ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Short Communication
Volume 3 Issue 6 - 2016
Simultaneous Prediction Intervals for Dose-response Curve Quality Control
Nan Song1, Peng Ding2, Kousick Biswas3 and Zhibao Mi3*
1University of Pittsburgh, USA
2Ningxia Health and Family Planning Inspection Agency, China
3VA CSPCC, VA Maryland Health Care System, USA
Received:May 24, 2016 | Published: June 23, 2016
*Corresponding author: Zhibao Mi, VA CSPCC, VA Maryland Health Care System, Perry Point, MD 21902, Tel: 410-642-2411 ext. 6019; Fax: 410-642-1167; Email:
Citation: Song N, Ding P, Biswas K, Zhibao Mi (2016) Simultaneous Prediction Intervals for Dose-response Curve Quality Control. Biom Biostat Int J 3(6): 00080. DOI: 10.15406/bbij.2016.03.00080

Abstract

Although dose-response curves have been widely used as efficacy readouts in the life sciences, methods are needed to improve quality control for bioassay dose-response curves. In this report, we propose constructing simultaneous prediction intervals dose-response curves as a quality control estimate of future generated curves with a predetermined level of probability. In the absence of curve fitted parameters, sample means, variances, and covariances of the responses at various doses were used to construct an ellipsoid prediction region using a multivariate technique and a prediction band using the Studentized maximum modulus technique. Based on our simulation results, the prediction region is more applicable when there are response correlations among the doses, whereas the prediction band is more applicable in the absence of response correlations.

Keywords: Dose-response curves; Prediction region; Prediction band; Bioassay; Quality control

Introduction

Dose-response curves are widely used to assess pharmacological, radiological, or toxicological effects in medical research and clinical applications. A dose-response design can provide evidence of causal effects between exposure/treatment and responses; however, such conclusions heavily rely on the quality of the dose-response curves. Thus there is considerable interest in developing analytical tools to control curve quality.

In common practice, dose-response curves are assumed to follow a parametric family, such as linear, exponential, or sigmoid distributions. An advantage of this parametric approach is that the curve information can be easily described by a single metric, such as half maximal effective concentration (EC50) or half maximal inhibitory concentration (IC50), by fitting a linear or non-linear model [1-5]. Usually, the curve fitting approach is suitable for dose-response data with smooth connecting points and curve pattern homogeneity; however, in many situations, dose-response curves generated using human specimens vary dramatically because of data heterogeneity. In such cases, it is not appropriate to use a model fitting approach to summarize dose-response curves. Instead, when curves are not approximated by a parametric model, it is preferable to develop empirical methods to summarize the dose-response data. A frequently used metric to summarize such dose-response curves is area under the curve (AUC) [6-8]. A commonly used method to compute AUC is based on the trapezoid rule to estimate the area under curve by connecting data points on the dose-response curve with straight line segments and then using the area under the polygon to approximate the actual area under the curve calculated by integration. This method has intuitive appeal and is easy to implement; however, it may underestimate the area when the curve is concave upward or overestimate it when the curve is convex upward. Furthermore, sometimes, dose-response curves with different curve patterns may share the same AUC values. Thus, there is a need to use whole dose-response curves instead of summarized curve metrics to assess their quality. Herein we propose constructing simultaneous dose-response curve prediction intervals using the whole dose response curves as an analytical tool for dose-response quality control.

Quality control systems are implemented to control every step that might introduce assay variation. For every new test condition, dose-response curves are generated to calibrate variations, and the test condition is adjusted correspondingly to meet the predefined standard criteria. The prediction region/band constructed from a group of standardized dose-response curves can then be used to predict whether dose-response curves generated under the new testing condition belong to the group. This approach serves as a quality control validation measure for both systemic errors, such as experimental condition change, machine calibration, and protocol modification, and random errors, such as sample preparation and technician operations.  A commonly used model-free method to build prediction intervals for dose-response curves is based on independent responses for each individual dose. This approach is simple and easy to implement; however, it may lose data information and lead to misinterpretation of the results.  To improve the method using individual prediction intervals, we developed two methods that use simultaneous prediction intervals.  Specifically, we took a multivariate approach to construct simultaneous prediction regions and we extended simultaneous confidence bands into simultaneous prediction bands using the Studentized maximum modulus technique and applied them to dose-response curves [9-10]. 

Generally, the dose-response data can be expressed as a certain function of the responses (y) and the doses (x), i.e. y=f(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEai abg2da9iaadAgacaGGOaGaamiEaiaacMcaaaa@3BC9@ . Let y i1 , y i2 , y i3 ,..., y i n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamyAaiaaigdaaKqbagqaaiaacYcacaWG5bWaaSba aKqbGeaacaWGPbGaaGOmaaqcfayabaGaaiilaiaadMhadaWgaaqcfa saaiaadMgacaaIZaaajuaGbeaacaGGSaGaaiOlaiaac6cacaGGUaGa aiilaiaadMhadaWgaaqcfasaaiaadMgacaWGUbqcfa4aaSbaaKqbGe aacaWGPbaabeaaaKqbagqaaaaa@4B70@  be a group of dose response data collected at dose x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamyAaaqcfayabaaaaa@394C@ , i=1,2,3,...,k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6ca caGGUaGaaiOlaiaacYcacaWGRbaaaa@4072@ , respectively, and y ij 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaiabgwMiZkaaicdaaaa@3CBC@  where j=1,2,3,..., n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOAai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6ca caGGUaGaaiOlaiaacYcacaWGUbWaaSbaaKqbGeaacaWGPbaajuaGbe aaaaa@4241@  represents study subjects.  Without extracting the dose-response curve by a single parameter, such as EC50, or AUC, interval estimates for prediction of the dose-response curves can be computed either by considering the responses y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEaa aa@3782@ as correlated k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@ -dimensional variables across the doses x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@  or the response y ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaaa@3A3C@  given x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamyAaaqcfayabaaaaa@394C@ as one dimensional variables. For k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@ -dimensional responses, a prediction region is built using a multivariate approach, whereas for one dimensional responses, a prediction band is built using the Studentized maximum modulus technique [10,11].

Prediction region: The concept of a multivariate prediction region is a simultaneous interval estimated by constructing a region that has ( 100 α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai aaicdacaaIWaGaeqySdegaaa@3A52@ )% probability of containing the next dose-response curve, or more generally, containing the sample means M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytaa aa@3756@ of the next r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36EE@ dose-response curves.  It is assumed that the next one or r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36EE@ dose-response curves are independent not only of one another but also of the n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ standard or previous dose-response curves. Assume Y = ( y 1 j , y 2 j , ... , y k j ) ~ M V N ( μ , Σ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywai abg2da9iaacIcacaWG5bWaaSbaaKqbGeaacaaIXaGaamOAaaqcfaya baGaaiilaiaadMhadaWgaaqcfasaaiaaikdacaWGQbaajuaGbeaaca GGSaGaaiOlaiaac6cacaGGUaGaaiilaiaadMhadaWgaaqcfasaaiaa dUgacaWGQbaajuaGbeaacaGGPaGaaiOFaiaad2eacaWGwbGaamOtai aacIcacqaH8oqBcaGGSaGaeu4OdmLaaiykaaaa@5171@ with unknown mean vector μ = ( μ 1 , μ 2 , ... , μ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0JaaiikaiabeY7aTnaaBaaajuaibaGaaGymaaqcfayabaGa aiilaiabeY7aTnaaBaaajuaibaGaaGOmaaqcfayabaGaaiilaiaac6 cacaGGUaGaaiOlaiaacYcacqaH8oqBdaWgaaqcfasaaiaadUgaaKqb agqaaiaacMcaaaa@48E0@ and covariance matrix Σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4Odmfaaa@377B@ . In practice, μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 gaaa@383A@ s and Σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4Odmfaaa@377B@ are often estimated by y ¯ i = 1 n i j = 1 n i y i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH9aqpdaWcaaqaaiaa igdaaeaacaWGUbWaaSbaaKqbGeaacaWGPbaajuaGbeaaaaWaaabCae aacaWG5bWaaSbaaKqbGeaacaWGPbGaamOAaaqcfayabaaajuaibaGa amOAaiabg2da9iaaigdaaKqbagaacaWGUbWaaSbaaKqbGeaajuaGda WgaaqcfasaaiaadMgaaeqaaaqcfayabaaacqGHris5aaaa@4ADD@ and the sample covariance matrix S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uaa aa@375C@ , respectively. The ( 100 α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai aaicdacaaIWaGaeqySdegaaa@3A52@ )%  prediction region for the next r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36EE@ dose-response curves is

nr n+r [ M Y ¯ ] ' [ S ] 1 [ M Y ¯ ]= (n1)k nk F( 1α,k,nk ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGUbGaamOCaaqaaiaad6gacqGHRaWkcaWGYbaaamaadmaabaGa amytaiabgkHiTiqadMfagaqeaaGaay5waiaaw2faamaaCaaabeqaai aacEcaaaWaamWaaeaacaWGtbaacaGLBbGaayzxaaWaaWbaaeqajuai baGaeyOeI0IaaGymaaaajuaGdaWadaqaaiaad2eacqGHsislceWGzb GbaebaaiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiaacIcacaWGUbGa eyOeI0IaaGymaiaacMcacaWGRbaabaGaamOBaiabgkHiTiaadUgaaa GaamOramaabmaabaGaaGymaiabgkHiTiabeg7aHjaacYcacaWGRbGa aiilaiaad6gacqGHsislcaWGRbaacaGLOaGaayzkaaaaaa@5E3A@                  (1)

Where M=( m 1 , m 2 ,..., m k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai abg2da9iaacIcacaWGTbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGG SaGaamyBamaaBaaajuaibaGaaGOmaaqcfayabaGaaiilaiaac6caca GGUaGaaiOlaiaacYcacaWGTbWaaSbaaKqbGeaacaWGRbaajuaGbeaa caGGPaaaaa@45AF@  and m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGPbaabeaaaaa@3803@  the mean of responses at dose x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaaaaa@380E@ of the testing curves and r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36EE@ is the number of testing curves; Y ¯ =( y ¯ 1 , y ¯ 2 ,..., y ¯ k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaraGaeyypa0JaaiikaiqadMhagaqeamaaBaaajuaibaGaaGymaaqc fayabaGaaiilaiqadMhagaqeamaaBaaajuaibaGaaGOmaaqcfayaba Gaaiilaiaac6cacaGGUaGaaiOlaiaacYcaceWG5bGbaebadaWgaaqc fasaaiaadUgaaKqbagqaaiaacMcaaaa@463F@ and y ¯ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgaaeqaaaaa@3827@  is the mean of the responses for the standard or historic curves at dose x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaaaaa@380E@ . Usually, r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36EE@ = 1. The left-hand side of equation (1) has the T 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaCa aaleqabaGaaGOmaaaaaaa@37B9@ - distribution [12].

Prediction band: Since no parametric model and dose response function are hypothesized for the dose-response curve y = f ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEai abg2da9iaadAgacaGGOaGaamiEaiaacMcaaaa@3BC9@ , then y n i = f n i ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamOBaKqbaoaaBaaajuaibaGaamyAaaqabaaajuaG beaacqGH9aqpcaWGMbWaaSbaaKqbGeaacaWGUbqcfa4aaSbaaKqbGe aacaWGPbaabeaaaKqbagqaaiaacIcacaWG4bWaaSbaaKqbGeaacaWG PbaajuaGbeaacaGGPaaaaa@44CA@ can be estimated by the sample means of the responses y ¯ n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGUbqcfa4aaSbaaKqbGeaacaWGPbaabeaa aKqbagqaaaaa@3B35@ at the non-decreasing serial doses x 1 < x 2 < x 3 < ... < x k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaGaeyipaWJaamiEamaaBaaajuai baGaaGOmaaqcfayabaGaeyipaWJaamiEamaaBaaajuaibaGaaG4maa qcfayabaGaeyipaWJaaiOlaiaac6cacaGGUaGaeyipaWJaamiEamaa BaaajuaibaGaam4Aaaqcfayabaaaaa@4736@ . The simultaneous 1 α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai abgkHiTiabeg7aHbaa@39CB@ prediction bands for y n i + 1 = f n i + 1 ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamOBaKqbaoaaBaaajuaibaGaamyAaaqabaGaey4k aSIaaGymaaqcfayabaGaeyypa0JaamOzamaaBaaajuaibaGaamOBaK qbaoaaBaaajuaibaGaamyAaaqabaGaey4kaSIaaGymaaqcfayabaGa aiikaiaadIhadaWgaaqcfasaaiaadMgaaKqbagqaaiaacMcaaaa@4804@

y ¯ n i t 1 α 2 ,k,nk s n 1+ 1 n i f n i +1 ( x i ) y ¯ n i + t 1 α 2 ,k,nk s n 1+ 1 n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaad6gadaWgaaadbaGaamyAaaqabaaaleqaaOGaeyOe I0IaamiDamaaBaaaleaacaaIXaGaeyOeI0YaaSaaaeaacqaHXoqyae aacaaIYaaaaiaacYcacaWGRbGaaiilaiaad6gacqGHsislcaWGRbaa beaakiaadohadaWgaaWcbaGaamOBaaqabaGcdaGcaaqaaiaaigdacq GHRaWkdaWcaaqaaiaaigdaaeaacaWGUbWaaSbaaSqaaiaadMgaaeqa aaaaaeqaaOGaeyizImQaamOzamaaBaaaleaacaWGUbWaaSbaaWqaai aadMgaaeqaaSGaey4kaSIaaGymaaqabaGccaGGOaGaamiEamaaBaaa leaacaWGPbaabeaakiaacMcacqGHKjYOceWG5bGbaebadaWgaaWcba GaamOBamaaBaaameaacaWGPbaabeaaaSqabaGccqGHRaWkcaWG0bWa aSbaaSqaaiaaigdacqGHsisldaWcaaqaaiabeg7aHbqaaiaaikdaaa GaaiilaiaadUgacaGGSaGaamOBaiabgkHiTiaadUgaaeqaaOGaam4C amaaBaaaleaacaWGUbaabeaakmaakaaabaGaaGymaiabgUcaRmaala aabaGaaGymaaqaaiaad6gadaWgaaWcbaGaamyAaaqabaaaaaqabaaa aa@6C64@        (2)

Where t 1 α 2 ,k,nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDam aaBaaajuaibaGaaGymaiabgkHiTKqbaoaalaaajuaibaGaeqySdega baGaaGOmaaaacaGGSaGaam4AaiaacYcacaWGUbGaeyOeI0Iaam4Aaa qcfayabaaaaa@4249@ is the upper  α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaHXoqyaeaacaaIYaaaaaaa@38EF@  point of the Studentized maximum modulus distribution with parameters k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E7@ and nk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgk HiTiaadUgaaaa@38C7@ , and s n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Cam aaDaaajuaibaGaamOBaaqaaiaaikdaaaaaaa@397B@  is the pooled estimate of the variance σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aaWbaaeqajuaibaGaaGOmaaaaaaa@3953@ . For r1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabgc Mi5kaaigdaaaa@3970@ , the simultaneous 1α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTiabeg7aHbaa@393E@  prediction band for f n i +r ( x ¯ r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamOBaKqbaoaaBaaajuaibaGaamyAaaqabaGaey4k aSIaamOCaaqcfayabaGaaiikaiqadIhagaqeamaaBaaajuaibaGaam OCaaqcfayabaGaaiykaaaa@4125@ ,

y ¯ n i t 1 α 2 ,k,nk s n 1 r + 1 n i f n i +r ( x ¯ r ) y ¯ n i + t 1 α 2 ,k,nk s n 1 r + 1 n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaad6gadaWgaaadbaGaamyAaaqabaaaleqaaOGaeyOe I0IaamiDamaaBaaaleaacaaIXaGaeyOeI0YaaSaaaeaacqaHXoqyae aacaaIYaaaaiaacYcacaWGRbGaaiilaiaad6gacqGHsislcaWGRbaa beaakiaadohadaWgaaWcbaGaamOBaaqabaGcdaGcaaqaamaalaaaba GaaGymaaqaaiaadkhaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaamOB amaaBaaaleaacaWGPbaabeaaaaaabeaakiabgsMiJkaadAgadaWgaa WcbaGaamOBamaaBaaameaacaWGPbaabeaaliabgUcaRiaadkhaaeqa aOGaaiikaiqadIhagaqeamaaBaaaleaacaWGYbaabeaakiaacMcacq GHKjYOceWG5bGbaebadaWgaaWcbaGaamOBamaaBaaameaacaWGPbaa beaaaSqabaGccqGHRaWkcaWG0bWaaSbaaSqaaiaaigdacqGHsislda Wcaaqaaiabeg7aHbqaaiaaikdaaaGaaiilaiaadUgacaGGSaGaamOB aiabgkHiTiaadUgaaeqaaOGaam4CamaaBaaaleaacaWGUbaabeaakm aakaaabaWaaSaaaeaacaaIXaaabaGaamOCaaaacqGHRaWkdaWcaaqa aiaaigdaaeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaaaaaeqaaaaa@6ECF@         (3)

Under the assumptions of y ij ~N( μ i , σ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaiaac6hacaWGobGaaiik aiabeY7aTnaaBaaajuaibaGaamyAaaqcfayabaGaaiilaiabeo8aZn aaBaaajuaibaGaamyAaaqcfayabaGaaiykaaaa@4529@  and that the mean and variance are unknown at dose x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaaaaa@380E@ , then we estimate the y ¯ ni MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaad6gacaWGPbaabeaaaaa@391A@  and s n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGUbaabaGaaGOmaaaaaaa@38CB@  as follows:

y ¯ ni = j=1 n i y ij n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGUbGaamyAaaqcfayabaGaeyypa0ZaaSaa aeaadaaeWbqaaiaadMhadaWgaaqcfasaaiaadMgacaWGQbaajuaGbe aaaKqbGeaacaWGQbGaeyypa0JaaGymaaqaaiaad6gajuaGdaWgaaqc fasaaiaadMgaaeqaaaqcfaOaeyyeIuoaaeaacaWGUbWaaSbaaKqbGe aacaWGPbaajuaGbeaaaaaaaa@4A38@                                                                                        (4)

s n 2 = 1 i=1 k ( n i 1 ) ( i=1 k ( n i 1 ) s i 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Cam aaDaaajuaibaGaamOBaaqaaiaaikdaaaqcfaOaeyypa0ZaaSaaaeaa caaIXaaabaWaaabCaeaadaqadaqaaiaad6gadaWgaaqcfasaaiaadM gaaKqbagqaaiabgkHiTiaaigdaaiaawIcacaGLPaaaaKqbGeaacaWG PbGaeyypa0JaaGymaaqaaiaadUgaaKqbakabggHiLdaaamaabmaaba WaaabCaeaadaqadaqaaiaad6gadaWgaaqcfasaaiaadMgaaKqbagqa aiabgkHiTiaaigdaaiaawIcacaGLPaaacaWGZbWaa0baaKqbGeaaca WGPbaabaGaaGOmaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaadUga aKqbakabggHiLdaacaGLOaGaayzkaaaaaa@590F@   (5)

Where s i 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbaabaGaaGOmaaaaaaa@38C6@ is the sample variance at each dose x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaaaaa@380E@ .     

Simulations: To illustrate the methods using the prediction regions and the prediction bands, we simulated dose-response curve data using multivariate normal parameters estimated. For each simulated data point, thirty training dose-response curves and ten test dose-response curves were used to compare the testing results generated by the two prediction methods. Two scenarios were simulated, one with covariance as estimated from the original data and the other with covariance among the responses across all doses assumed to be zero. The simulation tests for each method were repeated 1000 times and the average numbers of testing curves falling in the prediction bands/regions and the corresponding variances are listed in Table 1.  When using the prediction regions, the testing results were robust, whereas when using the prediction bands, the results varied with covariance. When the covariance among the responses was smaller, the test using prediction bands was more efficient than the prediction regions.

Method

With Covariance

Without Covariance

Mean

SD

Mean

SD

Prediction Band

9.809

0.461

9.792

0.493

Prediction Region

9.992

0.089

9.262

0.995

Table 1: Test results using simulated data (x1000).

Conclusion

Prediction interval estimation is an important statistical tool for dose-response curve quality control [13-15]. Based on standard or previous curves one can construct an interval estimate for future generated curves with predefined criteria. These interval estimates can be used to adjust newly produced curve(s) by calibrating experimental conditions to avoid systematic and random errors. Commonly used analytical methods for dose-response data are either based on parametric EC50 or empirical AUC. Both of these summarized metrics are problematic when dose response curves are irregular and do not follow certain parametric distribution. In this report we developed two simple methods, ellipsoidal prediction regions and simultaneous prediction bands, to predict testing dose-response curve(s) as quality control analytical tools for dose-response experimental designs. Both methods involve the construction of simultaneous interval estimates for a group of dose-response curves to predict that individual testing dose-response curves belong to the group of curves. These simultaneous prediction interval estimates can be easily and quickly derived for these decreasing dose-response curves, and do not rely on a parametric modeling. These simple methods offer an alternative to nonlinear regression techniques that are model dependent and computational intensive. Sometimes the proposed methods are more robust for those dose-response curves not belonging to a known family. The prediction region is preferred when the correlation of the responses among the series of doses is strong, whereas the prediction band is suitable for those dose-response curves where the correlation is weak or there are no correlations. Despite the advantages, there are restrictions to using these methods. For the prediction region, multivariate normal distribution is required [16], while for the prediction band, the responses at each dose point need to be normally distributed. When compared with the prediction band method, the prediction region method is more efficient when there are response correlations among the doses.

Acknowledgements

The authors gratefully thank Dr. Yao Zhang (Philadelphia, PA) for his technical comments.

**Nan Song and Peng Ding are equally contributed for publication of this manuscript.

Conflict of Interest

No author reports any conflict of interest.

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