Measures of relationships

•Cross tabulation and percentage difference•Gamma•Spearman’s rank order correlation co-efficient(Rho)

This section examines the relationship between variables.

E.g. Children grow, their weight increasesDemand increases if price decreasesApplication of fertilizers increases

crop yieldThere may be, Direct-indirect relationshipPositive-negative relationship

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?

Cross tabulation and percentage differenceIt is used for Nominal variables

which are purely qualitative and can be categorized only.

E.g. Age, income, brand, religion, language etc..

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 tableEach 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.

Preference for brand X and Age (Hypothetical)

The strength of relationship is determined by the pattern of differences between the values of variables.

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 ϒG= (ns – nd )/ (ns + nd )ns the number of similar pairsNd the number of dissimilar pairs

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 and Y are asked to rank the product characteristics that they would consider while buying a scooter.

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

.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

ns= 10; nd= 0ns= 0; nd= 10ns= 08; nd= 02

G= (10 - 0) / (10 + 0) = +1G= (0- 10) / (0 + 10) = -1G= (8 – 2) / (8 + 2 ) = +.60

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.

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)

Rho or e = 1- [ (6∑D2 ) / n(n2 – 1) ]

D= difference between X, Y ranks assigned to the object.

n= number of observation

e= 1- [ (6∑D2 ) / n(n2 – 1) ]=1-[(6*40)/ 5(52- 1) ]= -1.0This value – 1 indicates that there

is perfect negative association between the two sets of ranks.

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.

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.

Karl pearson product moment correlation coefficient r

Most common measureThis measure expresses both

strength and direction of linear correlation.

This measure expresses both strength and direction of linear correlation.

•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

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

r= 0.53The 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.