Sankhya B (November 2010) 72:202–218 DOI 10.1007/s13571-011-0013-0
Statistical inference in nonlinear regression under heteroscedasticity Changwon Lim · Pranab K. Sen · Shyamal D. Peddada
Received: 4 June 2010 / Revised: 9 September 2010 / Accepted: 13 September 2010 / Published online: 10 May 2011 © Indian Statistical Institute 2011
Abstract Nonlinear regression models are commonly used in toxicology and pharmacology. When fitting nonlinear models for such data, one needs to pay attention to error variance structure in the model and the presence of possible outliers or influential observations. In this paper, an M-estimation based procedure is considered in heteroscedastic nonlinear regression models where the standard deviation is modeled by a nonlinear function. The methodology is illustrated using toxicological data. Keywords Asymptotic normality · Dose-response study · Heteroscedasticity · Hill model · M-estimation procedure · Nonlinear regression model · Toxicology · Weighted M-estimator
C. Lim (B) · S. D. Peddada Biostatistics Branch, NIEHS, NIH, 111 T. W. Alexander Dr, RTP, NC 27709, USA e-mail:
[email protected] S. D. Peddada e-mail:
[email protected] P. K. Sen Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, 338 Hanes Hall, CB#3260, Chapel Hill, NC 27599, USA e-mail:
[email protected] P. K. Sen Department of Biostatistics, University of North Carolina at Chapel Hill, 3101 McGavran-Greenberg, CB#7420, Chapel Hill, NC 27599, USA
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1 Introduction Toxicologists routinely use nonlinear models in dose-response studies to determine various toxicity characteristics of a chemical or a drug. For example they are often interested in estimating the dose corresponding to 50% of maximum response of a chemical (known as ED50 ), the “slope” of the curve, the maximum tolerated dose (MTD), etc. (Velarde et al. 1999; Avalos et al. 2001; Pounds et al. 2004). The usual strategy is to fit a nonlinear regression model such as the Hill model (Gaylor and Aylward 2004; Sand et al. 2004; Crofton et al. 2007) and estimate the parameters of the model using standard ordinary least squares estimation (OLSE). These point estimates are used to perform statistical inferences on various linear and nonlinear functions of the parameters (Nitcheva et al. 2005; Piegorsch and West 2005; Wu et al. 2006). Several assumptions are made regarding the data and the model when performing inferences on the parameters of a nonlinear model (Seber and Wild 1989). Among them, an important assumption is that the variance of the response variable Y is constant across all observations (i.e., homoscedasticity). This assumption is often violated in toxicological studies. For example, at higher doses some animals may experience toxic response resulting in large variation in Y. In addition to non-constant variance (heteroscedasticity), outliers and influential observations are common in toxicological data. Under such situations, the statistical inference based on OLSE may be inaccurate. For instance the Type I errors and the coverage probabilities of confidence intervals (CIs) may not attain the nominal levels (Carroll and Ruppert 1988; Kutner et al. 2005). A common strategy to handle heteroscedasticity is to perform Box-Cox power transformations (Morris et al. 2002). An equally popular strategy is to perform iterated weighted least squares estimation (IWLSE) (Gaylor and Aylward 2004; Barata et al. 2006). In this paper a similar strategy is adopted since in many applications, especially in toxicology, it is possible to consider a parsimonious nonlinear function to model the variance. Recently, motivated by the work of Davidian and Carroll (1987), the strategy of modeling the variance using a pre-specified nonlinear model has been considered (Lim et al. 2010, unpublished manuscript). They proposed the weighted M-estimator (WME) by introducing M-estimation methods to deal with potential influential observations and outliers, which have been studied extensively in the literature (Jureˆcková and Sen 1996; Maronna et al. 2006; Sanhueza et al. 2009). The proposed WME was established for estimating the regression and variance parameters simultaneously. A log-linear model was used as the variance model for analyzing data. However, the error variance by the model would keep increasing as the dose level increases, which may not be realistic at the highest dose for some data sets. Therefore, in this paper, we propose a different variance model which has better characteristic. In Section 2 the WME proposed by Lim et al. (2010, unpublished manuscript) is defined along with its asymptotic result, and we propose a variance model which is adapted to the WME for better estimation. In Section 3
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extensive simulation studies are conducted to examine the performance of the WME with the proposed variance model. Section 4 illustrates a numerical application of the WME with the proposed variance model by applying a Hill model proposed by Hill (1910) to toxicological data.
2 Proposed methodology Let yi = f (xi , θ) + σi i , i = 1, . . . , n
(1)
denote the nonlinear regression model, where yi are the observable random variables, xi = (x1i , x2i , . . . , xmi )t are known regression constants, θ = (θ1 , θ2 , . . . , θ p )t is a vector of unknown parameters, f (, ) is a nonlinear function of θ of specified form; and the errors i are assumed to be independent random variables with mean 0 and variance 1. It is assumed that σi = σ (zi , τ ) for i = 1, . . . , n, where σ (, ) is a known function, zi = (zi1 , . . . , ziq )t are known vectors, possibly dependent on the xi , and τ = (τ1 , . . . , τq )t is a vector of unknown parameters. Lim et al. (2010, unpublished manuscript) proposed the weighted Mestimator (WME) and used a log-linear model for standard deviation at dose xi , which is log σi = τ0 + τ1 xi for i = 1, . . . , n. A limitation of the log-linear model is that log standard deviation is linearly increasing in dose, which may not always be true. In fact it is more reasonable to expect the standard deviation curve to have a shape similar to the mean response curve. Thus, as the dose increases the standard deviation increases according to a sigmoidal shape. Hence, we propose the following nonlinear model for describing standard deviation at dose xi σ (xi , τ ) = τ0 +
τ1 , i = 1, . . . , n. 1 + e−τ2 xi
(2)
As seen in Fig. 1, for different choices of parameters τ , the standard deviation model proposed in Eq. 2 is flexible to capture various commonly seen monotonic shapes of standard deviations. Our primary interest is in the estimation of regression parameters of nonlinear model (1), and the model (2) provides a simple parsimonious model for standard deviation. The WME of (θ t , τ t )t is defined by the following minimization problem (Lim et al. 2010, unpublished manuscript):
θˆ n τˆ n
= Argmin
n i=1
h
2
yi − f (xi , θ ) σ (zi , τ )
+ log σ (zi , τ )
: θ ∈ 1 ⊆ p , τ ∈ 2 ⊆ q ,
(3)
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(a)
0
1
2
(d)
0
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2
(g)
0
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0 5 1 2 0 15
02 06
05
0 1 0 3 100
(b)
(e)
(h)
205
0
1
02 06
2
0 15
0
1
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02 08 0
0
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0 9 2 0 05
(c)
0
(f)
(i)
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05 12 05
2
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0 2 0 0 15
0 01 1
0 05
Fig. 1 The proposed model of standard deviation for different patterns of parameters
where h(·) is a real valued function, 1 and 2 are compact subsets of p and q , respectively, and log σ (zi , τ ) is added within the sum, which is analogous to the maximum likelihood estimation when the errors are normally distributed. A distinct advantage of the proposed method over the classical IWLSE is that we do not require large sample sizes at each dose, as needed in IWLSE. We only require the total sample size to be large. Thus the proposed methodology is applicable even if the sample size at each dose is 1. The parameters associated with the nonlinear model (2) are estimated simultaneously with the regression parameters of model (1) using the estimating equations corresponding to the minimization problem (3).
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Define fθ (xi , θ) = (∂/∂θ) f (xi , θ), σ τ (zi , τ ) = (∂/∂τ )σ (zi , τ ), and ψ(z) = (∂/∂z)h2 (z). Then, the estimating equation for the minimization in Eq. 3 is given by: n
λ(xi , yi , θˆ n , τˆ n ) = 0
(4)
i=1
where
λ(xi , yi , θ, τ ) =
k(zi , τ )ψ (i ) fθ (xi , θ) k(zi , τ ) {ψ (i ) i − 1} σ τ (zi , τ )
,
(5)
and k(zi , τ ) = 1/σ (zi , τ ). Then, we obtain the asymptotic normality of the WME from the following theorem (Lim et al. 2010, unpublished manuscript): Theorem 1 (Lim et al. 2010, unpublished manuscript) Under some regularity conditions, −1 √ θˆ n − θ (6) ˆ 2 n −→ N p+q (0, I p+q ), τˆ n − τ − ν n (θ , τ ) where
−1 1 −1 1 ˆ ˆ 1 ˆ ˆ ˆ ˆ , 5n θ n , τˆ n 3n θ n , τˆ n 5n θ n , τˆ n n n n and ν n (θ , τ ), ˆ 3n θˆ n , τˆ n and ˆ 5n θˆ n , τˆ n are def ined in the Appendix. ˆ =
(7)
3 Simulation studies 3.1 Study design Since the Hill model is commonly used for modeling dose-response data in toxicological studies, in the simulation study presented in this paper the following model was used yij = f (xi , θ) + σi ij = θ0 +
θ1 xiθ2
θ3θ2 + xiθ2
+ σi ij, i = 1, . . . , 7, j = 1, . . . , 4, (8)
where yij are the response observations, xi are the dose levels, θ0 is the intercept, θ1 is the maximum effect of a drug, θ2 is the slope, and θ3 is ED50 , the dose level producing 50% of the maximum effect. In the above model, it is assumed ij ∼ N(0, 1) for all i, j. Patterned along the lines of some real data published in the National Toxicology Program’s technical report (NTP 2007, 72, 1-G4), the levels xi were taken to be 0, 0.1, 0.3, 1, 3, 10, 30, and (θ0 , θ1 , θ2 , θ3 ) was taken to be (0.12, 2.98, 1.85, 5.67). Also, based on the data
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provided in NTP (2007), the following nonlinear model was used for describing standard deviation at dose xi τ1 σ (xi , τ ) = τ0 + , i = 1, . . . , 7, (9) 1 + e−τ2 xi where (τ0 , τ1 , τ2 ) = (−0.58, 1.21, 0.15). There are two parts to the simulation study. At first, for illustration purposes, two data sets are generated and the parameters are estimated using OLSE, IWLSE, WME methods. One of the data sets is generated according to homoscedastic error with σi = 0.1 for all i (Data 1), while the other data set is generated according to heteroscedastic error using the above variance model (Data 2). Furthermore, in order to explore the effect of outliers on the estimation methods, an outlier at each of the largest two dose levels is generated using a mean shifted error distribution, that is ij ∼ N(±15, 1) changing signs with probability 0.5 for i = 6, 7, j = 4. The second component of our study is to compare OLSE, IWLSE and WME using 10,000 data sets generated according to various patterns of the variances. For each individual component of the estimator vector, the three procedures are compared in terms of: (i) bias, (ii) mean squared error (MSE), (iii) coverage probability at a nominal level of 0.95, (iv) length of 95% CIs. The three estimators are also compared in terms of the joint coverage probability of the 100(1 − α)% confidence elipsoid
θ) ˆ −1 (θˆ − θ) ≤ pF p,n− p (α) (θˆ − θ)t Var( (10) θˆ ) ˆ where p is the number of parameters and Var( centered at an estimator θ, is the appropriate variance estimator. Three types of data sets are generated from the Hill model: (i) Data 1: homoscedastic data with σi = 0.2 for all i, (ii) Data 2: heteroscedastic data with σi having the values of 0.017, 0.003, 0.005, 0.041, 0.074, 0.278, 0.266, obtained from a real data example (NTP 2007) using IWLSE, and (iii) Data 3: heteroscedastic data with σi s given by the model (9). It is noted that for Data 1 and 2, p = 4, while p = 7 for Data 3. For each type of error variance, data sets are generated using two types of error distributions: (i) ij ∼ N(0, 1) for all i, j (0% contamination), and (ii) ij ∼ N(0, 1) for i = 1, . . . , 5 and 0.65N(0, 1) + 0.35N(±5, 1) changing signs with probability 0.5 for i = 6, 7 (10% contamination). Outliers are set to only occur at the largest 2 out of 7 dose levels with 35% chance, which means that 10% of all data points are outliers. In all our simulations, we considered 7 dose groups and two different patterns of sample sizes per dose group. The first pattern of 4 observations per dose corresponds to moderate total sample size of n = 28, whereas the second pattern of 20 per dose group corresponds to large (total) sample size of n = 120 observations. In the case of n = 28, we used 10,000 simulation runs to compare various estimators in terms of bias, MSE and the coverage probability. These comparisons did not change substantially when we reduced the number of simulations to 1,000 runs. To reduce computational burden in the case of n = 140 (large sample size), we therefore used 1,000 simulation runs to make
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Table 1 Estimate and standard error for parameters of the models for Data 1 and 2 using OLSE, IWLSE and WME methods
Data 1 (homo.)
Data 2 (hetero.)
θ0 θ1 θ2 θ3 θ0 θ1 θ2 θ3
(Intercept) (Emax ) (slope) (ED50 ) (Intercept) (Emax ) (slope) (ED50 )
True values
OLSE Estimate
S.E.
IWLSE Estimate
S.E.
WME Estimate
S.E.
0.12 2.98 1.85 5.67 0.12 2.98 1.85 5.67
0.178 2.481 11.28 3.291 0.162 2.223 12.60 3.201
0.090 0.201 20,950 566.4 0.195 0.416 256,500 4,232.6
0.179 2.482 11.26 3.293 0.146 2.110 12.34 3.177
0.029 0.245 15,250 415.4 0.001 0.460 46,800 690.5
0.171 2.482 4.662 3.742 0.132 2.536 2.107 4.520
0.029 0.300 9.700 1.816 0.008 0.581 0.526 1.315
comparisons among the various estimators in terms of bias, MSE and the coverage probability. 3.2 Results The results of OLSE, IWLSE and WME and their standard errors for the two simulated data sets are summarized in Table 1. Although the fitted curves using OLSE and IWLSE seem reasonable based on the data (Fig. 2), their standard errors are very large. Due to the two outliers at the largest two dose levels (x = 10, 30), the OLSE fitted Hill curves reached their upper asymptotes (θ0 + θ1 ) before x = 10 thus resulting in a very large estimate for θ2 (slope parameter) and its standard error. On the other hand, the influence of the outliers is successfully reduced by WME and the estimated values of the parameters are much closer to the true values (especially θ2 ) with
(a)
(b)
Fig. 2 Example of model predictions by OLSE (solid line), IWLSE (dashed line), WME (dotted line) methods for a homoscedastic data (Data 1) and b heteroscedastic data (Data 2) with two possible outliers, one at each of the two highest dose levels
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relatively smaller standard errors. Furthermore, note that although Data 1 and 2 were generated according to homoscedastic and heteroscedastic errors, respectively, the WME performs well with smaller standard errors in both cases. Thus the WME performs well not only for heteroscedastic data but also for homoscedastic data. The results of 10,000 simulated data sets for comparing the performance of OLSE, IWLSE and WME are summarized in Tables 2, 3, 4, 5, 6 and 7. When n = 28 and there are no outliers, bias and MSE are not very different among OLSE, IWLSE and WME for homoscedastic data (Table 2), while bias and MSE of OLSE are larger than those of IWLSE and WME for heteroscedastic data (Tables 3 and 4). For homoscedastic data (Data 1) with outliers, similar results are observed overall as for Data 1 without outliers although the bias and MSE are increased for all three estimators. However, it is noted that bias and MSE for θ3 as well as total MSE of IWLSE are larger than those of OLSE and WME in the presence of outliers. The coverage probabilities of WME are close to the nominal level (0.95), as are those of OLSE for Data 1, while IWLSE has lower coverage probabilities. Both WME and OLSE methods have similar lengths for their CIs. In the presence of outliers, the joint coverage probability of OLSE is decreased severely, while that of WME is decreased slightly. However, the volume of the confidence elipsoid of WME is large compared to that of OLSE, possibly because the number of parameters ( p) in WME is larger than those in OME ( p = 7 vs. p = 4). As a consequence there is potentially greater uncertainty associated with WME than OLSE, resulting in a larger confidence ellipsoid. For heteroscedastic data (Data 2 and 3)
Table 2 Simulation results based on 10,000 replications for OLSE, IWLSE, and WME under the assumption of homoscedastic error (Data 1) with 0% and 10% contaminations (n = 28)
θ0 θ1 θ2 θ3 Joint
OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME
BIAS MSE 0% contamination
COV
LEN
BIAS MSE COV 10% contamination
LEN
3.723* 3.093* −1.652* 0.043 0.046 0.046 0.005 0.005 0.002 0.225 0.241 0.233 0.053 0.060 0.056
0.94 0.77 0.92 0.96 0.89 0.95 0.94 0.88 0.94 0.99 0.95 0.98 0.93 0.63 0.94
0.056 0.045 0.058 0.185 0.166 0.191 0.272 0.244 0.282 0.685 0.637 0.712 0.007* 0.002* 0.010*
−0.003 −0.005 −0.005 0.176 0.203 0.191 −0.073 −0.084 −0.086 0.844 1.050 0.916 0.748 1.150 0.882
0.073 0.047 0.059 0.295 0.371 0.397 0.347 0.355 0.393 1.213 1.878 1.677 0.075* 0.484* 0.408*
0.003 0.005 0.003 0.035 0.043 0.038 0.077 0.087 0.080 0.406 0.538 0.430 0.521 0.672 0.551
0.003 0.005 0.004 0.159 0.253 0.186 0.132 0.152 0.139 3.463 15.34 5.086 3.759 15.76 5.415
0.99 0.79 0.93 0.95 0.94 0.97 0.92 0.87 0.92 0.98 0.99 0.99 0.85 0.59 0.90
BIAS bias; MSE mean square error; COV coverage probability of 95% CI; LEN length of 95% CI; Joint sum of individual squared biases, MSEs *actual value = entry value × 10−4
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Table 3 Simulation results based on 10,000 replications for OLSE, IWLSE, and WME under the assumption of heteroscedastic error (Data 2) with 0% and 10% contaminations (n = 28)
θ0 θ1 θ2 θ3 Joint
OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME
BIAS MSE COV 0% contamination
LEN
BIAS MSE COV 10% contamination
LEN
−1.387* −0.233* −0.340* 0.068 0.028 0.030 −0.030 −0.006 −0.014 0.307 0.112 0.147 0.100 0.013 0.023
0.038 0.002 0.004 0.132 0.131 0.191 0.185 0.088 0.131 0.493 0.410 0.589 114.8** 0.005** 0.457**
−5.292* −0.349* −0.436* 0.289 0.097 0.059 −0.127 −0.016 −0.026 1.360 0.277 0.292 1.951 0.087 0.089
0.075 0.002 0.004 0.348 0.247 0.349 0.346 0.095 0.182 1.609 0.569 0.990 17114** 0.028** 5.440**
0.084* 0.006* 0.022* 0.046 0.025 0.034 0.053 0.014 0.021 0.795 0.256 0.346 0.895 0.295 0.402
1.00 0.83 0.90 0.82 0.88 0.95 0.87 0.86 0.92 0.83 0.91 0.97 0.74 0.61 0.93
0.186* 0.006* 0.027* 0.387 0.102 0.143 0.121 0.017 0.032 14.06 0.932 1.118 14.57 1.051 1.293
1.00 0.84 0.87 0.91 0.85 0.91 0.91 0.86 0.93 0.97 0.92 0.97 0.80 0.61 0.90
BIAS bias; MSE mean square error; COV coverage probability of 95% CI; LEN length of 95% CI; Joint sum of individual squared biases, MSEs *actual value = entry value × 10−3 **actual value = entry value × 10−9
(Tables 3 and 4), neither OLSE nor IWLSE have coverage probabilities closer to the nominal levels, while the coverage probability of confidence interval based on WME is closer to the nominal level. These results show that when Table 4 Simulation results based on 10,000 replications for OLSE, IWLSE, and WME under the assumption of heteroscedastic error (Data 3) with 0% and 10% contaminations (n = 28)
θ0 θ1 θ2 θ3 Joint
OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME
BIAS MSE 0% contamination
COV
LEN
BIAS MSE COV 10% contamination
LEN
−0.003 −0.001 −0.001 0.299 0.213 0.203 −0.098 −0.071 −0.070 1.218 0.801 0.757 1.583 0.692 0.620
1.00 0.81 0.94 0.85 0.92 0.96 0.91 0.86 0.93 0.94 0.98 0.99 0.77 0.61 0.93
0.073 0.007 0.009 0.329 0.350 0.398 0.338 0.213 0.251 1.422 1.218 1.369 6.086* 0.003* 0.013*
−0.007 −0.002 −0.001 0.862 0.514 0.502 −0.192 −0.100 −0.110 3.419 1.470 1.537 12.47 2.435 2.626
0.143 0.007 0.009 1.042 0.659 0.805 0.620 0.248 0.319 5.610 2.042 2.608 531.7* 0.050* 0.161*
25.29* 10.14* 7.807* 0.348 0.188 0.179 0.115 0.058 0.052 6.960 2.606 2.225 7.423 2.852 2.457
37.60* 10.67* 8.123* 2.109 0.717 0.767 0.202 0.080 0.075 52.39 9.015 8.288 54.70 9.812 9.130
1.00 0.79 0.91 0.92 0.87 0.92 0.94 0.81 0.87 0.98 0.98 0.99 0.72 0.53 0.84
BIAS bias; MSE mean square error; COV coverage probability of 95% CI; LEN length of 95% CI; Joint, sum of individual squared biases, MSEs *actual value = entry value × 10−5
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Table 5 Simulation results based on 1,000 replications for OLSE, IWLSE, and WME under the assumption of homoscedastic error (Data 1) with 0% and 10% contaminations (n = 140)
θ0 θ1 θ2 θ3 Joint
OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME
BIAS MSE 0% contamination
COV
LEN
BIAS MSE COV 10% contamination
LEN
−0.0005 −0.0006 −0.0004 0.004 0.004 0.004 0.002 0.002 0.003 0.017 0.016 0.018 0.0003 0.0003 0.0003
0.95 0.94 0.95 0.95 0.94 0.94 0.95 0.94 0.95 0.96 0.94 0.95 0.94 0.91 0.99
0.049 0.048 0.048 0.156 0.153 0.158 0.242 0.238 0.245 0.563 0.554 0.571 0.002* 0.001* 0.001*
−0.0006 −0.0006 −0.0007 0.024 0.021 0.023 −0.0014 −0.0001 −0.0013 0.106 0.092 0.100 0.012 0.009 0.010
0.065 0.049 0.049 0.211 0.270 0.286 0.319 0.358 0.359 0.767 0.989 1.012 0.014* 0.015* 0.021*
0.0006 0.0006 0.0006 0.007 0.007 0.007 0.016 0.016 0.016 0.083 0.084 0.084 0.106 0.107 0.107
0.0006 0.0007 0.0006 0.019 0.017 0.018 0.033 0.032 0.032 0.236 0.211 0.221 0.288 0.261 0.272
0.99 0.94 0.95 0.89 0.97 0.98 0.92 0.96 0.95 0.92 0.99 0.99 0.89 0.92 0.99
BIAS bias; MSE mean square error; COV coverage probability of 95% CI; LEN length of 95% CI; Joint sum of individual squared biases, MSEs *actual value = entry value × 10−6
sample size is small, WME performs as well as OLSE for homoscedastic data without any outliers. For heteroscedastic data, WME outperforms IWLSE and better than OLSE in presence of outliers. Table 6 Simulation results based on 1,000 replications for OLSE, IWLSE, and WME under the assumption of heteroscedastic error (Data 2) with 0% and 10% contaminations (n = 140)
θ0 θ1 θ2 θ3 Joint
OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME
BIAS MSE COV 0% contamination
LEN
BIAS MSE COV 10% contamination
LEN
0.401* 0.178* 0.838* 0.011 0.004 0.007 27.22* 14.55* 0.439* 0.046 0.012 0.023 0.0023 0.0001 0.0006
0.036 0.002 0.003 0.116 0.130 0.164 0.179 0.087 0.115 0.419 0.385 0.491 17.2** 0.003** 0.061**
−8.559* −0.138* −1.067* 0.036 0.006 0.001 −0.0102 −0.0004 −0.0027 0.171 0.016 0.023 0.0307 0.0003 0.0005
0.068 0.002 0.004 0.226 0.232 0.292 0.334 0.094 0.160 0.832 0.497 0.783 2246** 0.015** 0.759**
0.201* 0.007* 0.041* 0.009 0.005 0.006 0.016 0.002 0.005 0.130 0.040 0.057 0.155 0.047 0.068
1.00 0.93 0.91 0.79 0.94 0.97 0.85 0.94 0.90 0.77 0.94 0.95 0.75 0.92 0.97
0.487* 0.007* 0.047* 0.031 0.015 0.025 0.042 0.002 0.006 0.479 0.062 0.010 0.552 0.079 0.131
1.00 0.94 0.90 0.84 0.94 0.93 0.89 0.95 0.95 0.83 0.96 0.99 0.81 0.92 0.94
BIAS bias; MSE mean square error; COV coverage probability of 95% CI; LEN length of 95% CI; Joint sum of individual squared biases, MSEs *actual value = entry value × 10−4 **actual value = entry value × 10−11
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Table 7 Simulation results based on 1,000 replications for OLSE, IWLSE, and WME under the assumption of heteroscedastic error (Data 3) with 0% and 10% contaminations (n = 140)
θ0 θ1 θ2 θ3 Joint
OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME OLSE IWLSE WME
BIAS MSE 0% contamination
COV
LEN
BIAS MSE COV 10% contamination
LEN
−0.0007 −0.0002 −0.0002 0.050 0.030 0.031 −0.015 −0.009 −0.010 0.187 0.105 0.108 0.038 0.012 0.013
1.00 0.94 0.95 0.80 0.95 0.96 0.93 0.95 0.96 0.87 0.97 0.97 0.83 0.90 0.99
0.070 0.007 0.008 0.230 0.311 0.325 0.339 0.223 0.232 0.842 0.966 1.002 0.327** 0.002** 0.002**
−0.0040 −0.0004 −0.0004 0.226 0.090 0.078 −0.099 −0.025 −0.031 0.838 0.239 0.250 0.763 0.066 0.069
0.133 0.007 0.007 0.529 0.537 0.590 0.613 0.257 0.298 2.079 1.338 1.610 61.9** 0.009** 0.016**
0.006* 0.002* 0.001* 0.039 0.025 0.026 0.032 0.012 0.012 0.447 0.236 0.244 0.518 0.273 0.282
0.013* 0.002* 0.002* 0.208 0.077 0.087 0.081 0.016 0.016 3.039 0.459 0.508 3.328 0.552 0.611
1.00 0.93 0.93 0.88 0.96 0.96 0.95 0.94 0.96 0.98 0.99 0.99 0.87 0.90 0.97
BIAS bias; MSE mean square error; COV coverage probability of 95% CI; LEN length of 95% CI; Joint sum of individual squared biases, MSEs *actual value = entry value × 10−2 **actual value = entry value × 10−7
When n = 140 and data are homoscedastic without outliers (Table 5), all estimators perform equally well in terms of bias, MSE and the coverage probability. However, when there are outliers, the coverage probabilities of OLSE are decreased, while those of both IWLSE and WME are not. For heteroscedastic data (Data 2 and 3) (Tables 6 and 7), OLSE tends to have larger bias and MSE than IWLSE and WME do. Furthermore, the confidence intervals centered at WME and IWLSE tend to attain the nominal level better than those centered at OLSE although the OLSE intervals tend to be wider. These results show that when sample size (especially the number of observations at each dose level) is large, all three estimators perform equally well for homoscedastic data, while WME performs as good as IWLSE, and better than OLSE for heteroscedastic data.
4 Application to hexavalent chromium data 4.1 Description of the data This section illustrates the proposed WME methodology with real data from a National Toxicology Program (NTP) study. As stated in the NTP technical report (NTP 2007, pp. 11–12), chromium is a naturally occuring element in the earth’s rocks, having six oxidation states. The most common states are the metallic (Cr), trivalent (CrIII), and hexavalent chromium (CrVI). Chromium compounds are widely used in pigment production, metal finishing, leather
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tanning, and wood preservation. Chromium exposure occurs through food, water, and air, or through direct skin contact. CrVI compounds have been shown to cause lung cancers in humans when inhaled. Studies have shown increased incidences of lung cancers in people who work in chromium industries and are exposed to high levels of chromium compounds (NTP 2007, pp. 14–15). CrVI was detected in ground water in California (NTP 2007, pp. 11–12). Then, due to concerns about its presence in drinking water sources and its potential adverse health effects such as cancers, CrVI was assigned by the California Congressional Delegation and the California Environmental Protection Agency for toxicity and carcinogenicity testing (NTP 2007, pp. 17). In response to that, the NTP conducted 3-month and 2-year studies, where rodents were exposed to CrVI administered in drinking water as sodium dichromate dihydrate. The WME methodology is applied to several data sets from a toxicological study, as a part of the NTP study, that was designed to examine the relationship between concentrations of CrVI, as sodium dichromate dihydrate, in drinking water and accumulation of total chromium in tissue for three species (rats, mice, and guinea pigs). Groups of four Fischer 344 rats, four B6C3F1 mice, and four Hartley guinea pigs were randomly assigned to one of six concentrations of sodium dichromate dihydrate in their drinking water. All animals were between 6 and 10 weeks in age. Control groups were given water without added sodium dichromate dihydrate. The dose concentrations were 0, 2.87, 8.62, 28.7, 86.2, 287, and 862 mg sodium dichromate dihyrate/L (to yield 0, 1, 3, 10, 30, 100, and 300 mg chromium/L). When animals were sacrificed, total chromium concentrations in blood, kidneys, and femurs were measured. 4.2 Analysis and results The proposed WME method is illustrated using the blood and kidney data sets for mouse where chromium concentration (y) is modeled. The Hill model (8) is used for fitting the data, where the values of x, dose concentration are 0, 1, 3, 10, 30, 100, 300 and y is total chromium tissue concentration. There are 7 values of x and 4 observations at each x (n = 28). For each of the data set the parameters are estimated and their standard errors are computed using OLSE, IWLSE and WME methods. The results are summarized in Table 8, and the data and the fitted curves are plotted in Fig. 3. Visually the scatter plots in Fig. 3 seems to suggest that there is some amount of heteroscedasticity in the data. The sample standard deviation of blood data set for the seven dose groups were 0.034, 0.027, 0.030, 0.059, 0.054, 0.351, and 0.252, respectively. Thus they ranged from about 0.027 to 0.351, which indicates potential heteroscedasticity in the data. When the sample standard deviation was fitted against doses, we found that the nonlinear model (2) seems to fit the standard deviations reasonably well, although at the high dose there appears to be small decrease in the standard deviation. Similar result was observed for the kidney data set. The standard deviation function corresponding to the two data sets are provided in Fig. 4.
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Table 8 Estimate and standard error for parameters of the models for chromium mouse blood and kidney data using OLSE, IWLSE and WME methods
Blood
Kidney
θ0 θ1 θ2 θ3 θ0 θ1 θ2 θ3
(Intercept) (Emax ) (Slope) (ED50 ) (Intercept) (Emax ) (slope) (ED50 )
OLSE Estimate
S.E.
IWLSE Estimate
S.E.
WME Estimate
S.E.
0.367 0.715 12.77 33.13 0.335 3.444 12.21 32.07
0.045 0.102 185,400 47,700 0.195 0.417 105,500 18,450
0.363 0.669 12.80 32.83 0.348 3.181 12.14 31.82
0.009 0.111 179,900 41,570 0.008 0.416 65,080 10,050
0.360 0.759 5.389 38.02 0.106 7.191 0.842 167.4
0.011 0.170 37.10 63.42 0.010 3.833 0.089 166.1
Also visually, it appears that there is a decrease in blood chromium levels at the highest dose group. We performed a non-parametric test for an umbrella order in the mean values of blood chromium using the order-restricted inference methodology developed in Peddada et al. (2003, 2005). Thus we tested the null hypothesis of no difference in means across dose groups against the alternative that the mean response increases with dose until 100 mg/L and then drops at 300 mg/L, and found the p-value to be 0.001. Note that the non-parametric procedure of Peddada et al. uses bootstrap methodology by bootstrapping the residuals and hence is robust to heteroscedasticity. Thus it appears that the data in the highest dose group are potential outliers. The same result was observed for the kidney data set ( p-value = 0.001). Although the fitted curves using OLSE and IWLSE may seem reasonable based on blood data (Fig. 3a), their standard errors are large (Table 8). Due to the possible outlier at x = 100 (highest point), the fitted Hill curves reached their upper asymptotes (θ0 + θ1 ) before x = 100 using the least squares estimation, which makes the estimates of θ2 (slope parameter) and their
(a)
(b)
Fig. 3 Chromium concentration in (a) blood and (b) kidney for mice using OLSE (solid line), IWLSE (dashed line), WME (dotted line) methods
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(a)
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(b)
Fig. 4 The fitted curve of standard deviation against the dose for a blood and b kidney for mice using the variance function (2)
standard errors large. On the other hand, the influence of the outliers is successfully reduced using the WME and the estimated values of the parameter and their standard error are much smaller. Even though the point estimate for θ3 (ED50 ) using the three estimators are similar to each other, the WME estimates the ED50 with a smaller standard error. For kidney data (Fig. 3b), the fitted curve using WME is very different from those using OLSE and IWLSE. Here, again the highest two points at x = 100 may be possible outliers, and because of them, the Hill curves reached their upper asymptotes before x = 100 using OLSE and IWLSE, by which the point estimates of θ2 and their standard errors become very large (Table 8). Furthermore, the observations at x = 10 were completely ignored even though their variability was very small. On the other hand, the fitted curve using WME successfully passed through the points at x = 10 and since the influence of the possible outliers at x = 100 was reduced, the shape of the fitted curve was very different from those using OLSE and IWLSE. Thus, the point estimate for θ3 (ED50 ) using WME is completely different from those using OLSE and IWLSE.
5 Concluding remarks In this paper, an M-estimation methodology has been developed for analyzing nonlinear regression models that are subject to heteroscedastic variance structure. The error standard deviation is assumed to be a nonlinear function of unknown parameters. In addition to accounting for potential heteroscedasticity in the data, the proposed WME methodology is robust to outliers and influential observations. In terms of MSE as well as the coverage probability, the WME performs as well as the OLSE for homoscedastic data
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without any outliers, better with outliers, and for heteroscedastic data it performs better than IWLSE. It is worth noting that the coverage probabilities of WME are much closer to the nominal level than those of IWLSE without having wider CI. From the result of the simulation studies, one can see that bias and MSE for ED50 parameter (θ3 ) are quite large compared to other parameters. This phenomenon can be observed for all the scenarios considered in our simulation study. Surprisingly, this is true even when the total sample size is as large as 140, suggesting that the ED50 parameter can be difficult to estimate in some instances. From our experience with real data, we have seen situations where the estimate of the slope parameter can be very large, which results in very large standard error estimates. We believe that dose-spacing plays a major role when estimating parameters of nonlinear models, especially the ED50 and the slope parameters of a Hill model. It may be useful to develop sequential desings where the doses are chosen such that various parameters are estimated with greater efficiency. Future research in this direction would be very useful, especially in toxicology. In many applications, including toxicology, it is common to assume that the standard deviation is a monotonic function of mean response, and we modeled the standard deviation as a sigmoidal function of dose. The theoretical framework provided in this paper can be easily modified to deal with other parametric shapes for the standard deviation function, such as an umbrella shape. Acknowledgements This research was supported, in part, by the Intramural Research Program of the NIH, National Institute of Environmental Health Sciences [Z01 ES101744-04]. We thank Drs. Anastasia Ivanova and Gregg Dinse as well as the editor and the referee for many important comments which helped improve the presentation of the manuscript.
Appendix: Notations in Theorem 1 We state the definitions of the matrices and related quantities using in Theorem 1 as follows: (i) 31 (θ, τ ) 0 , 3 (θ, τ ) = 0 32 (θ, τ ) where 31 (θ, τ ) = limn→∞ n1 31n (θ , τ ), 32 (θ, τ ) = limn→∞ n1 32n (θ, τ ), 2 31n (θ , τ ) = σψ1
n
u(xi )k2 (zi , τ )fθ (xi , θ )ftθ (xi , θ),
i=1 2 32n (θ , τ ) = σψ2
n
v(xi )k2 (zi , τ )σ τ (zi , τ )σ tτ (zi , τ ),
i=1 2 2 σψ1 u(x) = Eψ 2 ()(< ∞), σψ2 v(x) = Var{ψ()}(< ∞), and =
y− f (x,θ) . σ (z,τ )
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(ii)
5 (θ, τ ) =
1 (θ, τ ) 0 , 0 2 (θ, τ )
where 1 (θ, τ ) = limn→∞ n1 1n (θ , τ ), 2 (θ, τ ) = limn→∞ n1 2n (θ , τ ), 1n (θ , τ ) = γ2
n
k2 (zi , τ )fθ (xi , θ)ftθ (xi , θ ),
i=1
2n (θ , τ ) =
n i=1
2γ1 + γ3 − 1 1 − γ1 t σ (z , τ )σ (z , τ ) + (z , τ ) , τ i τ i τ i σ 2 (zi , τ ) σ (zi , τ )
τ (zi , τ ) = (∂ 2 /∂τ ∂τ t )σ (zi , τ ), γ1 = E{ψ()}( = 0), γ2 = Eψ ()( = 0), and γ3 = E{ψ () 2 }( = 0). (iii) ν n (θ , τ ) =
−1 n 1 γ1 − 1 2n (θ , τ ) k(zi , τ )σ τ (zi , τ ). n n i=1
References Avalos, M., Mak, C., Randall, P. K., Trzeciakowski, J.P., Abell, C., Kwan, S.-W., and Wilcox, R. E. 2001. Nonlinear analysis of partial dopamine agonist effects on cAMP in C6 glioma cells. Journal of Pharmacological and Toxicological Methods 45:17–37. Barata, C., Baird, D.J., Nogueira, A.J.A., Soares, A.M.V.M., and Riva, M.C. 2006. Toxicity of binary mixtures of metals and pyrethroid insecticides to Daphnia magna Straus. Implications for multi-substance risks assessment. Aquatic Toxicology 78:1–14. Carroll, R.J., and Ruppert, D. 1988. Transformation and weighting in regression. New York: Chapman and Hall. Crofton, K.M., Paul, K.B., DeVito, M.J., and Hedge, J.M. 2007. Short-term in vivo exposure to the water contaminant triclosan: Evidence for disruption of thyroxine. Environmental Toxicology and Pharmacology 24:194–197. Davidian, M., and Carroll, R.J. 1987. Variance function estimation. Journal of the American Statistical Association 82:1079–1091. Gaylor, D.W., and Aylward, L.L. 2004. An evaluation of benchmark dose methodology for non-cancer continuous-data health effects in animals due to exposures to dioxin (TCDD). Regulatory Toxicology and Pharmacology 40:9–17. Hill, A.V. 1910. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. Journal of Physiology 40(Suppl):iv–vii. Jureˆcková, J., and Sen, P.K. 1996. Robust statistical procedures, asymptotics and interrelations. New York: Wiley. Kutner, M.H., Nachtsheim, C.J., Neter, J., and Li, W. 2005. Applied linear statistical models, 5th edn. New York: McGraw-Hill/Irwin. Maronna, R.A., Martin, D.R., and Yohai, V.J. 2006. Robust statistics: Theory and methods. New York: Wiley. Morris, J.B., Symanowicz, P., and Sarangapani, R. 2002. Regional distribution and kinetics of vinyl acetate hydrolysis in the oral cavity of the rat and mouse. Toxicology Letters 126:31–39. National Toxicology Program 2007. NTP toxicity studies of sodium dichromate dihydrate (CAS no. 7789-12-0) administered in drinking water to male and female F344/N rats and B6C3F1 mice and male BALB/c and am3-C57BL/6 mice. Toxicity Report Series vol. 72, 1–G4. U.S.
218
Sankhya B (November 2010) 72:202–218
Department of Health and Human Services, Public Health Service, National Institutes of Health, RTP, NC. Nitcheva, D.K., Piegorsch, W.W., West, R.W., and Kodell, R.L. 2005. Multiplicity-adjusted inferences in risk assessment with non-quantal data. Journal of Biopharmaceutical Statistics 15:17– 31. Peddada, S.D., Lobenhofer, L., Li, L., Afshari, C., Weinberg, C., and Umbach, D. 2003. Gene selection and clustering for time-course and dose-response microarray experiments using order-restricted inference. Bioinformatics 19:834–841. Peddada, S.D., Harris, S., Zajd, J., and Harvey, E. 2005. ORIOGEN: order restricted inference for ordered gene expression data. Bioinformatics 21:3933–3934. Piegorsch, W.W., and West, R.W. 2005. Benchmark analysis: Shopping with proper confidence. Risk Analysis 25:913–920. Pounds, J.G., Haider, J., Chen, D.G., and Mumtaz, M. 2004. Interactive toxicity of simple chemical mixtures of cadmium, mercury, methylmercury and trimethyltin: Model-dependent responses. Environmental Toxicology and Pharmacology 18:101–113. Sand, S., von Rosen, D., Eriksson, P., Fredriksson, A., Viberg, H., Victorin, K., and Filipsson, A.F. 2004. Dose-response modeling and benchmark calculations from spontaneous behavior data on mice neonatally exposed to 2,2 ,4,4 ,5-pentabromodiphenyl ether. Toxicological Sciences 81:491–501. Sanhueza, A.I., Sen, P.K., and Leiva, V. 2009. A robust procedure in nonlinear for repeated measurements. Communications in Statistics—Theory and Methods 38:138–155. Seber, G.A.F., and Wild, C.H. 1989. Nonlinear regression. New York: Wiley. Velarde, G., Ait-aissa, S., Gillet, C., Rogerieux, F., Lambre, C., Vindimian, E., and Porcher, J.M. 1999. Use of transepithelial electrical resistance in the study of pentachlorophenol toxicity. Toxicology in Vitro 13:723–727. Wu, Y., Piegorsch, W.W., West, W., Tang, D., Petkewich, M.O., and Pan, W. 2006. Multiplicityadjusted inferences in risk assessment: Benchmark analysis with continuous response data. Environmental and Ecological Statistics 13:125–141.
