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ESTIMATION OF CORRELATIONS BETWEEN [；

TRUNCATED CONTINUOUS AND POLYTOMOUS VARIABLES

by I Wai-ehung LUI [•

A 1::

Thesis |

submitted to

(Division of Statistics)

the Graduate School

The Chinese University of Hong Kong 丨

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of the Requirements for the Degree of |

Master of Philosophy |

(M Phil. )

I June, 1994 |

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THE CHINESE UNIVERSITY OF HONG KONG

GRADUATE SCHOOL

The undersigned certify that we have read a thesis, entitled

"Estimation of Correlations between Truncated Continuous and Polytomous

Variables" submitted to the Graduate School by Lui Wai C h u n g ( 當辟愈）

in partial fulfillment of the requirements for the degree of Master of

Philosophy in Statistics. We recommend that it be accepted.

Dr, S. Y. Lee

Supervisor

u) - y', ^ — ,

Dr. W. Y. Poon

Supervisor

Dr. K. H. Li

Dr. S. Y' Cheung

Prof. P. C. Chang

External Examiner

DECLARATION

No portion of the work referred to in this thesis has been

submitted in support of an application for another degree or

qualification of this or any other university or other institute of

learning.

ACKNOWLEDGEMENT

I would like to express my deep gratitude to my supervisors, Dr.

Sik-yum Lee and Dr. Wai-yin Poon, for their supervision and

encouragement during the course of this research program. It is also a

pleasure to extend my gratitude to all the staff of the Department of

Statistics, especially to Mr. Michael Leung and Dr. P. S. Chan for their

kind assistance.〜

ABSTRACT

In this thesis, a method of estimating correlations for the

model with truncated continuous and polytomous variables is developed.

First, maximum likelihood method is used for estimation with one

continuous truncated variable and one polytomous variable. The model is

then extended to several polytomous variables. To avoid heavy

computational time in obtaining these maximum likelihood estimates, the

Partition Maximum Likelihood method is proposed. The asymptotic

properties of the estimates are also studied. Finally, the

computational aspects is described and a simulation study is conducted

to investigate the performance of the estimates.

CONTENTS

Page

Chapter 1 Introduction. 1

Chapter 2 Estimation of the model with one truncated

continuous variable and one polytomous variable 6

§ 2.1 The model

§ 2.2 Likelihood function of the model

§ 2.3 -Derivatives of F(侈）

§ 2.4 Asymptotic properties of the model

Chapter 3 Estimation of the model with one truncated

continuous variable and several polytomous variables.... 22

§ 3.1 The model

§ 3.2 Partition Maximum Likelihood (PML) estimation

§ 3.3 Asymptotic properties of the PML estimates

Chapter 4 Optimization procedures and Simulation study 43

§4.1 Optimization procedures

§ 4.2 Simulation study

Chapter 5 Summary and Conclusion. ,54

Tables 56

References 76

Chapter 1. Introduction

Most of the current statistical literature on sampling

concerns unrestricted samples. In most real-life situations, however,

researchers are likely to find that their samples are ,truncated,.

Suppose we have a random sample with size n. The data are observed only

if its value is greater and/or smaller than a pre-assigned value, and it

is missed otherwise. Someone use the term ’truncated, to describe this

kind of data while someone use the the term 'censored, instead. In this

thesis, we use the term ' truncated' . Example can be found in a life

test, where the experimenter decides to stop the experiment before all

of the units on test have failed. Truncation is one kind of data

missing pattern which is non-ignorable which leads us to treat it

carefully. Moreover, truncation is a common topic since applications

can be found in quality control, life testing, biometrics, economics,

business, agronomy, manufacturing, engineering, medical and biological

sciences, management sciences, social sciences, and most areas of the

physical sciences.

Truncated samples with unknown sample size were first

encountered quite early in the development of modern statistics by Sir

Francis Galton (1897). His objective was to test the suitability of

trotting records, provided by the Wallace Year Book, Vols 8-12

(1892-1896), a publication of the American Trotting Association.

Afterwards, Karl Pearson (1902), Pearson and Lee (1908), R. A. Fisher

1

(1931, 1936) gave more theoretical analysis of the truncated samples

with unknown sample size. Later on, Stevens (1937) considered the

estimation of the mean and standard deviation from truncated sample with

known sample size in normal distribution. He applied the results to the

truncated time-mortality curve. Cohen (1950) used the method of maximum

likelihood to estimate the parameters of normal populations from singly

and doubly truncated samples with known sample size. In four later

papers Cohen (1955, 1957, 1959, 1961) extended the results given in his

1950 papers. More examples can be found in Ha Id (1949), Gupta (1952),

Hartar and Moore (1966), Schineider (1986) and Cohen (1991).

When continuous latent variables are only observable in

categorical form, they are called polytomous variables. In many

applications, particularly in behavioral and social science,

investigators frequently encounter dichotomous or polytomous data. For

instance, in behavioral studies, a subject is often asked to answer the

question on scale like

approve approve don't know disapprove disapprove • strongly strongly

It is an example in which a continuous variable underlies a polytomous

observed variable. When analyzing such variable, some statisticians may

assign integer value to each category and proceed in the analysis as if

the data had been measured on an interval scale with the desired

2

distribution. Many statistical methods seem to be robust against such

deviation from the distributional assumption, however, sometimes it may

lead to erroneous result. Olsson (1979a) showed that due to the biased

estimates of correlation, the application of factor analysis to such

kind of discrete data will lead to erroneous conclusions. Hence, as

expected, the applications of principal component analysis, multiple

correlations, canonical correlation analysis and structural equation

models may also lead to incorrect result because these statistical

methods may also depend largely on the estimation of correlation. So,

it needs to develop a method to estimate the "true" polychoric

correlation coefficients which are more reliable.

Assuming the normality of the underlying distribution, Pearson

(1901) introduced the tetrachoric correlation coefficient to estimate

the true correlation from a 2x2 contingency table. Lancaster and Hamdan

(1964) extended it to the polychoric case. Martinson and Hamdan (1971)

developed a two-step maximum likelihood method to estimate the

polychoric correlation coefficients. In their method, the thresholds

are first estimated by cumulative marginal proportions, and then the

polychoric correlation is estimated with the thresholds fixed at their

estimates. Olsson (1979b) developed the full maximum likelihood

approach to estimate the correlation and thresholds. Lee and Poon

(1986) extended the model to p-dimensional contingency table and used

the generalized least squares estimation. Statistical methods based on

different assumptions in analyzing polytomous data have been developed.

3

Examples are Lee, Poon & Bentler (1990, 1992), Poon and Lee (1992), Poon

and Leung (1993). The analysis of polytomous data related to missing

data was encountered by Lee & Chiu (1990). F u r t h e r m o r e， L e e & Tang

(1992) studied the estimation of polychoric and polyserial correlation

with incomplete data.

The main purpose of this thesis is to develop a method to

estimate the parameters in the model with truncated continuous variable

as well as polytomous variables. These parameters include the

correlations among the variables and the thresholds of the polytomous

variables. In Chapter 2, we treat the model with one truncated

continuous and one polytomous variable, and the method of maximum

likelihood is proposed. Asymptotic properties in this model are also

given. In Chapter 3, an extended model with one truncated continuous

and several polytomous variables is considered. To avoid the heavy

computational time in evaluating the multivariate distribution

functions, the Partition Maximum Likelihood (PML) estimation is used

(see, Poon and Lee 1987). The idea is to divide the r-dimensional model

into r(r-l)/2 sub-models to obtain Partition Maximum Likelihood

estimates. Statistical properties of the Partition Maximum Likelihood

estimates are also established. Chapter 4 describes the computational

aspects of the estimators. In order to find the estimated value, the

modified Davidon-Fletcher-Powell (DFP) algorithm as well as Fisher

Scoring algorithm, which are iterative optimization methods, are used.

In the latter part of Chapter 4, it describes a simulation to study the

4

behaviour of our estimates. The summary and conclusion are given in

Chapter 5.

•

‘

�

�

5

Chapter 2. Estimation of the model with one truncated continuous

variable and one polytomous variable.

2.1 The model

Let X, Y be random variables. Assume that (X, Y)' distributes

as bivariate normal with mean vector \x and correlation matrix P = (〜），

(i,j = 1,2) • Since our main concern is to estimate the correlation of

the model, without loss of generality, and for simplicity, y is assumed

to be a zero vector in the following passage. Also, note that p^^ = P2 2

= l , and let p ^ = p which is what we interest in.

Suppose that the random variable X is continuous and is right

truncated at a known truncation point ， say c. That is, we only observe

those X-values which are less than or equal to c, and miss those

X-values which are greater than c.

Moreover, suppose that the random variable Y cannot be

observed directly. We can only observe it through a polytomous random

variable Z, which is defined as

Z = k if a, ^ Y < a, , (2.1)

k k+1

for k = 1，...，h w i t h 、 = - ⑴ ； ah + 1

= +oo.

6

We call { a , a , ah + 1

> the thresholds of Z. Note that

these thresholds are also unknown except a^ and .

Let 分 be the parameter vector in this model, then the

dimension of is h, and it is given by,

= { p’ a2, ... , a

h }, (2.2)

Now, suppose we have a random sample from ( X ， Z ), with

sample size n. Also assume that m of these n vectors having observed

X-value. Without loss of generality, let the last (n-m) X’ s ’

...’ X be truncated and denote X . = ( X X )，. Also let

z = ( Z , … ’ Z )，be the corresponding observed polytomous data, -mis m+1 n

Also denote XQ b s

= ( Xr X

2 X

m ), and Z

Q b s = { Z ^ . . .

7 Z

m ),

Note that the number of observed and missing X-value, m and n-m

respectively, are known after the sample is drawn. Moreover, XQ b s >

Z , Z are the observations which are available while ^ are those ~obs ~mis 〜mis

truncated X—value which are greater than c. Denote X = ( )’

and Z = ( Z' , Z>

. ) ， .

2.2 Likelihood function of the model

In this section, we will derive the likelihood function of

the model. Suppose L (侈 | X, Z ) be the likelihood function, then it

can be expressed as the following.

7

I j

<x f (芒，g丨它）

~obs ~mis ~obs 〜mis 〜

= f ( X [ , Z l I 侈）• f( X • , Z • I 侈）、-obs ~obs ~ -mis -mis ~

= f ( Z I X , 旮） • f C X ^ 丨旮）丨钞） ~obs

1

-obs, ~ ~obs' ~ -mis' -mis ~

(2.3)

We have decomposed the likelihood function in (2.3) into three

parts. We shall discuss these three parts separately as the following.

Part I.

Consider the first term f( Z , | X , , ) in the likelihood

~ODS ~ODS ~

function. Let 〜 b e the number of observation in ZQ b s

= ( ..., Z ^ )，

that are equal to k ( k = 1, ... , h ). S o , 、 + n2 + . . . + 〜 = m .

(•’\ J. T_ Also denote X, be the i element among the n. observations such that

k K th

the corresponding polytomous variable Z is in the k category ( that is

Z = k ) . Then by the independency of the observations,

f( Z , I X , , ) ~obs ~obs 〜

- f ( z., •.., Z I X ” . . . ， X , 办 ) 1 m l . m -

m

= y ] f( Z I X 办)

j=l J J

~

8

= n Sk

P“ 〜 = k I X p ) ) k=l i=l

(2.4)

y

By our assumption, (X, Y) has a bivariate normal

distribution. By standard normal properties, the conditional

distribution of Y given X ,say Y| is given by Y lx = N( pX, 1-p 2 ) where

’ S ， denotes ' is distributed as ’.

Therefore, for any i=l, ...,n and k=l, ...， h

Pr( Z . = k 丨 x j1

) ) l k

二 Pr( Y. < ak + 1

I 亡）)

^ r a

k+i ) . r \ ^

p X

k1 }

1 = $ - $

I (1 - p2

)1 / 2

J ^ (1- P2

)1 / 2

^

= $ ( a 二 ^ ) 一少（〜：丄)

$ ( a *. ) if k=l r

^»i

-$ ( a

k ! l , i }

- $ (

V i } i f

1 米

1 - $ ( a^ . ) if k=h h, l

(2.5)

v( i )

r t

1 ,2

来 a - p . Xk | 1 t

where a. • = — a n d = r 7 7 5

~.exp( ) dt

k’1

(1 - p2

)1 / 2

. (2tt) 1 / 2

2 -co

9

which is the distribution function of standard normal N(0,1) . Also

来米

note that a , . ) = 0, a, • ) =1. So, we get 1,1 h+1,l

f( ？obs 丨？obs' ~ )

h n

k r 来米 -

= n n $

( a

k +i , i ) 一

$ (

a

k , i ) k=l i=l

L

' J

•

( 2 . 6 )

Part II.

For the second term f( X , | ^ ), by the independency of the

~obs ~

observations, it can be seen that

f (

w ？ )

m

= n f

( x

i I ? ) i=l

m —1/2. 1 2 = ( 2 7 1 ) • exp{ - X

i >

i=l 已

L

m

= ( 2 T T ) 'm / 2

• exp{ - X .2

} •

i=l

(2.7)

Part III.

米

For the third term f( X • ， Z . | ^ ), let n, be the number

~mis ~mis 〜 k

10

of observations in Z • = (Z Z )' that are equal to k

〜mis m+1 n 来来来

(k=l,...,h). So n1 + n

2 +. . . + n ^ = n-m .

来

Consider that for any i = 1 〜 a n d for any k = l,...,h ,

Pr { Z . = k and X . > c } l l

= P r { ak ^ Y ^ < a

k + 1 and X ^ > c >

r V i 「 +0 0

= 02( x , y; p) dx dy

J

an

J

c k

( 2 . 8 )

1 f

2 2 、 , 、 I x - 2pxy + y i

where 0o( x , y; p) = exp < o f

2

2ti il-p ) ^ 2 n-p ) )

denotes the bivariate normal density function. For simplicity, we

denote it as 0 (x, y) in the following passage. Therefore, we have

f( X . , Z . 丨侈） ~mis -mis ~

n

= n f( x

i , 、丨侈） j=m+l

J J

:

来

h n

= I T ffk P r

< = k and X. > c } k=l i=l

1 1

l * r a

i 丄 1 r* + c o

h n r f k + 1「 -i = j ] Tfk 诊2

(X

, y) d x d y

k=l i=l L . … J

J

a. c k

11

*

h r I r k+i r -i k x

= n 1 y) dx dy V

k= l L L *

Z

J ) . J

a, ^ c k

(2.9)

Combining the results in (2.6), (2.7) and (2.9), we finally

derive the likelihood function in this model, which is given by,

L ( 侈丨 X ， Z ) 〜〜〜

= L ( p , a0, … ， a I X , Z )

乙 n 〜〜

« n nk

f ^ a

kI i , i ) 一

$

( ) x

k=l i=l L

, J

m

( 2 7 r ) - m / 2 • exp{ - - i - [ X ^ > x

i=l

睾

, r a

w i r +co n

i, h f r 「 k+1

[T \ 09( x , y) dx dy \

k=l ^ L . • J J

• J

a, J

c k

(2.10)

In order to find the maximum likelihood estimate, we would

like to maximize the likelihood function L( ^ I X, Z ) which is

expressed in (2.10). Equivalently, by ignoring the constant terms, we

12

would like to minimize the negative log-likelihood function F(它）.

F(^) <x 一 log L(^) 〜〜

h nk

« - [ I log 卜（ ) 一“ ) •

k=l i=l

h

* r r 〜+ 1

r+ w

i

- ^ n* log 02( x , y) dx dy ,

k=l J

a c (2 .11)

where log represents natural logarithm throughout this

passage.

To further express the term, notice that since

r \ + i r +0 0

沴(x， y) dx dy J

a. J

c k

= $2( + o o , a

k + 1 ； p) 一少

2( + 0 0 ’ a

k； p) 一 a

k + 1 ； p) + $

2( c’ a

k； p)

( 2 . 1 2 )

i 广 x p y 2 2 \ , 、 I u - 2puv + v I , ,

where $0( x , y ；p) = exp < ^ Y dvdu 2

2n (l-pZ

) . ^ 2 il-p ) )

「一 -co -00 denotes the distribution function of bivariate normal. For simplicity,

we denote it as y) in the following. So, F(侈）can be expressed as Z ~

13

F(^)

h n

k r 来来 ’

< x , [ [ log [ 染k + 1

’ J " 〜 i ) •

k=l i=l

h

- ^ n* log $2(+oo,a

k + 1) 一 $

2(+oo,a

k) 一 +

$

2( C

’a

k )

k=l

(2.13)

2.3 Derivatives of F(^).

A

To find maximum likelihood estimate ^ of 它，we are required to

minimize the negative log-likelihood function F(^) in (2.13) which has

been derived in the previous section . Since the optimum solution

cannot be solved algebraically in closed form, the modified

Davidon-Fletcher-Powell (DFP) algorithm, which will be discussed in

Chapter 4, is used. Due to the need of the first partial derivatives of

F ⑷ in this algorithm, we compute them as the following.

5F( p, a 〜 ) In order to find , we first calculate

dp

.) (u, v; p)

！^~ . By Johnson and Kotz (1972), = 02(u,v;p)

dp dp “

So, we can see that

14

a$(a* .) k，l

dp

来

* a a

k i K > 1

dp

* 9

( a

k " p X

k 1 = • ) •

k’1

aP I (i- p

2

)1 / 2

J

r - p2

)1 / 2

- (ak- p x p n + m -

PV

1 / 2

卜制 l

:_ i- p .

Y( i )

* p a

k " X

k = . ) •

2 3 / 2

K’1

(1- p2

) .

(2.14)

5F( p, a , . . .， ah )

According to (2.12) and (2.13), is dp

given by

5F( p, a2, . . . , a

h )

dp

I > r

来 * i - l r 帅 . ) .) =

" E I 染 k + l ’ i H( a

k ’ i ) . 7 7 一 ^

k=l i=l J d p 9 P

(h

* 『「a

k + l 「+ c

° r

l - - y n k 0

2( x , y) dx dy x

I lc=l J

a. J

c k

15

a r V i r + c o

] —— 0 (x, y) dx dy —

dp ^

J

a, J

c

h n

, / _ 1 p 尸「 * * 2 3/2

= - I I ^ 〜 1 , 1 ) - $ ( a

k’i ) J

1

" P } X

k=l i=l I

- 来 (1 1 来 (1 ) "I

^ r - v+ o o

, ° w - ” 〜( c

’ ° w + vc

,a

k ) •

k= ! L 、 ⑴ , \

+ 1) _ V

+ W

’a

k ) _ ¥ ‘ W + $

2( C

,a

k ) • .

(2.15)

来

As remarks, for the term k=l in the summation, since $ ( ak i

) «

* r i ) 0 in F ⑷ ， t h e term 0(a . ) • (pa -X ) in the derivative vanishes.

〜 iC y 1 JC XV 来

Similarly, for the term k=h in the summation, since $ ( ak + 1

丄）=1 in F(它），

* r i) the term 0(a, , . ) • (pa. ) vanishes in the derivative.

k+l,i k+1 k:

Now, we are trying to find the partial derivatives of F(它）

with respect to the unknown thresholds. For any t=2 h , consider

that

a$(a* .) t, l

a a

t

来

9 a . . * t i

=

余( a

t’i ) • — a a

t

16

, Y

( i ) 来

d

( a

t " p X

t 1 = ‘ • • —

t,1 Q

I 2.1/2 , da

K

(1- p )

* 2 - 1 / 2 =4>{oc

t 丄）•（1- p ) •

( 2 . 1 6 )

Moreover, by referring to Johnson and Kotz (1972), we know

(u, v; p) / u - pv x that = 0(v) • $ — — - and if u=+oo, then

atL \ L / c» - p ) 乂

抛(+00’ V； p ) / +00 - p v X

=d>(v) . $ f a 2.1/2 I

dw ^ (1- P ) }

- 0 ( v ) • $(+00)

= 0 ( v ) .

(2.17)

Therefore, we have

a r a

t + i r + c o

—— 0 (x, y) dx dy

da ‘ 、 a

t c

d 「 _ =—— $

2(+oo,a

t + 1) 一 $

2(+oo’a

t) - $

2( c’a

t + 1) + $

2( c , a

t)

a a

t L

、

. ( c - pa^ x -

=一 d>{oc^) • 1- <E> and t “ 2、1/2

L ^ (1- p ) ; J

(2.18)

17

a ra

t r + c o

—— 02( x , y) dx dy

doc. s % t c

d r

•

a a

t L

r ( c

~ pa

+. \ -

= 0 ( a ) • 1 - — L 乂（ 1 _ p ) 乂

J

.

(2.19)

Finally, for any t=2,...,h ,

aF(^)

a a

t

f h n

a

f p r * * 1

=一 I $ ( \

+i , i ) —

$ (

a

k,i) a a

t l

k =i

i =i “ ^

h

" I \ l Q

g [ V+ M

，a

k+1 ) "

$

2( + 0

°’a

k)

k=l

- Vc

，a

k + 1 ) + VC

’ V ] }

o / n

o t-i 厂来来 _

= — — 1 “ I l o

g $

( a

t , i ) _ $

( a

t—l，i ) 5 a

t t i = 1

n

" I 地卜（a

t +l ^ i ) 一

$ ( ) .

i=l '

18

来厂 “

- n ^ log $2(+oo,a

t) - $

2(+oo,a

t__

1) - $

2(c’<x

t) + ^

2( c , a

t_

1)

* r i - n log $

2( + o o , a

t + 1) - $

2(+oo,a

t) —

$

2(

c

’a

t + 1) + 尘 ‘

" * 、 n 2,-1/2

b 来来

i=l $ ( a. . ) - $ ( a ) t,l t-l,l

" * 、 “ 2,-1/2 n 0(a. .) . (1- p ) t t, l

+ z ；； i = 1

电(a

tll,i) 一$

(a

t : i )

- r c - pa^ X -

* L

^ (l- p ) } 1

- n “

$ (+oo,at) 一 $

2(+oo,a

t_

1) - $

2( c , a

t) +

$

2

( c , a

t - l}

_ r c - pa^ X -

来 L ^ (1- p ) 夕 J

+ n r $

2(+oo’a

t + 1) — $

2(+oo’a

t) — $

2( c , a

t + 1) + $

2(c’a

t) •

(2 .20)

来

As a remark in this equation, note that when t=2’ )=0 来

and $2(+oo, a

t_

1) = 0 . Also, when t=h, $ ( a

t + 1 丄）=1 and ^(+00,

a

t + 1)

=

l .

19

At this stage, we have find the h partial derivatives of F(^).

As we have mentioned before, the minimum solution of the negative

log-likelihood function F ⑷ cannot be obtained algebraically in closed

form, so iterative procedure, which requires the first partial

derivatives, is used to obtain the maximum likelihood estimates . The

procedure that we used will be introduced in Chapter 4.

2.4 Asymptotic properties of the model.

The maximum likelihood estimate (MLE) of = ( p , 〜， . . . ，〜），

A

分 is consistent. Moreover, if is the true parameter value of 它，then

under mild regularity conditions, it follows from the well known

asymptotic theory (See, e.g., Rao 1973) that the asymptotic distribution

of n1 / 2

- 一旮）is multivariate normal with zero mean vector and the

covariance matrix is given by the inverse of the information matrix.

That is

, f 「, a F ( 办）、(d F W 1 r

1

—I I 〜〜 I ⑷ = j E • -

V. L /s/ � J ‘

(2 .21)

八

and the estimate of the information matrix, I(^) is given by

20

^ , f m

「, aF(x.,z.) 、 f aF(x.,z.) ,y

] 工 ⑷ ：丄， y ————"“

+

〜 n [ ¥ ] y ]

j

n

f, SF(Z.) 、 f aF(z.) x'l )

i H . - — ^ 、 J V J

i=m+l L

~ ~

( 2 . 22 )

The derivatives in (2.22) has been derived in the preceding

passage indirectly during the finding the partial derivatives. Hence，

the asymptotic covariance matrix can be estimated and hence the 八

estimated standard errors of can also be obtained.

21

Chapter 3. Estimation of the model with one truncated continuous

variable and several polytomous variables.

3.1 The model

In Chapter 2, we have studied the model with one truncated

continuous variable and one polytomous variable. Now, we shall extend

the model containing several polytomous variables.

Suppose X, Y1, Y

2,.••’ Y

p

a r e (

P+ 1 )

standardized random >

variables, with We also assume that (X’ Y1 > Y

2’ ... ’ Y

p) has a

(p+1) dimensional multivariate normal distribution with zero mean vector

and correlation matrix P. Denote P as

>

/ 1 p i

p = p n

k. ~ j

(3.1)

>

where p = (p p p ) is the vector denoting the correlation 〜 1 Z p

>

between X and Y = ( Y” Yp) , and IT = ( p

a b) denotes the matrix

of correlation of Y with p , being the correlation of Y and Y . ~ a.b a D

Similar to Chapter 2, we assume that the random variable X is

continuous and is right truncated at a known point c. Also, suppose for

any a=l p , is latent and is observed by where

22

Z = k(a) if a . r ^ < a

W a、

+i

a a,k(a) a a,k(.aj+l

(3.2)

for k(a)=l h(a) with 〜x = -co ；〜

h⑷

+ 1 = And let

t oc = { a … . ’ a , , 、 } b e the vector of the unknown thresholds of Z . ~a a,2 a,h(a)

a

Let be the parameter vector in the model, then

9

？ = {

’ … ， V P

2 V " " P

p，p-1; a

i , 2 ,..., a

l’h(l) ;a

p , 2 ’…

'a

p,h(p)

(3.3)

and its dimension is given by,

P

dim (5) = p + p(p-l)/2 + ^ ( h(a)-l )

a=l

P

= p ( p - l ) / 2 + V h(a).

a=l

(3.4)

Now, suppose we have n identical and independent random

observations of the form ( X。)，Z(」)’...’ Z ? ) , and suppose that m

of these n observed vectors having observed X-values. Similar to

Chapter 2’ we let the last (n-m) X, s, X ,...’ X be missing by

(1) (ni) truncation and the remaining m X，s, X ’ … ， X are observed.

Let n , 、 represents the number of observations corresponding k(a)

23

to Z(a) = k(a), while 〜 ⑷ represents the number of observations

corresponding to Z(a) = k(a) and Z(b) = k(b), and 〜 r e p r e s e n t s the

f

number of observations corresponding to Z = k where Z = (Z^, . . . , Z^) >

and k = (k(l) k(p)). Then, we have

h(a) h(a) h(b)

I n

k ( a ) E E n

k ( a ) , k ( b )

k(a)=l k(a)=l k(b)=l

h(l) h(p)

= Z …• I n

k = n

.

k(l)=l k(p)=l ~

(3.5)

Furthermore, let m, t 、represents the number of observed X, s such that

k (.a j

the corresponding variable Z(a) is equal to k(a). Denote these observed

X’s by X 冗),...’ J

. And let mk ( a )

represents the number of

missing X such that the corresponding variable is equal to k(a).

Then,

\ ( a ) +

\ ( a ) = n

k ( a ) ,

h(a)

[m

k ( a )

k(a)=l

24

h(a)

I \ ( a ) = n

"m

• k(a)=l

(3.6)

Also, let mk represents the number of observed X’ s such that

the corresponding variables Z equal k. Denote these observations by

乂⑴ X(竺）. And let m, represents the number of missing X such k , …

，

k k

that the corresponding Z equal k. Then, we have � �

m

k +

\ = n

k ’ 〜〜〜

h(l) h(p)

I I m

k =

m

' k(l)=l k(p)=l ~

h(l) h(p)

l … I \ = n

"m

' k(l)=l k(p)=l ~

(3.7)

3.2 Partition Maximum Likelihood (PML) estimation

y To estimate the parameter vector = { p ^ . . . , p ; P

2 1, • • •,

25

P p ,p- r

a

i ,2 ,...,

a

i , h ( i ) •’a

P’

2 ’... '

a

p,h(p) } i n t h i s m o d e l

, w e a p p l y

the Partition Maximum Likelihood (PML) estimation method.

For every a=l, • • . ’ P ， P is estimated based on the random

Si sample from the truncated continuous - polytomous sub-model

9

corresponding to (X,Z ) which we have discussed in Chapter 2.

Let 8 = (a , p ). According to (2.13) in Chapter 2, the a

negative log-likelihood function for this sub-model is given by

F (13 ) a -a

, , v m h(a) k(a)

« 一 I I ^ g [ $ ( o c Ja ) + 1

. ) - $ ( ) •

k(a)=l i=l

h(a)

- [ \ ( a ) l 0 g

[ $

2( + C 0

’a

a, k (

a)

+l ) "

$

2( + C 0

’a

a’k(

a) )

k(a)=l

* $

2( C

,a

a , k ⑷ + 1 ) + $

2( C

’a

a , k⑷） • •

(3.8)

Y( v )

* a - p • X. f x

, * a,u ^a k(a) where a = ^ ,

u

’v

(1 - p 2

)1 / 2

. a

26

Moreover, by (2.15) and (2.20) in Chapter 2’ the partial

derivatives of F with respect to the parameters in g are given as

3L 议

follows:

5F O ) a ~a

dp a

, , v m h(a) k (a)

「 . 来来 ..1 2 3/2 L L k(a)+l,l k(a)’i J a

k(a)=l L

『， * 、， v ⑴ 、 _

( a

k ( a )+l , i

}

• ( p

a 'a

a , k ( a )+l 一

X

k ( a ) )

* (i) 、 1 I

- 勞(\ ( 幻 , 1 ) • (

p

a 'a

a , k ( a ) - X

k ( a ) ) J |

一 h

ya )

[ j ; f ⑴,、,跡 1) H k ⑷ ）

: : V ^ M i l ? 1 -

-$

2( c

,a

a , k ⑷ + 1 ) + $

2( c

'a

a , k ( a )}

- •

(3.9)

dF (/3 ) a〜 a

da , ( x a,1(a)

27

" W i 少（ ) • d 力一1 / 2

= _ [ ；；

1 = 1

龟(a

l(l),i ) - “ a

l(a)-l,i )

〜 ⑷ “ a

na; , i ) •

( 1

- ~2 )

-1 / 2

+ I ；；

i = 1

^ ( a

l ( a )+l , i

}

" $ ( a

l ( a ) , i 3

. ' 、「， J

c

- 〒a’ l ( a ) 1 1

m

l ( a ) - l • 勞 ( 〜 ’ 丄 ⑷ ） . ^ H( 1

. 2) 1 / 2

J , ^a

~ (

、「， “ C

“ P

aa

a >l ( a ) 丫 1

1(a) ^ a,1(a) [ I (i- p

2

)1 / 2

) J O.

+ — ： — —

$

2( + C 0

'a

a , l ( a ) + l

)

-$

2( + M

'a

a , l ( a ))

-$

2( c

'a

a , K a )+l

) + $

2( C

'a

a , l ( a ))

’

(3.10)

for any 1(a) = 2,..., h(a).

To estimate the polychoric correlation p ,, for a’b 愁 1 P

with a>b, the bivariate sub-model corresponds to { Z ^ Z ^ ) is considered.

Let = ( & ，， % ’ ， Pa b ). Suppose d

k ( a ) k ( b ) denotes the

probability such that Z =k(a) and Zf e=k(b). Then

d

k(a),k(b)

28

= P r ( Z = k(a) and Z^ = k ( b ) ) a b

=P r

( a

a , k ( a ) ^ Y

a < a

a , k ( a )+l a n d a

b , k ( b ) ^ Y

b < a

b,k(b)+l )

= a

a , k ( a )+l ’

a

b,k(b)+l ) “ $

2 ( a

a , k ( a ) , a

b,k(b)+l )

一 a

a , k ( a )+l ’

a

b , k ( b ) ) + a

a , k ( a ) ， a

b , k ( b ) ) .

(3.11)

Let L be the likelihood function in this sub-model, then by ab

the independency of the observations,

L , O J ab -ab

h ( a ) h ( b )

( A ,

n

k(a),k(b)

"w n n (

d

k ( a ) , k ( b ) ) . k(a)=l k(b)=l

(3.12)

The negative log-likelihood function in this sub-model is given by

F , O , ), where ab -ab

F , (/3 J ab -ab

h(a) h(b)

" " I I n

k ( a ) , k ( b ) 'l 0 g ( d

k(a),k(b) ) •

k(a)=l k(b)=l

(3.13)

The partial derivatives of F

a b(§

a b) w i t h

respect to the

29

parameters in ^ , are derived as follows.

aF (3 ) To find ~ , we first know from Johnson and Kotz

^ a b

(u,v;p)

(1972) that =沴(u’v). So dp 乙

a F

a b

(

ga b

)

a p

a b

h(a) h(b) ^ y ^

n

k(a)’k(b) • o q

k ( a )>k ( b )

k(a)=l k(b)=l d

k(a),k(b) P

a b

h(a) h ( b )(

^ ^ n

k(a),k(b) •

k(a)=l k(b)=l^ d

k(a),k(b)

_ 2( a

a , k ( a )+l '

a

b , k ( b )+l

)

" 々2( a

a’k(

a)，

a

b,k(b)+l)

\

一 t( a

a, k (

a)

+ l’

a

b,k(b)) +

〜( a

a, k (

a)’

a

b , k ( b ) ) J ^ J

(3.14)

To find the partial derivatives of Fa b(§

a b) with respect to

^^k(a) k(b)

the thresholds, we first find in three cases separately. da , ( x

a,1(a)

30

Case I: l(a)=k(a).

Since by Johnson and Kotz (1972),

(u,v;p) , v-pu >

= 0 ( u ). $ ,

du ^ (1-p ) y

9 d

k ( a ) , k ( b )

〜 ’ K a O

M 2(

、， l (a) ’

a

b ， k a ^ + 1 ) + 〜〜 ⑷ 〜刚 )

= 一 — + —

a a

a , l ( a ) a < X

a,l(a)

f 、 [ J

a

b,k(b)+l - p

a b 'a

a , l ( a )飞

” ( 〜 ’ 丄 ⑷ ） . — (x

.p 2

}i / 2 j

- ab

f a

b , k ( b ) 二 p

a b 'a

a , l ( a ) ) + 2 1/2

1 ( ^t ) J J *

(3.15)

Case II: l(a)=k(a)+l.

a d

k ( a ) , k ( b )

a a

a , l ( a )

— 一 •

da “ 、 doc r

, a,1(a) a,1 la)

31

乂 . [ J a

b,k(b)+l - p

a b 'a

a , l ( a ) ) = 0 ( a

a , K a )}

' H ( x_

p 2

}l/2 J

L p

a b

_f a

b , k ( b ) - ^ b ^ a ^ C a ) )

一 ~ 7 " ； 2 ,1/2 J (

^ ) ；

J '

(3.16)

Case III: l(a)^k(a) and l(a)^k(a)+l.

In this case, !

= 0. doc ,

(,

a,1(a)

(3.17)

Finally,

dF , (|3 ,) ab ~ab

doc , ( a,1(a)

0 r h(a) h(b)

o

I I n

k ( a ) , k ( b ) 'l 0 g ( d

k(a),k(b) 3

a a

a , l ( a ) L k(a)=l k(b)=l J

. r h ( b )

o

= I n

l ( a ) , k ( b ) 'l 0 g ( d

l ( a ) , k ( b ) )

a,1(a) L

k(b)=l

h(b) -I

+ I n

l ( a ) - l , k ( b ) 'l o g ( d

l ( a ) - l , k ( b ) )

k(b)=l

32

h(b)

=_ x

n

l ( a ) , k ( b ) . c

^Q

l(a)>k(b)

k(b)=l d

l ( a ) , k ( b ) a a

a , l ( a )

h(b)

一 V n

l(a)^L，k(b) . o a

l ( a ) — l , k ( b )

k(b)=l d

l(a)-l’k(b) 5 a

a , l ( a )

h(b) f r ^ D

./v 、

r

n

l ( a ) , k ( b ) J a

b , k ( b )+l -

P

a b a

a , l ( a ) ]

k(b)=l a

l ( a ) , k ( b ) { L a b

(a

b ’ k ( b ) - p

a b 'a

a , l ( a )

1

( ^ a b ) -y

h(b) f 「 /v n

./V

r

n

l(a)-l,k(b) .f ,

a

b,k(b)+l - p

a b a , 1 ( a ) ]

- I . — “ � ) • \ ( 2 } l / 2 J k(b)=l

a

l(a)-l,k(b) [ L

ab

卿 J tt

b,k(b) - p

a b 'a

a, l ( a ) I

一 7 T 2 ,1/2 ‘ 1

( K b ) -y

h(b) r n n 1

=_ V . n a ) - l , k ( b ) -

n

l(a),k(b) • )•

L , , a，丄taj

k(b)=l L a

i (a) - l , k ( b ) 1 ( a ) , k ( b )」 V.

f a

b,k(b)+l _ p

a b 'a

a , l ( a )、 2 ,1/2

L 1

( 工卞北）；

(a

b , k ( b ) 一 p

a b 'a

a , l ( a ) \ 1

一 d) — •

f , 2 、l/2 1

( ^ a b ) ) - . y

(3.18)

33

Similarly, for any b=l,...p , and 1(b)=2,...,h(b) ’

ab ~ab

a a

b’l(b)

h(a) f r ^

n 1

r n

k(a)’l(b)-l k(a),l(b) . (

k(a)=l L d

k(a),l(b)-l d

k ( a ) , l ( b )」 V

“ (a

a >k ( a ) + l - P a b ^ b ^ C b ) 1

L 1

(丄卞让）

；

\

(a

a ’ k ( a ) - p

a b 'a

b , l ( b ) ) 1 一 ® — — •

r , 2 ,1/2 』

y

(3.19)

Similar to Chapter 2， the minimum of Fa( g

a) and

F

a b( g

a b)

cannot be solved algebraically in closed form. Hence some iterative

methods that mentioned in the next chapter will be used to obtain the

solution. Note that the iterative methods require the first partial

derivatives of the objective function and this is why we derive them in

the above.

In the Partition Maximum Likelihood estimation, we separate

the huge p+1 dimensional model into P + 1

C2 = p(p+l)/2 sub-models. In

order to obtain estimates of these smaller sub-models, we only need to

34

compute single and double integrals instead of the complicated multiple

integrals which will occur if full maximum likelihood estimation method

is used. So, a lot of computer time can be saved. However, the

thresholds estimates are not unique, there are p sets of threshold

estimates for each a based on p different sub-models. We use the mean

of these estimates as our final thresholds estimates since the

difference among them are very small based on our experience in

simulation studies.

3.3 Asymptotic properties of the PML estimates

y >

For each a=l,...’p ’ let g = (a^ , p ) be the vector that

〜a 〜a oi minimizes the function F (|3 ). Also, for a , b = l , . . . , p， a > b , let

a ~a t > y

g , = ( oc , a^ , p , ) be the vector that minimizes F (/3 ) • Lab ~a ab

a D

~a D

:，》 .，》 » Moreover, denote i) = ( , . . . , 3 ; 13,,..., 3 - ) and

〜〜丄〜P 〜乙丄〜丄

V = ( g:’..•’ 玲 ’ ；？。 ; ， . . . ，玲： ” ，） .

T h e n b

y standard maximum

likelihood theory that under mild regularity conditions, 5 is a

consistent estimator of v , where T) represents the vector of true

parameter value of i).

According to mean value theorem, for each a==l,...,p ,

35

dF (g ) dF O ) a2

Fo(/)

a

_ = a

~a o

+ ^ _ • (g - )

八 A ^ A 〜ao 5/3 dp dl3 a 俘。

~a -a

(3.20)

来

where 13 is the true value of the parameter vector 13 , and 13 is a ~ao ~

a

vector that lies between 玲 and . ~a 〜ao

Also, by mean value theorem, for each a,b=l p, a>b,

dF (g ) aF L ) d2

F ,(13 ,) a b ^ a b

J

= ab义abo) + ab '-ab .

(§ )

—

~ab 〜abo 5

§ab a

§ab a

§ab ^ a b (3.21)

来

where 侈ab〇 is the true value of the parameter vector g

a f e , and g

a f e is a

vector that lies between g , and ga b o

.

Combining the equations of the form in (3.20) for a=l p

and of the form in (3.21) for a,b=l,...,p ； a>b, we will obtain the

following matrix equation,

36

一 r i r o • o •

；

0

o

dF (g ) dF O ) a2

Fa(/) . 〜 a L a a

~a o

o a

o m 9

〜a〜ao dB a/3 di3 a/3 . 〜a 〜a 〜a • + 0 0 • • • • •

• 參

2 *

dF ^(g J dF K

) d F

a h

(

5a b

) o 〜

a

§ab 5

§ab ‘ 呼 a b ^ a b

0 • •

• . o • • • • • o •

—J L— 」 L

(3.22)

a F

( H ) f a F

l ' a F

p a F

2i

a F

P:

P- l 1’

Denote — , . • • ’ '» » . •., 1

% d

S2i %

f P- i

；

and let H be the diagonal block matrix with appropriate diagonal blocks

* -1 ^ a ^ l3

* which are equal to H O ) = n V - (for a=l p) and H

a b( g

a b) =

a d!3 5 玲 ’ ~

〜a 〜a

-1 a 2 p w O — ( f o r a,b=l,...p, a>b) respectively.

~ab -ab

Hence, the following equation is established.

37

aF(7)) dFiin ) * 〜〜O

TT 广

= + n-H (刃一7) ) • 〜〜o

dt) dT\

(3.23)

d¥{.r\)

Due to the definition of tj, we know that = 0, and 〜

a

2

hence

. , ,/ 0 aF(i7 )

1/2 广、 ru

*、-l -1/2 n • (T)一7) J = - C H ) -n •

~ dr\

(3.24)

Now, for every i=l n , k(a)=l h(a) , a=l,.‘ .， p ’

let AF (t), k, i) denote the augmented vector which has p(p+l )/2 ~ 〜〜

sub-vectors containing partial derivatives from two categories. The

first p sub-vectors are given by either

_ n y

( D d 「（

a

a,k(a)+l p

a A

k x

- l Q

g $

z T T T i 一 dl3 ^ (1 - P

0 ) 乂

~a :fm

a*

a . . . - p 'X. n ( a,k(a) a k 、

s> “ 2、 1/2

V (1 - p ) a . J

if the sample* s corresponding variable X is observed, or

38

a 厂

- ― “ 丄叩 $

2(+W

'a

a,k(a)+l

)

一 $2

(+⑴,

a

a，k(

a))

op L

-a

- ^ Z ^ ' V k C a ) . !3

+ $

2( C

，a

a, k (

a) ) .

if the sample's corresponding variable is missing， for any a=l,...’p.

Secondly, the sequent p(p-l)/2 sub-vectors are given by

d

~ — ~ l o g

(d

k(a),k(b)) a

§ab

d log Pr{ Z = k(a), Z, = k(b) >

… a

D a

§ab

where a，b=l，...p， a>b .

By defining this AF ( 5 , k, i), we have

a r c , ) ？)、

— = - l … I I ^ ( V ！5, i ) . a

5 k(l)=l k(p)=l i=l

(3.25)

Finally, we get

h(l) h(p) n

k

n1 / 2

- ( r 5 o) = ( H

*r l

-n

"1 / 2

' I ... I I ^

k(l)=l k(p)=l i=l

(3.26)

39

Now, since for i=l,,..’ 〜， k ( a ) = l , . . . , h(a) , a=l,..., P ,

(X⑴，Z,(i), ) is a sequence of identically and independently distributed k

random vectors, it can be shown that by the central limit theorem, the

h(l) h(p) n

k

asymptotic distribution of n一1 / 2

. ^ … ^ \ AF k , i) is

k(l)=l k(p)=l i=l

multivariate normal with mean zero and covariance matrix

In addition, as g ’ j3 , are consistent estimates of 3 and

〜a 〜aD

<3 ’ the diagonal blocks in H will converge in probability to the

corresponding matrices Ha( g

a Q) and H

a b( g

a b o) respectively. That is,

* p 来一 1 P 一1

H ——-——> H and hence (H ) > H .

1 /P Therefore, n • (tj-t) ) is asymptotically distributed as

multivariate normal distribution with zero mean and covariance matrix

H一1

n H- 1

where H is a diagonal block matrix with diagonal blocks equals o “

H (|3 ) for a=l, . . . ,p and H , (6 , ) for a’b=l,...,p , a>b. at 〜ao 3LD 〜aDo

Since our main concern is about the correlations, we let (r =

(n n ‘ o . , o ) be the vector of the unknown correlations.

...，Pp, y

2 V ' H

p , p - 1

Suppose J is a selection matrix such that Jt? = o; , then the Partition

Maximum Likelihood (PML) estimator ^ of <r is given by o = J ^ and the

1 / 2 � • asymptotic distribution of n •(〔-〔 ) is hence multivariate normal with

40

-1 -1 , mean zero and covariance matrix JH Q H J •

o

To find an estimate of the H~ , we actually need to find the

estimates for the blocks H (/3 ) and H , (3 K) . For the truncated

a 8LD ~SLD

continuous - polytomous sub-models, Ha( g

a) is estimated by H

a( g

a) where

it is given by

^ m r, aF (X. , Z . ) 、 / a F a ( x ” z ) x H , 、 r I a l i a i l

H (/3 ) = ) — • + a

；

u

\ dB ^ a/3

n

「 a F (z.) v ( 5F ( z . ) V" a l a l

. • ^ a/3 ) L d!3 i=m+l

L

~a J

(3.27)

for a=l,...,p. For the polytomous - polytomous sub-models, H

a b( g

a b)

c a n

be obtained as a by-product in the final stage of the iteration in

Fisher,s Scoring optimization procedure. The details will be discussed

in the following chapter.

Since (X,⑴，Z,⑴) is a sequence of identically and k -k

independently distributed random vectors, i = l , . • . ,〜， k ( a ) = l h(a)

a=l, . . . , p, the corresponding AF Cv, k, i) is also a sequence of

i.i.d. random vectors. Hence, we use the usual sample estimate to

estimate Q as the following.

41

Let AF denote the mean value of AF ( 5 ’ k, i), that is

x 「 h(l) h(p)

n

k •

af = — [ … r t ^ n

n

L k(l)=l k(p)=l i=l J •

(3.28)

Then the estimate of Q is given by,

:

T h(l) h(p) n

k ,“

“ 一 I . • I I (AF(^k,i)-AF) . [AF(5,k,i)-AF)

n一

1

k(l)=l k(p)=l i=l . • 參

(3.29)

八

Furthermore, since E[ AF ( 5 , k, i)] = 0, we can approximate Q by

『 h ( l ) h(p) n

k ,“

n = 一 V .. y Y AF(^,k,i)-AF(^,k,i)'

/ / • L4 〜〜〜〜〜〜

n一

1

[ k(l)=l k(p)=l i=l ..

(3.30)

42

Chapter 4 . Optimization procedures and Simulation study.

4.1 Optimization procedures

As mentioned in Chapter 2 and Chapter 3, the maximum

likelihood estimates of the parameter vector for each bivariate

sub-model is obtained by minimizing the corresponding negative

log-likelihood function. However, in practice, the minimum of the

negative log-likelihood function cannot be obtained in closed form.

Hence, some iterative algorithm (See, e.g., Lee & Jennrich, 1979) should

be used. We shall describe how to apply the modified

Davidon-Fletcher-Powell (DFP) algorithm as well as the Fisher Scoring

algorithm in analyzing two different kinds of sub-models as follows.

The procedure for minimizing (3.8) for the sub-model with

truncated continuous - polytomous pair which based on modified

Davidon-Fletcher-Powell (DFP) algorithm has been implemented by writing

in FORTRAN IV with double precision. DFP algorithm, which is also

referred as the variable metric method, is a line search algorithm (See,

e.g. , Luenberger 1973). Let f (x) be the objective function, then the

basic steps of the algorithm is as follows.

Given a symmetric positive definite matrix SQ ,and a starting point x

q ’

then starting with k=0,

Step 1 Set d = ~S

k 'g

k w h e r e

is the gradient vector of

43

the objective function f evaluated at x and is a

symmetric positive definite matrix.

Step 2 Minimizes f(x. + a-d, ) with respect to a^O to obtain

〜iC ~K

� \

( i i i )

Bk = a

k ^ k a n d

( i v )

i k+i '

Step 3 Set gk = |

k + 1 - I

k and

s … A ^ ： - 咖、

U p d a t e k a n d

k+1 k , ^ ,c ^

Bk 3k % s

k 2k

return to Step 1.

(4.1)

In 1970, Broyden, Goldford, Fletcher and Shanno suggested the

so called BGFS formula. The global convergence of the BGFS method with

inexact line searches which satisfy the conditions suggested by

Goldstein (See, e.g., Fletcher 1979) has been proved. The two

conditions suggested by Goldstein in the minimization procedure are

given as,

(i) f

k_ f

k + l ~ ""p

.ik’Sk ’ f o r s o m e

P€

(0

，1 / 2

)

(4.2)

which preserves the positive definite property of S and hence the

function value decreases monotonically in every iteration. Due to the

advantages of the method, we replace Step 3 of the procedure by the BGFS

44

formula,

c ‘ f , 9k'

s

k3k ) Bk Sk' B A W i A ’ S

W 1 = s. + 1+ JC+1 K , J 一 , — ^ ,〜 1

Bk 3k

;

Bk 3

k Bk 3

k •

(4.3)

In the modified Davidon-Fletcher-Powe11 (DFP) algorithm, the

positive definite matrix S, is updated in each iteration. Although this

algorithm ensures the decreasing of the objective function in

iterations, unlike the Fisher Scoring algorithm, the final Sk in the

iterations does not provide a good estimate for the Hessian matrix

(See, Lee & Jennrich, 1979).

The procedure for minimizing (3.13) for the sub-model with

polytomous - polytomous pair has been developed by Poon and Lee (1987),

and a program based on Fisher Scoring algorithm written in FORTRAN IV

with double precision has also been implemented. The basic step of the

th Scoring algorithm at the i step is given by,

M . = - < I . -1

? . (4.4) ~ l l

where ^ is a step-size parameter which takes the first value in the

sequence { 1’ 1/2， 1/4,••. > that reduces the function value, is the

gradient vector and is the information matrix at the it h

step with

= E ( g . g . , ) . Actually, we only need the first partial derivatives to

obtain the information matrix. The Fisher Scoring algorithm not only

produces the maximum likelihood estimate, but also an approximation of

45

its asymptotic covariance matrix and hence its standard errors in the

sub-models. So, the Fisher Scoring algorithm produces the consistent

estimate of H for each polytomous - polytomous sub-model in ab ^ab

Chapter 3.

As we can see that in either the DFP algorithm, or the Fisher

Scoring algorithm, only the first partial derivatives with respect to

the parameters are needed in the iterations. And these derivatives have

been derived in previous chapters.

In general, both of the algorithms are robust to the starting

value of the parameter vector. However, a good starting value would

reduce the time of convergence. Hence, we use the 'sample, correlation

based on the truncated or the polytomous data in each sub-model to be

the starting value for the parameter of correlation. For those

sub-models which involve the truncated continuous variable, we replace

the missing value by the truncation point value to calculate the

starting point. This approach uses all the data in the calculation of

the starting value- Although this starting point may possess bias,

based on our experience in the simulation study, it is a good starting

value since the procedure converges quickly to the solution. For the

starting values of the thresholds in the sub-models, we use the inverse

of the standard normal distribution evaluated at the cumulative cell

proportion of the polytomous variable.

46

Furthermore, the program is said to be converged and the

iterative procedure will stop if the root mean squares of the gradient

vector is less than a pre-assigned small number, say e.

4.2 Simulation study

Based on the algorithm discussed in the previous section, a

computer program written in FORTRAN IV with double precision has been

implemented to obtain the Partition Maximum Likelihood (PML) estimates

associating with the model that has been discussed in Chapter 3.

To study the behaviour of the estimate, different situations

which includes different sample sizes, different correlations matrices

and different truncation points of the continuous variable are used in

the simulation study.

The study is based on simulated data drawn from a multivariate

normal distribution with the dimensions of X and Y are one and three

respectively. The mean vector of the distribution \jl is chosen to be the

zero vector, and the correlation matrix P are taken as follows,

(I) Small correlations between variables:

47

l.o i r L Q “

p l.o 0.1 1.0 ^ =

p1

p l.o 一 0.1 0.1 1.0 p

2

p1 2

p 1.0 0.1 0.1 0.1 1.0 L 广 1 3 23 J L J

(II) Reasonably large correlations between variables:

“1.0 1 [ 1.0 . p 1.0

= 0.5 1.0

P =

p1

p 1.0 “ 0.5 0.5 1.0 p 2 p 1 2 p 1.0 0.5 0.5 0.5 1.0

> 严 3 1 3 23 L J

Moreover, the known truncation points of X are taken as:

(A) c=1.2816 is the 90 percentile of a standard normal

distribution which gives about 10% of truncated data.

(B) c=0.0000 is the 50 percentile of a standard normal

distribution which gives about 50% of truncated data.

For the three polytomous variables Z^, Z^ and Z^, we assume

each of them has three categories and we choose different kinds of

thresholds values for them. For variable Z^, we consider a symmetric

distribution and approximately equal amount of data in the categories ,

which means about one-third for each category. For the variables Z^ and

Z^, we consider asymmetric distributions which skew at the opposite

48

directions. The ratios of the data in categories of Z2 are roughly 20%,

30% and 50%; while 5€%, 30% and 20% are roughly the ratios for Z ^

Finally, the thresholds values of the polytomous variables are given as

follows:

a

l = { a

l , l = "" ’ a

l , 2 =

- 0 .5

, a

l’3 =

°'5

, a

l , 4 = + C

° }

a

2 = { a

2 , l =

- ⑴ ，a

2 , 2 = - 0 .8

， a

2’3 = , a

2 , 4 =

+⑴}

a

3 = { a

3 , l =

-“ ’ a

3 , 2 =

0.0 , a

3 , 3 =

°'8

, a

3 , 4 = }

.

In addition, five different sample sizes are in

considerations. (1) n=50 , (2) n=100 , (3) n=200 , (4) n=400 and (5)

n=800 .

With two sets of correlation matrices, two sets of truncation

points , one set of thresholds vectors and five sets of sample sizes,

there are totally twenty different combinations. For each combination,

50 replications are performed where the multivariate normal variates are

generated by the subroutine DRNMVN of IMSL (1975) with the specified

mean vector and correlation matrix. The simulated continuous random

vector , Y0, Y

0) ' is transformed to the polytomous vector Z=(Z ,

〜 1 Z 〜上

Z , ' based on the pre-assigned threshold values. Then the

parameters are estimated by our PML method. The convergence criterion e

is taken to be 0.0005. The simulated results concerning the

correlations and the thresholds estimates are reported from Table I.A.I

to Table II.B.5. { Note that Table I.A.I refers to the simulation study

49

with correlation matrix(I), truncation point(A) and sample size(l) and

so on. }

In each of the tables, the following statistics are reported.

(i) The mean values of the estimates:

50

1

so 人 1

k=l

^ f k") th where 〜 represents the i element of the estimated

parameter vector in the k ^ replication.

(ii) The root mean square errors:

50 1 / ?

( 1 n

2 1 RMSE. = \ - ^ ― V ( - ) —

1 1 50 丄 1 1 f k=l

th>

where 办.represents the i element of the true parameter

vector.

(iii) The sample standard errors of the estimates:

50 1 / ?

� H r K � ) 2 } k=l .

(iv) The average of estimated standard errors of the estimates:

50

50 1 八（k)

S.E. . = V (estimated standard error o f 、 ) 1

50 • k=l

(v) The ratio of the sample standard errors to the average of

estimated standard errors of the estimates:

S.D..

R . = — — ^ • E< • • •

i

We would expect that S.D. . is close to S.E. and thus the 、‘ l i

ratio R . would be close to one. l

From the tables, the following phenomena are observed.

(1) The mean values of the estimates are very close to the true

parameter values and the root mean square errors (RMSE), the sample

standard errors (S.D.) and the estimated standard errors (S.E.) are

reasonably small in all situations, especially when the sample size is

large.

(2) By comparing the tables with different sample sizes, as

expected, when the sample size increases, the RMSE, S.D. and S.E. of all

the estimates decreases and the estimates are much more accurate in all

situations.

(3) By comparing the tables with different truncation points of

51

the continuous variable, we can see that when the truncation point

increases which means less data are truncated, the following are

observed. The RMSE, S.D. and S.E. of the estimates of polyserial

correlations are smaller due to the gain of the information about the

continuous data. However, there is no change about the estimates of the

polychoric correlations since they are not affected by the truncation

point by applying the Partition Maximum Likelihood method.

(4) By comparing the tables with different correlation matrices,

we can see that when the population correlations increase, the RMSE,

S.D. and S.E. of all the estimates decrease since higher correlations

give more information between variables.

(5) In the 10% truncated case, the estimates of the polyserial

correlations are better than those of polychoric correlations by

comparing their RMSE,, S.D. and S.E. However, this phenomenon vanishes

in the 50% truncated case which give no significant difference between

the two types of correlation estimates.

(6) In all situations, the estimates of the correlations , either

polyserial or polychoric, are better than those of the thresholds

estimates.

(7) Within the estimates of the thresholds, we can see that those

estimates involving the polytomous variable with symmetric thresholds

52

(i.e. Y.) are better than those of the other estimates involving the

polytomous variable with asymmetric thresholds (i.e. Y or Y^). That

means in all situations, p has smaller RMSE, S.D. and S.E. than p2 or

p . Also, p1 2 and p

1 3 has smaller RMSE, S.D. and S.E. than p

2 3-

(8) In all situations, the Ratios fall into the range (0.8, 1.2).

It indicates that the estimates of the standard errors are acceptable.

53

Chapter 5. Summary and Conclusion

In this thesis, we develop a method in estimating the

correlations between the truncated continuous and the polytomous

variables. By using the method of Partition Maximum Likelihood (PML)

estimation proposed by Poon and Lee (1987), the (p+1)-dimensional model

is divided into p(p+l)/2 bivariate sub-models which can be classified as

two different kinds. The first kind involves one truncated continuous

variable and one polytomous variable while the second kind involves

variables which are both polytomous. The likelihood functions of these

sub-models have been found and the estimates of the parameters are

obtained through the modified Davidon-Fletcher-Powell (DFP) algorithm.

It follows from the statistical theories that these maximum likelihood

estimates have nice asymptotic properties. The asymptotical results are

also reported. Based on the results of our simulation study, we observe

that the estimates are very accurate in various conditions, including

different correlation matrices, truncation proportions and sample sizes.

We can also see that the results are still pretty good even when the

sample size is as small as 50 and proportion of truncation is as large

as 50%.

Certainly, this thesis gives only the starting point of the

problem, there are still a lot of practical problems that are needed to

be studied. The most trivial extension is to consider continuous

variable with doubly truncation, that is truncated at both sides. It is

54

believed that similar procedures can be applied and similar results will

be obtained. We can also consider the extension of the truncated

continuous variable to multi-dimensional. Based on similar ideas

provided in this thesis, we believe that new results on these topics can

be achieved in future.

55

Table II.A.2

( n = 100 )

( c = 1.2816 ) (10% truncated)

Parameter TRUE EST. RMSE S.D. S.E. RATIO

p 0.100 0.137 0.148 0.145 0.164 0.884

p 0.100 0.095 0.177 0.179 0.168 1.067 h

2

p 0.100 0.093 0.171 0.173 0.164 1.055

p 0.100 0.052 0.167 0.161 0.180 0.895 " 1 2 p 0.100 0.142 0.182 0.178 0.175 1.018

p 0.100 0.081 0.181 0.182 0,180 1.008

a -0.500 -0.516 0.189 0.190 0.187 1.017 1 , 2 a 0.500 0.530 0.209 0.209 0.188 1.109 1,3

a -0.800 -0.782 0.186 0.188 0.200 0.938 2,2

a 0.000 0.023 0.193 0.193 0,178 1.087 2,3

a 0.000 -0.022 0.192 0.192 0.177 1.083 3,2

a3 3

0.800 0.825 0.216 0.216 0.203 1.066

TRUE = TRUE PARAMETER VALUE

EST. = MEAN OF ESTIMATES

RMSE = ROOT MEAN SQUARE ERROR

S.D. = SAMPLE STANDARD ERROR

S.E. = MEAN OF ESTIMATED STARDARD ERROR

RATIO = RATIO OF S.D. TO S.E.

56

Table I.A.2

( n = 100 )

( c = 1.2816 ) (107. truncated)


p 0.100 0.114 0.127 0.128 0.110 1.161

p 0.100 0.105 0.111 0.112 0.115 0.969

p 0.100 0.110 0.132 0.133 0.117 1.133

p 0.100 0.135 0.131 0.128 0.126 1.104

p 0.100 0.090 0.123 0.124 0.127 0.973

p 0.100 0.109 0.129 0.130 0.130 1.001

a -0.500 -0.495 0.139 0.140 0.131 1.070 1,2

a 0.500 0.485 0.135 0.136 0.131 1.036 1,3

a -0.800 -0.794 0.115 0.117 0.141 0.827 2,2

a 0.000 -0.011 0.131 0.132 0.126 1.051 2,3

a 0.000 0.002 0.145 0.147 0.125 1.172 3.2

a 0.800 0.836 0.154 0.151 0.143 1.051 3.3







57

Table II.A.2

( n = 100 )

( c = 1.2816 ) (10% truncated)


p 0.100 0.102 0.078 0.079 0.079 0.999

p o.100 0.096 0.090 0.090 0.081 1.109

p 0.100 0.095 0.073 0.074 0.081 0.905 3 .:、、

p 0.100 0.113 0.083 0.083 0.089 0.923

p 0.100 0.109 0.078 0.078 0.090 0.868

p 0.100 0.103 0.102 0.103 0.093 1.108

a -0.500 -0.491 0.084 0.085 0.093 0.916 1,2

a 0.500 0.515 0.113 0.113 0.093 1,216 1,3

a -0.800 -0.803 0.114 0.115 0.100 1.151 2,2

a 0.000 -0.014 0.094 0.094 0.089 1.062 2,3

a 0.000 -0.014 0.086 0.086 0.088 0.971 3.2

a 0.800 0.781 0.100 0.099 0.099 0.999 3.3







58

Table I.A.2

( n = 100 )

( c = 1.2816 ) (107. truncated)


p 0.100 0.112 0.049 0.048 0.055 0.865

p 0.100 0.100 0.053 0.054 0.057 0.941

p 0.100 0.087 0.055 0.054 0.057 0.934

p 0.100 0.093 0.058 0.058 0.064 0.918

p 0.100 0.105 0.076 0.076 0.063 1.201

o 0.100 0.095 0.064 0.065 0.065 0.986 p

2 3

a 一 0 . 5 0 0 -0.499 0.067 0.068 0.065 1.035 1,2

a 0.500 0.506 0.072 0.072 0.066 1.101 1,3

a -0.800 -0.781 0.074 0.073 0,070 1.039 2,2

a 0.000 0.019 0.070 0.068 0.063 1.085 2,3

a 0.000 -0.002 0.063 0.063 0.063 1.011 3.2

a 0.800 0.796 0.074 0.074 0.071 1.056 3.3







59

Table I.A.2

( n = 100 )

( c = 1.2816 ) (107. truncated)


p 0.100 0.098 0.031 0.032 0.039 0.805

p 0.100 0.095 0.044 0.044 0.040 1.101

p 0.100 0.094 0.044 0.044 0.041 1.079

p 0.100 0.110 0.043 0.042 0.045 0.931

p 0.100 0.102 0.049 0.050 0.045 1.110

p 0.100 0.096 0.043 0.044 0.046 0,941

a -0.500 -0.506 0.040 0.040 0.046 0.861 1,2

a 0.500 0.488 0.050 0.049 0.046 1.056 1,3

a

2 2 -0.800 -0.798 0.037 0.037 0.050 0.748

a 0.000 0.002 0.041 0.041 0.044 0.930 2,3

a 0.000 0.001 0.046 0.046 0.044 1.044 3.2

a 0.800 0.803 0.059 0.060 0.050 1.192 3.3







60

Table I.B.3

( n = 200 )

( c = 0.0000 ) (507. truncated)


P i 0.100 0.136 0.167 0.164 0.179 0.917

p2 0.100 0.099 0.204 0.206 0.182 1.133

p 0.100 0.083 0.167 0.167 0.186 0.902

p 0.100 0.052 0.167 0.161 0.180 0.895

p 0.100 0.142 0.182 0.178 0.175 1.018 "13

n 0.100 0.081 0.181 0.182 0.180 1.008 h

23

a -0.500 -0.517 0.189 0.190 0.187 1.018 1,2

a 0.500 0.530 0.210 0.210 0.189 1.109 1,3

a -0.800 -0.782 0.186 0.187 0.199 0.938 2,2

a 0.000 0.022 0.192 0.193 0.179 1.078

a 0.000 -0.022 0.191 0.192 0.178 1.079

a 0.800 0.826 0.217 0.218 0.204 1.069 3,3







61

Table I.B.3

( n = 200 )

( c = 0.0000 ) (507. truncated)


p. 0.100 0.117 0.141 0.142 0.120 1.179

p 0.100 0.107 0.118 0.119 0.127 0.936

p3 0.100 0.112 0.151 0.152 0.127 1.197

p… 0.100 0.135 0.131 0.128 0.126 1.014

p1 3 0.100 0.090 0.123 0.124 0.127 0.973

p2 3 0.100 0.109 0.129 0.130 0.130 1.001

a -0.500 -0.494 0,139 0.141 0.131 1.074 1,2

a 0.500 0.485 0.136 0.137 0.132 1.037 1,3

a -0.800 -0.794 0.116 0.117 0.141 0.830 2,2

a 0.000 -0.011 0.131 0.132 0.126 1.044

a3 2

0.000 0.002 0.145 0.147 0.125 1.173

a 0.800 0.836 0.154 0.151 0.144 1.049 3,3







62

Table I.B.3

( n = 200 )

( c = 0.0000 ) (507. truncated)


oA 0.100 0.097 0.089 0.090 0.086 1.051

p 0.100 0.091 0.095 0.095 0.088 1.086

p 0.100 0.096 0,080 0.080 0.089 0.907

p 0.100 0.113 0.083 0.083 0.089 0.923

p 0.100 0.109 0.078 0.078 0.090 0.868

p 0.100 0.103 0.102 0.103 0.093 1.108

a -0.500 -0.491 0.085 0.085 0.092 0.922 1,2

a 0.500 0.516 0.113 0.113 0.094 1.212 1,3

a - 0 . 8 0 0 一 0 . 8 0 3 0 . 1 1 4 0 . 1 1 5 0 . 1 0 0 1 . 1 4 9 2,2

a 0.000 -0.014 0.094 0.094 0.089 1.057 2,3

a 0.000 -0,014 0.086 0.086 0.088 0.975 3.2

a 0.800 0.781 0.101 0.100 0.100 1.002 3.3







63

Table I.B.3

( n = 200 )

( c = 0.0000 ) (507. truncated)


pA 0.100 0.109 0.051 0.051 0.061 0.837

P 2 0.100 0.100 0.054 0.055 0.062 0.879

p3 0.100 0.094 0.058 0.058 0.063 0.923

p… 0.100 0.093 0.058 0.058 0.064 0.918

p 0.100 0.105 0.076 0.076 0.063 1.201

p 0.100 0.095 0.064 0.065 0.065 0.986

a -0.500 -0.499 0.067 0.068 0.065 1.035 1,2

a 0.500 0.506 0.072 0.073 0.066 1.101 1,3

a -0.800 -0.782 0.074 0.073 0.070 1.039 2,2

a 0.000 0.019 0.070 0.068 0.063 1.079 2,3

a 0.000 -0.002 0.063 0.063 0.062 1.015 3.2

a 0.800 0.796 0.074 0.075 0.071 1.060 3.3







64

Table I.B.3

( n = 200 )

( c = 0.0000 ) (507. truncated)


p 0.100 0.099 0.038 0.038 0.043 0.884

p 0.100 0.099 0.049 0.049 0.044 1.109 2

p 0.100 0.097 0.047 0.048 0.045 1.069

p 0.100 0.110 0.043 0.042 0.045 0.931

p 0.100 0.102 0.049 0.050 0.045 1.110

p 0.100 0.096 0.043 0.044 0.046 0.941 "23

a -0.500 -0.506 0.040 0.040 0.046 0.861 1,2

a 0.500 0.488 0.050 0.049 0.046 1.057 1,3

a -0.800 -0.798 0.037 0.037 0.050 0.750 2,2

a 0.000 0.002 0.041 0.041 0.045 0.930 2,3

a 0.000 0.001 0.046 0.046 0.044 1.047 3.2

a 0.800 0.803 0.059 0.060 0.050 1.192 3.3







65

Table II.A.2

( n = 100 )

( c = 1.2816 ) (10% truncated)


pA 0.500 0.537 0.108 0.102 0.121 0.843

p 0.500 0.508 0.115 0.115 0.128 0.900

p 0.500 0.519 0.137 0.137 0.127 1.079 3

p 0.500 0.467 0.145 0.142 0.146 0.979

p 0.500 0.524 0,134 0.133 0.139 0.960

p 0.500 0.496 0.145 0.146 0.146 1.002

a -0.500 -0.514 0.178 0.179 0.178 1.003 1,2

a 0.500 0.531 0.190 0.190 0.183 1.040 1,3

a -0.800 -0.797 0.197 0.199 0.194 1.024 2,2

a 0.000 0.061 0.188 0.180 0.172 1.048 2,3

a 0.000 -0.010 0.176 0.178 0.169 1.049 3.2

a 0.800 0.814 0.222 0.223 0.198 1.127 3.3







66

Table II.A.2

( n = 100 )

( c = 1.2816 ) (10% truncated)


p, 0.500 0.509 0.083 0.083 0.082 1.011

p2 0.500 0.497 0.091 0.092 0.088 1.044

p3 0.500 0.494 0.085 0.085 0.090 0.943

p… 0.500 0.520 0.111 0.110 0.100 1.103

p 0.500 0.492 0.094 0.094 0.103 0.916

p 3 0.500 0.497 0.101 0.102 0.107 0.948

a -0.500 -0.494 0.125 0.127 0.126 1.008 1,2

a 0.500 0.487 0.128 0.128 0.127 1.006 1,3

a 一 0 . 8 0 0 -0.801 0.122 0.123 0.136 0.903 2,2

a 0.000 -0.013 0.111 0.111 0.121 0.920 2,3

a3 2

0.000 -0.012 0.140 0.141 0.120 1.175

a 0.800 0.825 0.143 0.142 0.140 1.016 3 f







67

Table II.A.2

( n = 100 )

( c = 1.2816 ) (10% truncated)


p 0.500 0.506 0.065 0.065 0.059 1.103

p 0.500 0.496 0.065 0.066 0.063 1.049

p3 0.50Q 0.509 0.059 0.059 0.061 0.961

p 0.500 0.507 0.072 0.073 0.071 1.023

p 0.500 0.510 0.066 0.066 0.071 0.931

p 0.500 0.508 0.073 0.074 0.075 0.984

a -0.500 -0.496 0.076 0.076 0.088 0.861 1,2

a 0.500 0.511 0.105 0.106 0.091 1.168 1,3

a -0.800 -0.799 0.090 0.091 0.096 0.945 2,2

a 0.000 -0.017 0.093 0.092 0.085 1.081 2,3

a 0.000 -0.020 0.082 0.081 0.084 0.954 3,2

a

3 3 0.800 0.777 0.100 0.098 0.097 1.010







68

Table II.A.4

( n = 400 )

( c = 1.2816 ) (10% truncated)


p 0.500 0.501 0.035 0.035 0.042 0.836

p 0.500 0.495 0.039 0.039 0.044 0.904

p 0.500 0.493 0.043 0.043 0.044 0.962

p 0.500 0.491 0.042 0.041 0.051 0.801

p 0.500 0.500 0.047 0.048 0.051 0.933

p 0.500 0,495 0.052 0.052 0.053 0.982

a -0.500 -0.500 0.062 0.063 0.063 0.998 1,2

a 0.500 0.506 0.066 0.067 0.064 1.041 1,3

a -0.800 -0.790 0.065 0.065 0.068 0.964 2,2

a 0.000 0.016 0.063 0.061 0.060 1.013 2,3

a 0.000 -0.001 0.062 0.063 0.060 1.049 3.2

a 0.800 0.795 0.072 0.072 0.069 1.047 3.3







69

Table II.A.4

( n = 400 )

( c = 1.2816 ) (10% truncated)


pA 0.500 0.497 0.029 0.029 0.030 0.964

p 0.500 0.496 0.031 0.030 0.031 0.999

p 0.500 0.498 0.034 0.035 0.031 1.121

p 0.500 0.509 0.032 0.031 0.036 0.853

p 0.500 0.504 0.039 0.039 0.036 1.085

p 0.500 0.505 0.033 0.033 0.037 0.892 h

23

a -0.500 -0.505 0.037 0.037 0.044 0.840 1,2

a 0.500 0.490 0.047 0.046 0.045 1.025 1,3

a 一0.800 -0.796 0,043 0.044 0.048 0.912 2,2

a 0.000 -0.005 0.039 0.039 0.043 0.918 2,3

a 0.000 -0.007 0.044 0.044 0.042 1.049 3.2

a 0.800 0.801 0.051 0.052 0.049 1.062 3.3







70

Table II.B.4

( n = 400 )

( c = 0.0000 ) (50% truncated)


p 0.500 0.520 0.131 0.131 0.137 0.952

p 0.500 0.507 0.123 0.124 0.141 0.882

p3 0.500 0.498 0.145 0.147 0.149 0.984

p 0.500 0.467 0.145 0.142 0.146 0.979

p 0.500 0.524 0.134 0.133 0.139 0.960

p 0.500 0.496 0.145 0.146 0.146 1.002 h

23

a -0.500 -0.513 0.180 0.182 0.179 1.016 1,2

a 0.500 0.532 0.198 0.198 0.189 1.046 1,3

a -0.800 -0.797 0.199 0.201 0.194 1.037 2,2

a 0.000 0.061 0.193 0.185 0.178 1.036 2,3

a 0.000 -0.009 0.180 0.181 0.170 1.066 3.2

a 0.800 0.817 0.229 0.231 0.203 1.138 3.3







71

Table II.B.4

( n = 400 )

( c = 0.0000 ) (50% truncated)


p 0.500 0.518 0.093 0.092 0.093 0.989

p 0.500 0.501 0.096 0.097 0.099 0.982

p 0.500 0.497 0.102 0.103 0.103 0.995

p 0.500 0.520 0.111 0.110 0.100 1.103

p 0.500 0.492 0.094 0.094 0.103 0.916

p 0.500 0.497 0.101 0.102 0.107 0.948

a -0.500 -0.497 0.127 0.128 0.125 1.022 1,2

a 0.500 0.483 0.130 0.131 0.131 0.993 1,3

a

2 2 -0.800 -0.803 0.124 0.125 0.136 0.919

a 0.000 -0.016 0.113 0.113 0.125 0.902 2,3

a 0.000 -0.016 0.140 0.140 0.120 1.168 3.2

a 0.800 0.820 0.144 0.144 0.143 1.013 3.3







72

Table II.B.4

( n = 400 )

( c = 0.0000 ) (50% truncated)


p 0.500 0.500 0.071 0.072 0.067 1.079

p 0.500 0.491 0.070 0.070 0.069 1.008

p 0.500 0.508 0.060 0.060 0.069 0.873

p 0.500 0.507 0.072 0.073 0.071 1.023

p 0.500 0.510 0.066 0.066 0.071 0.931

p 0.500 0.508 0.073 0.074 0.075 0.984 h

23

oc -0.500 -0.495 0.077 0.077 0.089 0.871 1,2

a 0.500 0.512 0.106 0.106 0.093 1.137 1,3

a -0.800 -0.798 0.090 0.091 0.096 0.950 2,2

a 0.000 -0.016 0.094 0.094 0.089 1.060 2,3

a 0.000 -0.019 0.083 0.081 0.085 0.962 3.2

a 0.800 0.779 0.101 0.100 0.099 1.014 3.3







73

Table II.B.4

( n = 400 )

( c = 0.0000 ) (50% truncated)


p 0.500 0.499 0.039 0.039 0.047 0.824

p 0.500 0.493 0.046 0.046 0.048 0.949

p 0.500 0.500 0.048 0.048 0.050 0.969 3

p 0.500 0.491 0.042 0.041 0.051 0.801

p 0.500 0.500 0.047 0.048 0.051 0.933

p 0.500 0.495 0.052 0.052 0.053 0.982

a -0.500 -0.500 0.062 0.063 0.063 1.005 1,2

a 0.500 0.506 0.067 0.068 0.066 1.029 1,3

a -0.800 -0.791 0.065 0.065 0.068 0.962 2,2

a 0.000 0.016 0.062 0.061 0.063 0.971 2,3

a 0.000 -0.001 0.062 0.063 0.060 1.054 3.2

a 0.800 0.795 0.073 0.074 0.070 1.053 3.3







74

Table II.B.5

( n = 800 )

( c = 0.0000 ) (50% truncated)


p 0.500 0.500 0.030 0.031 0.033 0.914

p 0.500 0.498 0.032 0.032 0.034 0.928

p3 0.50Q 0.499 0.039 0.039 0.035 1.110

p 0.500 0.509 0.032 0.031 0.036 0.853

p 0.500 0.504 0.039 0.039 0.036 1.085

p 0.500 0.505 0.033 0.033 0.037 0.892

a -0.500 -0.506 0.037 0.037 0.044 0.835 1,2

a 0.500 0.488 0.047 0.046 0.046 1.000 1,3

a -0.800 -0.797 0.043 0.044 0.048 0.911 2,2

a 0.000 -0.006 0.040 0.040 0.044 0.899 2 y 3

a 0.000 一 0 . 0 0 8 0.045 0.045 0.042 1.066 3.2

a 0.800 0.799 0.053 0.053 0.050 1.078 3.3







75

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82

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