-. ..4Y,_5#C,'_-/_-#,//7

3 1176 00134 2527

•NASA Contractor Report ]59]]7

NASA-CR-159117

Iq"79 o o 00'72'0

THE ALTERATIONOF PROFILEANALYSISTO

ACCOMMODATETESTINGFUNCTIONS

RaymondH. Myers

RAYMONDH. MYERSBlacksburg, Virginia 24060

..

AoL-", '"".LANGLEYRESEARCHCE[LII'__R.

NASAPurchase Order L-72497A LIBRARY,tIAS#;

JuIy 1979 _._,,,,_.ot.j._,.,'I_=LU._

NationalAeronauticsandSpaceAdministration

LangleyResearchCenterHampton,Virginia23665

https://ntrs.nasa.gov/search.jsp?R=19790020780 2018-03-03T12:28:26+00:00Z

t

THE ALTERATIONOF PROFILEANALYSISTO\

ACCOMMODATETESTING FUNCTIONS

The purpose of this task is to develop a methodology for testing for

differences between several pilot describing functions, where the data

points represent averages at various frequencies. Typically a functions

represents a treatment or treatment combinations which are to be compared.

Data are taken in replicates and an output is measured for each frequency.

It is not unusual to experience 4-8 functions representing a like number

of experimental conditions in a real-time piloted aircraft simulation.

Typlcally, as many as 8-16 frequencies may be involved.

Consider the following plot for the special case of two functions.

_22 B24

function _2__/

value _1(degrees) _ 4

frequencies, rad/sec

Here, the Uij represents the population mean for the jth frequency under

the ith function. The problem is to determine whether or not, simultaneously,

the means for function i are equal to the means for function 2, across

frequencies. The extension to more than two functions is also of con-

siderable interest.

Specifically the goals of the task are as follows:

(1) Develop and describe the methodology for testing for differences

between functions.

2

(ii) Determine how to approach the problem of assessing the power of

the test. Supply charts or tables for power.

(iii) Use the power results to recommend how to design such experiments;

for example determine an effective number of frequencies; in addition,

what is the maximum number of functions that will allow a reasonably

sensitive test. Also what is a reasonable number of replicates?

(iv) Discuss software considerations.

Basic Assumptions for the Experiment

In this section we shall discuss the distributional and data structure

proposed for the test on k functions (k_2). Consider initially two func-

tions, in the context of Figure I. Suppose that we have two random vectors

Xll "] x21

x12 I x22_i = ." .' and _2 = "

! • I

x2t.J

th thwhere x. is the scalar output for the i function and the p frequency.

ip

These are basic output measurements. It is assumed that each of the

vectors follows a multivariate normal distribution with common variance-

covariance matrix Z, the latter a txt matrix. The practical implication

here is that within each function the observations are correlated and the

3

correlation structure is the same for each of the two functions For the

first function nI independent vectors are taken and for the second function

n2 vectors are observed•

The extension to more than two functions is obvious. It is assumed

that there are k-multidimensional vectors xI, x2,...,_

Xll x21 Xkl

x12 x22 Xk2

• • •• • •

Xlt , x2t Xkt

The assumption is •made that x. is multivariate normal with mean--i

(i=1,2,... ,k) and variance-covarlance matrix E.

Hypothesis

Consider again the case of two functions illustrated in Figure I.

A reasonable hypothesis that accomplishes the goals outlined here, is

given by

H0: _ii = B21

_12 = _22 (i)

_13 = B23

_14 = B24

or, in terms of vectors,

H0: _i =_2 '

where _. is the mean vector for the ith function, one can easily see

that this is the hypothesis that the experimenter needs to test. Namely,

the null hypothesis states that the mean value of the two functions are

equivalent at each frequency. In the general case of k functions and t

frequencies the hypothesis is

H0: _i = _2 = _3 = "'" = _k (2)

where each vector is t-dimensional. We need a test procedure for testing

the joint statement given in (2). In addition, we need guidelines on

the experimental design that will adequately give evidence in favor of the

alternate hypothesis when it is appropriate.

Profile Analysis

The problem posed here appears to be very similar to Profile Analysis

[2] which is a multivariate technique in which a "repeated measures"

design is used and the treatments are broken into groups. Basically,

measurements on individuals (or individual items) are conducted for each

of say t treatments, with the observations correlated from treatment to

treatment. The purpose of the experiment is to determine if there are

differences in treatment means. When the treatments are put into an

additional classification variable called "groups" then the procedure

becomes what is called Profile Analysis. Consider, again Figure I and

allow the functions to take the role of the groups and frequencies take

the role of the treatments. This would appear to be an application of

Profile Analysis.

5

In a Profile Analysis the test on treatments is not valid unless no

interaction exists between treatments and groups. Thus, initially a test

for parallelism (no interaction) is usually conducted before the test on

treatments is attempted. The hypothesis of the test on parallelism is

given by

H0: _iI - BI2 = _21 - _22

_13 - _12 = _23 - _22

BI4 - _13 = _24 - _23m

The test involves a Hotelllng's T2 [2] and is quite easy to extend to

more than two groups.

Alteration of Profile Analysis to Accommodate

Testing Functions

One can easily see (easier to visualize in the case of two functions

or groups) that if groups and interactions between groups and frequencies

(functions and frequencies) are notstatistically significant, one can

interpret this as implying that functions are not significantly different

point by point or frequency by frequency. One mode of verfication of this

is to note that if two functions are parallel between frequencies the only

way that the two function averages can be the same is for the two functions

to coincide. On the other hand, function averages might be very close

together but because of non-parallelism (interaction) the two functions

may be far from coincidental. Thus neither a significance test on groups

(function) averages or interaction will suffice for our purposes. Rather

it is the joint hypothesis of the two that is relevant.

6

Searching into the structure of the problem more closely suggests

that the condition of coincidental functions we have described here is

identical to the hypothesis in (I) or (2) for the general case. Thus it

can be said that we have a profile analysis in which we are attempting

to simultaneously detect parallelism and group (function) effects but we

are doing it with a single hypothesis and thus a single test and not with

two tests.

Test Statistic and Procedure

For the hypothesis in (I), we are simply testing the equality of

two mean vectors. For the jth function, n. independent vectors (replicates)J

are obtained and we have two averages _i' which are joint estimates of

the mean vectors _i and _2 (t-dimensional). The sample data is used to

compute an estimate of the variance-covariance E. Thus we have for the

first function, nI independent vectors, and n2 independent vectors for

the second function. The estimate of _ is found by obtaining sample

variances and covariances for each function and pooling over functions.

Thus the estimates have nI + n2 - 2 degrees of freedom. The test statistic

is given by [2]

T2 = (Xl-Xm)S-l(_l-X2) (3)

where

S = _(_i + l___)n2

7

The statistic in equation (3) follows Hotelling's T2 distribution with

nI + n2 - 2 degrees of freedom. If the same number of replicates are

taken for each function, say n, then of course nI + n2 - 2 is replaced

by 2(n-l). The statistic

(nl+n2-t-l) T2 (4)(nl+n2-2)t

follows a F distribution with t and nI + n2 - _ - i degrees of freedom•

This fact allows us to handle the power quite easily.

The extension to more than two functions is quite easily done•

Consider the hypothesis of equation (2), which, written in expanded form

is given by

H0: _iI = _21 = _31 = "'" = Bkl

U12 = _22 = B32 = "'" = _k2

_ _e

_It _2t _3t "" = _kt

m m n

Once again, there are vectors of averages _i, _2"''' _ ' t-dlmenslonal•

This data is used to compute a pooled estimate of the varlance-covarlance^

matrix £. In addition a txt matrix (hypothesis matrix) is given by

kh = E i i

rs j=l n--jTrj Tjs - N Gr Gs "

" Here T is the sum of the data on the rth frequency for function J.kjr

N = r nj • G is the grand total of all observations on the rthJ=l r

frequency. The test statistic is the largest elgenvalue of the matrix.

HE-l, where E is the error sum of squares matrix, which is given by

^

E= r. • d.f.

where d.f. are the degrees of freedom for estimating the variance-k

covariance matrix (d.f. = i=Zlni - k).

Examples of the computation for both the special case and the general

case will be given in a later section. Charts for the use of the largest

root statistic can be found in Morrison [2]. A computer program for the

computation of the test statistlc will be found in the Appendix.

Power of Tests

The power associated with the case of two functions is quite easy to

handle from a theoretical point of view. In a practical sense there are

some difficulties that are not insurmountable. The test statistic in

(3) follows Ft,n1+n2_t_I_ under the hypothesis of equality of means

stipulated in H0. One can immediately notice that the denominator

degrees of freedom automatically imply an experimental restriction,

namel Z that nI + n2 > (t+l). The test procedure, of course, involves

rejection of H0 if the calculated test statistic in (4) exceeds the

tall percentage point of the F-distribution. If _i _ _2 the testupper

statistic follows the non central F-distrlbution [i] with degrees of

freedom t and nI + n2 - t - i, and with non centrality parameter -

62 = nln2

nl+n2 (_i-_2)'E-I(__i-__2)•

Thus for a specific dlfference_l -_2 between the mean vectors and a

specific variance covarlance matrix , the power is given by the prob-

ability _ •

Pr[F_,nl+n2_t_l,_2 > F_,t,nl+n2_t_ I]

',nl+n2_t_l,_ 2 is a non central F varlate and F ,t,nl+n2_t_l iswhere Ft e

the upper ath percentage point of the F distribution. Thus the probability

distribution of the non central F is needed. Thlsdlstrlbutlon is

discussed in detail in Graybill [i]. Power charts are offered in

Morrison [2] and are quite easy to use. An explanation of their use is

given in the text.

In order to compute the power one must know E, or at least have an

estimate of it. For the studydescribed here, hypothetical values for

the variance-covariance matrices are used, these values generally taken

from previous studies at NASA Langley. A power study is described in

the following section. This study was done to arrive at appropriate

values of sample sizes (number of replicates) and number of frequencies

that provide good power for specific values (in terms of order of magnitude)

of the differences (_l-U_2) that are being detected between the functions.

The power computation for the test for more than two functions is

extremely difficult and is wrought with practical difficulties that

actually prohibit its use. Approximations are available [3]. However,

the approximations involve many quantities that must be estimated from

the data. The non central F probabilities are monotonic increasing

functions of the non centrality parameter which, if oneinvestigates

the structure of 62, is very pleasing. That is, 62 represents "how

l0

false" the null hypothesis is before we would wish to reject H0 with a

high probability, and as 62 grows large we would expect the power to

increase monotonically. However, the structure of the approximations

of the power of the "largest root test" is such that several non centrality

parameters appear and the power is not a monotonic function of them which

suggests that for at least some cases the approximation is not particularly

good.

It would seem reasonable then that sample sizes and other recommenda-

tions should be done for testing two functions at a time. In addition,

the user can assess his own power in a particular case by computing the

power through the use of tables of the non central F. This will be

reasonable aslong as one d__oe___ssno____tstudytoo many functions simultaneously.

This does not mean that one must restrict the number of functions in the

test, but rather in a power assessment it would only be accurate if the

number of functionswere kept lo____w,say four or less. It should be

emphazised here that the value, t, the number of frequencies, ni, the

number of replicates are more crucial than k, the number of functions.

Example of Test Computation

In this section we present some hypothetical data and numerical

examples of the computation for cases of two functions and three functions.

The output here is not what NASA Langley personnel has at its disposal

but it does represent a representative type format for the output. For

the case of two functions there are t = 5 frequencies and nI = n2 = 4.

5 T A 1 1 5 1 1 C A L A N A L Y 5 i 5 5 Y S T h. M 8€5] TUESDAY, MANCH 7, 1978 I

Ot_5 GkL}UP YI Y2 Y3 Y4 Y5

i I 12.a 9.5 lb.4 13._ 15.22 I 13.3 10.2 I_.7 13.9 15.__i 1 l.i.l I0.4 14.9 14.2 14.4

I I_.8 10.1 15.2 14.u 14.95 2 IO.b 9.3 12._ 12._ 11.bb 2 9.3 8.7 12.1 12.1 12.4! _ 9.9 9.0 ll.b 11.8 12.0

,_ 10.1 _,_ 11.9 11.9 11.9

p.A

6 I A [ I 6 I 1C A L A N A L Y 5 I 3 5 Y _ f E M 8:53 TUESDAY, MARCH 7, 1978 2

ANALYSIS OF VARIANCE P_UCEOURE

CLA@5 LEVEL INFORHATION

CLASS LEVELS VALUE5

G_UUP 2 1 2

NUHUt_ OF Ub_ERVATIUN_ i_ UAIA @_I : 8

Fa

5 T A T I 5 1 I C A L A N A L Y S ! 5 5 Y 5 T L M 8:53 TUESDAY, HA_H 7p 197B ]

ANALT_J5 UF VANIANCE P_UCEDURE I

kPE,_DE=_I VA_tAoL_= 11

UU_CL uF 3UPI UP SUUAK_ HkAN 5UUARE F VALU_ PN • F N'SQUANE C=V=

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_u_ b Io_75U0UU U.d_1_5000 5lD UEV YI MEAN

O_CT_U IOTAL I l_,_elSuuu 0._7U37_19 11,=3750000

OUNCE DF AI_UVA 3_ F VALUE PR • F

_OUP l 11olt1_bguU 17.3_ 0.0001

I-=

5 1 A 1 I 5 T I L A L A N A L Y S I 5 S Y S T E H 8=b3 TUESDAYe HATCH 7w 1978

ANALYSIS UF VARIANCE PRUCEUURE

iOU_C_ UP SUM UF b_UA_5 _AN SQUARE F VALU_ PH • F H-SUUA_ C.V.

'UO_L 1 _._2UVVUO0 2._2000000 22.0U O.U_5_ U,7_b71_ 5.Q912

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JUu_C_ UP A_UVA S_ F VALU_ PN • F

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I A T I b T I C A L A N A L Y 5 I S S Y 5 T E H 8:53 TUESDAY, MARCH 7r 1978 5

ANALYSI_ UF VARIANCE PROCEOURE

U_P_o_O_I_I VA_IAeL_: 15

SUU_C_ bF EU_ UF SuUAHE_ MEAN SUUAHE F VALUE PH • F E-SQUA_ CoV,

,4Uu_L 1 IB.911_bUOU 18.91125UOU 203._3 O.UOO! 0.971Jb_ _.2_b9

E_v_ b U.b_7buuO0 U.Uq291bO7 510 U&V Y3 _EAN

C_._CTtu I_IAL / i_._bo7bU_O 0.50_Hd_Sb 15.51_b0000

5UU_t UP A,wUVA b_ F VALU_ PR • F

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I A 1 1 3 1 I [ A L A N A L Y 5 I 5 5 Y 5 T E M _:b3 TUESDAY, HATCH 7, 1978 6

ANAL_Eb OF VAHIANCE PNUCEOUHE

SOURCE UF _UM UF _UAN_ M_AI_ SWUANE F VALUE PR > F R'SQUARE C,V.

HOU_L 1 7.b0bbVUUU 7.bVSOU00U 19_.17 U.UUU1 U.g7UV2b 1.522_

E_O_ b b._3bOOUUO U.O391bbb/ S[_ UEV Y_ MEAN

CUK-ECT_iJ [OTAL l 7,e4UUOuOU 0.1g7_VbTO 13.OOUUO000

50U_C_ UF AI_UVA _5 F VALUE PH > F

GRUUP ! ?,bubuOvuO 19Q.11 U.UU01

T A I I S T I C A L A N A L Y 5 I S S Y S T E M 8:53 TUEGUAYp MA_H 7w 1978 7

At_ALY_IS OF VAHIAr,ICF. la_UCEDURE

UFPL._IO_;_[ VA_IAt_LE: Y_

,SUu_CF.. UF 3UH UF 5GIUArc_3 MkAN SQUARE F VALUE PH > F R-SUUAHE C.V.

:,;UO_-L 1 Id. VUOUUOOU l_.OOOOOOoO 120.61 O.OUOt O.952bJJ 2.8662

I:,_U-_ b V._'_SUO000 0.1_916667 GTD UEV Y5 F1EAN

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SOU_I_E L)P ANUVA _ F VALUE PR • F "

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b T A I 1 5 T I C A L A N A L Y S I 8 S Y 5 T E M 8=53 TUESDAY# _A_CH 7p 1978 8A=WALY_I5 OF VARIANCE PROCEDURE

= ERROR SS&CP r4ATRIX

OP=b YI Y2 Y3 ya Y5

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T A l 1 S T i _ A L A N A L Y 5 I S S Y S T E M 8:53 TUESUAYr MARCH 7w 1978 g

A_WALYSIS UF VARIANCE P_UCEUUHE

H = AI_UVA 55&CP MATRIX FUN: GROUP

ur:| YI Y_ Y5 Y_ Y5

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".,'r'-%..'Y'_

5 A 1 1 b i L C A L A N A L Y S I 5 _ Y 5 [. E _i 8:55 TUESDAY, MARCH 7p 1978 2

ANALYSIS UP VARIANCE PRUCEDURE

CLA_5 LEVEL INFORMATIUN

CLASS LEVELS VALUES

G_UUP 3 1 _ 3

NUM_ OF b_SEHVArlUNS LN UA_A SET = 12

5 I A 1 1 _ T L C A L A N A L Y S [ S 5 Y $ T E M B:53 TUESDAY, NARCM 7, 1978 3

ANALY_]8 OF VARIANCE PROCEDURE

50u-C_ UF _UF, UF 5_UAHt3 FI_AN SQUARE F VALUE" PN > F R-SQUARE C.V.

;_UDEL @ bl.q31bOh67 25.71683333 ib6.91 U.UOUl 0.9721@I 3.20_5

_kKUR 9 I._lbVVVVO 0.Ib38_8_9 5T0 uEV YI MEAN

CU_ECIEU IUTAL 11 b@.gUbbbbb7 O,QUQ_3195 Z2.b3333333

SO_C_ OF A_OVA 55 F VALU_ P_ > F

G_UuP _ _l._316bbbl I_6.91 0.0001

%

T A I I _ T L _ A L A N A L Y S Z 5 S Y S T E H 8_53 TU_SUAY, HARCH 7t 1q78 Q

ANALT_LS UP VANIANC_ P_OC_DuH_

U_P_=_0L_I VAWIAOLd= Y_

SOurCe IJF 5U_ UP 5_U_W_ P_AN 50UARE F VALUE PW > F R-SUUA_ C,V.

r_OOEL _ 19,IbUUUUUO 9.88UUuUO0 9_.bO O.UOUI U,95#|_8 3.|39t

b,_.tJ_ 9 U.qbuUUVO0 0.10555556 5TI) UEV Y2 _EAf4

Co.-£UI_u IU1AL 11 du.itOuOUOO 0,32_8931_ IO,35UUO000

50u_CL OP AJ.UVA 55 f VALUL PR • F

G_u_H _ I_.ibOUUUOU 9_.fiO U,UUU1

f_UI

,° .'_"..•

•" o. • •

o° ._

5 ; A 1 I S r L _ A L A N A L Y 3 I 5 S Y 5 T E H 8:5] TUESDAYe HAdCH 7p 1978 5

A,_ALY515 OF VARIANCL PNOCEOUHE

UU_C_ OF SUM UF 5UUkk:5 MtAN SfJUARL F VALUE PH • F _-SUUA_b C,V,

UU_L _ _U._]_bbbl 2b._lOSJS_ 225._1 O.UUOl O.g_O_IU 2.2825

_U_ 9 I.uu/buuuU 0.1119_ 510 U_V Y5 _EAN

O_CI_u luIAL 11 _].q_lbOb7 O.3JQb_IOO t_.bb83J333

U_CE uP AJ_UVA b5 F VALU_ P_ > F

o_

3 T A T Z ,5 r I c A L A N A L Y S I 8 8 Y 8 T E H. _:53 TUESUAYp MA_CH 7w 1978 15

Ar,IALYS.[S UF VA_.,[ANCE PHUCI..DUI.+E

U.t:.+"t."Jde._+;l V_IA_LI;: Yq

,_OL,=',CE UF 5Ur,; uF 5UUA_P.b t,IP..Ai_ 5t,1UAHE F VALUF PH > F R-SUUAHE. C.V.

•luUE.L ," 3d._l lbb6b ! 16.20583333 z148.78 U.UOuI u._guu72 1.3557

;-_,_b_ '_ u-3'JSisOuoO U.U3611111 8TU DEV Y_ HEAN

CO_[-.C IIr U TUI_L II 3e.73bboobT. U.19OUd9 '_ l_.01bbbb67

_UU_._ UP" A6,UVA bb F VALUE PN > F

G,_OUP _ 3d. _ 1 1 bbe6 1 _q8.7 t_ 0.0 0 U1

ha

5 T A 1 1 5 T I C A L A N A L Y 5 I 5 S Y 5 T E M 8:53 TUESUAYw HATCH 7t 197_ 7

ANALYSIS UP VAHIANC_ PRUCEUUHE

O_k_.Iu_,,l _AMIA_Lt_ fb

SUu_Ct UF 5Ut4 UP 5UUA_t5 I.*_AN 5UUAHE F" VALUE PN > F R-SUUA_t C.V.

;_OdtL d _.oUtbb_b/ 11.900_33_ 113._4 V.UUU1 0.951613 2.3201

g_u_ 9 v.94bUVuvO U.10500VOO 510 UEV Y5 MEAN

C,J._ZCItO TUIAL 11 d4.lqbbObb/ 0.3_0_703 13.gbbbbbb7

5OUNCE UF A_wUVA b5 F VALUE PR > F

GROUP d _._Olob_b/ 113._ O.OU01

O_

T A I l _ T I C A L A N A L Y 5 I 5 3 Y 3 T b H _:_3 TUESDAY, HAHCH 7t 1978 8

ANALYSIS UF VAHIANCE PRUC_DUR_

: _HUH _S&CP MATHIX

DF=q YI Y2 Y3 Y4 Y_

Y! 1.415uuuvu u. IuUUUUUV -U.477_UUUU u._q75uuo0 -u._qZSOObOY_ Volut_uvuuu Uog_UUUUUU -u.UbSUVUUO U.IqUUUUUU °V._UUUUOUY5 -u._ll_vUVU -U.UobuO(J_O I.UU75UUUO U.UU7bUUUU -v.21/buvOuY_ u._7_uuvu U.I_UUUUuU U.VUT_UUOU U.5_SUUUUU -U._OOUOUUY_ -u.q_/_vuvu -U._OUUVOU -U._II_UUUU -U._3UUUUUU U.9_SUVVOU

PARTIAL CUN_ELATIU_ CULFFI_I_,_T5 FHUI4 THE ERRU_ 536CP HAT,IX / PHU_ > :RI

uF=_ YI Y_ Y3 Y_ Y5

TI _.UuUOUU U.5915_q -0.5_[10! U._29bHQ -U._15_7v. Ovuo 0.0118 O._b_O 0._15_ O._

Y_ O._5_Q 1.000000 -O.U_bQQU OeSQ_qOU -U.5_9_5

Y_ -O.3_|?Ul -O.vb_U I.UUUUU_ U.OI_IO7 -O._gv_v._o_v V._b5_ U.UUvV O._llJ O.5_b_

Y_ O.Q_Y_H_ U.5_1_0 U.OLSIU7 I.OOOUUO -_.Q)SU_IU._lbd U.5_5 O._Tl_ U.UOOU O._u

Y_ -U._dlS_7 -O.b_qq_5 -U,_e_gu_ -0._15081 I.OUOUUU "o._d U.U_5_ U._59 O._u O.UUVO

T A I I S T L C A L A N A L Y S I U S Y _ T _ M _:SJ TUESDAYw MARCH 7e lg7b

ANALY_I_ OF VARIANCE PRUCEOUHE

H = Av_UVA _&CP MATRIX FOR: GRUUP

_F=_ YI Y2 Y3 Y_ Yb

YI 51.Q3|bbb6! 3u.b3uuVOUO 50.SIQl_bb? QU.58583333 31,660_333Y_ 3v._3vuuuvv Ig./6uvvuvo 30.l_oooooO 25.u30uovou Ib.b3UOOOuOY3 bu._lulb_7 3U.I4UUVUUU bO._2lbobb7 39._bu83333 31.970_3333Yg _u.5_b83333 _b. U3OUOVOU 39.gbOE3333 3e. Qllbbbb7 _3,bghBbbblYi 3t.bbU83333 Xu.b3uuuuuu 31.97083333 _).bgbbbb67 _3._O|bbbb7

C--_ALIcMI3T. IC _uuIb Ar_U V_UFU_5 UF: _ liwv_bE * ,. WHERE H = AhUVA 556C.P MATRIX FUR: G_UUH E = ER_UR SS&CH r4AT_IX

C_A_ACI_IaIIC P_C_NT CMARACI_HI_IIC V_CTUk V'tV=INUU(YI Y2 Y3 Y_ Y_

_bu.elO_3_b_ _o._3 O.b_SuOo_b 0.31_02_93 0.8_38821 0.891u_337 1.I0_67913

b._bb2971d 1.3! -U.)911o_4_ 0._89_b9_3 -U._79_0_95 1.18_9_U_ "U.5_blUbgOo

U.UUUUUUuU U.uu -U.U7_UlgU_ 1.0711142_ U._SlOb3ob -1._1133ba7 U.176_b957

u.ubuuvuuu u.uu O._bS_13bU -O.b_Ibbg_U O.8OOblI_b -U._2b_O3bU -O.bBlbb_O_

u.uuuuuuvv u.Ou O._13/blu_ -U.38162081 -O.03_l_O_/ -o.5,egeeQ7 -O.3bblSb8_

5 T a I ( 5 T I C A L A N A / y 5 I S 5 Y 5 T E M 8:55 TU_5OAYp MARCH .7, 1978 10

ANALY313 uF VARIANC_ PRUCEL)UF(_

;,IAI_UVA ItSl (_RI|_HTA FUH THP_ HYPOTHFST@ OF NO OVFHALL GROUP EFFECT -'r

H = A;_UVA 5S&CP MATRIX FOR: GRUUP

= UtP. VArIAbLE5 = 5 _'t ") t:_u U = RAU_ UF H = 2

............ . ,.i = HI,';tP,(J) = 2 "'i/_-r.._.-...;, _:,¢.; ..,-..._ i.;: .bLAUS(P-U)-I) " 1.0 " !

' J_ -- .b[,_r--P-1) = 1.5 X_,o") -I_. I

I_m_IIWmmmmmmII_mm w_--Immmm_Im_miIImmlwm_.mm mm m_.omimwmmmmmlmi mI_iIm Imm_mmmImmmm_m iiiImmii_mImi

rIUIr.LLI(_G-LAt,LI-y II_A_I- = Ih(t_-*m-lmtlj = _Sb.4b(_'/.l_bO (SF_ PTLLAIW,5 rAbLF_ #_)

I" APHROXIt.IATIuI_ : €:[b*N+I)*T_([_.**'I*H)I[b*5*(2H+_.I)) nlTH 5(2F;+5+1) AND _(5*lW+l) OF

h(IV,_} = l(}_.b9 PRUc$ • F = U.Ot)01

PILLAI'b ThttCt V : I_((H*II'_V{H+*-)) : 1.d5997_)/40 (SEF PILLAI'b TABLE #_)

k A_'P'UXlF;AIIu,,_ = 1_,_+5+1)/(,_,_.5+1) * V/iS-V) nITH 5(_14.5.1) AND 5(_N+5+1) UF

FitO_l "_) = 15.9_I Pk(U_ • F = O.O00t

,IL_b' LI_I[_(IuN L = b_l [_.)/bt.l i_i+h.) = O,OUO.IObq.l (SE_ |_KA VOL 53 P' 3(47)

,', : "(,Wt-.b(e'-b_+l,I J*L(_(,L) : 5b.bSbb

.............. .........PI,w_r CA*,UO_ICAL VAIiLA'_LP_ YI*'LU5 AN F UPPI'R tlbUNt) _'

h [,,9) = Ot),b.gb (UPPER :O_U) ....... ]__.

............];• ,}

,.o,

32

First, all univariate F-statistics are given comparing function values

for separate frequencies; In addition, the _ and H matrices are given.

The Hotelling's T2 value is given for the case of two functions and the

largest eigenvalue of E-IH is given for the case of three functions.

In both cases, the test statistic is significant. These examples are

shown in the Appendix.

Recommendation of Experimental Design

Through Power Study

An extensive Power Study was made in which the power of the test was

computed for varying number of frequencies, number of replicates, correla-

tion structure, and assigned differences between the function means. A

set of plots are provided in order to give the reader an indication of

what magnitude of power can be expected for various combinations of the

parameters and to devise recommendations of what are reasonable values

for the number of replicates and frequencies, given a particular correla-

tion structure. It should be emphasized that regardless of power con-

siderations there is a certain restriction that must hold. For the case

of two functions, if we call n the number of replicates for each frequency-

function combination, the denominator degrees of freedom for the F

statistic will be non-positive unless

2n > t - i (5)

Thus equation (5) represents a necessary restriction in the case of two

functions. In the general case it would be adviseable to keep the same

33

restriction, i.e., that given by equation (5) because it is unlikely that

the quality of the test (i.e., power) will be good at all unless n

satisfies equation (5). In addltionsince is is recommended tocompute

the power on the basis of Hotelling's T2 (tests on two functions) one

cannot assess the power unless equation (5) holds.

We attempt here to give some rationale regarding our power study.

The most difficult variable to cope with was the covarlance structure.

Keep in mind that the correlations represent association between observa-

tions from one frequency to another within functions. The following

covariance or correlation structures were studied:

(a) Low correlation, correlation constant (0.i, 0.2)

(b) High correlation, (0.6-0.9)

(c) Mixed correlation between high and low (0.1-0.95) .

In addition, the power (through the noncentrallty parameter) is a function

°f _i -_2' the difference between the means of the two functions. Here,

we are assuming that there is a typical difference which is a proportion

of_, where c is the same at each frequency. Proportions of 0.3, 0.5,

0.7, and 1.0 are used in this study. We feel thatit is reasonable to

assume that 0.70 - 1.00 is a reasonable range in the difference between

the two functions which one would like to detect. It is clear from the

plots that for any value less than 0.7_, experimentation is much too

demanding if one wishes to obtain high power. The significance level of

- the test was fixed at _ = 0.I0. Plots of power against frequency are

shown in the Appendix for various correlation structures. Curves are

shown for various proportions of o which one is attempting to detect.

34

The purposeof the plots is to attempt to show the role of t, the number

of frequenciesand n, the number of replicates,on the power. The plots

indicate the following:

(a) All correlationsequal - low correlations(0.I,0.2). The

role of t, the number of frequenciesis displayeddramaticallyin the

plots shown. As t grows large, the power is reduced,and for t > i0 the

only conditionunder which power is moderatelygood is when n > 10.

Thus one can assume that the testwill be ineffectiveunless t < i0;

but if t must exceed i0, the number of replicatesshould be 10-12.

(b) High correlation(0.7-0.9). Here it is very clear that despite

the number of frequencies,in order to obtain any effectivenessin the

test, at least 12 replicatesshouldbe used. Of course,it is probably

an unusualexperimentalsituationfor which all correlationswould be

high.

(c) Correlationsmixed betweenhigh and low. This conditionlikely

representsthe most typicalexperimentalconditionthat one confronts.

Again, however, it appearsas if small numbers of replicateswill not

give power values that are acceptableif one desiresa quality test.

For 8-10 frequencies,n = i0 is barely acceptablewhereas for t > i0,

a sample size of at least 12 is necessary.

ConclusionsL

A test procedure is given for testing equality of k different pilot

describing functions. The method for handling the power for the case of

two functions is described. It is felt that this is not an unreasonable

35

assessment of power as long as the number of functions is four or less.

For rather typical types of correlation structures a power study is made

in order to determine reasonable number of replicates and frequencies.

Two other recommendations come to light, in addition to those involving

the experimental design. These are given in the following two paragraphs.

In using the techniques described here the NASA Langley personnel

should be very careful to sample to determine what correlation category

prevails. The power is very much dependent on the correlation. In fact,

if possible it might be adviseable to take data initially to determine

the approximate correlation structure, then supplement with the required

data according to the experimental design recommendations.

The user should not be attached,to a particular significance level.

We used c = 0.I0 for the power study. However, because of the inherent

lack of power in tests such as these one should be prepared to consider

that a test which is significant at the 0.15 or even 0.20 level is

evidence in favor of differences between functions. This is particularly

true if one is forced to use a large number of frequencles.

36

CORRELATION= ,1

S T A T 1S T I C A L A h A L Y S I 8 S Y S T E M 1_I19 MUNDAYe A_IL 3w 1q78 17NI=5

PLO_ OF PO_ER*FREq SYMBOL IS VALUE OF PC

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PLOT OF PO_'_ER*FRL(_ SYMBOL IS VALUE OF PCPC),'_ER '

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NOTE: 20='IS HIDDEN

S T A T I S I" I C A L A N A L Y S I S. S Y S T F M l_J:19 MONOAY, APRIL 3, 1978 19r,ll=7

PLOT OF pUv_EI_*FHEQ SYHHOL 15 VALUE OF PC

!.t

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PLOT OF PU_ER*FRE(_ SY_F_0L IS VALUE OF PC

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45

CORRELATION= ,2

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V.l " d 2 ,a!Il

UoU '€'t

:

F_0

,,,dTt.: d L,_b ,_1,'dC,,

,_1=1 1

P'LUI ,:t- e'or.r._*t-_k_ 5YI4UUL 1,5 VALU_. OF PC

PU.'_ ,_

1.0 .

" ! L: !

U._ + 1: i

U.I + l•' l

!/ !

/ 10.5 * I

?' 7 1 *'_

U.U .7

7

U,5 _ _ 7b 7

5 ._5

O._ * b 5b ", b _ 5

U.l + _.i

V,_) +

I A I I b 1 I C ,_ L ,_ N A L. Y 5 J. ,b 3 Y 5 r I- r4 1_:_, a MOhUAY, API_IL 3e 1_7b 2Q

r'LUI ul- P'U,'_r._*P_t.Q 5yIcjI_UL 15 VALOP.. IJF PC

PU_',_<

loU *ii

O ,,_'/ . L: j 1: 1: 1

: I 1i

• 1u.I + 1

i

• 1I

O.b * i 1; I i• I: I

_)._ + I

7

v.- : 7 7: b _ _ 7

'5 b ,_,J

-' b b" b

3 _ 3 3! _ _ 3 3 _ 3

o.l +1!o

O.'J +!!

N'JI'_: | LJI)O r'IuLaL._

54

HIGHCORRELATION

5 r A T I S T I _. A L A Fj A L Y S I S S Y S T .FP,: |/3:_._MONDAYw APRIL 3, 197B 23NI=5

PLOT OF POWE_*FRI:fa SYMBOL IS VALUE OF PCPO_'_E_ !

1.0 +!

O.q +!!!

o,6 +1

!o.7. .

!!!

O.b +!!!

o.S •!

! ,.i,0.. +

!!!

0.3 +!!! !

0.9 • 1! I: 7 7' 5 b 5

0.I + a _ 2!!!

0.0 +!!!

FREQ

r,JOTE: u LI_5 HII)DE,'_

S I A T I S T I C A L A N A L Y S I 5 $ Y 5 T E H t_;2Li MONDAY, APkIL 3p 1978 2uN1=6

PLOT OF pDY,EI_*FREQ 5YMF_(IL IS VALUE OF PC"

p0_.,El,_,

l.q *

!O.q +

!

!

{'}.7. .!

0.€} +!!!

0.5 *, u_, o_

0._ .

0.3 +' 1'. %'. !

0.2 •T 7 7 I! 5 7 7 I

0.1 + 2 ? 2 P.!!

[0.0 +

.l

.... ......... +......... 4- ......... + ......... + ......... "1'......... + ......... . ......... + ..... _--_+ ..... _--+ ......... +''_'_ .... .'"5 b ? _. 9 lO II IP 13 11,1 15 Ib

FRF[_

,'tOTEr 5 (_S hIDDEN

S f A T I _ 1 ! C A L A f_ ^ L Y $ ! $ S ¥ 8 T E M $u:2a r_ONOAY, AP_IL 3, 1978 25,N1"7

PLOT ()F PO¢,FH*FRP.O SYMHOL IS V_LUF. OF PC.

l.O +

O.q +!!!

_.8 +!!,

0.7. +

!

!!!

0.5 ..' u,

!0._ .

!' !!

0.3 + I

: 1: 7 I

0.2 + ; I: 7 I! 5 5 5 7 7 1

0.1 • P 2 _ 2 _ 2 2!!!

0.0 +

!

III _ II1--11111 + II!!i1111 _!!!'!!!ii_ I!11111_!€ !1 ! . ! I!1 I _ IIi Illlll _ I!1 !II ! II _ ! I !1 !1111 _ i111 IIII I _ I lllll i !1 + I i!!1 ! ill_ Illtlli I! _!i

a b 6 ? n 9 10 11 1_ 13 lU 15 lb

FREO

"OTE: b 085 HIf)OEN

S T A T I"S T I C A L A N A L Y S I S S Y S T _ v 1q:2_ MONDAY, APRIL 3e 197_ _NI=_

PLOT OF PO_"ER*FREO SYMBOL IS VALUE OF PC

Pn_ER !!

!

1.0 .!!I

O.q .I

I

!

0._ +i

!

t

0.7 *!I

!0.6 +

!

I

i

i

.t

0._ + Ii

0.3 + 1! 1

! 7 !0._ + 7

! 5 7 7 I I' 5 5 _ 7 7 I 7i 3 3 3 3 _ 5 5 5

o.I + _ _ 2 _ ? 2 2 2 PiiI

0.0 +!i

!

5 b 7 _ q 10 11 l? 15 I_ 15 tb

FREQ

NOT_ : b tJh5 HIDi)EN

i • t !

S T A T I S T I C A L A N ,_ L Y s I $ s Y s T £" _ l_J'2_J MONDAY, APk_'J'L3, 197_ 27H| =9

PLOT OF POW£R*FREQ SYMR()L IS VALUE OF P.C

! •

1.0 +!!I

0.9 *!I

O.P, +

!0.7 +

!I

0.6 +I!!

0.5 +l

' 1

0." + 1I

' !! I0.3 +! 7 1: 7•' 7 !

0.? + 7 I 1; 5 5 7 7 I 1 I

" 5 5 5 7 7 7 7 7.' 3 " 3 S 3 5 5 5 5 5 5

0.1 ,*. ; 2 2 p 2 _ 2 . 2 2 2 2!!

0.0 +I

!

_J 6 6 7 _ 9 10 II I? 13 1o 15 lbFREO

'_OrE: b O_S .IOOE_

S! A rI_TICAL A NALY SIS S YSTE M I_:2_tONDAY, APRIL 3, 1978 28NI=IO

PLOT OF pO_E_*F_F_ SYMBOL IS VALUE OF PC

I

!t.O +

!t

!N.9 .

!i

]Ao_ +

!$I

0,7 .I

I

!0.5 +

t

0

i

0.5 * 1! oI 11

O.a . 11 1!! 1

0.3 + 7 1! 1! 7 7 1 1 1! 7 1 1

7 70.2 _ 5 b 5 7 7 7 7

i s s s 7 _ 7, _ _ 3 _ _ 3 3 b 3 5 _ 5

0.I + ? 2 2 _ _ 2 _ ? 2 2 2!

II

•0.0 +i!! +.... ..... +..

b b 7 g q 10 11 12 13 la 15 16

FREO

_(}TE= 6 0_5 HIm)EN

5 I A T I S ! ! C A L A N A L Y S ] S S Y S T E M 1_'.2_ MONDAY, APRIL 3, 197_ 29NI=II

PLOT OF pO_FR*FRE(O SY_._NL IS VALUE OF PC"

pO,F_k) .'

I1.0 +

!I

O.q +I

i!

0._ +m

!

0.7 .i

i

!O.h +

I

: Io

0.5 + o,! 1

i

0.(I + 1., i i

! 7 I Io.3 + I I

, 7 7 1l 7 ]! 7

n._ + b 5 7 7 7 7 7, 5 5 7 7; _ b L; _ 5 5 5; _ P 3 3 _ 3 3 5 3 3 3 3

0.I ;, 2 2 2 2 2 2 ;> 2 2 2!Io

0.0 '+!

€I

a % _ 7 _ Q I0 11 12 13 I. Ib I_

F"PE(,l

P.iIIFi ) (IH5 HIUI)_N

5 T A T I S lI C A L A N A L Y S I 5 S Y S T E _ t_:_u HONDAY, APRIL 3, 1978 30NI=12

PLOT OF pOI_FR*FREO SY_HOL IS VALUE OF PC

PO_E_ iot

1.0 +!o

l

O.q +!o

I

O.A 4oof

0.7 +l

!

!O.b *

o

0.5 4 .1

! IO.u • 1 1

i 1! 7 I I! I

0.3 + 7 7! 7 1! 7! 5 7 7 7 7

i " 5 5 7 5 53 5 5 5 5 5

! _ _ _ 3 3 3 3 3 3 3 3 30.1 + 2 ? 2 2 2 2 2 2 2

oIi

0.0 •!

v

!

b b 7 A 9 I0 II I? 13 la 15 16

FREG

hOI_: _ Otis-I,)DFN

63

MIXEDCORRELATION

S T A I I S I I C A L A N A L Y S ] S S Y 5 T E _ IQ:35 MONDAY, APRIL 3, 197_ 23NI:5

PLOT OF D{IvdER,',FREN SYMHOL IS VALUE OF PC..

PO_',EP !

1.0 +

O.Q +

0.._ 4

0.7 +

l

0.(_ +

!

0.5: o_

0._ +

0.3 +

' I

0.2 + I•' 7 1.' 7

O.l + P 2 2

0.0 +

(4 5 h 7 ) g I0 II l; l) I_ 15 Ib

FRE{_

_'-_OTE" LJ U_S HIL)OFN

e v _V

j,

8 T A T I S T [ _ A L A N A L Y $ Z S S Y S T F M 1U:35 MONUAY, APNIL 3, 1978 .?.."N1=6

PLOT OF PO_ER*FREQ SYMBOL IS V_LUE OF PC-

_O_E_ !Il

1.0 +!

!

O,g *I

l

0.8 +!!

o.0,,7" .!

!

l

O,b +!oo

0.5 + o_0 _jm

I

;

O,tJ +I

0.3 + I! 1o

, I0._ + 7 I I

, 7; 5 _ 7 7 7, _ _ 5 5 5

nol _ d _ i/ 2 2o

!

n.O +!

;

I •

"..'. o.... ..... ,o..... - .... € ......... + ......... .-- ....... 4........ --. ......... .---" ..... +'''" ..... .'''' ..... +''''''''-. ......... .'"b o / 8 9 10 11 12 13 la lb lb

FREG

_{._TE: 3 U_S HIDI)ErJ

S T A T I S ] I C A L A h A L Y 5 I S S Y 5 T F _ I_:35 MONDAY, APRIL 3, Ig7_ 2_NI-7

PLOT ()F pO_,,ER*FRE(a SYMF_I(IL18 VALUE OF PC.

POP,ER !

lI.(_ +

l

O.q *!

!1_,_ +

,!!

0.7 +!

O.b .

0.5 +s

II

0._ +' %

' I0.3 +

-' 1 I: 1 1: 7

0.2 + 7' 5 7 7 7 7 I! 5 % . 5i _ _ 3 5 _ 5 5

0.I + 2 2 _ _ 2 2 2

0.0 .!!o

"---,F ......... $ + + . + + 4- ..... +--- ...... .---- ..... + ......... + ......... "IF--% h 7 H _ 10 . 11 I? 13 I" 15 Ib

FRE{,I

N{IIE: " OH5 HI{.JOF_I%

5 T A T I S T I C A L A r_ A L Y 5 I S S Y S T F M 1_:35 MONDAY, APRIL 3, 1Q78 2b_1=8

PLOT OF pOwER*F_EO SYMBOL I$ VALUE OF PC

PO_ER !t

I

1.0 *!io

O.q +I

i

0.8 .!i

!0.7 *

te!

_ob .!!o

O.b .!€

! !

! ! 1!1 1 1

0.3 + 1I

_ 7! 7 7 1

0.2 . 7 7 7 I1 5 5 I! 5 5 5 S 7 7! 3 _ _ _ 3 3 5 5 5

0.1 . 2 2 2 2 2 2 2 ?II!

0.0 +t

e

g

b h 7 8 q I0 11 12 13 I_ 15 15 ;:

FRFO

_OTE: U[IilS_IIJDE N

S T A T I S T I C A L A N A L Y 5 I 5 S Y S T E M IU:]5 HON|)AY, APRIL 3, lq7_ 27N I=Q

PLOT OF PO_ER*FkE(_ SYMBOL IS VALUE OF PC

PO_'.ER :!

I

1.0 +!ii

O.q +i

!

t

0.8 +Ii

0

0.7 +!

o

I

!!I

0.5 +• O0.' I: I

0._ + 1' I' Io

_.3 +: 7 i, 7 7 I! 7 7 7 1

0.2 + 5 I! 5 S _ 5 7 7 I' 5 5 7 7' P 3 3 3 t, 7, 3 3 ._ 5 5

0.I + _ _ _ ? P P ?. ?. 2 ?.i

I

0.0 +LI

!... . . . + 4- . "€' + + . •

a b h 7 _ 9 I0 11 lP 13 1" I) Ib

FRE(_

NOTE: b aJb5 .It>OE_

L •

, 5 T { f I S T I C A L A N A L Y S I S S Y S T E M 1a:35 MONDAY, APRIL 3, 1o7_ 78NI-IO

PLOI OF pfit_,ER*FRF(J SYMBOL IS VALUE OF PC

pOt,,FR 'I

I

1,0 .!!

!

!!

!O.M +

t

0

0.7 +!!t

O,b *;

!: ! 1

0.5 + I o,,, I ,,o!

, 100_ 4. 1

!Li 7 I

O. 3 + 1 1' l 7 I I, 7 7 I!

0.? .; _ % S 7 7 I, 5 S r_ 7 7; " S 5 '_ 5 7 7i _ ? 3 3 ;? 3 3 3 3 3 5 b

0.I + p p _ 2 _ 2 0 ? P!

!t

0.0 +I!• 'ot

........ .. ......... +.'.... .... ..... .... o.......---. ......... . ....... .-. ......... ... '

a 5 6 7 g 0 I0 11 12 13 I/a 15 Ib

FREQ

NOIE; _ O_S MII)I)EN

$ [ a I I F I I C a L A N a L Y $ I S S Y S T E M Ia:35 MONOaYo aPrIL 3, 1978 29NI:ll

•PLOT OF PO_ER*FRE_ SYMBOL 15 VALUE OF PC

P(InE_ !!

1.0 +i

!

0.9 +Ii

!i

i

0.7 +I

;t

0._ +: I I: Ii• !

n.5 +' Ii

0,_ .' I l' 7 l! 7 7 I

0.3 + l: 7 7 1: !; 5 5 7 7

0.9 + 5 5 5 7 7: S 7 7

; 2 P 2 3 _. _ 3 _ x 3 3 .50.I + 2 2 2 _ 2 2 2i

l

,0.0 +

t

i

"''. €' " + "'------- + ---------- 4 ----- .... - .----- ---------'t -- .... ------,--------------,-------------,-------------,.----.------.--,----4 b 7 _ q IO II I? 13 ta 16 Ib

Fk'EQ

'JOTE: q 0_5 hII)OEN

$ T A T I S T ! C A L A h A L Y $ I $ $ Y S T E M 11"35 P,ON|)AYw APRIL 3, tg78 30NI=12

PLOT OF pO_ER*FRE(; SYMBOL IS VALUE OF PC

PO,':Ew !oI

1.0 +i

o

t

O.q +i

!i

!I!

0,7 .!o , |

: I0._ + 1

', .iB

: !0.5 +

!

! 1 1(_._ • + I

: 7 7 1! 7 7: !

0,3 + 7 7 1!

! 7 7.' 5 5 5 5 7 7

0.2 + 5 5 7 7! 5 5 5 5! 3 _ _ 3 5 5: 2 ; 2 2 ? P 3 _ 3 3 3 3

o.1 + 2 2 P 2 2 2!J

O.O +D

!

1.._ ......... 4_......... + ......... . ......... + ......... + ......... + ......... . ......... + ......... + ......... + ......... 4 ........ _)---',"6 _6 7 _ 9 I0 11 12 13 14 16 "

FRE(I

NOTF: ? ,)H5 Hl;Jf_f'J

72

Bibliography

i. Graybill, Franklin A., Introduction to Linear Statistical Models,Duxbury Press, 1976.

2. Morrison, Donald, Multivariate Statistical Methods, Second EditionMcGraw-Hill, 1977.

3. Posten, Harry, "Power of the Likelihood-Ratio Test of the General

Linear Hypothesis in Multivariate Analysis," Ph.D. dissertation,

VPI&SU, 1960.

APPENDIX

Computer Program and Wrlteup

There are three subroutines supplied in the appendix for calculating

various multivariate statistics for many hypothesis testing situations.

Some notation:

H - The hypothesis sum of squares matrix.

E - The error sum of squares matrix.

NP - The dimension of H and E.

N-UH- The hypothesis degrees of freedom.

NUE - The error degrees of freedom.

The first subroutine ROY requires as input:

H, E - Square matrices of order NP stored columnwise in the vectors

H and E respectively.

NUH, NUE - Degrees of freedom are also supplied by the calling

program.

RTS A work vector of length NP.

N - A work vector of length NP*NP.

Output:

RTS - Contains the ordered eigenvalues of HE-I.

C _max _ where %max is the largest elgenvalue of HE-I.theta = .l+%max_

LS, RM, RN are S, M, N respectively for use in Heck's charts.

The second subroutine NILKS needs as input H, E, NP, NUH, and NUE as

_ previously described, where H and E are square NP*NP matrices stored

columnwise in a vector of the same name.

A-2

Output:

IE1for comparison with Wall's Tables.

WI = IE+H I

W2 = -(NUE-(NP-NUH+I)/2)*LN(WI) for comparison with chi-square

tables, in large samples where W2 has NP*NUH degrees of

freedom.

W3 = F-ratlo having INUM numerator and IDENOM denominator degrees

of freedom when and exact F-test is available otherwise

W3=0 and INUM=IDENOM=0.

The third subroutine HOTEL can calculate M T2-statistlcs. The

necessary input includes

X - a vector of length M*NP containing M sets of mean vectors

of length NP stored columnwise.

S - Sigma - where X(J), (J=I,M) has covariance-matrix

(I/A(J))*Sigma.

nln 2

A(J) - A vector of length n containing nl+n 2 (where the n.l are

the number of replicates in the Ith function), stored

columnwise.

T2 - is a work vector of length M.

Output:

S : S-inverse

T2 - contains the M values of Hotelllng's T2 for direct comparison

with tables

IEH - comes from matrix inversion routine

= 0 No error occurred during inversion

= -i No result because of wrong input parameter n or the

matrix is not positive definite

= k if there is a loss of significance.

D_TF 03/31/7_ PAGE

SU_(IU 1 I r','E ROY (H, F.,'_,rJP,_IUH,RUE, H TS,I HET A, LS oi_M,RN)CC SUH_ULJI]_tJES wEI'JUIIREI):ONWL)OT ANt)DETGF. N StlPPLIEO _Y USER.CC '._ATWIC[:S H L)U_ TU HYPOTHESIS ANt) E Ot.IETO EQ#OR, r)Ir,_FNSION NPe AND -C 3EGREES (DF FmF.E{)O_._,,'JU_F{)_'HYPOIHESIS AND NIJE FOR {-RRUR, ARE SUPPLIED IN ":.C T,,E L_ALLIt,I(;PI'4tll,_''V .C H A'_i._E APE SNt'A"(t r/AT_'ICE.S OF {_P.i:FRNF' STrI_ED COLUT,.T,;I:_ISF.C RT5 ,_.f::) _'.AwE_ ;'.URn,VECTI)_,_ {IF DI_F",JSI[J_,S Np AND NP*NP, PESPECTIVFLY.C vAl !JE-:5,.'E-IU_'JE-_)ARF THE vFCIOR F,'IS UF r,RI)E_E,.] LArF_'_T _'O[JTS L,F H,E-INVERSE,C T-_-.Tt,-(LA_(;ESI _;;(JI)I(I.+I. AF'GFS[ _'(),'JT),A{,,DPARA_FIFF;S 13, R.,,4,AI':DRN FURC .;'.;.FE;_{....JCE-ILl Hk('._.'SC,-IA:_IS(')_(.JPIHE F'Ik'ST LF..TTER.OF FACH).C

D Iv.F..;,_I,}N it(t),E( I),n(1 ),RIS(I )_:)tJ_Lk PRECl.'il(i,_ H,E,RIS,I,ETA,X,,',I: t',JLIH-:,_P) 15, t 5, @f)

]5 LS :',._]r,.".:1T_: ,t%

.,]h._-_.PP% '_"=(Fl :)_T( ]A,-IS(qll_-f,_P)-I ))/2.

._'J-(FLI_AI(:,Jl.J_ -'_p-) ) )/_.•rt.LL [)F_,.jrj'/(l.peH,FePTSe..,_)I-_r.IA:QI,_(1)I(!.UO+_IS(1) ]_ETL/RhE '_f)

S_!HROuT I_F rVIL_S(M,E,NPe',IUH,NUE,wI,¢_2,n'_,INU_,IDENI)t,_,IEw)

C SJ_.ROUTI_,'ES REUUI_ED: DMFS r) FF_O:4 5SP LIH_ARY.CC ,_AT_ICF.S _ ()LIF TO HYPOTHESIS AND F. !)tJETO EPHOR, f)Ie"ENSION NP, ANDC Z)E(;,_EF5 OF F_F_EDUM, _ItJMFOR HYPUTHESIS A;',ii)_UE FrlR ERR[JR, ARE PR[IVIDEDC ;]y THE. CALLING P_'0GPA_,4.C _ AND E- A_E SI[tREI) COLLJMN-_ISE .AS UPPER TRIANGULAR IVATRICES.cC OUTPUI I _',_CLUDES:C ,',I-UEI(F)/F),F.T(E_.H) FOH CDv_PARISO'_ _'_ITH ¢."ALL'S TABLES,C :_,P='(rv,LIE'[::P'hUM.t]/P.)*L"J(,:I) F[}P CO_P_RIS_}N _TTH Ct_I-SQtJARF. HAVINGC NP*',U_. [)FG'_EES OF FREEOO_ IN LAPGF SA:'PLES" {'LN[.) IS NATLIk'AL LOG).C h3=F-;vATI[J _,AVI_,G irrUPt NU_,_ERA.I[)R ArH.) I!_FNn',_,')ENO_.'INAI(_RDF(;REESC OF F_EF..UU'_ ;_E,_ LxACT F-TEST_ A_E AVAILAHLE: nlHERv, ISF ._3=0.0,C I,,,U'._:O_, A,,_r) IIJENO,_i=O.C

DIMENSION t_(l },l'(l]D_IJ._L_" PPEC[.%IO_'J H,E,T,v_I_n2,v_,/I_,OETE,DFTT,PROr)TOL=.O00001

')5 2u I=I,Lt:_( ] )+t-.(1)

20 '_(I}-'I '"CALL I).','FSD(L ,F!R, TOL, ]EE)

D_TE 03/31/7_ PA{;E 3

CALL I)uFSD(F_,NP,TOL, IEH)PR Of.)= ] ,I)0LI=IL ? .-,.)

LI:LI+LP

_ull--l.On1.1=1LP=;

_;"V,._="(L I}*_UD-]=LI+L?

30 L,'==L e _"I"= TT-P_{]D**_.'_l:UI'.IF/I)E TT,'.w:I}FLUA I (NuF)- {DFLOA I {NP-,'qUH. I ))/2. DOt"@ =-,'._-_*OL UG {_';l )IF(hU,.EO.I) 60 ;0 50I f.(,'J|JH.h(,)._) GU TU _0I ;"(',V._..J. 1) GO TO 70IF('_ ....._] L,U lO (_(1$3 T{_ qO

So I",'l>'_=';PI:')E..JU-*=hLl_+%UH-NP•'_- (I.,)0-,',I)*DF LOA [ (IDENOte) / (_ ]*DFLOA T (INtJM) ) >._._ TO I_O

hO I,'_d'-'=P*NPI OFNh'._=_P* (,_LIP +N_IH-NP- I ].',_=(l..m-')bqF_r[¢_l ))*DFLOAT(IDENOM)/(OSORT(_I),OFLOAT(INuM));2 10 I_O

70 ] "_,1:._='_.HI"_F"_0v,=',;UF•'_._:(I.[).I-_'_I)*DFLOAT(IDENO_)I(e,I,OFLOAT(INu_) ],;3 [_ I ';0

PO ] '_11"=_ *:_!JHIO:.',r;'.;=P*(_t_L_ I ],',_=(I.O0-DS(_F_I(_I)]*OFLOAT(IDENOM)/(f)SqRI(w! )*I)FLOAI (INUH)),_ TU I .t11

qO /_ ._ "" 0. I) I|! 'Jd v =

130 IF(ILE.L_.O) GO I'U I._5G_I ILl ] d5

135 IF'(IFH.F_,_.O) GO I(] IQOGO TU t_5

It_O ] E,',=i_G:) TO ISO !_

150 ,,?ETURNEND

S*.J,_,_bUT TNE NUTEL CS,"_, ;_P, x, A, T@,_, IEH)CC SJp:_,qlTIr;F.S WEfZLIINF.!): f)SINV IN 5SP LI_RARYCC S IS A S_'.'vlrUIC _:ATRIx OF ()ROER NP STORED COLII'_N-&'_ISE IN UPPERC TR I._*_{,iL.',i_F lift'4.C .',A'.{)Te' AHL vk(.ItJ_tS ,iF L{",'_GTM.._.C .':l:i _ -'&_', VDCT(J'€ {)F LFNGIH JP.C AS.N!I"E VFC|Hw.'I x(j) TI{ HAVE CtIVARIANCE F'ATRICFS (I./A(J))*SIf;MA, FORC J=t,v. T,E ;._AIPlx S Ai,{DVECTURS X{J) AN[ SUPPLIEr) BY THE CALLIN{, PROGRAMC AS _'.ELL AS T_E f)I"_F_,_SI0tJNP AND ,'4. THE VECTORS x(J) ARE SIOkED COLUMN-C ?,ISE ];,JT,'_EVECI(}R X (iF LENGTH NN,F,I.C _\ ,-FTIJW'] 5 I._, w_'PL.'_CED HY 31;_vFRSF, EACH _(J) 13 REPLACED _Y 'THEC C:IR;-:[SP()"JF)I;,_(;C{,tFFICIENTS {IF A LI,_kA_ {)[SCRI_'IINANT FIINCTION, A;_OC T_E VFCIL,_ 12 C(_'_TAIq_ VALUES {)F THE HOIL-.LI.ING'S T-SQLIA_F. STATISTICSC F3R I)IRECI CO.-_PARISO{J nJTH SPECIAL AID TAqLES (SEF dENSER AND HO_wE, 1987)C

O I_Er,'Sl(h_ 5( I ), X ( ] ),a(I), T2( 1),,'_(1 ):')3U_LE'PRECISION S,x,A, T_._,SIJM,SUMtIOL=. Ot)O00 lCALL DSINv(S,r,P, IOL, IEH)LP:IL -_:',P03 _) i:l,m% I.':u.:}hL1:1 ""_"._;.) _% J=I.P.,L._D:) _h _,:L_,J5 J"*l =_ (K)*S(Lt) * x (J)IF(J--) lb,?O,15

_1) S.J_ :StJ'._+.%U."|?.S LI:L.I+l

TP( I ):5H.,t*A ( I }L .)=L_;.._ .

_0 L):LS+NP.-0LI:1L@:hP)9 50 I=I,ML _:L *_,F'Od u_l J=LI,L_SJ_'=O.:)O03 _b K=LI,L@LJ:J'L_LW:K-L ._

C_LL LUC (LJ,LK, JK,NP,NP, I ) ! |5 U'_: SU_"_+_ (J_.)*x (K) j'.."

D'_'TF 0_I_]17_ P/IbP_ b

35 :.' ( L J ) =SLr"a0 C_r_IIhOE

U3 U¶_ JJ=LI,L2JL=JJ-LJ

!J% _ J JJ: (.Jl)L:L+ILI :L| +"t_

50 L2:L? _r;''p

k '_I)

Sl_=_tjlJl'i=,,l:. f)NNHnT(M_.wp, pXL,X) NPfl0 "_703T'_E.';SILI', ^(1),H(I},XL(1),X(1) I'Jf41lO _R0")[]t)_l_ P14_Cj511)tl AwHpXLwXw_UqV N_IIO LlhO'_:1 NPOrJ hlO'3J 1Oq} J:2, ._1 hq[JO _;U.:,_*(J-1 ) t,;14Or) P,JO30 10,) l:l,J h,-,ll(I hdOL:L I I rll?_)fl 6!_()_,:,., + | ,_tN|)[.I 5_{)

•",..=o N_(lf) 7 I 0CALL OElf;EN(_,X,'_,'_V) Nh"}(I 720L:O NRf]O 77003 I10 J:l,_l "N_O[] 180L:L+J NI_UI) 790

11" __t ,;,:1.,/ .:.-I('_;'3(.(L])) _I_!_!_ _Ot) _-,.=,1 NP0I) _11_ I33 115 J:t,,'-: N_)O 820b:) I15 I:l, '4 t,l#tJO 8_0<=K+l NRn0 _0

llb _(<]:X(K)*XL(J) h_Ot) _bO

_;P=O r,ll_() (1 qOCj]i1 12(_ j=I,M N_qil() 910X_l:V,(I-l) N_O(} 920L:V.(,i.l)+[ NPO0 q30x (L):o.DO NRO(3 qaO.)3 1,_0 ,(:1, _ N_OiJ 9bn_1:'_1+1 NNU|I rq60x!p=h_+l h_O0 _7(}

l_O x(LJ:x(.L)e_i(N1 ),_A(NP} N_4I_tl q_OL ': .I f_lll) N_O•),j 1_,0 d=l,_ N_'(JfJlOO0_O 1_,,) l:l,J N}4UtllOIONI : l -_" NR()I'_ 1020,2:'4, ( J- I ) _RLHI I 0 ._,O

L :L * I ,,141,(11 gl'4(} ,

A (L ):r,•"0 hm'LHI 10%n i 1['_3 130 K=I,. '4 NPflUIIJbO

OATE 03131178 PAGE h

N?=_.+ t N_001080130 _(L )=_(L) +X(NI)*H(N_) NR{)[)IOqO

"_LL _J_lSL'_(_, x, _I,vV) NROnl130L=u NkO() I I _40T).)lu0 I=1,:" N_OUI150L:L + I N_dU 11 h0

IU0 XL(I)=A(L) N_()O1170I)t,ISA I=I,M NHUUI_.IO

_j 16(i J=1,,'4 _'(_ODIP._NI= l-)' N_('I(})_u0

L= "* (J-I) + I rJ._()(l1_€,()& ( L ) "'l i . l)l) f,llllr) 1 _hO.) 15 ' -I, (,,t_..)/7,)NI:' 1+'-I• rr.(Zl,l?H,1

150 _(L.I=_(L)+.(hI)*x(N_) _J_()rllAO0L:() NHIJllI .4I(J. = . ,+l(_+(} I ._0' 1 '_ J-'l,' ,< lJ'_t,)

).l_v=n.D 0 l'.J_{Jfl) _d(lI)2)170 I:t,"! N_(IL'I'_%(IL=L . I (_rJ(k13h0

17n 5 ,;'.' V =,";d"_ V +_' ( L ) * ._( L ) _l_(-iO I 37017'5 S.i_.v=,'i'J_i(Su:'W) m NRO013_0

_)L)1at, 1=| _v, N_O(l139n•(:'_ + I N(If)OIdt)O

180 x(K)=,.(_)/Su,.:v N_rlrll_I0

: _ (ll_bO

+

',_EMBEP NA'_E EI{;EN PAGE 00(I!C EIGE 10C .................................................................. EI{;E 20C EIGE 3oC SU_RnUI" INE EIGFN EIGE _f)C FI(;E 50C P_ RHOSE E I(,F_ 60C C{"-'UtJTE EIGENVALUES A_l.'_EIG.FNVFCTrJRS OF A R_AL SY_F'I'RIC EIGK 70C MAT_,'IX E ]{,F. 80C F [GE geC oSA(;f EIG_ I00C C_LI. [I(;FI,_(A,R,N,.,,'V) Ir.JGE I10C EIGF. I?OC r)ESC_IPII_),, UF PAkA.',,'kIERS EI(';E l.'r,OC A - {1_It;l;,;AL.'AAl..,Ix(SY_'Ir_ET_elC'),DESTPi]YEO II'JC{fl',:PUIATION. EIGE laOC w_S.JLTA'JT EIGK_vALLtES ARE OFVEI_NpED I'_ DIAGONAL I)F F:IC-F_.150C r"AIRIX _ i".t i;ESCEIJ_)I'JGOw(_F.R. FI(;{- 1,_0C _ - Fv_S:JLI_:UT :_AIRI( _i_ ki.{;F._¢EC,I,_)_'S(STO_'FD C()LU:,'h",ISE, F IG__ 170C Ir, SA',,:I- SF ,_ .F,',<C_ AS FI_,E_.;VAL_IES} EIr;E laOC N - O_'_)_.pqiF _AI._ICES A ahU n Ei(:E 190C ,"V- l!,_PU,i,COO!: Elr,E 20nC 0 C'J'.:plIT'FEIGEtjVALU[S A_iD EIGENV_.CTGRS EIGE 210C l Cl'i',IPlJTk_EIL;Z:_VALiJES IINLy (P r'_EFD N(II GE EIGF. P20C D|.'_E_','SI.3'uEi)'_iJT ,"_UST SIILL AP_EAF IN CALLING FIGE 230C SEOtJE,'_CF ) F Ir;E ?uOC EIGF. _SOC wE ',IAwKS E Ii;F P.oOC ORI,;I_',AL '4ArRIX a _A,JST HE REAL SYr_.r._ETPlC {STORAGE _"r)DE=I) F]Gk _70C .'_'IRIX A CAN'IOT HE IH IHl- _A',AE LIICATIO;,J IS _._ATRIX R ETGF ,_,_0C E I{1_-_?9nC StI_R_.IUT,II.ES _.t'lr) FUNCTIO:,I SU"JPQOGRAt,'S REn,iIIF_E_) F 1GE "500C "_O',E EIGF _110 coC F lr,k 3_0C t-_EIHUi) FI(;E _._0C L)IaG_):._ALIZATIOr,I _F,I,HUi){JRIGI_IATEI) 14y JACOHT AND ADAPIFD FIGE 3LiOC HY VOW'.'NFUN_A_*'I F'(IR LAWGE CU'._PUTERS AS FOUNI) IN "MATME_ATICALFIGE 350C _FIHUUS FUR DIr;IIAL CO_P.L,T{-nS• , EDITFD !_Y A. RALSIUt,, AND EIGE 3h0C H.S. _',IlF, j(iwN vIILFY AhO 51_f_S, {'_E_'_yOq'K, lqb2, ('.HAPTER 7 EIGE 370C _"I(:E 3anC .................................................................. EI_;E 390C k I{;_-_on

SJd_'{,L_TI._;EEIGEN(A,_,N,,_V) EIGE _InDI_FI, SIO._ A(1),R(1) EIGF /JPO

C EIf;E _30C

i,,<,<,,L,,; cis .............vERSION OF THIS ROUT'INK IS r)ESIRfr), THE EIGE aaOC C IN COLIJ"_N I _H(]"JL{)8E RF.POV[".I)FR(]e_ THF UOUqLF PRECISION EIGE a70C SIAIE_.Ehl ,';_]Cff FI_LLfi._:S. EIGE q_OC EIGE _QOC O01.1HLE P_ECISIO;_ A,R,A_"OR_,A_J_''X, TH_,X,Y,SIFJX,ST_JX2,COSx., F I[;E '_00C I ChSXP, S I'dCS,wA_t,E EIC,E 510C E I(';E5_0 i"C T,HF C :_tl,_T ALSO HE RE._Ov-'r) FRO'._ DnUHLF. PRECI'_TON STATE_FNTS EIGE 5%0C AH_.'FAPING IN UTHER RUtJTINF.S USEI) Iri Clf'J.J_NCTION nITH IH]S EIC,,F.5t._OC POUT I._L. EIGF_ 550

THE O(.IU_ILFHRLCISI(_h vE_SI(IN uV THIS 51)HRf]U'rT"J["MUST ALSO EIGE 570C()t,.TAIN L)(IUNLF_PRF__CISIfiN FORTRAN FUNCTIONS° Rt}RI IN SIATEMENTSEIGE 5bO

PAGE 0002

•._E_B E_ _jA:,_F [-IBENC _0, _,_, 15, A_'J')l_ MUST _F. CHANGED TO r)SQRT. AHS IN STAIEMENT EIr,E 500C _ '._USI HE C-HA'JC.EI)It) DAMS. T_F_ CONSTANT IN ,_TAIE_ENI S SHOULI) EIC,E bOOEI{;E hlnC _F_ L_"J_;E[) Tu 1.01)-1_, EIC, E 620C ........ FIt,E _30r. .... ,,,.................................................. EI(;E b40C E Ir.;k65OC C,E',_ERAI("IOENIIIY _ATr_Ix EIGE 6hOC EIr,E _70

TF(._v-I} lO,?b,lO Elr,k _.qO1C, I_,=-:'. {:Ir.E 700

[:,_?0 J-l,_ E'It,E 7lh1 _:l,,.:_ FIGE 7,?0h'_ P,_ 1=1,(', EIGE 130IJ=l':+l EIG_F 7tJ_n (IJ)-'-_.O EI{;E 750IF(l-J) 20,lb,PO FIGE 7bn

15 R(IJ)=t.O FI{;E 77n_0, cor..IIqtJk F.I(;F7_0

CC CO_PUIE I.NIIIAL A.',,DFINAL 'WOR:V.S(Ar,oR',_AN[) A_JOR_X} EIt;F 790E I r:.E AOOC EI.{:F. _10

DO 55 I=I, _'! EI[:.E_303{J __4 j=l,I.; EI(;F _"0IF(I-.I) _0,35,_0 FIr,F 8bO

3n IA=I_ (J,J-J)/?.A;_,';-"-'.'.,'_(,_'._.A(IA]*A(I") ETGE R_,O

MSOIF(A, .);;,".') 1{_%,I_%, u() FIGE 8_0_,) _.,j(_.'=1.ul_,._,J,_l(,','.:!w V').',._._..x=.'..r.(_:',,,WA:Jg;F_/FLoAT(N) FI(;E 900El(;F.911"_

CC ]_JIII."LIZE I;"OlCAIOQS AND COMPUTE THRESHOLO, THR EIGE oPOklr,k g-_OC EI{;E 040

l_'_0=0 F.TGE 050I wi_" Ar'JOR_'_ F.IGE q_O

45 TF_,.,=THRIFLOAT(N) EIGE qTO50 L=I EIt;E 96055 "=L*I " F.IGE 99A

C EI{;EIO00C Ct)_,_PIITESIN A_,_DCO_ EI(;EIOIOC E Ir,F..10Pn

e,n _q=(_.,,'._-F,')l_ EI(;ElO3nLi}=(L*L-L)I_ EI(;EI04OL_i=L . ',,,;

P.;) IF( AH.S(A(LM))-TH,q; |30,_,hS =''_t(;EIo-_A- t;EI 0_005 INO:I EIGEIO70

LL=L+L{_ F.I!;E 1080u,,":,,__,.-'._,_ EI(;E 1_1_ i.Ix :n -5_' ( A(LL)'A (M_'_)) EIGE1100

_ Y=-_(L")/ 5QRT(&(LM)*_'(L M).X*X) EI{;FlIIO }IF(x) 7n,lb,75 EIGht120

70 Y--Y7€') SI:,,X=YIS{_RT{_.O*(I.O.( 5;aOT(I.O-Y*Y)))) F.IGE1130

Sl_,,xP=Slr,_x,Sl_,_x EI{;FI t_O7_ C(.,_ = b_J_I(I.O-SINX_) EIGEIISO

COSX?=COSX*CUSX F.I{;EIlbn

PAGE OOO3

S I:JCS =SI:Jx*cosx F..IGE 1170E If;E 11P, nCC _Ol'"T_ L AND _ COLUMNS E]GI'-IIqO

E I(;E.IP_noC Tl q:r.!"{L-I ) F I_F.Iel 0

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_(: IF(I -',_)_5,115,q0 EIf,EIA_hO_.% i,..,=[+ ,A,.. E I{;F.1270

r;,.ilq_ uh EI{'-EI;)HNqO I't--'.'+ 1,:J E IC.EIPqO_% IF(I-L) 100,10%,10% EIGEIz'O0

11".0]l =I+L_J EI(;FI _'lh{;f! lo 110 EIr;FI_20

105 IL=L.Iq EI(_E 133nII0 X=z,(IL)*COS_-A(I,,I)*SINX EI{;EI _"'('}

.,.,(l",',): A ( IL ) ,,SINX +A (],".i) *COSx E IT,El 350A(LL):X F l_.;EI ._hO

115 l:¢_'q-I) l.;0,1_b,IPO EIGFt._,TN.I_D II.I.-=IL'.I.I EIGFI _#')

1 "_'-I-:,€I E Ir,F.l_qnx'_ (IL,..'),,CuSx-R ( I_'q)*S I_x F Ir;.EI.'._()nq (I..'_):.v(ILR)*SI_W+R(l_,_N) *CtlSX E IC,F..1al 0,.,'(It._)"x E Ir,t-T1 _.0

12'i CU,_II:'.,t_E EIGI::I_,Ox=?. O*t.,(Lt'..)*5 I'._C5 E I(:,EIu'.lOy :A (LL )*C.LISXP. A (;.IVl)*ST P,;%_-X E I¢;EluS(}x:a [LL) *S l..',.:x,?.A (.,,".a)*C(ISxp+x E IC;FIuhO i.-..A (L.I)= (A (LL )-a 1..I'I))*SI*,ICS+A (Lr-1)* (COSXP-SINxP') E 1(:,I:It_.7N oA(I L ).=Y F I[;F:I;,_('IA(.Y._)=W EI"-F 1_'}n

C E l(:;k1_neC IF.')IS F[Jm CO','PLFTI(')F• EI_,F.ISInC E !t;{":1S?oC TEST FtJ_ ,M : LAST COLU'Wi',_ EIGF IbShC EIGFI%"_O

1311 IF(M-_) 165,1"0,135 EIG[IbSn135 _=r,_. 1 E IGE 15(",0

GU TO an _l(..EISTnC EI[;E 15_0C TEST F()R L = Se.Ctlr_DFP9 '4 LAST COLU:4N El[;EI%'..)nC EIr,E 1P,oO

Tan IF[L-(h-I)) I_5,150,Ia5 EIGEIhlOI_5 LzL+t E I(;E1(_PO

_(1 It} b5 EI(,Etb_oIS0 IF(IF]O-I) 160,155,160 EIGEIbaO155 1,'J()=0 E I{.;E1650

C,O TO bo El(:.E1bbnC EIr'EIbTOC CO'.'_RARE THRESh{ILl) _Ir_ FII',_ALN(IR'_ EIL;FIbHOC EIGFIhgN

lhO 'IF(IF,_-AAr._R._X)105,Ih5,0% EIC,E1700C E I(i_:1710C SORT FIGENVALtIES AF,!.,)LIGENVECI[.)WS EIGEITPnC EI(;EI730

la_ I[;=-N EI[;EI 7_n

PAGF 000_MEt_HE_ NA_,E EIC,F.N

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170 x=._(Lt.) EI(;EI_OA(LL) :A (,.b_-1J F.IC.Fl_(_nAf"',,] :x F.l f:E 1_,50IFC':V-I) 175p]/_, 175 FI(',F:I_hO

17_ 3 ) I_,} ",=l,'J EI(,F..I_/nIL{.:I_.'+V, tIGklP,_.n

x:,v(lL,.,') i-_I{;k1900_(II R):_(I;.'R) EI(;ElOlO

l_n _ (I'._):x EIGf IO?n1_b C.)'_II.';UE EI(;Lln_O

_F.l;j_:, EI(':F_]r),j(_F.;_ EI L,6I gbO

!

C Or_SD I nC ................................................................ .. DISI)_ 20C DMSD 30C 8 il H !'aCII.J}' ] _'J,F_["}rAFS {') [)MSI) LIOC OHSD 50C PU_PHSF [')MSD _flC FACTOR A {;IVFN SY_EIRIC POSITIVE DEFINITE MATPIX DMSD 70C {)F,SD 8,IC USA(,E O_S{) 9nC CALL DMFSI)(AeN,LPS,IER) D_SI) 100

C I;EL;CRIPII(,h ('IFPAwAU.FTENS I)M,'.;D120C A - {)LI_H'.LFPRI'[.IL-_I_{,;UPPER TRIArJG!.!LAR PART OF GIVEN De'iSf) 130C SY_¢F.I-_IC PU.'IIIIVE DEFINITE :'_Hy N COEFFICIEhT D'_SL) lqOC _ATRIX. D,ISD 150C (H', _I:1_1_",I A C(I_JTAI',ISTHE RE.RLILTAMT UPPER DMSI) 'I._0C TRIAN,;IILAR MATRIX IN D(}UMLF PRECISION. DMSD 17(_C N - IMP. q,J_IHER (l_" ,'d'l_S(C(ILII;_NS} IN 31VEN MATRIX. DMSL) I_0C I:PS - SI:_C,LE Pwl'.ClSlrV; I'_PIJT CPNSI'D.F, T :_HICH IS IISED F)_]SO |gOC AS PF-LATI_E TI}LERAHCE I-0_ TEST O_ L_}SS {;F D'_SI_ @()nC bl_:.;k IC,:r,'CE. _)r._sn21hC IER - _5S_ILTING E_Q{'),RPARA_:F.TFP r.lr_)FOAS F()LL(J_,.'i IH,ISI)_2hC I_:0 - _\.'1 FF,'R1R DrASI) 2'_C Ik_=-1 - ',d ,_E_ULT ,_ECAU_F. rip ,,'Rr_',l[;JLPI.IT PAPAMF_- D_4SI) a_flC T[-R :_ {)R HFCA_ISF Sr)'.'F.q&_ICA_D I,':;N(IN- D_A.SD @%0C P{)SIrlv_ (_alUlX ^ IS N_T P[_SITIvlF l)h._l)@_(IC I.)[.FI:_]TL_ PII3SIHLY [)IIF lq LOSS OF L_Ir,NI- (_SI) P.lnC FlCahLl- ) Dr_SD ?HOr. ILR=K - _AW,,ilt,{',wHICH IN_)Ir.AIFS I.{)5S r)F SII;NIFI- I)HSD 2_f) _'C CA,_CK. THE _AI)ICA_Ji) F r}P'AFI)AT FACTdJRT?A- I)_ISL)_on('. lid'; _TF.M K*I _.A3 STILl_ PI)SIIIVI-. HUT I,i1) [')'_,_RI')_1.") MC LU_L;F_ (;.WEATiF-RIHAI..IAHS(FP_*ACP,.IeK.I)). [}v_:_l)(.,_0

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C SUR_(:!JTI,N_S At,_DFUNCTIO"_ SUbPrOGRAMS RFrJUIREO D_S[) auOC r_Ll_',E DF'S[) _50I_ DMSD LI6 ('}C _'_FTffOi') r)t,,SD_70C SOLU1](Ihl IS Dfl"_E LISI_IG IHE SI_IIARE-w{}rlT _F.TH{)D {}F CH(ILESKY. OMSi') IJSOC IHF GIVEN '_AIRIX IS _EP_E_ENIEr) AS PP(.}I)IICT()F T_(I TRIAN{;ULAROMSD _g_C r,_&I,_ll;E_, -EwE r,E LI-FI _ANO F_CTIIR IS TwF TRANSP(ISE OF DMSI.) 500C I'-E _ETu_r,Eh RIGHT HA:,H) FACT()R. {]t_SD _|hC DMSD ";20C .................................................................. DMSI) 53,")C DF_Sf_ 5_0

Stl_uuTIr,le I)'_FSI)(A,,,I,EPS,]ER) I)MSD 5.':)nC ()MS_) 5bOC DMS() S7n

DIr4ENbIDN A(1) DMSD 5_(_

v

PAGE O00bmEmFIE_'Ja.'.,E D:,_FSO

DOIJ_LE PRFCISION DPIVwDSU_eA D.'_SD 5goC DmSD 6on

IFST ON ,'.RDNG INPUT PA;_AMEI_R N DI,:SD blOC IF{'_,'t} IP,l,1 r)mSD 6?0

I I_,4=0 Dt.'.SD_,30(' Dr.,S D #)_0C IF,I IIALIZE OIAI;O',JAL-LgOP O'-_51)650

w_PlV:0 Dt'_5O _0I'_n It Krt,N D,v,SI) 070

I;_[,,:wp I V D;"51) hq0Lk'lat_: f_ -- I L)MSI) 700

C r),mSI) 710C CALC=II ATE T(}LEPArJCE D.v.S()7PO

T;.JL:A."S (kP.S*SNGL (A (KP IV )}) I)MSD 7_nC Dr,_sD lu0C SIAWT FACTOklZAIIUN-LOOP OVER K-TH P(IL'd I,,MSO 7bO

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