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Spearman Rank Correlation A measure of Rank Correlation Group 3
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The Spearman Correlation <ul><li>Spearman’s correlation is designed to measure the relationship between variables measured on an ordinal scale of measurement. </li></ul><ul><li>Similar to Pearson’s Correlation, however it uses ranks as opposed to actual values. </li></ul>
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Assumptions <ul><li>The data is a bivariate random variable. </li></ul><ul><li>The measurement scale is at least ordinal. </li></ul><ul><li>X i , Y i is independent of X j , Y j where i ≠ j </li></ul>
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Advantages <ul><li>Less sensitive to bias due to the effect of outliers </li></ul><ul><li>- Can be used to reduce the weight of outliers (large distances get treated as a one-rank difference) </li></ul><ul><li>Does not require assumption of normality. </li></ul><ul><li>When the intervals between data points are problematic, it is advisable to study the rankings rather than the actual values. </li></ul>
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Disadvantages <ul><li>Calculations may become tedious. Additionally ties are important and must be factored into computation. </li></ul>
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Steps in Calculating Spearman’s Rho <ul><li>Convert the observed values to ranks (accounting for ties) </li></ul><ul><li>Find the difference between the ranks, square them and sum the squared differences. </li></ul><ul><li>Set up hypothesis, carry out test and conclude based on findings. </li></ul>
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Steps in Calculating Spearman’s Rho <ul><li>If the null is rejected then calculate the Spearman correlation coefficient to measure the strength of the relationship between the variables. </li></ul>
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Hypothesis: I <ul><li>(Two-Tailed) </li></ul><ul><li>H o : There is no correlation between the Xs and the Ys. </li></ul><ul><li>(there is mutual independence between the Xs and the Ys) </li></ul><ul><li> H 1 : There is a correlation between the Xs and the Ys. </li></ul><ul><li>(there is mutual dependence between the Xs and the Ys) </li></ul>
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Hypothesis: II <ul><li>B. (One-Tailed - Lower) </li></ul><ul><li>H o : There is no correlation between the Xs and the Ys. </li></ul><ul><li>(there is mutual independence between the Xs and the Ys) </li></ul><ul><li>H 1 : There is a negative correlation between the Xs and the Ys. </li></ul>
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Hypothesis: III <ul><li>C. (One-Tailed - Upper) </li></ul><ul><li>H o : There is no correlation between the Xs and the Ys. </li></ul><ul><li>(there is mutual independence between the Xs and the Ys) </li></ul><ul><li>H 1 : There is a positive correlation between the Xs and the Ys. </li></ul>
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Test Statistic <ul><li>For small samples (N < 40): </li></ul><ul><li>T= Σ d i 2 = Σ [R(X i ) - R(Y i )] 2 </li></ul><ul><li>For large samples: </li></ul><ul><li>(Reject using the appropriate Z critical value) </li></ul>
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Test Statistic <ul><li>In the case of a large sample: </li></ul>
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Decision Rules <ul><li>Two-tailed: </li></ul><ul><li>Reject H 0 if T≤ S α/2 or T > S 1- α/2 . </li></ul><ul><li>Do not reject otherwise. </li></ul><ul><li>B. One-tailed - Lower: </li></ul><ul><li>Reject H 0 if T > S 1- α . </li></ul><ul><li>Do not reject otherwise. </li></ul><ul><li>C. One-tailed- Upper: </li></ul><ul><li>Reject H 0 if T≤ S α . </li></ul><ul><li>Do not reject otherwise. </li></ul>
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<ul><li>In the case of few ties (less than 5% of the sample): </li></ul><ul><li>Where d i is the difference in the ranks of each pair and N is the number of pairs </li></ul>Spearman’s Rho
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Spearman’s Rho <ul><li>If there are numerous ties: </li></ul>
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Example 1 <ul><li>The ICC rankings for One Day International (ODI) and Test matches for nine teams are shown below. </li></ul><ul><li>Test whether there is correlation between the ranks </li></ul>Team Test Rank ODI Rank Australia 1 1 India 2 3 South Africa 3 2 Sri Lanka 4 7 England 5 6 Pakistan 6 4 New Zealand 7 5 West Indies 8 8 Bangladesh 9 9
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Example 1 Answer: Team Test Rank ODI Rank d d 2 Australia 1 1 0 0 India 2 3 1 1 South Africa 3 2 1 1 Sri Lanka 4 7 3 9 England 5 6 1 1 Pakistan 6 4 2 4 New Zealand 7 5 2 4 West Indies 8 8 0 0 Bangladesh 9 9 0 0 Total 20
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Example 2 <ul><li>A composite rating is given by executives to each college graduate joining a plastic manufacturing firm. The executive ratings represent the future potential of the college graduate. The graduates then enter an in-plant training programme and are given another composite rating. The executive ratings and the in-plant ratings are as follows: </li></ul>
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<ul><li>At the 5% level of significance, determine if there is a positive correlation between the variables </li></ul><ul><li>Find the rank correlation coefficient if the null is rejected </li></ul>Graduate Executive rating (X) Training rating (Y) A 8 4 B 10 4 C 9 4 D 4 3 E 12 6 F 11 9 G 11 9 H 7 6 I 8 6 J 13 9 K 10 5 L 12 9
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