Nonlinear regression models arise when definite information is available about the form of the relationship between the response and predictor variables. Such.
Communications in Statistics—Theory and Methods, 38: 138–155, 2009 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0926 print/1532-415X online DOI: 10.1080/03610920802074836
A Robust Procedure in Nonlinear Models for Repeated Measurements ANTONIO SANHUEZA1 , PRANAB K. SEN2 , AND VÍCTOR LEIVA3 1
Department of Mathematics and Statistics, Universidad de La Frontera, Temuco, Chile 2 Department of Biostatistics, University of North Carolina, North Carolina, USA 3 Department of Statistics, CIMFAV, Universidad de Valparaíso, Chile Nonlinear regression models arise when definite information is available about the form of the relationship between the response and predictor variables. Such information might involve direct knowledge of the actual form of the true model or might be represented by a set of differential equations that the model must satisfy. We develop M-procedures for estimating parameters and testing hypotheses of interest about these parameters in nonlinear regression models for repeated measurement data. Under regularity conditions, the asymptotic properties of the M-procedures are presented, including the uniform linearity, normality and consistency. The computation of the M-estimators of the model parameters is performed with iterative procedures, similar to Newton–Raphson and Fisher’s scoring methods. The methodology is illustrated by using a multivariate logistic regression model with real data, along with a simulation study. Keywords Consistency; M-estimators; M-tests; Normality; Uniform asymptotic linearity; Wald-type tests. Mathematics Subject Classification Primary 62J02; Secondary 62G35.
1. Introduction Repeated measurements occur frequently in many scientific fields where statistical models are employed. For instance, in agriculture, agriculture-crops yield in different fields over different years; in biology, growth curves; in education, student Received May 23, 2007; Accepted March 24, 2008 Address correspondence to Antonio Sanhueza, Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54-D, Temuco, Chile; E-mail: asanhueza@ ufro.cl
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progress under various learning conditions; in medicine, successive periods of illness and recovery under different treatment regimens; and so on. The primary characteristic of repeated measurements that distinguishes them is the fact that more than one observation on the same response variable is available on each observational unit. The analysis of nonlinear models for repeated measurements generally consists of estimating the unknown parameters and testing hypothesis about these parameters. In many cases, estimators of the parameters in univariate and multivariate nonlinear models are based on classical methods of estimation, such as maximum likelihood (ML) and least squares (LS) procedures. However, estimators and test statistics based on these methods are usually non robust to outliers or departures from the specified distribution of the error term in the model. Thus, from considerations of robustness (again outlier or for the error contamination in a nonlinear model), the classical procedures are not very desirable and robust estimation procedures, such as M-procedures (ML type), are better in this respect. In the univariate linear model, a variety of robust methods based on M-, L-, and R- estimators have been proposed; see Jurecková and Sen (1996). The M-estimators include ML and LS methods as special cases and also allow for the construction of estimators and tests that are robust against departures from normality. For the multivariate linear model, robust M-estimators have been proposed for estimating its unknown parameters and for testing hypotheses about these parameters. Maronna (1976) proposed a simultaneous M-estimation procedure for the multivariate location and scatter parameters, under the assumption of elliptical symmetry of the underlying distribution. Singer and Sen (1985) developed two types of M-estimators for multivariate linear models, coordinatewise M-estimators and an extension of the Maronna-type M-estimator. The objective of this study is to consider M-procedures for estimating and testing the parameters of the nonlinear regression model for repeated measurements. The proposed M-procedures are formulated along the lines of generalized least square procedures. It may be seen as an extension of the M-procedures that was studied in univariate nonlinear models (see Sanhueza and Sen, 2001, 2004), and also as a generalization of the robust methods for the multivariate linear model (see Maronna, 1976). Under some regularity conditions, the asymptotic properties of the M-procedures are presented, such as uniform asymptotic linearity, normality and consistency. The computation of the M-estimators of the model parameters is performed by using iterative procedures, similar to Newton–Raphson and Fisher’s scoring methods. The methodology is illustrated by means of an analysis of repeated measurement data from a study of the combined effects of hepatotoxins in which between and within subject measurements are collected. Also, a simulation study is carried out assuming a multivariate nonlinear model. In Sec. 2, we define the M-estimators of the parameters of the general multivariate nonlinear model. In Sec. 3, we develop the asymptotic properties of the M-estimators. In Sec. 4, we consider hypothesis testing of interest on the parameters of the considered model by using robust M-tests. Also, we present an iterative computational method for computing the M-estimators. Finally, Sec. 5 illustrates a numerical application of the developed M-procedures by using a multivariate logistic regression model with real data. Furthermore, a small simulation study is conducted in order to examine the performance of the developed M-estimators.
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2. Definition of the Robust Estimators We consider the nonlinear regression model for repeated measurements: Yi = fi + ei i = 1 n
(1)
where Yi = Yi1 Yip is a p × 1 vector of repeated measurements; fi = fxi1 fxip is a p × 1 vector of nonlinear functions of of specified form; xij = xij1 xijm is a m × 1 vector of known regression constants (may be time-dependent covariates); = 1 s is a s × 1 vector of unknown parameters; and ei = ei1 eip is the error vector, which is assumed to be independent and identically distributed with cumulative distribution function (cdf) G, defined on p , with mean vector zero and covariance matrix . The model incorporates both between and within unit measurements, where the within designed unit is the same for all n units. In the following, we use the notation y2A = y A y, where A is a positive definite (p.d.) matrix. Definition 2.1. The M-estimator of given in Eq. (1) is defined as: ˆ n = Arg min
n
hui 2−1
∈⊆ s
(2)
i=1
where hui = hui1 huip with h· being a real valued function, uij = Yij − fxij , for j = 1 p and = Varhui . In particular, if we let hz = z in Eq. (2), we have the LS method for estimating . In the conventional setup of robust methods (see Hampel et al., 1986; Huber, 1981, and Jurecková and Sen, 1996), we primarily use bounded and monotone functions h·. In this respect, the so-called Huber-score function corresponds to: 1 √ z 2 hz = 1 k2 2 kz − 2
if z ≤ k
(3) if z > k
for suitable chosen k, with 0 < k < . Remark 2.1. The minimization in Eq. (2) is equivalent to the robust estimating equations given by: n
Yi ˆ n = 0
i=1
where Yi = Xi −1 ui
Xi = fi = f˙ xi1 f˙ xip
(4)
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f˙ xij = fxij fxij fxij = s 1
ui = ui1 uip uij = 2 huij h uij and h u =
d hu du
Here · is called the score function. We may rewrite Eq. (4) by using matrix notation as: ∗ ˆ n = X∗ ˆ n ∗ ∗ ˆ n = 0
(5)
where
X∗ = X1 Xn np×s ∗ = diag np×np and
∗ = u1 un np×1 In order to study the asymptotic properties of the M-estimator defined in Eq. (2), we make the following sets of regularity assumptions concerning: [A] the cdf G, [B] the score function ·, and [C] the function f· defined in Eq. (1). [A] The cdf G is absolutely continuous with an absolutely continuous density function g· having a finite Fisher information, that is:
+ −
g v gv
2 dGv < where g v =
d gv dv
[B1] The score function · is non constant, absolutely continuous, and differentiable with respect to j , for j = 1 s. [B2] u = 0 and u2 < , u < and u2 < , with d u = du u. [B3] lim→0 sup≤ u + − u = 0, where is defined in Eq. (2) and lim→0 sup≤ u + − u = 0. [C1] The function fx is continuous and twice differentiable with respect to ∈ , where is a compact subset of s .
[C2] lim→0 sup≤ fx + fx + − fx fx = 0 and j k j k
2
2 lim→0 sup ≤ fx + − fx = 0, for j k = 1 s, where j k
j k
is given in Eq. (2).
3. Asymptotic Theory for M-Procedures Here we present the uniform asymptotic linearity of the √ M-estimator defined in Eq. (4). Also, the existence of a solution of Eq. (4) that is a n-consistent estimator
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of is proven, which admits an asymptotic representation. Finally, we present the asymptotic normality of the developed M-estimators. The proofs of the following theorems are presented in Appendix. Theorem 3.1. Under the regularity conditions [A], [B1]–[B3], and [C1]–[C2] given in Sec. 2, we have: n 1 t 1 sup √ Yi + √ − Yi + n t = op 1 as n → n n i=1 n t≤C (6) where n = fv dv.
n
−1 i=1 Xi
W Xi , with W = diag and =
v
Theorem 3.2. Under the regularity conditions [A], [B1]–[B3], and [C1]–[C2] given in Sec. 2, we have: n 1 K t 1 √ + − Y Y + + t √ √ i i n = op 1 n n i=1 n n t≤C vecK≤D (7) sup
as n → , where K is a positive definite matrix and n is as defined in Eq. (6). Theorem 3.3. Under the regularity conditions [A], [B1]–[B3] and [C1]–[C2] given in Sec. 2, there exists a sequence ˆ n of solutions of Eq. (4) such that: √ nˆ n − = Op 1 as n → where ˆ n = +
1 n −1 n n
n i=1
(8)
1
Yi + op n− 2 .
Theorem 3.4. Under the regularity conditions [A], [B1]–[B3], and [C1]–[C2] given in Sec. 2 and ch1 i ni=1 i −1 −→ 0 as n −→ , we have: n 1 Yi −→ Ns 0 √ n i=1
(9)
where = limn→ n1 ni=1 i and = Varui = kl are p.d. matrices, with i = Xi −1 −1 Xi , for k l = 1 p, and kl = uik uil . Corollary 3.1. Under the regularity conditions [A], [B1]–[B3], and [C1]–[C2] given in Sec. 2, we have:
√ n ˆ n − −→ Ns 0 −1 −1 where = limn→ n1 n is a p.d. matrix, with n = W = diag , with = v fv dv.
n
−1 i=1 Xi
(10) W Xi , and
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Corollary 3.2. Under the regularity conditions [A], [B1]–[B3], and [C1]–[C2] given in Sec. 2, we have: − 21
√ n ˆ n − −→ Ns 0 Is
(11)
where = n
n −1 n n −1 Xi ˆ n , with n = ni=1 Xi ˆ n −1 W n = ni=1 Xi ˆ n −1 = n1 ni=1 −1 Xi ˆ n , = diagˆ1n ˆ np , = 1 n ui ˆ n ui ˆ n and hui ˆ n h ui ˆ n , W i=1 n 1 n ˆ nj = n i=1 uij ˆ n . Corollary 3.3. Under the regularity conditions [A], [B1]–[B3], and [C1]–[C2] given in Sec. 2, we have:
−1
ˆ n − −→ 2 s n ˆ n −
4. Testing Hypotheses and Computational Algorithm In order to construct hypotheses (linear and nonlinear) for the parameters of interest, we develop robust M-tests and find their asymptotic distributions. We also present an iterative algorithm for computing the M-estimators of the parameters of the considered model. 4.1. Test of Linear Hypotheses We may write = 1 2 , where 1 is a r × 1 vector and 2 is a s − r × 1 vector and also 12 11
−1 =
21
22
Thus, by using the Cochran’s theorem, we have:
11
ˆ 1n − 1 −→ 2 r n ˆ 1n − 1
where ˆ 1n is the M-estimator of 1 . We consider the linear hypothesis H0 1 = 0 again H1 1 = 0. For testing H0 11 ˆ 1n , which under H0 follows we can define the Wald-type M-test as W = n ˆ 1n
2 (asymptotically) the r distribution. 4.2. Tests of Nonlinear Hypotheses We consider the nonlinear hypotheses H0 a = 0 again H1 a = 0, where a· is a real-valued (nonlinear) function of . By using Corollary 3.3 and the delta method, we have that: a˙ ˆ n − 2 ˙a ˆ n
1
√ n aˆ n − a −→ N 0 1
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where a˙ =
a.
Also, by using the Cochran’s theorem, we obtain:
a˙ ˆ n −1 aˆ n − a −→ 2 1 n aˆ n − a a˙ ˆ n
Then, for testing the null hypothesis we can use the Wald-type M-test given in this case by:
a˙ ˆ n −1 aˆ n W = n aˆ n a˙ ˆ n
which under H0 follows asymptotically the 2 1 distribution. Alternatively, we may consider the q-dimensional nonlinear hypotheses H0 d = 0 again H1 d = 0 where d· is a vector-valued function, i.e., d s −→ q , such that, D = d/ exists. − 21 √nˆ n − in Corollary 3.2, we may From the asymptotic normality of
prove that:
− 21 √
n dˆ n − d −→ Nq 0 Iq
D ˆ n Dˆ n
and
D ˆ n −1 dˆ n − d −→ 2 q n dˆ n − d Dˆ n
Thus, for testing the nonlinear hypothesis, we may use the Wald-type M-test given in this case by: D ˆ n −1 dˆ n W = n d ˆ n Dˆ n
which under H0 has approximately the 2 q distribution. We also may define a modified Wald-type M-test given by: F=
N − s W q SSEˆ n
where N = np is the total number of observations and SSEˆ n =
n
h ui ˆ n −1 hui ˆ n
i=1
which under H0 has approximately the q N − s distribution. We could also define a likelihood ratio-type M-test based on:
− SSEˆ nfull N − s SSEˆ nrest Q= q SSEˆ full n
where ˆ nrest is a restricted estimator obtained by minimizing Eq. (2) subject to the constraint d = 0, and ˆ nfull is the unrestricted estimator under the full model. Both estimators are computed using the unrestricted variance–covariance estimator , calculated under the full model. Under H0 , with known, we may prove that Q has approximately the q N − s distribution.
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4.3. Computational Algorithm Here, we present an iterative algorithm for computing the parameter M-estimates of the model given in Eq. (1). This algorithm is suggested by the asymptotic result given in Theorem 3.3 and is implemented by the following three steps. Step 1. Estimate by ˆ 0 , set l = 0. A natural choice for ˆ 0 would be to use the weighted least square (WLS) estimator, which is also calculated by using an iterative algorithm. Step 2. Estimate the covariance matrix . A natural approach for estimating would be to use the following expression l = n1 h∗ ˆ l h∗ ˆ l , where:
hu11 · · · hu1p h∗ = hu1 hun = hu1n · · · hunp n×p Step 3. Using the estimate in Step 2, re-estimate by solving X∗ ∗l ∗ ∗ ∗l l = 0, where X and are given in Eq. (5), and = diag l = In×n ⊗ l . This could be done by using the result in Theorem 3.3, which produces the following equation as a standard Newton–Raphson iteration: ∗
−1 ∗ l ∗l ∗ l ˆ ˆ nl+1 = ˆ nl + n ˆ nl X ˆ l −1 l with U∗l = diag W where n ˆ nl = X∗ ˆ l U∗l X∗ ˆ l ,
1
l −1 l l −1 l l W = In×n ⊗ W W = diag ˆ ˆ = ˆ 1 ˆ p = n 1n×1 ⊗ Ip×p vec and
ˆ ˆ · · · u1p u11 = ˆ ˆ u1n · · · unp
(12)
Here, 1n×1 is the vector of ones. Thus, we consider this estimate as a new preliminary estimate ˆ l and then go to Step 2. This procedure should be iterated until it converges. If the sample size is not small, often one or two iterations work out well.
5. Simulation and Illustration We illustrate the computational procedure of the methods proposed in Sec. 4 for repeated measurement data. Our intention is not to present an exhaustive analysis of the data, but simply to illustrate an application of the developed M-procedures. We consider a multivariate logistic regression model, which has proven to be useful for analyzing a sigmoid dose-response relationship as discussed by Volund (1978). Also, we generate bootstrap samples from the data and compute the standard errors (SE) of the estimates. By doing so, we compare the accuracy of the developed M-estimators with respect to bootstrap estimators. Finally, we present the results of
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a small simulation study conducted to examine the performance of the developed M-estimators and compare it to the WLS estimators for different sample sizes. 5.1. Numerical Example The considered data for our example are taken from an in vitro toxicity study of isolated hepatocyte suspensions presented in Genning et al. (1989). The effect of the combination of carbon tetrachloride (CC14) and chloroform (CHC13) on cell toxicity, which is measured by the lactic dehydrogenase (LDH) enzyme leakage, was studied using a 4 × 4 factorial design with four independent replications. The dose levels of CC14 and CHC13 were 0, 1.0, 2.5, and 5.0 mM and 0, 5, 10, and 25 mM, respectively. The 16 different treatment combinations were randomly assigned to flasks within each replication and studied over time. The percent LDH leakage from the cells in each flask was measured at 0.01, 0.25, 0.5, 1, 2, 3, 4, 5, 6, and 8 hours post-treatment. The primary hypothesis of this analysis is that CHC13 antagonizes the toxicity of CC14 at early time points post-treatment. To test this hypothesis, the data corresponding to measurements taken no later than three hours post-treatment were considered in the subsequent analysis. We fit the data by using the multivariate logistic regression model given by: Yij =
1 + eij i = 1 64 j = 1 6 1 + exp−xij
(13)
where Yij is the percent LDH leakage at the jth time point for the ith flask of cells; xij = xij1 xij8 is the vector of design variables, where xij1 = 1, xij2 = CC14i is the dose of CC14 for the ith flask of cell, xij3 = CHC13i is the dose of CHC13 for the ith flask, xij4 = tij is the jth time point for the ith flask cell, xij5 = CC14i ∗ CHC13i , xij6 = CC14i ∗ tij , xij7 = CHC13i ∗ tij , and xij8 = CC14i ∗ CHC13i ∗ tij , with “∗” denoting the interaction effect. The vector = 1 8 consists of unknown parameters, where 1 , 2 , 3 , 4 , 5 , 6 , 7 , and 8 are associated with untreated cells, the effects of CC14, CHC13 and time, the interactions between CC14 and CHC13, CC14 and time, CHC13 and time, and CC14, CHC13 and time, respectively. The vector ei = ei1 ei6 consists of errors which are assumed to be i.i.d. r.v.’s with mean vector zero and unknown covariance matrix, which will be estimated. In order to compute the M-estimator of the parameter vector in the model given in Eq. (13), we minimize the expression given in Eq. (1), where the vectorial function hui is given by: hui = h Yi1 −
1 1 − h Y i6 1 + exp−xi1 1 + exp−xi6
where hz is the Huber-score function defined in Eq. (3), such that: 1 if z ≤ k
z if z ≤ k
z = and z = 0 if z > k
k signz if z > k
where the constant k can be obtained in two different ways. We could use the fact 2 ˆ −1 e ≤ k = q, such that k = q p . We may −1 e ∼ 2 p and thus e that e ˆ ch min
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also consider different values of the constant k for each coordinate. In this case, we use a common value for the univariate case, which corresponds to kj = 15 ˆj , with j = 1 6 where ˆj is an estimate of the standard deviation of Yj . The minimization in Eq. (2) is carried out by using the algorithm given in Sec. 4. This algorithm requires calculation of the derivative of fxij =
1 1 + exp−xij
with respect to , which is given by: exp−xij fxij = x f˙ xij = 1 + exp−xij 2 ij Then, the matrix Xi is: Xi =
exp−xi1 2 xi11 1+exp−x i1
fxi =
exp−xi6 2 xi61 1+exp−xi6
··· ···
exp−xi1 2 xi18 1+exp−xi1
exp−xi6 2 x 1+exp−xi6 i68
In Step 1 of the algorithm, we compute the initial of the parameters, Yestimates which are found by transforming Yij to Yij∗ = log 1−Yij . We see that Yij∗ ≈ ij xij , so that starting values for the iterative algorithm could be the univariate LS estimates. Next we estimate the parameters of the model given in Eq. (13) by using the M-procedure and later a bootstrap M-procedure, i.e., considering bootstrap samples from the original sample. 5.1.1. Real Sample. The WLS estimates are given in Table 1. These WLS estimates were computed by using an iterative algorithm similar to the one presented in Sec. 4, but with the function hz = z. The final results for the M-estimator of are presented in Table 2.
Table 1 WLS estimates and their corresponding 95% confidence intervals (L, U) Parameter
Estimate
SE
L
U
1 2 3 4 5 6 7 8
−21920 00975 00094 00914 −00069 00511 00236 −00010
00761 00237 00052 00478 00019 00139 00027 00009
−23406 −00510 −00007 −00024 −00106 00240 00182 −00027
−20424 01440 00195 01852 −00033 00783 00290 −00007
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Sanhueza et al. Table 2 Robust M-estimates and their corresponding 95% confidence intervals (L, U) Parameter
Estimate
SE
L
U
1 2 3 4 5 6 7 8
−21828 00695 00066 00946 −00052 00777 00259 −00022
0.0754 0.0242 0.0052 0.0532 0.0019 0.0154 0.0031 0.0010
−23305 00221 −00036 −00097 −00088 00476 00199 −00042
−20350 01168 00168 01989 −00015 01078 00319 −00003
In general, we note that the estimated SE of the developed M-estimators are slightly bigger than the estimated SE of the WLS estimators. Also, these M-estimators produce more conservative results for testing the parameters in the model. However, we cannot have a general picture of this pattern by just using a specific random sample of observations. In this respect, we need to conduct a simulation study in order to compare the efficiency of the developed M-estimators related to the WLS estimators, which is presented in the next subsection. One of the interests of this analysis is to investigate the relationship between CHC13 and CC14, which is characterized by the combination parameter: = 5 + 8 Following the work by Carter et al. (1988), it can be concluded that when is not different from zero, the interaction between CHC13 and CC14 is additive; when is positive, the interaction is synergistic; and when is negative, the interaction is antagonistic. We can formulate the hypotheses: H0 = 0 again H1 = 0, which can be written as: H0 C = 0 again H1 C = 0, where C = 0 0 0 0 1 0 0 1 , i.e., = C . In order to test the null hypothesis, we compute the Wald-type = C C −1 , and
is the covariance matrix ˆ where
ˆ
M-test as: W = 2 ˆ 1, with of . The numerical value for W is 15.21, which is bigger than 095 p-value = 00001. Thus, we say that the relationship between CHC13 and CC14 . ˆ is not additive. The 95% confidence interval for is given by: ± 196
The numerical value for this interval is −00111 −00037, which indicates that the relationship between CHC13 and CC14 is antagonistic. 5.1.2. Bootstrap Samples. We examine the bootstrap SE of the parameters of the nonlinear model given in Eq. (13) and compare them to the estimated SE of the developed M-estimates. We take B = 500 simple random samples from the data and for each sample compute the parameters estimates by using the procedure described in Sec. 4. These estimates are denoted by ˆ b , for b = 1 500. The bootstrap estimate of the covariance matrix is obtained by the sample covariance matrix 1 B ˆ ¯ ˆ ¯ of the B = 500 replications which is given by SB = B−1 b=1 b − b b − b ˆ jb B where ˆ b = ˆ 1b ˆ 8b , ¯ = ¯ 1 ¯ 8 , and ¯ j = b=1 B , for j = 1 8. The bootstrap estimates and their SE are given in Table 3. We note that the estimated SE of the M-estimates are smaller than the bootstrap SE.
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Table 3 Bootstrap estimates and their corresponding 95% confidence intervals (L, U) Parameter
Estimate
SE
L
U
1 2 3 4 5 6 7 8
−21930 00698 00065 01132 −00052 00806 00261 −00024
0.0755 0.0415 0.0096 0.0710 0.0027 0.0314 0.0056 0.0018
−23409 −00114 −00123 −00260 −00105 00191 00151 −00059
−20451 01510 00253 02524 00001 01421 00371 00011
5.2. Simulation Study We conducted a simulation study designed to investigate the performance of the developed M-estimates under different sample sizes. We compute the 95%, 90%, 80%, and 70% confidence interval coverage rates based on estimated SE of the Mestimates. We compare the estimated SE of these M-estimates to the true SE, which are the squared roots of the diagonal elements of n1 X b−1 Xb. We also compare the estimated SE of the M-estimates with the estimated SE of the weighted least squares estimates. Simulation was carried out assuming the nonlinear model Yi = 1 1 − exp−2 xi + ei i = 1 n j = 1 5 where = 1 2 = 100 05 , xi = 05 1 2 4 8 and ei ∼ N5 0 . The covariance matrix was taken to be 4 8 7 9 8 8 25 21 225 25 49 315 28 = 7 21 9 225 315 81 27 8 25 28 27 100 The simulation study considered different scenarios for the sample size. Specifically, sample sizes for n = 30, n = 50, and n = 100 observations were analyzed. Values of n lesser than 30 did not produce coherent results in the simulation and are not presented here. Tables 4 and 5 present the results of this simulation based on 500 replications per each n scenario. Table 4 shows the 95%, 90%, 80%, and 70% confidence interval coverage rates based on the estimated SE of the developed M-estimates. The estimated SE of the M-estimates performed well in these cases, with coverage rates generally falling within one or two percentage points of the nominal levels. Table 5 shows the means and estimated SE of the M- and WLS estimates based on the complete data and using the procedure presented in Sec. 4. Here, the means are the average of the 500 estimates. The estimated SE are empirical means of the estimated SE obtained
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Sanhueza et al. Table 4 Confidence interval coverage rates using M-estimator Sample size
True confidence
1
2
95% 90% 80% 70% 95% 90% 80% 70% 95% 90% 80% 70%
94.80 90.80 79.80 70.00 93.60 89.20 78.40 69.20 94.60 89.20 81.00 71.00
94.80 89.40 77.40 66.00 95.60 92.00 82.00 72.80 94.00 88.80 80.40 71.60
n = 30
n = 50
n = 100
given in Corollary 3.2. Table 5 also shows the true from the diagonal elements of
SE. Estimates of were unbiased as evidenced by the fact that the mean bias for any component of ˆ was no more than 0.4% of its true value. For the three different sample ˆ sizes, the use of the M-estimator resulted in lower mean bias in the estimated SE of .
Table 5 Means and SE of the M-estimator and WLS estimator Sample size n = 30
Estimator
Statistic
1 = 100
2 = 05
M
Mean True SE Estimate Mean True SE Estimate Mean True SE Estimate Mean True SE Estimate Mean True SE Estimate Mean True SE Estimate
998405 16309 16399 1000010 16309 15387 999055 12633 12704 999634 12633 12132 999929 08933 09052 1000246 08933 08756
0.4998 0.0106 0.0106 0.4998 0.0106 0.0098 0.5002 0.0082 0.0084 0.5003 0.0082 0.0079 0.5001 0.0058 0.0059 0.5003 0.0058 0.0057
WLS
n = 50
M
WLS
n = 100
M
WLS
SE
SE
SE
SE
SE
SE
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In other words, the estimated SE of the developed M-estimates were closer, on average, to the true SE than the corresponding elements of the WLS estimates. For the considered sample sizes, the results of the simulation study suggest that the developed M-estimator performs well and better than the WLS estimator in the analyzed model. In general, the test statistics based on M-estimators are more conservative than those based on WLS estimators.
6. Concluding Remarks The methodology presented in this article allows one to use robust procedures for estimating and testing hypotheses of the parameters in nonlinear models for repeated measurement data. In general, the results presented are useful in practical applications for statistical modeling of repeated measurements data, where the detection of outliers may be complicated. From the numerical example and simulations, we showed that the M-estimators behave better than the WLS estimators. We feel, however, that there are some issues that need to be studied before these methods are utilized in a more widespread way. For instance, we need to study the robustness properties of the the M-estimators through their influence functions and breakdown points.
Appendix Proof of Theorem 3.1. We consider the jth element of the vector Yi denoted by: j Yi =
p p l=1 m=1
ml fxil uil j = 1 s j
where ml is the m l element of −1 . We use the linear Taylor expansion and have: s 1 1 j Yi + n− 2 t − j Yi = √ tk j Yi k n k=1 p 1 ht +√ tk Y + √ k j i n k=1 n − Y k j i Then, for each j = 1 s, we have:
n
1 ht
sup √ j Yi + √ − j Yi n i=1 n t≤C
1 nspq ml + fxil fxil tk
n ijkl=1 j k
ns
1
ht tk j Yi + √ − j Yi
≤ sup
k k n t≤C n ik=1
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Sanhueza et al.
ns
1
1 nspq ml
+ sup tk j Yi + fxil fxil tk
k n iklm=1 j k t≤C n ik=1
where
ns
1
1 nspq ml
tk j Yi + fxil fxil tk
= op 1 sup k n iklm=1 j k t≤C n ik=1 (14) On the other hand,
2 ht ht f xil + √ uil + √ sup
n n t≤C k j ht ht ht − f xil + √ f + √ uil + √ j n k n n
2 − fxil uil + fxil fxil uil
k j j k
2 ht ht
− uil ≤ sup uil + √ f xil + √ k j n n t≤C
2 2
ht + sup
− f xil + √ fxil
uil k j n t≤C k j
ht
+ sup uil + √ − uil
n t≤C
ht ht
× f xil + √ f xil + √ j n k n
ht ht
+ sup f xil + √ f xil + √ n k n t≤C j
− fxil fxil
uil j k
Thus, by using conditions [B3](i)–(ii) and [C2](i)–(ii), we have:
2 ht ht
sup f xil + √ uil + √ n n t≤C k j ht ht ht − f xil + √ f xil + √ uil + √ j n k n n
2
− fxil uil + fxil fxil uil
−→ 0 ∀i ≤ n k j j k
Robustness in Nonlinear Models
and supt≤C n1 ns ik=1 tk Also,
k j
153
!
Yi + √htn − j Yi −→ 0 as n −→ .
k
n s
1
ht
Var sup tk j Yi + √ − j Yi
k k n t≤C n i=1 k=1
n s
ht C2
Var sup j Yi + √ − j Yi
≤ 2 n i=1 n k k k=1 t≤C
=
C2K −→ 0 n
Therefore,
n s
1
ht sup
tk j Yi + √ − j Yi
= op 1 k k n t≤C n i=1 k=1 Finally, from Eqs. (14) and the last equation, we have:
n
j Y i + sup
t≤C i=1
ht √ n
√ n
− j Yi
nspq
+
f ml fl tk
j m k
iklm=1
n
= op 1
Proof of Theorem 3.2. We may write: n K t 1 1 − Yi + n t Yi + √ + √ √ n n i=1 n n n t 1 Yi + √ − Yi =√ n i=1 n K 1 t t − Yi + √ + n t + Yi + √ + √ n n n n Then: n 1 K t 1 sup − Yi + n t Yi + √ + √ √ n n i=1 n n t≤CvecK≤D n 1 t 1 Yi + √ − Yi + n t ≤ sup √ n n i=1 n t≤C n 1 K ∗ ∗ − Yi Yi + √ + sup √ n i=1 n vecK≤D where ∗ = +
√t . n
Now, from Theorem 3.1, we have:
n 1 t 1 sup √ Yi + √ − Yi + n t = op 1 n n i=1 n t≤C
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Sanhueza et al.
Then, we can prove: n 1 K ∗ ∗ − Yi Yi + √ sup √ = op 1 n i=1 n vecK≤D
(15)
Proof of Theorem 3.3. From Theorem 3.1, we have that the following system of equations: n
1
j Yi + n− 2 t = 0
i=1
has a root tn that lies in t ≤ C with probability exceeding 1 − , for n ≥ n0 . Then 1 1 ˆ n = + n− 2 tn is a solution of Eq. (4) satisfying n 2 ˆ n − ≤ C ≥ 1 − , for 1 n ≥ n0 . Now, inserting t −→ n 2 ˆ n − in Eq. (6) we prove the theorem. Proof of Theorem 3.4. Let us consider an arbitrary linear compound n 1 Yi ∈ p Zn∗ = √ n i=1
and have n n 1 1
Xi −1 ui = √ Zi Zn∗ = √ n i=1 n i=1
where Zi = Xi −1 ui are independent random variables with Zi = 0 and VarZi = Xi −1 −1 Xi = i2 . We may write Zn∗ = ni=1 cni Zi∗ , where cni = √in and Zi∗ = Zi . Then, by using the Hájek-Šidak Central Limit Theorem n
c Z∗
ni i (see Sen and Singer, 1993), we show that i=1 1 −→ N0 1 and we have that 2 2 ni=1 cni n n 1 1 √ Z −→ N0 , i.e., √n i=1 Yi −→ N0 . Therefore, n i=1 i n 1 √ i=1 Yi −→ N0 . n
Acknowledgments The authors wish to thank the Editor and referees for their helpful comments that aided in improving this article. This study was partially supported by DIUFRO DI08-0061, DIPUV 29-2006, and FONDECYT 1080326 grants, Chile.
References Carter, W. H., Gennings, C., Staniswalis, J. C., Campbell, E. D., White, K. L. (1988). A statistical approach to the construction and analysis of isobolograms. J. Amer. College Toxicol. 7:963–973. Genning, C., Chinchilli, V. M., Carter, W. H. (1989). Response surface analysis with correlated data: a nonlinear model approach. J. Amer. Statist. Soc. 84:805–809. Hampel, F. R., Ronchetti, E., Rousseeuw, P. J., Stahel, W. (1986). Robust Statistics: The Approach Based on Influence Functions. New York: Wiley. Huber, P. J. (1981). Robust Statistics. New York: Wiley.
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155
Jurecková, J., Sen, P. K. (1996). Robust Statistical Procedures, Asymptotic and Interrelations. New York: Wiley. Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter. Ann. Statist. 4:163–169. Sanhueza, A, Sen, P. K. (2001). M-methods in generalized nonlinear models. In: Charalambides, Koutras, Balakrishnan, eds. Probability and Statistical Models with Applications. London: Chapman-Hall/CRC, Cap. 23:359–375. Sanhueza, A., Sen, P. K. (2004). Robust M-procedures in univariate nonlinear regression models. Braz. J. Probab. Statist. 18:183–200. Sen, P. K., Singer, J. M. (1993). Large Sample Methods in Statistics: An Introduction with Applications. New York: Chapman and Hall. Singer, J. M., Sen. P. K. (1985). M-methods in multivariate linear models. J. Multivariate Anal. 17:168–184. Volund, A. (1978). Application of the four-parameter logistic model to bioassay: comparison with slope ratio and parallel line models. Biometrics 34:357–365.
