Designs for Clinical Trials

Transcription

1 Designs for Clinical Trials

2 Chapter 5 Reading Instructions 5.: Introduction 5.: Parallel Group Designs (read) 5.3: Cluster Randomized Designs (less important) 5.4: Crossover Designs (read+copies) 5.5: Titration Designs (read) 5.6: Enrichment Designs (less important) 5.7: Group Sequential Designs (read include.6) 5.8: Placebo-Challenging Designs (less important) 5.9: linded Reader Designs (less important) 5.: Discussion

3 Design issues

4 Design issues Research question Statistical optimality not enough! Use simulation models! Confounding Objective Control Statistics Design, Variables Ethics Variability Cost Feasibility

5 Parallel Group Designs R Test Control Control Yij = µ + ε i i =, K, n j j =, K,k ij Easy to implement ccepted Easy to analyse Good for acute conditions ε ij ( ), ~ N σ

6 Where does the varation go? Placebo Drug X 8 week blood pressure Y = Xβ + ε nything we can explain Unexplained

7 etween and within subject variation DP mmhg Placebo Drug X Female Male aseline 8 weeks

8 What can be done? Stratify: Randomize by baseline covariate and put the covariate in the model. More observations per subject: aseline More than obsservation per treatment Run in period: Ensure complience, diagnosis

9 Parallell group design with baseline Compare bloodpressure for three treatments, one test and two control. R Test Control Control aseline 8 weeks Model: Yijk = µ + { i=} τ j + ξk + ε Observation i =, ijk Treatment j =,,3 Treatment effect τ j Subject k =, K,n ( ξ iid, ) j Subject effect k N σ s Random error ε ijk iid N (, σ )

10 Change from baseline The variance of an 8 week value is Var ( Y ) ( ) jk = Var µ + { i=} τ j + ξk + εijk ( ξ + ε ) = Var( ξ ) + Var( ε )= = Var k ijk k ijk σ s +σ Change from baseline = Y Y = ε ( τ + ) Z jk jk jk jk j ε jk The variance of change from baseline is σ Usually σ < σ s σ < σ + σ s

11 aseline as covariate R Test Control Control aseline 8 weeks Model: Y ij Treatment effect aseline value Random error = µ + τ + βx + ε ε j τ j xi ijk iid i N ij (, σ ) Subject Treatment i =, K,n j j =,,3

12 Crossover studies ll subject gets more that one treatment Comparisons within subject R Wash out Sequence Sequence Period Period Within subject comparison Reduced sample size Good for cronic conditions Good for pharmaceutical studies

13 Model for a cross over study ξ Obs=Period+sequence+subject+treament+carryover+error Yijk γ i = effect of sequence i =, γ +γ = i( k ) = effect of subject k =, K, ni within sequence i iid N, π j = effect of period j =, π { } τ t = effect of treatment t, ε randomerror iid N, σ ijk = = + γ i + ξi( k ) µ + π + τ + δ λ + ε j t ( ) j r +π = ijk τ + τ = ( σ ) s

14 by Crossover design [ EY ]= ijk µ µ µ µ + π τ γ + π + τ + λ γ + γ + + π τ γ + π + τ + γ τ (( µ µ ) + ( µ µ )) (( π π + τ τ λ ) + ( π π + τ τ λ )) τ ( λ + λ ) = Effect of treatment and carry over can not be separated! =

15 Model: Sum to zero: Matrix formulation Yijk λ π τ Matrix formulation ( µ γ ) T π τ ( µ, µ, µ µ ) T β = µ =, = + γ i + ξi( k ) µ + π + τ + ε + λ = +π = +τ = µ = Xβ X j = t ijk

16 Matrix formulation Parameter estimate: ( X T X) X T Y ˆ β = σ ˆ = ( T Y Y ) T βx ˆ T ( ˆ β ) ( X T X) σˆ Cov = ( X T X) = Estimates independent and Var ( ˆ τ ) =.5 ˆ σ

17 lternatives to * Compare to Same model but with 3 periods and a carry over effect Yijk = + γ i + ξi( k ) ξ γ i π j µ + π + τ + δ λ + ε i( k ) = effect of subject k =, K, ni within sequence i iid N, = effect of sequence i =, = effect of period j =,,3 { } τ t = effect of treatment t, ε randomerror iid N, σ ijk = j t ( ) π γ j r + γ = +π = τ + τ = ijk ( σ ) s

18 Parameters of the, design Yijk = µ + γ i + ξi k + π j + τ t + ε [ ] ( ) ijk E Y ijk = µ ijk µ µ µ 3 µ µ µ 3 µ = µ + γ + π + τ µ = µ + γ + π + τ + λ µ 3 = µ + γ + π 3 + τ + λ µ µ + γ + π + = = µ + γ + π = µ + γ + π 3 τ µ + τ + λ µ 3 + τ + λ λ + λ = π + π + π 3 = τ +τ = λ λ = +

19 Matrix again = λ τ π π γ µ β X ( ) = X T X Effect of treatment and carry over can be estimated independently!

20 Other sequence 3 period designs 3 4 ( ) Var τˆ ( ) Var λˆ N/.5.. N/ 3 Corr ( ˆ τ, ˆ λ ) N/ 4

21 Comparing the, and the, designs ( ) Var ˆ τ =.5 ˆ σ ( ) ˆ ˆ Var τ =.9σ Can t include carry over Carry over estimable treatments per subject 3 treatments per subject Shorter duration Longer duration Excercise: Find the best treatment 4 period design

22 More than treatments Tool of the trade: Define the model ( ) X T X C C C C C C D C D C C D C D Investigate C C C C C Watch out for drop outs!

23 Titration Designs Increasing dose panels (Phase I): SD (Single scending Dose) MD (Multiple scending Dose) Primary Objective: Establish Safety and Tolerability Estimate Pharmaco Kinetic (PK) profile Increasing dose panelse (Phase II): Dose - response

24 VERY careful with first group! Titration Designs (SD, MD) Dose: Z mg X on drug Y on Placebo Dose: Z mg X on drug Y on Placebo Dose: Z k mg X on drug Y on Placebo Stop if any signs of safety issues

25 Titration Designs Which dose levels? Start dose based on exposure in animal models. Stop dose based on toxdata from animal models. Doses often equidistant on log scale. Which subject? Healty volunteers Young Male How many subjects? Rarely any formal power calculation. Often on placebo and 6-8 on drug.

26 Titration Designs Not mandatory to have new subject for each group. X mg X mg X 3 mg X 4 mg X 5 mg X Y mg Gr. Gr. Gr. Gr. Gr. 3 Gr Slighty larger groups to have sufficiently many exposed. Dose in fourth group depends on results so far. Possible to estimate within subject variation.

27 Factorial design Evaluation of a fixed combination of drug and drug placebo placebo active placebo P placebo active active active The U.S. FD s policy ( CFR 3.5) regarding the use of a fixed-dose combination The agency requires: Each component must make a contribution to the claimed effect of the combination. Implication: t specific component doses, the combination must be superior to its components at the same respective doses

28 Factorial design Usually the fixed-dose of either drug under study has been approved for an indication for treating a disease. Nonetheless, it is desirable to include placebo (P) to examine the sensitivity of either drug give alone at that fixed-dose (comparison of with P may be necessary in some situations). ssume that the same efficacy variable is used for studying both drugs (using different endpoints can be considered and needs more thoughts).

29 Factorial design Sample mean Y i N( µ i, σ/n ), i =,, n = sample size per treatment group (balanced design is assumed for simplicity). H: µ µ or µ µ H: µ > µ and µ > µ j=, n Y Yj T j = ; j, : ˆ = σ Min test and critical region: min ( T T ) C :, : >

30 Group sequential designs large study is a a huge investment, $, ethics What if the drug doesn t work or is much better than expected? Could we take an early look at data and stop the study is it look good (or too bad)?

31 Repeated significance test Let: ( ~ N, σ ) Y Test: H : µ = µ ij Test statistic: µ j ˆ ˆ mk µ µ = ˆµ = σ mk mk i= Z mk j Y ij For k =, KK If Z k Cα Stop, reject otherwise Continue to group k + If H Z k Cα Stop, reject H otherwise stop accept H

32 True type I error rate Repeat testing until H rejected Tests Critical value P(type I error)

33 Pocock s test Suppose we want to test the null hypothesis 5 times using the same critical value each time and keep the overall significance level at 5% k K If Z C ( α K ) For =, K fter group K Choose C p ( α, K ) k p, Stop, reject otherwise Continue to group k + Stop, reject H If Z C ( α K ) Such that k p, otherwise stop accept H H P( Reject H at any analysis k = K K) = α

34 Pocock s test Tests Critical value P(type I error) ll tests has the same nominal significance level group sequential test with 5 interrim tests has level ( (.43) ). 58 α' = φ =

35 Pocock s test Z k Reject H.43 ccept H stage k -.43 Reject H

36 O rian & Flemmings test Increasing nominal significance levels k K If Z C ( α, K ) K k For =, K fter group K : Choose C p ( α, K ) : k p / Stop, reject otherwise Continue to group k + If Z C ( α K ) k p, Stop, reject H otherwise stop accept H Such that P( Reject H at any analysis k = K K) = α H

37 O rian & Flemmings test Critical values and nominal significance levels for a O rian Flemming test with 5 interrim tests. Test (k) C (K,α) C (K,α)*Sqrt(K/k) α Rather close to 5%

38 O rian & Flemmings test Z k Stage K

39 Comparing Pocock and O rian Flemming O rian Flemming Pocock Test (k) C (K,a)*Sqrt(K/k) α C P (K,a) α

40 Comparing Pocock and O rian Flemming 6 4 Zk Stage k

41 Group Sequential Designs Pros: Efficiency Gain (Decreasing marginal benefit) Establish efficacy earlier Detect safety problems earlier Cons: Smaller safety data base Complex to run Need to live up to stopping rules!

42 Selection of a design The design of a clinical study is influenced by: Number of treatments to be compared Characteristics of the treatment Characteristics of the disease/condition Study objectives Inter and intra subject variability Duration of the study Drop out rates

43 ackup ack up: Var ( Y ) ( ) ( ) Y = Var Y + Var Y ( ) ( σ + σ + σ + σ ) = s s n n