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# Measures of relationships

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### Measures of relationships

1. 1. Measures of relationships •Cross tabulation and percentage difference •Gamma •Spearman’s rank order correlation co-efficient(Rho)
2. 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. 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. 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. 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. 6. Preference for brand X and Age (Hypothetical)
7. 7.  The strength of relationship is determined by the pattern of differences between the values of variables.
8. 8. 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
9. 9.  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.
10. 10.  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
11. 11.  .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
12. 12.  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
13. 13.  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.
14. 14. 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)
15. 15.  Rho or e = 1- [ (6∑D2 ) / n(n2 – 1) ]  D= difference between X,Y ranks assigned to the object.  n= number of observation
16. 16.  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.
17. 17. 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.
18. 18.  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.
19. 19. 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.
20. 20. •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
21. 21. 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
22. 22.  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.