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In this chapter, we delve into nominal level measures of association, which are essential for understanding the strength of relationships between two nominal variables. We will explore key metrics such as Phi, Cramer’s V, and Lambda, learning how to compute and interpret these values using SPSS. While Phi is used for 2x2 tables, Cramer’s V applies to larger tables. Lambda offers a proportional reduction of error measure, providing insights into the predictive power of relationships. Understanding these metrics empowers better data analysis in bivariate tables.
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Chapter 13 Association Between Two Variables Measured at the Nominal Level
Nominal Level Measures of Association • It is always useful to compute column percentages for bivariate tables. • But, it is also useful to have a summary measure – a single number – to indicate the strength of the relationship. That’s what we’ll learn about in this chapter.
Nominal Level Measures of Association • For nominal level variables, there are two commonly used measures of association: • Phi or Cramer’s V • Lambda
Nominal Measures: Phi • Phi is used for 2x2 tables. • The formula for Phi:
Nominal Measures: Cramer’s V • Cramer’s V is used for tables larger than 2x2. • Formula for Cramer’s V:
SPSS: Phi and Kramer’s V • SPSS has both instructions combined into one • You need to know which one applies • Phi if it’s a 2 x 2 table • Kramer’s V for any other cross tabulation
Let’s ask SPSS to calculate a few chi square based measures • Class and happiness • Ager3 and grass • Ager3 and attend • Attend and grass • Attend and happy
Nominal Measures: Lambda • Like Phi and Kramer’s V, Lambda is used to measure the strength of the relationship between nominal variables in bivariate tables. • Unlike Phi, Lambda is a PRE(proportional reduction of error) measure and its value has a more direct interpretation. • While Phi is only an index of strength, the value of Lambda tells us the improvement in predicting Y while taking X into account.
Nominal Measures: Lambda • Formula for Lambda:
Lambda as PRE measure • E1 = errors made in predicting the dependent variable without knowing the independent variable = N – largest row total • E2 = For each column, subtract the largest cell frequency from the col. total and add those values • This will become more clear when we look at an example
Association and Bivariate Tables • To compute λ, we must first find E1 and E2: • E1 = N – largest row total = 44 – 22 = 22 • E2 = For each column, subtract the largest cell frequency from the col. total = (27 – 17) + (17 – 12) = 10 + 5 = 15 Lambda = (E1-E2)/E1 = (22-15)/22 = 7/22 = .32
Nominal Measures: Lambda • Lambda is a PRE measure. • A Lambda of .32 means that knowing authoritarianism (X) increases our ability to predict efficiency (Y) by 32%.
The Limitations of Lambda • Lambda gives an indication of the strength of the relationship only. • It does not give information about pattern. • To analyze the pattern of the relationship, use the column %s in the bivariate table. • When the mode is the same in each column of the independent variable, lambda will be zero even if a relationship exists. Thus we request both lambda and Kramer’s V/Phi.
