EXAMPLE 12.4

ANOVA model for the Facebook friends study. In the Facebook friends example, there are five profiles that we want to compare, so I = 5. The population means μ₁, μ₂, . . . , μ₅ are the mean social attractiveness scores for the profiles with 102, 302, 502, 702, and 902 friends, respectively. The sample sizes n_i are 24, 33, 26, 30, and 21. It is common to use numerical subscripts to distinguish the different means, and some software requires that levels of factors in ANOVA be specified as numerical values. In this situation, it is very important to keep track of what each numerical value represents when drawing conclusions. In our example, we could use numerical values to suggest the actual groups by replacing μ₁ with μ₁₀₂, μ₂ with μ₃₀₂, and so on.

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The observation x_1,1, for example, is the social attractiveness score for the first participant who observed the profile with 102 friends (Profile 1). The data for the other participants assigned to this profile are denoted by x_1,2, x_1,3, . . . , x_1,24. Similarly, the data for the other four profile groups have a first subscript indicating the profile and a second subscript indicating the participant assigned to that profile.

According to our model, the score for the first participant in Profile 1 is x_1,1 = μ₁ + ϵ_1,1, where μ₁ is the average score for all undergraduates after viewing Profile 1 and ϵ_1,1 is the chance variation due to this particular participant. Similarly, the score for the last participant in Profile 5 is x_5,21 = μ₅ + ϵ_5,21, where μ₅ is the average score for all undergraduates after viewing Profile 5 and ϵ_5,21 is the chance variation due to this participant.