Supplementary materials for this article are available at 10.1007/s13253-016-0268-z.
A Reference Population-Based Conformance Proportion Hsin-I Lee, Hungyen Chen, Hirohisa Kishino, and Chen-Tuo Liao Conformance proportion is a useful index in agricultural, biological, and environmental applications, which is defined as the proportion that a characteristic of interest falls in an acceptance region of a specification. In practice, however, the acceptance region may not be available when employing the regular conformance proportion. In this article, we propose a new index in which the acceptance region is obtained from a reference population. Furthermore, we develop approaches to constructing confidence limits for the proposed conformance proportion using the concept of a fiducial generalized pivotal quantity. We first consider the simple situation that the characteristic is assumed to be a univariate normal random variable. Then we extend it to the one-way random effects model. The proposed method is evaluated through simulation study and real data analysis. It is shown that our proposed new index and its statistical inference are easy to implement and reasonably satisfactory for most applications. Supplementary materials accompanying this paper appear on-line. Key Words: Agricultural management; Environmental assessment; Generalized pivotal quantity; Tolerance interval; Variance component.
1. INTRODUCTION Conformance proportion π ∗ is the proportion that a characteristic Y falls within an acceptance region of a specification. That is, π ∗ = [L ≤ Y ≤ U ], where L and U are known values; and L may be negative infinity and U may be positive infinity. Considering Y being a univariate normal distribution N (μ, σ 2 ), we have
U −μ π = σ ∗
−
L −μ , σ
where (·) denotes the cumulative probability density function of the standard normal distribution. The problem of computing confidence limits for π ∗ has been studied by Wang and Lam (1996), Iyer and Patterson (2002), Perakis and Xekalaki (2002, 2015), and Patterson
Hsin-I Lee and Chen-Tuo Liao (B) Division of Biometry, Institute of Agronomy, National Taiwan University, Taipei, Taiwan (E-mail:
[email protected]). Hirohisa Kishino and Hungyen Chen Graduate School of Agricultural and Life Sciences, The University of Tokyo, Bunky¯o, Japan. © 2016 International Biometric Society Journal of Agricultural, Biological, and Environmental Statistics, Volume 21, Number 4, Pages 684–697 DOI: 10.1007/s13253-016-0268-z
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et al. (2004). Among which, Patterson et al. (2004) developed fiducial generalized confidence intervals (FGCIs) for π ∗ based on fiducial generalized pivotal quantities (FGPQs) that play the key role in the fiducial generalized inference (FGI). The FGI has been thoroughly investigated by Hannig and his coauthors, the interested reader is referred to Hannig et al. (2006), Hannig (2009, 2013), and Hannig et al. (2016). In specific, Hannig et al. (2006) proved that the FGCI for π ∗ asymptotically has the correct frequentist coverage and Patterson et al. (2004) showed that it sufficiently maintains empirical coverage probabilities close to the nominal level with finite sample sizes by a simulation study. Patterson et al. (2004) further investigated the FGCI for the regular conformance proportion under the one-way random effects model. Similarly, they conducted detailed simulation studies to show that the FGCI has satisfactory performance with finite sample sizes. Recently, Lee and Liao (2012) considered the regular conformance proportion under the normal variance components model. They developed a confidence interval construction based on the modified large sample method presented by Graybill and Wang (1980) and compared it with the FGPQ-based method. Also, Lee and Liao (2014) provided another Student’s t statistic-type method for constructing confidence limits for the regular conformance proportion under balanced and unbalanced normal random effects models. They also compared their method with the FGPQ-based method and a Bayesian method by a simulation study. Overall, the FGPQ-based method is found to usually perform equally well or better than the other methods. The most attractive merit of the FGPQ-based method is its appealing small sample size properties that have been verified through a variety of applications and simulation studies (Weerahandi 1995, 2004). When employing the regular conformance proportion π ∗ , acceptance limits (L and U ) of a specification are assumed to be fixed and known. However, L and U are often undefined or not available in practical use. We thus propose a new conformance proportion whose acceptance region of a specification is obtained from a reference population. Let YT denote the characteristic of interest for a test population and YR denote that for its reference population. Both YT and YR are considered as continuous random variables. For a given probability β, there must exist a region [δ L , δU ] such that Pr (δ L ≤ YR ≤ δU ) = β. That is, δ L and δU are two quantiles of the reference population and functions of parameters involving the reference population. Then we define π = Pr (δ L ≤ YT ≤ δU ) as the reference populationbased conformance proportion for YT with respect to the β-content of YR , abbreviated as the β-content RPB conformance proportion. Thus, the β-content RPB conformance proportion involves all the parameters in both distributions of the reference and test populations. Figure 1 displays a β-content RPB conformance proportion. The β-content RPB conformance proportion is conceptually similar to the proportion of similar response, presented in Rom and Hwang (1996), for assessing the relative efficacy of two treatments. It is also similar to the areas between curves for the analysis of malnutrition (Bohning et al. 1992). If the test population of Y A is capable or qualified, then a high β-content RPB conformance proportion is expected. Thus, one is often interested in the hypothesis testing: H0 : π ≤ π0 versus H1 : π > π0 , where π0 is a specified threshold value. A (1 − α) lower confidence limit for π , denoted by πˆ L , is desired for performing the hypothesis testing. Under the significance level of α, the H0 is rejected if πˆ L > π0 . In this study, we develop the FGCI for π under two situations. Section 2 considers that both YT and YR are distributed as univariate
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Test population (YT) Reference population (YR)
δU
δL
Figure 1. The area with slant lines represents the β-content RPB conformance proportion.
normal distributions. Section 3 considers that YT and YR are fitted by two one-way random effects models. Two real data examples are given to illustrate potential applications of the proposed method. Some concluding remarks are provided in Sect. 4.
2. UNIVARIATE NORMAL DISTRIBUTION We first discuss the simple case that both populations are univariate normal distributions. It is assumed that YT ∼ N (μT , σT2 ) and YR ∼ N (μR , σR2 ), where “∼” is read as “is distributed as.” For a given β, we have Pr (δ L ≤ YR ≤ δU ) = Pr
δ L − μR δU − μR ≤Z≤ σR σR
= β,
where Z ∼ N (0, 1). We consider the symmetric limits around μR that δ L = μR −z (1+β)/2 σR and δU = μR + z (1+β)/2 σR . Here z q denotes the 100(q)th quantile of the standard normal distribution, so δ L and δU are actually respective the 100((1−β)/2)th and 100((1+β)/2)th quantiles of N (μR , σ R2 ). Subsequently, we have the β-content RPB conformance proportion as π = Pr (δ L ≤ YT ≤ δU ) = Pr μR − z (1+β)/2 σR ≤ YT ≤ μR + z (1+β)/2 σR μR − z (1+β)/2 σR − μT μR + z (1+β)/2 σR − μT − . = σT σT Note that the maximal value of π is equal to β if σT = σR and 1 if μT = μR . Obviously, the β-content RPB conformance proportion is a function of the parameters μT , σT , μR , and σR . There may be no satisfactory conventional interval estimation method for such a complex parametric function, so we simply apply the concept of a FGPQ to obtain the required lower
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confidence limit πˆ L . As mentioned earlier, the FGPQ-based method is particularly beneficial for analyzing small samples, probably due to the fact that it relies on explicit probability statements instead of large sample theory. We review the concept of a FGPQ below. Let y be the realized value of the observable random vector Y and ξ be the vector of model parameters. Also, let θ be a function of ξ for which a confidence limit is sought. A function Rθ = Rθ (Y ; y, ξ ) of Y , y, and ξ is called a FGPQ for θ if it satisfies the following two conditions: (FGPQ1) Rθ has a probability distribution that is free of ξ . (FGPQ2) For every allowable y, Rθ ( y; y, ξ ) = θ . Let Rθ,α denote the 100(α)th quantile of the distribution of Rθ . Since Pr (Rθ,α ≤ Rθ ) = 1 − α and Rθ ( y; y, ξ ) = θ , Rθ,α can be used as the (1 − α) lower confidence limit for θ . The required quantile can be conveniently obtained using Monte-Carlo sampling algorithm. 2.1. FGPQ-BASED M ETHOD Suppose YT =
nT
YT,i /n T , ST2 =
i=1
nT (YT,i − Y T )2 /(n T − 1) i=1
and YR =
nR i=1
YR,i /n R ,
S 2R
nR = (YR,i − Y R )2 /(n R − 1), i=1
where n T and n R are the sample sizes. Also, let WT =
Y T − μT ∼ tn T −1 , VT = (n T − 1)ST2 /σT2 ∼ χn2T −1 √ ST / n T
and WR =
Y R − μR ∼ tn R −1 , VR = (n R − 1)SR2 /σR2 ∼ χn2R −1 , √ SR / n R
where t f and χ 2f separately denote the Student’s t and the chi-square distributions with f degrees of freedom. Define √ RμT = y T − WT sT / n T , RσT =
(n T − 1)sT2 /VT ,
and √ RμR = y R − W R s R / n R , RσR =
(n R − 1)s 2R /VR ,
where y T , sT2 , y R , and sR2 are the observed values of Y T , ST2 , Y R , and SR2 , respectively. Clearly, RμT and RσT are free of unknown parameters, hence they both satisfy condition
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(FGPQ1). Also, RμT = μT and RσT = σT if Y T and ST2 are replaced by their observed values y T and sT2 in RμT and RσT , hence they both fulfill condition (FGPQ2). Thus, RμT and RσT are separately FGPQs for μT and σT . Similarly, RμR and RσR can be verified to be FGPQs for μR and σR , respectively. By the invariance property of FGPQs like that of maximum-likelihood estimators, a FGPQ for the β-content RPB conformance proportion is immediately obtained as Rπ =
RμR + z (1+β)/2 RσR − RμT RσT
−
RμR − z (1+β)/2 RσR − RμT RσT
.
(1)
The required distribution of Rπ can be generated by the following Monte-Carlo algorithm: Step 1: Choose a large simulation sample size G, say G = 10, 000. For g equal to 1 through G, carry out the following two steps. Step 2: Generate mutually independent random variates WT,g , VT,g , WR,g , and VR,g from tn T −1 , χn2T −1 , tn R −1 , and χn2R −1 , respectively. Step 3: For a given set of observed values (y T , sT2 , y R , s 2R ), calculate Rπ,g using Eq. (1). The required (1 − α) lower confidence limit for π is obtained as the 100(α)th quantile of the collected values Rπ,1 , Rπ,2 , . . . , Rπ,G .
2.2. S IMULATION S TUDY A statistical simulation study is conducted to evaluate the performance of the proposed FGPQ-based method. The setting of this simulation study is based on the following three scenarios. First, we fix YR ∼ N (0, 1) and β = 0.95. Then, under Scenario I, YT ∼ N (1, 0.81); Scenario II, YT ∼ N (1, 1); and Scenario III, YT ∼ N (1, 1.21). Moreover, we set α = 0.1, n T = 10, 20, 50, 100 and n R = 10, 20, 50, 100. For each simulation setting combination, we generate mutually independent variates Z T ∼ N (0, 1), VT ∼ χn2T −1 , Z R ∼ N (0, 1), √ and VR ∼ χn2R −1 . A set of realized statistics is thus yielded by y T = μT + Z T σT / n T , √ sT2 = VT σT2 /(n T − 1), y R = μ R + Z R σ R / n R , and sR2 = V R σR2 /(n R − 1). We obtain the 0.9 lower generalized confidence limit for π using the Monte-Carlo algorithm. The procedures are repeated 10,000 times, i.e., 10,000 sets of the realized data are generated for each simulation setting combination. The simulated coverage probability is computed as the proportion that the resulting 0.9 lower generalized confidence limits are less than the true value of π . The simulation results are displayed in Table 1. From Table 1, the FGPQ-based method for estimating the generalized confidence limit for π appears to generally maintain the simulated coverage probabilities close to the nominal level of 0.9. Few slightly conservative confidence limits happen when n A or n B is not large enough (n A = 10; n B = 10), meaning that their simulated coverage probabilities are slightly greater than the nominal level. This supports that the proposed testing procedure for the β-content RPB conformance proportion satisfies the basic requirement of a statistical hypothesis testing that the empirical type I error rate can be controlled at the nominal significance level.
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A Reference Population- Based Conform ance Proportion Table 1. Simulated coverage probabilities for the 0.9 lower generalized confidence limit for π . nT
nR
Scenario I
Scenario II
Scenario III
10 10 10 10 20 20 20 20 50 50 50 50 100 100 100 100
10 20 50 100 10 20 50 100 10 20 50 100 10 20 50 100
0.9366 0.9223 0.9203 0.9172 0.9288 0.9192 0.9084 0.9090 0.9241 0.9126 0.9080 0.9062 0.9233 0.9150 0.9083 0.9011
0.9409 0.9259 0.9238 0.9228 0.9315 0.9221 0.9124 0.9120 0.9266 0.9149 0.9096 0.9089 0.9267 0.9168 0.9099 0.9037
0.9429 0.9293 0.9277 0.9257 0.9352 0.9244 0.9154 0.9152 0.9287 0.9180 0.9122 0.9108 0.9293 0.9195 0.9112 0.9070
Table 2. The estimates of the air quality characteristics in Example 1.
SO2 CO O3 PM10
yT
yR
sT2
2 sR
πˆ
πˆ L
4.12 6.07 45.48 38.64
2.50 6.17 33.29 27.86
0.0496 0.0790 2.2690 1.8879
0.0129 0.0386 1.2375 1.3144
1.75 × 10−10 0.8026 1.51 × 10−11 2.64 × 10−10
3.96 × 10−12 0.7381 2.11 × 10−13 5.47 × 10−12
2.3. I LLUSTRATIVE E XAMPLE Example 1. To monitor or assess the air quality of a city, its government usually builds monitoring stations to measure important quality characteristics. Chen (2016) evaluated the air quality of the capital of Taiwan, Taipei city, in comparison with Hualien County that is the least polluted area in Taiwan. A dataset consisting of 100 readings of four air quality characteristics, sulfur dioxide (SO2 ), carbon monoxide (CO), ozone (O3 ), and suspended particulates (PM10 ), for Taipei city and Hualien County was provided in the dissertation. We would adopt the data to illustrate the application of the RPB conformance proportion. Thus, we have sample sizes n T = 100 and n R = 100. The sample means, sample variances, together with the resulting estimates of the 0.95-content RPB conformance proportion (πˆ ) and 0.9 lower confidence limit for π (πˆ L ) are displayed in Table 2. Also, the estimated density curves for individual characteristics are shown in Fig. 2. From Table 2, the estimated 0.95content RPB conformance proportions and the lower confidence limits for SO2 , O3 , and PM10 are found to be extremely small indicating that there is a significant difference in the pollution level between these two areas.
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SO2
Taipei Haulian
0.0
0.0
0.5
1.0
1.0
2.0
1.5
3.0
Taipei Haulian
2.0
CO
2.0 2.5 3.0 3.5 4.0 4.5 5.0
5.0
5.5
6.0
6.5
7.0
PM10 Taipei Haulian
0.00
0.00
0.10
0.10
0.20
0.20
0.30
Taipei Haulian
0.30
O3
30
35
40
45
50
25
30
35
40
45
Figure 2. The estimated density curves for the air quality characteristics in Example 1.
3. ONE-WAY RANDOM EFFECTS MODEL Consider YT and YR being fitted by the following models: YT,jk = μT + F j + εT,jk ,
(2)
YR,jk = μR + F j + εR,jk
(3)
for j = 1, . . . , J , k = 1, . . . , n T , and k = 1, . . . , n R . YT,jk and YR,jk , respectively, denote the responses of the test and reference populations at random effect F j and replications k and k ; μT and μR stand for their expected values; F j ∼ N (0, σ F2 ); εT,jk ∼ N (0, σe2T ); and εR,jk ∼ N (0, σe2R ). Also, F j , εT,jk , and εR,jk are assumed to be mutually independent for any j, k, and k . T T Write Y T = (1/J n T ) Jj=1 nk=1 YT,jk , Y T,j = (1/n T ) nk=1 YT,jk , S12 = n T Jj=1 J n T 2 2 2 (Y T,j − Y T )2 and S22 = j=1 k=1 (Y T,jk − Y T,j ) . Likewise, S3 and S4 , respectively, 2 2 denote the mean squares corresponding to S1 and S2 under Eq. (3). The degrees of freedom and expected values of the mean squares are displayed in Table 3, where Si2 ’s are mutually
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Table 3. The corresponding degrees of freedom (DF) and expected mean squares (EMS) for Eqs. (2) and (3). Source YT
DF
Fj Error
YR
Fj Error
EMS
f1 = J − 1
σ12 = σe2T + n T σ F2
f 2 = J (n T − 1)
σ22 = σe2T
f3 = J − 1
σ32 = σe2R + n R σ F2
f 4 = J (n R − 1)
σ42 = σe2R
independent with Vi = f i Si2 /σi2 being chi-square random variables with f i degrees of freedom for i = 1, 2, 3, 4. Let σT2 and σR2 denote the variances of YT,jk and YR,jk , respectively. Then, we have σT2 = σ F2 + σe2T =
1 1 1 1 σ22 + σ12 + 1 − σ2 − σ2 nT + nR nT + nR nT + nR 3 nT + nR 4
and σR2 = σ F2 + σe2R =
1 1 1 1 σ42 . σ12 − σ22 + σ32 + 1 − nT + nR nT + nR nT + nR nT + nR
Moreover, the sample mean vector Y T , Y R is distributed as a bivariate normal distribution with mean vector [μT , μR ] and covariance matrix τT2 τTR = , τTR τR2 where 1 2 1 2 σF + σ J J n T eT 1 1 1 1 1 2 σ + σ2 − σ 2, σ2 + = − J (n T + n R ) 1 J nT J (n T + n R ) 2 J (n T + n R ) 3 J (n T + n R ) 4 1 1 2 τR2 = σ F2 + σ J J n R eR 1 1 1 1 1 σ 2, = − σ12 − σ22 + σ32 + J (n T + n R ) J (n T + n R ) J (n T + n R ) J nR J (n T + n R ) 4 τT2 =
and τTR = =
1 2 σ J F 1 (σ 2 − σ22 + σ32 − σ42 ). J (n T + n R ) 1
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3.1. FGPQ-BASED M ETHOD 4 4 2 2 2 For easy presentation, rewrite σT2 = R,i σi , where i=1 cT,i σi and σR = i=1 c 4 2 2 cT,i ’s and cR,i ’s are the corresponding coefficients. Also, rewrite τT = i=1 h T,i σi , 4 4 2 2 2 τR = i=1 h R,i σi , and τTR = i=1 h TR,i σi , where h T,i ’s, h R,i ’s, and h TR,i ’s are also the corresponding coefficients. Let Z 1 and Z 2 be mutually independent standard normal
variables. Following the definition of the bivariate normal distribution for Y T , Y R , we have
Y T = μT + Z 1 τT and Y R = μ R + φ Z 1 + 1 − φ 2 Z 2 τR , where φ = τTR /τT τR . Define the FGPQs below
RτT
4 4 2 h f s h R,i f i si2 T,i i i = max 0, , RτR = max 0, , Vi Vi i=1
i=1
RτTR =
4
h TR,i f i si2
i=1
Vi
, Rφ =
RτTR , RτT RτR
where si2 are the observed values of Si2 for i = 1, 2, 3, 4. Furthermore, define
RμT = y T − Z 1 RτT , RμR = y R − Rφ Z 1 + 1 − Rφ2 Z 2 RτR , where y T and y R are the observed values of Y T and Y R , respectively. It can be verified that RτT , RτR , RτTR , Rφ , RμT , and RμR all satisfy the two conditions of a FGPQ, hence they are FGPQs for the corresponding parameters. Similarly, FGPQs for σT and σ B are given as
RσT
4 4 2 2 c f s c f s T,i i R,i i i i = max 0, and RσR = max 0, . Vi Vi i=1
i=1
Consequently, a FGPQ for the β-content RPB conformance proportion can be immediately obtained by the same formula as Eq. (1). Generalized confidence limits using the FGPQ for the β-content RPB conformance proportion can be similarly obtained by the Monte-Carlo algorithm presented in Sect. 2.1. 3.2. S IMULATION S TUDY We also conduct a simulation study to evaluate the proposed method under the one-way random effects models as Eqs. (2) and (3). The parameters for the simulation study are fixed as follows: μT = 1; μR = 0; σe2R = 1; σe2T = 0.81, 1, 1.21 and σ F2 = 1/9, 3/7, 1, 7/3, 9, leading to that the intraclass correlation ρ = σ F2 /(σ F2 + σe2R ) = 0.1, 0.3, 0.5, 0.7, 0.9. Also, β = 0.95, α = 0.1, J = 5, 10, 25, 50, 100, 200, and n T = n R = 5, 10, 25, 50, 100. For a given combination of values for J , n T , n R , μT , μR , σe2T , σe2R , and σ F2 , we can have the
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corresponding values of σ12 , σ22 , σ32 , and σ42 through Table 3, hence those of τT , τR , τTR , and φ. Then, we generate mutually independent variates Z 1 ∼ N (0, 1), Z 2 ∼ N (0, 1), and Vi ∼ χ 2fi for i = 1, 2, 3, 4. A set of realized statistics is yielded by y T = μT +
Z 1 τT , y R = μ R + φ Z 1 + 1 − φ 2 Z 2 τR , and si2 = Vi σi2 / f i . We obtain the 0.9 lower generalized confidence limit for π using the Monte-Carlo algorithm. The procedures are repeated 10,000 times. The simulated coverage probability is defined as the proportion that πˆ L ,i is less than the true value of π and the simulated expected length is computed 10000 by i=1 (1 − πˆ L ,i )/10000, where πˆ L ,i denotes the ith resulting 0.9 lower generalized confidence limit for π . The simulation results under σe2T = 0.81, 1, 1.21 are quite similar, so we only report those of σe2T = 0.81 in Table 4, the remaining results are available from the authors upon request. The FGPQ-based method can generally control the simulated coverage probabilities close to the nominal value as J is large enough (J ≥ 25), but it becomes more liberal as ρ increases. The simulated expected length gets shorter as the sample size increases, meaning that the estimation improves its precision with the increase of sample size. Under a fixed total sample size J × n T or J × n R , we would recommend a large J rather than n T or n R . It is straightforward to verify that the confidence limit for π asymptotically has the correct frequentist coverage by the results presented in Hannig et al. (2006). The simulation setting of the larger sample size (J = 100, 200) is used to judge how the asymptotic takes effect. As shown in Table 4, the improvement on coverage probability becomes very limited when J > 50. 3.3. I LLUSTRATIVE E XAMPLE Example 2. A safety analysis is usually required to assess whether the level of a compositional component on a genetically modified (GM) crop differs meaningfully from the normal level of its traditional comparator that has a history of safe consumption. The βcontent RPB conformance proportion might be applied to such a safety analysis. A dataset of a composition analysis for GM oilseed rape and its conventional counterpart was provided in Yang (2007). A total of 56 components were investigated for this composition analysis. The field trial was conducted at six locations for two years and both the GM crop and its comparator had four replications at each location. For enhancing the reliability of data analysis, we combine the levels of year and location into 12 distinct levels of environment and treat them as random effects. Let YT,jk and YR,jk , respectively, denote the responses of the GM crop and its comparator at the jth level of environment and replications k and k , respectively. The data under consideration can be fitted by the one-way random effects models as Eqs. (2) and (3). Here J = 12, n T = n R = 4, F j ’s stand for the effects of environment, and εT,jk ’s and εR,jk ’s denote the error terms. We choose one of the 56 components, Phytic acid, as our illustration. The data of Phytic acid of the GM oilseed rape and its conventional counterpart are available in Table S1 of the supplementary materials. Recall that the corresponding degrees of freedom and expected mean squares for the responses are displayed in Table 3. Thus, we have σT2 = 18 σ12 + 7 2 1 2 1 2 1 2 1 2 1 2 7 2 2 8 σ2 + 8 σ3 − 8 σ4 and σR = 8 σ1 − 8 σ2 + 8 σ3 + 8 σ4 . Moreover, the variances and
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Table 4. Simulated coverage probabilities (C.P.) and expected lengths (E.L.) for the 0.9 lower generalized confidence limit for π . ρ = 0.1 J
n
5 5 5 5 5 10 10 10 10 10 25 25 25 25 25 50 50 50 50 50 100 100 100 100 100 200 200 200 200 200
5 10 25 50 100 5 10 25 50 100 5 10 25 50 100 5 10 25 50 100 5 10 25 50 100 5 10 25 50 100
ρ = 0.3
ρ = 0.5
ρ = 0.7
ρ = 0.9
C.P.
E.L.
C.P.
E.L.
C.P.
E.L.
C.P.
E.L.
C.P.
E.L.
0.8767 0.8757 0.8725 0.8691 0.8578 0.8935 0.8870 0.8811 0.8796 0.8727 0.8937 0.8944 0.8890 0.8869 0.8882 0.8939 0.8962 0.8935 0.8926 0.8903 0.8944 0.8969 0.8957 0.8921 0.8951 0.9018 0.8929 0.8971 0.8957 0.8966
0.256 0.219 0.186 0.169 0.158 0.227 0.197 0.172 0.160 0.152 0.193 0.174 0.158 0.150 0.145 0.175 0.162 0.151 0.146 0.142 0.163 0.153 0.146 0.142 0.140 0.154 0.148 0.142 0.140 0.138
0.8692 0.8620 0.8454 0.8373 0.8274 0.8862 0.8774 0.8670 0.8608 0.8496 0.8906 0.8851 0.8760 0.8726 0.8717 0.8913 0.8923 0.8819 0.8833 0.8751 0.8924 0.8932 0.8908 0.8825 0.8825 0.8969 0.8910 0.8925 0.8895 0.8886
0.216 0.181 0.154 0.142 0.135 0.187 0.162 0.143 0.135 0.129 0.159 0.144 0.133 0.128 0.124 0.146 0.136 0.128 0.124 0.122 0.136 0.129 0.124 0.122 0.120 0.130 0.125 0.121 0.120 0.119
0.8629 0.8495 0.8327 0.8265 0.8201 0.8792 0.8690 0.8573 0.8509 0.8460 0.8859 0.8782 0.8674 0.8672 0.8671 0.8880 0.8857 0.8811 0.8798 0.8745 0.8912 0.8894 0.8852 0.8780 0.8804 0.8935 0.8892 0.8878 0.8876 0.8874
0.173 0.145 0.127 0.119 0.114 0.147 0.130 0.117 0.112 0.109 0.126 0.117 0.109 0.106 0.104 0.117 0.111 0.106 0.103 0.102 0.111 0.106 0.103 0.101 0.100 0.106 0.103 0.101 0.100 0.099
0.8571 0.8444 0.8297 0.8285 0.8246 0.8758 0.8614 0.8536 0.8504 0.8473 0.8800 0.8760 0.8621 0.8647 0.8690 0.8862 0.8799 0.8804 0.8757 0.8751 0.8877 0.8856 0.8850 0.8778 0.8822 0.8941 0.8867 0.8871 0.8869 0.8876
0.127 0.110 0.099 0.095 0.092 0.108 0.099 0.092 0.089 0.087 0.095 0.090 0.086 0.085 0.084 0.090 0.086 0.084 0.083 0.082 0.086 0.084 0.082 0.081 0.081 0.084 0.082 0.081 0.080 0.080
0.8545 0.8432 0.8325 0.8330 0.8324 0.8727 0.8592 0.8547 0.8522 0.8470 0.8801 0.8726 0.8652 0.8650 0.8724 0.8858 0.8774 0.8796 0.8759 0.8758 0.8863 0.8834 0.8828 0.8774 0.8826 0.8933 0.8865 0.8877 0.8881 0.8873
0.077 0.072 0.069 0.067 0.067 0.070 0.067 0.065 0.064 0.064 0.065 0.064 0.062 0.062 0.062 0.063 0.062 0.061 0.061 0.061 0.062 0.061 0.061 0.061 0.060 0.061 0.061 0.060 0.060 0.060
Note that n = n T = n R in the table.
4 12 4 covariance of Y T = 12 j=1 k=1 YT,jk /48 and Y R = j=1 k=1 YR,jk /48 are given by 1 1 1 2 2 2 2 2 2 2 2 2 2 τT = 96 (σ1 +σ2 +σ3 −σ4 ), τR = 96 (σ1 −σ2 +σ3 +σ4 ), and τTR = 96 (σ12 −σ22 +σ32 −σ42 ). From the dataset, we have y T = 2.076, y R = 2.037, f 1 = f 3 = 11, f 2 = f 4 = 36, s12 = 0.232, s22 = 0.037, s32 = 0.262, and s42 = 0.033. Plugging these point estimates into π to yield πˆ = 0.943, a 0.9 lower confidence limit for π is obtained as 0.917 which can be compared with a threshold value π0 . This could provide useful information for the safety assessment. The R code for this example is available in the supplementary materials. Since the estimates of σe2T and σe2R (s22 = 0.037 and s42 = 0.033) are close to each other, it is reasonable to further assume that σe2T = σe2R = σe2 . Thus, we have σ12 = σ32 = σe2 + 4σ F2 1 and σ22 = σ42 = σe2 , leading to σT2 = σR2 = 18 σ12 + 38 σ22 + 18 σ32 + 38 σ42 , τT2 = τR2 = 96 (σ12 +σ32 ) 1 2 2 2 2 and τTR = 96 (σ1 − σ2 + σ3 − σ4 ). Plugging the point estimates into π to yield πˆ = 0.948, a 0.9 lower confidence limit for π is obtained as 0.940.
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0.2 0.0
0.1
Density
0.3
Scenario I Scenario II Scenario III Scenario IV
0
2
4
6
8
10
12
Figure 3. The density curves for the test populations in Table 5.
4. CONCLUDING REMARKS Even though we consider only the univariate normal distribution and the one-way random effects model in this study, the proposed method can be extended to the general setting of normal variance components model presented in Lee and Liao (2012). For example, Kang and Vahl (2014) investigated the equivalence test on a GM crop compared with a group of commercial non-GM reference varieties. Let YT and YR , respectively, denote the responses of the GM crop population and the reference varieties population. Under the multienvironment trials as suggested by Kang and Vahl (2014), we have that YT ∼ N (μT , σT2 ) and YR ∼ 2 2 N (μR , σR2 ), where σT2 = σ S2 + σ B(S) + σ E2 and σR2 = σV2 + σ S2 + σ B(S) + σ E2 . Here σV2 , σ S2 , 2 2 σ B(S) , and σ E denote the variance components for the reference varieties, the site effects, the block effects within sites, and the random errors, respectively. We will present this specific focus in a future communication. The proposed FGPQ-based method is dependent on the normal assumption. A simulation is conducted to investigate the robustness against the departure from the normal assumption. Let YR be the normal distribution N (3, 1) in which 99.97% of the reference population falls in the range [0, 6]. We consider YT being a gamma distribution, denoted by Gamma(γ1 , γ2 ), with expected value equal to E(YT ) = γ1 γ2 and variance V (YT ) = γ1 γ22 , where γ1 and γ2 denote the shape and scale parameters, respectively. We set four scenarios for the test population displayed in Table 5 and plot their density curves in Fig. 3. Consider the 0.95-content RPB conformance proportion, so δ L = μR − 1.96σR = 1.04 and δU = μR + 1.96σR = 4.96. And the true value of π for YT ∼ Gamma(γ1 , γ2 ) can be immediately obtained from the corresponding probability density function. Fix n T = 10, 20, 50, 100 and n R = 10, 20, 50, 100. For a given n T and Gamma(γ1 , γ2 ), generate a random sample with size n T from the gamma distribution and calculate its sample mean
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Table 5. Four scenarios of gamma distributions for test populations in Sect. 4. Scenario
γ1
γ2
E(YT )
V (YT )
I II III IV
16 8 4 2
1/4 1/2 1 2
4 4 4 4
1 2 4 8
Table 6. Simulated coverage probabilities for the 0.9 lower generalized confidence limit for π under scenarios in Table 5. Scenario I
Scenario II
Scenario III
Scenario IV
nT
nR
π = 0.8350
π = 0.7711
π = 0.7078
π = 0.6122
10 10 10 10 20 20 20 20 50 50 50 50 100 100 100 100
10 20 50 100 10 20 50 100 10 20 50 100 10 20 50 100
0.9279 0.9033 0.8964 0.8909 0.9222 0.9098 0.8949 0.8878 0.9285 0.9152 0.9069 0.8989 0.9280 0.9195 0.9124 0.9071
0.9494 0.9357 0.9263 0.9198 0.9514 0.9462 0.9346 0.9345 0.9591 0.9549 0.9519 0.9529 0.9646 0.9643 0.9638 0.9642
0.9654 0.9596 0.9516 0.9538 0.9729 0.9697 0.9686 0.9678 0.9839 0.9821 0.9836 0.9845 0.9900 0.9891 0.9938 0.9960
0.9692 0.9641 0.9631 0.9571 0.9756 0.9749 0.9741 0.9711 0.9906 0.9908 0.9927 0.9915 0.9957 0.9965 0.9974 0.9980
y T and variance sT2 . For a given n R and YR ∼ N (3, 1), generate y R and sR2 from the corresponding normal and chi-square distributions. Then obtain 0.9 lower confidence limit for π using the Monte-Carlo algorithm of Sect. 2.1. The simulated coverage probabilities are reported in Table 6. From the table, the proposed FGPQ-based method is found to be quite conservative as the test population deviates from the normal assumption. Thus, we suggest the practitioner to check the normal assumption first when employing the method. Meanwhile, we suggest the power transformation by Box and Cox (1969) for the non-normal data. Although the proposed FGCI for π asymptotically has the correct frequentist coverage, however, there is room for improvement with small sample sizes, particularly for the more complex linear models. The approach proposed by Cisewski and Hannig (2012) for FGCI construction for normal linear mixed models can be a useful tool for future study.
ACKNOWLEDGEMENTS The authors thank the editor, an associate editor, and three referees for their constructive comments that resulted in a much improved article. The work of Liao was supported in part by the Ministry of Science and Technology of ROC (contract MOST 104-2118-M-002-006-MY2).
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[Received February 2016. Accepted October 2016. Published Online November 2016.]
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