Statistical principles of clinical trial

ST 520 D Zhang ST 520: Statistical Principles of Clinical Trials and Epidemiology Daowen Zhang zhang@stat.ncsu.edu http://www4.stat.ncsu.edu/∼dzhang2 Slide TABLE OF CONTENTS ST520, D Zhang Contents Introduction 1.1 Brief Introduction to Epidemiology 1.2 Brief Introduction and History of Clinical Trials 29 Phase I and Phase II Clinical Trials 42 2.1 Phase I clinical trials (from Dr Marie Davidian) 44 2.2 Phase II Clinical Trials 99 Phase III Clinical Trials 133 Randomization 159 Some Additional Issues in Phase III Clinical Trials 201 Sample Size Calculations 209 Comparing More Than Two Treatments 234 Slide TABLE OF CONTENTS ST520, D Zhang Causality, Non-compliance and Intent-to-treat 275 8.1 Causality and Counterfactual Random Variables 275 8.2 Noncompliance and Intent-to-treat analysis 282 8.3 A Causal Model with Noncompliance 288 Survival Analysis in Phase III Clinical Trials 300 9.1 Describing the Distribution of Time to Event 301 9.2 Censoring and Life-Table Methods 310 9.3 Kaplan-Meier or Product-Limit Estimator 320 9.4 Two-sample Log-rank Tests 326 9.5 Power and Sample Size Based on the Log-rank Test 336 9.6 K-Sample Log-rank Tests 352 9.7 Sample-size Considerations for the K-sample Log-rank Test 355 9.8 Analyzing Data Using K-sample Log-rank Test 357 10 Early Stopping of Clinical Trials 360 10.1 General Issues in Monitoring Clinical Trials 360 Slide TABLE OF CONTENTS ST520, D Zhang 10.2 Information Based Design and Monitoring 366 10.3 Choice of Boundaries 382 10.4 Power and Sample Size in Terms of Information 388 Slide CHAPTER 1 ST 520, D Zhang Introduction Two areas of studies on human beings: EPIDEMIOLOGY and CLINICAL TRIALS EPIDEMIOLOGY: Systematic study of disease etiology (causes and origins of disease) using observational data (i.e data collected from a population not under a controlled experimental setting) • Second hand smoking and lung cancer • Air pollution and respiratory illness • Diet and Heart disease • Water contamination and childhood leukemia • Finding the prevalence and incidence of HIV infection and AIDS Slide CHAPTER ST 520, D Zhang CLINICAL TRIALS: The evaluation of intervention (treatment) on disease in a controlled experimental setting • The comparison of AZT versus no treatment on the length of survival in patients with AIDS • Evaluating the effectiveness of a new anti-fungal medication on Athlete’s foot • Evaluating hormonal therapy on the reduction of breast cancer (Womens Health Initiative) Slide CHAPTER 1.1 ST 520, D Zhang Brief Introduction to Epidemiology I Cross-sectional study: data are obtained from a random sample at one point in time This gives a snapshot of a population • Example: Based on a survey or a random sample thereof, we determine the proportion of individuals with heart disease at one time point This is referred to as the prevalence of disease The prevalence can be broken down by age, race, sex, socio-economic status, geographic, etc • Important public health information can be obtained Useful in determining how to allocate health care resources • However such data are generally not very useful in determining causation Slide CHAPTER ST 520, D Zhang • If exposure (E) and disease (D) are binary (yes/no), data from a cross-sectional study can be represented as D E ¯ E ¯ D n11 n12 n1+ n21 n22 n2+ n+1 n+2 n++ ¯ = unexposed; D = disease, where E = exposed (to risk factor), E ¯ = no disease D Here all counts n11 , n12 , n21 , n22 are random variables (n11 , n12 , n21 , n22 ) ¯ ¯ P [D ¯ E]) ¯ ∼ multinomial(n++ , P [DE], P [DE], P [D E], Slide CHAPTER ST 520, D Zhang • With this study, we can obtain estimates of the following parameters of interest n+1 prevalence of disease P [D] (by ) n++ n1+ ) exposure probability P [E] (by n++ n11 ) n1+ ¯ (by n21 ) prevalence of disease among unexposed P [D|E] n2+ prevalence of disease among exposed P [D|E] (by Slide CHAPTER ST 520, D Zhang • relative risk ψ of getting disease between exposed and un-exposed: P [D|E] ψ= ¯ P [D|E] ⋆ ψ > ⇒ positive association ⋆ ψ = ⇒ no association ⋆ ψ < ⇒ negative association • Estimate of ψ from a cross-sectional study: ψ= n11 /n1+ P [D|E] = ¯ n21 /n2+ P [D|E] Slide 10 CHAPTER 10 ST 520, D Zhang Table 3: Inflation factors as a function of K, α, β and Φ α=0.05 α=0.01 Power=1-β Power=1-β K Boundary 0.80 0.90 0.95 0.80 0.90 0.95 Pocock 1.11 1.10 1.09 1.09 1.08 1.08 O-F 1.01 1.01 1.01 1.00 1.00 1.00 Pocock 1.17 1.15 1.14 1.14 1.12 1.12 O-F 1.02 1.02 1.02 1.01 1.01 1.01 Pocock 1.20 1.18 1.17 1.17 1.15 1.14 O-F 1.02 1.02 1.02 1.01 1.01 1.01 Pocock 1.23 1.21 1.19 1.19 1.17 1.16 O-F 1.03 1.03 1.02 1.02 1.01 1.01 Pocock 1.25 1.22 1.21 1.20 1.19 1.17 O-F 1.03 1.03 1.03 1.02 1.02 1.02 Pocock 1.26 1.24 1.22 1.22 1.20 1.18 O-F 1.03 1.03 1.03 1.02 1.02 1.02 Slide 395 CHAPTER 10 ST 520, D Zhang • Example with dichotomous endpoint: Let π1 and π0 be the population response rates for treatments and ∆ = π1 − π0 Want to test H0 : ∆ = vs HA : ∆ = at level α = 0.05 using a 4-look O’Brien-Fleming boundary (Φ = 0) ⋆ we will reject H0 the first time when |T (tj )| = | p1 (tj ) − p0 (tj ) p¯(tj ){1 − p¯(tj )} ≥ 4.049/ Slide 396 n1 (tj ) j, j = 1, , + n0 (tj ) | CHAPTER 10 ST 520, D Zhang The boundaries are given by Table 4: Boundaries for a 4-look O-F test j bj nominal p-value 4.05 001 2.86 004 2.34 019 2.03 043 Slide 397 CHAPTER 10 ST 520, D Zhang ⋆ Suppose π0 = 0.3 and we would like to have power 90% to detect π1 = 0.45, how we design the study? ⋆ The fixed sample size design requires nF S =   1.96 + 1.28  2 3×.7+.45×.55  2×.375×.625 15  × × 375 × 625 = 434 ⋆ The inflation factor for α = 0.05, power=0.9, K = and Φ = is IF=1.02 The maximum sample size using a group sequential test is 1.02 × 434 = 444, or 222 patients for each treatment ⋆ Since 222/4 = 56, we recruit 56 patients for each treatment first and then interim analysis If we dont reject H0 , then recruit additional 56 patients to each treatment and interim analysis, etc Slide 398 CHAPTER 10 ST 520, D Zhang Information based monitoring • π0 = 0.3 and π1 = 0.45 are needed to derive the sample size They might not be the case in practice • Suppose we would like a level 0.05 test to have power 0.9 to detect ∆ = 0.15, then the information needed for a fixed sample size design is Zα/2 + Zβ ∆A = 1.96 + 1.28 15 = 466.6 • So the MI for a 4-look O-F design is 466.6 ì 1.02 = 475.9 ã So the information required at the jth interim analysis j × 475.9 = 119 × j, j = 1, , 4 Slide 399 CHAPTER 10 ST 520, D Zhang • The information available at the jth interim analysis is approximately −2 ˆ [se{∆(t)}] p1 (t){1 − p1 (t)} p0 (t){1 − p0 (t)} + = n1 (t) n0 (t) −1 • This implies that we should interim analysis when p1 (tj ){1 − p1 (tj )} p0 (tj ){1 − p0 (tj )} + n1 (tj ) n0 (tj ) −1 = 119×j, j = 1, , and use the test statistic and the boundary values given before • The information-based monitoring will maintain the overall type I error prob and desired power to detect a difference of interest even if the nuisance parameter values may be different than what we assume Slide 400 CHAPTER 10 ST 520, D Zhang II Average information • For the same design characteristics, which boundary is better? Pocock or O-F? • Pocock design has higher IF, but it is easier to stop using Pocock design • Compare them using average information (similar to average sample size) needed for the alternative • If H0 is true, the chance that H0 will be rejected will be small (α is usually taken to be 0.05) So the chance the trial will be stopped is small too (at most α) So the average information under H0 will be very close to the MI Slide 401 CHAPTER 10 ST 520, D Zhang • For example, if K = 5, α = 0.05, power = 90% to detect an alternative of interest, then Maximum Average Designs information information (HA ) 5-look Pocock I F S × 1.21 I F S × 68 IF S IF S 5-look O-F Fixed-sample I F S × 1.03 where IF S = Zα/2 + Zβ ∆A I F S × 75 is the information for fixed sample size design • Pocock designs required smaller sample size on average if HA is true Slide 402 CHAPTER 10 ST 520, D Zhang • Remarks: ⋆ If you want a design which, on average, stops the study with less information when there truly is a clinically important treatment difference, while preserving the level and power of the test, then a Pocock boundary is preferred to the O-F boundary ⋆ By a numerical search, one can derive the “optimal” shape parameter Φ which minimizes the average information under the clinically important alternative ∆A for α, K, and power (1 − β) For example, when K = 5, α = 05 and power of 90% the optimal shape parameter Φ = 45, very close to the Pocock boundary (Wang and Tsiatis, 1987, Biometrics) ⋆ However, the designs with smaller average information under HA requires more information if the null hypothesis were true ⋆ Most clinical trials with a monitoring plan seem to favor more “conservative” designs such as the O-F design Slide 403 CHAPTER 10 ST 520, D Zhang Statistical Reasons Historically, most clinical trials fail to show a significant difference; hence, from a global perspective it is more cost efficient to use conservative designs (such as O-F design) Even a conservative design, such as O-F, results in a substantial reduction in average information, under the alternative HA , before a trial is completed as compared to a fixed-sample design (in our example 75 average information) with only a modest increase in the maximum information (1.03 in our example) Slide 404 CHAPTER 10 ST 520, D Zhang Non-statistical Reasons In the early stages of a clinical trial, the data are less reliable and possibly unrepresentative for a variety of logistical reasons It is therefore preferable to make it more difficult to stop early during these early stages Psychologically, it is preferable to have a nominal p-value at the end of the study which is close to 05 The nominal p-value at the final analysis for the 5-look O-F test is 041 as compare to 016 for the 5-look Pocock test This minimizes the embarrassing situation where, say, a p-value of 03 at the final analysis would have to be declared not significant for those using a Pocock design Slide 405 CHAPTER 10 ST 520, D Zhang III Steps in the design and analysis of group-sequential tests with equal increments of information Design Decide the maximum number of looks K and the boundary Φ K does not have to be very large Table 5: O’Brien-Fleming boundaries (Φ = 0); α = 05, power=.90 Maximum Average K Information Information (HA ) IF S IF S I F S × 1.01 I F S × 85 I F S × 1.02 I F S × 77 I F S × 1.02 I F S × 1.03 Slide 406 I F S × 80 I F S × 75 CHAPTER 10 ST 520, D Zhang Compute I F S , then translate it to the sample size or number of events Find the inflation factor IF (α, K, Φ, β) and get M I = I F S × IF (α, K, Φ, β) Also calculate the maximum sample size or maximum number of events Slide 407 CHAPTER 10 ST 520, D Zhang Analysis Conduct data analysis after equal increment of M I/K information −2 ˆ This can be achieved by monitoring [se{∆(t)}] , although in practice, this is not generally how the analysis times are determined At the j-th interim analysis, the standardized test statistic ˆ j) ∆(t , T (tj ) = ˆ se{∆(tj )} is computed using all the data accumulated until that time and the null hypothesis is rejected the first time the test statistic exceeds the corresponding boundary value Note: The procedure outlined above will have the correct level of significance as long as the interim analysis are conducted after equal increments of information However, in order for this test to have the desired power to detect ∆A , it must be computed after equal increments of statistical information Slide 408 CHAPTER 10 ST 520, D Zhang M I/K where MI = Zα/2 + Zβ ∆A IF (α, K, Φ, β) If the initial guesses on the nuisance parameters were correct, then we would have the right power Otherwise the study may be underpowered or overpowered We should monitor ˆ j )}]−2 [se{∆(t to see if it deviates significantly from the required information j × M I/K This helps detect the problem and fix it at the early stage Slide 409 ... 28 CHAPTER 1.2 ST 520, D Zhang Brief Introduction and History of Clinical Trials • Definition of a clinical trial: ⋆ A clinical trial is a study in human subjects in which treatment (intervention)... I and Phase II Clinical Trials 42 2.1 Phase I clinical trials (from Dr Marie Davidian) 44 2.2 Phase II Clinical Trials 99 Phase III Clinical Trials 133 Randomization... for the evaluation and marketing of new drug treatments Slide 41 CHAPTER 2 ST 520, D Zhang Phase I and Phase II Clinical Trials Phases of Clinical Trials: • Preclinical (drug discovery): experimentation