Spearman Rho Correlation

Transcription

1 Spearman Rho Correlation Learning Objectives After studying this Chapter, you should be able to: know when to use Spearman rho, Calculate Spearman rho coefficient, Interpret the correlation coefficient, Test hypothesis for significance of relationship.

2 Spearman s Rho Correlation Purpose: To measure relationship between two variables Requirement: Dependent variable - Ordinal (rank) Independent variable - Ordinal (rank) Introduction Similar to chi-square, Spearman s rho is also a non-parametric statistic which does not require that data in the population to be normally distributed. You may use Spearman s rho to measure relationship between two true ranked variables such as preference for academic programs among new students. In other instance you may have ranking measurement data which is originally interval or ratio scale. Generally you apply Spearman s rho for this interval or ratio-scaled variables if the the data does not meet the assumption of normality. Otherwise you may use another statistic called Pearson Product-Moment correlation which will be discussed in Chapter 12. Spearman s rho involves both descriptive and inferential elements. The descriptive element implies calculation of Spearman s rho coefficient(r s ) and decribe the nature of relationship between the two variables. The inferential element is to test the significance of the relationship between the variables.,. Calculation of r s Regardless on the scales of measurement of the data, either a true ranking or measurement data, calculation of r s is based on difference (d) between ranks of the two variables assigned for each observation. Formula 11.1 can be used to calculate r s.

3 Formula 11.1 The Spearman s rho coefficient ranges between -1 to +1. The former indicates a perfect negative relationship while the later indicates a perfect positive relationship Describing r s Once you have derived the Spearman s rho coefficient, you may describe the nature of relationship between the to variables. You may describe the strength and direction of the relationship. Use the Guildford s rule of thumb to elaborate on the strength of the relationship. Testing the Significance or r s You may apply testing of hypothesis for Spearman s rho if you are interested to test the significance of relationship between two variables. This section provides both the manual as well as SPSS procedures to test for the hypothesis. Steps in Hypothesis Testing (Manual): 1. State the hypothesis Since relationship between two variables can be either positive or negative, you have the option to apply a oneor two-tailed tests. One-tailed test: If you are test for a positive relationship H O : 0 H A : > 0 If you are test for a negative relationship H O : 0 H A : < 0

4 Note: The above null hypothesis for the one-tailed test can also be written as: Two-tailed test: H A : 0 2. Test statistic Use the value of the Spearman s rho coefficient that you have calculated earlier as your test statistics 3. Critical value Refer to Table in Appendix for the critical value. 4. Decision Your decison on whether to reject or fail to reject the null hypothesis is based on the following decision criteria: Decision criteria» Reject null hypothesis if Spearman s rho coefficient is bigger than critical value (CV)» Fail to reject null hypothesis if Spearman s rho coefficient is smaller or equals to critical value Criteria r s > CV r s CV Decision Reject the null hypothesis Fail to reject the null hypothesis 5. State your conclusion If you reject the null hypothesis, you can conclude that there is a significant relationship between the two variables. Steps in Hypothesis Testing (SPSS): 1. State the hypotheses You may refer to the preceeding topic on the way to write the null and alternative hypotheses.

5 2. Test statistic Extract the value for the test statistic from the SPSS output. You may refer to Appendix on the procedure to run one sample t-test. 3. Significance of the test statistic The SPSS output displays the exact probability of a Type 1 error (the p level) or the actual significance of the test statistic. This p value is printed as sig. (2-tailed) 4. Decision Decision Criteria for SPSS output:» Reject null hypothesis if p is smaller than alpha» Fail to reject null hypothesis if p is bigger or equal to alpha Criteria Decision p < Reject the null hypothesis p Fail to reject the null hypothesis 5. State your conclusion If you reject the null hypothesis, you can conclude that there is a significant relationship between the two variables. Example 1: You have ranked your students based on their interest in statistics and performance in the subject.. You are interested to know if there is any relationship between the two variables; interest and performance in statistic. Calculate Spearman s rho coefficient and describe the nature of relationship between the two variables. Then test the hypothesis that there is a positive relationship between the two variables at.05 level of significance. Student X Y

6 ANSWER (Manual calculation) 1. Calculation of r s Student X Y d d d = 0 d 2 = Describing r s Based on Guildford s rule of thumb, there is a positive and high relationship between students interest and performance in statistics 3. Testing hypothesis a) Hypothesis H A : > 0 b) Test statistic r s =.786

7 c) Critical value Critical r s =.714 d) Decision Since r s is bigger than critical r s, therefore reject the null hypothesis. e) Conclusion There is a significant relationship between students interest in statistics and performance in the subject at.05 level of significance. ANSWER (SPSS) Please refer to Figure 10.1 for the SPSS output of the Spearman s rho analysis. 1. Deriving r s Based on the SPSS output, r s =.786 (This value is reported as Spearman s rho correlation coefficient) 2. Describing r s Based on Guildford s rule of thumb, there is a positive and high relationship between students interest and performance in statistics 3. Testing hypothesis a) Hypothesis H A : 0 b) Decision and conclusion. Your decision on whether to reject or fail to reject the null hypothesis is based on p value which is given as sig. (2 tailed) in the output; below the r s. Since p (.036) is smaller than.05, therefore rejct the nul hypothesis. Conclude that there is a

8 significant relationship between students interest in statistics and performance in the subject at.05 level of significance. Figure 11.1: SPSS output of Spearman s rho Example 2: A researcher is interested to determine reationship between work stress (X) and quality of work life (Y). To ascertain his postulation, he collected data among a selected group of support staff. Data for the two variables, measured as intervalscaled is, follows. X Y Calculate Spearman s rho coefficient and describe the nature of relationship between work stress and quality of work life. Test the hypothesis at.01 level of significance. ANSWER (Manual calculation) 1. Calculation of r s

9 X Y r x r y d d d = 0 d 2 = Describing r s Based on Guildford s rule of thumb, there is a negative and moderate relationship between work stress and quality of work life. 3. Testing hypothesis a) Hypothesis H A : 0 b) Test statistic r s = c) Critical value Critical r s =.881

10 d) Decision Since r s is smaller than critical r s, therefore fail to reject the null hypothesis. e) Conclusion There is no significant relationship between work stress and quality f work life at.01 level of significance. ANSWER (SPSS) Please refer to Figure 10.2 for the SPSS output of the Spearman s rho analysis. 1. Deriving r s Based on the SPSS output, r s = (This value is reported as Spearman s rho correlation coefficient) 2. Describing r s Based on Guildford s rule of thumb, there is a negative and moderate relationship between work stress and quality of work life. 3. Testing hypothesis a) Hypothesis H A : < 0 b) Decision and conclusion. Your decision on whether to reject or fail to reject the null hypothesis is based on p value which is given as sig. (1 tailed) in the output; below the r s.. Since p (.038) is bigger than.01, therefore fail to rejct the nul hypothesis. Conclude that there is no significant relationship between work stress and quality f work life at.01 level of significance.

11 Figure 10.2: SPSS output of Spearman s rho