The object of the analysis of the Happiness data (Davis, 1977; originally published in Clogg, 1982) is to gain insight into the relationship between happiness as reported by participants in a survey, and the number of their siblings and the number of years of schooling they had completed. The main interest lies in the three-way interaction between the two predictors and the dependent variable Happiness. Of the two-way interaction terms, that between the two predictors is of limited interest because it bears no relation to the prediction of the criterion because it is the design interaction. Its relevance very much depends on the sampling frame.
The three-way contingency table has as its variables: Happiness reported by participants in a survey (3 categories), Number of siblings (Siblings - 5 categories) and the Years of schooling completed (Schooling - 4 categories). It is thus a frequency table containing in the cells the number of people with a particular combination of categories.
|1.||<12||Less than 12 years|
|2.||12||12 years (equivalent to primary + secondary school)|
|3.||13-16||13-16 year (some) college|
|4.||17+||17+ (more than undergraduate)|
|1.||Happy-||Not too happy|
A three-way data array X = (x(i,j,k)) has the following form
|-----|i=1 |-----| |i=2 |-----| | |.. | | | |.. | | |____|i=I=4 k=K=3 | |____| k=2 |_____| k=1 j=1,.,J=5
The actual data file has the following form:
j=1,.,J=5 |-----|i=1 | |i=2 | |.. k=1 | |.. |_____|i=I=4 |-----|i=1 | |i=2 | |.. k=2 | |.. |_____|i=I=4 |-----|i=1 | |i=2 | |.. k=3 | |.. |_____|i=I=4
Thus the first mode (i) is nested in the third mode (k) and there are 4 (Schooling) times 3 (Happiness) rows and 5 (NumSiblings) columns.
Categorical data in contingency tables are usually divested of the influence of the marginal proportions which represent the model of independence. Thus the original frequencies are transformed in chi-terms ( = observed - expected)/sqrt(expected). These chi-terms are the basis for the correspondence analysis. For details see Kroonenberg (2008). Applied multiway data analysis. Hoboken NJ: Wiley (Chapter 17).
[Download the zipped Happiness Data]