Bivariate statistics: correlation

Transcription

1 Research Methods for Political Science Bivariate statistics: correlation Dr. Thomas Chadefaux Assistant Professor in Political Science 1

2 Bivariate relationships: interval-ratio variables 2

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5 GDP and Electricity consumption 5

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12 Covariation and correlation We can express the association between two interval-ratio variables in terms of their covariation or correlation We start with covariation and then discuss correlation 12

13 Remember: variation Example: German election results 2013: seats per party Party Seats CDU 255 SPD 192 Linke 64 Grüne 63 CSU 56 13

14 Variance Remember: Variance= (sum of square distances from mean) / (N 1) 14

15 Covariance: seats and percentage Party Seats obtained Percentage of votes CDU SPD Linke Grüne CSU

16 Covariance 16

17 Step 1. Calculate the product of the deviation of seats and the deviation of percentages. 17

18 Step 1. Calculate the product of the deviation of seats and the deviation of percentages. 18

19 Step 2. Calculate the sum of those products =

20 Sum of products = 4515 N = 5 Step 3. Divide this sum by N - 1 Covariance = 4515 / (N-1) =

21 Variance and covariance Remember the formula for variance: variance = (+, +) 0 = (+, +)(+, +) covariance = (+, +)(4, 54)

22 Interpreting covariance Covariance depends on the scale of the measurements used. But we can tell something from the sign: If cov(x, y) = 0, then these variables are not related If cov(x, y) > 0, then they vary in the same direction. As x increases, so does y. If cov(x, y) < 0, then they vary in the opposite direction. As x increases, y decreases. 22

23 More Intuition for the covariance 23

24 Correlation The maximum covariation depends on the scale. Can we transform this score in something which ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation)? 24

25 Correlation The maximum covariation between two variables is equal to the product of their standard deviations:! "! # So if we divide the covariance by! "! #, we will get a measure of how much the variable co-vary, compared to how much they could possibly covary, and this measure will be between 0 and 1 25

26 Correlation The Pearson product-moment correlation r is found by dividing the covariance by the product of the standard deviation of the two variables:! = #$% &,( ) & ) ( 33

27 The Germany example Covariance was Standard deviation seats (x) = 91.8 Standard deviation percentages (y) = 12.3! = #$% &,( ) & ) ( = =

28 Calculating r (sample formula) r = (% & %)(* & +*) (, 1). /. 0 35

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30 Interpreting r values Value Description Explanation r = -1 Perfect negative correlation If x increases, then y decreases r < -0.5 Strong negative correlation If x increases, then y decreases r < -0.3 Moderate negative correlation If x increases, then y decreases r < -0.1 Weak negative correlation If x increases, then y decreases r = 0 No correlation Variation x and y are not related r > 0.1 Weak positive correlation If x increases, then y increases r > 0.3 Moderate positive correlation If x increases, then y increases r > 0.5 Strong positive correlation If x increases, then y increases r = 1 Perfect positive correlation If x increases, then y increases These are rules of thumb. Always look at your data! 37

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32 SPSS Analyze... Correlations... Bivariate 39

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34 Significance 41

35 Significance We can test whether a correlation is statistically significant by using the t- distribution with N 2 degrees of freedom.! " = " $%& '%" ( 42

36 Signifiance: PD and FF R =.387 N = 2549 t=21.18 As our obtained value > critical value, we conclude that the correlation coefficient is statistically significant. 43

37 Spearman s rho Correlation of ranks Useful if relationship is not linear or if assumptions about normality are not met First, rank both variables from small to large and write down rank numbers (1 for the smallest value, N for the largest value) Then calculate r for the rank numbers. 44

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40 Spearman s rho in SPSS 47

41 Spearman s rho: output 48

42 Watch out! Pearson s r and Spearmans rho provide a measure of association, not proof of causation. We do not take third variables into account. The statistics do not indicate the direction of the relationship. 49