Signature
p_j = exp(theta_j) / (1 + exp(theta_j))
| Inputs | Definition | Unit |
|---|---|---|
theta_j | Log-odds of response in basket j, the quantity the hierarchical model treats as exchangeable across baskets | log-odds |
p_j | Response probability in basket j | probability |
|---|
Function
Partial pooling of response rates across basket trial baskets
Maps the responders and evaluable patients in each basket of a basket trial, together with a common mean response and a between-basket variance, to a response estimate for each basket that borrows information from the other baskets. The result lies between analysing each basket separately and pooling every patient. The basket estimates can then be reweighted to the mix of tumour types expected in practice before they enter a response-based economic model. A full hierarchical model does the same jointly on the log-odds scale.
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Implementations
Excel
Basket response probability from a log-odds cell
With the log-odds in a cell named LogOdds, the formula returns the response probability. LN, the natural logarithm, reverses it.
=EXP(LogOdds)/(1+EXP(LogOdds))
Assumptions
Logit link between basket response and the hierarchical scale
The hierarchical model places its normal distribution on the log-odds scale, so a log-odds and a probability are not interchangeable. Because the transformation is not linear, the mean of the back-transformed probabilities differs from the back-transformed mean of the log-odds.
Worked examples
Prior centre of minus 0.8473 as a basket response probability
Murphy and colleagues centred the normal prior for the mean log-odds of response on minus 0.8473, which corresponds to a response probability of 0.30.
theta_j = -0.8473; p_j = 0.3000
Log-odds of zero as a basket response probability of one half
A log-odds of 0 corresponds to a response probability of 0.50, the common mean in the article's illustrative example.
theta_j = 0; p_j = 0.50
Common errors
Log-odds mean entered as a basket response probability
Entering a log-odds mean of minus 0.8473 into HE-FM-BSKT-001 as if it were a probability gives a negative sampling variance, about minus 0.157 for a basket of 10, and a partially pooled estimate outside the possible range. The common mean must first be converted to the probability scale, 0.30 in this case.
Sources
Murphy and colleagues on the logit hierarchical model for histology response
Murphy P, Claxton L, Hodgson R, Glynn D, Beresford L, Walton M, Llewellyn A, Palmer S, Dias S. Exploring heterogeneity in histology-independent technologies and the implications for cost-effectiveness. Medical Decision Making. 2021;41(2):165-178. Methods section on predictive response using the Bayesian hierarchical model, which treats the log-odds of response in each histology as exchangeable and normally distributed, recovers the response probability as exp of theta divided by one plus exp of theta, and centres the prior for the mean on a response probability of 0.3, a log-odds of minus 0.8473.
Canonical Identity
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