The Emax model is one of the key nonlinear regression models which may reflect the underlying biological response (Ting, 2006). In its 4-parameter form this model is usually written as: 3.6 where response is the response of interest, E0 is the basal effect or placebo response when Dose is equal to 0, Emax is the maximum achievable effect a drug can have over the placebo response, ED50 is the dose which produces 50% of the effect, and S is the slope. It is not uncommon to fix the slope as 1 in the analysis of Emax model. Ting (2006) discussed the Emax model in (3.2) as a particular case of Emax model in dose response modelling which is referred to as the hyperbolic Emax model. The Emax model with the slope factor (slope is not fixed at 1) is referred as sigmoidal Emax model. The steepness of dose response relationship is determined by the slope parameter in Emax model. Although both models can be justified biologically in the relationship of drug receptor, the use of hyperbolic model is primarily for empirical reasons. Both a hyperbolic and sigmoidal model can be used to fit dose response, the choice of hyperbolic is primarily for empirical reason. The study with hyperbolic Emax model (Slope =1) was designed prior to my joining the study team and was the protocol specified analysis. In the model fitting of proposed Bayesian Emax model with the slope parameter (sigmoidal Emax mode) from simulated data, computationally errors often occur in WinBUGS, regardless of the choices of non-informative or informative prior (for example the slope follows a uniform distribution between -2 and 2 or Normal distribution with mean of -1 and standard deviation of 0.5). While with the slope parameter fixed at 1, there were little computational errors. The bias of the model parameters ED90 were also small (the estimated ED90 is within 20% of true ED90).
Emax model using Linear Contrast Method Thomas (2006) extended the linear contrast to the sigmoid Emax model and designed a set of linear contrast to describe difference shapes of Emax dose-response curve. Emax model is a flexible model, with varying the model parameters E0, Emax, ED50 or slope or Hill parameters, to approximate many families of dose response models with monotonic response (Thomas 2006), such as the linear, log-linear, or exponential model. The linear contrasts are then generated from the dose response of different curves of possible Emax models, example Emax dose curves are displayed in Figure 3.3. The six dose response profiles in the dose response plots are then described by 6 linear contrasts which are pre-specified in the analysis. A statistical test is constructed on the smallest P-value from a set of contrasts. Bayesian methods are recommended and weighting of the contrasts can be implied by prior distribution (Thomas, 2006). Figure 3.3 Example dose response curves for building linear contrasts. Trend Test in Dose Response Analysis If an assumption is made that an increase in dose will result in an increase in a response variable, possible methodologies in dose response analysis are trend-based or monotonic response analyses. A trend test can be considered as a special case of linear contract method, where the weight is chosen so the trend test is more powerful to detect associations when the assumed trend is true. It is often expected that response increases or decreases monotonically with dose, but not necessarily linear.
Where ED90 is defined as the dose to achieve 90% of maximum DAS28 response. In this calculation the maximum response is estimated from the maximal DAS28 effect at all doses. The 90% (ED90) of maximum response is then calculated as the lowest nominal dose in which is closest to the estimated dose that achieves 90% of maximal efficacy. A factor of 1.1513 maps the dose level (dose level 1-10) to dose in mg/kg on the scale of interest (0.001–30 mg/kg). The slope of the Emax model has been fixed at 1 so the slope is removed. The planned clinical doses are 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1, 3, 10, and 30, which are the nominal doses with a rate of half log increase. Therefore, a factor of “exp(1.1513)” converts clinical doses to dose levels with 1 to 1 mapping as illustrated in below table. β3: Dose level in Eqn 4.x Clinical doses= Exp(1.1513*( β3-7) 1 0.001 2 0.003 3 0.010 4 0.030 5 0.100 6 0.300 7 1.000 8 3.000 9 10.000 10 30.000 The model parameter β3 covers a broader range of doses, it also prevents the clinical dose from being negative by taking the exponential transformation. The estimate of β3 is then approximated at the lowest nominal dose level that is close to ED90. If the estimate of β3 is outside range, i.e. the maximal estimate of β3 in the dose range is used i.e. the patients will be assigned to maximal dose at 30 mg/kg. The Bayesian dose response model for the GSK123456 PoC study was ,
θ1 ~ N(0, σ12) and δ1 ~ N(0, σ12) where DAS28jk is the observed individual change in DAS28 score from baseline to day 56 at Dosej for kth subject (Dose is placebo when j=1). DAS28 at Day 56 change from baseline follows a Normal distribution with mean (θj + βXk) for each Dosej and with variance of σ2, the Dosej is assumed to be spaced equally. θj is the estimated treatment effect at Dosej, Xk is the baseline DAS28 score fitted as covariate and β is the coefficient for covariate. Furthermore, θj has a linear relationship with neighbouring θj−1 with slope of δj−1(slope parameter). θ1 is the untreated or placebo response when the drug dose is equal to 0 and δ1 is the slope between placebo and the first dose. Both θ1 and δ1 follow Normal distributions. Bayesian NDLM method was used in the model estimation and it calculated the estimate based on sampled data from posterior distribution of multivariate normal distribution. Bayesian methods (Weir, 2007) offer a robust estimation of parameters in complicated model (Carlin, 2010). IN the retrospectively analysis, the priors of the effect of both the first dose θ1 and δ1 follow a Normal distribution with the mean following a vague prior of N(0, 100) and variance of distribution IG(0.001,0.001)). The prior distributions on σ2, σθ2, and σδ2 are vague inverse-gamma distribution IG(0.001,0.001)). WinBUGS code is available in Appendix 11.1.2. The results of modelling fitting of NDLM model is presented as red dashed line in Figure 6.2. Results of Retrospective Analysis Using NDLM U-shaped dose response curve was an unexpected finding from the Phase 2a PoC clinical trial of GSK123456. To investigate whether NDLM model can improve the model fitting and statistical power, the data from the GSK123456 study are retrospectively analysed using a NDLM approach. The NDLM model is applied to DAS28 change from baseline data at 56. Baseline DAS28, as a
of 17% signals that the NDLM model is over-sensitive and inflate the false positive so appropriate adjustment to the test statistics or decision criteria may be needed. The analysis of the Emax and NDLM models on the Emax curve and linear curve also show the same trend. The probability of success is 98% for both Emax and NDLM models if the true response curve follows Emax data and 96% and 95% for log linear curve for Emax and NDLM models respectively. For U-shaped dose response curves, the power measured as the probability of success of the Emax model is 26% and power using the NDLM model is 92%. The NDLM model significantly improves the probability of success compared to the Emax model in the fixed design model. Table 6.6 Probability of final success and failure in fixed design using Bayesian Emax model and NDLM model in Scenario 1. Assumed True Dose Response Bayesian Emax model Bayesian NDLM model Prob. of final success Prob. of final failure Prob. of final success Prob. of final failure Placebo like flat curve 0.06 0.94 0.17 0.83 Emax with EC50=3 0.98 0.02 0.98 0.02 LogLinear Curve 0.96 0.04 0.95 0.05 U-Shaped curve 0.26 0.74 0.92 0.08 Adaptive Design Simulation Fixed design assumes that all 6 doses are allocated to a fixed number of patients. No adaptations were adopted in this design. In the adaptive simulation, study may stop for futility or success or subjects are allocated to certain dose based on the dose response of the subjects already enrolled in the study.
Table 6.9. Comparisons of the operating characteristics of four types of dose response curve using Bayesian NDLM model in the adaptive design settings. Weak prior was used in the calculation with adaptive allocation. True Dose Response Dose Level (mg/kg) 0.03 0.3 3 10 20 30 Mean allocation Flat placebo like curve 7.3 7.3 7.3 7.3 7.3 7.3 Emax like Curve 4.6 4.7 4.7 4.6 4.7 4.7 Log Linear Curve 4.9 4.9 4.9 4.9 4.9 5.0 U-Shaped Curve 5.3 5.3 5.3 5.3 5.4 5.4 Proportion of simulation being selected as ED90 Flat placebo like curve 18% 17% 18% 13% 13% 14% Emax like Curve 1% 2% 15% 24% 33% 25% Log Linear Curve 2% 3% 6% 13% 28% 48% U-Shaped Curve 4% 38% 44% 7% 6% 2% Proportion of simulation being selected as MED Flat placebo like curve 0% 0% 0% 0% 0% 0% Emax like Curve 24% 18% 33% 15% 6% 2% Log Linear Curve 25% 16% 16% 16% 15% 6% U-Shaped Curve 22% 51% 16% 1% 0% 0% Proportion of simulation being selected as MAX Flat placebo like curve 16% 17% 19% 15% 14% 18% Emax like Curve 0% 1% 11% 20% 35% 33% Log Linear Curve 1% 2% 4% 10% 27% 56% U-Shaped Curve 2% 32% 46% 10% 8% 3% Probability of dose compared to control Flat placebo like curve 51% 51% 52% 50% 50% 51% Emax like Curve 58% 66% 87% 91% 94% 93% Log Linear Curve 58% 64% 74% 83% 90% 95% U-Shaped Curve 61% 89% 93% 79% 72% 56%
Table 6.10 Probability of success and failure at interim and final analysis in adaptive design settings, the mean subjects are summarized in the last column across 5,000 simulations. A) Bayesian Emax model in the analysis dose response models; B) Bayesian NDLM model in the analysis dose response models A True Dose Response Bayesian Emax model Prob. of early success Prob. of early failure Prob. of final success Prob. of final failure Mean subjects Placebo like flat curve 0.00 0.23 0.04 0.73 61.2 Emax with EC50=3 0.74 0.00 0.26 0.01 54.8 Log Linear Curve 0.58 0.00 0.40 0.02 57.0 U-Shaped curve 0.06 0.09 0.10 0.75 61.5 B True Dose Response Bayesian NDLM model Prob. of early success Prob. of early failure Prob. of final success Prob. of final failure Mean subjects Placebo like flat curve 0.00 0.00 0.12 0.88 63.9 Emax with EC50=3 0.53 0.00 0.44 0.03 57.4 Log Linear Curve 0.45 0.00 0.47 0.08 57.9 U-Shaped curve 0.41 0.00 0.47 0.12 58.6
Table 6.13 Probability of success and failure at interim and final analysis in adaptive design settings, the mean subjects are summarized in the last column across 5,000 simulations. A) Bayesian Emax model in the analysis dose response models; B) Bayesian NDLM model in the analysis dose response models. A True Dose Response Bayesian Emax model Prob. of early success Prob. of early failure Prob. of final success Prob. of final failure Mean subjects Placebo like flat curve 0.00 0.33 0.06 0.61 55 Emax with EC50=3 0.77 0.02 0.17 0.04 42 Log Linear Curve 0.66 0.02 0.25 0.08 45 U-Shaped curve 0.06 0.23 0.11 0.60 54 B True Dose Response Bayesian NDLM model Prob. of early success Prob. of early failure Prob. of final success Prob. of final failure Mean subjects Placebo like flat curve 0.00 0.01 0.12 0.87 64 Emax with EC50=3 0.51 0.00 0.41 0.08 56 Log Linear Curve 0.45 0.00 0.43 0.12 56 U-Shaped curve 0.36 0.00 0.44 0.20 58 Table 6.14 and table 6.15 display the additional operating characteristics of the model fitting to the data that were analysed using the Emax model and NDLM model respectively. The mean
Table 6.18 Probability of success and failures at interim and final analysis after calibrating type I error rate in the half adaptive design (Scenario 3) A: Bayesian Emax model (Type I error rate was calibrated at 5%) True Dose Response Bayesian Emax model Early success Early failure Final success Final failure Total Success Mean subjects Placebo like flat curve 0.00 0.27 0.05 0.68 5% 60.7 Emax curve 0.79 0.00 0.20 0.01 99% 55.6 Log Linear Curve 0.66 0.00 0.31 0.03 97% 57.7 U Shaped curve 0.06 0.11 0.13 0.70 19% 61.9 B. Bayesian NDLM (Type I error rate was calibrated at 5%) True Dose Response Bayesian NDLM model Early success Early failure Final success Final failure Total Success Mean subjects Placebo like flat curve 0.00 0.21 0.05 0.74 5% 62.8 Emax curve 0.70 0.00 0.18 0.11 88% 57.6 Log Linear Curve 0.62 0.02 0.20 0.25 82% 57.8 U Shaped curve 0.53 0.00 0.21 0.17 74% 58.0
The correlation coefficient of change from baseline of early and late response can be given by 7.7 If the correlation of post-baseline response (Y2 or Y3) and baseline is zero, Eqn 7.7 becomes Thus, the correlation of DAS28 change from baseline has a linear relationship to the correlation of raw DAS28 score with slope and intercept of . The correlation in change from baseline response is dependent of the correlations in raw score as well as the correlation of post-baseline and baseline and is equal to , when there are no correlations between the raw scores Y1, Y2, and Y3. Let ϒ2 be the ratio of σY2 and σY1, and ϒ3 to be the ratio of σY3 and σY1.ϒ2 = σY2 /σY1 and ϒ3 = σY3 /σY1, so the denominator can be written as . After replacing σY1, σY2, and σY3 with ϒ2 and ϒ3, the a, b, c, and d the above result is transformed to Let a=; b=; c=; and d=.
endpoint should be fully captured by surrogate endpoint. Symbolically, let Tj and Sj be random variables of true (DAS28 at Day 56) and surrogate endpoints (DAS28 at Day 14), respectively, for the jth subject and let Zj be an indicator variable for treatment, so Prentice’s method (1989) can be illustrated using below formulas (Buyse 1998, 2000), Sj = µS + αZj + εSj 7.7 Tj = µT + βZj + εTj 7.8 Tj = µ + ϒSj + εj 7.9 where α and β are the slopes for the treatment Z on surrogate endpoint S and true endpoint T. ϒ is the slope of the true endpoint (T) on surrogate endpoint S as linear regression. Sj, Tj and Zj are surrogate response, true response, and treatment in jth patients. µS, µT, and µ are the intercepts for the linear regressions from (7.7), (7.8), and (7.9) respectively. ej is error term for effect of true effect on surrogate endpoint. Both eSj and εTj, are Normally distributed correlated errors which satisfy , and Lastly, the true response is examined using linear regression model with both treatment and surrogate endpoint as shown below. Tj = µT + βsZj + ϒzSj + εTj 7.10 where S is the estimate of the effect of Z on T after adjusting surrogate endpoint S. s is the parameter estimate for treatment effect Z; ϒz is the slope estimate of effect of surrogate endpoint S on true endpoint T after adjusting treatment Z. To achieve surrogacy using Prentice’s criteria,
multiple-trial or meta-analysis setting. In the next subsection, validation methods will be discussed multiple-trial setting. Validation of surrogate endpoints in multiple-trial setting In the multiple-trial setting, the surrogate endpoint and final true endpoint are collected in multiple trials and both individual patient data and trial data are assumed to be available for the analysis. Both fixed effect models and random effect models with treatment as an independent variable are suggested in the literature (Buyse and Molenberghs 1998) in the validating the surrogate endpoint, the method is called meta-analytic approach (Buyse and Molenberghs 1998). The two-stage fixed effect model is illustrated as below. At the first stage, the surrogate endpoint and true final endpoint are assumed to be jointly Normally distributed with treatment effect model presented in both surrogate endpoint and true endpoint in a linear model. Sij = µSi +αiZij + εSij 7.12 Tij = µTi + βi Zij + εTij 7.13 where µSi and µTi are the intercepts for the linear regressions respectively. Here, αi and βi are the corresponding treatment effect with εj being the error term for effect of true effect on surrogate endpoint. εSij and εTij, are correlated errors , where At second stage, it assumes the µSi, µTi, αi and βi are a mixture of fixed and random effects, so ,