5.2 Statistical Model

[1] We developed a statistical model for discriminating composed from random triads. The model used features of the triads’ spacing and doubling. (See §5.1.) For each feature, the model estimated an optimal weight. Larger weights were assigned to features that were more useful for discrimination. Positive weights were assigned to features that were more common in composed triads. Negative weights were assigned to features that were more common in random triads.

[2] The features may be represented as X₁,X₂,.... The weights may be represented as β₁,β₂,.... A triad’s weighted features are added together to represent the probability that the triad is composed rather than random:

Prob (composed) = Λ ( β₁X₁ + β₂X₂ + ... )

[3] (Here Λ is the cumulative logistic distribution function.) This way of modeling probabilities is known as logistic regression analysis.⁽⁸⁰⁾ When logistic regression is used to discriminate between two types of objects, as here, it is called a logistic discrimination model.

[4] In its pure form, logistic discrimination is used for observations that are independent of one another. Our composed and random triads, however, are not independent but paired. To accommodate this pairing, we used a conditional logistic model that is appropriate for matched-pair data.⁽⁸¹⁾

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