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Regression Analysis for Survival Data
Published in Trevor F. Cox, Medical Statistics for Cancer Studies, 2022
The probability density function of the gamma distribution is , where is the Gamma function. The survival function is , where is the incomplete Gamma function. The hazard function is and does not have a “nice” formula. Replace λ by , and so
Bayesian Hierarchical Models
Published in Andrew B. Lawson, Using R for Bayesian Spatial and Spatio-Temporal Health Modeling, 2021
would represent a basic model with intercept to capture the overall rate and prior distribution for the intercept and the random effect could be assumed to be and The hyperprior distribution for the variance parameters could be a distribution on the positive real line such as the gamma, inverse gamma, or uniform. The uniform distribution has been proposed for the standard deviation () by Gelman (2006). Here for illustration, I define a gamma distribution:
Precision Medicine
Published in Shein-Chung Chow, Innovative Statistics in Regulatory Science, 2019
where and denotes the density of a normal variable. For Bayesian approach, a beta distribution can be employed as the prior distribution forγ, while normal prior distributions can be used for and . In addition, a gamma distribution can be used as a prior for . Under the assumptions of these prior distributions, the conditional posterior distributions of can be derived. In other words, assuming that
Patient characteristics associated with all-cause healthcare costs of alopecia areata in the United States
Published in Journal of Medical Economics, 2023
Wei Gao, Arash Mostaghimi, Kavita Gandhi, Nicolae Done, Markqayne Ray, James Signorovitch, Elyse Swallow, Christopher Carley, Travis Wang, Vanja Sikirica
Power coefficients, representing the relationship between the sample mean and variance, were assessed using a modified Park Test. For example, in the gamma distribution, the variance is the square of the mean. The Park test was conducted by first running a generalized linear model with the specified distribution to compute expected values and squared errors for each observation, conditional on covariates. The natural log of the expected values and errors were then calculated, and a second regression was used to fit the variance model. A power estimate of 0 (no relationship between mean and variance) would suggest that a Gaussian distribution is appropriate. Power estimates of 1, 2, and 3 would suggest the appropriateness of Poisson, Gamma, and inverse-Gaussian distributions, respectively. Based on the power coefficients estimated from each of these models (Supplementary Material), a Gamma distribution was selected as the most appropriate distribution to model healthcare costs.
Evaluation of phase I clinical trial designs for combinational agents along with guidance based on simulation studies
Published in Journal of Applied Statistics, 2023
Shu Wang, Elias Sayour, Ji-Hyun Lee
In terms of parameters β, their paper assumes that β follows a Gamma distribution with mean
Effect of non-normality on the monitoring of simple linear profiles in two-stage processes: a remedial measure for gamma-distributed responses
Published in Journal of Applied Statistics, 2022
Paria Soleimani, Shervin Asadzadeh
This subsection deals with investigating the effect of t and gamma distributions on the performance of t-distribution kurtosis is significantly different from normal distribution for small degrees of freedom. Moreover, concentrating on gamma distribution, it is remarkable that for a fixed scale parameter, increasing the shape parameter makes the skewness and kurtosis criteria closer to normal distribution. Besides, the kurtosis of gamma distribution with the small shape parameter is significantly larger than the kurtosis of normal distribution. Table 1 provides the non-normal data effect on the in-control ARL (ARL0) of the t-distribution. It should be noted that the ARLs are calculated with 10,000 simulation runs.