Students Opportunities: Conferences: Free Opportunities: Free webinars for members Free student membership Professional Growth: Affordable trainings at conferences Get mentoring from professionals Students Opportunities: Scholarships Travel grants Student paper awards Conferences: Joint Statistical Meeting Regulatory-Industry Stat. Workshop Nonclinical Biostatistics Conference Learn more by visiting the website: https://community.amstat.org/biop/home

The Use of Percent Change as an Endpoint in Clinical Trials 2019 JSM Jitendra Ganju Ganju Clinical Trials, LLC jganju@yahoo.com and Kefei Zhou Jazz Pharmaceuticals

Percent Change vs Change Reasons for using % change - Follow-up value depends on baseline value - Clinically relevant metric Question: Are these reasons good enough to prefer % change over absolute change? Goal of talk: to explain why change is still usually better than % change

Percent Change in Practice Blood measures, pain scores, bone mineral density It is well-known that when baseline mean not large relative to std dev distribution of % change unstable We give more reasons to prefer change over % change Vickers, BMC Medical Research Methodology, 2001

Real Example: Osteoporosis Ph 2 Study From study protocol, available at: https://static-content.springer.com/esm/art%3A10.1007%2Fs00198-015-3392-7/MediaObjects/198_2015_3392_MOESM1_ESM.pdf

Power or Clinical Relevance? Is ‘clinically relevant’ a good enough reason? Suppose power for change >> power for % change power for change > power for % change power for change ≈ power for % change. Don’t know power for imputation adjusted population or power for subgroups unknown power for change < power for % change But wait…

Implication of What is Not Considered Different populations and outlets for results How well do we understand % change power? Implications for small pharma. Example: % change result positive for primary popln but negative for subgroup. Absolute change result positive for both. Not a trivial matter! Observed data Imputation Subgroup Prescribing info ✓ ? FDA review Publication

Notation Baseline: X, Follow-up: Y, Z = 𝑌−𝑋 𝑋 Will look at Y as an additive response (e.g. Y = 10+X) and as proportional to baseline (e.g. Y = 0.75*X) T = treatment (A, B) Δ = Group B mean – Group A mean

Marsaglia’s Remarkable Work Well known result: Z = 𝑌−𝑋 𝑋 not stable if 𝜇 𝜎 for X small Marsaglia shows that 𝜇 𝜎 for Y also affects distribution. Gives precise conditions for normal Z (amazing result) Example: X and Y normal, 𝜎=1, Corr(X, Y)=0.5 𝜇 𝑋 =5 𝜇 𝑋 =5 𝜇 𝑌 =7 𝜇 𝑌 =3 Z more normal-like when Y = X – k than when Y = X + k, k > 0 Conditions not included because excessive notation required Marsaglia G. Ratios of normal variables. J. Statistical Software (2006), and JASA (1965)

Simulations Corr(X, Y) ≈ 0.5 N = 50/group Power (two-sided alpha = 5%) Z = T two sample t test, Z Z = T + X covariate adjusted, Z|X Y = T + X covariate adjusted, Y|X (same as Y – X = T + X)

Normal Data, Additive Effect Case 1 (% change not normal) Case 2 (% change normal) Δ=0.5 Δ=0.5 Marsaglia conditions not met for Y Marsaglia conditions met Details All variances = 1 Case 1: 𝜇 𝑋 = 5, 𝜇 𝑌𝐴 = 7, 𝜇 𝑌𝐵 = 7.5 Case 2: 𝜇 𝑋 = 5, 𝜇 𝑌𝐴 = 3, 𝜇 𝑌𝐵 = 3.5 81 81 78 79 73 42 Z Z|X Y|X Z Z|X Y|X

Marsaglia conditions not met for X and Y Normal Data, Bad Case Marsaglia conditions not met for X and Y Δ=0.5 Details All variances = 1 𝝁 𝑿 = 3, 𝝁 𝒀𝑨 = 4, 𝝁 𝒀𝑩 = 4.5 81 40 25 Z Z|X Y|X

Chi-square Data Additive Proportional Mean diff = 2 25% reduction from baseline Details Additive: X ~ 𝜒 2 7 , YA ~ 𝜒 2 7 , YB ~ 𝜒 2 9 Proportional: X ~ 𝜒 2 7 , YA ~ 𝜒 2 7 , YB ~ 0.75∗𝜒 2 7 85 81 70 69 60 58 Z Z|X Y|X Z Z|X Y|X

Poisson* Data Additive Proportional Mean diff = 2 10% reduction from baseline Details Additive: X ~ 1+Pois(18), YA ~ 1+Pois(18), YB ~ 1+Pois(20) Proportional: X ~ 1+Pois(20), YA ~ 1+Pois(20), YB ~ round(0.9*(1+Pois(20))) *1 added to avoid 0 in denominator 80 76 74 70 64 57 Z Z|X Y|X Z Z|X Y|X

Mixture of Poissons* Proportional 30% reduction from baseline Details 90% of data from Pois(20), 10% from Pois(80) (non-parametric methods perform much better) Details X and YA : with prob. 0.9 and 0.1, resp., from Pois(20) and Pois(80) YB ~ 0.7 of mixture of Pois(20) and Pois(80), with p = 0.9 and 0.1 *1 added to avoid 0 in denominator 72 68 63 Z Z|X Y|X Corr(X, Y) ≈ 0

Main Points Power is paramount. Difference more stable than ratio. Make ancova on Y default for inference. Make the case for % change Power of % change with covariate > w/o covariate. Covariate more important when Marsaglia conditions not met If highly non-normal data then non-parametric methods better. Comparisons should be between ranked % change and ranked raw data .