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Measures of Association for Nominal & Ordinal Variables Chap. 10 Healey’s Hypothesis Testing III: ANOVA

Learning Objectives Understand what “association” means Learn when to use different measures of association Learn to interpret the results of association tests Learn how to conduct tests of association using SPSS

What does “association” mean? Concerns the nature of the relationship between two (or more) variables Information complements tests of significance (e.g., chi square) Measures of association quantify the importance or “strength” and, in some cases, the direction of the relationship Hypothesis tests of significance detect non-random events, but not any other information about the relationship. Directionality is only for variables that are measured at ordinal and above. With nominal variables, we can only talk about the pattern across categories.

Three Characteristics of Relationships between Variables the existence of an association the strength of the association the direction or pattern of the association (i.e., positive or negative) Some of this information comes from examining bivariate tables, for example, from looking at the column percentages. Give examples of positive and negative associations. The existence of a association is a necessary, but not sufficient, condition for making claims about causality. That is, without some degree of association, causality is unlikely. Proof of causality is always difficult.

When to use which measure of association? The answer depends of what level of measurement your two variables are. Interval/ratio Ordinal Nominal Correlation convert interval to ordinal (i.e., ordinal-ordinal) Point biserial (not discussed) gamma (also Somer’s d, Kendall’s tau-b & tau-c) convert ordinal to nominal (i.e., nominal-nominal) Cramer’s V, phi, lambda One strategy for dealing with mixed levels (e.g., interval/ratio-ordinal or ordinal-nominal) is simplify the data by reducing the higher level variable to the level of the lower level variable. However, only do this if it is logical to do so. That is the recoded variable makes conceptual sense and is interpretable.

Nominal-Nominal Measures: Chi Square Based Cramer’s V and phi are chi square-based measures Measures range from 0 to 1, with 1 being the strongest association Use phi if 2 x 2 table only; Cramer’s V can be used with any size table Limits: measures have not direct interpretation

Nominal-Nominal Measures: “PRE” based Lambda is proportional reduction of error (PRE) measure Measure ranges from 0 to 1, with 1 being the strongest association Lambda has a meaningful interpretation indicates the improvement of predicting the dependent variable PRE-based measures of measures indicate the improvement of predicting the dependent variable based on knowing the value of the independent, relative to predicting the dependent without knowledge of independent. For example, a lambda of .37 means the error is reduced by 37% by knowing the category of the independent variable. The ease of interpretation is one why PRE measures, like lambda, are sometimes preferred over chi square-based measures.

How to interpret nominal measures of association If the value is between: The strength of the relationship is: 0.00 and 0.10 weak 0.11 and 0.30 moderate Greater than 0.30 strong

SPSS example of nominal-nominal measure of association Research topic: Gender and political involvement. Research question: Are men more likely to have voted in the last parliamentary election? Statistics to ask for: Cramer’s V, phi, lambda.

Ordinal-Ordinal Measures Gamma is also a proportional reduction of error (PRE) measure Measure ranges from -1 to +1 the sign of the coefficient indicates direction of the relationship Gamma has a meaningful interpretation indicates the improvement of predicting the dependent variable PRE measures for ordinal variables predict the rank of pairs of cases, rather than categories of the dependent variable (i.e., nominal PRE measures). Somer’s d, Kendall tau-b, and Kendall tau-c are also used. Use Kendall’s tau-b if the table contains equal numbers of rows and columns, tau-c otherwise. Gamma gives more information than nominal measures of association: It tells you the direction of the relationship. A positive gamma means a positive relationship between variables, meaning variables change in the same order (going up or down together). High scores on one variable are associated with high scores on the other. A negative gamma indicates a negative relationship. Variables change in opposite directions (IV goes up, DV goes down and IV goes down, DV goes up). (Illustrate this with two tables.) High scores on one variable are associated with low scores on the other.

How to interpret ordinal measures of association If the value is between: The strength of the relationship is: 0.00 and 0.30 weak 0.31 and 0.60 moderate Greater than 0.60 strong

Direction of Relationship Positive Association Negative Association Variable X Variable Y Low Mod High X Variable X Variable Y Low Mod High X Positive (1.0): ranked in same order, scores tend fall on the diagonal from upper left to lower right. Negative (-1.0): ranked in opposite order, scores tend fall on the diagonal from lower left to right right.

SPSS example of ordinal-ordinal measure of association Research topic: The relationship between education and “world citizen” self-identification. Research question: Do the more educated have stronger world citizenship identity? Statistics to ask for: gamma, d, kendall’s tau-b and tau-c.

Ordinal-Nominal Measures Treat the ordinal variable as nominal (may involve collapsing categories) This becomes an nominal-nominal comparison, which you test using Lambda (or Cramer’s V and phi) In the ordinal variable has large number of values, collapsing them is an option if it makes sense theoretically.

SPSS example of ordinal-nominal measure of association Research topic: Gender and support of authoritarian government. Research question: Are men more likely to believe that “strong leaders” are good? Recode interval v148 into nominal (bad/good). Statistics to ask for: chi square, Cramer’s V, phi, lambda.

Interval-Ordinal Measures Convert interval variable into ordinal by grouping the interval variable This becomes an ordinal-ordinal comparison, which you test using Gamma In this case, we need to reduce the level of measurement of the interval data to ordinal by creating grouping the data into a limited number of ranked groups (max. 5 or 6).

SPSS example of interval-ordinal measure of association Research topic: Age and support of authoritarian government. Research question: Are older people more likely to support “strong leaders”? First, convert interval to ordinal. Statistics to ask for: gamma, d, kendall’s tau-b and tau-c.