Chapter 10 The Analysis of Frequencies

The expression “cross partition” refers to an abstract process of set theory. When the cross partition idea is applied to the analysis of frequencies to study relations between variables, we call the cross partitions crosstabs or sometimes crossbreaks. This kind of analysis to be shown is called contingency analysis, or contingency table analysis. Table 10.1, 10.2

Data and Variable Terminology A distinction was made between active and attribute variable, the former meaning an experimental or manipulated variable and the latter a measured variables. Remember that a attribute variable is not always a categorical variable or qualitative variable. An attribute is any property of any object, whether the object is measured in an all-or none way or with a set of continuous measures.

Crosstabs: Definitions and Purpose A crosstab is a numerical tabular presentation of data, usually in frequency or percentage form, in which variables are cross partitioned. Crosstabs enable the researcher to determine the nature of the relations between variables, but also have other side purposes: They can be used to organize data in convenient form for statisitcal analysis.

Simple Crosstabs and Rules for Crosstab Construction The rules are (1) categories are set up according to the research hypotheses; (2) categories are independent and mutually exclusive; (3) categories are exhaustive; (4) each category is derived from one and only one classification principle; (5) all categories are on one level of discourse. In general, we will report the levels of the independent variable in column and the outcome responses of the dependent variable as rows in the contingency table.

Calculation of Percentages Percentages are calculated from the independent variable to the dependent variable. Figure Table 10.3 (p.225) Use percentages but not frequencies to highlight the relation of variables in crosstabs. Why? A common base.

Calculation of Percentages In Table The Payette-Clarizio problem is pointed toward the misclassification of children as eligible or ineligible for learning disability treatment. A hypothesis implied by the problem is: Decisionmakers are biased in their decisions about female children. This is a statement of the “If p, then q” kind: If female, then they are most likely eligible for learning disable considerations.

Calculation of Percentages Why not calculate the percentages the other way: from the dependent variable to the independent variable? Why not calculate the percentages over the whole table?

Statistical Significance and the Chi- square Test Look at Table Do they really express a relation between gender and learning disability eligibility? Or could they have happened by chance? Are they one pattern among many patterns of frequencies that one would get picking numbers from a table of random numbers, such selection being limited only by the given marginal frequencies?

Statistical Significance and the Chi- square Test We may say here that “degrees of freedom” defines the latitude of variation continued in a statistical problem. In the problem above, there is one degree of freedom because the total number of cases is fixed, 100, and because as soon as one of the frequencies is given, the other is immediately determined.

Levels of Statistical Significance The 0.05 level means that an obtained result that is significant at the 0.05 level could occur by chance no more than five times in 100 trials. The 0.05 level was originally chosen—and has persisted with researchers—because it is considered a reasonably good gamble. It is neither too high nor too low for most social scientific research.

Levels of Statistical Significance There is a newer trend of thinking that advocates reporting the significance levels of all results. Another school of thought advocates working with what are called “confidence intervals.” Rozeboom (1960) advocates the use of confidence intervals and the reporting of precise probability values of experimental outcomes. However, Brady (1988) states that such precision is generally meaningless in the social and behavioral sciences because of the inaccuracy of measurements.

Levels of Statistical Significance A statistically significant result does not imply personal or practical significance. It is Cramer’s V, a measure of association based on the chi-square value. The formula is:

Types of Crosstabs and Tables There are three types of tables: one- dimensional, two-dimensional, and k- dimensional. Theoretically, there is no limit to the number of variables that can be considered at one time. The only limitations are practical ones: insufficient sample size and difficulty of comprehension of the relations contained in a multidimensional table.

One-dimensional Tables There are two kinds of one-dimensional tables. One is a “true” one-dimensional table; it is of little interest to us because it does not express a relation. Only one variable is used in the table. Social scientists sometimes choose to report their data in tables that look one- dimensional but are really two-dimensional, such as Table 10.7.

Two-dimensional Tables Two-dimensional tables or crosstabs have two variables, each with two or more subclasses. Table 10.8, 10.9, 10.10,

Three- and k-Dimensional Tables The analysis of three or more variables simultaneously has two main purposes. First, is to study the relations among three or more variables. The second purpose is to control one variable while studying the relation between the other two variables.

Specification Specification is a process of describing the conditions under which a relation does or does not exist, or exists to a greater or a lesser extent. In the above analysis (Table 10.13, 10.14), the data were specified: it was shown, by introducing the social-class variable, that the relation between level of aspiration and success in college was stronger in one group (middle class) than in another group (working class).

Specification This is similar to the phenomenon of interaction discussed in chapter 9. Strictly speaking, “interaction” is a term used in experimental research and analysis of variance. The position taken in this book is that interaction is a general phenomenon of great importance occurring in both experimental and nonexperimental research.

Crosstabs, Relations, and Ordered Pairs A relation is a set of ordered pairs. Table 10.15, 10.16, Figure 10.4.

The Odds Ratio Odds are computed as the ratio of the probability that the event will occur to the probability that it will not occur. Odds ratio The chi-square statistic is still the preferred method; however, it is unable to give the type of information that odds ratios can give.

Multivariate Analysis of Frequency Data Many frequency data analysis, however, are of three and more variables. It is so- called “multi-way contingency tables with frequency data,” which can be handled by log-linear model.

Computer Addendum Figure 10.10~10.13