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The posterior density (shown in red) is more peaked and shifted to the left compared with the prior
distribution (shown in blue). The posterior distribution combined the prior information about with
intro — Introduction to Bayesian analysis 3
the information from the data, from which y = 0 provided evidence for a low value of and shifted
the prior density to the left to form the posterior density. Based on this posterior distribution, the
posterior mean estimate of is 2=(2 + 40) = 0.048 and the posterior probability that, for example,
< 0.10 is about 93%.
If we compute a standard frequentist estimate of a population proportion as a fraction of the
infected subjects in the sample, y = y=n, we will obtain 0 with the corresponding 95% confidence
interval (y �� 1.96
p
y (1 �� y)=n; y + 1.96
p
y (1 �� y)=n) reducing to 0 as well. It may be difficult
to convince a health policy maker that the prevalence of the disease in that city is indeed 0, given
the small sample size and the prior information available from comparable cities about a nonzero
prevalence of this disease.
Are there influential studies in your data?
Panel-data ERMs
Extended regression models (ERMs) were a big new feature last release. The ERM commands fit models that account for three common problems that arise in observational data—endogenous covariates, sample selection, and treatment—either alone or in combination.
In Stata 16, we introduce the xteregress, xteintreg, xteprobit, and xteoprobit commands for fitting panel-data ERMs. This means ERMs can now account for the three problems we mentioned above and for within-panel correlation. These new commands fit random-effects linear, interval, probit, and ordered probit regression models. They allow random effects in one or all equations, and they allow random effects to be correlated across equations.
Researchers from all disciplines who work with observational (nonexperimental) data are interested in ERMs and will be excited about the new panel-data versions of these commands. However, different disciplines talk about these models differently.
Above, we referred to the problems ERMs solve as endogenous covariates, sample selection, treatment, and within-panel correlation. While this terminology is common in some disciplines such as economics, other disciplines may use other terms.
• Instead of panel-data and within-panel correlation, researchers may ask for models for multilevel (two-level) data that account for within-group correlation.
• Instead of endogenous covariates, researchers may ask for methods of dealing with unobserved confounding or unmeasured confounding.
• Instead of sample selection, researchers may be concerned about trials with informative dropout, nonignorable nonresponse, or outcomes missing not at random (MNAR).
• Instead of treatment, researchers may ask about methods for causal inference or estimating average treatment effects (ATEs).
The important message is that all disciplines are interested in ERMs, but they often speak different languages.
Study-specific effect sizes
Precision of effect sizes
Overall effect size
Detect potential outliers
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