1. 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:

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

3. 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

4. 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

5. Real Example: Osteoporosis Ph 2 Study
From study protocol, available at:

6. 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…

7. 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

8. 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

9. 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)

10. 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)

11. Normal Data, Additive Effect
Case 1 (% change not normal) Case 2 (% change normal)
Δ= 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 = 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

12. 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

13. Chi-square Data Additive Proportional
Mean diff = % 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

14. Poisson* Data Additive Proportional
Mean diff = % 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

15. 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

16. 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 .