-i -- __ __________________

TECHNICAL REPORT ARBRL-TR-02481

MAXIMUM LIKELIHOOD PROGRAM

FOR SEQUENTIAL TESTING DOCUMENTATION

Ann E. McKaigJeirry Thomas

March 1983

A h V US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLAND

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Ann E. McKaigJerry Thomas

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Maximum Likelihood EstimatesNewton-Raphson ProcedureQuantile ResponseSensitivity Testing

2(k ADSTRACT (Citnuei do tern.. 014 N eoem7 and tddentiy by bloc* mawmb. alQuantal response problem- encountered by the Bal liffic Research Laboratory, suchas armor plate penetration and primer testing evaluations, have been a subject ofinterest for some time. A computer program developed by A. R. DiDonato and M. P.Jarnagin, Jr., has been modified and installed on the Ballistic Research Labora-tory's CDC computer. The program provides maximum likelihood estimates of themean and standard deviation of the response distribution Pnd the standard devia-tions of these estimates. The response function is assumed to have a cumulative

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normal distribution. A nonparametric algorithm is incorporated for the cases iwhich maximum likelihood estimates do not exist.

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TABLE OF CONTENTS

Page

I. INTRODUCTION .................... ........................... S

II. THEORY ........................ .............................. 6

III. PROGRAM DESCRIPTION ................. ....................... 10

A. DiDonato-Jarnagirt Procedure ....... ................. ... 10

1. Likelihood Function ......... ................... ... 10

2. Comnuting Procedure ................................. 12

3. Output Description ........ .................... ... 15

B. Nonparametric Algorithm ............. ................... 15

1. Computing Algorithm ....... ................... .... 15

2. Output Description ........ ................... ... 16

IV. APPLICATIONS (EXAMPLES) ........ ... ..................... ... 16

V. SUI'XARY ................. ............................. .... 20

VI. %CKNOWLEDGEMENTS ........... ......................... ... 20

APPENDIX A. EXPECTED VALUES NEEDED FOR ASYMPTOTIC VARIANCESOF AND a ......... ..................... ... 21

APPENDIX B. PARTIAL DERIVATIVES NEEDD FOR NEWTON-RAPHSONSPROCEDURE .......... ...................... ... 29

APPENDIX C. INPUT NEEDED AND PROGRAM LISTING .............. ... 37

APPENDIX D. COMPUTER OUTPUT FOR EXAMPLES ..... ............. 57

DISTRIBUTION LIST ............................... ........... 63

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Jo

I. INTRODUCTION

The Army has used sensitivity testing for many years, especially in theareas of primer testing and armor penetrating projectiles. In sensitivitytesting, only one observation per sample is obtained. This observation ischaracterized by a quantal response, i.e., go or no-go, explosion or non-explosion, penetration or nonpenetration, The purpose of most sensitivityexperiments is to determine the amount of stimuli needed in order to achievea 50% probability of success.

For armor penetrating projectiles, the stimulus is velocity. So insensitivity testing, the aim is t- determine the V5 0 , i.e., the velocity neededto have a 50% probability of penetrating the armor.

There are many methods or schemes for collecting data in sensitivitytests. Some of the more popular ones are tne Bruceton up-and-down method, theLanglie design, the Robbins-Monro design and the Alexander design. For adiscussion of these designs see Reference 1.

As mentioned earlier, we are interested in estimating V 0 and getting an

estimate of its variability. The most commonly used technique of obtainingparameter estimates is to assume that the critical stimuli levels are normallydistributed and to use maximum likelihood estimation (MLE) techniques toestimate 11 (V5 0 ) and a. By critical stimuli levels, i.e., critical penetration

velocities, it is meant those levels (velocities) at or above which a penetrationwould always take place. These estimates are not dependent upon the sensitivitydesign used to collect the data.

There are many computer programs that compute MLE's. However, most ofthese programs, at one time or another, have a convergeý-- .oblem. That .s.initial values of the estimates cannot be Cound that r in the prop":nconverging on estimates of the mean critical stimulus le , i, and the stai, .-ard deviation of the critical stimuli levels, a.

DiDonato and Jarnagin2 used a Newton-Raphsorn procedc ransformedparameters and claim that it converges globally to the "be. estimates (whenestimates exist),regardless of what initial values were used. Unique MLE'swill exist as long as the data meet the following two restrictions: 1) min-imum level, ai, at which a success is observed < maximum level, b., at which

a failure is observed, and 2) average aiis (penetrations) > average bi's

rIonpenetrations). They documented a program written for an IBM computerusing their procedure. DiDonato and Jarnagin's program was modified andinstalled on the Ballistic Research Laboratory's CDC computer.

ID. Rothman, M.J. Alexander and J.M. Zimmerman, "The Design and Analysis ofSensitivity Experiments," NASA CR-62026-Vol. I, May 1965.

2 A.R. DiDonato and M.P. Jarnagin, Jr., "Use of the Maximum Likelihood MethodUnder Quantal Responses for Estimating the Parameters of a Normal Distributionand its Application to an Armor Penetration Problem," Naval Weapons Laboratory,November 1972, NWL Technical Report TR-2846.

S

IThis report discusses the statistical theory needed to estimate the mean

(Vs0 ) and the standard deviation of the response distribution and to obtain

the standard deviations of these estimates. A discussion of the DiDonato andJarnagin computing procedure used to obtain these is provided. A nonparametricalgorithm has been incorporated with the DiDonato and Jarnagin program toprovide an estimite of the mean (Vs0 ) and standard deviation of the response

distribucion when the data do not meet the requirements for the DiDonatoand Jarnagin procedure. Examples are provided for each of these algorithms.A computer listing, the appropriate inputs and corresponding output of the pro-gram are .ncluded in the appendices. The reader who is more interested in learn-ing to run the program rather than in the theory is referred directly toAppend.- Y.

II. THEORY

"1e principle of maximutm likelihood was first introduced and extensivelydeveloped by R. A. Fisher. To aid in deriving the likelihood function, can-sider the results of an experiment (such as an armor plate penetration problem)in which the levels of the stimulus are not completely under control. There isgenerally only one observation per level, and the response for each level iseither a "success" or a "failure." Call the probability of a successfulresponse Pi, i=l, ... , n and the probability of a failure qj, j=l, ... , m.

Then it follows from elementary statistical concepts that the probability ofthe observed set of responses may be given by the product

L= plP2 ... Pnqlq2 ... qm"

recorded at stimulus levels al, a 2, "". a nand m failures recorded at levels

bl, b2 , , and assuming the critical levels of stimulus are normally dis-

tributed, then the probability of occurrence of this set of events may bewritten in the more usual notation,

n mL, = TJ Pi 11 qj()i=l j=l

(a -1j)/o

where pi p[(ai-u(/°] = i f exp(_t2/2)dt' (2)

and

6

qj E q[(bj-p)o =1 exp(-t 2/2)dt. (3)

The method oi maximufr likelihood now consists of choosing, as estimatesof the unknown population values of U and a, those particular values thatrender L, as large as possible. Since likelihood functions are products, andsims are usually more convenient to work with than products, the usual practiceis to maximize the natural logarithm of the likelihood function rather than thelikelihood function itself, i.e.,

Sn m

Lt. ,cf) in pi + I ft qj. (4ni=l j=l

The logarithm of the likelihood function has its maximum at the same poin,as does the likelihood function.

The maximum-likelihood estimators of p and a are obtained by settingthe partial derivatives of the likelihood function equal to zero, i.e.,

3 L •L

0, L_ = 0 (5)

The likelihood equations can then be solved iteratiXely using some techniquesuch as Newton-Raphson, to obtain estimates, V and a.

Previously it was assumed that there was a given set of observed responses,n successes at levels a2, ... , a and m failures at levels bl, b2 , . bm,

Suppose that for a set of distinct levels of stimuli, Xk, there is a statement

associated with each level that it was either a "success" or a "failure." Cr-n-

sider the discrete random variable 6k (k = 1,2,...,N), where N equals the number

of levels observed and each 6k may take on one of two possible values, i.e.,

0, if a failure occurs at level 9k6k = k

1, if a success occurs at level Z

If the probabilities of success at the resulting levels are p. and the

probabilities of failure are q = -k' then the likelihood of observing some

sct of responses 6k (k = 1,2,..., N) at levels Zk (k 1,2,... ,N) isk k klk

NLo= H Pk k q k (6)

k=l

71

and the natural logarithm of the likelihood function becomes,

NL° =lH [6k Un Pk + (1"6 k) tn qk]. (7)

km 1

It is this form of the likelihood function that will be used to constructthe elements of the covariance matrix utilizing the theory of large sample dis-

tributions. The variances of p and a may be obtained using the relationshipsbelow between the expected values of the first and second-order partial deriva-tives of L° (Appendix A),

S2 3L° 2 N Zk22L°; ý7 1 _- = J q (8)

-\ a2 / 2 k=l Pk k

12 22= j 2-E 32 =E a kl (kqk

and

0E ( 0 (LL 1 N vjkzk 2 (10)-E (3- -] 3a - ] o-- k1 kqk(0

Additionally, i

AAaa .(13)0A

Am 2 L 0 A L0 0) (LO A~0(

3A. Med and F. Graybill, Introduction toi the Theory. of Statistics, McGraw-Hill

Book Company, Inc., 1963, pp. 235-239.

!-

where A 1 and A!" are the asymptotic variances of i1 and a, respectively, and

A'0 is the asymptotic covariance of U and a. It should be noted that A"" is

the asymptotic variance of the estimate of V while Ae is the asymptotic

variance of the standard deviation, a, of the response distribution.

For convenience of notation, equations (8) - (13) may be rewritten.Recall that k = 1,2,..., N, and divide N into n successes (penetrations) andm failures (nonpenetrations), so

N n + m.

The n successes recorded at stimulus levels al, a2, . a and m failures

I recorded at levels b1 , 2 may be rewritten as

si (ai - V)/O, where i = 1,2,..., n (15)

and

t.= (b - i)/o, where j = ,2,. m, (16)3 .

which are their standardized normal equivalents. The s. are associatedwith successes and the t. are associated with failures. 1

Also,

x. z(s.) = (l:" (-si2/2), when a success is observed (17)

and

Sz(t) = (lV-) exp 1 /2), when a failure is observed.(I8)j J

Equations (8) - (10), which rel -.it the expected values of the second-orderpartial derivatives taken ovo- t entire set of observations, can each bepartitioned into the sum ove set of successes plus the sum over the setof failures.

^2The coefficients of a • ", A , A I may be expressed as

A22 m y 2

""A ÷ .2 ( 19)*2A piqi PjO q

2 2n s. x. M t.y.

i=l Piqi =1 Pjqj

-F- 2 2-.-2' 2

and 2 n S. X. 2 m 2

a Aac Y 1 + .(21)i=l Piqi j=l Pjqj

III. PROGRAM DESCRIPTION

This section first presents an overview of the DiDonato-Jarnagin procedure,including: i) the reparameterization of the likelihood function, ii) the

AA

actual procedure used to obtain ^ and a, and iii) a description of the outputsheet. Should a more detailed mathematical description of the analyres berequired, the reader is referred to the DiDonato and Jarnagin study.

A nonparametric algorithm which has been incorporated into the originalprogram and its associated output sheet are also summarized. This procedureclosely resembles a procedure presently utilized by the Materie.l Testing Direc-torate, Aberdeen Proring Ground, Mar/land.

The complete program is available for use on the BRL CDC computer system.Both the DiDonato-Jarnagin procedure and the nonparametric algorithm arewritten in FORTRAN IV.

A. DiDonato-Jarnagin Procedure

1. Likelihood Function. Recalling the likelihood function for anobserved set of n successes at stimulus levels a 1, a2 , ... , an and m failuresat stimulus levels bI, b2, ... , b , we have

n mSL(i%,o) = p i + Zn qj. (22)

Pi4 j=l

DiDonato and Jarnagin then apply the following transformations to replacethe parameters V and a with new parameters a and B.

Il= i> 0 [o = lIB]. (24)

In addition, the original stimulus levels are reexpressed as,

si a.-ct (2S)

t. = b.B-c. (26)

so that pi and qj are transformed in terms of the new parameters to

10

pi = p(si) j z(t)dt (27)

qj ý q(t • - z(t)dt (28)

2where z(t) = (ly2-W) exp(-t /2) (29)

* For ease in computing the partial derivatives of the logarithm of the

likelihood function, the computations of the needed partial derivatives ofpi and q are included as Appendix B. The partial derivatives of the logarithm

of the likelihood function, L, used in the Newton-Raphson procedrre are

m nlSDL (yJ/qJ) - l (xi/pi) (30)

3L n mO Ll =1 1 jillai(xi/pi) - Jlb.(yj/qj) (31)

= 2 j7 (yj/qj)[(y/q) - tj] (32)

n7 (xi/Pi)[(xi/pi) + Si]

a g = 2b (yj/qj)[(yj/qj) - tj] (33) [n

+ . ai(x i/Pi)[(xi/pi) si]

i=l

where x.i z(si) = (IVW) exp(-si 2 /2) (35)

1t.) = exp(-t. /2). (36)yj - t11

The {a. = stimulus level at which there isa response, penetration) and

{b. = stimulus level at which there is a nonresponse, nonpenetration),

where i=l, ... , n, j=l, ... , m, represent input to the program. The quantitiesai and b. may take on any arbitrary real values although they are almost always

positive. The standard deviation, a, and its reciprocal, $, are by naturealways positive.F In -heir study, DiDonato and Jarnagin point out that to insure theex.-tence of a unique point at which L attains a maximum, the a. and b. mustsatisfy certain necessary and sufficient conditions, i.e., 1 3

a min < b [zone of mixed results] (37)

where amin = min(ai) and bmax max(bj),

and I y b. < _ a (38)m n

2. Computing Procedure. The DiDonato-Jarnagin procedure first insuresthat the necessary and sufficient conditions indicated in (37) and (38) aresatisfied by the input. If either condition is not satisfied, then nomaximum likelihood estimates exist for p and a and the program is routed tothe nonparametric algorithm. If both conditions are satisfied, an initialpoint (a ,0o ) is obtained (unless the user supplies initial estimates,

Appendix C) using the following relationships,

Iin m 2 1/2a L b')/a. ( +; a. + b. v (39)

ST n m j=l = m j=l

S 1/0 n a 2 m v2 (40)1o : (w°o :[n--m I•'ai + b'2 )40

I=l j=l j

n m

where v = n a + Y b.).Im j=l ~(1

After choosing an initial point, the ordinary Newton-Raphson (N-R) pro-cedure is applied in order to reduce La and La to zero simultaneously and producethe N-R increments Aa and AO and a new point (cl,B1). The two iterative equa-tions utilized by the estimation scheme are

AaLact + A$L4 = -La (42)

AciLaW + AM$ = -LO. (43)

12

Solving this linear system of equations yields the increments Aa and A$

as follows:

Am = (L1 La -[a LLB)/A (44)

Aa= (La La- LO Lcaa)IA (45)

2where A B Laa LBa- L2ý > 0. (46)

The increments Aa and LO are the changes in the old values of a ard 0 calcu-lated at each iteration of the N-R procedure.

If a sequence, ((,xk), 0 determined by iteration and thethak and k denote the k approximations to a and •, then the iterations are

terminated when

Ik< EI and JA5kI < e2 B2 . (47)

The parameters c1 and 2 are defined as follows:

Cl = 2.5 x 10- 4 =1/2 2 " (48)

At this point, additional notation is introduced in order to rewrite equations(19) - (21) and (30) - (34) as actually used in the DiDonato-Jarnagin proce lure.

Referring to the parameter input description in Appendix C, it can beseen that only those a. which are different are listed along with an integer

1n(i) which denotes the number of times ai appears as input. The integer

m(j) denotes the number of times each different bj appears as input. Noting

that the expressions for La, L., Laa, W6s, LBO are linear sums, quantities

such as (xi/pi) and (yj/q.) need only be computed for the different ai or b.,

then multiplied by n(i) or m(j), respectively.

If the kth different ai is denoted by a(k), the rth different bj is

denoted by b(r), and K and R denote the total number of different aiand b.,

respectively, then

K Rn= n(k), m= miu(r) (49)

k=l r"=1

and n and m have their usual meaning. The equations referenced above can nowbe written as:

13

R Ka m m(r) (y /q ) . n (k) /k) (50)

r=1 l k=1

K RLa n(k) a(k) (xk/k - I m(r) b r(y r/qr) (i

R= - m (r)Yr /q r) (Yr/q r t r)

r=1

Kn )n(k) (x/p)'kk Sk (52)

k= 1

KT.~ n(k)a(k C-k/Pk)(xk/Pk + sk)

k= 1

+ .m(r)b(r) (yr /qr) (yr/~ r r (r= 1

R 2 r(/q)Yq

r= 1

-Kn(k)a 2(k)(xk/Pk)(xk/Pk +Sk) (54)

k= 1

"ý2 Ka Ap~i nY )( /P)xk/k

k= 1

R+ I m(r)(y 1~pr)(Yr /q r) (55)

r= 1

"ý2 Ka Alia =)n(k)s (x /~(xk/q~

k-i

+ m(r)t ryrr r rY/~)(6

r= 1

14

tK

"^2 K 2cr Auc n (k)St. (xk/k x/kk=l :

R 2+ 7 m(r)t r2 (Y/Pr)(Y /qr) (57)

r=1

where sk = aCk) 8-a (58)&k

t = b(r) B-c . (59)

3. Output Description. The output for examples 1 and 3. (Section 4)having MLE's generated under this procedure is shown in Appendix D Beginningin the upper left-hand corner, each output sheet lists a sequence of

alphanumeric characters which mc.y indicate a title, identification number, etc.This is followed by a listing of the different a. and b and the number of

times each value occurs. The maximum likelihood estimates, p and a, identi-fied as 'MU' and 'SIGMA,' respectively, are listed toward the top center.

A A'MU,' or p, is the estimate of V so while 'SIGMA,' or a, is the estimate of

the standard deviation of the response distribution. Directly below theestimates, the elements of the covariance matrix are recorded, followed byA A

a and a identified as the 'STD DEV MU' and '3TD DEV SIGMA,' respectively.

These quantities are followed by the initial approximations of a°0 ('ALPH0'),

80o ('BETA0'), 10 ('MU0), and ao (ISIGMA0'). The Newton-Raphson increments

at each iteration, Ac, A8, written as 'DELTA ALPHA' and 'DELTA BETA,' aswell as the associated value of L, are listed next. The last line lists thef,. 1 estimates of the likelihood function L,, the determinant A, a, and aana are identified as 'MAXIMUM,' 'DELTA,' 'ALPHA,' and 'BETA,' respectively.

B. Nonparametric Algorithm

1. Computing Algorithm. As previously mentioned, this algorithm isinitiated should either of the necessary and sufficient conditions imposedupon the input data not be satisfied. This routine will compute a maximumof three (3) estimates for th• ,,ean and staidard deviation using a methodof averaging particular levels of response and nonresponse. To facilitateexecution of the algorithm, the original inpat arrays have been expanded toinclude all ai's and bi's, including replicated values. Each array is rank

ordered from the smallest to the largest level.

The first estimate of the mean is obtained by averaging the lowest valueof a response aad the hig'est value of a nonresponse. Then the two lowestresponses and the two largesi nonresponses are averaged to obtain a second

is

estimate for the mean. This iterative procedure may be continued until alleven number combinations of the ai's and b.'s are exhausted. However, it

iJhas been elected to allow the algorithm to generate only a possible maximumof three (3) estimates (requiring a total of six (6) input levels). At eachiteration an estimate of the standard deviation of the distribution iscalculated as follows:

1/2 x [range between levels] (60)

where 'range between levels' is defined as the difference between

max [B0(MTOTB), A0(I)j - mrin [B(J), A0(1)] (61)

and AO(l) = smallest response value of input array (62)

SA00I) = response ,alue at kth iteration k=l 3 (63)

BA(J) = nonresponse value at kth iteration, k=l, ... ,3(64)

BO(MTOTB) = largest nonresponse value of input array. (65)

2. Output Description. The output generated for example 2 (Section 4)is shown in Appendix D. At the top of the output sheet, the proeram printsout a description of the first necessary and sufficient condition that isnot satisfied. Directly under this is an ordered listing of the input arraysof responses and nonresponses, including replications. Finally, for eachiteration, the estimate number and the number of rounds required for each

A

estimate of v and a fo.lcwed by the values for U and a, written as 'MU' and'SIGMA,' respectively, are shown.

IV. APPLICATIONS (EXAMPLES)

The three cases used for illustration are taken from armor plate pene--tration studies. The actual output of the computer program for each of theexamples is included as Appendix D. The data set for the first examplesatisfied the necessary and sufficient conditions outlined in Section 111;therefore, the output will be generated from the DiDonato-Jarnagin procedure.Example 2 is a 'no zone of mixed results' case which utilized the nonparametricalgorithm. The third example satisfied both conditions required for MLE's

to exist for V1 and a, yet diverged using computer generated starting valuesin a different estimation program. This case is shown for two differentsets of starting values for which the results shew convergence.

16

For each example the initial estimates were generated by the program;in addition, the third example was run a second time with user supplied start-ing values for a° and B

0 0

EXAMPLE 1

in firing 10 rounds at a given armor plate the following observationswere recorded:

VELOCITY (mis) RESPONSE

1008 Penetration

976 Nonpenetration

98:4 Penetration

973 Penetration

947 Penetration

943 Penetration

924 Nonpenetration

931 Nonpenetration

960 Penetration

942 Nonpenetration

Initial estimaces computed were a = 37.367 m/s (Vo = 956.625 m/s),

ý° - .039 L!Rs (ao = 25.464 m/s) The N-R iteration procedure qas then applied

4 times to generate final estimates of a = 32.123 m/s (p 948.823 m/s),

.034 m/s (a 29.537 m/s). Employing the solutions obtained, the

approximate asymptotir variances of p and a were obtained. The covariancematrix is given by

183.506 -SO.6S3

- 50.6S3 358.946

Thus,aA2 = 183.506 m/s (a. = 13.546 mfs) and a^2 = 358.946 m/s (a^ 18.946 m/s).

EXAMPLE 2

Seven rounds of a given projectile were fired at a given armor plate. Thefollowing observations were recorded:

17

VELOCITY (m/s) RESPONSE

944 Nonpenetration

973 Penetration

961 Nonp• etration

982 Penetfation

966 Penetration

949 Nonpenetration

970 Penetration

Once the data were sorted, it was clearly seen that there was 'no zone of mixedresults.' The program then routed to the nonparametric algorithm. To illus-

A A

trate the method, :r.t's calculate the second estimate of MU (U) and SIGMA (a).The second estimate required a total of 4 levels or rounds (2 lowest responses,2 highest nonresponses). From the sorted input lists, the two lowest responseswere 966 m/s and 970 m/s; the two highest nonresponses were 949 m/s and961 m/s. From the routine,

M4JO = MUIi + (A(2) + B(2))/2.0

Lstimate 1 for MJ = 963.5, so

MUO = 963.5 + (970 + 949)/2.0

MUO = 1923

W = 2.0 (W = KOIJNT; KOUN7 = iteration .,.

MU = MOJ0/W = 1923/2.0 = 961.5 m/s

SIGMA = (RRANGE - SRANGE)/2.0

Where RRANGE = MAX(BO(MTOTO•, AO(1))

= MAX(961, 970)

= 970

SRANGE = MIN(B0(J), AO(l))

= MIN(949, 966)

= 949

Thus, SIGMA = (970 - 949)/2.0

= 10.5 M/s

18

---------.

EXAMPLE 3

Again, 10 rounds were fired at a given armor plate and yielded thefollowing results:

VELOCITY (m/s) RESPONSE1295 Nonpenetration

1296 Penetration

1298 Nonpenetration

1301 Nonpenetration

1303 Nonpenetration

1304 Nonpenetration

1307 Nonpenetration

1310 Penetration

1311 Penetration

1314 Nonpenetration

a) Initial estimates generated by the program were o= 210.224 m/s

(o = 1304.405 m/s), So = .161 m/s (CO = 6.205 m/s). Applying the N-R scheme

3 times produced final estimates of t = 50.657 m/s (p = 1317.893 m/s),

B .038 m/s (a = 26.016 m/s). The following asymptotic variances were= 'obtained:

OA = 690.476 m/s (ci = 26.277 m/s)

anda = 2142.252 m/s (Ga = 46.283 m/s).ai

b) Initial approximations of •o = 26.000 m/s, B = .006 m/s were input

by the user. These values generated initial values for U0 and a of4333.333 m/s and 166.667 m/s, respectively. The N-R procedure had to be applied

AAA A

5 times. Final estimates for a, B, p, a were 50.658 m/s, .038 m/s,

1317.892 m/s, and 26.016 m/s, respectively. Applying these solutions yielded

the following asymptotic variances for p and 0;

G^ = 690.443 m/s (a- = 26.276 m/s)

and2G^ = 2142.0S7 m/s (GA = 46.282 m/s).

The above results clearly show convergence.

19

V. SUMM4ARY

For problems of a quantal response nature, a discussion of the maximumlikelihood estimation theory and large-sample distribution theory requiredto obtain the estimates of the mean (Vso) and standard de/iation of a cumula-

tive normal response function has been given.

The A. R. DiDonato and M. P. Jarnagin, Jr. maximum likelihood estimationprogram has been modified and installed era the Ballistic Research Laboratory'sCDC computer. Instructions needed to properly execute the program have beenprovided. This program has successfully converged to provide maximum likeli-hood estimates for all data sets tried by the authors. A nonparametricalgorithm is also available for those cases where maximum likelihood estimatesdo not exist.

VI. ACKNOWLEDGEMENTS

The authors wish to express their appr~ccation to the following individualsof the Experimental Design and Analysis Branch of the BRL: William E. Bakerand Barry A. Bodt for their programming technical support and Jill II. Smith andJock 0. Grynovicki for reviewing the text and providing technical assistance.

I-

20

APPENDIX A. EXPECTED VALUES NEEDED FOR ASYMPTOTIC VARIANCES OF i AND a

21

APPENDIX A. EXPECTED VALUES NEEDED FOR ASYMPTOTIC VARIANCES OF ji AND a

A

The variances of vi and a may be obtained in accordance with maximum like-lihood theory and large-sample theory from the expected values of the second-order partial derivatives of the function LOP

NL 0 = nL = Y. [6k tn P k + (1 - S (A ln qk1. (Al)

k=l

Use of the following general relationship will simplify the conputationof the expected values,

ae 2S L T =- -jBj,(A2)

where the partial derivative is taken with respect to some parameter 0. Here,0 would be equivalent to v and/or a.

Define zk =(1/,2T) expVk/ 2 , where vk = (Yk-V)/O. (A3)

and Pk f (1/yr2.)exp L 2 dt - qk

Then,

aPk "Zk qqk Zk 3Pk "Vk Zk

and

a qk V k Zkz kz

The maximum likelihood equations with respect to p and a may be writtenas

23

k (Pkk).n ' An q% ]

k I

N -Zk zk

I k=l Pk oqk

3L N z z

§.- •. [ 6k • k1 + l6 k1 (AS)qkk= 1

Lo 1 [ (1.6k) k_ zk (ASi~3 17 S k "Pk 6

k= q

k=1 k

, N

Now, consider the fact that •[g k•,=• (k" 1 hr

tions~ (AS an aa( k~ hrkk~l

E(6k) =f(1) =kadE{1- 6 k) = f(O) = q;then the expected values of equa-

LoI k k 1 k

--") k=l k qk k Pk(M

I N

=-k k" 'kk

ok="tions AS) ad (M)becom

E - ) = 0. CAT)

24

rr'DL 0 N VkZk VkZk

E Do a Y . [(I-0) q q'k (1C) •Pk0 k k=lP

N.

ii,k=1

aL.

E (0.-) = O. (AM)

Using equations (A2) and (AS) - (A6) and the definitions developed in

the previous paragraph, the expected values of the second-order partial

derivatives of L° may be easily computed as follows:

2a2L 3Lo 2|- 0 [---lu E [ --0-

N zkE ,. [(- - k k

SI N k2 k2o k! qk Pk

- 2I{-k]pq 2 2

2N 2 k2 k

N z NF L 2 kk + 2E [2 1 1(1 2+ 'k~ Ak=l q"Pkl

O" qk Pk :

25

4kI

3L 2A A

2L

30

=E I i y. [(1_6k') _6k ]crk -I qk k P

2k kki2~E 0 k= l 1 k Vq k P

2 2 2 2

Nz 7 v z

E1' (-k2 vk k 62 k k= E 2 [ 1 -2-qT-2+ 2

cy k=l qk Pk

0ak1

1 N 2 2 222

a L 0 v[(1-0 k 2 v Vk zka k=1 kP

- 01

2 22 2au-- T k='l k k=l P

a2Lo3L 3Lo

-E 0 1- E-

f N z k z k= •k~ q k tk

* _ N 1 'k~ k Vk~k~k-- 1kk=l kq kk Pk

N 2 2"2 kZk 2 VkZk

E M- _6k)+ k 22 k,=1 q. Pk

2VkZk

- 26k(1-6k! 1 Ik

N 2 2_ [{I=()2 zk 2 VkZk

- 01

26

L

a 2Lo N VkZk N v(kzkS-E [U -a -2 Y ' . - + (All)

a k=l k k=l Pk

Therefore, equations (A9) - (All) may be rewritten as

0 1 - k (A12)

2 2 2SL N v z

kk-E [ ! I (A13)aa a •o2k=l Pkqk

anda 2L 1 N v kzk2

a- k=l Pk1k

27

APPENDIX B. PARTIAL DERIVATIVES NEEDED FOR NEWTON-RAPHSON PROCEDURE

29

-1 I - -,=- -~ - - -- - - -

APPENDIX B. PARTIAL DERIVATIVES NEEDED FOR NEWTON-RAPHSON PROCEDURE

We wish to compute the partial derivatives of the pi and q. as an aid

in computing the partial derivatives of the function,

n mL = InL. = £n p + " In qj. (BI)

i=l j=l

Let the pi and qj be expressed in terzus of the new parameters as follows:

1I a.8-a2= 1 exp[-t2/2]dt, recall s. = ai0-a (B2)

I• and+co

qj= i2n exp[-t 2 /2]dt, recall t = b a-(1 (B3)

_+ bjb 0--

tJ

Also,

x- (1//2-n) exp(-s2 /2) (B4)

and

y. (1/127) exp(-t2 /2) (BS)3J

Let = (-I//-F) exp[-(a8-a)c2 /2] = -x. (B6)1 1

then, a2pi a(-x.) (ai A-a)

=- 3a exp[-(a.0-c) /2]

2

am 2= -s.. (B7)

2a pi a(-xi) ai(a.0-a)2

____=exp[-(a $-a) 2/21

= aPi x 0B8)

31

- 2

3$2- ' - (B9

2aPi 2

Similarly for the q.,

Lt !i= (1/vr2-) exp[-(b.0-00 2 /2] =y. (Bil)

then, B . y. (b.0-ci) 2! i= 'I= 3 expf-(b O-a) /2]

3aa act

2 j t. (B12)am2 jy'j'

a q. ay. -b.(b.B-Ci) 2= ~ expII-(b.$-ai) /2]

32 - b ty. (B13)Baa0 3 J J-

aq. -b.2-a 3 exp[-(b B-ci) /2] 1 b.y., (B14)as /37 3 3

and a q. b. 2(b.6-ca) 2as 2 27 expr-(b.8-00) /2]

23

Using che above results, we have

n 11

31.= -y. q. + (En qx1 , ( i

2a (t p, II j-,aci i=1

ny a Bq3q

L Bci qj 30

j=i

j=l p

2 2 X

j~l 3a qj P~ i p

LacBy = v/q)(.q.Vt ~(.p) B7

33

1 '3 -

2 mn X

ji=1

a)y. 3q.

j~l 2

+ ~ 1+ 1

i=1 2

2 2m -b.t.y. b.y. n 1. 1.x .x

_j 1 j +1 1 + 1 1 )i~l qi p.

mbc8 7 b(yjfq.)II(y./q) -t.j

n+ a (x /+)(x/ si1. (Bi8)

Similarly,

n mLa -(tn pi) +7-(in qj)

n ap.aq.I -L ) + L 3

il Pi 3 jlq

n In

a a(x./p.) - 7 b.(y.Iq.), (B19)i= 1 1 1 1 j 1 3

34

n a. x..

ae 2 i~l Pi j=1 s l

3x. ap.

30 2 ai=1 Pi

~y. aq.

y ii 2~q(~~ ~

2a. x.n -a s x.i 1 1

m ~-b.t.y. by2

b 2

m 2L8B -)S (y.i/q.3)f(yjfq.) -t.]

nay 2 (x /p1)II(xifPi) +s1J. (B20)

35

APPENDIX C. INPUT NEEDED AND PROGRAM LISTING

37

APPENDIX C. INPUT NEEDED AND PROGRAM LISTING

The A. R. DiDonato and M. P. Jarnagin, Jr. maximum likelihood estimatingprogram is available on the Ballistic Research Laboratory's CDC computersystem.

Prior to executing the program, a data file must be created and storedon MFA (mainframe A). A list of the parameters that must be read in as inputand the format and instructions needed to create the data file are providedhere.

The three (3) card jobstream that must be created atid submitted to MFZ(mainframe Z) in order to run the program is also given.

It is important to remember that the data file must be made accessibleto the NFZ while in the interactive facility (IAF) mode.

The following list briefly describes each of the parameters which mustbe read into the program prior to execution and which also comprises thecalling sequence for the main computing subroutine EPPA.

CALL EPPA (IDENT,K,L,ALPHO,BETO,FNAA,FNB,B)

IDENT - Array dimensioned at 8 locations with up to 80 charactersallowed per job. The Hollerith character content of IDENTis printed on the top line of the output. Note: IDENT is notprinted when the nonparametric procedure is employed.

K- Twice n, where n is the number of numerically different a.'s(responses). n > 2. 1

L- Twice m, where m is the number of numerically different b.'s(nonresponses). m > 2.

ALPHO.BETO - User supplied starting values a , B , respectively, to thealgorithm. Setting BETO 4 0 allows Nhe user to have the algorithmcompute the starting values ao = 1oI0o and 00 = 1/0o.

FNA,A - Arrays dimensioned at k = 2*n. FNA(i) specifies the number ofA(i) values used, i = 1,2, .... ,n. FNA(nl), FNA(n÷2), ... ,

FNA(2n) and A(n~l), A(n+2), ... , A(2n) is used by EPPA asworkspace.

FNB,B - Arrays dimensioned at I = 2.m. FNB(i) specifies the number ofB(i) values used, i = 1,2, ... ,m. FNB(m+l), FNB(m.2), ... FNB(2m)and B(m+l), B(m+2), ... , B(2m) is u3ed by EPPA as workspace.

39

STEP 1. CREATING THE DATA FILE

Data should be created and stored as an MFA permanent file.

Card 1 Columns Format

IDENT 1-80 8AlO

Card 2 2 inputs

K 1-5 i5

L 6-10 Is

Card 3 2 inputs

ALPHO 1-10 FlO.2

BETO 11-20 F10.2

Cards 4,5

Input may consist of a maximum of 8 data points per card. Morethan one data card may be required to input each parameter. Thesecards contain data read in with an F10.2 format.

Columns 1-10 11-20 ............ 71-80

Card 4 FNA(1) FNA(2) FNA(8)

Card S A(l) A(2) A(8)

Cards 6,7

Input may consist of a maximum of 8 data points per card. Morethan one data card may be required to input each parameter. Thesecards contain data read in with an F10.2 format.

Columns 1-10 11-20 ............ 71-80

Card 6 FNB(1) FNB(2) FNB(8)

Card 7 B(i) B(2) B(8)

STEP 2.

Prior to running the program, the data file must be made accessible tothe MFZ while in IAF (/) mode with

PERMIT, PFN, MFZ

where

PFN = filename under which the input data is created

J4

STEP 3.

To run the program, create and submit the following 3 card jobstream to MFZ.

JOBNI•4E, STMFZ.

ACCOUNT, XXXXXX.

BEGIN, NMn, MUIR, PFI= ., (RJE=RJE

where

PFI = file name under which the input data is stored,

UN = user name identification for above file,

RJE = RJEXXXX where XXXX is a 4-digit code designating a particularRJE as the output device; if omitted, the central site servesas the output destination.

41

irz+ thEA 4 .

.31 3j - 3.. 2.)

*-4 4J- 1.1-. zL jA Ai 4)

Z-- -3j r,.4 r J -

.. n %. n - s.- , 1:nI z A. Z j D-

S. A31 1.3 J3-

K" Z ,J -Z,0 - -- 4 &',J -

T 4 nAjJ- 4 £.j ~J~ AJ41

I) Ai Q 4

, •v .. " . 1 -,J' I,, X.I '

Z c 3. 1,- -4 * ( 4i

:D*---* 10 Z . 4 JE -j 1D . -- 0 ~ z• m)---,-*-* x > A

o * 0. 1 .- .Z jJ 3 -- 3.1 .Aa

4 *o• ) .-4 U')r.-•; '. 3 i .

a. 49 --4 -4 -4 -0 0 ~-)

-4 -42 M .. u . "u-4 J

z DT u Q 4 3 1j.I . n j--4- z x M a .r AJ >-:0X . J4-1'J M C

- ;:4U &1 * r J 0- 4 -44 *

.a L A £ 3- T ~ X X r4&-j 0 ** O Z .a *L 2.3Ia. I.

U.-- - - -9 AU .4CD .n n p n 4C U. Ql -- 1 0

Z4 jJJ2 ID 'V ID 2. 1--

ID14 LI 1 IL.L & 2. )Z 4 4 i x 23 .44.aJJ x -*e x *e 2. -- A1 01- 24 -i

13Z -410 mi 0-AA u u AiJ-

C-IL - -- 4J.-4 IV3.& IV *ZZ-IX X XW-) ~~~ ~ ~ -1.-XI W~bff U.ILA 41 43 Cf L W 1,-

3ZZ-4- V N JlJ 4~Jli~ ~ rx Li 3Z42'

U I

X4.

T 4

.3 -A. IJ4 Z

-14,- A - o 0 0

4)i Ji 4

4 . -4 T IV it . TnUA. A A4 r' 0 1--d Z A ' t

4j w 4A 4 0 l' j a.d - a

b.4 *ý -4-4t 3wAj J1 i ) I .4 a.0 Z DItT 04 0 t?.%.0 t I =u u ci :AJ *j c qc x I Z -A ( I U.L (n u o u z t 0 tL 1. I~j.

.aJ -~-0~ 40 -4 43 4h4*~

0'~~44Z Z ~ *--*Z * V

AAJzd

r z r

z 4%.X2AJ .3

-4i

rz

13 ID X

*4 4. j: .ji ~ ~ ~ 4I D LLa I-' 1

-4 .01 b

+- : 7- M I

4- .j 4

aJ. .I4E r 3--1.I-.349f

it I-A-4 If It r

O2 a *'EZ * _j X~ -'J1- CL)~ ' "4TL0- .J)Z4~~ *tLX*T' a~ £4

tn n.s L.n 0 Lei~ 0 ~ * E -?ZZ4T 4f~ &I~,.i4Q'~Z *~-4ZQ.4 , *

~ .lh~f.4I2 .44~*

- L

* 33

3 AJ

rJ 7ip A. L

A.L

Z .1.

J)I I -

V4 0 j 1

L~~ A 0. z.0 -

A .LI 1 0A 0 -4_j 1) N 4p 3~

-4 -4 _*fl LI J 4AJ

f- n 3

- 1 0 - m

4 I' Il-4.aU If< iAn fI .

* 3 -4 -4

r * - t4 ** NO

3 ',r~r~.4I:A~0~ - I45

- -4

PL

'-lr

'C

?4

'I

t&. I

r

_j _j* -4A * *14-4

It ,A" A ,,.ýi I 'A , m '.l.-) : ' .• '.1 -4 ,t .- -

.,,.

4b ,, J -

- ZIZ X ) Z

A- I- z LX tc- ýzz" 1ý

LnI-L -- Q >u U X: fqx )V .1 )c M- U.- U.1 c

-~ -4 ~ *JJ46

AA'

-4 -

AJ z

4-

ra -444

n NN

0 4

Ic --

-4 -V n

t 4O &i & N

n ; * f -t 11*0A* -Z) a -J -'J'4 n - 4 %J'

& 0. -. .4D. P

r 7.4 4 - 4 4 -- 4 N -

.n 'I LI J) 0 -0LA 0t m

-0- ml- M- in a4 -r -1.-$

4. 4V47

0-4 \

4j

21 -4

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X 1 .4 X t1 1I tZ 1 1-X I 1

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_jL 0 0-4C4

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9 v - 0 LiO< .40 t. r. 3 J

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If A0 -) - 1

vc- -4 VJ o-

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X _Lx I AC D 4n o -C -.-1 j * -4 I 1~ 4 .0x J I

o. r- a- 4c A i.A :D aL C4 -, Z x 4 0

l 10 ..J 1 'n is0 A4 i

0Z I'J Z A A- A1 .2

4) D.- 11 1. aJO x 4 m0

~~- fl. -4* .(Al. r-i 000 44

.A J).- -4 LI Ir.. a. AAIL0 n-2 a A 6- .. a

A* tv1 -Al v *.i cm mm m fI.-So v

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If1 4I n c "n 'M InD 0 NDO 40 4 ''jW.

If t A i 4 I1 . ~-0 le .. 3 N N IfI 1 W

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so

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T - J j 4DJ 0 MV

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ZIA-ý kSA )A JA J A

Aj~u w q -

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- *J 35 t 4 0 * 01

0 0 30 m J0 P £vý -q' 1Xt 91 % 9_'U N -- M . *P 1- D'3, - XI CH > 11 11 -0 0 11 .

-s -4 e ~ *V 0 3 9A41 fITL )- r 1w7WZA 1C 10NL90 1 1 C~ f -4- - .<l > AM 06 - 3..l' .X3 XJ C' Q.I- -

M)% %f \* *%. 0 0* -4

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A)

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Ai 0

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A X m 4) -4 W * 4 4 -. 4-4

1104 x 1 l &I zL t oZ'i 1 . Z ( - f w- 0 .114./)N~.-J I VCZ X X 31- I41 X~ X W Zq S fl-i'OCt- Agi

IA A

53

- - -

n xD ir. a "4jJ

Zf~A Z i

A t A t Li 4

r J-xn~ D A &

ýw - 4i _~j .0

Aj aJ3J A&)U * -4i -

I t- 1) LL. Ll =a -,

1 A aiJ fZ rt-Z-lr -

*L -4 - .4 -.A. LAJ 4C I- -r

Ai n-e J 4- Z jI

=~~~L * ij J '.j E rA A A )- > JD A

It * z z 1.1) x -4 -44 zb-4~ xJ~ I. A aAj

*~ T: - A'J-.lJ--a .r Ai ~ A

7 E Ai n~~ AJ~ -' 'J'DA -. Z

I)anT LMX )t a p- J

r 0 -- I D 0 -4~~ (,$-Z3z r r AJ D -jý- T

j IT' T '31 0 ~ %- -4o -4 6 A i 4 E C J Z -

x Z~ 'tbA*D z LxZI j ZxZt

CA -3A 4h %Jw 0C

b" I4b- rU Z ~ 'D A

S4

X 'a

-- -- A

1 5

ea 3Da 0 Z

U- nl 1.- 3 1.

ke I34z L-433n

Fl 4 in

0i Z

3-4Z-.33-o .Zi43

o I,-

S• /)Z • • •ss

• l. . .? a-I I • I-E E I . -• I I II - ••

I.E

T 'E

D > f) f -

p -4

II~~~1 7 l I '•

56

........

APPENDIX D. COMPUTER OUTPUT FOR EXAMPLES

57

I--

tL

ILE

0 0 4100 '.0 -0

r on

1, L 00 CO

N

a 0 V% on

'0%0% 0'

N I,~ 00

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01 rl, i

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0 : 'a 0 oc I

4, 00 N

0- W0 0i05-' 00 so 0ý40PQ' I44 0 0f V%4pq 4.4 5-. 40 a410 . 0j" .0

0% VS0 00 0

4 Its or 0 00 0it 1.S 0* 0-

in V% N00N *0

s-s w0 w-

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w IS .41 0% llO

5-1 - VS 0 .v- 0 % 0.o14 Ill lotVON

4bI 4D 10.4tN00 10 I 0. & %^

02 0 ol, N I, %#

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0 0 0

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w 0 0 9.- vs *afa*s

0 N 0 400'40 a

~~ 0 0.-'N

N 8. 0 09961

DISTRIBUTION LIST

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12 Administrator 1 CommanderDefense Technical Info Center US Army Communications RschATTN: DTIC-DDA and Development -ommandCameron Station ATTN: DRDCO-PPA-SAAlexandria, VA 22314 Fort Monmouth, NJ 07703

Commander I CommanderUS Army Materiel Development US Army Electronics Research

and Readiness Command and Development CommandATTN: DRCDMD-ST Technical Support Activity5001 Eisenhower Avenue ATTN: DELSD-LAlexandria, VA 22333 Fort Monmouth, NJ 07703

Commander 1 CommanderUS Army Armament Research US Army Missile Command

and Development Command ATTN: DRSMI-RATTN: DRDAR-TDC Redstone Arsenal, AL 35898Dover, NJ 07801

1 Commander2 Commander US Army Missile Command

US Army Armament Research ATTN: DRSMI-YDLand Development Command Redstone Arsenal, AL 35898

ATTN: DRDAR-TSSDover, NJ 07801 1 Commander

US Army Tank Automotive RschCommander and Development CommandUS Army Armament Materiel ATTN: DRDTA-UL

Readiness Command Warren, MI 48090ATTN: DRSAR-LEP-LRock Island, IL 61299 1 Director

US Army TRADOC SystemsDirector Analysis ActivityUS Army ARRADCOM ATTN: ATAA-SLBenet Weapons Laboratory White Sands Missile RangeATTN: DRDAR-LCB-TL NM 88002Watervliet, NY 12189

1 CommanderCommander US Army Research OfficeUS Army Aviation Research ATTN: DRXRO-MA, Dr. Robert Launer

hnd Development Command P.O. Box 12211ATTN: DRDAV-E Research Triangle Park, NC 277094300 Goodfellow Blvd.St. Louis, MO 63120 1 Commander

US Army Combat DevelopmentDirector Experimentation CommandUS Army Air Mobility Research ATTN: ATEC-SA, Dr. Marion R. Bryson

and Development Laboratory Fort Ord, CA 93941Ames Researcit CenterMoffett Field, CA 94035

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