Bios 6648: Design & conduct of clinical research

Transcription

1 Bios 6648: Design & conduct of clinical research Section 2 - Formulating the scientific and statistical design designs 2.5(b) Binary 2.5(c) Skewed baseline (a) Time-to-event (revisited) (b) Binary (revisited) (c) Skewed (d) Change-from-baseline Bios pg 1

2 designs (d) Change from baseline Outline: 1. Motivation and data structure 2. Approaches to defining when an endpoint is measured at baseline and follow-up 3. Other applications 4. Statistical design (sample size and CI evaluations) 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 2

3 (d) Change from baseline 1. Motivation and data structure Why measure change? Usually: within-individual change is less variable. For example, consider the CHEST data: Subj Trt 6 Minute walk distance ID Grp Baseline 12-weeks Change (b) Binary 2.5(c) Skewed baseline Bios pg 3

4 (d) Change from baseline 1. Motivation and data structure CHEST Trial summary statistics Study Placebo Riociguat Visit Mean (sd) Mean (sd) Baseline ( 74.7) ( 81.9) 16-weeks (122.2) (119.2) Change -5.5 ( 84.3) 38.9 ( 79.3) 2.5(b) Binary 2.5(c) Skewed baseline Approaches to analysis: Compare 6MWD after 16-weeks Compare 16-week improvement in 6MWD Linear regression: 16-week 6MWD conditional on baseline walk distance. Bios pg 4

5 (d) Change from baseline 2. Approaches to outcome definition Evaluate difference at last time measurement time: Outcome: Final measurement on each subject. Functional: θ1f and θ 0F denote the mean outcome with active and control therapy at the final measurement time. Contrast: θ = θ1f θ 0F. Example (CHEST) T-test of 16-week walk distance: 2.5(b) Binary 2.5(c) Skewed baseline ˆθ = ˆθ 1 ˆθ 0 = = se(ˆθ) = = %CI = ± se = ( , 62.08) p value = Result: At 16-week mean walk distance with Riociguat is meters farther than placebo (95% CI: to meters; p = 0.055). Notice inconclusive result. Bios pg 5

6 (d) Change from baseline 2. Approaches to outcome definition Evaluate change over follow-up period: Outcome: Change in outcome measurement over the time (final minus baseline) Functional: θ 1 = θ 1F θ 1B (mean change with active treatment) θ 0 = θ 0F θ 0B (mean change with control treatment) Contrast: θ = θ1 θ 0. CHEST: T-test of change in walk distance over 16 weeks: 2.5(b) Binary 2.5(c) Skewed baseline ˆθ = ˆθ 1 ˆθ 0 = 38.9 ( 5.5) = se(ˆθ) = = %CI = ± se = (23.05, 65.78) p value = Result: Over 16-weeks Riociguat treatment improves mean walk distance by meters (95% CI: to meters; p < ) more than the improvement with placebo. Bios pg 6

7 (d) Change from baseline 2. Approaches to outcome definition Last measurement time adjusting for baseline: Outcome: Outcome at the last measurement time (adjusting for baseline value) Functional: outcome at last measurement time Contrast: Difference in mean adjusted for baseline levels (θ in the following regression model): 2.5(b) Binary 2.5(c) Skewed baseline θ kf = β 0 + θtx + β 1 θ kb where Tx is the indicator for active treatment. CHEST: Fit the linear regression model: Y if = β 0 + θtx i + β 1 Y ib Estimate Std. Error t value Pr(> t ) (Intercept) e-02 exmpl[, "RioTx"] e-05 exmpl[, "base6"] e-47 That is: ˆβ0 = ˆθ = ˆβ 1 = Bios pg 7

8 (d) Change from baseline 2. Approaches to outcome definition Last measurement time adjusting for baseline: CHEST Result (con t): Point estimate: Among two populations with the same baseline walk distance, after 16 weeks a population taking riociguat will end up walking 46.1 meters farther than a population taking placebo. 95% CI: (25.35, 66.84) P-value: Comparison of approaches: 2.5(b) Binary 2.5(c) Skewed baseline Approach Estimate 95% CI p-value 16-week difference (-0.55, 62.08) Change from baseline (23.05, 65.78) Adjusting for baseline 46.1 (25.35, 66.84) Bios pg 8

9 (d) Change from baseline Graphical depiction of the CHEST results 16 Week WD P R Placebo Riociguat 2.5(b) Binary 2.5(c) Skewed baseline P R Baseline WD Bios pg 9

10 (d) Change from baseline Notes on previous graph: Boxplots on left represent the data used for a t-test of 16-week differences. Regression lines are the result of the above linear regression analysis. Vertical distance between regression lines is 46.1 meters (i.e., the effect of riociguat adjusted for baseline walk distance). Bloxplots on bottom show no confounding (similar distribution of baseline WD). This example shows how we can increase power by adjusting for a precision variable. I now illustrate the general behavior in a series of graphs 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 10

11 Precision variables (in linear regression) A precision variable reduces noise" (extraneous variation) so that the relationship between outcome and the primary explanatory variable is more precise. A precision variables must be: unrelated to the primary explanatory variable. an independent predictor of outcome. Adjusting for a precision variable increases precision for the comparison of interest. 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 11

12 Example (Precision variable) The nature of a precision variable can be illustrated using scatterplots. Let: Y denotes outcome X denotes primary explanatory variable (2-categories: H and L) Z denotes a covariate (precision variable) We are interested in the relationship between X and Y. In the following plots: The relationship between X and Y is fixed. There is no relationship between X and Z (the precision variable is unrelated to the explanatory variable). The relationship (correlation) between Y and Z is increasing. 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 12

13 Example (Figure 1a) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 13

14 Example (Figure 1b) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 14

15 Example (Figure 1c) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 15

16 Example (Figure 1d) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 16

17 Example (Figure 1e) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 17

18 Example (Figure 1f) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 18

19 Example (Figure 1g) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 19

20 Example (Figure 1h) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 20

21 Example (Figure 1i) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 21

22 Example (Figure 1j) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 22

23 Example (Figure 1k) Response (Y) L H (b) Binary 2.5(c) Skewed baseline L H Covariate (Z) Bios pg 23

24 Example (Precision variable) We would like to compare the results of an analysis if we ignore the precision variable (the crude" difference) with the results after we adjust for the precision variable (the adjusted difference"). (Z,Y) Crude Difference Adjusted Difference Correlation Estimate SE Estimate SE Fig 1a Fig 1b Fig 1c Fig 1d Fig 1e Fig 1f Fig 1g Fig 1h Fig 1i Fig 1j Fig 1k (b) Binary 2.5(c) Skewed baseline Bios pg 24

25 Example (Precision variable) Notice: Crude and adjustments are identical Standard error (SE) of the adjusted estimate is smaller than the standard error of the crude estimate (Note: smaller SE gives more power) The precision of the adjusted estimate increases with the correlation between Y and Z The precision variable is explaining" some of the variation (reducing the noise) in the primary comparison 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 25

26 (d) Change from baseline 3. Other applications Other situations in which the primary analysis is adjusted for baseline values: Common to adjust for stratification variables May adjust for scientific interpretability May adjust for comparability to previous studies It is important to pre-specify all adjustments as part of your primary analysis. 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 26

27 (d) Change from baseline 4. Implications for power and information The above examples and graphical illustration illustrate general principles: Variance (precision) of various approaches to defining with change from baseline data. It is particularly clear when σ0 = σ 1 (= σ) and N 1 = N 0 (= N): 2.5(b) Binary 2.5(c) Skewed baseline Variance when comparing only the last time point: var(ˆθ) = 2σ2 N Variance when comparing change from baseline: var(ˆθ) = 4σ2 (1 ρ) N Variance when comparisons are adjusted for baseline: var(ˆθ) = 2σ2 (1 ρ 2 ) N In all of the above, ρ is the correlation between baseline and follow-up measures. (1) (2) (3) Bios pg 27

28 (d) Change from baseline 4. Implications for power and information The above relationships can be used to prove: If ρ < 0.5, then it is more powerful to analyze follow-up differences (i.e., DO NOT compare change from baseline). If ρ > 0.5 then it is more powerful to analyze change from baseline than follow-up differences. it is always more powerful to us regression to adjust for baseline: This is also known as Analysis of covariance" (ANCOVA). The CHEST paper refers to it as the least-squares" estimate. 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 28

29 (d) Change from baseline 4. Implications for power and information Relative sample size of the analytic approaches: Change from baseline relative to follow-up only: 2.5(b) Binary 2.5(c) Skewed baseline 4σ 2 (1 ρ) 2σ 2 = 2(1 ρ) ANCOVA relative to follow-up only: 2σ 2 (1 ρ 2 ) 2σ 2 = (1 ρ 2 ) Bios pg 29

30 (d) Change from baseline 4. Implications for power and information Relative sample size of analytic approaches as function of correlation: Relative sample size Sample size relative to follow up only analysis Follow up only Change from baseline ANCOVA 2.5(b) Binary 2.5(c) Skewed baseline Correlation Bios pg 30

31 (d) Change from baseline 4. Implications for power and information How to evaluate information and sample size with change from baseline : Follow the same approaches described for other * Potential inference * Power and hypotheses discriminated Evaluate sensitivity to assumptions about correlation * As with other design parameters it may be necessary to evaluate sensitivity to design parameters. * See example below (b) Binary 2.5(c) Skewed baseline Bios pg 31

32 Evaluating information in a change from baseline design Use the form for V N given above (equations (1)-(3)) as we have previously done: Potential inference given V * Critical value = 1.96 N : V N * 95% CI at critical value = ( ) V 0, 3.92 N 2.5(b) Binary 2.5(c) Skewed baseline Find the power given V N : z β = θ + θ V N z α Sample size needed for power β: N = [ ] 2 zα + z β V θ + θ Bios pg 32

33 Evaluating information in a change from baseline design Power as a function of sample size, variance, and correlation Total sample size σ Correlation = Correlation = Correlation = Correlation = For example: With 350 subjects (175 per group) an ANCOVA analysis has 97.6% power if σ = 100 and and correlation is moderately high (ρ = 0.7). 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 33

34 (d) Change from baseline 4. Implications for power and information Frequent asked questions (FAQ) about the ANCOVA analysis: The ANCOVA model described above fits parallel lines. What happens if the lines are not parallel? * Non-parallel lines represents interaction; treatment works better (or worse) for low baseline values. * Interactions are explored in subsequent trials. What happens if the relationships are not linear? * Not a problem as long as baseline distribution is the same in both treatment groups (assured by randomization). * The line represents the first order approximation to the curve (i.e., is it treading up, down, or flat?). 2.5(b) Binary 2.5(c) Skewed baseline Bios pg 34

35 (d) Change from baseline 4. Implications for power and information Example: ANCOVA FAQ s in the CHEST trial No evidence for substantial non-linearity: 16 Week WD Relationship between baseline and 16 week walk distance Placebo Riociguat 2.5(b) Binary 2.5(c) Skewed baseline Baseline WD Bios pg 35

36 (d) Change from baseline 4. Implications for power and information Example: ANCOVA FAQ s in the CHEST trial Separate lines in each treatment group are nearly parallel: 16 Week WD Relationship between baseline and 16 week walk distance Placebo Riociguat 2.5(b) Binary 2.5(c) Skewed baseline Baseline WD No problem with interaction. Bios pg 36

37 (d) Change from baseline 4. Implications for power and information Example: ANCOVA FAQ s in the CHEST trial Separate lines in each treatment group are nearly parallel: 16 Week WD Relationship between baseline and 16 week walk distance Placebo Riociguat 2.5(b) Binary 2.5(c) Skewed baseline Baseline WD No problem with interaction. Bios pg 37