INFLUENCE ANALYSIS ON EXPONENTIAL NONLINEAR MODELS WITH RANDOM EFFECTS

2003,23B(3):297-308

INFLUENCE ANALYSIS ON EXPONENTIAL NONLINEAR MODELS WITH RANDOM EFFECTS Zhong Xuping (

~ Jt-f-)

Zhao Jun (

~1k.

1

)

Department of Mathematics, Yangzhou University, Yangzhou 225001, China

Wang Haibin (

..!.~ii.)

Wei Bocheng ( 4'itf-A )

Department of Mathematics, Southeast University, Nanjing 210096, China

Abstract This paper presents a unified diagnostic method for exponential nonlinear models with random effects based upon the joint likelihood given by Robinson in 1991. The authors show that the case deletion model is equivalent to mean shift outlier model. Fromthis point of view,several diagnostic measures, such as Cook distance, scorestatistics are derived. The local influence measure of Cook is also presented. Numerical example illustrates that our method is available. Key words Cook distance; exponential nonlinear models; fixed effects; local influence; random effects 2000 MR Subject Classification

1

62F25

Introduction

Mixed models are frequently used in determining sampling designs and quality-control procedures, in evaluating the performance of products, in interlaboratory studies and in statistical genetics. In recent years, there is an enormous literature on linear mixed models and their influence analysis, such as [1-8]. However, the published works on nonlinear mixed models are not too much and scattered, see [9-10]' especially we have not seen the work on the diagnostics in exponential nonlinear random effects models. In this paper, we present a unified diagnostic method for exponential nonlinear models with random effects, based upon the joint likelihood given by Robinsonltl, and several diagnostic measures are derived. Section 2 reviews some properties for exponential nonlinear models with random effects. Section 3 discusses influence analysis related to these models. We investigate case deletion model (CDM), mean shift outlier model (MSOM) and their properties. Our results show that case deletion model is equivalent to mean shift outlier model for diagnostics purpose. Then we derive several diagnostic measures used in practice. Section 4 covers local influence analysis on exponential nonlinear models with random effects. We extend the results of [5,7,8] to exponential nonlinear random effects models. An example is given in Section 5 to illustrate all the related results. lReceived March 20, 2000. Project supported by NSFC (19631040), NSFJ and the grand ofYZU

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Exponential Nonlinear Models with Random Effects

We assume that Y is an n x 1 observed vector of Yi, U is a q x 1 unobservable vector (random effect) with u ,..., N(O, (12~), in general ~ is a known matrix, and the conditional probability density function of Y with respect to u can be written as PI (y, 0, ¢>Iu) = exp{ ¢>[OTY - b(O) - c(y, ¢>)]},

(2.1)

o= f(X, (3) + Zu,

(2.2)

where 0 is the natural parameter defined in a natural parameter space e c H"; the dispersion parameter ¢> defined in an open subset = (1-2 which is usually treated as nuisance parameter, f(X,(3) = f((3), b(·), c(·,·) are some specified function, X, Z are x P and x q design matrixes with and as the i-th row respectively and Z is a matrix with elements of 0 or 1; (3 is a P x 1 unknown parameter vector having fixed value (fixed effect) defined in B. We also assume YIlu, Y21u, "', Ynlu are conditional independent, then b(O) = L.i b(Oi), c(y, ¢» = L.i C(Yi, ¢», and we call the above model as exponential family nonlinear mixed models. If u = 0, then the above model is just the one discussed by Wei[12]; if 0 = X(3 + ZU,the above model is the one discussed in [11], by the above assumptions, the probability density function of random vector u can be written as

n

n

xi

zT

(2.3) From the discussion of [12], we have

E(ylu) = p, = b(O), Var(ylu) = ¢>-IV = rIb(O).

(2.4)

From the above assumptions, the log joint likelihood function can be written as

l(y,u;(3)

= ¢>[OT Y -

lIT

q

b(O) - c(y,¢»] - "2log(27f¢>-I) - "2logl~l- "2¢>u ~-IU.

(2.5)

From ley, u; (3) = 1((3) + l(uly), where 1((3) is the marginal log likelihood with respect to Y, and l(uly) the conditional log-likelihood of u with regard to Y, then we have

1((3)

= log

I

PI(y,O,¢>lu)P2(U)du,

l(uly) = log {PI (y, 0, ¢>lu)P2(u)/

I

PI (y, 0, ¢>lu)p2(u)du} .

By the method of Laplace approximation for the marginal likelihood (see also [11,13]), (2.6)

where C = (q/2) log 21l' - (1/2) log ID*I, D* is the matrix whose ij-th element is -a21(y, u;(3)/ aUiaujlu=u, and ii is the solution to equation Ol(y,u;(3)/au = 0, By the discussion of [11], we have a log ID*I/a(3 = op(l). Denote I (y, ii; (3) as lp ((3), which is so-called profile likelihood function. We will obtain the estimation of (3 based upon Ip((3) (see also [4,11,13]), then we can use the following equation to obtain the estimation Ol(y, u; (3) = 0 Ol(y,u;(3) = 0; (2.7) a(3 . au

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We assume that the similar regular conditions of [4,11,22] are satisfied for our model, then there is a unique maximum point for lp((3) in the compact set B, then should satisfy ip(S) = 0. Denote u = u(g), which is the predictive value of u by the definition of [4], we also call u as a estimate of u, and by a little calculation, we can obtain

g

g

lp((3) = l(y, Uj (3) -T-

= ¢[O Y - b(O) - c(y, ¢)]

q I I I T I Iog(21l"¢- ) - 2 log I~I- 2¢u ~- U, +2

(2.8)

where 0 = f((3) + Zu, u = u((3). Lemma 2.1 Under the notation and assumptions stated above, the Score function and the observed information matrix for model (2.1)-(2.2) can be

i p((3 ) = 8l~~) = ¢DT e,

-l~((3) = - ~~a~j = ¢DTn- 1 D -

(2.9)

¢[eT][W],

(2.10)

where D = 8f((3)/8(3T,e = y - JL(O), 0 = f((3) + Zu, u = u((3) is the solution to the equation 8l(y, u; (3)/8u = 0, n = V-I + Z~ZT, W = 8 2 f((3)/8(38(3T is the n xp xp array, [.][.] is defined in [12]. Proof Differentiating (2.8) to (3, we can get

i p

((3) = 8l p((3) = 8l(y, u; (3) = 8l(y, u; (3) 8(3 8(3 8(3

From u satisfy 8l(y, u; (3)/8u

+

8l(y, Uj (3) 8u = A.D T( _ ) BuT 8(3 'I' Y JL.

= 0, we can obtain ZT(y -

JL)

= ~-Iu, then (2.11)

Differentiating the above formula to (3, we get

(2.12) then

By

which results in (2.10).

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3 Influence Analysis on Exponential Nonlinear Random Effects Models To consider the effect of dropping the i-th case from the data set, it requires the use of simple, inexpensive updating formulas. In this Section, we provide such formulas, while these formulas are remarkably similar to the updating formulas used in standard linear regression models[ 141. Now we first focus on introducing the computing method to get the estimate Taking a first order approximation to the equation (2.6), we obtain

S.

Therefore from (2.9) and (2.10), the Gauss-Newton iteration procedure can be expressed as

This iteration equation can be written as

{3i+ 1 = {(D T 12- 1D)-IDT 12- 1R }i, where R = D{3 + T, T rv N(O, 12). 3.1 On diagnostical models A fundamental approach of influence diagnostics is based on the comparison of parameter estimates S, a- 2 with parameter estimates S(i)' a-fi) that correspond to so-called case deletion model (CDM) with i-th case deleted: (3.1)

where B(i), f(i) ({3), Z(i) and E:(i) are matrices with the i-th case deleted from Y, f({3), Z, E:, respectively. It is convenient to write partitioned matrices in which the i-th case is isolated. Let 12(i) be matrix 12 with the i-th row and column deleted. Then the matrix 12 may be written as (here i = 1)

vT )

12 = (Vii Vi 12(i)

Lemma 3.1 Under the notation stated above, the estimate S(i) in the case deletion model (3.1) can be expressed as

• (3(i)

• (D T 12- 1D)-1 D T 12- 1<5i €i ~ (3 1- h ii

where <5i is an n x 1 vector with 1 at i-th position and zeros elsewhere, €i () = f(S) + Zit, it = ft(S), and bu is the diagonal element of H, where

H = I - 12-1

(3.2)

'

= Yi -

fj,i, fj,i

= b((}i),

+ 12-1 D(D T 12- 1D)-1 D T 12- 1.

Proof From iii) (Si) = O. The first order approximation of score function at

S gives

No.3

Zhong et al. INFLUENCE ANALYSIS ON EXPONENTIAL NONLINEAR MODELS

(3'(i)

~

(3'

T

T

n- D n-l' + {D (i)H(i) (i) )-ID (i)H(i) e(i)} (3' 1

It is easily shown that the inverse of the partitioned matrix

n-1 -_ where s,

= Vii -

(

1/ s, -(1/ Si)n~)Vi n~)

301 (3.3)

n is

-(1/ Si)Vrn~)

)

+ n~)ViVrn~) / s,

,

Vrn~)Vi' Using the results

Tn

1

D(i)H~)

D(i)

= DTnH- ID -

d-di7'/ i i Si,

DT n-l' D T Hn-1,e - d-iei - / Si, (i)H(i) e(i) =

where

where d; is the i-th row of D (see also [5]), then we have

where

h- t.t..

= dY!'(D Tn- 1D)-I,]. = s~o'Tn-l D(DTn- 1D)-1 D Tn- 1o· t

1.

t

t

t"

Substituting the above expressions into (3.3) and by a little calculation result in (3.2). The case deletion model is the basis for constructing effective diagnostic statistics, and it is the most important one in practice because it is very straightforward and easy to compute. Another commonly used diagnostic model is so-called mean-shift outlier model (MSOM, [14], p.20). MSOM can be represented as

() = f((3) + Zu + on,

(3.4)

where 'Y is an extra parameter to indicate the presence of outlier. It is easily seen that the nonzero value of'Y implies that the i-th case may be an outlier because the case (Xi, Zi,Yi) no longer comes from the original model. This model is usually easier to formulate than case deletion model. To detect outliers, one may either estimate the parameter 'Y or make a testing of hypotheses H o : 'Y = 0, using MSOM (we shall discuss this later). The estimators of (3, 'Y and a 2 in (3.4) are denoted by Smi':Ymi and a~i' respectively. It is well known that in linear regression models, CDM and MSOM are equivalent in following sense: the least square estimates (LSE) of parameters are equal for CDM and MSOM[14]. It was Storer and Crowley[15] who conjectured that Smi = S(i) may hold in broader class of regression models. [16] have solved this problem for many commonly encountered models, but not including mixed models. Now we shall show that Smi = S(i) holds for model (2.1). Theorem 3.1. Under notation and definitions stated above, we have , , , ( DTn- 1D)-1 D Tn- lo iei (3(mi) = (3(i) ~ (3 1_ h (3.5) ii ' which means that CDM and MSOM are equivalent. Proof It follows from (2.2) that the joint log-likelihood of CDM and MSOM are respectively

l(i)(y,u;(3) = C - I>p[()j(Yj - b(()j) - c(Yj,¢)] j#.i

~¢uT:E-Iu,

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l(mi)(y, u; (3) = C - I>/>[Oj(Yj - b(Oj) - C(Yj, ¢)] - ¢[OiYi - b(Oi) - C(Yi, ¢)]) Hi

1 T -1 -2a 2U ~ u,

Oi

= Ji((3) +ZiT U+'Y.

Then S(mi), U(mi) and 'Y(mi) satisfy

m(mi) (y, Uj (3) 8(3

=0

,

m(mi) (y, Uj (3) 8u

= 0,

m(mi) (y, u; (3) 8'Y

----'.----'c;:-_--'-

= O.

From m(mi)(Y' Uj (3)/8'Y = 0, we get

Yi - b(fi((3) +

zT U + 'Y) = 0:

Substituting the above formula into m(mi)(y,uj(3)/8(3, we can obtain

i .( (m.)

'(3) - m(mi)(y,Uj(3) -. 8(3

Y, U,

+ ddYi _ b(fi((3) + zT U + 'Y)]

= m(i) ~~ Uj (3) =

i(i) (y,

u; (3),

which means Semi) = S(i)' and this means that the estimators of parameters in CDM and MSOM satisfy the same equation. Combining Lemma 3.1, we get the desired result. 3.2 Influence diagnostics Once we get the joint log-likelihood and its corresponding estimates, many diagnostic measures are immediate consequences, now we list them below. 3.2.1 Score statistics of outlier We can get a score statistic[17] to detect the outlier based on mean-shift outlier model. In fact, for model (3.4), one can make a testing of hypothesis: H o : 'Y

= OJ

HI: 'Y =f O.

If H o is rejected, then the i-th case may be a possible outlier because this case may not come

from the original model. We now derive a score statistic for H o. Theorem 3.2 For MSOM, the score statistic for the testing of hypothesis H o : 'Y

= 0 is

where ei and h-u are given in Lemma 3.1, Ti = ei/(av1 - h ii), Proof The profile likelihood of L(mi) (y, Uj (3) is lp((3, 'Y), The Score statistics for H o can be (see [17]) sc: = {(m p((3,'Y))TJ"Y"Y(mp((3,'Y))}1 .. • 8'Y 8'Y (13,0"2)' where In is the right corner matrix of J(f/"Y)' J((3,'Y) is as following

lp((3, 'Y)

= c + ¢ f)BiYi i=1

b(Bi) - C(Yi, ¢)] -

~¢uT~-IU,

Zhong et al. INFLUENCE ANALYSIS ON EXPONENTIAL NONLINEAR MODELS

No.3

where B = 1((3) + Zu + c5n, u = ~ZT(y - J-L(B)). By Lemma 2.1, we obtain

8~~) = V(Z~~ + c5 = V( _Z~ZT8~~) + c5 i)

Jfj,fj

303

= ¢>DTn-1D.

i),

then

then dIp((3, "() d"(

= c5,Tn- 1V- 1[y -

J-L(f((3) + Zu + c5n)]

+¢>c5[n- 1 Z~~-l~ZT[y - J-L(f((3) + Zu + c5n)] =

_d

2I

¢>c5[n- 1(V-l +

Z~ZT)[y - J-L(f((3)

+ Zu + c5n)]

= ¢>[Yi - J-Li (fi ((3) + z[ u + "()], p ((3 ,"() = ~c5Tn-lc5. d"(2 'I' , ,.

On the other hand

_8

2I

p((3,"()

8(38"(

~DTn-lc5.

=

'I'

"

then

DTn - 1D DTn-lc5i) J(fj,,)

=

J" = ¢>{c5[n- 1c5i

¢> ( c5[n-1 D c5[n-1c5 i -

'

c5[n-1 D(DTn- 1D)-l D Tn- 1c5d,

a little calculation can result in our desire results. 3.2.2 Generalized Cook distance That is the norm of /J - /J(i) with respect to certain weight matrix M > 0, i.e.

It is very natural to choose M = J((3) =

a- 2 (D Tn- 1 D),

(see (2.10)); then

CD = (/J - /J(i))T (DT~-l D)(/J - /J(i)) . i 0'

After some calculations from Lemma 3.1 we can get

3.2.3

Residuals and studentized residuals

ei = Yi -

A

•

A

J-L((}i) = Yi - b((}i)'

ri =

ei

a~. 1 - h ii

The statistics Ti is called the studentized residuals. Theorem 3.2 shows that the score statistic SCi is just the square of studentized residual that is an adequate diagnostic statistic as often used in linear regression diagnostics.

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3.2.4 Leverage That is the diagnostic elements of the hat matrix H (see [14)). It is well-known that 8fJi/8Yi is referred to as the leverage, which is the instantaneous rate of change in the i-th predicted value with respect to the response value[18]. By the discussion in Section 2, we can get

y = p,(O),

where 0 = f(fi) + Zu, u = u(fi). Theorem 3.3 Under the assumptions stated above, we can get

::T ~ I - V- 1+ HV- 1.

Proof

then

By [24]

then ::T

~ n- D (D Tn1

1

D)-lD T n- 1 v- 1

+ n- 1 Zl:ZT

= n- 1 D(DT n- 1 D)-l D T n- 1 v - 1 + n- 1(n _ V- 1 ) = 1- V- 1

+ HV- 1 •

Theorem 3.3 show that 8fJi/8Yi ~ l-li- 1 (Oi)(I-h ii ) ; if b(O) = (1/2)0 2 , then 8fJi/8Yi = hii. [5] take hii / s, (see the proof of Lemma 3.1) as generalized leverage for linear models with random effects, but we think that hii as leverage is more reasonable. The two leverage have the relation -1 1- h ii = Si (1- hii/S i).

4

Local Influence Measure

The local influence approach was presented by [19] and developed further by several other authors (see [20,21)). In this section we first review the basic idea and formulas of local influence approach, and then apply them to exponential nonlinear models with random effects. To study the sensitivity of uncertainties in the data or model, one can proceed by specifying a perturbation scheme through an n x 1 vector W with components Wi attaching to case i. Here

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w is admitted to vary over a neighborhood of wo, a null point at which the perturbed loglikelihood satisfies l(O,wo) = l(O). Assuming that given w, l(O,w) is maximized at O(w), while l({}) is maximized at 0, then O(wo) = O. From the arguments of [19), the likelihood displacement surface is given in the form as

a(w)

=(

w ), LD(w)

(4.1)

with LD(w) = 2{l(O) - l(O(w))} as the solution locus. The normal curvature of a(w) at Wo along the line w = Wo + -rd takes the form (4.2) where 6. = 8l (0, w)j 8{}8w T, evalued at 0 and wo, d is an n x 1 unit vector, and r is a scale parameter (see also Wei [12], p124). Cook[16] showed that the maximum curvature direction can provide important diagnostic information. Moreover, as pointed out by [20), Fii is also an appropriate diagnostic measure. Now we consider some common perturbation schemes to our models. The key point is to find F. 4.1 Mean Shift Perturbation One common way to describe the uncertainties in the mean is to perturb y to y + w, which is identical with perturbing the vector of the observed responses in normal case. Assuming for simplicity that ¢ is known or replaced by ¢, from (2.6) the relevant part of the perturbed log-likelihood is

l(y, u; (3, w) = ¢[OT (y + w) - b(O) - c(y + w, ¢))

+ ~ 10g(2?T¢-1)

T L:- 1 u, -"21 log IL:l- I"2¢u where Wo = 0 yields the non-perturbed log-likelihood. By the similar discussion in Section 2, we can get the corresponding profile likelihood as following

lp((3, w)

= ¢[OT(y + w) - "211og 1"'1 LJ

where

-

b(O) - c(y + w, ¢))

+ ~ log(2?T¢-1)

1 ,j, -T L:-1-u, "2'1'U

0 = 1((3) + Zii, u = L:ZT(y + w - f-t(O)),

then

because

So we get

:::T

= n- 1 ZL:ZT.

Substituting the above formulae into 8 2l p((3,w)j8(38w T, we can get

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Then In the direction of the i-th unit vector of Rn, the curvature of the influence graph of w derived 2from (4.2) is given by -Fii = ¢;b-2 ((h )si h ii . 4.2 Case Weights Perturbation The case-weights perturbation is often the basis for the study of local influence; deleting a case is identical with attaching a zero weight to that case. Let w denote an n x 1 vector of case weights, then from (2.2) the perturbed log-likelihood can be denoted as ••

...

A

n

l(y,uj(3,w) = LWiPl(Yi,{1,¢;jU) +logp2(u) i=l

where Wo = (1,1"", l)T yields the non-perturbed log-likelihood. By a similar discussion to (2.6) of Section 2, we can get

~ -T q lIlT lp((3, w) = L.J wilB Y - p(B) - c(y, ¢;)] + "2 log(27f¢;- ) - "2 log I~I - "2¢;u ~-lU, i=l

where e = f((3) + ZU, u = ~ZTW(y - pee)), then

lp(:~W) = ¢;DTW(y -

pCB)),

because

then

2l 8 p((3,w) I . = A.D Tn- lv- 1 8· 8(38Wi wo/3 '/" tet· = ¢Dv-ln- l D(DTn- l D)-l DTn-lv- l D,

8e I . - V-lei _ 8Wi wo,/3 n

-F

n- 1 V - l )8·tet·

A

A

•

•

where D = diag(el,'" ,en). In the direction of the i-th unit vector of Rn, the curvature of the influence graph of derived from (4.2) is given by .. A -2"-2 A A2-Fii = ¢;Si b (Bi)ei b-u-

w

We can also use the maximum curvature direction as a diagnostic measure (see example given in the next section ). 4.3 Perturbation of Covariate in Random Effects Consider perturbing the data for the k-th explanatory variable of Z, by modifying the data matrix Z as Zw = Z + wd[, where d k is a q-vector with 1 at k-th position and zeros elsewhere. In this situation, the perturbed joint log-likelihood can be written as

ley, u; j3, w) = ¢;[B T Y - b(B) - c(y, ¢;)] + where B = f(j3) + Zwu, Wo

lp((3,w)

~ log(27f¢;-1) - ~ log I~I - ~¢;uT~-lu,

= 0 yields the non-perturbed joint log-likelihood.

= ¢;(B-T y -

Then we have

q 1 lIT b(B) - c(y,¢;)) + "2log(27f¢;- ) - "2logl~l- "2¢;u ~-lU,

Zhong et al. INFLUENCE ANALYSIS ON EXPONENTIAL NONLINEAR MODELS

No.3

where 0 = f((3)

+ ZwfJ"

307

U = ~ZJ(y - f.J,(0)). Then

Olp((3, w) = D T ( _ (0)) 8(3 Y f.J, , In the direction of the i-th unit vector of H", the curvature of the influence graph of w derived from (4.2) is given by

-Fi i = ¢u%si2hii + ¢e;dr~zTn-l D(DTn- 1D)-l DTn- 1 Z~dk

+21'I-'8,T n- 1 D(DTn-l D)-l DTn- 1 zx« k u k e',.

5

Example

In this section, we use the Plasma Concentrations ofIndomethacin data ([9], p18, Table 2.1) for diagnostics analysis. The data is taken from a study of the pharmacokinetics of indomethacin following blous intravenous injection of the same dose in six human volunteers. For each subject, the pharmacokinetics of indomethacin were measured at 11 time points ranging from 15 minutes to 8 hours post-injection. [9] used nonlinear repeated model to analyze the data, we reanalyze the data by using nonlinear mixed model (2.1). Now we use the following model Yij

= f(Xij, (3)

+ Ui + Cij,

i = 1, ... ,6, j = 1, ... , 11,

where f(x, (3) = ef31 exp( -e f32x) + ef33 exp( -e f34x). [9] showed that case 23 is an outlier. Our results are shown in Figures 1 to 3 by index plots. Figure 1 is the index plot of Cook distance, which shows that case 23 is most influential, and case 1 also has a relative large influence. The plot of score statistics is given in Figure 2, and the plot of the maximum curvature direction of case-weighted perturbation is shown in Figure 3. They state the same results. The residual of ei and the stundentized residual r, have also the same diagnostic results, we omit here. 30

4 3.5

25

3 20

2.5

U 15

c 2

u

00

1.5

10

0.5 0

\ 0

J\

10

20

JIA

30

40

case Fig.1

5

/\r-..

50

1\

60

70

0

0

10

20

case Fig.2

60

70

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1.0,----------------, 0.9 0.8 0.7 0.6 <:>0

0.5 0.4

0.3 0.2 0.1

o

o

1\

10

20

30

40 case Fig.3

50

.60

70

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