December 6, 2021
3:30: Introduction and background theory (30 minutes)
4:00: Proportional hazards applications with Shiny app (25 minutes)
4:25: Intro to nonproportional hazards (NPH; 5 minutes)
4:30: Software and piecewise model (15 min)
4:45: Average hazard ratio (AHR; 20 minutes)
5:05: Break (10 minutes)
5:15: NPH design with logrank test (25 minutes)
5:40: Weighted logrank and combination tests (40 minutes)
6:20 Summary and questions (10 minutes)
All opinions expressed are those of the presenters and not Merck Sharp & Dohme Corp., a subsidiary of Merck & Co., Inc., Kenilworth, NJ, USA.
Some slides need to be scrolled down to see the full content.
For minimum effort and maximum benefit, you will probably at least want to use the Shiny interface for the gsDesign R package at https://rinpharma.shinyapps.io/gsdesign/. This is also available at https://gsdesign.shinyapps.io/prod/, but this site is only licensed for a small number of simultaneous users.
data/
: contains design files for examples; also simulation resultsvignettes/
: reports produced by Shiny app to summarize designssimulation/
: R code and simulation data for the last part of courseAlternate hypothesis: Type II error \(\beta= 1  \hbox{power}\)
\[\infty\le a_k<b_k, k=1,\ldots,K1,\] \[a_K\le b_K\] If \(a_K = b_K\) then total Type II error is \[\beta = \sum_{k=1}^{K} l_k = \sum_{k=1}^{K} \text{Pr}(\{Z_k < a_k\} \cap_{j=1}^{k1} \{a_j \le Z_j \le b_j\}\mid H_1)\]
Test each treatment for superiority vs the other
Usually not of interest in pharmaceutical industry
Give up if experimental arm not trending in favor of control?
Give up if experimental arm trending worse than control
Approaches to calculate decision boundary:
The error spending approach: specify boundary crossing probabilities at each analysis. This is most commonly done with the error spending function approach (Lan and DeMets 1983).
The boundary family approach: specify how big boundary values should be relative to each other and adjust these relative values by a constant multiple to control overall error rates. The commonly applied boundary family include:
Main idea:
Modified HaybittlePeto procedure 1:
Bonferroni adjustment:
Advantages:
Definition:
For 2sided testing, Wang and Tsiatis (1987) defined the boundary function for the \(k\)th look as \[ \Gamma(\alpha, K, \Delta) k^{\Delta  0.5}, \] where \(\Gamma(\alpha, K, \Delta)\) is a constant chosen so that the level of significance is equal to \(\alpha\).
Two special cases:
For 2sided testing, the Pocock procedure rejects at the \(k\)th equallyspaced of \(K\) looks if \[Z_k > c_P(K),\] where \(c_P(K)\) is fixed given \(K\) such that \(\text{Pr}(\cup_{k=1}^{K} Z_k > c_P(K)) = \alpha\).
total number of looks(K)  \(\alpha = 0.01\)  \(\alpha = 0.05\)  \(\alpha = 0.1\) 

1  2.576  1.960  1.645 
2  2.772  2.178  1.875 
4  2.939  2.361  2.067 
8  3.078  2.512  2.225 
\(\infty\)  \(\infty\)  \(\infty\)  \(\infty\) 
We will reject \(H_0\) if \(Z(k/4) > 2.361\) for \(k = 1,2,3,4\) (final analysis).
Weakness:
Overly aggressive interim bounds
High price for the end of the trial.
\(c_P(K) \to +\infty\) as \(K \to + \infty\).
Requires equally spaced looks.
total number of looks(K)  \(\alpha = 0.01\)  \(\alpha = 0.05\)  \(\alpha = 0.1\) 

1  2.576  1.960  1.645 
2  2.580  1.977  1.678 
4  2.609  2.024  1.733 
8  2.648  2.072  1.786 
16  2.684  2.114  1.830 
\(\infty\)  2.807  2.241  1.960 
Example:
Procedure name  Boundary  Advantages  Disadvantages 

HaybittlePeto  K1 at interim analyses and 1.96 at the final analysis  simple to implement  
Pocock  a constant decision boundary for Zscore  (1) requires the same level of evidence for early and late looks at the data, so it pays larger price for the final analysis ; (2) requires equally spaced looks 

Oâ€™BrienFleming  constant Bvalue boundaries, steep decrease in Zboundaries  pay smaller price for the final analysis  too conservative in the early stages? 
Key aspects of the design as documented in the protocol accompanying Gandhi et al (2018).
Poisson mixture cure model we consider:
\[S(t)= \exp(\theta (1  \exp(\lambda t)).\]
Note that:
More details in book.
See the following link for the Moderna COVID19 design replication: https://medium.com/@yipeng_39244/reverseengineeringthestatisticalanalysesinthemodernaprotocol2c9fd7544326
Can you reproduce this using the Shiny interface?