Measures of relationships
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This document discusses different methods for measuring relationships between variables, including cross tabulation, percentage difference, Gamma, and Spearman's rank order correlation coefficient. It provides examples of each method and explains how to calculate and interpret them. Cross tabulation involves creating a two-way table to examine relationships between categorical variables. Gamma and Spearman's rank order correlation examine relationships between ordinal variables by comparing pairs of ranks. Pearson's correlation coefficient measures both the direction and strength of linear relationships between variables.
![ Rho or e = 1- [ (6∑D2 ) / n(n2 – 1) ]
D= difference between X,Y ranks
assigned to the object.
n= number of observation](https://image.slidesharecdn.com/measuresofrelationships-140506114618-phpapp02/85/Measures-of-relationships-17-320.jpg)
![ e= 1- [ (6∑D2 ) / n(n2 – 1) ]
=1-[(6*40)/ 5(52- 1) ]
= -1.0
This value – 1 indicates that there is perfect
negative association between the two sets
of ranks.](https://image.slidesharecdn.com/measuresofrelationships-140506114618-phpapp02/85/Measures-of-relationships-19-320.jpg)
Measures of relationships
- 1. Measures of relationships •Cross tabulation and percentage difference •Gamma •Spearman’s rank order correlation co-efficient(Rho)
- 2. This section examines the relationship between variables. E.g. Children grow, their weight increases Demand increases if price decreases Application of fertilizers increases crop yield There may be, Direct-indirect relationship Positive-negative relationship
- 3. In examining the relationship between variables, we have to consider certain questions: Is there relationship between variables? What is the degree and direction of their relationship? Is the relationship a casual one? Is the relationship statistically significant?
- 4. Cross tabulation and percentage difference It is used for Nominal variables which are purely qualitative and can be categorized only. E.g.Age, income, brand, religion, language etc..
- 5. Procedure, Two way table is prepared. The value of one variable are put along one side of the table. And the values of the other variable along the other side of the table Each variable is categorized into two or more categories; and the variables are cross-tabulated for those sub categories. The percentages are computed for them. On the basis of the percentages, conclusions are drawn.
- 6. Preference for brand X and Age (Hypothetical)
- 8. The strength of relationship is determined by the pattern of differences between the values of variables.
- 9. Gamma The ordinal level variables have rank order differences . The most common ordinal variables are Gamma & Spearman’s rank order correlation coefficient. Gamma or G or ϒ G= (ns – nd )/ (ns + nd ) ns the number of similar pairs Nd the number of dissimilar pairs
- 10. It is based on pair by pair comparison It uses information about one variable to tell us something about a second one. E.g. suppose two respondents X andY are asked to rank the product characteristics that they would consider while buying a scooter.
- 11. After counting the number of similar and dissimilar pairs, the level of association between the two sets of ranks is compared by the ratio of the preponderant type of pairs. This value is ϒ
- 12. .Characteristics Respondent X Respondent Y Fuel efficiency -F 1 1 Price - P 2 2 Mechanical Efficiency - M 3 3 Riding comfort - R 4 4 Style- S 5 5 F 1 5 P 2 4 M 3 3 R 4 2 S 5 1 F 1 1 P 2 2 M 3 3 R 4 5 S 5 4
- 14. ns= 10; nd= 0 ns= 0; nd= 10 ns= 08; nd= 02 G= (10 - 0) / (10 + 0) = +1 G= (0- 10) / (0 + 10) = -1 G= (8 – 2) / (8 + 2 ) = +.60
- 15. A coefficient of +1 indicates the perfect positive association between the variables in terms of pair by pair comparison. G= -1 indicates the perfect inverse association. G= + 0.60 combination of similar & dissimilar ones.
- 16. Spearman’s rank order correlation co-efficient Rho or e This is the oldest of the frequently used measures of ordinal associations. It is a measure of the extent of agreement or disagreement between two sets of ranks. It is nonparametric measures and so it dose not require the assumption of a bivariate normal distribution. Its value ranges between -1 (perfect negative association) and +1 (perfect positive association)
- 17. Rho or e = 1- [ (6∑D2 ) / n(n2 – 1) ] D= difference between X,Y ranks assigned to the object. n= number of observation
- 19. e= 1- [ (6∑D2 ) / n(n2 – 1) ] =1-[(6*40)/ 5(52- 1) ] = -1.0 This value – 1 indicates that there is perfect negative association between the two sets of ranks.
- 20. Correlation analysis It involves three main aspects: 1. Measuring the degree of association between two variables 2. Testing whether the relationship is significant 3. Establishing the cause and effect relationship if any.
- 21. When two variables move in same direction, their association is termed as positive correlation. When they move in opposite direction, their association is termed as negative correlation.
- 22. Karl pearson product moment correlation coefficient r Most common measure This measure expresses both strength and direction of linear correlation. This measure expresses both strength and direction of linear correlation.
- 24. •Examine the relationship between period of education (X) and religious prejudice (Y) for a sample of six respondents. X Y 3 1 6 7 8 3 9 5 10 4 2 2 38 22
- 25. X Y X2 Y2 XY 3 1 9 1 3 6 7 36 49 42 8 3 64 9 24 9 5 81 25 45 10 4 100 16 40 2 2 4 4 4 38 22 294 104 158
- 26. r= 0.53 The correlation coefficient ranges from - 1.0 to +1.0. -1.0 indicates perfect negative correlation & +1 indicated perfect positive correlation. + or - 0.5 moderate positive or negative correlation.