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Bsc Chemistry ____________________________________________________________________________________________________ Subject PSYCHOLOGY Paper No and Title Paper No. 2: Quantitative Methods Module No and Title Module No. 20: Analysing relationships: non parametric correlation methods Module Tag PSY_P2_M20 TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Spearman’s rank order correlation coefficient (rho) 3.1 The concept of spearman’s rank order correlation 3.2 Numerical Application of Spearman Rank Order Correlation 3.3 Significance of ρ 4. Kendall’s rank order correlation coefficient- tau 4.1 The concept of Kendall’s Rank Order Correlation 4.2 Numerical Application of Kendall’s Rank Order Correlation coefficient (τ) 4.3 testing significance of kendall’s tau 4.4 Using Kendall’s tau in partial correlation 5. Some other correlational methods 5.1 Tetrachoric “rt” 5.2 The phi coefficient (Ø): 6. Summary PSYCHOLOGY Paper No. 2 Quantitative Methods Module No. 20 Analysing relationships: non parametric correlation methods ____________________________________________________________________________________________________ 1. Learning Outcomes After studying this module, you shall be able to Know some popular non parametric methods to analyze relationships Learn the rank order correlation coefficients: spearman’s Rho and Kendall’s tau Know about some of the other correlational methods like tetrachoric and phi coefficients Evaluate the values of non-parametric correlation coefficients. Analyze the significance of the calculated values of correlation coefficients etc. 2. Introduction Non Parametric methods of correlation It is always of interest to a researcher to find out the degree of association between variables of interest. The usual method of correlation is the Pearson’s product moment correlation, but that requires the data to fulfill the assumptions of parametric statistics. We always have data with continuous variables or underlying continuity is assumed along a numerical scale. There are non-parametric methods of correlation that are suitable for data that may not be apt for data otherwise. These non- parametric methods of correlation are used when: The data is in ordinal or nominal scale of measurement or in descriptive categories only. No assumptions about the population from which the samples are drawn are made. Or if any assumptions are there they are very few The non-parametric correlation methods are easier to calculate and apply to the data. Let us explain this by taking an example, it is hypothesized that a person’ ability to be a leader is related to his success in sports events played in school. The leadership ability was measured using a scale while the rank held during the sports event played in school was taken from the records maintained in school and is in ordinal scale. The Pearsons’ product moment correlation is applicable to data in interval or ratio scales so, ideally not suitable. Thus, non-parametric correlation coefficients have to be applied. Some of the most popularly used non parametric correlation methods are: Spearman’s Rho, kendall’s Tau, Phi. The current module discusses these methods and also mentions a few others which are relatively less prevalent. PSYCHOLOGY Paper No. 2 Quantitative Methods Module No. 20 Analysing relationships: non parametric correlation methods ____________________________________________________________________________________________________ 3. Spearman’s Rank Order Correlation - Rho 3.1 The concept of spearman’s rank order correlation The Spearman’s rank order correlation is a non parametric correlation coefficient that is applicable to data which which does not confirm to the parametric norms. The variables such as truthfulness, sporting spirit, marketing ability etc. are complex variables and thus difficult to measure; assigning quantities to them is not possible. .However, ranking them is easier and therefore, statistical procedures have been devised to analyze such data. To analyze relationship among such variables methods are available and one such is Spearman Rank Order correlation. The correlation coefficient is known as “rho” and symbolized as “ρ”. The basic logic behind rank order correlation developed by Charles Spearman, is that the variables being measured are in at least ordinal scale. The data has to be thus, ranked in two ordered series. Once the data has been ranked in the differences between the rank obtained by each individual on both the variables is calculated. The disparity between the ranks for each individual is indicative of association that might be there between the two variables. As obvious if there would be a high difference between the ranks then it would directly mean a lower association between the variables. If the difference between the ranks is closer to zero it would point at a near perfect association between the variables. Once the ranks are calculated for the obtained data and Pearson’s correlation equation is applied to it. Charles Spearman(1863-1945) developed a method to calculate correlation of qualitative variables and named it Spearman’s rank correlation coefficient. PSYCHOLOGY Paper No. 2 Quantitative Methods Module No. 20 Analysing relationships: non parametric correlation methods ____________________________________________________________________________________________________ 6 Ʃ d2 ρ = 1 - -------------- --------eqn. (1) N ( N2 – 1 ) Where, d is the difference in the ranks between the two variables N is the sample size This formula needs correction when the ranks are tied i.e. two or more individuals have the same score on the same variable (we shall take up such examples in the coming sections). However, if the number of tied ranks is not many one may still use the formula in equation (1) otherwise it is advisable to use the formula given below. 3 2 (N – N) – 6 Ʃ d – (Tx + Ty ) / 2 ρ = -------------------------------------------------------- ------eqn. (2) 3 2 3 √ (N – N) - (Tx + Ty ) ((N – N) + Tx Ty Where, d is the difference between the ranks T is the correction factor for tied ranks and calculated as, T = Ʃ (t3 – t); t is number of all the tied ranks in a group of a particular variable. N is the sample size 3.2 Numerical Application of Spearman Rank Order Correlation 3.2.1 Untied ranks The data given below in table I gives the rating given by two political party leaders about the six personal qualities they feel are important to win an election. The political leaders’ ratings are subjected to rank order correlation to know about the degree of association among their responses. PSYCHOLOGY Paper No. 2 Quantitative Methods Module No. 20 Analysing relationships: non parametric correlation methods ____________________________________________________________________________________________________ Qualities Leader A Leader B d d2 1 2 5 -3 9 2 3 6 -3 9 3 1 1 0 0 4 4 2 2 4 5 6 3 3 9 6 5 4 1 1 0 32 Using the formula given in equation (1) one can calculate the value of ρ, by following the given steps. The first three columns of the table give the six qualities and the ranks given by the leaders. Fourth column gives the differences between the ranks as “d” for each quality. Calculate d2 ρ = 1 – {(6 X 32) ÷ 6(62 – 1)} = .086 3.2.1 Tied ranks A study was taken up to see if there is really a significant relationship between creativity and how quickly one completes the plot of a short story. The creativity scores were obtained using a standardized test while time taken to complete the story plot was recorded in seconds. the sample comprised of 15 artists from the art school. The rank order correlation was calculated to understand the relationship between the two variables. The obtained data is given in table II below. Table II: the data in table shows the scores obtained by the subjects on creativity and time taken to give an end to a story. Individual X Y ( time Ranking Ranking d d2 (creativity) taken) of X of Y 1 185 110 6 7.5 -1.5 225 2 203 98 1 5 -4 16 3 188 118 4 10.5 -6.5 42.25 4 195 104 3 6 -3 9 5 176 112 8.5 9 -5 25 PSYCHOLOGY Paper No. 2 Quantitative Methods Module No. 20 Analysing relationships: non parametric correlation methods ____________________________________________________________________________________________________ 6 174 124 10 14 -4 16 7 158 119 12 12 0 0 8 197 95 2 3 -1 1 9 176 94 8.5 1.5 -7 49 10 138 97 14 4 10 100 11 126 110 15 7.5 7.5 56.25 12 160 94 11 1.5 9.5 90.25 13 151 126 13 15 -2 4 14 185 120 6 13 -7 49 15 185 118 6 10.5 -4.5 20.25 455.50 703.5 As the variables X and Y are continuous in nature both have to be ranked as shown in the table. The variable X has been ranked with the highest score 203 given the rank 1, the next highest 197 given the rank 2. (Take a note, the observation 176 is repeated twice and therefore, (8+9)/2= 8.5 is given as rank order for both scores of 176). The variable Y is scored in seconds and thus, the two smallest scores (94) are actually the highest and ranked 1.5 each. As the data has relatively high number of tied ranks in variable Y, the rank order correlation should be calculated using equation (2). 3 3 Tx = (3 – 3) + (2 – 2) = 30 3 3 3 Ty = (2 – 2) + (2 – 2) + (2 – 2) = 18 (153 – 15) - (6 X 703.5) – (30 + 18) / 2 ρ = -------------------------------------------------------------------------- √ (153 – 15)2 – (30 + 18) (153 – 15) + (30 X 18) 1225.5 / 3335 = .367 3.3 Significance of ρ The Spearman rank order correlation value can be tested for significance, with the null hypothesis that the two variables under study are not associated in the population. The alternative hypothesis PSYCHOLOGY Paper No. 2 Quantitative Methods Module No. 20 Analysing relationships: non parametric correlation methods ____________________________________________________________________________________________________ can be there is an association (two tailed) and there is a positive or there is a negative correlation (one tailed).
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