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Prior Distributions for Variance Parameters in Hierarchical Models Gelman, Andrew Uploaded 03-28-2004 Keywords Bayesian inference hierarchical model multilevel model noninformative prior distribution weakly informative prior distribution Abstract Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new folded-noncentral-$t$ family of conditionally conjugate priors for hierarchical standard deviation parameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inverse-gamma family of "noninformative" prior distributions. We suggest instead to use a uniform prior on the hierarchical standard deviation, using the half-$t$ family when the number of groups is small and in other settings where a weakly informative prior is desired.

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Rich state, poor state, red state, blue state:What's the matter with Connecticut? Gelman, Andrew Shor, Boris Bafumi, Joseph Park, David Uploaded 11-29-2005 Keywords availability heuristic ecological fallacy hierarchical model income and voting multilevel model presidential elections public opinion secret weapon varying-slope model Abstract We find that income matters more in ``red America'' than in ``blue America.'' In poor states, rich people are much more likely than poor people to vote for the Republican presidential candidate, but in rich states (such as Connecticut), income has a very low correlation with vote preference. In addition to finding this pattern and studying its changes over time, we use the concepts of typicality and availability from cognitive psychology to explain how these patterns can be commonly misunderstood. Our results can be viewed either as a debunking of the journalistic image of rich ``latte'' Democrats and poor ``Nascar'' Republicans, or as support for the journalistic images of political and cultural differences between red and blue states---differences which are not explained by differences in individuals' incomes. For decades, the Democrats have been viewed as the party of the poor, with the Republicans representing the rich. Recent presidential elections, however, have shown a reverse pattern, with Democrats performing well in the richer ``blue'' states in the northeast and west coast, and Republicans dominating in the ``red'' states in the middle of the country. Through multilevel modeling of individual-level survey data and county- and state-level demographic and electoral data, we reconcile these patterns. Key methods used in this research are: (1) plots of repeated cross-sectional analyses, (2) varying-intercept, varying-slope multilevel models, and (3) a graph that simultaneously shows within-group and between-group patterns in a multilevel model. These statistical tools help us understand patterns of variation within and between states in a way that would not be possible from classical regressions or by looking at tables of coefficient estimates.

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A default prior distribution for logistic and other regression models Gelman, Andrew Jakulin, Aleks Pittau, Maria Grazia Su, Yu-Sung Uploaded 08-03-2007 Keywords Bayesian inference generalized linear model least squares hierarchical model linear regression logistic regression multilevel model noninformative prior distribution Abstract We propose a new prior distribution for classical (non-hierarchical) logistic regression models, constructed by first scaling all nonbinary variables to have mean 0 and standard deviation 0.5, and then placing independent Student-$t$ prior distributions on the coefficients. As a default choice, we recommend the Cauchy distribution with center 0 and scale 2.5, which in the simplest setting is a longer-tailed version of the distribution attained by assuming one-half additional success and one-half additional failure in a logistic regression. We implement a procedure to fit generalized linear models in R with this prior distribution by incorporating an approximate EM algorithm into the usual iteratively weighted least squares. We illustrate with several examples, including a series of logistic regressions predicting voting preferences, an imputation model for a public health data set, and a hierarchical logistic regression in epidemiology. We recommend this default prior distribution for routine applied use. It has the advantage of always giving answers, even when there is complete separation in logistic regression (a common problem, even when the sample size is large and the number of predictors is small) and also automatically applying more shrinkage to higher-order interactions. This can be useful in routine data analysis as well as in automated procedures such as chained equations for missing-data imputation.