Whether we use maximum likelihood estimation (Chapter 4) or some other method (Chapter 3) for fitting models to data, one issue is how to fit data from multiple units. Those units are typically individuals, but we can also observe higher-level clustering of individuals (e.g., students in different schools). One issue we already gave some discussion to in Chapter 4 is how to model the data from individual participants. We know from the previous chapter that we can obtain a joint log-likelihood for multiple participants by multiplying the individual likelihoods (or, equivalently, adding log-likelihoods), but this leaves open multiple ways of fitting the data from multiple individuals. This chapter addresses different ways of fitting multiple participants. We first highlight how the manner of obtaining an “ average” fit can affect the conclusions we draw from data, and then consider different ways in which multiple participants can be modelled. Later in the chapter we discuss ways of identifying and fitting clusters of participants, and discuss how individual differences can be accounted for in computational models.

It Matters How You Combine Data from Multiple Units

Suppose you are an affirmative-action officer at a major university and you learn that of the nearly 13,000 applicants to your institution's graduate programs, 8,442 were males and 4,321 were females. Suppose furthermore that 44% of the males but only 35% of the females were admitted. Red alert! Inscapes; this clear evidence of a gender bias in admissions? Your suspicions are confirmed when you conduct a statistical test on these data to detect whether there is a relationship between gender and admission rates and find χ2 (1) = 110.8, with a p-value that's nearly indistinguishable from zero. It seems obvious that the next action is to identify the culprit or culprits – that is, the departments that discriminate against women – so that corrective action can be taken. We did not make up these numbers; they represent the real admissions data of the University of California at Berkeley in 1973 (Bickel et al., 1975). And as you might expect, those data (quite justifiably) caused much concern and consternation, and the university embarked on an examination of the admission records of individual departments. A snapshot of the outcome of this further examination, taken from Freedman etal.