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Chapter 13 (Ch. 11 in 2nd Can. Ed.) Association Between Variables Measured at the Nominal Level: Phi, Cramers V, and Lambda

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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.

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Nominal Level Measures of Association For nominal level variables, there are two commonly used types of measures of association: 1. Chi-square based statistics: Phi or Cramers V 2. A PRE (Proportional Reduction in Error) measure known as Lambda

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Chi-square Based Measure: Phi ( ) Phi is used for 2x2 tables. The formula for Phi:

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Example of Nominal Measure: Calculating Phi for #12.1 (#11.1 in 2 nd ) Step 1. Using the 5 step method explained in Chapter 11, calculate for the table. = (critical value = 3.841) There is a significant relationship between Authoritarianism and Worker Efficiency ( )

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Example: Calculating Phi for 12.1(11.1) Step 2: Calculate Phi… What does this mean?* Phi = 0.33 There is a strong association between authoritarianism and efficiency. *Keep this table handy! ValueStrength 0.0 > 0.10Weak 0.10 > 0.30Moderate > 0.30Strong

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Nominal Measures: Cramers V (V) Cramers V is used for tables larger than 2x2. Formula for Cramers V: Where (min r-1, c-1) is the minimum value of either the # rows-1 or the # columns-1.

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Limitations of Phi and V Phi is used for 2x2 tables only. For larger tables, use Cramers V. Phi (or V) is an index of the strength of the relationship only. It does not identify the pattern. Phi or V cannot be used to compare the strength of one relationship to another. To analyze the pattern of the relationship, see the column % in the bivariate table.

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Nominal Measures: Lambda (λ) Like Phi and V, Lambda is used to measure the strength of the relationship between nominal variables in bivariate tables. It can also be used to compare one relationship to another. Unlike Phi and V, Lambda is a PRE measure and its value has a more direct interpretation. PRE = Proportional Reduction in Error = how much less error do you make in your prediction of the dependent variable (y) when you take into account the values of the independent variable (x) Lambda tells us the improvement in predicting Y while taking X into account.

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Lambda (λ) cont. Formula for Lambda: Where E 1 = (N – the largest row total) and E 2 = (For each column, subtract the largest cell frequency from its column total and then add the differences together)

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Calculating Lambda (λ)- #12.1 (11.1 in 2 nd ): To compute λ, we must first find E 1 and E 2 : E 1 = N – largest row total = 44 – 22 = 22 E 2 = For each column, subtract the largest cell frequency from the col. total and add together = (27 – 17) + (17 – 12) = = 15 Efficiency (y) LowHighTotal Low High17522 Total Authoritarianism (x)

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Calculating Lambda (cont.) Lambda is a PRE measure. A Lambda of.32 means that knowing something about authoritarianism (X) reduces our error in prediction of efficiency by 32%. We can also say that knowing about X increases our ability to predict efficiency (Y) by 32%.

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More about Lambda (λ) The values of Lambda vary from 0.0 – +1.00, where 0.0 is very weak and 1.00 is very strong. Lambda can be used to compare the strength of two or more bivariate relationships. Suppose that you calculate: 1. Prestige X Happiness, λ =.55 and 2. Income X Happiness, λ =.20, …then you could say that Relationship 1 is stronger, and that Prestige has a greater effect on Happiness than Income.

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More about Lambda (λ) Lambda, like Phi and V, tells you about the strength of the relationship only. It does not give information about pattern. To analyze the pattern of the relationship, use the column % in the bivariate table. Note that when row totals are very unequal, lambda can be zero even when there is an association between the variables.

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Answering the Three Questions when working with Nominal Level Variables. 1. Is there an association? can be answered by looking at the % and by doing a test of significance like Х How strong is the association? can be answered by using a measure of strength like Phi, Cramers V, or Lambda. 3. What is the pattern of the association? can be assessed by looking at the percentages.

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Practice Question: Healey #13.1 (11.2 in 2 nd Ed.) Work with a partner on question #13.1 (a,b,c) and find an answer to all three of the above questions Answers are on the last slide. Compare the three relationships, and use Lambda to evaluate which relationship is the strongest. Write a brief summary of your findings.

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Answers to Healey #13.1 (11.2 in 2 nd Ed.) #13.1 (11.2) a Phi =.00 Lambda =.00 #13.1 (11.2) b Phi =.09 Lambda =.00 #13.1 (11.2) c Phi =.25 Lambda =.14

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