The terminology “hierarchical model” is quite general and can imply anything from simple use of a prior distribution to a highly organized data hierarchy (students nested in classes nested in schools nested in school systems nested in states nested in countries).
Hierarchical models are often used in the following commonly-encountered settings:
members of a “cluster” share more similarities with each other than with members of other clusters, violating the typical independence assumption of generalized linear models (like linear or logistic regression) – examples of clusters include members of a family or students in a class
hypotheses of interest include context-dependent associations, often across a large number of settings – e.g., does success of a new mode of instruction depend on the individual teacher
it is necessary to borrow information across groups in order to stabilize estimates or to obtain estimates with desirable properties – e.g., we want to make state-specific estimates of election candidate preference by country of origin, but some states may have few immigrants from a given country
The generalized linear model framework accommodates many popular statistical models, including linear regression, logistic regression, probit regression, and Poisson regression, among others.
Two popular GLM’s we will use in class include the linear regression model and the logistic regression model.
Linear regression is perhaps the most widely-used statistical model. This model is given by \[y_i=\beta_0+\beta_1x_{1i}+\cdots+\beta_px_{pi}+\varepsilon_i,\] where \[\varepsilon_i \sim N\left(0,\sigma^2\right).\]
If the parameter \(\beta_j>0\), then increasing levels of \(x_j\) are associated with larger expected values of \(y\), and values of \(\beta_j<0\) are associated with smaller expected values of \(y\). \(\beta_j=0\) is consistent with no association between \(x_j\) and \(y\).
Logistic regression is a type of generalized linear model, which generalizes the typical linear model to binary data. Let \(y_i\) take either the value 1 or the value 0 (the labels assigned to 1 and 0 are arbitrary – that is, we could let 1 denote voters and 0 denote non-voters, or we could exchange the labels – we just need to remember our coding).
The logistic regression model is linear on the log of the odds: \[\log\frac{\pi_i}{1-\pi_i}=\beta_0+\beta_1x_{1i}+\cdots+\beta_px_{pi},\] where \(\pi_i=Pr(y_i=1)\).
If the parameter \(\beta_j>0\), then increasing levels of \(x_j\) are associated with higher probabilities that \(y=1\), and values of \(\beta_j<0\) are associated with lower probabilities that \(y=1\). \(\beta_j=0\) is consistent with no association between \(x_j\) and \(y\).
For some intuition behind hierarchical models, we’ll check out this neat tutorial by Michael Freeman at University of Washington.
Suppose an expert suggests the following average 9-month salaries for starting assistant professors (0 years experience) by discipline: (biomedical) informatics (102K), English (52K), sociology (60K), biology (80K), and statistics (105K), with a similar impact of 3.5% higher salary per year of experience all disciplines.
Specify a model and corresponding parameter values consistent with this information.
Note: this approximate model for faculty salaries is not a very good one!
A firm specializing in security at sports events has conducted an experiment comparing three mechanisms to reduce the crowd size awaiting entry at any one time. The firm compares three strategies: full screening with bag searches, a clear bag strategy that eliminates the need for searching every bag, and enhanced full screening (with extra screening stations). The outcome of interest is a summary variable representing the average size of the crowd, averaged over the 20 minutes with the biggest crowd size. (It is desirable to minimize crowd size at entry because at this point there has been no screening for safety.) Your data include crowd size for over 200 major college and professional sporting events, with roughly 1/3 of the events using each strategy.
Specify a statistical model (using equations if possible) for evaluating the null hypothesis that the three strategies are associated with equal crowd sizes. Explain how you would determine which strategy is the best (assuming that is based only on crowd size).