Ordinal Measures of Association for Surveytype Data Christoph

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: www. arl. iup. edu 1

References Discovering Statistics Using SPSS by Andy Field, Sage Publications, 2005. ISBN 0 -7619 -4452 -4. – 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. , 2005. ISBN 013 -199527 -8 2

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

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. 4

Two-level 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 or male female. Not true for Neighborhood: A, B, C and Other. 5

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? 6

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

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

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

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. 10

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 Q 07_USE (library use per week) – Less than 1 time per week – 1 -3 times per week – 4 or more time per week Q 09_VALUABLE (library is a valuable resource) -Strongly disagree -Disagree -Agree -Strongly Agree 11

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

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

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. 15

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

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

Three Ordinal Measures 18

Where P= # of concordant pairs Q = # of discordant pairs TR = # of ties in the row variable TC = # of ties in the column variable TY = # of ties in the dependent variable Note: In this presentation, the dependent variable is always on top of the table, so TY = TC 19

Two types of relationships Directional: Looking at how one variable (independent variable) influences another variable (dependent variable) Example: Q 01_GENDER affects Q 09_VALUABLE Symmetric: Looking at how two variables are related, where the variables can not be classified as independent and dependent. Example Q 09_VALUABLE and Q 10_PERSONNEL 20

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) 21

Perfect Positive Association With 150 Individuals All three ordinal measures = 1 Case 1 Q 09 library is a valuable resource Q 07 library use per week Strongly Disagree Agree Strongly Agree less than 1 time 0 40 0 0 1 -3 times 0 0 80 0 4 or more times 0 0 0 30 P = 6800 Q = 0 T C = TR = 0 22

Perfect Positive Association With 150 Individuals All three ordinal measures = 1 Case 2 Q 09 library is a valuable resource Q 07 library use per week Strongly Disagree Agree Strongly Agree less than 1 time 40 0 1 -3 times 0 0 80 0 4 or more times 0 0 0 30 P = 6800 Q = 0 T C = TR = 0 23

Perfect Negative Association With 150 Individuals All three ordinal measures = -1 Case 3 Q 09 library is a valuable resource Q 07 library use per week Strongly Disagree Agree Strongly Agree less than 1 time 0 0 0 40 1 -3 times 0 0 80 0 4 or more times 0 30 0 0 P = 0 Q = 6800 T C = TR = 0 24

Positive Association With 150 Individuals Case 4 Q 09 library is a valuable resource Q 07 library use per week Strongly Disagree Agree Strongly Agree less than 1 time 20 20 0 0 1 -3 times 0 0 80 0 4 or more times 0 0 0 30 P = 6800 Q = 0 TC = 0 TR = 20*20 =400 25

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An extreme example showing why I don’t recommend using only Gamma Case 5 Q 09 library is a valuable resource Q 07 library use per week Strongly Disagree Agree Strongly Agree less than 1 time 0 40 0 0 1 -3 times 0 80 0 0 4 or more times 0 29 1 0 P = 120 Q = 0 TC = 6680 TR = 29 27

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Interpretations for Gamma Suppose that gamma =. 6 for Q 07 vs Q 09 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. 29

Interpretations Continued Or since (gamma+1)/2 =(. 6+1)/2=. 8 is the proportion of nontied 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. 30

more typical data (n=148) for Does a patron’s gender influence how much they value the library? Case 6 Q 09 library is a valuable resource Q 01 gender Strongly Disagree Agree Strongly Agree Female 3 6 28 61 male 4 5 21 20 P = 944 Q = 2106 TC = 1850 TR = 3336 31

Displaying Data In Tables for the Directional Case: Does a patron’s gender influence how much they value the library? directional independent=Q 01_GENDER (row) dependent=Q 09_VALUABLE (column) Recommendations: Include row percentages, i. e. percentages of responses to Q 09 by gender. 32

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

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

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

SPSS Steps for Generating the Graphs 36

Somers’ d= -. 237 larger values of Q 07_GENDER tend to correspond with smaller values of Q 09_VALUABLE. 37

Somers’ d= -. 237 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. 38

Does a patron’s gender influence how much they value the library? Case 7 Q 09 library is a valuable resource Q 01 gender Strongl y Disagre e Female 38 6 3 51 male 4 20 21 5 P = 1919 Q = 2391 Agre Strongly e Agree TC = 590 TR = 3566 39

Non-monotonic relationship! 40

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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. 42

Displaying Data As Tables for the Symmetric Case: Show the overall percentages, rather than row percentages. Variables = Q 09_VALUABLE (row) and Q 10_PERSONNEL (column) 43

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

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SPSS Steps for Generating the Graph 46

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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 724 -357 -4530 or visit our website at www. arl. iup. edu 48