NONPARAMETRIC BENCHMARK ANALYSIS IN RISK ASSESSMENT: A COMPARATIVE STUDY BY SIMULATION AND DATA ANALYSIS

Abstract

We consider the finite sample performance of a new nonparametric method for bioassay and benchmark analysis in risk assessment, which averages isotonic MLEs based on disjoint subgroups of dosages, and whose asymptotic behavior is essentially optimal (Bhattacharya and Lin (2010)). It is compared with three other methods, including the leading kernel-based method, called DNP, due to Dette et al. (2005) and Dette and Scheder (2010). In simulation studies, the present method, termed NAM, outperforms the DNP in the majority of cases considered, although both methods generally do well. In small samples, NAM and DNP both outperform the MLE.

1. Introduction

Given a set of N independent observations for the purpose of estimating some parameter or functional, consider the seemingly naive procedure of (1) dividing the data into r disjoint subsets, (2) estimating the quantity based separately on each subset, and (3) averaging over the r estimates, perhaps after a bias correction. Here r is a nondecreasing function of N, r → ∞ as N → ∞. For parametric estimators such as the MLE based on i.i.d. observations and under standard assumptions, this naive estimator does not improve upon the MLE. But as long as r/N → 0, or, better still, as $r/\sqrt{N}\to 0$ (in the absence of bias correction), it achieves the same asymptotic efficiency as the MLE. As one can check, with a little effort, that in nonparametric curve estimation based on i.i.d. data, a similar comparison prevails between the kernel based estimate and its naive counterpart.

It may then come as a surprise that, in our present context of bioassay or benchmark analysis in risk assessment, the procedure outlined above improves upon, often dramatically, the nonparametric isotonic MLE considered in Bhattacharya and Kong (2007). Indeed, with a proper choice of r, the new estimator has asymptotically optimal rates of mean integrated squared error (MISE) and asymptotic variance as shown in Bhattacharya and Lin (2010) (See Section 2). It may be pointed out that the kernel-based methods do not improve with this procedure of subgrouping and averaging, although some of them may have fine asymptotic properties (See, e.g., Müller and Schmitt (1988), Dette et al. (2005), and Dette and Scheder (2010)). It will be shown in this paper, by extensive simulation and data analysis, that the method proposed here performs quite favorably in comparison with other leading nonparametric methods, including the new method due to Dette et al. (2005) and Dette and Scheder (2010) termed DNP.

It may be noted that although the problem of bioassay is an old one, effective nonparametric estimation here is relatively new. The importance of good estimation of effective dosage is still a very important problem in biology, medicine and pharmacy. In addition, concerns about the impact of pollutants in the environment on various species, including humans, have added a great sense of urgency to the development of effective means for the estimation of benchmarks in risk assessment (See, e.g., Piegorsch and Bailer (2005), Nitcheva et al. (2007), US EPA (1997)).

Consider quantal dose-response experiments in bioassay where one records 1 for the response of a subject to a drug or a chemical agent and 0 for non-response. Let F(x) be the probability of response to a dose level x. The function F(.) is called the dose-response curve, and it is assumed to be monotone increasing. The effective dosage ζ_p or ED_p for a targeted response (probability) p is defined as

$${\zeta }_p={\mathit{ED}}_p=F^{-1}(p),0\le p\le 1;F^{-1}(p):=\inf \{x:F(x)\ge p\}.$$ (1.1)

A standard measure of effectiveness of the chemical, or drug, in bioassay is ED_0.5, often denoted as ED₅₀ (See, e.g., Morgan (1992)). In environmental studies, one is generally interested in finding a level of the pollutant such that, when a subject is exposed to it, the probability of response is a small value of p such as 0.05, 0.10, etc (Piegorsch and Bailer (2005), chapter 4).

Suppose that n_i subjects are given a dosage x_i (i = 1, …, m), where x₁ < x₂ < … < x_m−1 < x_m, with the total number of observations N = Σ_1≤i≤m n_i. One may assume, without loss of generality, that 0 = x₁ < … < x_m = 1. The number of responses observed at dosage x_i is r_i (i = 1, …, m). The likelihood function for the estimation of F(x_i), 1 ≤ i ≤ m, is

$$L(p_1,\dots ,p_m)=\prod _{i=1}^m{p}_i^{r_i}{(1-p_i)}^{n_i-r_i}(0\le p_1\le \dots \le p_m\le 1);[p_i:=F(x_i)].$$ (1.2)

The maximum likelihood estimator (MLE) of (p₁, …, p_m), under the monotonicity constraint, is given in Ayer et al.(1955) by the following PAV, or pool-adjacent-violators algorithm (Also see Barlow et al.(1972), p.73, and Cran (1980)):

$${\stackrel{\sim }{p}}_i=\underset{0\le u<i}{\max }\underset{i\le \upsilon <m}{\min }\frac{{\sum }_{j=u}^{\upsilon }{r}_j}{{\sum }_{j=u}^{\upsilon }{n}_j}(1\le i\le m).$$ (1.3)

One needs to estimate the entire dose-response curve in order to obtain a consistent estimator of ED_p. This was obtained by linear interpolation in the interval (x_i, x_i+1) in Bhattacharya and Kong (2007), where precise conditions for consistency of such an estimator and for proper confidence intervals for ED_p were derived. This method is referred as the B-K method, using the terminology in Dette and Scheder (2010). It is quite different from the kernel based nonparametric methods for quantal bioassay, such as those of Müller and Schmitt (1988), Park and Park (2006), Dette et al. (2005) and Dette and Scheder (2010). A description of these methods may be found in the last two articles. Simulation studies in Dette and Scheder (2010) show that their method, which is an immediate extension of that of Dette et al. (2005), outperforms other kernel based methods, as well as the B-K method, in most cases.

For the DNP method, one first estimates F(x) by a local linear estimator F̂(x) using a symmetric kernel K and a bandwidth λ. The p − th quantile ED_p = F⁻¹ (p) is then estimated as

$${\widehat{\mathit{ED}}}_p={\mathit{\int }}_0^1{\mathit{\int }}_{-\infty }^p\frac{1}{{\mathrm{\lambda }}_d}{K}_d\left(\frac{\widehat{F}(x)-u}{{\mathrm{\lambda }}_d}\right)\mathit{du\; dx}.$$ (1.4)

Here K_d is a symmetric kernel with the same properties as K (e.g, K_d = K). But λ_d goes to zero faster than λ.

We now introduce the PAV-based new nonparametric adaptive method NAM. For simplicity, let n_i = n for all i = 1, ⋯, m, and let the dosages be equidistant.

As in Bhattacharya and Kong (2007), define F̃(x) of F(x) as follows:

$$\stackrel{\sim }{F}(x)=\{\begin{array}{ll}{\stackrel{\sim }{p}}_i& \mathrm{if}x=x_i\\ {\stackrel{\sim }{p}}_i+\frac{{\stackrel{\sim }{p}}_{i+1}-{\stackrel{\sim }{p}}_i}{x_{i+1}-x_i}(x-x_i)& \mathrm{if}{x}_i<x\le x_{i+1}.\end{array}$$

The inverse of F̃ is the estimate of ED_p as given by:

$${\stackrel{\sim }{\mathit{ED}}}_p=\{\begin{array}{ll}{x}_1& \mathrm{if}p\le {\stackrel{\sim }{p}}_1\\ x_i+\frac{p-{\stackrel{\sim }{p}}_i}{{\stackrel{\sim }{p}}_{i+1}-{\stackrel{\sim }{p}}_i}(x_{i+1}-x_i)& \mathrm{if}{\stackrel{\sim }{p}}_i<p\le {\stackrel{\sim }{p}}_{i+1}\mathrm{for\; some\; i}\\ x_m& \mathrm{if}p>{\stackrel{\sim }{p}}_m,\end{array}$$ (1.5)

if p̃_i+1 > p̃_i and, more generally, by F̃⁻¹ (p) = inf {x : F̃ (x) ≥ p}.

From now on, we will assume, for simplicity, that there are m equidistant dosages and the same number n of i.i.d. 0 − 1 valued observations at each dosage

Let p̂_i denote the observed proportion of 1’s at dosage x_i. Divide the observed proportions and dosages into r groups, each with approximately s(n) = m/r dosages, and consider the following application of the PAV algorithm to each of the r groups of levels below:

$$\begin{array}{c}[\mathrm{Group}1]:(x_1,{\widehat{p}}_1),(x_{r+1},{\widehat{p}}_{r+1}),(x_{2r+1},{\widehat{p}}_{2r+1}),\dots ,(x_m,{\widehat{p}}_m);\\ [\mathrm{Group}2]:(x_1,{\widehat{p}}_1),(x_2,{\widehat{p}}_2),(x_{r+2},{\widehat{p}}_{r+2}),(x_{2r+2},{\widehat{p}}_{2r+2}),\dots ,(x_m,{\widehat{p}}_m);\\ \dots \dots \\ [\mathrm{Group}r]:(x_1,{\widehat{p}}_1),(x_r,{\widehat{p}}_r),(x_{r+r},{\widehat{p}}_{r+r}),(x_{2r+r},{\widehat{p}}_{2r+r}),\dots ,(x_m,{\widehat{p}}_m).\end{array}$$ (1.6)

Note that Group 2 through Group r − 1 each has s(n) + 2 levels, while Groups 1 and r each has s(n) + 1 levels. Also, except for the smallest and the largest levels (with proportions p̂₁ and p̂_m), the sets of levels covered by them are disjoint. Together, they comprise the set of all m dosages.

By linear interpolation, each Group j (j = 1, …, r) provides an estimate F̃_j of the dose-response curve F on [0, 1], and an estimate ζ̃_p,j of F⁻¹. Note that while F⁻¹ is defined on [F(0), F(1)], ζ̃_p,j and ζ̃_p below are defined on [F̃(0), F̃(1)]. Compute

$$\stackrel{\sim }{F}:=(1/r)\sum _{1\le j\le r}{\stackrel{\sim }{F}}_j,{\stackrel{\sim }{\zeta }}_p:=(1/r)\sum _{1\le j\le r}\stackrel{\sim }{{\zeta }_{p,j}},$$ (1.7)

and choose the values of r for which the estimated MISEs of F̃ and ζ̃ are the smallest. These are the NAM estimates of F and F⁻¹. We use the bootstrap (Efron and Tibshirani (2003), Hall (1992)) for estimating MISEs. For the asymptotically optimal error rates of F̃ and ζ̃, we refer to Bhattacharya and Lin (2010).

Remark 1.1

In benchmark analysis with environmental risk data, the majority of cases use 4, 5, or at most 6 dosages (Nitcheva et al. (2007)). Even with such small m, the NAM method performs very well, as simulation studies with m = 5 show (See Section 3).

In the next section, we state the asymptotic optimality results which form the theoretical basis for the finite sample comparisons carried out in Section 3 and data analysis in Section 4.

In Section 3.1, we compare the B-K, DNP, NAM methods and the MLE, using data from four parametric models, namely, Logistic, Probit, Beta and Weibull models. Both DNP and NAM outperform the MLE when the sample size n is not large. The NAM is seen to yield narrower confidence intervals than the DNP in the majority of cases. Although these intervals vary from sample to sample, the overall relative comparisons do not significantly change. Due to lack of space, we omit these calculations with alternate samples. Section 3.2 is devoted to the computation of true bias corrected MISEs in a class of models considered by Dette and Scheder (2010), with m = 10, n = 5. This provides a validation for the data adaptive NAM.

In section 4, three data examples are used for the application of the different methods to real data, and the methods are compared by the estimated MISEs.

As suggested in Dette and Scheder (2010), the bandwidth λ for the DNP is estimated using a formula of Rice (1984). Bootstrapping and bias correction for confidence intervals are carried out using the Matlab. We have also successfully implemented the computation of the MLEs for Logistic, Probit, Beta and Weibull models on the Matlab.

2. Asymptotic Optimality of NAM

As stated earlier, the new adaptive method NAM presented in this article achieves asymptotic optimality in the class of nonparametric estimators of the dose-response curve F and the effective dosage F⁻¹, under rather mild design restrictions. The basic intuition underlying this method has been described in the Introduction. We state the precise asymptotic results in this section. Their proofs are provided in Bhattacharya and Lin (2010).

One may recall the definition of the mean integrated squared error (MISE) of a curve estimator such as the NAM estimator F̃:

$$\mathit{MISE}(\stackrel{\sim }{F})={\mathit{\int }}_0^{1}E{\left(\stackrel{\sim }{F}(x)-F(x)\right)}^2\mathit{dx}.$$ (2.1)

We use the notation ≍ to indicate that the ratio of its two sides is bounded away from zero and infinity.

As before, let p̂_i = r_i/n denote the sample proportion of responses to the dose level x_i (i = 1, …, m). For the sake of simplicity, it is assumed that n_i = n for all i, the dosages x_i are equidistant and lie in [0, 1] (after a linear scaling). We will write N = mn for the total number of observations.

Theorem 2.1

Assume that the dose-response curve F is twice differentiable with F″ bounded, and f = F′ is positive on [0, 1]. Then the following hold.

1. MISE (F̃) = O (N^−4/5) as N → ∞, if m ≍ n^1/4, provided r = 1 (or, r = O(1)).

2. If m/n^1/4 → ∞, m = o(n^3/2/(log n)^5/2), then MISE (F̃)=O(N^−4/5), provided r ≍ (m⁴/n)^1/5.

Remark 2.1

It is known that, under given assumptions on F, the optimal order of the MISE in nonparametric curve estimation, is O(N^−4/5) (See, e.g., Györfi et al (2002), chapter 5, and Eubank (1999), pp. 16, 129, 172, 196, 258).

We next turn to the crucial estimation of the effective dosage curve ζ̃ : p → ζ̃_p (See (1.7)), the main object of interest in this article.

Theorem 2.2

Under the same hypothesis on F as in Theorem 2.1, the following hold.

1. If m ≍ n^1/4 then with r = 1, one has MISE(ζ̃) = O (N^–4/5).

2. If m/n^1/4 → ∞, but m/n^2/3 ↛ ∞, then MISE (ζ̃) = O (N^−4/5), with the choice of r satisfying r ≍ (m⁴/n)^1/5.

Finally, for obtaining desired levels of confidence intervals for ζ_p = F⁻¹(p), the following result on the asymptotic distribution of ζ̃_p is important. For its statement, define δ² (p) ∈ [1/2, 1] by

$${\delta }^2(p)\equiv \sum _{i=1}^{m-1}{(x_{i+1}-x_i)}^{-2}\{{(x_{i+1}-x)}^2+{(x-x_i)}^2\}{1}_{I_i}(p)$$ (2.2)

where I_i = [p_i, p_i+1) for 1 ≤ i ≤ m − 2, I_m−1 = [p_m−1, p_m], x = ζ̃_p. Next define, ${\overline{\delta }}^2(p)=1/r{\sum }_{j=1}^r{\delta }_j^2(p)$, where ${\delta }_j^2(p)$ is the quantity as in (2.2), but defined for the j − th subgroup of dosages defined in (1.6), in place of the whole group of m dosages.

Theorem 2.3

Assume the same hypothesis as in Theorem 2.1. In addition, assume m/n^1/4 → ∞. Then the following hold.

1. If m = o(n^3/2/ log^5/2 n), then with r ≍ (m⁴/n)^1/5,

   $$\frac{\sqrt{\mathit{rn}}({\stackrel{\sim }{\zeta }}_p-E{\stackrel{\sim }{\zeta }}_p)}{\overline{\delta }(p)}\stackrel{ℒ}{\to }N\left(0,\frac{p(1-p)}{f^2(p)}\right)$$ (2.3)

   as n → ∞.
2. If m = o(n^2/3 / log log n), then with r ≍ (m⁴/n)^1/5/(log log n)^6/5,

   $$\frac{\sqrt{\mathit{rn}}\left({\stackrel{\sim }{\zeta }}_p-{\zeta }_p\right)}{\overline{\delta }(p)}\stackrel{ℒ}{\to }N\left(0,\frac{p(1-p)}{f^2(p)}\right)$$ (2.4)

   as n → ∞.

Remark 2.2

Note that (2.3) implies that the asymptotic variance of ζ̃_p around Eζ̃_p is O(N^−4/5), the optimal order. This holds for a very broad range of m. When centered at ζ_p, the asymptotic variance of ζ̃_p is only slightly larger, namely, O(N^−4/5(log log N)^6/5). This is typically the case for optimal curve estimation. Comparing the ranges of values of m in (a) and (b), one may find it advisable to make a bias correction when m is large relative to n.

3. Comparison of Confidence Intervals

In actual bioassay experiments, it is rare that both m and n are large. In environmental risk assessment, m is almost never more than 6. It is, therefore, not clear a priori if asymptotics provide a valid guide for finite sample comparisons with small or moderate values of n and/or m. By extensive simulations, we show in this section that, indeed, the theoretical results in Section 2 are good indicators of the performance of NAM in realistic situations.

In this section, 95% confidence intervals are obtained by applying different methods to data simulated from different models as described below.

For data simulated from the Logistic model $F(x)=\frac{1}{1+(\exp (-\alpha -\beta x))}$, we take α = −20, β = 10. The confidence intervals for the MLEs of different quantiles are obtained by fitting the logistic model to data. Denote by (α̂, β̂) the MLE of (α, β). Let υ₁₁, υ₂₂ be the asymptotic variances of α̂ and β̂, respectively, and υ₁₂ the covariance of the two. Also write d_p = log (p/(1 − p)), ED_p = (d_p − α)/β. Then the MLE $\widehat{{\mathit{ED}}_p}$ for ED_p is given as (d_p − α̂)/β̂, and the 95% CI for ED_p is given by using the Fieller method as $\left[{\widehat{\mathit{ED}}}_p-1.96\sqrt{\widehat{w}}/\widehat{\beta },{\widehat{\mathit{ED}}}_p+1.96\sqrt{\widehat{w}}/\widehat{\beta }\right]$, where $\widehat{w}={\upsilon }_{11}+2{\upsilon }_{12}{\widehat{\mathit{ED}}}_p+{\upsilon }_{22}{({\widehat{\mathit{ED}}}_p)}^2$(See Piegorsch and Bailer (2005), pp. 30, 39-40).

We take samples from the Probit model F(x) = Φ((x − μ)/σ) with μ = 0.5 and σ = 0.3.

Simulations from the Beta distribution $F(x)={\int }_0^x({(B(\alpha ,\beta ))}^{-1}{t}^{\alpha -1}{(1-t)}^{\beta -1})\mathit{dt}(0\le x\le 1)$ are carried out with α = 2 and β = 3.

The last set of simulations are from the Weibull model F(x) = 1 − exp(−(x/α)^β) with α = 2 and β = 1.5.

The computations of the MLEs of the parameters of the models are carried out on the Matlab, starting with some properly chosen easily computable consistent estimators as initial values.

Bootstrapping is used for constructing bias-corrected confidence intervals for the DNP and the NAM estimates and the MLEs of the last three models. For the NAM, the value of r in each case is the one among r = 1, 2, 3 for which the bootstrap estimate of MISE (ζ̃_p) is the smallest.

Simulation studies were carried out for computing confidence bands for the cases m = 5, 10, and n = 5, 10, 25,50, with samples from each of the four models-Logistic, Probit, Beta and Weibull, as mentioned above. However, to save space, graphs of confidence bands are provided for the DNP, NAM and MLE, only for data obtained from the Probit and Weibull models, and omitting n = 50. Graphs and tables for all the four models, and for all the four sample sizes, may be found in our website www.math.arizona.edu/~lizhen/about.html. In the graphs, the blue line represents the true curve of quantiles, the green lines represent 95% confidence intervals for the MLE obtained by bootstrapping. The red lines with circles represent the confidence intervals obtained by NAM with r = 2 for m = 5 and r = 3 for m = 10, and the dark dashed lines represent the confidence bounds of DNP estimates. When the bias is not corrected, the confidence bands of the DNP and NAM still include the true curve in almost all cases. Here we include only the bias corrected figures for the DNP and NAM, and, of course, Tables 1-12 are the same whether the bias is corrected or not.

Table 1.

The Length of Confidence Intervals for DNP and NAM(r=2) for m=5,n=5

DNP:	0.1487	0.2320	0.2779	0.2871	0.2944	0.3224	0.3601	0.4471	0.4750
NAM:	0.2331	0.2479	0.2861	0.3009	0.3627	0.3681	0.3795	0.3597	0.2624

Table 12.

The Length of Confidence Intervals for DNP and NAM(r=3) for m=10,n=25

DNP:	0.1969	0.5101	0.7398	0.7254	0.6308	0.6309	0.8897	1.0056	0.8166
NAM:	0.1608	0.3385	0.4566	0.5475	0.5700	0.5768	0.6072	0.7201	0.8915

A number of significant features are revealed by the graphs and tables. First, both the DNP and the NAM tend to perform better than the MLE for small values of n such as n = 5. The MLE also seems to be more bias prone in small samples than the two nonparametric methods. Secondly, for m = 10 and n = 5, DNP and NAM are more or less on par in performance. This makes sense because the DNP was originally devised for large m and small n (such as n = 1) (See Dette at al. (2005)). For larger n (n = 10 or more), the NAM outperforms the DNP in almost all cases. A succinct comparison of the two nonparametric methods, by the lengths of confidence intervals, for all four models (Logistic, Probit, Beta and Weibull) is provided in Table 13. Here an entry (i, j) means that for i dose levels the NAM has shorter intervals than the DNP, and for j dosages the DNP has shorter intervals, for the particular model and particular values of (m, n) as indicated.

Table 13.

Comparison by Number of Shorter Confidence Intervals

		Model
m	n	Logistic	Probit	Beta	Weibull
5	5	(5,4)	(2,7)	(5,4)	(6,3)
5	10	(6,3)	(9,0)	(9,0)	(8,1)
5	25	(9,0)	(9,0)	(9,0)	(9,0)
5	50	(9,0)	(9,0)	(9,0)	(9,0)
10	5	(4,5)	(2,7)	(5,4)	(6,3)
10	10	(7,2)	(8,1)	(7,2)	(7,2)
10	25	(8,1)	(9,0)	(7,2)	(8,1)
10	50	(9,0)	(8,1)	(9,0)	(9,0)

Although we do not show the confidence intervals for the B-K method (r = 1) in order to avoid cluttering up of the graphs, our computations show that they are wider than those of the DNP and NAM in most cases.

An entry (i, j) in Table 13 indicates that for the estimation of i effective dosages (for the particular model, and particular combination of (m, n) for the cell), the NAM provides shorter confidence intervals than the DNP, and for j of them the DNP has shorter confidence intervals.

3.1. True Bias-Corrected MISEs in A Class of Models

In this short subsection, we compute bias-corrected MISEs or integrated variance, for the DNP and NAM estimates for a class of models considered by Dette and Scheder (2010) with m = 10, n = 5. These computations in Table 14 show that NAM corresponds to r = 3 in the population, as it does in the samples. The models are listed as follows:

Table 14.

Integrated variance with m=10, n=5.

	B-K	r=2	r=3	DNP	NAM
Linear Model:	0.0140	0.0109	0.0051	0.0088	0.0051(r=3)
Normal1 Model:	0.0192	0.0150	0.0077	0.0130	0.0077(r=3)
Normal2 Model	0.0028	0.0021	0.0010	0.0014	0.0010(r=3)
Weibull Model:	0.0263	0.0210	0.0127	0.0210	0.0127(r=3)
Cauchy Model:	0.0051	0.0045	0.0027	0.0030	0.0027(r=3)
Beta Model:	0.0076	0.0055	0.0028	0.0039	0.0028(r=3)
Logistic Model:	0.0036	0.0025	0.0015	0.0017	0.0015(r=3)

1. Linear model: $F(x)=\{\begin{array}{lll}2x& \mathrm{if}& 0\le x\le 0.3\\ 0.4x+0.48& \mathrm{if}& 0.3\le x\le 0.8\\ x& \mathrm{if}& 0.8\le x\le 1\end{array}$

2. Normal Model 1 : F(x) = Φ ((x − μ)/σ), with μ = 0.5, σ = 0.5.

3. Normal Model 2: F(x) = Φ ((x − μ)/σ), with μ = 0.5, σ = 0.1.

4. Weibull Model: F(x) = 1 − exp(−x^γ), with γ = 0.52876.

5. Cauchy Model: F(x) = 1/2 + 1/π arctan ((x − μ)/σ), with μ = 0.15, σ = 0.05.

6. Beta Model: $F\prime (x)=\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}{(1-x)}^{\beta -1}{x}^{\alpha -1}$, with α = 2, β = 3.

7. Logistic Model: F(x) = 1/(1 + exp (5 − 15x)).

4. Data Examples

4.1. Data Example I: Remission Cancer Data

In this subsection, in a real data example, estimates of ED_0.5 (also denoted as ED₅₀) are obtained by fitting parametric models and also using nonparametric methods such as the B-K, NAM and DNP. The source of the data is Lee (1974). This remission cancer data set has 27 observations (binary 0-1 variable with 1 for remission of cancer). There are 14 ‘dose levels’ between 8 and 38 for the explanatory variable labeling index (LI). This labeling index measures proliferative activity of cells after a patient receives an injection of tritiated thymidine, representing the percentage of cells that are labeled. The number of observations are 1 for 6 levels, 2 for 3 levels and 3 for 5 levels. We are interested in estimating the quantile curve and, in particular, the effective dosage ED_0.5.

Lee (1974) fitted the logistic model to the data, Dette and Scheder (2010) fitted Cauchy and Weibull models. The maximum likelihood estimators for ED_0.5 are 26.05, 23.65 and 26.09 respectively under the three models. The estimated curves are given by $\widehat{F}(x)=\frac{1}{1+\mathit{exp}(-3.777+0.145x)}$ for the Logistic model, F̂(x) = 1 − exp (− (0.00028x)^2.3954) for the Weibull model $\widehat{F}(x)=\frac{1}{2}+\frac{1}{\pi }\arctan \left(\frac{x-23.6474}{6.2391}\right)$ for the Cauchy model.

For the nonparametric estimates of ED_0.5, Dette and Scheder (2010) computed the DNP estimate as 20.35229, and the B-K estimate as 8. After checking repeatedly, our calculations show that the B-K estimate is actually 17.8462. Also, for an estimate of ED_0.5 using the NAM method proposed in this paper, we obtained 21.2 with r = 2 and 25.1667 with r = 3. Since the number of dosages is much larger here than the numbers of responses at the individual level, we would prefer r = 3 for the NAM. Note that the NAM estimate is then closer to all the MLE’s than the DNP estimate. The estimates of ED_0.5 are given in Table 15 for the different methods. The estimates of ED_p with p = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0,8, 0.9, 1 for both parametric and nonparametric methods are illustrated in Figure 13. Asymptotic comparisons are not feasible here using bootstrapping, since the numbers of responses n_i are very small. This data example is a follow up of the calculations carried out in Dette and Scheder (2010), and it is only meant to illustrate that the NAM may be applied even with such small sample sizes.

Table 15.

The estimates of ED₅₀ using different methods.

Methods:	B-K	r=2	r=3	DNP	Logistic	Cauchy	Weibull
${\widehat{ED}}_{50}$:	17.8462	21.2	25.1667	20.35229	26.5	23.65	26.09

Figure 13.

4.2. Data Example II: Data on Insect Mortality

The Logistic model together with the nonparametric methods B-K, NAM (r=2, 3) and DNP, are used to fit the data in this example which records kills of Tribolium confusum following 5-hour exposures to known concentrations of Carbon Disulphide. The data are due to Strand and may be found in Bliss (1935). This data set has 8 dosages from 49.06 mg/litre to 76.04 mg/litre, with around 30 insects for each dosage.

The estimates of ED_p for p = 0.1, 0.2, …, 0.9 are obtained and the estimated quantile curves are illustrated in Figure 14.

Figure 14.

The MISE is calculated for the Logistic model, Probit model, Weibull model and for each of the nonparametric methods, through bootstrapping. The NAM with r = 3 yields the smallest estimated MISE 0.9130 compared to 1.0413 of the Probit model and 1.9374 of DNP as shown in Table 16.

Table 16.

Estimated MISEs of the estimates of the B-K, DNP, NAM and MLEs.

Methods:	B-K	r=2	r=3	DNP	NAM (r=3)	Logistic	Probit	Weibull
MISE:	2.2090	1.3258	0.9130	1.9374	0.9130	1.2355	1.0413	1.2077

4.3. Data Example III: the Toxicity of Rotenone

In this example using data from Martin (1942), we are investigating the relation between the dose of rotenone (mg/l) and the response of exposed insects. The percentage of insects dead or seriously affected out of about 50 insects are recorded for each of the 5 dosages from 2.6mg/l to 10.2mg/l. The parametric estimates are obtained by fitting the Logistic, Probit and Weibull to data together with the nonparametric estimates of the B-K, NAM and DNP. The results are shown in Figure 15.

Figure 15.

Comparisons are carried out among the different methods by their estimated MISEs through bootstrapping as shown in Table 17. The Weibull models yields the smalled estimated MISE 0.1626 followed by that of the NAM estimates which is 0.1888.

Table 17.

Estimated MISEs of the estimates of the B-K, DNP, NAM and MLEs.

Methods:	B-K	r=2	DNP	NAM (r=2)	Logistic	Probit	Weibull
MISE:	0.4070	0.1888	0.3512	0.1888	0.2066	0.1908	0.1626

5. Concluding Remark

Simulation studies and data analysis carried out for this article show that the new method NAM for risk assessment in bioassay and environmental studies perform remarkably well for small, moderate and large sample sizes that are generally used in practice. This also demonstrates that the desirable asymptotic properties of the method, derived in Bhattacharya and Lin (2010), are good guides for its finite sample behavior.

Figure 1. [Probit].

Figure 2. [Probit].

Figure 3. [Probit].

Figure 4. [Probit].

Figure 5. [Probit].

Figure 6. [Probit].

Figure 7. [Weibull].

Figure 8. [Weibull].

Figure 9. [Weibull].

Figure 10. [Weibull].

Figure 11. [Weibull].

Figure 12. [Weibull].

Table 2.

The Length of Confidence Intervals for DNP and NAM(r=2) for m=5,n=10

DNP:	0.2159	0.3341	0.4126	0.4264	0.3729	0.3423	0.3422	0.4062	0.2959
NAM:	0.2144	0.2835	0.3498	0.3716	0.3611	0.3404	0.3265	0.4057	0.2604

Table 3.

The Length of Confidence Intervals for DNP and NAM(r=2) for m=5,n=25

DNP:	0.2382	0.2785	0.2533	0.2119	0.2286	0.2034	0.2014	0.2269	0.2443
NAM:	0.1575	0.2329	0.2091	0.1857	0.1884	0.1817	0.1797	0.1939	0.1954

Table 5.

The Length of Confidence Intervals for DNP and NAM(r=3) for m=10,n=10

DNP:	0.1916	0.2217	0.2194	0.2188	0.2293	0.2444	0.2735	0.2962	0.3149
NAM:	0.1534	0.1904	0.2139	0.2072	0.2229	0.2519	0.2580	0.2700	0.2671

Table 6.

The Length of Confidence Intervals for DNP and NAM(r=3) for m=10,n=25

DNP:	0.1380	0.2559	0.2370	0.2489	0.2447	0.1917	0.1216	0.1506	0.2003
NAM:	0.1227	0.1853	0.2108	0.2197	0.1695	0.1396	0.1215	0.1169	0.1274

Table 7.

The Length of Confidence Intervals for DNP and NAM(r=2) for m=5,n=5

DNP:	0.6969	1.0458	1.1129	1.1841	1.2247	1.1968	1.1445	1.0868	0.9663
NAM:	0.9469	1.0721	1.1421	1.1569	1.1716	1.1863	1.1274	0.9948	0.6963

Table 8.

The Length of Confidence Intervals for DNP and NAM(r=2) for m=5,n=10

DNP:	0.2949	0.5102	0.6378	0.7579	0.8538	1.0105	1.0884	1.2061	1.2235
NAM:	0.1982	0.4228	0.5281	0.6325	0.7405	0.9112	1.0303	1.1790	1.2932

Table 9.

The Length of Confidence Intervals for DNP and NAM(r=2) for m=5,n=25

DNP:	0.1768	0.4657	0.5843	0.6694	0.8776	0.8382	0.7878	0.8514	1.3161
NAM:	0.1219	0.3443	0.4422	0.5409	0.6491	0.6968	0.7236	0.8510	0.8382

Table 10.

The Length of Confidence Intervals for DNP and NAM(r=3) for m=10,n=5

DNP:	0.3918	0.7013	0.8197	0.8952	0.9395	0.9487	0.9247	0.8603	0.8180
NAM:	0.6207	0.6573	0.8481	0.8931	0.9406	0.9233	0.8597	0.8192	0.7286

Table 11.

The Length of Confidence Intervals for DNP and NAM(r=3) for m=10,n=10

DNP:	0.2442	0.5456	0.7215	0.9547	1.1352	1.2921	1.3359	1.2411	1.1389
NAM:	0.2163	0.4406	0.5700	0.7428	0.9899	1.1956	1.2320	1.3621	1.9545

Table 4.

The Length of Confidence Intervals for DNP and NAM(r=3) for m=10,n=5

DNP:	0.1238	0.1962	0.2452	0.3028	0.3492	0.3638	0.3565	0.3171	0.2571
NAM:	0.2030	0.2497	0.3244	0.3479	0.3625	0.3627	0.3691	0.3409	0.2962