilk's criterion and a
chi-squared approximation provided for moderate to large samples by
Bartlett can be used to test that hypothesis (Anderson,1984). This is

(2.9a) x2 - - (N - (2p + 11)/6 ) logIRrI

where p is the number of variates, erl is the determinate of ARA' and R
is the correlation matrix corresponding to S.

Book's work on reformulating factor analysis in the form of a
random model has spurred (the development of more complicated models)and
situations. Bock and Bargmann (1966) presented a method for analyzing a
sample covariance matrix<3n order to assesépthe latent sources of
variance and covariance within multivariate normal data. This
"structural” analysis of the sample covariance matrix has a two fold
purpose. The first purpose is to statistically test the feasibility of

/a hypothesized model/and the second is to provide estimates of variance

components associated with the latent variables of this model.

V .m-

12

This analysis is an alternative to Type III (Mixed) ANOVA Model.
The method of maximum likelihood estimation is used to test the model
and estimate the latent variance-covariance components. The model for

the observed score vector of p tests is given as
(2.10s) yu-u-i-Afi14- e

where u is the vectorKnean of the 13' tests, @is a p x 9 matrix of 44..

@Coefficients'connecting the "observed and theVlatent variables,

“wt/flw

g is an m x 1 vector of latent scores for subject i having an m x m
W

4- _- ”V“ " ii". I”..- ’
#'

covariance matrix 0 and {gig is a p x 1 vector of measurement errors
with a p x p covaW
The model implies that vector x“ has a multivariate normal

distribution 41th mean vector u and covariance matrix 2y where
(2.11a) 2: - A no + i:

In this model the latent variables are considered independent of each

___ _.._ __._
k. ..m;——

other with <5 considered to be a diagonal matr x\

_ .'¢‘WM~-mw

 

The likelihood function of the general model proposed by Bock and
Bargmann for p measures on N individuals sampled randomly from a

multivariate normal population is
12/2 -1/2 _ 3 - , -1 _
(2.12s) L nf_1(2«) lzyl exp{ 2 (11 2,) 3,, (x1 2,) 1

Taking the natural log of the function, differentiating and setting the

derivitive equal to zero will yield a maximum likelihood estimate

13

(2.13a) Log(L) - -(Np/2) 10g21r - (N/2) longyI - (N/2) cm 2'13 )

Assuming the elements of E; are functions of a scalar variable x,

the first derivitive with respect to x is

(2.1%) mm - <N/2) tr { a; < 2‘13 2" - 2'1) 1

6x 6x

The second derivitive with respect to scalars x and y is

(2.15a) mam-Numb Sim-gummy
6x 6y 2 6x 6x6y

+Htr(:}i§.2}+utr{W§:Z_}
2 6y 6x 2 6x5y

where w - 2'13 2'1

Because the scalars/ire not directly estimable, the Newton

Raphson algorithm as used. ,Tgis algorithm reauired the first and

 

 

 

 

 

second derivitives of the Log likelihoodménd ma tel ield
wv- .’
var w_ ,_ e values. .\(:9’

___._...— ‘v-fv

The likelihood equations and computational schemefhbove has been

distinct—easee'

worked out for/three structural models.//They were hree
V ’

   

for the model.
w.

2.16 2 - A 0 A' + W
( a) x, / ‘ /

ase I. r The latent variables are Yncorrelated W
M

——¥ _7 u" «q... ’ME‘K” V

s ec d and unscaled,§and\the error variances are assumed homogeneous‘Qy
t-wv .e_ , -- ~———____.~
; 'f—"V
( i - 0’21 1. Ci’ “*

...—--\, ..., ,uv/ " r 1 .
Case 11. The 1 variables are uncorrelated A is coggletely

specified angiagégaled, andflthe error variances 3F§g§§3299d

 

 

14

Wéi-diag [it ,W aunt!” ] ).

Case 6‘ The latent variables are assumed uncorrelated, A is W,

mwfl'wwm' h“ I .

/ W, but scaled by an unknown, but estimable, matrix of g

:Cdé‘lin i Q%#7 Wt variances» are 1assumed homogeneous: J1

- a'zl ).

 

 

 

Bock and Bargmann chose the likelihood statistic

|s|§

(2.17a) A - - -*-
|z|§

to test the hypothesis that the population covariance matrix has a Case

I, II or III structurevvs an unrestricted structure. The distribution

«4"
—\ ,- _'~'-

of S in large samples may be approximated by a chi square.
(2.18a) x? - -2 log) - N log{|Z|/ISI}

The degrees of freedom for this statistic is equal to the difference in

the number of parameters in the restricted and unrestricted models.
FThese three cases are only a small number of the many possible

covariance \4tructure modelsf‘ thatlcan be hypothesized for the general

_\ v
model (2.6a).

 

Wiley (1161) developed liset of 16 models that can be hypothesized

W

by applyingdifferent combinations of restrictions ,tomthe threermai-n

wuentsm of the general model. Stud in othe '

7 A... n. MU...M- :Wm,u,mnmw

ariables to also be correlated

The constraints

‘ ‘11“ a“. ”MP-Q!“ ‘1

Wm

 

 

about the model, he (a/llowed the

Q“ and on of the elements of L

   

 

 

 

 

 

A

W

he proposed for the parameter matrices /of the model /formed 16 /possible

 

\l/ 15

models (4 x 2 x 2). The(E:fiatrix could be constrained four different
ways/while‘é and“! could have one of two different shapes.

The four forms of restriction on A are

(1) General ( A ) - all elements are to be estimatedrv

LM ‘1.- mes-mm

r"

(2) General ( A ) - most elements are to estimated excgp§,jhrmcaxtain~

“aw-NM! nfimmmi“ vvvvvv
rfl‘.‘ m."""’

specified ones s

q I "Hfl‘k

 

 

'(3) Completely specified (2“ and scaled by n unknown but Westimable‘/////

 

matrix7of scaling weights ( P )

"no-u...

Completely specified (A and unsealed
\ \

 

The covariance matrix of the latent variables ( ¢ ) has two

restrictions.

(l)The latent variables are uncorrelated i.e. ¢ is a diagonal matrix.

(2)The latent variables are correlated i.e. d is a symmetric matrix.

*1

The matrix of errors can take on one of two forms.
(l)The errors areihetengeneous, {a general diagonal matrix.\/

(2)The errors are homogeneous (ya; I

a. _ “-1:
final, .uv- "‘ "'t ' ‘" ‘« Hwagmww “'w"

16

These models cover Book and Bargmann's three cases and also a number of
Joreskog's confirmatory factor analysis models.
Wiley,Schmidt and Bramble (1975) developed the maximum likelihood

estimators for eight of these models. They used only restrictions<:;Lnd

(QQfor A.
_, ./

.0“,

(4414/4444 4 44 44,2 4 4444»
"WM” (7 fi 4 17 i p ./ 212/ rw/ €441!

Mul level Covariance Structure Models

77777 //‘: M41441: . [’g fie, [#7 /,_

#395, 3 I249}: "I'Z MV/ / 3? a 1%

Book's mode-Iriac the ability to separate lassroom effects rom

y :11"\

 

individual effects\for data gathered in a natural classroom setting. If
tests are given to classes, the variance-covariance matrix of these ‘-
tests will be affected by £Ehe classroom effect{ 4
Assuming classes were sampled at random, Schmidt's Multivariate
Random Model gives estimates of the population variance-covariance
matrices of the tests at both class and individual level. An

individual's score is composed of a number of parts.

(2.1b) y.“ -u+§1 +2”
, where x is a vector of p scores for person in roup i
.4 J 13 8
«TOO u is a vector of p grand means
1/: ’yzi) ”,44 91 is a vector of p effects due to being in group i
I .,
g ,4 7/20 14 g“ is a vector of p effects for person j in group i
;¥Q. '
The cova ance matrix of this model would be :
+ —< >4? /" 4
4. 437,) ‘7’"!
b ”ff-Q"
(2.2b) 2y - Z + 6

2y is the variance-covariance of p measures for y
2 is the variance-covariance due to class effects

9 is the variance-covariance due to individual effect.

The two variance-covariance matrices contain information about the class
level effects and the individual level effects. These two matrices can

be expressed as a function of matrices relating observed to latent

, I

3 2.
:4 w“ d

4 g '
l ’ s
‘ \

18

variables and to errors of measurement.
In order to estimate the underlying latent covariance structure
and error that compose the two matrices, Schmidt (1971) applied Bock's

model to multilevel data. The model included class effects.

(2.3b) yum - pk + all + b1.j + on + duh + emm
where i-1,...,m groups, j-l,,..,nfli) students/group, k-l,...,p
measures, n-l, . . . ,N students, "I: is the overall mean of the kth

variable, a1 is the effect of group 1, bid is the effect of

person j in group i, on is an interaction betwaen measure 1: and
class i, d“: is the interaction between student i in group j and
measure 1:, and eun is measurement error for person j in group i on
test k for this occasion.

Notice that the effect at the class level occurs in two terms

(2.4b) 6 - a1 + on

\

and the effect at individual level is found in another two terms.

(2.5b) 63k - bi: + duh

Substituting these variables in the model give the following equation.
(2.51)) )4 kn - uk + 611: + £3 + e ‘\\:

Assuming that u, 6 , 5 and e are uncorrelated, the covariance matrix of

the y‘s is given by

19

(2.71)) 2 -0+¢+‘F

where 0 is the covariance matrix of Q, r is the covariance matrix of
i and i is the covariance matrix of g. The model now has three
components, 0 which contains the effects at class level, 1 which
contains the effects at individual level and ‘1' which contains the
measurement errors.

The interactive random variables 2&1 and {51: could be

visualized as combinations of some latent random vectors 2 and g.

(2.8b) -A¢.0.

(2.9b) fi -Ag

The error matrix i can be rewritten as the linear combination of two
components, a within (‘7') and a between group matrix (in) . From

these two assumptions the covariance matrix of y is

(2.1%) 2 -A¢A'+A¢A'+w +1!
y one to a

where A is a pxr matrix of weights relating the observed mean level
variables to the vector of r latent variables.

This implies that the basic model for the structural analysis of
covariance component matrices of the multivariate random model is given
by
(2.11b) 2' - AR‘A‘ + Wu

(2.12b) Z-Afl’A-i-‘I'w

20
where All is matrix of weights relating the observed mean-level
variables to the latent variables Q, A is matrix of weights relating the
observed individual variables to the latent variables g,, o. and o
are the covariance matrices of the between and within latent effects 9,
and g, and i;, and i; are the diagonal covariance matrices of the two
error matrices. Each of these variance-covariance matrices, 2.4and 2,
correspond to those considered by Joreskog (1967). The primary
difference is that these models, which represent a set of equations, are
themselves intended to be simultaneously estimated.

A class of models can be generated by varying the restrictions on
the six parameter matrices of this model, A‘, A, Q;, w, W. and 0;. The
classifications proposed by Wiley (see last section) can be fit to
these parameters.

The two forms of matrices that the latent variance covariance

matrices, é’ and Q, can assume are:
I

(l) The latent variables are uncorrelated i.e. éb is a diagonal
matrix.
(2) The latent variables are correlated i.e. ¢b is a symmetric

matrix.

The two error matrices, W and W , can have one of the
8 W

following two structures:

(l)The errors are heterogeneous, a general diagonal matrix.

(2)The errors are homogeneous ( 021 ).

21

The four forms of restriction on A are

(1) General ( A ) - all elements are to be estimated.
(2) General ( A ) - most elements are to estimated except for certain

specified ones

._ /,4m\
If A s reparameterized into/T A/bhere(£ is a matrix of scaling factor ,

two different sets of restriction can be applied.

(3) Completely specified (A)\and scaled by an unknown but estimable
matrix of scaling weights ( P ).\

(4)Completely specified (A) and.unscaled. \

4A.
x’ ‘t

R

There are (4 x 4 x 2 x 2 x 2 x 2)i25S)possible models‘which can
V!

be formed from them.

22

3. Extension of o. and w to causal models.

Once the latent covariance matrices, é and ¢., are estimated,
the matrices themselves can be used to test causal models. Specifically
once scaling factors have been specified and measurement error removed,
these residual covariance components contain all of the relevant
information necessary to analyze a given data structure and hypothesized
causal models can be tested.

Joreskog (1971) has developed a procedure for estimating the
parameters of causal models using maximum likelihood estimation
- (LISREL). His procedure estimates the parameters for two components of
casual models, namely the measurement model (based on his work mentioned
in section A) and the structural model. The measurement model estimates
the underlying latent constructs of the model while the structural model
specifies the causal relationships among the latent variables. These
two components, in turn, are used to describe the causal effects and the
amount of unexplained variance among the observed var ables.

Wisenbaker (1980) extended Joreskog's model t multilevel
situations. His work simultaneously estimated par eters of causal
models at both the between and within levels.

The focus in this dissertation is on the estimation of o‘ and
Q. One natural extension of this work is to develop the algorithm
necessary for directly estimating the parameters of Wisenbaker's causal

model when groups are unbalanced.

CHAPTER III: REVIEW OF RELEVANT WORK

1. Schmidt's Structural Model

The focusvof this dissertation is the estimation of the latent
covariance matrices Q. and tb. Schmidt (1969) developed a general
procedure \for estimatingNthese latent covariance matrices“ for
multivariate normal data: Assuming classes \Vto be drawn at random, the

random multivariate model is :

(3.18) X '1 +a+§

where y“ is the observed set of individual level variables for p values
and y” is ‘a p x 1 vector of general means. The term 31 is a random
vector of schools and g“ is a random vector of errors. Both of these
are considered to be distributed multivariately normal with zero mean
vectors and covariance matrices 23. and 2.. This would imply that the

covariance structure for this model would be:
(3.2a) 2 - 2 + 2
Y C 0
Usually 2. and 2. are estimated by using the expectations of the

mean squares of a Multivariate Analysis of Variance (MANOVA). In the

random multivariate model it can be shown that

23

24

(3.3a) E( S(w)/[kn-k] ) - 2.
and
(3.4a) E( S(b)/[k-l] ) - 2.-+ n 2..

By using these formulae, 2. and 2.4can be estimated from the

following equations

A

(3.5a) 2. - S(w)/[kn-k]

A

(3.68) E. - (l/nH S(b)/[k-1] - S(W)/[kn-k] }.

Unfortunately this method can yield non-positive definite estimates of
the matrix 2..

Schmidt used the principle of maximum likelihood to estimate
these two matrices. The data in a random model would consist of m

factor levels each containing n subjects with p measures on each

 

subject.
Dependent Variables (measures)
Subjects
Factor 1 ,2 3 . . - n
Levels 1
2
l

25

Basing the likelihood function on the general notions of Tiao and Tan,
(1965), the data can be visualized as m independent observations from a
np-dimensional multivariate normal distribution. The general linear

model for any y is given by

(3-78) I - l 9 g + l 8

(b
4.
IO

where l 8 g is a vector of pn means (p means repeated over n times),

26
g1 is vector of pn effects (p effects repeated over n times) and g
is a vector of pn errors. Both 51 and g are considered to have come

from multivariate normal distributions

(3.8a) 5 ~ N( 0. 2‘) 2 ~ N( O. 2.)

The covariance matrix for this model is

(3.9a) B-E -ll'@2 +132
Y n I e

P

This appears as

 

 

Fz+z. . . . . . . 2 -1
. C l
2:

.
t3, ......2.+2'j

The covariance between observations within a factor level is given by
(3.10a) Cov (XJ' Y'i) - 2‘I i fj

The density function of y is then

l-l/Z

(3.1141) f<y> - new” Iznp exm-j [(1 - 182)'2n:(1 - 1mm

from which the likelihood function follows

-m/2

(3.12a) 1.04. E...) - <2«>‘““”"|znp| eprj (my, - lem'znjwi - 1mm

27
The matrix 2n must be expressed in terms of 2. and 2.. The relationship
is given in (3.9a), from which the following inverse and determinate

follow.

(3.13s) |2 |- I: |“'1 + |z + nz|
up 0 o a

(3.14s) 2'1 - I e 2’1 - 11' e (2 + n2 )'1 2:" 2'1
up 0 o a a o

The Likelihood can be simplified as

(3.14...) M44. 2.. z.) - <21r>""“"’zlz,|""""”zexp{-;1 [tn-{2:131
+ m ma: + n2 )‘131
C l I
+ m crux. + nz‘f‘dv - m5! - pm 1 1
n II o
where S. - 1/mn 2.1-1 21% (YL1 ' Y1) (Y1.1 ' Y1)
a I
S. - n/m ZN (y1 - >')(y1 - y)
and y” is a pxl observation vector for the jth person in the ith

group. The log of the likelihood is

(3.15a) Log(L(p, 2., 2.) - - ffiogax) + “immglzJ
Q -l
- 2 logIZ. + nBJ - :Imn tr{2.S)
+ m tr((2. + nza)'ls‘)

+ mn tr{(2. + n2.)'1(y-u)(y-u)'11

The effective part of the log likelihood function for the estimation of

2. and 2. is given by

28

(3.16a) Log L(g, 2., E.) - Ill:zmilogIZJ- 2’ logIE. + nZ‘I

- ’5‘ cuzllm - 2 crux:ll + n2.)-18a}

By expressing g",in.this manner, Schmidt was able to obtain the
following maximum likelihood estimates for 2.4and 2a.

A

(3.17s) 2.!— [n/(n-1)]S.

A

(3.1841) 2. -.% (s. - [n/(n-l)]S.)

This gives estimates of the between and with-in covariance
matrices but says nothing about the latent constructs or the measurement
error associated with the observed values. The equation for a single

observation with latent constructs is

(3.19a) y -u+a+b +c +d +e
ijkn k 1 13 1k 13k iJkn
i-l,...,m groups j-l,...,n(i) students/group k-l,...,p measures
n-l,...,N students

where u is the mean of the kth variable, a is the effect of group i, b
is the effect of person j in group i, c is an interaction between
measure k and class i, d is the interaction between student 1 in group j
and measure k, and e is measurement error for person j in group i on
test k for this occasion. Notice that the class effect occurs in two

terms .

29

(3.20a) Qn - a1 + on

The effect due to individuals exists in two terms.

(3.21s) fink - b1.j + (Ink

The model can be written in terms of the effects at each level.

(3.228.) -u+Q +§ +g

1mm '1 u: Jk ijkn
Assuming that g, Q, g and g are uncorrelated, the covariance matrix of

the y‘ s are

(3.23a) 2y -0+ 1' +\F

where 0 is the covariance matrix of Q, r is the covariance matrix of
6 and I is the covariance matrix of g. The model now has three
components, 0 which contains the effects at class level, 1 which
contains the effects at individual level and W which contains the
measurement errors.

The vectors Q and g could be visualized as combinations of latent

random vectors 9 and 9,.

(3.21m) Qk1 - A69

(3.25a) £11k - A Q

30

The error matrix Q can be rewritten as the linear combination of two
components, a within ( i; ) and a between group matrix ( W. ). From

these two assumptions the covariance matrix of y is

(3.26s) 2 -AQA'+AQA'+W +1!
y III I I
where A is a pxr matrix of weights relating the observed mean level
variables to the vector of r latent variables.
This implies that the basic model for the structural analysis of

covariance component matrices of the multivariate random model is

(3.278.) 2 -A¢A + W
a sea a

(3.28a) 2 - A Q A + W”

where A. is matrix of weights relating the observed mean-level variables
to the latent variables 9, A is matrix of weights relating the observed
individual variables to the latent variables 9,, Q. and Q are the
covariance matrices of the between and within latent effects g, and g, and
i., and i; are the diagonal covariance matrices of the two error
matrices.

Substituting the structural model for 2. and 2. into the
likelihood function in (3.16a) gives the maximum likelihood appropriate for
the structural analysis. Taking partial derivitives of the log
likelihood in respect to Q;, Q, A., A, Q., and W; and setting them
equal to zero will yield maximum likelihood estimates of those parameter

matrices. These equations proved to be to complicated too be solved

31

algebraically and Schmidt used the modified method of Davidson as an
algorithm to estimate the matrices.

Formulating the maximum likelihood equation by considering the
data as 3 random vectors from a multivariate normal distribution of up
measures constrained the model to have the same number of individuals in
each group (i.e. there must be 2 measures for n students). When groups
have unequal numbers of students, the likelihood function developed in
(3.16a) will no longer hold true. In education, researchers are often
in the position of collecting data for groups of unequal sizes. To
obtain maximum likelihood estimates of the structural matrices in this
situation requires either a new analytic strategy or the development of
an alternative likelihood function. However finding maximum likelihood
estimates of the covariance matrices of an unbalanced design has proved

to be difficult.

32

2. Unbalanced Designs

In multivariate analysis very little has been done to exam the
effects of unequal group size on the estimation of the covariance
matrix, although there has been much exploration in estimating variance
components in the this design for the univariate case. Since Anderson
(1984) feels that a number of statistical problems arising in
multivariate populations are straightforward analogs of problems arising
in univariate populations and the suitable method for handling these
problems are similar; parsimony would suggest looking at previous
developments regarding the univariate case.

Searles (1971) points out the following problems which must be

faced when dealing with unbalanced designs:

"The property of unbiasedness itself merits questioning in
the case of variance component estimators. This is so
because with unbalanced data from random models the concept
of repetitions of similarly structured data and associated
repetitions of estimators is often not appropriate --- more
data, maybe, but not necessarily with the same pattern of
unbiasedness. Replications of data can not be thought of

as mere resamplings of the data already available."

and

"even in the simplest of cases the effect of the n-pattern

on properties of estimators is apparently itself a function

33

of the variance components being estimated. The effects of
unbalancedness therefore appear to differ according to the

values of the true variance components."

This last statement refers to the fact that the MINQUE procedure
and those based on it rely on the researcher choosing the “true" ratio
of the between variance component to the within variance component of
the variable.

Welsh (1937) was the first to point out how unequal number of
subjects in each group can affect the estimation and testing of
statistical hypotheses. Henderson (1953) proposed three methods for
estimating variance components for the unbalanced random design, using
the expectations of the Random Anova Model.

Rao (1971) advanced a new method for estimating variance
components called MINQUE, a minimum quadratic unbiased estimator.
Ahrens, Kleffe and Tenzler (1981) state "this procedure provides some
kind of optimality and does not refer to the normal assumption" and
'MINQUE ... has been justified by heuristic arguments without reference
to the normal distribution". Formulas for the MINQUE have been
developed with increasing explicitness by Lamotte (1973, 1976) and
Ahrens (1978). MINQUE has also been developed for more difficult
designs (e.g. see Kleffe (1977))

MINQUE can at times give negative estimates of the variance
components. Rao (1972), in turn, developed MINQE which gives variance
estimates that are always positive but can be biased. It may be noted
that no properties are yet known about this estimator.

Searle (1972) devotes an entire review to the methods of variance

34
estimation in unbalanced random designs. The estimators reviewed are
all unbiased, their other properties are unknown. Most of these
estimation procedures can lead to negative estimates of the variance
component.

Chatterjee and Das (1983) developed a simple estimator of
variance components in the random model based on Weighted Least Squares
(WLS). They found that as the number of classes increase the proposed
estimator is seen as not only to be the best asymtoptically normal but
also to be asymtoptically equivalent to the maximum likelihood
estimates. A review of recent developments in WLS can be found in
Williams, Radcliffe and Speed (1975).

There is no agreement on what constitutes a good estimator of the
covariance when groups are unbalanced. As shown above there are many

different measures each with its own strengths and weaknesses.

CHAPTER IV: STATEMENT OF PROBLEM

The interest of this dissertation lies in the latent covariance

 

structure implied by the simple true score model. Based on the Simple

 

Multivariate Random Effects Model, the two variance components, between

'g

 

 

(2.) and within (2), are expressed as linear combinations of a set of

latent variables.

(4.1.) 2-A§A+W
a sea a

(4.2) 2 - A Q A + W

It is the latent covariance matrices Q. and Q that are of primary

interest. I In chapter 3, a maximum likelihood procedure developed by

fi-r—

Schmidt was presented for estimating the error matrices %; and i and
..ac.__.lWh‘_fi~“_"fd__v_fi,

/—‘———-’_f ‘— w—
the latent covariance matrices Qll and Q when A. and A are known.__.4i—l—7“

MW

 

 

 

 

 

... -QHH
E‘- h" ~ ‘4‘ fl.- ." --

...—....— 4-F"'

f However, this procedure can only be used when groups in the sample
, 1......)

contain the same number of individualsy If the number of individuals in
,___———————-—"” 1

each group is different, this estimation procedure would not be directly

 

 

 

’-

 

appropriate.

The focus of this dissertation is upon the estimation of the
group and individual level variances, with measurement error removed,
when group sizes vary in a sample.

A promising approach is the EM Algorithm. Developed as an
estimation procedure for handling data sets with missing data, it offers
a method of finding maximum likelihood estimates of parameters in

situations where classical maximum likelihood procedures fail.

35

36

The applicability of the EM Algorithm to latent structure models

is demonstrated in the next chapter.

CHAPTER V: ESTIMATION PROCEDURE

1. Expectation-Maximization (EM) Algorithm

The EM algorithm has gradually evolved as a method for estimating
the parameters of a model when a sample contains missing data. Early
works by Hartley (1958), Healy and Westmoratt (1956), Baum et al (1970),
and Brown (1974) among others contained specific uses of the EM
algorithm under different names. Dempster, Laird and Rubin (1975)
developed a more general form for the algorithm and provided a formal
proof that if the algorithm did converge, it would result in maximum
likelihood estimates.

Missing data cannot be directly measured but exists as function
of observed data. This could be censored or truncated data where the
value of the data is not the direct value of interest or it could be
viewed as being comprised of combinations of latent constructs which
form the observed data (Hartly and Hocking, 1971).

Assuming a sample, y, is drawn from a population of a known
distribution with unknown parameters ¢, then y (incomplete data) can be
pictured as a subsample of x (complete data) determined by the equation
y - y(x). .The complete data situation has a family of sampling
densities f(x|¢) depending on ¢, from which the corresponding family of
sampling distributions for the incomplete data, g(y|¢) can be derived.

The EM algorithm is aimed at finding the ¢ which maximizes g(y|¢)
given an observed y, but making essential use of the family f(x|¢).
There are many possible f(x|¢) that will generate a g(y|¢), making the

choice of f(xl¢) a major problem.
37

38
Each iteration of the EM algorithm goes through two steps, the
expectation step (E-Step) and the maximization step (M-Step). If the
complete data, x, comes from a distribution with parameter o, the steps

can be stated as follows:

1. E-step: Estimate the ggmnlg§g_§§§g sufficient statistics
conditional upon the inggmplggg_g§;§, y, and the parameter d. This step
provides the connection between the complete data, x, and the incomplete
data, y.

2. The M-step determines the parameter ¢ that maximizes the conditional
anplggg gag; sufficient statistics. This requires writing the Maximum

Likelihood equation for ¢ in terms of the complete data.

The sufficient statistics for the complete data are calculated
using the incomplete data and estimates of the parameters. (For the
first iteration these values of the parameters are given by the user.)
The sufficient statistics are then used to estimate the parameters.
This value is used to recalculate the sufficient statistics which in
turn are used to recalculate the parameter ¢. The iterations continue

until some chosen criterion for convergence is met.

39

2. Theory for the Restricted Model

It is the application of the E-M algorithm to the estimation of
the variance-covariance matrices of a latent multivariate model that is
the thrust of this dissertation. The model chosen was based on
Schmidt's latent multivariate model with two modifications. First, the
groups may or may not contain different numbers of subjects; Schmidt's
model allowed only groups of equal size. This modification, however,
makes Schmidt's estimation procedure inapplicable. Second, the group
level error term in Schmidt's model is not included in the present
model.

To estimate the parameters of this unbalanced model, the EM
algorithm was employed. The E-step requires the derivation of the
conditional sufficient statistics and the M-step requires the Maximum
Likelihood Equations of the parameters for the complete data.

The model of interest has the following structure

(5.1) X“ - u + Ala-Q1 + A g” + g“

whereXU is a p x 1 vector of observed variables for subject j in

group i (incomplete data)

2 is a p x 1 vector of grand means for p variables.

(Complete data)

0

1.1

1.1

40

is a p x q matrix connecting the p observed measures for

individuals to the q underlying group latent values.

is a q x 1 vector of q (where q 5 p) latent group effects

for group i. (Complete data)

is a p x r matrix connecting the p observed measures for

individuals to the r underlying individual latent values.

is a r x 1 vector of r (where r S p) latent individual

effects for person j in group 1. (complete data)

is a p x 1 vector of random error.

For purposes of the derivation of the conditional equations

necessary for this EM Algorithm, u will be considered to be a random

vector from a multivariate normal distribution with a mean vector of

zero and covariance matrix, 2“. ZLater in the derivation, 2: will be

defined as a zero matrix, yielding posterior estimates of the grand

means. This procedure is mentioned in Dempster g; a1 (1976) and

further elaborated in Raudenbush (1986).

The latent effects and the error are assumed to come from the

multivariate normal distributions

(5.2)

prams“) g-N(

lo

.9)

§~N(Qv¢a) 6~N(9."F)

41

It is the estimation of Q.,1Q and i that is of interest.

Assuming the latent effects are independent, then:

Cov{u.fi)- 0 COV(§.9_)- 0
(5.3) COV(u.e)- 0 COV(Q..{)- 0
COV(u.:.)- 0 COV(9...{)- 0-

Before finding the conditional sufficient statistics for the
maximum likelihood equations, it is important to have a clear
understanding of which variables comprise the missing data, the complete
data and the parameters of interest. The missing data is our observed
dataset X. The complete data consists of the three latent variables u,
Q and.g. The parameters of interest are the covariance matrices Q.,

Q and i.

E-Step

Development of the conditional expectations and dispertions for
u,,Q and g are delineated in this section. These expectations are
conditional on the observed data Y and the three parameter matrices

Q;, Q and i. The observed dataset has the following expression:
(5.4) I-llflg+(X®A‘)Q+(IN®4\)g+g.
where 1' is a N x 1 vector and X is an N x m pattern matrix

containing 1's and 0's. This matrix connects person N(i) with

group m(k).

42

The variance of the observed dataset, Y, is:

(5.5) 2-11'82+)Q('84\Q4\+I®AQ4\+I®W
y N u one

The joint distribution of X, g, Q and g is:

 

 

 

r- _ 7- fl
Y J 2’ Symmetric Matrix
(5.6) p l' O 2 N2 :
~ N N u u 1
a X' 8 Q A' 0 I 9 Q i
a a k a ‘
J 9 I e 01 o o I: o o ’
L .1 __ N ___J

with

Cov(Y, u ) - 1' o 2“
Cov(Y, 6 ) - X 0 Q.
Cov(Y, a ) - I 0 Q

Covm. Q ) - 0

COV(u. a ) - 0

CONE. a ) - 0.

By defining the matrices and vectors as

z- [1‘91 xoxa 19A]
_ ., P 1
p N Eu Symmetric Matrix
(5.7) T - 6 O— O I @Q
k a
a O O I 8 Q
h _ 4b N J

 

 

 

 

43

the joint normal prior distribution can be written as

Z 0
0

I Z 0 Z' + I 9 W
(5.8) ~ N
I 0 Z'

Raudenbush (1986) derived the formulas for the conditional expectation

and dispersion of matrices written in this form. The conditional

expectation and dispersion of T given Y are:

(5.9) mm - ("54ng + “212".

 

(5.10) pm!) - ( rm 49 «0‘12 + 0’1)’1

I e W)":

By substituting the original values of Z, I and O in (6.7) and

allowing 2‘: to become a matrix of zeros as previously mentioned, the

dispersion matrix becomes

 

I’N i-1 Symmetric Matrix
(5.11) Dmx) - 1 a any" I e (n x'w’lx + o")
k i a k i a a a
_ 1' 0 mil x o Arr") I" 9 (MFA + o'l)

 

Partitioning this matrix into a 2 x 2 matrix and applying the procedure

in Morrison (1972) for inverting such a matrix, leads to the following

values for the elements

1
D(T|Y)- 2 3
a 5 6

(5.12) 1-D(u|X) - w

(5.13) 2 - um, um - -1]: o ‘33?“

44

(5.14) 3 - mam - 1k 9 (o. - 14.13113.) + 141; e <53;in Q3113.
(5.15) a - D(g, pm - -1n 0 o 1'in
(5.16) 5 — D(g, SIX) - 1:411; o (1/n1) o 1'ij Q1A.Q.
- x o (1/n1) <5 1'ij Q1113.
(5.17) 6 - D(g|X) - 1’11; 9 (1/n1)(1/nJ) o 1 de Q14\ o

+ In a (an - <1 .vu A e) + xx' 9 (l/n1)QA'(M - (1/n1)Q1)AQ

where u - (A o 1' + it)"
, -1«1
Q1 - (4\.Q‘l4\II + (1/ni)M )
-1
w - [2:10,]
The conditional expectations of u, Q and a can be formulated by

substituting (6.7) into (6.10). This yields:

uIY u 2qu QiY
(5.18) E QIY - 2' - 1.. o “'0 (i - 2’)
a a i i
..le-j LQJ 1n 8 Q 4\'M(XLj - 4\ 9 - p)

 

 

 

M-Step.

The second step of the EM Algorithm requires expressing the
maximum likelihood equations of the parameters Q;, Q and 0 in terms
of the complete data. Assuming the values g, Q and g are known, the
expression of the likelihoods for our three parameters can be directly

stated as:

(5.19) L(Y, o.) - 11::1 (21r)""’2|414“|'1’2 exp {-4} e'ofe}

45

'1/2

(5.20) L(Y, e) - n“ (2«)“”2|e| exp {-zl-e'e‘le)

i-l

‘1/2

(5.21) L(Y, 1F) - II“ (2x)"”z|w| exp {-2 e'w’le)

1-1

The maximum likelihood estimate of Q. is derived by first finding the

log of (5.18).
(5.22) log(L(Y, e.)) - -kp/2 (103(k)) - flogIQJ - ézfue'ofe

Taking the derivitive with respect to Q. and setting it equal to

zero yields:
(5.23) o -éz" 99'
a i-l

In reality, only the conditional expectation of g, Q and g are known

(U., 6., 0*). By rewriting (5.22) as

>0>

t -321; (9* + 9 ‘ 9*)(9. + 9 - 61')’
“a ‘32-: (9.5”) + (9.)(9 - 6')' + (e - 94*)(9') + (e - 9*)(6 - 9*).

(5.24) e. - 22:“ (9'5") + mg')

The equation can be clarified by substituting By substituting (5.18) for
8. and (5.12) for D(6.). The maximum likelihood estimate becomes:

A - O l I c u
(5.25) Q.I Q. 3 _1[Q'4\.(Q1 Q1(A W)Qi)A.Q.]

where A - (i1 - #1.) (i1 - u*)'

46
Following the same likelihood procedure from steps (5.22) through

(5.24) above, gives the following maximum likelihood estimate for Q:

A
*

_ 1 *.
(5'26) ‘b u 2:1 rat-1 a1.) a” + 0(013)

Equation (5.25) can be clarified by substituting By substituting (5.18)
for e' and (5.15) for 0(6). The maximum likelihood estimate

becomes:

(5.27) e - <5 - QA'[(l/N)):_1QA'((1/n1)[Q1- 0541-10011 - M[B-(n1- l)M-1]M)AQ

where n - (1,5 - (4* - in: - 1.9} - u’)’

The maximum likelihood for W followed steps (5.22) and (5.23).

The maximum likelihood estimate is:

‘ _ 1 4
(5.28) 0 u :1 $31“ cue“

Replacing e by Y - (p + A + Aa) permits (5.25) to be rewritten as:

A

- l o c '
(5.29) w l X X (Yid (,4 + A‘s + xenon“1 (,4 + 1.9 + 1a))
replacing conditional values expanding the equation yields

(5.30) w '11): X ()413 - (p*+A.9*+Aa*))(Y - (p'+.\.e'+).e'))'
+ D03. + i.e. + if)
(5.31) 0(p'+x.e*+xa‘) - D(m*)+2n(p*, i.e')+2n(m*, Aa*)+D(A‘6*)

+ 20(189', Aa*)+D(.\a')

47

By substituting (5.12), (5.13), (5.14), (5.15), (5.16), (5.17) and

(5.18) into (5.30) the estimate of Q can be written as:

(5.32) v - w -:)f_lw<<1/n,>[c2,- 010140011 - MlB-(ni - 1)M"1M>w

where B - <3:1d - A 6' - u'm: - Ag" - m‘)’

48

3. The E-M Algorithm.

The implementation of this algorithm is now complete for this
latent model. The E-step uses the observed dataset Y and starting
values for the parameters Q;, Q and Q to find estimates of the
following sufficient statistics in (5.12) through (5.18). The M-step
finds estimates of the three parameters using the values from the first
step in (5.25), (5.27) and (5.32). These estimates are used in step 1
to reestimate the sufficient statistics in (5.12) through (5.18). Then
in the M-step, Q.,‘Q and W are estimated again using the new values
from E-step. The algorithm iterates between these two steps until some

criteria is reached.

Chapter VI: Design of Study
1. Design.

By applying the estimation procedure (described in Chapter Five)
to a set of data sampled from a population of known parameters, a check
was provided for the solution of the EM algorithm together with the
identification of its prOperties. Although the underlying parameters of
the sample data were known, this was not intended to be a simulation
study but, rather, an example of the algorithm's ability to estimate the
parameters of a latent model.

The EM algorithm, in operational terms, was used to estimate
covariance components from both unbalanced and balanced samples drawn
from the same multivariate normal distribution with known parameters.
The balanced case contained 30 subjects for each group, while the
unbalanced case averaged 30 subjects per group. The distribution of
subjects across the groups in the unbalanced case was as follows: 20%
of the groups included 10 subjects within each group, 20% had 20
subjects, 20% had 30 subjects, 20% had 40 subjects and, finally, the
last 20% had 50 subjects. .

The estimates of the balanced and unbalanced samples were both
studied while varying two factors, namely the number of groups in the
sample (the size) and the particular model being estimated (i.e. the
unrestricted model, the correctly restricted model and incorrectly
restricted model). The size of the sample consisted of two levels. The
small sample consisted of 25 groups and the large sample of 100 groups.

The difference in the number of classes gave an indication of the

49

50

properties of the EM Algorithm when the sample size varied from 25
groups with 750 subjects to 100 groups with 3000 subjects.

Each of the three models studied contained different sets of
parameters to be estimated. The first set involved the two covariance
components for the simple multivariate random model (Unrestricted
model); 2., the between groups covariance and 2, the within groups
covariance matrix. The Unrestricted model gave estimates of the between
and within covariance matrices of the multivariate random model.

The second set consisted of Q‘, the latent group covariance
matrix, Q,the latent individual covariance matrix and W, the error
covariance matrix from the latent multivariate model. These parameters
were derived from a latent measurement model based on the multivariate
random model. The latent group covariance matrix, Q‘, and the
latent individual covariance matrix, Q, were allowed to be full rank
(i.e. covariances were not constrained to zero) while the error
covariance matrix was constrained to a diagonal matrix.

The last set of parameters were from an incorrectly specified
latent multivariate random model. The parameters included were
Q2, the latent group covariance matrix, Q., the latent individual
covariance matrix and 1., the error covariance matrix. All three matrices
were restricted to diagonal matrices. Applied to data from a population
in which the parameter matrices contained non-zero covariance terms,
this model demonstrated the reaction of the EM algorithm to incorrectly
specified models.

Figure l is a diagram of the design. There were 12 cells, each

containing 10 replications.

51

FIGURE 1

Design of Study

W525
25 100
Classes Classes
I .

Balanced | Unbalanced I Balanced | Unbalanced

119519.]. I I l
I | I | I
Unrestricted I a | b I c | d I
I | I | I
I | I | I
Correctly I e | f I g | h I
Specified I | I | I
I | I | I
Incorrectly I i | j I k | 1 I
Specified I | I | I

1. Each cell contains 10 repetitions (different sets of data)
2. Cells a through f contain comparable datasets; the same can
be said for datasets g through 1.

52

2. Generation of Data.

.r
--...-"

Implementation of the experimental design required a method of
creating samples drawn from a population of known parameters. The data
had to fit the assumptions for the multivariate random latent model
specified in Chapter Five. The values of the parameters chosen for this
example are listed in Table 1.

Each subject's four observed scores, 1“, were a combination
of three latent group effects, QL , three latent individual
effects, 9” 1, four measurement errors, $1.1 , and four grand
means, u.. The most direct way to generate a dataset of observed
values containing these characteristics is to create four separate
vectors, one for each effect, for each subject and then to create the

observed values through the equation ,<. b

6.1 y
("7’73")
This is the equation from the random latent model in Chapter Five.
Unlike the other three vectors, the grand mean, u , will be identical
Eor all subjects. Each vector is representative of a sample vector from
normal distribution with mean zero and variance covariance matrix as
shown in Table l.

The SAS package contains a subroutine which generates independent
values from a univariate normal distribution with mean of zero and
variance of one. By repeated applications three vectors of dimensions
3 x 1, 3 x l and 4 x l were created for each subject. The vectors,

X( Q ), X( g ) and X( g ) each constitute a random sample of values

53

Table 1

Parameter Values Used in Data-Generationvf

The dimension of the observed variables is four (p-h).

The

dimension of the latent group variables is three (r-3) and the dimension

of the latent individual variables is three (s-3).

The pattern matrices connecting the latent to the observed

variables are:

I
P'P‘F‘h‘
u:uau:u1
0000
uauauau:

o
0000

L - A -

P‘F‘P‘P‘

The latent error, individual and group matrices are:

64 8 40
m-e- a 5 7
‘ 40 7 107

PS - Q -

OOOUI
OOO‘O
H
...;

25
10
15

OM - Q -

10
20
10

-O.
-0.

UIU'IUIU'I

0.5
-0.5
0.5
-0.5
15
10
35

The Expected values of the grand means of the four observed

variables were set to zero (g - Q).

54

drawn from multivariate normal distributions having means of 0 and
identity covariance matrices.

The three vectors X( Q ), X( 2,), and X( g ) are each created
from multivariate normal random distributions with mean vectors of
zero and identity variance-covariance matrices. Sample data from a
population with that parameter were obtained by multiplying each vector
by the cholesky of the known parameter matrix. The final sample
vectors were

me) -T(<».)*x<§)

X'(g)) -T(¢)*X(se)

10(1) -T(‘I')*X(_6.)
where T( Q. ) is the cholesky of Q., T( Q ) is the cholesky of
Q and T( W ) is the cholesky of W.

By using X'( Q ), X'( g ), X'( g ) and 5 together as shown in
equation 6.1, an observed sample data set from a population of known
latent parameters was created. Each data set was created through the
SAS normal random generation procedure using different seed numbers.
The data sets were then used by a computer program to estimate the

values of the parameters.

55
thsrtinaflalues 0}? Parameter. $251998... V
The EM algorithm requires starting values for each parameter
being estimated. The computer program written for this study estimates
two sets of parameters, first the unrestricted between and within
covariance matrices for the observed data, then in turn, the parameters
of a more restricted latent model. The calculations of the starting
values for the parameters of the latent model are based on the final
estimates of the unrestricted between and within covariance matrices.
Starting values for the between and within variance covariance

matrices were estimated from equations 6.2 and 6.3 as developed by

Schmidt (1971).

6.2 2- [n/(n-1)]3

A

6.3 2.:— l/m (S. - [n/(n-l)]S)

These estimates are maximum likelihood estimators under the case of
equal group size.

These equations are used as starting values for an unbalanced
design with one modification, replacing n by its harmonic-mean. In a
balanced design the use of harmonic n will give the maximum likelihood
estimates. When the design is unbalanced, the harmonic Q will give
weighted starting values for the parameters.

After 2 and )3. have been estimated, starting values for the
parameters of the latent model Q., Q and Q are found. Assuming

E-AQA and2-AQA+\Fthen
I III
‘ 'v'

6.4

6.5

6.6

56

A

o - (A'A )‘1'A'z A (A'A).1
I I I I I I I I .
o - (A'A)-1'A 2: A (11%)“.

11-12 -AQA4+2-AQA,
AI I .

 

--r

‘sh_

These values form the starting estimates of theflparameters for the

latent model.

57

4. The Computer Program.

The computer program was written using the §A§ Procedure "Froc_MatrixF.r
, v

1. Necessary Input: Y - a N x p matrix for p measures on N

individuals, A. and A. a which are pattern matrices connecting

the underlying latent variable with the observed-level variables and a

K x 2 matrix, 2, which specifies the number of students in each group.

2. Next the program creates two new matrices, YM, a K x p matrix of

means for each group, and SS, a Kp x p matrix containing the sum of

squares for each group.
Estimate of the parameters of a simple random model.

3. Using Schmidt's Maximum Likelihood Equations 6.2 and 6.3 the program

then estimates starting values for 2. and 2.
E-step

4. Using the equations given above the program next estimates the

conditional varibles W, Q, g*, and Q..
M-step

5. Using these values the program then recalculates estimates for the

parameters 2. and E.

58

6. A reiteration then occurs between steps 4 and 5 until the changes in

2. and 2 are less than 0.01

Estimate the parameters of the latent Restricted Model.

7. The next step in the program computes the starting estimates of
Q., Q and i from 2. and 2 from the two parameter matrices in
step 5 and estimates the sufficient statistics W, Q, M, Q. and

u. from these values (E-step) .
M-step
8. Reestimated values for the parameters Q‘, Q and i are then

obtained and, finally, the program iterates between steps 7 and 8 until

the changes in the parameters Q., Q and Q are less than 0.01.

CHAPTER VII: RESULTS

1. Design and Measures.

The EM Algorithm's ability to estimate covariance components in
both balanced and unbalanced latent multivariate random effects models
were demonstrated by estimating the parameters of independent samples
generated from a population with a known distribution. The balanced
samples contained 30 subjects for each group, and the unbalanced samples
averaged 30 subjects per group.

[The estimates of the balanced and unbalanced samples were studied
across two dimensions, namely the number of groups in the sample and the
type of model being estimated (i.e. the unrestricted model, the
correctly specified latent model and an incorrectly specified latent
model. Twenty elements were estimated in the unrestricted model, 10 for
the Phi matrix and 10 for the Psi matrix. Sixteen elements were
estimated in the correctly specified model; six for the Ph matrix, six
for the Om matrix and four for the Pa matrix. The incorrectly specified
model differed from the correctly specified model only in the number of
matrix elements being estimated. Only the 10 diagonal elements were
estimated, three for the Ph matrix, three for the Om matrix and four for
the Pa matrix.

Tables 2, 3 and 4 contain descriptive statistics of the estimates
of the individual items of the covariance matrices for the three models
over 10 repetitions for different situations. These tables include the
Expected Value of the parameter (E), the Mean, the Standard Deviation

(SD), the Bias, the Mean Square Error (MSE), the Bias divided by the

59

60

parameter's Expected Value (B/E), Ratio(l) and Ratio(2). Mean and
Standard Deviation are self explanatory. The Bias is the difference
between the Expected value of the parameter and its sample mean. The
Mean Square Error (MSE) is the averaged squared deviation of the
parameter estimates around their Expected value.

The ratio of the Bias to the Expected Value (B/E) of a variance
or covariance term converts the Bias into a percentage of the
parameter's Expected Value, giving it a relative value.

The difference between the MSEs of the balanced sample estimate
and the corresponding unbalanced sample estimate divided by the MSE of
the unbalanced sample estimate (Ratio(l)) yields a comparison of the
MSEs of the two types of datasets.

The last measure (Ratio(2)) is the difference between the MSEs of
an element of the incorrectly specified model and the correctly
specified model divided by the MSE of the element of the correctly
specified model. The ratio gives a comparison of the precision of the
two models.

The three tables contain the lower diagonal elements of the
different covariance matrices. In Tables 2 and 3, the elements of the
latent matrix at group level, Q., are labeled Ph(X), the elements of
the latent individual level matrix, Q , are Om(X) and the elements of
the error matrix from the latent models, Q ,are labeled Ps(X). The (X)
corresponds to the elements position in the lower diagonal. The latent

covariance matrices would be

 

 

 

.1 - v- --.

r1 1 11 ,1 i

l , 5

Ph( )- 2 3 0m( )- 2 3 j Ps( )-' 2 ‘
a 5 6 a 5 6; ' 3 :l

. 4!

b ...3

Ph(3) is the variance of the second latent measure and Ph(5) is the
covariance between the second and third elements of the Ph matrix.
Ps(2) would the variance of the second observed measure.

The elements of the unrestricted model are similarly labeled.
The elements of the between group covariance matrix, 2., are labeled
Phi(X) while the elements of the within covariance matrix, 2 , are
Psi(X). They are the same dimension as the error matrix of the latent
model, 4 x 4, but include the six covariance terms in their estimates.

Tables 5 through 7 contain aggregated statistics for each matrix
in the latent models under the different conditions. Table 8 lists the
Maximum Likelihood Ratio test of the estimates of the correctly and
incorrectly specified latent models. Tables 9 and 10 list aggregated
statistics for each matrix in the unrestricted models under the
different conditions. Table 11 has information about the iterations
necessary for the algorithm to converge.

With only 10 repetitions per cell, the power of any statistical
test would be low. Although some characteristics of the estimation
procedure may appear with this size sample, it should be recalled that
this was just a demonstration of the use of the EM algorithm for an
unbalanced latent model under different circumstances and not a

statistical study.

62

2. Results of Estimation Procedure

Table 2 contains information about the estimates of the
parameters for four different situations when the datasets are
comprised of 100 groups. The four conditions are: (l) applying the
correctly specified model to the balanced samples; (2) applying the
correctly specified model to the unbalanced samples; (3) applying the
incorrectly specified model to the balanced samples; and, lastly, (4)
applying the incorrectly specified model to the unbalanced samples.
The items in this table represent the statistics for cells g, h, k and
l in Figure l.

The B/E in Table 2 indicates the percentage of bias of the
estimates. The correctly specified latent model had values of the B/E
ranging from -0.9% to -13.7% for the estimates of the elements of the
Ph matrix for the balanced data and 15.1% to -11.5% for the unbalanced
dataset. The B/E of the estimates of the elements of the Om matrix
ranged from 8.7% to -l.3% for the balanced data and from 10.9% through
-9.8% for the unbalanced data and the Pa matrix had B/E values ranging
from 8.7% to -27.4% for the balanced data and 5.2% to -4l.5% for the
unbalanced data.

In the incorrectly specified model, the estimates of the
elements of the Ph matrix, the variance components, had B/E values are
almost identical to the corresponding elements in the correctly
specified model. The B/E of the estimates of the elements of the Om
matrix ranged from 6.4% to -10.4% for the balanced data and 6.6%

through -10.2% for the unbalanced data and the P3 matrix had B/E values

63
TABLE 2

Summary Statistics of the Latent Models for Balanced
and Unbalanced Data Sets with 100 Groups

Ph Matrix

Balance Unbalance
Expected Balance Unbalance Diagonal Diagonal -

(h) (k) (1) (1)
Ph(l) Mean 64.000 63.400 62.960 63.440 63.240
SD 9.330 10.710 9.290 10.530
MSE 87.449 115.906 86.653 111.523
Bias(B) -0.600 -l.040 -0.560 -0.760
B/E -0.009 -0.016 -0.009 -0.012
Ratio 1 0.325 0.287
Ratio 2 -0.009 -0.038
Ph(2) Mean 8.000 7.600 9.210 0.000 0.000
SD 1.880 2.090
MSE 3.712 5.995
Bias(B) -0.400 1.210
B/E -0.050 0.151
Ratio 1 0.615
Ratio 2
Ph(3) Mean 5.000 4.550 4.450 4.550 4.450
SD 0.640 0.600 0.650 0.560
MSE 0.635 0.696 0.648 0.650
Bias(B) -0.450 -0.550 -0.450 -0.550
B/E -0.090 -0.110 -0.090 -0.110
Ratio 1 0.097 0.003
Ratio 2 0.020 -0.067
Ph(4) Mean 40.000 39.060 38.720 0.000 0.000
SD 6.270 5.610
MSE 40.295 33.293
Bias(B) -0.940 -l.280
B/E -0.023 -0.032
Ratio 1 -0.l74
Ratio 2
Ph(5) Mean 7.000 6.040 6.560 0.000 0.000
SD 3.210 3.360
MSE 11.328 11.505
Bias(B) -0.960 -0.440
B/E -0.137 -0.063
Ratio 1 0.016

Ratio 2

64

TABLE 2 (Continued)
Ph Matrix

Balance Unbalance
Expected Balance Unbalance Diagonal Diagonal

Ph(6) Mean 107.000 98.420 99.560 98.190 99.210
SD 14.980 12.760 15.470 13.060
MSE 306.196 224.322 325.561 237.990
Bias(B) -8.580 -7.440 -8.810 -7.790
B/E -0.080 -0.070 v0.082 -0.073
Ratio 1 -0.267 -0.269
Ratio 2 0.063 0.061
0m Matrix
0m(1) Mean 25.000 25.580 25.900 26.600 26.640
SD 0.880 0.990 1.000 0.960
MSE 1.148 1.880 3.844 3.910
Bias(B) 0.580 0.900 1.600 1.640
B/E 0.023 0.036 0.064 0.066
Ratio 1 0.637 0.017
Ratio 2 2.348 1.080
0m(2) Mean 10.000 10.870 11.090 0.000 0.000
SD 0.830 0.900
MSE 1.530 2.130
Bias(B) 0.870 1.090
B/E 0.087 0.109
Ratio 1 0.392
Ratio 2
0m(3) Mean 20.000 19.740 19.990 18.780 18.750
SD 0.830 0.830 0.710 0.650
MSE 0.764 0.689 2.158 2.159
Bias(B) -0.260 -0.010 -1.220 -1.250
B/E -0.013 -0.001 -0.061 -0.063
Ratio 1 -0.098 0.000
Ratio 2 1.824 2.133
0m(4) Mean 15.000 15.020 14.840 0.000 0.000
SD 1.300 1.410
MSE 1.690 2.017
Bias(B) 0.020 -0.160
B/E 0.001 -0.011
Ratio 1 0.193

Ratio 2

65
TABLE 2 (Continued)
0n Matrix

Balance Unbalance
Expected Balance Unbalance Diagonal Diagonal

0m(5) Mean 10.000 10.390 9.020 0.000 0.000
SD 3.710 4.140
MSE 13.933 18.207
Bias(B) 0.390 -0.980
B/E 0.039 -0.098
Ratio 1 0.307
Ratio 2

0m(6) Mean 35.000 36.300 36.370 31.370 31.440
SD 2.900 3.180 0.670 0.690
MSE 10.288 12.198 15.090 14.558
Bias(B) 1.300 1.370 -3.630 -3.560
B/E 0.037 0.039 -0.104 -0.102
Ratio 1 0.186 -0.035
Ratio 2 0.467 0.193

Ps Matrix

Ps(l) Mean 5.000 3.630 5.260 12.120 12.070
SD 4.410 5.460 0.460 0.470
MSE 21.534 29.887 56.539 55.760
Bias(B) -l.370 0.260 7.120 7.070
B/E -0.274 0.052 1.424 1.414
Ratio 1 0.388 -0.014
Ratio 2 1.626 0.866

Ps(2) Mean 6.000 5.250 3.510 2.070 2.050
SD 3.650 4.170 0.260 0.250
MSE 13.948 24.278 17.229 17.399
Bias(B) -0.750 -2.490 -3.930 -3.950
B/E -0.125 -0.415 -0.655 -0.658
Ratio 1 0.741 0.010
Ratio 2 0.235 -0.283

Ps(3) Mean 11.000 12.280 11.530 6.260 6.250
SD 3.060 3.540 0.570 0.550
MSE 11.184 12.844 25.289 25.372
Bias(B) 1.280 0.530 -4.740 -4.750
B/E 0.116 0.048 -0.431 -0.432
Ratio 1 0.148 0.003
Ratio 2 1.261 0.975

Ps(4)

66
TABLE 2 (Continued)

0m Matrix

Balance Unbalance

Expected Balance Unbalance Diagonal Diagonal

Mean 12.000 13.040 13
SD 2.120 1
MSE 5.696 7
Bias(B) 1.040 1
B/E 0.087 0
Ratio 1 0

Ratio 2

.890
.960
.811
.890
.158
.371

15.390 15.480
0.610 0.610
13.141 13.828

3.390 3.480
0.283 0.290

0.052
1.307 0.770

67

ranging from 142% to -66% for the balanced data and 141% to -66% for
the unbalanced data.

Table 3 contains information about the estimates of the
parameters for the four different situations in Table 2 when the
datasets are comprised of 25 groups. Statistics for cells e, f, i and
j in Figure 1 are presented in this table.

The B/E of the corresponding items in Table 3 show higher
percentages of bias than those in Table 2. The correctly specified
latent model had values of B/E ranging from ~6.6% to -33.9% for the
estimates of the elements of the Ph matrix for the balanced data and
-0.6% to -33.9% for the unbalanced dataset. The B/E of the estimates of
the elements of the 0m matrix ranged from 7.9% to 1.1% for the balanced
data and 5.7% through -1.7% for the unbalanced data and the Ph matrix
had B/E values ranging from 14.3% to -43.6% for the balanced data and
3.7% to -13.0% for the unbalanced data.

In the incorrectly specified model the estimates of the elements
of the Ph matrix, the variance components, had B/E values almost
identical to the corresponding elements in the correctly specified
model. The B/E of the estimates of the elements of.the 0m matrix
ranged from 5.1% to -8.0% for the balanced data and 4.9% to -8.4% for
the unbalanced data and the P3 matrix had B/E values ranging from 136%
to -65% for the balanced data and 136% to -64% for the unbalanced data.

Table 4 contains information about the estimates of the
variables of the unrestricted model under the following conditions: (1)
for balanced samples containing 100 groups; (2) for unbalanced samples
containing 100 groups; (3) for balanced samples containing 25 groups;

and lastly (4) for unbalanced samples containing 25 groups. Although

68

TABLE 3

Summary Statistics of the Latent Models for Balanced
and Unbalanced Data Sets with 25 Groups

Ph Matrix
Balance Unbalance
Expected Balance Unbalance Diagonal Diagonal
(e) (f) (1) (J)
Ph(l) Mean 64.000 59.750 63.620 59.890 64.070
SD 16.730 20.280 16.790 16.830
MSE 299.962 411.439 300.673 283.254
Bias -4.250 -0.380 -4.110 0.070
B/E -0.066 -0.006 -0.064 0.001
Ratio 1 0.372 —0.058
Ratio 2 0.002 -0.312
Ph(2) Mean 8.000 6.260 7.290 0.000 0.000
SD 3.050 3.520
MSE 12.667 12.951
Bias -1.740 -0.710
B/E -0.218 -0.089
Ratio 1 0.022
Ratio 2
Ph(3) Mean 5.000 4.040 4.400 4.060 4.130
SD 1.580 1.530 1.580 2.080
MSE 3.520 2.741 3.478 5.167
Bias -0.960 -0.600 ~0.940 -0.870
B/E -0.192 -0.120 -0.188 -0.174
Ratio 1 -0.221 0.486
Ratio 2 -0.012 0.885
Ph(4) Mean 40.000 28.940 33.070 0.000 0.000
SD 14.760 18.120
MSE 353.773 381.695
Bias -11.060 -6.930
B/E -0.277 -0.l73
Ratio 1 0.079
Ratio 2
Ph(5) Mean 7.000 4.660 4.630 0.000 0.000
SD 4.160 6.170
MSE 23.390 44.310
Bias -2.340 -2.370
B/E -0.334 -0.339
Ratio 1 0.894
Ratio 2

69

TABLE 3 (CONTINUED)

Ph Matrix

Balance Unbalance
Expected Balance Unbalance Diagonal Diagonal

Ph(6) Mean 107.000 77.450 78.720 77.910 78.770
SD 22.380 26.990 23.190 27.490
MSE 1471.089 1617.081 1478.030 1641.181
Bias -29.550 -28.280 -29.090 -28.230
B/E -0.276 -0.264 -0.272 -0.264
Ratio 1 0.099 0.110
Ratio 2 0.005 0.015
Om Matrix
0m(l) Mean 25.000 25.380 25.580 26.270 26.220
SD 1.850 1.680 2.130 2.160
MSE 3.583 3.196 6.329 6.319
Bias 0.380 0.580 1.270 1.220
B/E 0.015 0.023 0.051 0.049
Ratio 1 -0.108 -0.002
Ratio 2 0.766 0.977
0m(2) Mean 10.000 10.770 10.540 0.000 0.000
SD 1.790 1.540
MSE 3.863 2.696
Bias 0.770 0.540
B/E 0.077 0.054
Ratio 1 -0.302
Ratio 2
0m(3) Mean 20.000 21.010 21.150 20.060 20.430
SD 2.400 2.310 2.230 2.400
MSE 6.893 6.806 4.977 5.965
Bias 1.010 1.150 0.060 0.430
B/E 0.051 0.057 0.003 0.021
Ratio 1 -0.013 0.199
Ratio 2 -0.278 -0.123
0m(4) Mean 15.000 15.170 14.970 0.000 0.000
SD 1.230 1.610
MSE 1.545 2.593
Bias 0.170 -0.030
B/E 0.011 -0.002
Ratio 0.678

NDF‘

Ratio

70
TABLE 3 (CONTINUED)
0m Matrix

Balance Unbalance
Expected Balance Unbalance Diagonal Diagonal

0m(5) Mean 10.000 10.500 9.830 0.000 0.000
SD 2.830 3.900
MSE 8.287 15.242
Bias 0.500 -0.170
B/E 0.050 -0.017
B /MSE 0.034 0.002
Ratio 1 0.839
Ratio 2

0m(6) Mean 35.000 37.780 35.910 32.190 32.070
SD 5.340 2.010 2.420 2.370
MSE 37.103 4.960 14.630 15.156
Bias 2.780 0.910 -2.810 -2.930
B/E 0.079 0.026 -0.080 -0.084
Ratio 1 -0.866 0.036
Ratio 2 -0.606 2.055

Ps Matrix

Ps(l) Mean 5.000 2.820 4.350 11.810 11.820
SD 4.250 4.700 1.450 1.430
MSE 23.343 22.559 53.632 53.725
Bias -2.180 -0.650 6.810 6.820
B/E -0.436 -0.130 1.362 1.364
Ratio 1 -0.034 0.002
Ratio 2 1.298 1.382

Ps(2) Mean 6.000 5.260 5.380 2.100 2.190
SD 3.560 3.560 1.190 1.230
MSE 13.282 13.101 18.316 17.642
Bias -0.740 -0.620 -3.900 -3.810
B/E -0.123 -0.103 -0.650 -0.635
Ratio 1 -0.014 -0.037
Ratio 2 0.379 0.347

Ps(3) Mean 11.000 12.570 11.410 6.430 6.520
SD 1.960 2.810 1.020 0.980
MSE 6.580 8.083 24.246 23.261
Bias 1.570 0.410 -4.570 -4.480
B/E 0.143 0.037 -0.415 -0.407
Ratio 1 0.228 -0.041

Ratio 2 2.685 1.878

71
TABLE 3 (CONTINUED)
Ps Matrix

Balance Unbalance
Expected Balance Unbalance Diagonal Diagonal

Ps(4) Mean 12.000 12.280 11.830 14.060 13.880
SD 2.330 1.290 1.220 0.430
MSE 5.516 1.696 6.204 4.112
Bias 0.280 -0.170 2.060 1.880
B/E 0.023 -0.014 0.172 0.157
Ratio 1 -0.692 -0.337
Ratio 2 0.125 1.424

72

TABLE 4

Summary Statistics of the Unrestricted Models for Balanced
and Unbalanced Data Sets for Both 25 and 100 Groups

Phi Matrix
100 100 25 25
Groups Groups Groups Groups
Expected Balance Unbalance Balance Unbalance
(e) (d) (a) (b)
Phi(l) Mean 143.500 141.962 141.230 127.691 139.317
SD 16.359 16.865 34.203 44.422
MSE 270.245 290.154 1447.539 1992.756
Bias(B) -1.538 -2.270 -15.809 -4.183
Bias/Expected(B/E) -0.011 -0.016 -0.110 ~0.029
Ratio 1 0.074 0.377
Ratio 2 4.356 5.868
Phi(2) Mean 46.500 48.521 47.472 51.845 56.904
SD 12.099 13.831 20.467 23.513
MSE 150.924 192.346 450.641 673.131
Bias 2.021 0.972 5.345 10.404
B/E 0.043 0.021 0.115 0.224
Ratio 1 0.274 0.494
Ratio 2 1.986 2.500
Phi(3) Mean 32.500 35.633 34.445 39.816 44.156
SD 11.939 12.684 19.156 21.152
MSE 153.446 165.087 426.423 598.365
Bias 3.133 1.945 7.316 11.656
B/E 0.096 0.060 0.225 0.359
Ratio 1 0.076 0.403
Ratio 2 1.779 2.625
Phi(4) Mean 49.500 49.980 49.060 53.453 53.344
SD 7.474 7.330 8.875 9.751
MSE 56.117 53.944 96.128 111.500
Bias . 0.480 -0.440 3.953 3.844
B/E 0.010 -0.009 0.080 0.078
Ratio 1 -0.039 0.160
Ratio 2 0.713 1.067
Phi(S) Mean 129.500 128.490 128.139 116.448 126.424
SD 14.652 15.129 33.778 42.182
MSE 215.815 230.945 1330.236 1789.834
Bias -1.010 -1.361 -13.052 -3.076
B/E -0.008 -0.011 -0.101 ~0.024
Ratio 1 0.070 0.346
Ratio 2 5.164 6.750

73

TABLE 4 (Continued)

Phi Matrix
100 100 25 25
Groups Groups Groups Groups

Expected Balance Unbalance Balance Unbalance

Phi(6) Mean 56.500 55.834 55.339 59.803 60.973
so 9.087 8.879 9.528 10.791
use 83.066 80.334 102.905 138.676
Bias -0.666 -1.161 3.303 4.473
B/E -0.012 -0 021 0.058 0 079
Ratio 1 -0.033 0.348
Ratio 2 0.239 0.726
Phi(7) Mean 120.500 119.725 119.638 109.674 118.120
so 13.156 13.843 33.908 40.686
use 173.748 192.454 1279.977 1661.644
Bias -0.775 -0.862 -10.826 -2.380
o/o -0.006 -0.007 -0.090 -0.020
Ratio 1 0.108 0.298
Ratio 2 6.367 7.634
Phi(8) Mean 39.500 41.359 41.109 45.143 49.719
so 10.461 12.173 21.005 23.745
ass 113.272 151.058 476.592 679.856
Bias 1.859 1.609 5.643 10.219
B/E 0.047 0.041 0.143 0.259
Ratio 1 0.334 0.426
Ratio 2 3.207 3.501
Phi(9) Mean 30.500 33.114 32.664 37.596 41.590
so 10.240 10.916 19.657 21.313
ass 112.450 124.362 442.346 590.897
Bias 2.614 2.164 7.096 11.090
B/E 0.086 0.071 0.233 0.364
Ratio 1 0.106 0.336
Ratio 2 2.934 3.751
Phi(lO) Mean 47.500 46.748 47.373 51.063 50.635
so 6.523 6.079 8.626 9.364
MSE 43.178 36.972 88.513 98.605
Bias -o.752 -0.127 3.563 3.135
B/E -0.016 -0.003 0.075 0 066
Ratio 1 -0.144 0.114
Ratio 2 1.050 1.667

74

TABLE 4 (Continued)

Psi Matrix
100 100 25 25
Groups Groups Groups Groups

Expected Balance Unbalance Balance Unbalance

Psi(1) Mean 73.750 73.985 74.058 73.822 73.404
so 1.682 1.659 4.438 4.283
ass 2.890 2.858 19.702 18.477
Bias 0.235 0.308 0.072 -0.346
8/3 0.003 0.004 0.001 -0.005
Ratio 1 -0.011 -0.062
Ratio 2 5.816 5.466
231(2) Mean 31.250 31.613 31.616 31.392 31.609
so 1.118 1.095 3.445 3.387
ass 1.396 1.348 11.890 11.615
Bias 0.363 0.366 0.142 0.359
B/E 0.012 0 012 0.005 0.011
Ratio 1 -0.035 -o.023
Ratio 2 7.515 7.617
931(3) Mean 6 250 6.533 6.513 6.341 6.817
so 0.609 0.576 2.235 2.399
ass 0.460 0.409 5.004 6.112
Bias 0.283 0.263 0.091 0.567
B/E 0.045 0.042 0.015 0.091
Ratio 1 -0.111 0.221
Ratio 2 9.882 13.958
931(4) Mean 13.750 13.925 13.894 14.022 14.362
so 0.767 0.728 1.601 1.545
ass 0.622 0.553 2.645 2.803
Bias 0.175 0.144 0.272 0.612
8/8 0.013 0.010 0 020 0.045
Ratio 1 -0.111 0 060
Ratio 2 3.251 4.069
231(5) Mean 43.750 44.095 44.216 43.565 43.741
so 1.396 1.323 2.512 2.201
ass 2.081 1.992 6.348 4.844
Bias 0.345 0.466 -0.185 -0.009
B/E 0.008 0.011 -0.004 0.000
Ratio 1 -0.043 -0.237
Ratio 2 2.050 1.432

Psi(6)

Psi(7)

Psi(8)

Psi(9)

Psi(10)

Mean
SD
MSE
Bias
B/E
Ratio
Ratio

Mean
SD
MSE
Bias
B/E
Ratio
Ratio

Mean
SD
MSE
Bias
B/E
Ratio
Ratio

Mean
SD
MSE
Bias
B/E
Ratio
Ratio

Mean
SD
MSE
Bias
B/E
Ratio
Ratio

NH

NH

'00"

NH

NH

75

TABLE 4 (Continued)

25

Psi Matrix
100 100 25

Groups Groups Groups
Expected Balance Unbalance Balance
34.750 34.987 34.936 35.350
0.857 0.864 2.306
0.797 0.785 5.718
0.237 0.186 0.600
0.007 0.005 0.017

-0.015
6.175
49.750 49.935 50.034 49.971
1.306 1.237 2.009
1.744 1.620 4.090
0.185 0.284 0.221
0.004 0.006 0.004

-0.071
1.346
16.250 16.690 16.710 15.915
1.122 1.084 2.278
1.474 1.410 5.314
0.440 0.460 -0.335
0.027 0.028 -0.021

-0.043
2.605
11.250 11.354 11.308 11.681
0.660 0.628 1.812
0.448 0.398 3.490
0.104 0.058 0.431
0.009 0.005 0.038

-0.111
6.796
30.750 30.860 30.861 30.662
0.905 0.878 1.210
0.832 0.785 1.473
0.110 0.111 -0.088
0.004 0.004 -0.003

-0.058
0.769

Groups
Unbalance

35.831
1.922
4.992
1.081
0.031

.127

5.360

0.328
2.072
4.664
0.578
0.012
0.140
1.880

.095
2.303
5.331
.155
.010
0.003
2.780

.121
.836
.214
.871
.077
.207
.584

\DOOCiPI-‘N

.770
.141
.302
.020
.001
.116
.660

OOOOHHO

76

the Unrestricted model was a linear combination of the latent variance
and covariance components of the correctly specified model, it was
directly estimated from the data.

For the estimates from data containing 100 groups, B/E ranged
from 9.6% to -1.6% for the balanced data and 7.1% to -12.7% for the
unbalanced data. The estimates from the datasets containing 25 groups
had B/E ranging from 23.3% to -11.0% for the balanced data and 36.4% to
-2.9% for the unbalanced data.

Table 5 contains the average B/E value for each matrix, for the
balanced and unbalanced data, for both the correctly and incorrectly
specified latent models.

In the correctly specified latent model, the matrices from large
group data (n-100 groups) showed lower average B/E than matrices from
small group data (np25 groups). The Om and P8 matrices had average B/E
percentages ranging between 2.9% and -4.9% versus 4.7% and -9.8%. The
Ph matrices had values of -2.3% to -6.5% versus -16.5% to -22.7%.

The findings regarding the incorrectly specified model were not
as consistent. The B/E for the Om matrix was lower for the small group
data in both the balanced and unbalanced datasets. The Ph and Ps
matrices, on the other hand, had lower B/E for the large group data.

The data from the balanced group would be expected to have the
best estimates with the smallest MSE. If this procedure can get esti-
mates of the unbalanced design, with only a small increase in the MSE,
than the estimation procedure would be practical. Table 6 contains the
average values of Ratio(l), the ratio of the difference between the MSE
of the balanced and unbalanced samples to the MSE of the balanced

sample, of each matrix of the two latent models for both sample sizes.

77

Table 5

Average B/E of the Latent Models for Balanced and
Unbalanced Data Sets for Both 25 and 100 Groups

25 Classes
(e) (f) (i) (J)

Incorrect Incorrect

Balance Unbalance Balance Unbalance
Ph -0.227 -0.165 -0.175 -0.146
(0.093) (0.121) (0.130) (0.130)
Om 0.047 0.024 -0.009 -0.004
(0.029) (0.030) (0.047) (0.050)
Ps -0.098 -0.053 0.115 0.120
(0.250) (0.078) (0.899) (0.894)

100 Classes
(3) (h) (k) (1)

Incorrect Incorrect

Balance Unbalance Balance Unbalance
Ph -0.065 -0.023 -0.060 -0.065
(0.047) (0.091) (0.045) (0.050)
Om 0.029 0.012 -0.034 -0.033
(0.035) (0.069) (0.087) (0.088)
Ps -0.049 -0.026 0.103 0.102
(0.185) (0.256) (0.936) (0.933)

(a), . indicates row of figures under letter refer to cell

in Figure 1.

78

Table 6

Average Ratio(l) of the Ph, 0m and Ps Matrices for the
Latent Models in Data Sets for both 25 and 100 Groups

25 Classes

Ph

Ps

100 Classes

Ph

Ps

Unbalance

0.208
(0.386)

0.038
(0.634)

-0.128
(0.395)

Unbalance

0.102
(0.326)

0.269
(0.245)

0.412
(0.245)

Incorrect
Unbalance

0.179
(0.227)

0.078
(0.089)

-0.103
(0.113)

Incorrect
Unbalance

0.007
(0.197)

-0.006
(0.019)

0.013
(0.028)

Ratio 1 - ( MSE(Unbalanced)-MSE(Ba1anced) )/( MSE(Ba1anced) )

79

In the correctly specified model for the samples containing 100
groups, the unbalanced samples had greater MSE than the balanced in all
three of the matrices averaging from 10% to 41% more. For samples
containing 25 groups, the two individual level matrices, Om and P3,
showed little or negative increase on the average. The P3 matrix had
an average drop of 13% for the MSE, while the Om matrix had only a
slight increase of 3%. The Ph matrix had an average increase of 20%.

The incorrectly specified model showed the same results for
samples containing 25 groups. The balanced samples containing 100
groups showed very little difference in the MSE from the unbalanced
samples containing 100 groups for all three matrices.

The estimates of the balanced samples improved more than the
unbalanced samples (in terms of MSE) as group size increases.

The imposition of a structure on the data, in turn, permits
specification of incorrect models. Table 7 contains the average values
of Ratio(2), the ratio of the difference between the MSE of the
correctly and incorrectly specified latent models divided by the MSE of
the correctly specified model, of each matrix for the balanced and
unbalanced samples for both sample sizes.

In the samples containing 25 groups, the balanced data shows
little difference between the MSE's of the incorrectly and correctly
specified models for the Ph and Om matrices, 0% and -4%. The P3 matrix
had an increase in the MSE of 112%. The unbalance sample had rises in
the MSE of 19% for the Ph matrix, 97% for the Om matrix and 125% for
the P8 matrix.

For the balanced and unbalanced samples containing 100 groups,

the Ph matrix showed little increase in the MSE between the correctly

80
Table 7

Average Ratio(2) of the Ph, Om and P3 Matrices for Balanced
and Unbalanced Data Sets for both 25 and 100 Groups

25 Classes
Incorrect Incorrect
Balance Unbalance
Ph -0.002 0.196
(0.006) (0.619)
0m -0.039 0.970
(0.507) (1.093)
Ps 1.121 1.258
(1.157) (0.647)
100 Classes
Incorrect Incorrect
Balance Unbalance
Ph 0.025 -0.015
(0.036) (0.067)
0m 1.546 1.135
(0.971) (0.971)
Ps 1.107 0.582
(0.604) (0.583)

Ratio 2 - ( MSE(Incorrect)-MSE(Correct) )/( MSE(Correct) )

81

and incorrectly specified models, while the 0m matrix increased 13% and
15%, respectively, and the P3 matrix showed increases of 58% and 110%.

The large rises in the MSE of the P3 matrices for the
incorrectly specified model can be explained by reviewing Tables 2 and
3. The P3 matrix of the incorrectly specified model is very biased
with a small sampling variance. It is this bias that causes the MSE to
greatly increase. Without knowing the true values of the variances and
covariances, the incorrect model would be tempting to accept because of
the small sampling variance that accompanies it.

The incorrectly specified model does well in estimating the
variances of Ph, but Om and P3 show problems with their estimates. In
Tables 1 and 2 the standard deviation of the Ps elements in the
correctly specified model vary between 2 to 10 times as large as the
corresponding elements for the incorrectly specified model. On the
other hand, the bias of the estimates of the elements of the P3 matrix,
in the incorrectly specified model, range between 2 to 10 times as
large as the bias for the corresponding elements in the correctly
specified model. The percentage of the MSE which was due to bias in
the incorrectly specified model in P3 (all sizes) ranged between 91% to
99.6%. The low standard deviation of the sample, but very incorrect
estimates, indicate a very consistent but extremely biased estimate.
The direction of the bias was not consistent across elements.

It is also important to test the model for fit. By using the
Maximum Likelihood Ratio (MLR), the correctly and incorrectly specified
models can be tested for fit. If the MLR is significant, it is an
indication that the model does not fit the data. Table 8 contains the

statistics of the maximum likelihood ratio for the correctly and

25

100

82

Table 8

Maximum Likelihood Ratio Test of the Pit
of the Correct and Incorrect Models

Classes
*
(b)
Balance
Mean 2.77
(1.76)
Minimum 1.29
Maximum 7.41

No. of samples (1)
significant at

p < .05
Classes
*
(h)
Balance
Mean 7.41
(7.05)
Minimum 0.82
Maximum 22.72

No. of samples (3)
significant at
p < .05

* df - 4
** df - 10

'k

(e)
Unbalance

4.21
(3.77)

1.58

13.13

(2)

*

(k)
Unbalance

15.16
(12.49)

2.46

34.25

(6)

*1?

(C)
Incorrect
Balance

129.77
(39.16)

81.12
210.99

(10)

*‘k

(i)
Incorrect
Balance

618.59
(99.09)

475.21
769.72

(10)

**

(f)
Incorrect
Unbalance

144.53
(54.53)

68.36
231.57

(10)

*1!

(1)
Incorrect
Unbalance

603.88
(89.86)

443.02
783.49

(10)

83

incorrectly specified models. The MLR's of the correctly specified
model are lower than those for the incorrectly specified model's by a
factor of more than 10. The tests of fit for all forty samples for the
incorrectly specified model had significant MLR's. Only 12 of those
samples were significant for the correctly specified model, nine of
which were from samples containing 100 groups.

The datasets containing 100 groups had MLR's four times the size
of those from datasets with 25 groups. If the dataset is very large,
the fit may be acceptable, but the MLR significant. This is a common
problem in covariate structural analysis. The same problem occurs in
Lisrel when using a very large sample.

The unrestricted model estimated only one covariance matrix for
the group level and one matrix for the individual level. Neither of
these matrices, Phi or Psi, were structured or constrained. This
particular model was estimated separately from the latent models.

The starting values of the unrestricted model were Maximum
Likelihood Estimates when the data was balanced. Schmidt's Maximum
Likelihood Equations for the between and within covariance matrices of
a multivariate random model were used as starting points. This
algorithm always converged at the end of the first iteration for
balanced datasets.

Table 9 contains the average B/E of Phi and Psi in the
unrestricted model. Both the balanced and unbalanced samples showed
little difference in the average B/E of either matrix when the data
contained 100 groups. The B/E averaged less than 2.4% of the expected
value. When the data was comprised of 25 groups, the average B/E of

the Psi matrix was less than 2.5% for both the balanced and unbalanced

Table 9

Average B/E of the Unrestricted Model for Balanced and
Unbalanced Data Sets for Both 25 and 100 Groups

(e)
100
Balance

Phi 0.023
(0.042)

Psi 0.013
(0.013)

(d) (a) (b)
100 25 25
Unbalance Balance Unbalance
0.013 0.063 0.135
(0.033) (0.127) (0.154)
0.013 0.007 0.025
(0.013) (0.016) (0.035)

samples. The Phi matrix, on the other hand, had values of 13% and 17%

for the balanced and unbalanced datasets irrespectively.

The increase

in the sample size, from 750 to 3000, had little effect on the B/E for

the Psi matrix.

The increase from 25 to 100 groups, however, reduced

the average B/E for the estimates of the elements of the Phi matrix in

the balanced and unbalanced samples from 13% and 17% to 2% and 1%.

Table 10 summarizes the values of Ratio(l) for the unrestricted

model.

The Phi matrix showed a higher average MSE for the unbalanced

samples than for the balanced samples in both small and large group

data. (33% and 8%, respectively).

The Psi matrix showed little

difference between the MSE's of the balanced and unbalanced samples for

samples of either size.

As the number of groups in a sample increase,

the MSE of the unbalanced data evidently approaches that of the

balanced data.

One last note, statistical theory states that as the sample size

increases the sample variance will decrease.

The SD should be about

Table 10

Average Ratio(l) of the Phi and Psi Matrices for Balanced
and Unbalanced Data Sets for Both 25 and 100 Groups

100 25
Unbalance Unbalance
Phi 0.083 0.330
(0.142) (0.116)
Psi -0.061 0.007
(0.039) (0.151)

twice as large for the 25 group as for the 100 group samples. This is
borne out for the Ph and 0m matrices, but not by the P3 matrix. The P3
matrix has approximately the same SD for both samples of 25 and 100

groups.

86

3. Results of the Process.

A section of the findings of this study apply to the process of
the EM algorithm Problems encountered in the procedure of the
algorithm in this environment may apply to other situations.

Table 11 contains information on the number of iterations the
estimation procedure took to converge for each of the cells in Figure 1.
The incorrectly specified model needed the most iterations to converge
for both data containing 100 groups and data containing 25 groups. The
means of the balanced and unbalanced samples were very
close, 82 and 86 for the data with 25 groups and 99 and 100 in the data
with 100 groups.

The correctly specified model averaged 74 and 76 iterations for
the balanced and unbalanced samples of 25 groups. For data containing
100 groups, the unbalanced sample averaged 38 iterations less than the
balanced sample, 99 vs 61.

The unrestricted model averaged fewer iterations than either of
the latent models for all conditions. The latent models, at the least,
averaged over 50 more iterations than the unrestricted model. The
samples containing 100 groups for the unbalanced sample averaged less
iterations than the unbalanced sample from the 25 group case, 5.8
against 10.3.

The unrestricted model for the balanced sample under both cases
always stopped after the first iteration. The starting value for the
algorithm was Schmidt's maximum likelihood estimators for the between
and within models. This confirmed that the algorithm was capable of

stopping at a maximum likelihood estimate.

25 Groups
Unrestricted

Correctly
Specified

Incorrectly
Specified

100 Groups
Unrestricted

Correctly
Specified

Incorrectly
Specified

87

TABLE 11

Iterations Required by Algorithm to Convergence

Mean Standard

1.0

74.6

82.2

1.0

98.8

Balanced
Design
Range

Deviation
- 1 l
53.43 23 198
19.42 50 105
- 1 1
62.84 27 192
3.62 92 104

HHHHHHHHHHHHHHHHHHH

Mean

10.

76.

86.

U)

Unbalanced
Design

Standard
Deviation
7.79

39.40

16.83

Range

1 28
37 159
53 102
5 6
27 117
96 104

Ph and Om were estimated as full matrices for the correctly
specified model and as diagonal matrices for the incorrectly
specified model.

88

Convergence was found to be a problem for two datasets and they
were not used in the final analysis. In one dataset under the
unbalanced case, the unrestricted model moved toward convergence for
seven iterations until only one element of the three covariance matrices
was slightly larger than criterion. At iteration eight, the estimates
of the parameters of the matrix
diverged from the expected values until the program automatically
stopped at the 250th iteration. The estimates of the parameters had
significantly diverged from the maximum likelihood values. WA slight_“
change in the values of the starting matrix of this dataset caused the
algorithm to converge in seven iterations.

In the Unrestricted model the Psi matrix converged very quickly.
The convergence of the model came only after the elements in the Phi
matrix reached the criteria.

In the correctly specified latent model the Ph matrix was the
first matrix for all of the elements to reach the convergence criteria.
The P3 matrix was the last in which all elements reached criteria.

_Finally, the starting values of the parameters affected the final

 

 

 

estimated values at which the algorithm converged. Specifically,

 

proximity of the starting values to the true values appeared to be
positively related to how closely the final estimated parameters would
be to the maximum likelihood estimates when reaching criterion.
Criterion was reached when all elements in the covariance matrices
changed less than .01. Using a set of data, an initial computer run was
done on the data using values close to the expected values as starting
values. These starting values caused the final estimates of the

parameters to be close to the Expected value. A second run of the

89
program was then done on this data using Schmidt's formula to find
starting values for the parameters. The run resulted with estimates of
the parameter that were not as close to the expected values as were
those from the first run. The criteria were ignored and the second was
allowed to continue for 95 more iterations. It then reached values

which were nearly identical to those from the first run.

CHAPTER VIII: DISCUSSION

1. “Summary and Conclusions..

wAlthough statistical procedures are available for estimating
treatment effects for students taught in classrooms, these procedures
are applicable, only, when every class has the same number of
students. The present study investigated a procedure that was
originally established to handle missing data (EM Algorithm) but-which
also provides a solution to the problem of estimating parameters in
multivariate analysis when samples contain unequal group sizes. The
focus of the present dissertation was on the estimation of latent group
and individual level variances and covariances with measurement error
removed when group sizes varied in a sample. Previous methods could
only find maximum likelihood estimates for this problem if the dataset
contained groups of equal size. The EM Algorithm offers a method for
finding maximum likelihood estimates of parameters in situations where
classical maximum likelihood procedures failed.

To estimate a set parameters, the EM Algorithm requires two
steps, an expectation step (E-step) and a maximization step (M-step).
The E-step is characterized by the formulation of the sufficient
statistics in terms of the observed data and the parameters. The
M-step consists of developing the maximum likelihood equations for the
parameters in terms of the conditional statistics. ‘93595—31VEE_
starting values for the parameters, the algorithm calculates the
sufficient statistics in the E-step. These values are used to estimate

the parameters in the M-step. The algorithm returns to the E-step to

90

91

recalculate the sufficient statistics based on the new values of the
parameters. The parameters are reestimated using these new values of
the sufficient statistics. The algorithm iterates between the E-step
and the M-step until a specified criteria is reached.

The estimate of balanced and unbalanced samples were both
studied while varying two factors, mainly the number of groups in the
sample (the size) and the particular model being estimated (that is to
say, the unrestricted model, the correctly specified model and the

incorrectly specified model). The unrestricted model was

(8.1) I“ -u+;z1 +3.“
I is a p x 1 vector of observed data.
u is a p x 1 vector of grand means.
1 is a p x 1 vector of group effects.

6 is a p x 1 vector of individual effects.

These variables were considered to have come from multivariate normal

distributions:

(8.2) x~N(0.2Y) 1~N(9.21)

u~N(Q.2u) 3~N<0.2¢)

The parameters of interest for this model were 2% and 2‘.
The correctly specified model was visualized as the application
of a structure to the unrestricted model. By assuming 1 -'A}§ and

g - Ag + £1, (8.1) becomes:

 

(8.3) I“ - g + A.§1 + Ag”
1 is a p x 1 vector of observed data.
u is a p x 1 vector of grand means.
A is a p x q matrix connecting the p observed variables

with q latent group variables.
Q is a q x 1 vector (q 5 p) of the latent group effects.
A is a p x r matrix connecting the p observed variables

with r latent group variables.

g is a r x 1 vector (r 5 p) of the latent individual
effects.
;1 is a p x 1 vector of the latent individual errors.
with X-N(Q.Z‘Y) §~N(.Q.45a)
u-N(Q.3u) 2~N(9..4>)
sl~N(Q.‘I')-

The parameters of interest in this model are 0., Q and W.

The incorrectly specified model differs from the correctly
specified model in the constraints placed on the two latent covariance
matrices. In the incorrect model, the latent covariance matrices,
and 0; and 0 are considered to be diagonal matrices. No such constraints
are placed on the latent matrices in the correct model.

‘g Tests of the m9g§1;5a§ed on the criteria of convergené37showed

this estimation procedure_tgfibe a satisfactory and effective method n

 

 

 

 

 

theory. wever, once the study had been completed, it was recognized
u e‘rgrmterm, ould gg(§”EZ;;33iti)for all

 

 

tha ‘a’ggdel.

\f 93
pract lication . This model can be described as follows:

_.\

\

z )
(8.5) X-g-O-AQ-i-Aa +£_l+_§_2_

i.) a i ‘13 i.) 1.) I
I is a p x 1 vector of observed data.
a is a p x 1 vector of grand means.
A is a p x q matrix connecting the p observed variables

with q latent group variables.
Q is a q x 1 vector (q 5 p) of the latent group effects.
A is a p x r matrix connecting the p observed variables

with r latent group variables.

3 is a r x 1 vector (r g p) of the latent individual
effects.
£1 is a p x 1 vector of the latent individual‘érrors.
$2 is a p x 1 vector of the latent groupvgrrors.
with x~N(9.2Y) 1-N(0.9.)
u~N(Q.2“) a~N(0.¢)
11~N<0.91) 32~N(0.wz)

The parameters of interest in this model are 0., 0, $1 and 92.
The EM Algorithm was developed for thi§_purpose and run on a
,\,__‘~________’_7 1__15 _,__1
trial set of data. The results were similar to those obtained in the

N4
old del See Table 12

Three issues whighithe users of the EM orithm st contend
with! are tie—enjem W and W

6./

 

 

 

Summary Statistics of the Four Parameter Latent Model for
Balanced and Unbalanced Data Sets with 100 Groups

Ph(l)

Ph(2)

Ph(3)

Ph(4)

Ph(5)

Ph(6)

Mean
SD

MSE
Bias(B)
B/E
Ratio 1

Mean
SD

MSE
Bias(B)
B/E
Ratio 1

Mean
SD

MSE
Bias(B)
B/E
Ratio 1

Mean
SD

MSE
Bias(B)
B/E
Ratio 1

Mean
SD

MSE
Bias(B)
B/E
Ratio 1

Mean
SD

MSE
Bias(B)
B/E
Ratio 1

94

TABLE 12

Ph Matrix

Expected Balance

0!)

64.000 64.040
3.710
13.766
0.040
0.001

8.000 7.160
3.990
16.704
-0.840
-0.105

5.000 4.380
1.600
2.987
0.620
0.124

40.000 39.430
10.440

109.355

-0.570

-0.014

7.000 7.040
6.880
47.336
0.040
0.006

107.000 102.600
15.970

276.552

-4.400

-0.041

Unbalance

(k)

64.

3

12.
.280

0.
-0.

0

100.
18.
384.
-6.
-0.
.390

280

.470

128

004
119

.130
.780
.689
.870
.109
.418

.340
.480
.674
.660
.132
.105

.720
.860
.027
.280
.007
.079

.000
.900
.610
.000
.000
.006

940
540
536
060
057

95
TABLE 12(Continued)

Summary Statistics - Ph Matrix

Balance Unbalance
Mean of B/E -0.046 -0.050
SD of B/E 0.056 0.059
Mean of Ratio 1 0.112
SD of Ratio 1 0.238
Om Matrix

Expected Balance Unbalance

0m(1) Mean 25.000 24.490 24.420
SD 1.020 0.900

MSE 1.329 1.184

Bias(B) -0.510 -0.580

B/E -0.020 -0.023

Ratio 1 -0.110

0m(2) Mean 10.000 11.780 11.670
SD 1.360 1.410

MSE 5.370 5.087

Bias(B) 1.780 1.670

B/E 0.178 0.167

Ratio 1 -0.053

0m(3) Mean 20.000 18.270 18.360
SD 1.080 1.510

MSE 4.492 5.269

Bias(B) -1.730 -l.640

B/E -0.087 -0.082

Ratio 1 0.173

0m(4) Mean 15.000 16.920 16.940
SD 1.540 1.370

MSE 6.468 6.059

Bias(B) 1.920 1.940

B/E 0.128 0.129

Ratio 1 -0.063

0m(5) Mean 10.000 12.950 13.220
SD 4.200 4.060

MSE 27.309 28.004

Bias(B) 2.950 3.220

B/E 0.295 0.322

Ratio 1 0.025

96

TABLE 12(Continued)

Om Matrix

Expected Balance Unbalance

0m(6) Mean 35.000 32.320 32.510
SD 2.710 3.380

MSE 15.325 18.313

Bias(B) -2.680 -2.490

B/E -0.077 -0.071

Ratio 1 0.195

Summary Statistics - Om Matrix

Mean of B/E 0.070 0.074
SD of B/E 0.155 0.160
Mean of Ratio 1 0.028
SD of Ratio 1 0.129
Psl Matrix

Balance Unbalance

Psl(l) Mean 5.000 6.710 6.700
SD 10.380 10.270

MSE 110.993 108.684

Bias(B) 1.710 1.700

B/E 0.342 0.340

Ratio 1 -0.021

Psl(2) Mean 6.000 15.380 15.670
SD 6.740 6.180

MSE 143.188 142.091

Bias(B) 9.380 9.670

B/E 1.563 1.612

Ratio 1 -0.008

Psl(3) Mean 11.000 25.330 26.330
SD 8.410 8.910

MSE 298.894 340.509

Bias(B) 14.330 15.330

B/E 1.303 1.394

Ratio 1 0.139

Psl(4) Mean 12.000 23.910 22.970
SD 7.540 6.250

MSE 214.461 172.775

Bias(B) 11.910 10.970

B/E 0.993 0.914

Ratio 1 -O.l94

97

TABLE 12(Continued)

Summary Statistics - Psl Matrix

Balance Unbalance
Mean of B/E 0.700 0.710
SD of B/E 0.527 0.564
Mean of Ratio 1 -0.021
SD of Ratio 1 0.137
P52 Matrix

Expected Balance Unbalance

Ps2(1) Mean 7.000 5.160 5.330
SD 8.170 8.170

MSE 70.511 69.848

Bias(B) —l.840 -l.670

B/E -0.263 -0.239

Ratio 1 -0.009

Ps2(2) Mean 6.000 7.130 7.370
SD 5.700 5.620

MSE 33.909 33.670

Bias(B) 1.130 1.370

B/E 0.188 0:228

Ratio 1 -0.007

Ps2(3) Mean 10.000 10.490 11.090
SD 7.470 7.860

MSE 56.068 63.100

Bias(B) 0.490 1.090

B/E 0.049 0.109

Ratio 1 0.125

Ps2(4) Mean 11.000 10.700 9.760
SD 6.380 5.620

MSE 40.804 33.293

Bias(B) -0.300 -1.240

B/E -0.027 -0.113

Ratio 1 -0.184

Summary Statistics - P52 Matrix

Balance Unbalance
Mean of B/E -0.009 —0.002
SD of B/E 0.189 0.211
Mean of Ratio 1 -0.019

SD of Ratio 1 0.127

98

The present study.usedfthe absolute change of the estimate! as,(Z:)
\ ‘ r (

_£heconvergence criteria. Raudenbuash (1986) used the change in the

 

 

 

 

likelihood for their criteriaélatfli gradie of

also been suggested (See no (1987)): However, each_hasfits problems:)

V

Theflfirst criteriawmay not reach the max likelihood solution since
r.

./”’
the starting point/definitel affects’the finishing point_in such a
situational Using the likelihood or its gradient can fail if theremis a

 

 

K

ghance.that the matrix bei“

‘ using estimate differences, will con!gzgg_ggE_a_gingg;§;_mgggixg;fl___i#3

The pattern of the estimation in the two sets of data which

failed to converge exposes a problem. The data irs onver ed toward

the ooggflsjthe¥ digerged. The slight change of one value in

the starting matrices/baused/éhese data sets/go converge, There are
W T

 

is singular. The a1gorithm,g¥}

 

some articles written on the convergence of the E-M algorithm (most
notably Wu (1983)) but these are for univariate cases. The
multivariate case becomes much more complex. Being a linear method,

the EM Algorithm goes on a slow line toward a convergence point. ££_i§,,_

much more susceptiblesto any_localgmaxima_gngLQymg_§han/Raphson-Newton
or any 0th?!39.565353933999919 rewihicomaxiusoishss--9_9___1£§._J.9R§R§x

toward_§hg maximum likelihood estimate.

_"-——~._, .... -

.3ggj13531§331tho restrictiflgcgf.£hggggégla The less_

fThelthifd.

rEEEEEEEEEEERRLQQedwonathe.medelutheubetterwthe,estimatignnappearsto

 

 

be. If the model is wrongly restricted, however, the EM Algorithm will

still converge yielding\bad (but attractive) result' with no indicationbfr

that a problem existsa\ It become imperativ that t e Maximum Likeli-

hood Ratio test or a similar test be used to test the fit of the model.

99

2. Future Exploration.
/ i
There is no clear choice-as the best criterion/for this method.
This convergence criterion for this algorithm used the absolute change
of the estimates. Alternative choices for the convergence criterion are
the change in the model's likelihood or the likelihood's gradient.

Each has its advantages. Si fina teria used in this study

“was affected by the starting values of the estimated matrices, the

”a...

 

other choices of criterion might yield closer consistent estimates.
owever if any 6f thewmatriceé)is singular, the likelihood will
approach infinity and likely fail to converge.

Future models can be expanded nglarger more regtrictiveéand
complicated models. The only problem facing these models is the.number;>
o£_iterations necessary for convergence. _As the models become more
complicated, more iterations are required to reach convergence. New
developments‘arising in the work on the E-M algorithm might shorten
j this process. The E-M algorithm however\must be derived separatelyifor

_each model to which it is applied limiting the generalization from one
model to another.

Another factor not examined here but of importance is the
unbalancedness of the sample. The degree to which the data contains
unbalanced groups may or may not affect the estimation procedure.

Using the unbalanced design a lower and upper limit of the MSE could
be found for the sample. Literature indicates that the relative size

of the matrices of the random model can affect the reliability of the

estimates.

100
Lastly the expansion of this model into the unbalanced design
is important for educational research. This procedure opens the way
for more complicated multilevel analysis such as the causal modeling of

Joreskog.

APPENDICES

APPENDIX A
EQUATIONS FOR THE ESTIMATION OF THE COVARIANCE COMPONENTS OF

THE TWO-PARAMETER MODEL USING THE EM ALGORITHM

The model of the two-parameter (unrestricted) model is:

- +
I u+11 51

13 J

I is a p x 1 vector of observed data.
g is a p x 1 vector of grand means.
1 is a p x 1 vector of group effects.

is a p x 1 vector of individual effects.

In

The EM algorithm is used to estimate the two covariance matrices
of this model, 2% and 22. Both matrices are assumed to come from

multivariate normal distributions:

1~N(Q,2)
7

£~N(QREE)

E-Step

The conditional expectations of 7 and E are:

101

MY 7" 2:11 W Q1?
E - = -
1|Y 1 1k 9 231(17- u)
where Q1 - (Z1 + (l/n1)2‘)-1

w - [2" 0,1"

i-l

M-Step

° The maximum likelihood equations of the parameters for the M-step

are I

A

2:1 - 21 - é):_1[27(Q1 - 01<A - 1003271
where A - (3'11 - 14")(21 ' ”3'
2¢ - z. - [(l/N) film/n,» 3J9. - Q1(A - wmilzc - B - 2e

where B - (X - 3.9. - a.) (X ' 3‘9. ' a.)'

APPENDIX B

EQUATIONS FOR THE ESTIMATION OF THE COVARIANCE COMPONENTS OF

THE FOUR-PARAMETER MODEL USING THE EM ALGORITHM

The four-parameter (unrestricted) model is:

:1 - g + AG) + Ag +-£J,-+ 62
8

ij "'i 1.1 1.) _1.1
X is a p x 1 vector of observed data.
3 is a p x 1 vector of grand means.
A is a p x q matrix connecting the p observed variables

with q latent group variables.
Q is a q x 1 vector (q s p) of the latent group effects.
A is a p x r matrix connecting the p observed variables

with r latent group variables.

g is a r x 1 vector (r 5 p) of the latent individual
effects.

;1 is a p x 1 vector of the latent individual errors.

32 is a p x 1 vector of the latent group errors.

The EM algorithm is used to estimate the four covariance matrices
of this model, 0.,10, ‘1 and $2. The matrices are assumed to come from
multivariate normal distributions:

9~N<0.¢.> 3~N(0.9)
11~N(0.91) 3.2~N<0.92).

103

104

E-Step
The conditional expectations of Q, g, gl_and g2 are:

ulY\ u' 2:111 Q1?
QIY L 9' - 12° 93525; - 8*)
1le 7' 32} 11‘ s 9201(3- 11)

aIY/ 3' 1 some: -Ae*-a‘-32‘)
1! i.) a

(5.18) E

where a - (A o 1' + 1:1)"

'1 '1

Q1 - (he’s); + *2 + (l/m)M )
-1
w - [2:101]
M-Step

The equation can be clarified by substituting By substituting (5.18) for

and (5.12) for D(8.). The maximum likelihood estimate becomes:

‘ i .
0 - o. - k :11 [3. 1.821 - Qi (A - W) (21)]

" 1
w - 92 - k 2:, [‘FZ(Q1 - 01(4 - 1009921

2

9 - 9 - 9w} lewvul/nincz1 - 05A - mi]

- ans-(n1 - 1)a‘1]a) A o

105

~11 - 1'1 - «‘XLIRIMI/mptcz1 - 0104 - 10011

- ans-(n1 - 1)a'1]a)i«1

where a - (j:1 - abet1 - u‘)’

8 - (x1d - A e' - in: - A‘s." - a)

APPENDIX C
COMPUTER PROGRAM FOR THE ESTIMATION OF THE THREE PARAMETER

LATENT MODEL IN SAS

SECTION 1 - PART 1

THIS SECTION CREATES THE SAMPLE DATASET FOR USE IN THE E-M ALGORITHM
STEPS. EACH DATA POINT CONSISTS OF THREE COMPONENTS, LATENT WITHIN
(OM), LATENT BETWEEN (PH) AND ERROR (PS). THE OBJECT IS TO USE
PATTERN MATRICES L AND LA TO CONVERT THE 3 X 3 LATENT MATRICES INTO
4 X 4 MATRICES OF OBSERVED VALUES. THE ERROR MATRIX IS ALWAYS A

4 X 4 MATRIX OF MEASUREMENT ERRORS OF THE OBSERVED VALUES.

1. SEED IS ANY RANDOM NUMBER USED TO CREATE RANDOM VALUES FROM A
RANDOM GENERATOR (NORMAL).
USED IN STUDY
22.939u25 199.939n25
10199 -
50199
80199
100199
110199

2. CIRCLE IS A COUNTER USED TO LOOP THROUGH THE PROGRAM CREATING
DIFFERENT DATA SETS FOR ANALYSIS.

3. {AI IS A Z X 2 MATRIX OF THE NUMBER OF STUDENTS IN THE
GROUPS. NOl HAS THE NUMBER OF SUBJECTS IN GROUPS - N02 HAS THE
NO OF GROUPS OF THAT SIZE.

FOR UNBALANCED 25 GROUPS: | FOR BALANCED 25 GROUPS:
PAT-10 5/20 5/30 5/40 5/50 5; I PAT-30 25;

FOR UNBALANCED 100 GROUPS: | FOR BALANCED 100 GROUPS:
PAT-10 20/20 20/30 20/40 20/50 20; I PAT-30 100;

4. QM IS THE PARAMETER OF THE WITHIN COVARIANCE MATRIX. OF THE
POPULATION .

5. EH IS THE PARAMETER OF THE BETWEEN COVARIANCE MATRIX OF THE
POPULATION.

6. 23 IS THE PARAMETER OF THE ERROR COVARIANCE MATRIX OF THE
POPULATION.

7. L IS A PATTERN MATRIX CREATING LINEAR COMBINATIONS OF THE LATENT
VARIABLES IN PH.

8. LA IS A PATTERN MATRIX CREATING LINEAR COMBINATIONS OF THE LATENT
VARIABLES IN OM.

I'M-Sbfl-l-X'al-l-fl-l-M-l-fi-fl-fl-il-3"I'M.*M-M-l-fl-fl-i-fl-fi-abfl-II-Ifib’l-fi-fifl-I-fl-fl-X-fl-

PROC MATRIX ;
SEED-10199;
CIRCLE-O;

106

-0 ‘0 -0 .0 .0 -0 -0 .0 ‘0 .0 .0 .0 .0 .0 ‘0 -0 ‘0 .0 .0 .0 .0 .0 .0 ‘0 .0 .0 -0 .0 .0 .0 .0 .0 .0 -0 -0 .0 .0 .0 -0 .0 -0

107

PAT-30 25;

OM - 25 10 15/ 10 20 10/ 15 10 35;

PH - 64 8 40/ 8 5 7/ 40 7 107;

PS - 5 0 0 0/ 0 6 0 0/ 0 0 ll 0/ 0 0 0 12;

L - 1 0.5 0.5/
l 0.5 -0.5/
1 -0.5 0.5/
1 -0.5 -0.5;
LA- 1 0.5 0.5/
l 0.5 -0.5/
1 -0.5 0.5/
1 -0.5 -0.5;

NOTE 'THESE ARE THE PARAMETER VALUES AND PATTERN OF SIZES' ;
PRINT PAT WITHIN BETWN ERR;
*

* SECTION 1 - PART 2

*

* THREE DIFFERENT VECTORS OF DATA ARE NEEDED, ONE FOR PH, ONE FOR OM

* AND ONE FOR PS. THESE ARE INDEPENDENT RANDOM VARIABLES.

* FOR LATER USE THE CHOLESKYS OF OUR PARAMETER MATRICES ARE NEEDED.

*

* 9. QHQLQM IS THE CHOLESKY OF DH.

*10. QHQLPH IS THE CHOLESKY 0F PH.

*11. QHQLPS IS THE CHOLESKY 0F PS.

*12. A 18 VECTOR OF 21330 VALUES GENERATED AT RANDOM FROM A POPULATION

* OF VALUES WITH A MEAN OF 0 AND A VARIANCE OF 1 FROM SAS

* SUBROUTINE NORMAL.

*13. 2 IS A VECTOR OF 3000 VALUES EQUAL TO THE INDIVIDUAL VALUES FOR

* OM.

*14. 21 IS A VECTOR OF 100 VALUES EQUAL TO THE GROUP VALUES FOR PH.

*15. 22 IS A VECTOR OF 3000 VALUES EQUAL TO THE INDIVIDUAL VALUES FOR

* PS.

*16. g, 91 AND g2 ARE THE VARIANCE-COVARIANCES FOR THE THREE 2 VECTORS.

* THESE MATRICES SHOULD BE IDENTITY MATRICES.

*

CHOLOM - HALF(OM);

CHOLPH - HALF(PH);

CHOLPS - HALF(PS);

BEGIN: CIRCLE-CIRCLE+1;

A - J.(21300,1,0);

I - 1;

L: A(I,1)-NORMAL(SEED);

I-I+1;

IF I<- 21300 THEN GO TO L;

2 - a(1:3000,1)||a(3001:6000,l)||a(6001:9000,1);

Zl- a(21001:21100,1)||a(21101:21200,1)||a(21201:21300,1);

22-a(9001:12000,1)||a(12001:15000,1)||
a(15001:18000,1)||a(18001:21000,1)

TOTMI-NROW(Z)-1

TOTMIG-NROW(Zl)-l

C - (Z'*Z)#/TOTMI

Cl- (Zl'*Zl)#/TOTMIG

02- (22'*22)#/TOTMI

108

NOTE 'THESE ARE THE VAR-GOV OF THE RANDOM DATA (NO TRANS)’ ;
PRINT C C1 C2;

* e
* SECTION 1 - PART 3 ;
* e
* BY MULTIPLYING RANDOM DATA FROM A POPULATION WITH MEAN 0 AND VARIANCE;
* 0F 1 BY THE CHOLESKY OF A MATRIX, A VECTOR IS CREATED WHICH WILL ;
* RECREATE THAT MATRIX. -
'k

*17. 1 IS THE PRODUCT 0F 2 AND CHOLW. ,
*18. 11 Is THE PRODUCT 0F 21 AND CHOLB. ,
*19. 12 IS THE PRODUCT OF 22 AND CHOLERR. ;
*20. n, 21 AND 02 ARE THE VARIANCE-COVARIANCES FOR THE THREE Y VECTORS-;
* THESE MATRICES SHOULD BE CLOSE TO THE PARAMETER MATRICES. ,
* 8
Y - z * CHOLW;

o - (Y'*Y)#/TOTMI;

Yl- 21 * CHOLB;

D1 - (Y1'*Yl)#/TOTMIG;

Y2- 22 * CHOLERR;

D2- (Y2'*Y2)#/TOTMI;

NOTE 'THESE ARE THE VAR-COV OF THE TRANSFORMED DATA’ ;

PRINT o D1 D2 ;

* ;
* SECTION 1 - PART 4 ;
* ;
* BY MULTIPLYING VECTORS 2 AND 21 To L AND LA, THE OBSERVED VALUES FOR ;
* FOR EACH INDIVIDUAL ARE CREATED. INSTEAD OF THREE MEASURES PER ;
* INDIVIDUAL THERE WILL BE FOUR. (THE ERROR MATRIX WAS CREATED IN ;
* TERMS OF ERRORS FOR EACH OBSERVED VARIABLES AND IS ALREADY 4 x 4.);
* ;
*21. x IS THE PRODUCT OF Y AND L. ;
*22. x1 IS THE PRODUCT OF Y1 AND LA. ;
* ;
x - Y * L'

X1- Y1 * LA' ;

*

* SECTION 1 - PART 5

*

* BY ADDING VECTORS X, YYl AND Y2 TOGETHER, A TOTAL SCORE IS ACHIEVED
FOR EACH INDIVIDUAL. THESE SCORES OBVIOUSLY CONTAIN THE THREE
VARIANCE COMPONENTS. ALL 30 INDIVIDUALS IN EACH GROUP RECEIVE
THE SAME GROUP VECTOR(X1). OTHERWISE EACH RECEIVES A DIFFERENT
VALUE FROM BOTH X AND Y2.

36361-363!-

*23. 112 BECOMES A 3000 x 4 VECTOR WHICH REPEATS THE SAME X1 VALUE FOR
* N(I) TIMES FOR EACH GROUP.

*24. X BECOMES THE SUM OF X AND YYl AND Y2.

*25. FIN IS THE VARIANCE-COVARIANCE MATRIX FOR THE FINAL SET OF DATA-
* ITS A 4 X 4 MATRIX BASED ON 3000 OBSERVATIONS.

*26. HE IS A VECTOR OF SIZE K CONTAINING THE GROUP SIZE FOR EACH GROUP.
*27. ED REPLACES X AS THE MATRIX OF DATA. THIS IS USED IS OTHER

* SECTIONS.

109

MMhO;

II—l;

NN-l;

JJ: MHFMM+1;

CC-J.(PAT(II,1),1,1);

DD-(CC @ X1(NN.)) ;

YYliYYl//DD ;

NN-NN+1 ;

NKFNK//PAT(II,1) ;

IF MM LT PAT(II,2) THEN GO TO JJ;

"MFG; II-II+1 ;

IF NN LT PAT(+, 2) THEN GO TO JJ ;

Fl-(YY1'*YY1)#/NROW(YY1) °

NOTE 'THIS IS THE VAR- COV MATRIX OF GROUP DATA FOR ALL IND' ;
PRINT Fl ;

RD-X+YY1+Y2 ;

FIN- (RD'*RD)#/TOTMI

NOTE 'THIS IS THE VAR- COV MATRIX OF THE DATA TO BE USED' ;
PRINT FIN ;

FREE MM NN II X X1 Y Y1 Y2 D 01 D2 E E1 FIN I YYl Z 21 22 CC DD ;
FREE F1 C C1 C2 A EE EEl EE2 TOTE WITHIN BETWN ERR TOTMI TOTMIG ;

END OF SECTION 1

AT THIS POINT IT BECOMES IMPORTANT TO REALIZE THAT ALL THE LINES
ABOVE DEAL ONLY WITH CREATING THE DATA FOR THIS ANALYSIS. THEY
CAN BE DROPPED IN USING THE EM ALGORITHM. TO USE THE REST OF THE
PROGRAM WITHOUT THE PRIOR LINES, THE FOLLOWING LINES MUST BE PLACED
AT THE TOP OF THE PROGRAM (REMOVING THE * FROM THE FRONT - SEE SAS
FOR THE FETCH COMMAND)

Ski-*I-I-I-l-l-fl-X-H-l-

*PROC MATRIX
*FETCH RD
*FETCH LA

SECTION 2 - PART 1

THIS SECTION USES THE EM ALGORITHM TO GET ESTIMATES OF THE
UNRESTRICTED MODEL. THE BETWEEN AND WITHIN VARIANCE-COVARIANCE
MATRICES ARE ESTIMATED WITH NO STRUCTURE APPLIED. THIS FIRST PART
TURNS OUT THE SUFFICIENT STATISTICS FOR THE SAMPLE DATA NEEDED

IN PART 2 AND IN PART 3.

E NUMBER OF GROUPS IN THE SAMPLE.

E NUMBER OF OBSERVED VARIABLES IN THE SAMPLE.
E TOTAL NUMBER OF OBSERVATIONS IN THE SAMPLE.
KP
KP

I
I
I
VECTOR OF THE GROUP MEANS. _

X KP MATRIX OF EACH GROUP‘S SUM OF SQUARE/NK.

*fifl-fil’I-I-fi’fl-Il-M'fl-H-fi-I-

U|§UNH

K S TH
E S TH
H 8 TH
. 13 IS A
SS IS SA

.0 .0 .0 .0 ‘0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 ‘0 .0 .0 .0 .0 .0 '0 .0 .0 .0 .0 '0 .0 .0 .0 .0 .0

110

ZEROl-J.(NROW(L),NROW(L),O) 3
ZEROZ-J.(NCOL(L),NCOL(L),O) ;
ZERO3-J.(NCOL(LA),NCOL(LA),0) ;

NH-(l#/SUM(INV(DIAG(NK))))*K
PSI-((RD'*RD)-B1)*1#/(N-K)
PHI-(1#/NH)*((Bl-(RD(+.)'*RD(+.)*(1#/N)))*1#/(K-1)-PSI)
NOTE 'HERE ARE THE STARTING MATRICES'

PRINT PHI PSI

FREE Bl B s NH A ;

KFNROW(NK) , *NO OF CLASSES - K ;
P-NCOL(RD) ; *NO OF OBSERVED VARIABLES - P ;
NiNK(+,) ; *NO OF TOTAL INDIVIDUALS - N ;
GRP-J.(NROW(NK),1,1) ; *VECTOR OF 1'8, X X 1 ;

D-O ;

Bl-O ,

DO I-l TO K ;

C -D+l ;

D -C+NK(I,)-1 ;

A -RD(C:D,) ;

B -A(+.)*(1#/NK(I.)) ;

Ya -YM//B' ; *GROUP MEANS ;

E -(A'*A)*(1#/NK(I.)) ;

SS -SS//E ; *ss FOR EACH GROUP ;
Bl-((B'*B)*NK(I,))+B1 ; *SS/K OF THE GROUP MEANS ;

END :

* ;
* SECTION 2 - PART 2 ;
* e
* STARTING VALUES ARE NEEDED FOR THE BETWEEN GROUP COVARIANCE MATRIX ;
* PHI AND THE WITHIN COVARIANCE MATRIX PSI. THE MLE FOR EQUAL N'S ;
* WILL BE USED WITH NK (NUMBER OF STUDENTS IN A GROUP) REPLACED BY THE ;
* HARMONIC MEAN OF NK. ;
* ;
* 6. NH IS THE HARMONIC NK OF THE GROUPS. ;
* 7. PHI IS THE BETWEEN GROUPS COVARIANCE MATRIX. ;
* 8. PSI IS THE WITHIN GROUPS COVARIANCE MATRIX. ;
* ;

SECTION 2 - PART 3

THIS PART CREATES THE CONDITIONAL VALUES FOR THE MEAN AND GROUP
EFFECT FOR PHI AND PSI ESTIMATES. THERE ARE FOUR IMPORTANT VARIABLES'
CREATED HERE. THEY ARE CREATED IN SUBROUTINES ALPHAB AND ALPHAU. '
THIS IS PART OF THE INTERATIVE LOOP, THE E STEP.

10. TH IS A K X P MATRIX OF GROUP EFFECTS.
11. Q IS A KP X P WEIGHTING FACTORS FOR THE GROUPS.
12. W IS A P X P WEIGHTING FACTOR CALCULATED FROM THE Q'S.

*fl-l-fl-fl-fl-fl-fl-l'I-fl'fl-ll’

9. U IS A P X 1 VECTOR OF CONDITIONAL MEANS. ;

BUDDYEO

BUD:BUDDY-BUDDY+1

IF NROW(PAT) EQ 1 THEN LINK ALPHAB
ELSE LINK ALPHAU

111

D0 I-I To K

C -(P*I)-P+I

D -P*I

A -PHI*Q(C:D,)*(YM(C:D,)-U)
TH-TH//A

END

FREE A

at-

SECTION 2 - PART 4

THIS PART CALGULATES THE MAXIMUM LIKELIHOOD VALUES FOR PHI AND PSI
USING THE DATA AND THE CONDITIONAL VARIABLES. (M-STEP). THIS PROGRAM
WILL KEEP LOOPING TO THE LAST PART UNTIL THE DIFFERENCES IN PHI
AND PSI, AND THE NEW ESTIMATES OF PHI AND PSI ARE LESS THAN .01.

13. ONEl IS THE DIFFERNECE BETWEEN PHI ON THE LAST ITERATION AND
THE NEW ESTIMATES OF PHI.

l4. TWOl IS THE DIFFERNECE BETWEEN PSI ON THE LAST ITERATION AND
THE NEW ESTIMATES OF PSI.

036*3636‘36301-01"

E-J(P,P,0)

F—J(P,P,0)

DO I-l TO K

C -(P*I)-P+1

D -P*I

A -TH(C:D,)+U

E -E+NK(I,)*((SS(C:D,))-(YM(C:D,)*A')

-(A*YM(C:D,)')+(A*A')+(PHI*Q(C:D,)*
(PSI*(1#/NK(I.))+(W*Q(C:D.)*PHI-2*W)))) ;

F -F+Q(C:D.)-(Q(C:D.)*((YM(C:D.)-U)*(YM(C:D.)-U)'+W)*Q(C:D.)) ;

END ;

FREE A

PHI -PHI-(PHI*((I#/K)*F)*PHI)

ONEI-PHI-PHI

PSI -((1#/N)*E)+W

TWOI-PSl-PSI

PHID-DIAG(PHI)

PSID-DIAG(PSI)

PHID-PHID<>ZEROI

PSID-PSID<>ZEROI

PHI - PHI-DIAG(PHI)+PH1D

PSI - PSI-DIAG(PSI)+PSID

PHI-PHI

PSI-PSI

FREE PSI Q TH PHI PHID PSID

IF BUDDY GT 250 THEN GO TO FINAL;

IF MAX(ABS(ONE1)) LT 0.01 AND MAX(ABS(TWOI)) LT 0.01
THEN GO TO FINAL ;
ELSE GO TO BUD ;

.0 .0 .0 ‘0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0

FINAL:PRINT BUDDY PHI PSI ONEl TWOl U
FREE PSl ONEl TWOl U BUDDY

-0 -0 .0 .0 .0 .0 .0 .0 ‘0 .0 .0 -0 -0

112

END OF SECTION 2
SECTION 3 - PART 1

THIS SECTION USES THE E-M ALGORITHM TO GET ESTIMATES OF THE
RESTRICTED MODEL. PH, OM AND PS ARE ESTIMATED WITH STRUCTURE
APPLIED TO THE MODEL. THIS FIRST PART MAKES USE OF PHI AND PSI
FROM THE LAST SECTION TO GET OPENING ESTIMATES OF PH, OM AND PSI.

PH IS THE LATENT GROUP LEVEL VAR-COV.

OM IS THE LATENT IND LEVEL VAR-COV.

PS IS THE OBSERVED VARIABLES ERROR MATRIX.
8 IS THE DIMENSION OF PH.

. R IS THE DIMENSION OF OM.

3F$$$$$$$$$$3$$$$$$
UI§UNH

Y1-INV(L'*L) ;
Y2-INV(LA'*LA) ;
PHiY1*L'*PHI*L*Yl ;

OM-Y2*LA ' *PS I*LA*Y2 ;
PS-PSI+PHI-L*PH*L'-LA*OM*LA' ;

NOTE 'THESE ARE THE STARTING VALUES IN THIS STEP' ,

PRINT PH OM PS ;

S-NCOL(L) ; *NO OF LATENT CLASS VARIABLES - S ;
RPNCOL<LA) ; *NO OF LATENT IND VARIABLES -R ;

SECTION 3 - PART 2

THIS PART CREATES THE CONDITIONAL VALUES FOR THE MEAN AND GROUP
EFFECT FOR PH, OM, PS ESTIMATES. THERE ARE FOUR IMPORTANT VARIABLES'
CREATED HERE. THEY ARE CREATED IN SUBROUTINES BETAB AND BETAU.

THIS IS PART OF THE INTERATIVE LOOP, THE E STEP.

I A P X l VECTOR OF CONDITIONAL MEANS. ,
S A K X P MATRIX OF GROUP EFFECTS. ;
A P X P WEIGHTING FACTOR CALCULATED FROM THE Q'S. ,

S A VAR-COV OF THE IND LEVEL MATRICES, OM AND PS. ,

°PPFF
gnogc

S
I
S A KP X P WEIGHTING FACTORS FOR THE GROUPS.
S
I

l .

30303603900361-363601-0

BUDDYl-O ;

BUD1: MMP(LA*OM*LA'+PS) ;

BUDDYl-BUDDY1+1 ;

MFINV(MM) ; *INV OF WITHIN VARIANCES - P x P;

IF NROW(PAT) EQ 1 THEN LINK BETAB ;
ELSE LINK BETAU

DO I-l TO K

C-(P*I)-P+1

D-P*I ;

B-PH*L'*Q(C:D,)*(YM(C:D,)-U) ;

TH-TH//B ; *COND THETA - KP x P;

END ;

FREE B ;

113

SECTION 3 - PART 3

THIS PART CALCULATES THE MAXIMUM LIKELIHOOD VALUES FOR PH,OM AND PS
USING THE DATA AND THE CONDITIONAL VARIABLES. (M-STEP). THIS PROGRAM
WILL KEEP LOOPING TO THE LAST PART UNTIL THE DIFFERENCES IN PH
OM AND PS AND NEW ESTIMATES OF PH, OM AN PS ARE LESS THAN .01.

11. ONE IS THE DIFFERNECE BETWEEN PH ON THE LAST ITERATION AND
THE NEW ESTIMATES OF PH.

12. TWO IS THE DIFFERNECE BETWEEN OM ON THE LAST ITERATION AND
THE NEW ESTIMATES OF OM.

'13. THREE IS THE DIFFERNECE BETWEEN PS ON THE LAST ITERATION AND
THE NEW ESTIMATES OF PS.

*$**$*%*$*$$*fi-

VE— J(P,P,0) ;

B - J(P,P,0) ;

XX! J(P,P,0) ;

HH-LA*OM*LA' ;

II-LfiPH*L' ;

DO I-l TO K ;

C-(P*I)-P+l ;

D-P*I ;

CC-(S*I)-S+1 ;

DD-S*I ;

FF-LflTH(CC:DD,)+U ;

GG-Q(C:D.)*W*Q(C:D.) ;

KK-II+((1#/NK(I,))*HH) ;

VE-VE+(Q(C:D.)-Q(C:D.)*( (YM(C:D.)-U) * (YM(C:D.)-U)' - W )*Q(C:D.)) ;

AFMfiNK(I,)*(SS(C:D,)-(YM(C:D,)*FF')-(FF*YM(C:D,)')+(FF*FF'))*M ;

B-B+((l#/NK(I,))*GG+M*L$PH*L'*Q(C:D,)) + A ;

XXPXX+(PS*A*PS+(NK(I,))*((1#/NK(I,))*HH*M*HH-2*KK*Q(C:D,)*W

+10<*(GG-Q(C:D.))*KK)) ;

END ;

ONE--((1#/K)*(PH*L'*VE*L*PH)) ; *EST OF PHI (GROUP LEVEL) - S X S;

TWO-(OM*LA'*(B*(l#/N)-M)*LA*OM) ;

*EST OF OMEGA (IND LEVEL) - R X R;

THREE-DIAG(W-PS*M*PS+II+(I#/N)*XX) ;

PH-PH+ONE ; *EST OF PHI (GROUP LEVEL) - S X S;

OMhOM+TWO ;

PS-DIAG(PS+THREE)

PHD-DIAG(PH)<>ZERO2

OMD-DIAG(OM)<>ZERO3

PSD-DIAG(PS)<>ZEROI

PH-PH-DIAG(PH)+PHD

OM-OM-DIAG(OM)+OMD

PS-PS-DIAG(PS)+PSD

FREE TH Q W B VE XX B II

IF BUDDYI GT 250 THEN GO TO FINALI

IF MAX(ABS(ONE)) LT 0.01 AND MAX(ABS(TWO)) LT 0.01 AND
MAX(ABS(THREE)) LT 0.01 THEN GO TO FINALI ;
ELSE GO TO BUDl ;

FINALI: PRINT BUDDYI PH OM PS ONE TWO THREE SEED ;

PRINT U ;

.0 .0 -0 .0 .0 .0 .0 -0 .0 .0 .0 .0 ‘0 ‘0

FREE NK RD YM 88 N K R S

'114

IF CIRCLE LT 2 THEN GO TO BEGIN

STOP

*

* HERE ARE THE SUBROUTINES
*

* ALPHAU

*

ALPHAU: TOT -J(P,P,O)
DO I-l TO K

Tl -INV((PSI*(1#/NK(I,)))+PHI)

TOT -TOT+T1
Q -Q//T1
END

W -INV(TOT)

U -W*Q'*YM

FREE T1 TOT

RETURN

*

* ALPHAB
*

ALPHAB: Tl -INV((PSI*(l#/NK(1,)))+PHI)

Q -GRP @ Tl
W -INV(T1*K)
U -W*Q'*YM
FREE Tl
RETURN

*

* BETAU
*

BETAU: W - J(P,P,0)
DO I-l TO K

A-INV( (L*PH*L') + (MM*(l#/NK(I.))) )

Q-Ql/A
WhW+A

END

FREE A
WhINV(W)
U-W*Q'*YM
RETURN

*

* BETAB
*

BETAB: A!INV( (LRPH*L')
O -GRP @ A

W -INV(A*K)

U-W*Q'*YM

RETURN

*

+ (MM*(1#/NK(1.)))

' *MATRIX OF Q

*COND VAR FOR U

' *COND U

' *COND U

)

-I(PXP;

"U'U

$3!-*$**$****$*$**fl-363636363636361-1-1-3636000001-0

PROC MATRIX
SEED - 101997
CIRCLE - O
PAT - 30 100

APPENDIX D

COMPUTER PROGRAM FOR THE ESTIMATION OF THE FOUR PARAMETER

LATENT MODEL IN SAS

SECTION 1 - PART 1

THIS SECTION CREATES THE SAMPLE DATASET FOR USE IN THE E-M ALGORITHM
STEPS. EACH DATA POINT CONSISTS OF THREE COMPONENTS, LATENT WITHIN
(OM), LATENT BETWEEN (PH) AND ERROR (PS). THE OBJECT IS TO USE
PATTERN MATRICES L AND LA TO CONVERT THE 3 X.3 LATENT MATRICES INTO
4 X 4 MATRICES OF OBSERVED VALUES. THE ERROR MATRICES ARE ALWAYS

4 X 4 MATRICES OF MEASUREMENT ERRORS OF THE OBSERVED VALUES.

THE NOMENCLATURE USED IN THIS PROGRAM IS THE SAME AS THAT
IN THE PROGRAM IN APPENDIX 3. REFER TO APPENDIX 3 FOR
DEFININTIONS.

l. SEED IS ANY RANDOM NUMBER USED TO CREATE RANDOM VALUES FROM A
RANDOM GENERATOR (NORMAL).

USED IN STUDY
lQQ_§EQy£S - 26298, 27309, 49329, 93369, AND 181449

3. RAT IS A Z X 2 MATRIX OF THE NUMBER OF STUDENTS IN THE
GROUPS. NOl HAS THE NUMBER OF SUBJECTS IN GROUPS - N02 HAS THE
NO OF GROUPS OF THAT SIZE.

FOR UNBALANCED 100 GROUPS: FOR BALANCED 100 GROUPS:
PAT-10 20/20 20/30 20/40 20/50 20; | PAT-3O 100;

4. QM IS THE PARAMETER OF THE WITHIN COVARIANCE MATRIX OF THE
POPULATION.

5. 23 IS THE PARAMETER OF THE BETWEEN COVARIANCE MATRIX OF THE
POPULATION.

6. PSI IS THE PARAMETER OF THE WITHIN ERROR COVARIANCE MATRIX OF
THE POPULATION.

6. 2S2 IS THE PARAMETER OF THE BETWEEN ERROR COVARIANCE MATRIX OF
THE POPULATION .

.0 .0 .0 .0

0a - 25 10 15/ 10 20 10/ 15 10 35;
PH - 64 8 40/ 8 5 7/ 40 7 107;

Ps1 - 5 0 0 0/ 0 6 0 0/ 0 0 11 0/ 0 0
Ps2 - 7 0 0 0/ 0 8 0 0/ 0 0 10 0/ 0 0

0
0

F‘P‘
ran:

115

.0 -0 -0 -0 U0 .0 .0 -0 -0 -0 -0 ‘0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 ‘0 .0 .0 ‘0 .0 .0 '0 .0 .0 ‘0 -0

116

PRINT OM PH P81 P82 ;

L - 1 0.5 0.5/
1 0.5 -0.5/
1 -0.5 0.5/
1 -0.5 -0.5;
LA- 1 0.5 0.5/
1 0.5 -0.5/
1 -0.5 0.5/
1 -0.5 -0.5-

EOM - L#0M*L'+PSl ;

EPH - LA*PH*LA'+PS2 ;
PRINT EOM EPH ;
'A'

* SECTION 1 - PART 2

*

* FOUR DIFFERENT VECTORS OF DATA ARE NEEDED, ONE FOR PH, ONE FOR OM

* ONE FOR PS1 AND ONE FOR P82. THESE ARE INDEPENDENT RANDOM VARIABLES.

* FOR LATER USE THE CHOLESKYS OF OUR PARAMETER MATRICES ARE NEEDED.

*

CHOLOM - HALF(OM) ;

CHOLPH - HALF(PH) ;

CHOLPSI - HALF(PSl) ;

CHOLPS2 - HALF(P52) ;

BEGIN: CIRCLE-CIRCLE+1 ;

A - J.(21700,1,0);

I-l;

L: A(I,l)-NORMAL(SEED);

I-I+1;

IF I<- 21700 THEN GO TO L;

z - a(l:3000,1)||a(3001:6000,1)||a(6001:9000,l) ;

21- a(21001:21100,1)||a(21101:21200,1)||a(21201:21300,1) ;

22-a(9001:12000,l)||a(12001:15000,l)
||a(15001:18000,1)||a(18001:21000,1) ;

Z3-a(21301:21400,l)l|a(21401:21500,1)
||a(21501:21600,1)||a(21601:21700,1)

TOTMI-NROW(Z)-l

TOTMIG-NROW(Zl)-l

*

* SECTION 1 - PART 3

*

* BY MULTIPLYING RANDOM DATA FROM A POPULATION WITH MEAN 0 AND VARIANCE;
* OF 1 BY THE CHOLESKY OF A MATRIX, A VECTOR IS CREATED WHICH WILL ;
* RECREATE THAT MATRIX. ;
* ;

Y - Z * CHOLOM ,
Yl- Zl * CHOLPH ,
Y2- 22 * CHOLPSl ;
Y3- 23 * CHOLPS2 ;
*

* SECTION 1 - PART 4

'k

* BY MULTIPLYING VECTORS Z AND 21 TO L AND LA, THE OBSERVED VALUES FOR
* FOR EACH INDIVIDUAL ARE CREATED. INSTEAD OF THREE MEASURES PER

117

* INDIVIDUAL THERE WILL BE FOUR. (THE ERROR MATRIX WAS CREATED IN

*
X - Y * L' ,
X1- Y1 * LA' ,
Xl-XI+Y3 ;
X2-X +Y2 ;

FOR EACH INDIVIDUAL.
VARIANCE COMPONENTS.

I'M-#0363639!!-

MMhO;

II-l;

NN-I;

JJ: HHFHH+1;
CC-J.(PAT(II,1),1,1);
DD-(CC @ X1(NN,)) ;
YYldYYl//DD ;
NN-NN+1 ;
NKFNK//PAT(II,1) ;

IF MM LT PAT(II,2) THEN GO TO JJ
“MFG; II-II+1
IF NN LT PAT(+,2) THEN GO TO JJ

FREE MM NN II X1 Y Y1

FREE A OM PH P81 P82 TOTMIG

RD—X+YY1+Y2
FIN— (RD ' *RD)#/TOTMI

FREE X Y2 FIN YYl TOTMI Z 21 Z2 Z3

END OF SECTION 1

FOR THE FETCH COMMAND)

03000360000003!-

*PROC MATRIX
*FETCH RD
*FETCH LA
*FETCH L

*FETCH NK
*

*

SECTION 1 - PART 5

BY ADDING VECTORS X1 AND X2 TOGETHER, A TOTAL SCORE IS ACHIEVED

THESE SCORES OBVIOUSLY CONTAIN THE FOUR
ALL INDIVIDUALS IN EACH GROUP RECEIVE

THE SAME GROUP VECTOR (X1) AND A DIFFERENT VALUE FROM X2.

I Z 21 Z2 CC DD

AT THIS POINT IT BECOMES IMPORTANT TO REALIZE THAT ALL THE LINES
ABOVE DEAL ONLY WITH CREATING THE DATA FOR THIS ANALYSIS. THEY
CAN BE DROPPED IN USING THE EM ALGORITHM. TO USE THE REST OF THE
PROGRAM WITHOUT THE PRIOR LINES, THE FOLLOWING LINES MUST BE PLACED
AT THE TOP OF THE PROGRAM (REMOVING THE * FROM THE FRONT - SEE SAS

* TERMS OF ERRORS FOR EACH OBSERVED VARIABLES AND IS ALREADY 4 X 4.);

.0 .0 .0 .0 .0 -0 ‘0 .0 .0 ‘0 '0 .0 .0 .0 .0 .0 -0 .0

118
SECTION 2 - PART 1

THIS SECTION USES THE EM ALGORITHM TO GET ESTIMATES OF THE
UNRESTRICTED MODEL. THE BETWEEN AND WITHIN VARIANCE-COVARIANCE
MATRICES ARE ESTIMATED WITH NO STRUCTURE APPLIED. THIS FIRST PART
* TURNS OUT THE SUFFICIENT STATISTICS FOR THE SAMPLE DATA NEEDED
* IN PART 2 AND IN PART 3.

*

;ZEROI - J.(NROW(L),NROW(L),0) ;
ZERO2 - J.(NCOL(L),NCOL(L),0) ;
ZER03 - J.(NCOL(LA),NCOL(LA),0) ;
ZEROl - DIAG(ZEROI) ;
ZER02 - DIAG(ZER02) ;
ZER03 - DIAG(ZERO3) ;
KPNROW(NK) ° *NO OF CLASSES - K
P-NCOL(RD) *NO OF OBSERVED VARIABLES - P
NhNK(+,) *NO OF TOTAL INDIVIDUALS - N
GRP-J.(NROW(NR),1,1) *VECTOR OF 1'8, X X 1

S-NCOL(L) *NO OF LATENT CLASS VARIABLES - S
RFNCOL<LA) *NO OF LATENT IND VARIABLES -R
D-O

B -J(P,P,0)

Bl—J(P,P,0)

DO I-l TO K

C -D+1

D -C+NK(I,)-l

A -RD(C:D,)

AZ-A(+.)*(1#/NK(I.))

Yap YM//A2' ; *GROUP MEANS

EE - (A'*A)

SS-SS//(EE*(1#/NK(I.)))

B -B+EE-NK(I,)*A2'*A2

Bl-((A2'*A2)*NK(I,))+B1 ; *SS/K OF THE GROUP MEANS

$36-$36?!-

‘0 .0 .0 .0 ‘0 -0 .0 .0 '0 .0 .0 .0 ’0 -0 ‘0 .0 ‘0 '0 '0 ‘0

5

*
* SECTION 2 - PART 2
*
*

STARTING VALUES ARE NEEDED FOR THE BETWEEN GROUP COVARIANCE MATRIX
* PHI AND THE WITHIN COVARIANCE MATRIX PSI. THE MLE FOR EQUAL N'S
* WILL BE USED WITH NR (NUMBER OF STUDENTS IN A GROUP) REPLACED BY THE
* HARMONIC MEAN OF NR.
*
;NH-(1#/SUM(INV(DIAG(NK))))*K ;
PSI-((RD'*RD)-Bl)*1#/(N-K) ;
PHI-(1#/NH)*((Bl-(RD(+.)'*RD(+.)*(1#/N)))*1#/(K-1)-PSI) ;
FREE s NH A ;
'k

* SECTION 2 - PART 3
*

* THIS PART CREATES THE CONDITIONAL VALUES FOR THE MEAN AND GROUP ;

* EFFECT FOR PHI AND PSI ESTIMATES. THERE ARE FOUR IMPORTANT VARIABLES
* CREATED HERE. THEY ARE CREATED IN SUBROUTINES ALPHAB AND ALPHAU.

* THIS IS PART OF THE INTERATIVE LOOP, THE E STEP.
*

119

BUDDY-0
BUDzBUDDYhBUDDY+l
IF NROW(PAT) EQ 1 THEN LINK ALPHAB
ELSE LINK ALPHAU
E-J(P,P,0)
FhJ(P,P,0)
*
* SECTION 2 - PART 4
*
* THIS PART CALCULATES THE MAXIMUM LIKELIHOOD VALUES FOR PHI AND PSI
* USING THE DATA AND THE CONDITIONAL VARIABLES. (M-STEP). THIS PROGRAM
* WILL KEEP LOOPING TO THE LAST PART UNTIL THE DIFFERENCES IN PHI
* AND PSI, AND THE NEW ESTIMATES OF PHI AND PSI ARE LESS THAN .01.
*
DO I-1 TO K ,
C -(P*I)-P+l ,
o -P*I ;
G-Q(C=D.)*(W+(YH(C=D.)-U)*(YM(CID.)-U)')*Q(C=D.)-Q(C=D.) ;
s -E+1#/NK(I,)*G -

F -F+G ,
END .
FREE A ,
EE-(B-(N-K)*PSI) ,
ONEl-(PHI*((1#/K)*F)*PHI) ,
PHl -PHI+ONE1 ,
TWOl-((l#/N)*(PSI*E*PSI+EE)) ,
PSl -PSI+TWOl ;
PHlD-DIAG(PH1) ;
PSlD-DIAG(PSl) ,
PHlD-PHlD<>ZEROl ,
PSlD-PSlD<>ZEROl ,
PHI - PHl-DIAG(PH1)+PH1D ,
PSI - PSl-DIAG(PSl)+PSlD ,
FREE PSl Q W PHl PHlD PSlD G E F EE ,
IF BUDDY GT 25 THEN GO TO FINAL;

IF MAX(ABS(ONE1)) LT 0.01 AND MAX(ABS(TWOl)) LT 0.0l

THEN GO TO FINAL
ELSE GO TO BUD
FINALzPRINT BUDDY PHI PSI

FREE PSl ONEl TWOl U BUDDY
*

END OF SECTION 2

SECTION 3 - PART 1

I'M-0*!-

* THIS SECTION USES THE EM ALGORITHM TO GET ESTIMATES OF THE
RESTRICTED MODEL. PH, OM, PSl AND P52 ARE ESTIMATED WITH STRUCTURE
* APPLIED TO THE MODEL. THIS FIRST PART MAKES USE OF PHI AND PSI FROM
* THE LAST SECTION TO GET OPENING ESTIMATES OF PH, OM, PSl AND PS2.

3'.

.0 .0 ‘0 -0 .0 -0 .0 .0 -0 -0 -0

120

Y1-INV(L'*L)
Y2-INV(LA'*LA)
PHin*L'*PHI*L*Yl
OM-Y2*LA'*PSI*LA*Y2
PSl-PSI-LA*OM*LA'
PS2-PHI-L*PH*L'
*NOTE 'THESE ARE THE STARTING VALUES IN THIS STEP' ;

*PRINT PH OM PSl ;
*

* SECTION 3 - PART 2
'A'
* THIS PART CREATES THE CONDITIONAL VALUES FOR THE MEAN AND GROUP
* EFFECT FOR PH, OM, PSI AND PS2. THERE ARE FOUR IMPORTANT VARIABLES
* CREATED HERE. THEY ARE CREATED IN SUBROUTINES BETAB AND BETAU.
* THIS Is PART OF THE INTERATIVE LOOP, THE E STEP.
*
BUDDYI-O
BUD1: MMP(LA*OM*LA'+PSI)
AA-(L*PH*L'+P82)
BUDDYl-BUDDY1+1
MPINV(MM) ; *INV OF WITHIN VARIANCES
IF NROW(PAT) EQ 1 THEN LINK BETAB

ELSE LINK BETAU

.0 .0 .0 '0 .0 .0

P X P

* SECTION 3 - PART 3

*

* THIS PART CALCULATES THE MAXIMUM LIKELIHOOD VALUES FOR PH OM PSI P82
* USING THE DATA AND THE CONDITIONAL VARIABLES (M-STEP). THIS PROGRAM
* WILL KEEP LOOPING TO THE LAST PART UNTIL THE DIFFERENCES IN PH, OM,
* PSI AND PS2 AND THEIR NEW ESTIMATES ARE LESS THAN .01.

*

DO I-l TO K ;

C-(P*I)-P+1 ;

D-P*I

LV-PH*L' *Q(C: D, )*(YM(C: D, ) -U) ;

'I'H-T'W/Lv ;

END ;

FREE LV ;

CVE- J(P,P,0) ;

BVE- J(P,P,0) ;

DO I-l TO K ;

C-(P*I)-P+l ;

D-P*I ;

CC-(S*I)-S+l ;

DD-S*I ;.

Z-LiTH(CC: DD, )+U

AVE-NR(I, )*M*(SS(C: D, )- (YM(C: D, )*Z' )- (Z*YM(C: D, )' )+(Z*Z' ))*M ;
BVE-BVE+(Q(C: D, )*( (YM(C: D, ) -U) * (YM(C: D, ) -U)' +W)*Q(C: D, )- -Q(C:D,) );
CVE-I#/NK(I,)*( Q(C:D,)*W*Q(C:D,)-Q(C:D,) ) +CVE +AVE ;

END ;
E-(N-K)*M ;
ONE-((1#/K)*(PH*L'*BVE*L*PH)) ;
TWO-((1#/N)*OM*LA'*(CVE-E)*LA*OM) ;
THREE-DIAG((l#/K)*(PSZ*BVE*PSZ)) ;

121

FOURPDIAG((l#/N)*(PSl*(CVE-E)*PSl))

PH-PH+ONE

OMhOM+TWO

PSl-DIAG(PSl+FOUR)

PSZ-DIAG(P82+THREE)

PHD-DIAG(PH)<>ZER02

OMD-DIAG(OM)<>ZERO3

PSAFDIAG(PSl)<>ZEROl

PSB—DIAG(P82)<>ZEROI

PH-PH-DIAG(PH)+PHD

OMFOM-DIAG(OM)+OMD

PSl-PSl-DIAG(PSl)+PSA

PSZ-PSZ-DIAG(PS2)+PSB

FREE Q W CVE AVE BVE TH

IF BUDDYl GT 250 THEN GO TO FINALl

IF MAX(ABS(ONE)) LT 0.01 AND MAX(ABS(TWO)) LT 0.01 AND
MAX(ABS(THREE)) LT 0.01 AND
MAX(ABS(FOUR)) LT 0.01 THEN GO TO FINALl
ELSE GO TO BUDl

FINALl: PRINT BUDDYl PH OM P81 P82

FREE NK RD YM SS N K R 8

IF CIRCLE LT 3 THEN GO TO BEGIN

PRINT SEED

STOP

*

* HERE ARE THE SUBROUTINES
*

* ALPHAU

*

ALPHAU: TOT -J(P,P,0)

DO I-1 TO K

T1 -INV((PSI*(1#/NR(I,)))+PHI)
TOT -TOT+T1

Q -Q//'1'1

END

W -INV(TOT)

U -W*Q'*YM

FREE T1 TOT

RETURN

*

* ALPHAB

*

ALPHAB: T1 -INV((PSI*(1#/NK(1,)))+PHI)
Q -GRP @ T1

W -INV(T1*K)

U -W*Q'*YM

FREE T1

RETURN

*

* BETAU
*

BETAU: W - J(P,P,O)
DO I-l TO K

‘0 -0 .0 .0 .0 .0 ‘0 -0 -0 .0 .0 ‘0 .0 .0 .0 .0 ‘0 ‘0 -0 -0 .0 ‘0 .0 .0 -0 -0 -0 .0 .0 .0 .0 -0 .0

.0 .0 .0 -0 -0 .0 .0 .0 .0 .0 -0 -0 ‘0 .0 .0

122

A~INV( AA + (MM*(1#/NK(I.))) ) ;
Q—Q//A ; *MATRIX OF Q - KP X P;
W-W+A ;
END

FREE A

W-INV(W) ; *COND VAR FOR U -
U-W*Q ' *YM ; *COND U -
RETURN

*

* BETAB

*

BETAB: A.INV( AA + (MM*(1#/NK(1.))) )
Q -GRP @ A

w -INV(A*K)

U-w*Q'*YM ; *COND U - P X P;
RETURN :
*
ul-

"U'd
NM
'U’U

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