Copyright © 2014 by Nelson Education Limited Chapter 11 Introduction to Bivariate Association and Measures of Association for Variables Measured at the Nominal Level

Association Between Variables and the Bivariate Table (Crosstab) Three Characteristics of Bivariate Associations Chi Square-Based Measures of Association: Phi and Cramer’s V Lambda: A PRE Measure of Association Copyright © 2014 by Nelson Education Limited In this presentation you will learn about:

Copyright © 2014 by Nelson Education Limited Two variables are said to be associated when they vary together, when one changes as the other changes. If variables are associated, score on one variable can be predicted from the score of the other variable. The stronger the association, the more accurate the predictions. Introduction to Bivariate Association

Copyright © 2014 by Nelson Education Limited Association can be important evidence for causal relationships, particularly if the association is strong. Bivariate association can be investigated by finding answers to three questions: 1.Does an association exist? 2.How strong is the association? 3.What is the pattern or direction of the association? Introduction to Bivariate Association (continued)

Copyright © 2014 by Nelson Education Limited To detect association within bivariate tables: 1.Calculate percentages within the categories of the independent variable. 2.Compare percentages across the categories of the independent variable. Does an association exist?

Copyright © 2014 by Nelson Education Limited When the independent variable is the column variable (as is usually the case): 1.Calculate percentages within the columns (vertically). Column percentages are conditional distributions of Y for each value of X. 2.Compare percentages across the columns (horizontally). Does an association exist? (continued)

Copyright © 2014 by Nelson Education Limited In summary, to detect association (when the independent variable is in the column of a bivariate table) follow this rule: “Percentage Down, Compare Across” Does an association exist? (continued)

Copyright © 2014 by Nelson Education Limited Forty-four males and females were asked whether or not they watched the last CFL Grey Cup game. Association and Bivariate Tables: An Example

Copyright © 2014 by Nelson Education Limited o The table below shows the relationship between sex (X) and Grey Cup watching (Y) for 44 people. Is there an association between these variables? Association and Bivariate Tables: An Example (continued) Watched Grey Cup Sex ____ Male FemaleTotals No Yes Totals271744

Copyright © 2014 by Nelson Education Limited An association exists if the conditional distributions of one variable change across the values of the other variable. –With bivariate tables, column percentages are the conditional distributions of Y for each value of X. –If the column percentages change, the variables are associated. Association and Bivariate Tables: An Example (continued)

Copyright © 2014 by Nelson Education Limited To calculate column percentages, each cell frequency is divided by the column total, then multiplied by 100: ◦ (10/27)*100 = 37.04% ◦ (13/17)*100 = 76.47% ◦ (17/27)*100 = 62.96% ◦ ( 4/17)*100 = 23.53% Association and Bivariate Tables: An Example (continued)

Copyright © 2014 by Nelson Education Limited Grey Cup Watching by Sex, Frequencies (Percentages) Association and Bivariate Tables: An Example (continued) Watched Grey Cup Sex Male FemaleTotals No10 (37.04%)13 (76.47%) 23 Yes17 (62.96%) 4 (23.53%) 21 Totals27 (100.00%)17 (100.00%) 44

Copyright © 2014 by Nelson Education Limited The column percentages show sex of respondents by Grey Cup watching. o The column percentages do change (differ across columns), so these variables are associated. Grey Cup Watching by Sex, Percentages Association and Bivariate Tables: An Example (continued) Watched Grey Cup Sex Male Female No37.04%76.47% Yes62.96% 23.53% Totals100.00%

Copyright © 2014 by Nelson Education Limited The stronger the relationship, the greater the change in column percentages (or conditional distributions). –In weak relationships, there is little or no change in column percentages. –In strong relationships, there is marked change in column percentages. How Strong is the Association?

Copyright © 2014 by Nelson Education Limited One way to measure strength is to find the “maximum difference,” the biggest difference in column percentages for any row of the table. Note, the “maximum difference” method provides an easy way of characterizing the strength of relationships, but it is also limited. How Strong is the Association? (continued)

Copyright © 2014 by Nelson Education Limited The scale presented Table 11.5 can be used to describe (only arbitrary and approximately though) the strength of the relationship” How Strong is the Association? (continued)

Copyright © 2014 by Nelson Education Limited The “Maximum Difference” is: –76.47–37.04=39.43 percentage points. –This is a strong relationship. Grey Cup Watching by Sex, Percentages How Strong is the Association? (continued) Watched Grey Cup Sex ____ Male Female No37.04%76.47% Yes62.96% 23.53% Totals100.00%

Copyright © 2014 by Nelson Education Limited “Pattern” = which scores of the variables go together? To detect, find the cell in each column which has the highest column percentage. What is the Pattern of the Relationship?

Copyright © 2014 by Nelson Education Limited The majority of males watched the last Grey Cup game. The majority of females did not watch the last Grey Cup game. What is the Pattern of the Relationship? (continued) Grey Cup Watching by Sex, Percentages Watched Grey Cup Sex ____ Male Female No37.04%76.47% Yes62.96% 23.53% Totals100.00%

Copyright © 2014 by Nelson Education Limited If both variables are ordinal, we can discuss direction as well as pattern. In positive relationships, the variables vary in the same direction. –Low on X is associated with low on Y. –High on X is associated with high on Y. –As X increase, Y increases. In negative relationships, the variables vary in opposite directions. –As one increases, the other decreases. What is the Direction of the Relationship?

Copyright © 2014 by Nelson Education Limited The relationship is positive. –As education increases, library use increases. Library Use by Education, Percentages What is the Direction of the Relationship? (continued)

Copyright © 2014 by Nelson Education Limited The relationship is negative. –As education increases, TV viewing decreases TV Viewing by Education, Percentages What is the Direction of the Relationship? (continued)

Copyright © 2014 by Nelson Education Limited Conditional distributions, column percentages, and the maximum difference provide useful information about the bivariate association, and should always be computed and analyzed. But, they can be awkward and cumbersome to use. Measures of association, on the other hand, characterize the strength (and for ordinal- and interval-ratio level variables, the direction) of bivariate relationships in a single number. Measures of Association

Copyright © 2014 by Nelson Education Limited For nominal level variables, there are two commonly used measures of association: –Phi (φ) or Cramer’s V (Chi square-based measures) –Lambda (λ) (PRE measure) While more suitable for nominal-level variables, Phi, Cramer’s V, and Lambda can be also used to measure the strength of the relationship between ordinal-level variables in a bivariate table. Nominal Level Measures of Association

Copyright © 2014 by Nelson Education Limited Phi is used for 2x2 tables. Formula for phi: where the obtained chi square, χ 2, is divided by n, then the square root of the result taken. Chi Square-Based Measures of Association

Copyright © 2014 by Nelson Education Limited Cramer’s V is used for tables larger than 2x2. Formula for Cramer’s V: Chi Square-Based Measures of Association (continued)

Copyright © 2014 by Nelson Education Limited Phi and Cramer’s V range in value from 0 (no association) to 1.00 (perfect association). o Phi and V are symmetrical measures; that is, the value of Phi and V will be the same regardless of which variable is taken as independent. Chi Square-Based Measures of Association (continued)

Copyright © 2014 by Nelson Education Limited o General guidelines for interpreting the value of Phi and V are provided in Table o These are similar to the guidelines used for interpreting the maximum difference in column percentages in Table o The scale in Table is arbitrary and meant only as a general guideline. Chi Square-Based Measures of Association (continued)

Copyright © 2014 by Nelson Education Limited The following problem was considered in Chapter 11 of the textbook: A random sample of 100 social work graduates has been classified in terms of whether the Canadian Association of Schools of Social Work (CASSW) has accredited their undergraduate programs (independent variable) and whether they were hired in social work positions within three months of graduation (dependent variable). Chi Square-Based Measures of Association: An Example

Copyright © 2014 by Nelson Education Limited We saw in Chapter 10 that this relationship was statistically significant: o Chi square = 10.78, which was significant at the.05 level. However, what about the strength of this association? Chi Square-Based Measures of Association: An Example (continued)

Copyright © 2014 by Nelson Education Limited To assess the strength of the association between CASSW accreditation and employment, phi is compute as: o A phi of.33 indicates a strong association between these two variables. Chi Square-Based Measures of Association: An Example (continued)

Copyright © 2014 by Nelson Education Limited To help identify the pattern of this relationship, column percentages are presented in Table 11.11: The table shows that graduates of CASSW-accredited programs were more often employed as social workers. Chi Square-Based Measures of Association: An Example (continued)

Copyright © 2014 by Nelson Education Limited Phi is used for 2x2 tables only. –For larger tables, the maximum value of phi depends on table size and can exceed 1.0. –Use Cramer’s V for larger tables. Phi (and V ) is an index of the strength of the relationship only. It does not identify the pattern. To analyze the pattern of the relationship, see the column percentages in the bivariate table. Limitations of Chi Square-Based Measures of Association

Copyright © 2014 by Nelson Education Limited Logic of PRE measures is based on two predictions: 1.First prediction: Ignore information about the independent variable and make many errors (E 1 ) in predicting the value of the dependent variable. 2.Second prediction: Take into account information about the independent variable in predicting the value of the dependent. If the variables are associated we should make fewer errors (E 2 ). Proportional Reduction in Error (PRE) Measures

Copyright © 2014 by Nelson Education Limited Like Phi (and V ), Lambda (λ) is used to measure the strength of the relationship between nominal variables in bivariate tables. Like Phi (and V ), the value of lambda ranges from 0.00 to Unlike Phi (and V ), Lambda is a PRE measure and its value has a more direct interpretation. –While Phi (and V) is only an index of strength, the value of Lambda tells us the improvement in predicting Y while taking X into account. Lambda

Copyright © 2014 by Nelson Education Limited Formula for Lambda: Lambda (continued)

Copyright © 2014 by Nelson Education Limited Let's again consider the relationship between sex (X) and Grey Cup watching (Y) for 44 people. –Here, we will analyze the association between these variables with a measure of association, Lambda: Lambda: An Example

Copyright © 2014 by Nelson Education Limited To compute Lambda, we must first find E 1 and E 2 : –E 1 = n – largest row total = 44 – 23 = 21 –E 2 = For each column, subtract the largest cell frequency from the column total = (27 – 17) + (17 – 13) = = 14 Lambda: An Example (continued) Watched Grey Cup Sex ____ Male FemaleTotals No Yes Totals271744

Copyright © 2014 by Nelson Education Limited Using Formula 11.3 to compute lambda, we get: λ = /21 = 7/21 =.33 A lambda of.33 means that sex (X) increases our ability to predict Grey Cup viewing (Y) by 33%. According to the guidelines suggested in Table 11.14, a Lambda of 0.33 indicates a strong relationship. Lambda: An Example (continued)

Copyright © 2014 by Nelson Education Limited Lambda is asymmetric: Value will vary depending on which variable is independent. Need care in designating independent variable. 2.When row totals are very unequal, lambda can be zero even when there is an association between the variables. For very unequal row marginals, better to use a chi- square based measure of association. The Limitations of Lambda

Copyright © 2014 by Nelson Education Limited 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 percentages in the bivariate table. The Limitations of Lambda (continued)

Copyright © 2014 by Nelson Education Limited Correlation and causation are not the same things. Strong associations may be used as evidence of causal relationships but they do not prove variables are causally related. –Hence, association is just one of the criteria needed to establish a causal relationship between variables. Correlation vs. Causation