Technical Publication TP-72-25

WHAT DISTINGUISHES AN INDEPENDENTLY

OBSERVED VECTOR FROM AN ESTIMATED

MULTIVARIATE NORMAL POPULATION?

By

A. C. BITTNER, JR.Systems Integration Division

3 April 1972

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

NAVAL MISSILE CENTER

Point Mugu, California

I0 A

NAVAL MISSILE CENTER

AN ACTIVITY OF THE NAVAL AIR SYSTEMS COMMAND

E. E. IRISH, CAPT USNCommanding Officer

D. F. SULLIVANTechnical Director

CDR W. H. Nelson, Head, Human Factors Engineering Branch; Mr. E. P. Olsen, Head,Systems Integration Division; and Mr. J. J. O'Brien, Associate Laboratory Officer, havereviewed this report for publication.

Technical Publication TP-72-25

Published by ....................................... Editorial BranchTechnical Publications Division

Photo/Graphics DepartmentSecurity classification ................................. UNCLASSIFIEDFirst printing ........................................... 145 copies

CONTENTS

Page

SUMMARY ......................... ........................... 1

GLOSSARY ........................ ........................... 3

INTRODUCTION ...................... ......................... 5

APPROACH ........................ ........................... 6

DERIVATIONS ...................... .......................... 6

DISCUSSION ................... ........................... .. 10

REFERENCES .................... .......................... 10

APPENDIXTwo Working Theorems .................... ..................... 11

TP-72-253 April 1972

NAVAL MISSILE CENTERPoint Mugu, California

WHAT DISTINGUISHES AN INDEPENDENTLY OBSERVED VECTORFROM AN ESTIMATED MULTIVARIATE NORMAL POPULATION?

ByA. C. BITTNER, JR.

SUMMARY

Once a researcher has determined that a multivariate observation x is different from anestimated population, he still has an unanswered question. He wants to know, "How is it

different?" A method for answering this question is considered in this report.

Two results are shown and both involve the estimated population parameters K-the esti-

mated mean, an i-the estimated covariance matrix. The first result answers the question "Whichlinear combinations of the elements of the difference x -x- are significant?" The second

answers the question "Which elements of the difference xo - x_ are significant?" Both resultsare derived using S. N. Roy's Union-Intersection Principle; hence, one can set an overall a/signifi-

cance/type-i level for the totality of tests.

A brief discussion of an application is also presented.

Publication Unclassified.

Approved for public release; distribution unlimited.

GLOSSARY

xo A p-component (p by 1) vector observation, "o", independent of the vectorsx1P x2.... 9 xýn

n The number of independent observations which are independent of the vector x0

i n-= n i A p-component (p by 1) estimate of the population mean vector/yi=n

IV A p-variate (p by 1) population mean vector

X A p by p population covariance matrix

An unbiased estimate of the population covariance matrix based on m degreesof freedom

A, A

x The inverse of the matrix X

a The probability that the (null) hypothesis H0 will be rejected when it is true

H0 The hypothesis that the vector x and the set of vectors x1, X2,... ?6are from the same population

H1 The negation of the hypothesis H0

N(g, •) The multivariate normal population with parameters M andA

m The number of degrees of freedom of the matrix estimate

t A "students" t-test statistic

c A (1 by p) p-variate vector of constant coefficients

3

_ct The p by I transpose of the vector c

_ The (1 by p) p-variate vector which maximizes the function T2(_)

A 1 by p vector which has a I as the i-th element and zeros elsewhere

A p-variate (p by 1) vector of partial derivatives whose i-th component is

NO a~T 2(C-)a(Ci)

0 A p by I vector which has zeros for all the elements

si2 An unbiased estimate of the i-th variate from NW, •)

D A matrix conformal with x

4

INTRODUCTION

In a previous report by the author (reference 1), a probability density/distribution function

was developed for an observed (p by 1) vector (xo) from a multivariate normal distribution withestimated parameters (see A-l)* This result enables a researcher to ask the general question

"How unlikely is it that the observation x arose from N(y, X) where y is estimated by 3? andA -0

Sis estimated by X?" Specifically the statistic:

(_Xn ° _ 1 )t _ (_ - (1)

mpwhich is distributed as -- F with p and m-p+1 degrees of freedom (df), can be compared

m - p+lwith tables of the F distribution for probability of occurrence. If the probability of occurrenceis less than some specified amount (a), then the hypothesis (H0 ) that xo is from the same popu-

lation that generated K- could be rejected. In other words, the hypothesis (H,) that the popu-

lations which generated x. and x- are not the same would be accepted.

The rejection of the hypothesis that x and K are from the same population (H0 ) doesn't

show which of the variates of x were significantly different. Although individual tests of sig-

nificance could be constructed (e.g., t-tests), there is no control of the overall a level for the set

of comparisons. The reasons for this are twofold; there are p such tests, and the variates are

generally correlated. Hence an approach is needed for testing the individual variates of x while

controlling the overall a-level. The purpose of the present development is to delineate such a

procedure.

*The results A-1 and A-2 are working theorems which are given in the appendix.

5

APPROACH

The approach employed here uses S.N. Roy's Union-Intersection Principle (reference 2,

and reference 3). This principle allows one to fix the overall significance level for thetotality of tests of linear compounds of the difference x - R. Operationally, this is accom-plished by employing the same (a-level) criterion required for the test of the most unlikelylinear compound, E(xo - JR), to all particular tests of linear compounds. Since the individualvariates x0 i - _x (i=l, ... , p) can be tested by the linear compounds ei(_x - _) (i=1, - - -this procedure will yield a solution to the problem posed in the Introduction.*

Let us define the statistic T2(c) as follows:

T2 (c) =••[c(x° nY)]t [c^ t -(]t c ]ý (2)

where c is a fixed 1 by p vector. This equation is a special case of equation (1) with the p-vari-ate terms xo,, and f replaced by the corresponding univariate terms cxo, cx and c Z ct. Theseunivariate terms are those appropriate for linear compounds of the respective variates (see A-2).Under the hypothesis (H0 ) that xo is from the same population that generated K, equation (2)is distributed as F with I and' m degrees of freedom. Hence, the most unlikely T2(c_) valuewould occur for that c which maximizes (2).

DERIVATIONS

In the following, a theorem will be stated which contains both the conditions for maxi-mizing T2 (c), and a procedure for testing the significance of the totality of linear compoundswith a fixed overall significance level. Consider the following:

Theorem 1.0. If T2 (c) is defined as in equation (2), then its maximum value is

2c n A

T2 _-) = ý+ (-X-o X4 x-)t' -X-o- (3)

mpwhich is distributed as -- F with p and m-p+l degrees of freedom when H0 is true. This

m - p+lvalue of T2 (_) is obtained for

_= XoX•) - (4)

*ei (i=l,., p) is a 1 by p vector with a I in the i-th entry and zeros elsewhere.

6

Further, the totality of linear compounds of the form c(Xo - K) could be tested for significance

at the overall a-level by the test:

H1> mp (5)

There will be one nonzero eigenvalue of (10) since the rank of (Ro -D) (xo _?R)t is oneA -and the product of it and any conformal nonsingular matrix (e.g., 7-1) would have the same

rank. Since the trace (tr) of (10) equals the sum of its eigenvalues (of which there is exactlyone, T2(c), that is nonzero), it follows that

T 2(7) = tr Ln•-T,. (2;° _ -F (tx° _ _)t 011)

This can, by the commutative laws for traces, be rewritten

T = (2-) tr [ jo - x (_Xo - K)J (12)

n (-)o - 0)t -I(- - D) (13)

which is the first result (3) of Theorem 1.0.* The distribution of (3) or (13) is given by A-1;hence, the first result of the theorem is completed.

The second result of this theorem (F) follows upon substitution of the value of__ given in

equation (4) for c in equation (6). This yields a T2 (c) value of

knU~ -V~~ (- 'o-X'o--(-o - Dxt '-0 - Y) (14)

or equivalently

( - j X O ) (15)

Because this corresponds to the maximum value of T2((c), equation (5) gives the desired vector.

The last portion of the theorem can be seen when one recalls the nature of the Union-

Intersection principle. Specifically, by employing the significance criterion (a - level) of T2 C@)

for each test of the type T2(c), one is assured that the totality of such tests has a joint signifi-

cance level of a. Since p F a:pm-p+l is this criteria, as indicated by the first part of them- p-Il ap

theorem and A-1, the general test (5) follows directly.

*See reference 4 for discussion of the rank of the product of two matrices.

**Reference 4 also contains a discussion of various results concerning the traces of matrices.

8

Tests for the individual components can be derived from specialization of equation (5).Since testing a specific component for significance (i-th) is equivalent to testing T2 (.e) for

significance, one could substitute ei for c in equation (5) and obtain a specialized result. Thiswill be considered in the remainder of this section.

Substitution of ei for c in equation (5) yields

H1> mp_'ej Fap-l (16)

H0 M - P m1

which by equation (2) is equivalent toH1 mp_! e --- F (17)( _ )[9e4(_Xo n ex--)0 [ et- -e -- -x--) < m - p+l a:p,m-p+l ( 7

H0

Multiplying out each of the bracketed terms,

n- A H1 mpXon -'i•iYJ[ 1} I'xoi - m] > - l Fa:p,m~p+1 (18)

H0

Awhere xoi is the i-th entry of x -i is the i-th entry of x_, and is the i, i-th entry of .Since in general the diagonal entries of Xare variance estimates, [:Jii' can be written asI12 where

S.s?2 is the estimated variance of the i-th variate. Thus equation (18) can be written:

n \(Xoi _ -i)2 H1Xi)_ )> mp (9

-s2

DISCUSSION

In the above, two results were derived which answered the question "How does the obser-

vation xo differ from the population which generated _?" The first of these, Theorem 1.0,

allows one to ask if any particular linear combination of variates (c__o) distinguishes xo from

the population which generated R_. The second result, Corollary 1.1, allows one to ask if any

particular variate distinguishes xo from the R generating population. Both of these results have

obvious practical application. Let us briefly consider one.

In a multiple criteria experiment, an unforseen (random) event occurs which makes suspect

a single observation xo. The first question which faces the research is, "Is xo from the same

population as the set of other observations (_x , x2 . . ,x2) from the same condition?"

This question can be answered by employing the statistic:

T2= V~ i -I( (21)

en A In 2 (n-l)pwhere_ = x- = l (xi - K) (xi - 3-)t, and T is distributed as F with

i= 1 'n-p

p and n-p degrees of freedom.* Given that this statistic (21) is significant, the next question is,A

"Which variates ofx° differ from the population which generated R and 2 ?" This could be

answered by applying Corollary 1.1 with m equaling n-I. Guided by the costs of observations

and their number, the results of these tests would be useful for decisions regarding the inclusion

of x as part of the data of the experiment.

The above does not exhaust the set of possible applications. Hopefully this application

will suggest others to the reader.

REFERENCES

1. Naval Missile Center. Probability Density When an Independent Observation is From anEstimated Multivariate Normal Population, by A. C. Bittner Jr. Point Mugu, Calif., 15 Jul1971. (TP-71-33) UNCLASSIFIED

2. Morrison, D. F. Multivariate Statistical Methods, San Francisco, McGraw-Hill, 1967.

3. Roy, S. N. Some Aspects of Multivariate Analysis, New York, Wiley, 1957.

4. Marcus, M. and Minc, H. Introduction to Linear Algebra, New York, Macmillian, 1965.

5. Anderson, T. W. An Introduction to Multivariate Statistical Analysis, New York, Wiley,1958.

*The statistic shown in (21) is a variation of A-I. A proof for its distribution was shown in reference 1.

10

APPENDIX

TWO WORKING THEOREMS

The first theorem (A-i) was shown in reference 1 by the author. The second theorem

(A-2) was shown by Anderson (reference 5) in 1958.

A-i

If x0 is an observed p-variate vector from N(y, ),

1 n

i=lI

Ais a mean vector also from N(m,$) based on n independent observations, and m$ is the sum of

the matrix products of mn independent N(Q, $) p-variate vectors (4 1, ? 2 , ie.

mS= E Z-iZi=lI

then

T2= n (x An+- -0 --Xto - -

is distributed as:

mp F

mr-p-p-i

where F has p and m - p + 1 degrees of freedom.

11

A-2

If x is distributed according to NM, Z), then Z = Dx is distributed according to N(Dy,

D:D t ).

12

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0) 0 o

cn4 I* ~ ~ t

44 p

u , C?) 0. f4 C'I ?

o) 0 z0 0 0Z

,I Z 0 -

m V

-. w 4- o c -4 -,

CCi)

ca 0.. : P14'.:

I 0nIý N

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UNCLASSIFIEDSecuritv Classification

DOCUMENT CONTROL DATA - R & D

(Se-crity classification of title, body, of abstract -td indesing an'otatian must be entered when It-l oerall report is classified)

I . ORIGINA TING AC TI VITY (Corporate autfhor) [2.. REPORT SECURITY CLASSIFICATION

Naval Missile Center - UNCLASSIFIED

Point Mugu, California 93042 2b. GROUP

3. REPORT TITLE

WHAT DISTINGUISHES AN INDEPENDENTLY OBSERVED VECTOR FROM ANESTIMATED MULTIVARIATE NORMAL POPULATION?

4. DESCRIPTIVE NOTES (Type of report and inclusive dates)

5. AUTHOR(S) (First name, middle initial, last name)

Alvah C. Bittner, Jr.

6. REPORT DATE 7a. TOTAL NO, OF PAGES 7b. NO. OF REFS

3 April 1972 12 5Ba. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S)

b. PROJECT NO. TP-72-25

c. 9b. OTHER REPORT NO(S) (Any othernumbers thatmay be assignedthis report)

d.

10. DISTRIBUTION STATEMENT

Approved for public release; distribution unlimited.

1. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Naval Air Systems Command

13. ABSTRACT

Once a researcher has determined that a multivariate observation xo is different from an

estimated population, he still has an unanswered question. He wants to know, "How is itdifferent?" A method for answering this question is considered in this report.

Two results are shown and both involve the estimated population parameters i-the estimatedA

mean, and $-the estimated covariance matrix. The first result answers the question "Which linearcombinations of the elements of the difference xO - R are significant?" The second answers the

question "Which elements of the difference x° - R are significant?" Both results are derived

using S. N. Roy's Union-Intersection principle; hence, one can set an overall a/significance/type-1 level for the totality of tests.

A brief discussion of an application is also presented.

FORM 174 (PAGE 1)DD, ,NOV1473 UNCLASSIFIEDS/N 0101•807-6801 Security Classification

21

UNCLASSIFIEDSecurity Classification

KE SLINK A LIN LINK C

ROLE WT ROLE WT ROLE WT

ApplicationsMultivariate normal

Normal populationSignificance testStatisticsUnion-intersection principle

DD D, '.. 1473 (BACK) UNCLASSIFIED(PAGE 2) Security Classification

22