Slide 1 Ordinal Measures of Association for Survey-type Data Christoph Maier Coordinator of the ARL December 6, 2007 Stats For Lunch Please visit our ARL website:

Slide 2 References Discovering Statistics Using SPSS by Andy Field, Sage Publications, ISBN –Note: Not so much information on measures of association, but a great reference for intermediate and advanced statistical methods. SPSS 14.0 Statistical Procedures Companion by Marija J Norusis, Prentice Hall Inc., ISBN

Slide 3 Outline Slides 4-7 variables and bivariate relationships Slides 8-9 SPSS screens Slides 10-17Concordant, discordant, and ties Slides 18-21A look at three ordinal measures. Slides 22-31Six examples. Slides 32-47Displaying data in table and graphs. Slide 48Concluding Advice

Slide 4 Types of Variables (SPSS) Nominal examples: gender, neighborhood, how did you get to the library, etc. Ordinal examples: age group, variables using a Likert scale. Scale examples: time to get to library, cost of using the library.

Slide 5 Two-level Nominal Variables Nominal Variables With two levels can be considered ordinal when assessing association. Gender has two levels: male and female. Although one gender is not more positive or negative than the other, they still can be ordered: female male or male female. Not true for Neighborhood: A, B, C and Other.

Slide 6 Correlation vs. Association On each of the four scatterplots, characterize the relationship between the variables (scale). Which graphs show strong association between the variables? strong correlation?

Slide 7 Correlation vs. Association perfect positive association strong negative association Also perfect positive correlation perfect positive association strong non-monotonic relationship strong positive association at first then strong negative association

Slide 8 SPSS: Analyze  Descriptive Statistics  Crosstabs… Then select the Statistics button.

Slide 9 In SPSS, right-click on a statistic to see a short explanation.

Slide 10 Ordinal Measures Gamma Somers’ D Kendall’s tau-b Kendall’s tau-c All based on looking at all pairs of subjects and categorizing the relationship as concordant, discordant, or tied in some sense.

Slide 11 Example of Research hypothesis Do patrons who use the library more often attribute more value to the library as a resource to them? Responses in increasing order for Q07_USE (library use per week) –Less than 1 time per week –1-3 times per week –4 or more time per week Q09_VALUABLE (library is a valuable resource) -Strongly disagree -Disagree -Agree -Strongly Agree

Slide 12 Responses for seven patrons Gary 1-3 visits Agrees that library valuable. Dan 4+ visits Strongly agrees Chris 1-3 visits Agrees Lisa 4+ visitsStrongly disagrees Tom 4+ visitsAgrees Nate 1-3 visitsStrongly Agrees

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Slide 14 Relationship Between Gary and Chris Gary and Chris tied on both variables, so this pair is disregarded for all ordinal measures.

Slide 15 Relationship Between Gary and Nate and between Gary and Tom Gary and Nate in same row, so this pair is tied on row variable. Gary and Tom in same column, so this pair is tied on column variable.

Slide 16 Relationship Between Gary and Larry Larry has more visits than Gary (increase), but Larry values library less than Gary (decrease) So Gary and Larry constitute a discordant pair.

Slide 17 Relationship Between Gary and Dan Dan has more visits than Gary (increase), and Dan values library more than Gary (increase) So Dan and Paul constitute a concordant pair.

Slide 18 Three Ordinal Measures

Slide 19 Where P= # of concordant pairs Q = # of discordant pairs T R = # of ties in the row variable T C = # of ties in the column variable T Y = # of ties in the dependent variable Note: In this presentation, the dependent variable is always on top of the table, so T Y = T C

Slide 20 Two types of relationships Directional: Looking at how one variable (independent variable) influences another variable (dependent variable) Example: Q01_GENDER affects Q09_VALUABLE Symmetric: Looking at how two variables are related, where the variables can not be classified as independent and dependent. Example Q09_VALUABLE and Q10_PERSONNEL

Slide 21 My recommendation for ordinal variables for the Directional Case:  Gamma and  Somers’ D for the Symmetric Case:  Gamma and  Somers’ D or  Kendall’s tau-b ( Look at symmetric option of Somers’ D on the SPSS printout.) Options when the relationship is non-monotonic  Phi and Cramer’s V  Lambda (nominal measures)

Slide 22 Perfect Positive Association With 150 Individuals All three ordinal measures = 1 P = 6800 Q = 0 T C = T R = 0 Case 1 Q09 library is a valuable resource Q07 library use per week Strongly DisagreeDisagreeAgree Strongly Agree less than 1 time times or more times00030

Slide 23 Perfect Positive Association With 150 Individuals All three ordinal measures = 1 P = 6800 Q = 0 T C = T R = 0 Case 2 Q09 library is a valuable resource Q07 library use per week Strongly DisagreeDisagreeAgree Strongly Agree less than 1 time times or more times00030

Slide 24 Perfect Negative Association With 150 Individuals All three ordinal measures = -1 P = 0 Q = 6800 T C = T R = 0 Case 3 Q09 library is a valuable resource Q07 library use per week Strongly DisagreeDisagreeAgree Strongly Agree less than 1 time times or more times03000

Slide 25 Positive Association With 150 Individuals Case 4 Q09 library is a valuable resource Q07 library use per week Strongly DisagreeDisagreeAgree Strongly Agree less than 1 time times or more times00030 P = 6800 Q = 0 T C = 0 T R = 20*20 =400

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Slide 27 An extreme example showing why I don’t recommend using only Gamma Case 5 Q09 library is a valuable resource Q07 library use per week Strongly DisagreeDisagreeAgree Strongly Agree less than 1 time times or more times02910 P = 120 Q = 0 T C = 6680 T R = 29

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Slide 29 Interpretations for Gamma Suppose that gamma =.6 for Q07 vs Q09 When comparing pairs of subjects that do not have the same responses on one or both questions, the proportion of pairs of subjects where the subject who used the library more often also valued the library more as a resource was 60 percentage points higher than the proportion of pairs of subjects where the subject who used the library more often valued the library less.

Slide 30 Interpretations Continued Or since (gamma+1)/2 =(.6+1)/2=.8 is the proportion of non- tied pairs that are concordant. When comparing pairs of subjects that do not have the same responses on one or both questions, 80% of the pairs of subjects had one subject who gave more positive responses on both questions and 20% of the pairs of subjects had one subject who gave the more positive response on one question and the more negative response on the other question.

Slide 31 more typical data (n=148) for Does a patron’s gender influence how much they value the library? Case 6 Q09 library is a valuable resource Q01 gender Strongly DisagreeDisagreeAgree Strongly Agree Female male P = 944 Q = 2106 T C = 1850 T R = 3336

Slide 32 Displaying Data In Tables for the Directional Case: Does a patron’s gender influence how much they value the library? directional independent=Q01_GENDER (row) dependent=Q09_VALUABLE (column) Recommendations: Include row percentages, i.e. percentages of responses to Q09 by gender.

Slide 33 SPSS: Analyze  Descriptive Statistics  Crosstabs… Then select the Cells button. Under Percentages:  Row

Slide 34 Generate a graph of the counts See SPSS steps on slide 36

Slide 35 For the directional case, also graph the “row” percentages See SPSS steps on next slide

Slide 36 SPSS Steps for Generating the Graphs

Slide 37 Somers’ d= larger values of Q07_GENDER tend to correspond with smaller values of Q09_VALUABLE.

Slide 38 Somers’ d= so males tend to attribute less value to the library than females do. In fact when we look at all pairs of subjects of different genders, then the proportion of female/male pairs where the male subject values the library less exceeds by 23.7 percentage points the proportion of female/male pairs where the male subject values the library more.

Slide 39 Does a patron’s gender influence how much they value the library? Case 7 Q09 library is a valuable resource Q01 gender Strongly DisagreeDisagreeAgree Strongly Agree Female male P = 1919 Q = 2391 T C = 590 T R = 3566

Slide 40 Non-monotonic relationship!

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Slide 42 Conclusion Female patrons tend to have more extreme opinions (90.8% strongly agree or strongly disagree) on the value of the library than do male patrons (only 18% strongly agree or strongly disagree). Using the Cramer’s V of.721, we conclude that there is a strong relationship between gender and how much the patron values the library. The small values of the ordinal measures indicate negligible association between the variables.

Slide 43 Displaying Data As Tables for the Symmetric Case: Show the overall percentages, rather than row percentages. Variables = Q09_VALUABLE (row) and Q10_PERSONNEL (column)

Slide 44 SPSS: Analyze  Descriptive Statistics  Crosstabs… Then select the Cells button. Under Percentages:  Total

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Slide 46 SPSS Steps for Generating the Graph

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Slide 48 Concluding Advice Be very careful how you state your research hypotheses. directional or symmetric? Looking for a relationship or more specifically for association? Always graph your data and carefully study the graphs to see the nature of the relationship between the variables. Be specific when writing your conclusions, carefully explaining what the statistics mean in the context of your research. The ARL will be happy to help you along the way. Contact us at or visit our website at