On Exponential Growth and Half-Lives: A Comment on Bermingham

A recent paper (hereafter referred to as the Paper), Bermingham (2003), presented what appeared to be an analytical review of current situation of declining population growth rates in many parts of the world. The Paper suggested that the increasing growth rates before about 1970 constituted exponential growth, but that the currently declining growth rates were not exponential growth. Hence, the paper asserted that we should not use the terms exponential growth and doubling times in describing the current situation. Many of the suggestions in the paper are contrary to established mathematics. These suggestions are examined here in some detail.     

The Growth of Home Ownership: 1940-1980

Housing went through a major transition between 1940 and 1980 as the proportion of home owners nearly doubled. This article examines how that change took place. Compositional changes in the population and process changes led by post--World War II housing policies and programs are tested as explanations for the growth of home ownership. A process analysis and a decomposition of the change in home ownership between cohorts are used to evaluate each interpretation of change. A combination of both explanations accounts for the change in home ownership.     

Application of a nonlinear mathematical model to data on a successful theraputic intervention

A mathematical model is applied to happiness and mood data from a successful group therapy intervention with elderly nursing home residents (Rattenbury and Stones, 1989). The model permits estimations of acute environmental change within and across therapy sessions, and the comparison of such estimates with the changes in happiness from before to after the total intervention. The findings show intervention success (i.e., the change in happiness) to correlate with across session changes, and particularly those in the basal (presession) acute environment, but with no correlation between therapeutic success and within session changes. These findings suggest a positive facilitation to the chronic environment during the progression of therapy.     

A MCMC algorithm to fit a general exchangeable model

Consider the exchangeable Bayesian hierarchical model where observations y_i are independently distributed from sampling densities with unknown means, the means µi, are a random sample from a distribution g, and the parameters of g are assigned a known distribution h. A simple algorithm is presented for summarizing the posterior distribution based on Gibbs sampling and the Metropolis algorithm. The software program Matlab is used to implement the algorithm and provide a graphical output analysis. An binomial example is used to illustrate the flexibility of modeling possible using this algorithm. Methods of model checking and extensions to hierarchical regression modeling are discussed.     

Modified cramer-von mises and anderson-darling tests for weibull distributions with unknown location and scale parameters

The standard Cramer-von Mises and Anderson-Darling goodness-of-fit tests require continuous underlying distributions with known parameters. In this paper, tables of critical values are generated for both tests for Weibull distributions with unknown location and scale parameters and known shape parameters. The powers of the Cramer-von Mises, Anderson-Darling, Kolmogorov-Smirnov, and Chi-Square tests for this situation are investigated. The Cramer-von Mises test has most power when the shape is 1.0 and the Anderson-Darling test has most power when the shape is 3.5. Finally, a relation between critical value and inverse shape parameter is presented.     

A Bayesian Analysis of a Poisson Random Effects Model for Home Run Hitters

The problem of interest is to estimate the home run ability of 12 great major league players. The usual career home run statistics are the total number of home runs hit and the overall rate at which the players hit them. The observed rate provides a point estimate for a player's “true” rate of hitting a home run. However, this point estimate is incomplete in that it ignores sampling errors, it includes seasons where the player has unusually good or poor performances, and it ignores the general pattern of performance of a player over his career. The observed rate statistic also does not distinguish between the peak and career performance of a given player. Given the random effects model of West (1985), one can detect aberrant seasons and estimate parameters of interest by the inspection of various posterior distributions. Posterior moments of interest are easily computed by the application of the Gibbs sampling algorithm (Gelfand and Smith 1990). A player's career performance is modeled using a log-linear model, and peak and career home run measures for the 12 players are estimated.