Business Statistics CONTENTS The correlation coefficient The rank - PowerPoint PPT Presentation

𝜍 : ESTIMATES AND TESTS Business Statistics

CONTENTS The correlation coefficient The rank correlation coefficient Testing the correlation coefficient Non-linear relationships Old exam question Further study

THE CORRELATION COEFFICIENT Correlation coefficient 𝑡 𝑌,𝑍 ▪ 𝑠 𝑌,𝑍 = 𝑡 𝑌 𝑡 𝑍 Or written in full 𝑜 σ 𝑗=1 𝑦 𝑗 − ҧ 𝑦 𝑧 𝑗 − ത 𝑧 ▪ 𝑠 𝑌,𝑍 = 𝑦 2 σ 𝑗=1 𝑜 𝑜 𝑧 2 σ 𝑗=1 𝑦 𝑗 − ҧ 𝑧 𝑗 − ത Or using the sums-of-squares notation 𝑇𝑇 𝑌,𝑍 ▪ 𝑠 𝑌,𝑍 = 𝑇𝑇 𝑌,𝑌 𝑇𝑇 𝑍,𝑍

THE CORRELATION COEFFICIENT Correlation coefficient ▪ for two related numerical variables (paired data: 𝑌, 𝑍 = ) 𝑦 1 , 𝑧 1 , 𝑦 2 , 𝑧 2 , … , 𝑦 𝑜 , 𝑧 𝑜 ▪ −1 ≤ 𝑠 ≤ 1 ▪ indicator of linear association between two numerical variables Alternative names: ▪ Pearson correlation coefficient ▪ Pearson product-moment correlation coefficient ▪ named after Karl Pearson, 1857-1936

THE CORRELATION COEFFICIENT Scatter plots showing various situations

THE CORRELATION COEFFICIENT because for correlation, the Some points to observe variables are standardized ▪ There is only a “sign” relation between the correlation coefficient and the slope of the regression line ▪ if 𝑠 = 1 , the points fall on a straight line with slope>0 ▪ if 𝑠 = −1 , the points fall on a straight line with slope<0 ▪ Interchanging 𝑌 and 𝑍 will not change the correlation coefficient ▪ so 𝑠 𝑌,𝑍 = 𝑠 𝑍,𝑌 ▪ Rescaling 𝑌 or 𝑍 will not change the correlation coefficient ▪ in particular, 𝑠 is not sensitive to changes in units of 𝑌 or 𝑍 ▪ Correlation coefficients are sensitive to outliers

THE CORRELATION COEFFICIENT Note: ▪ correlation implies no causality ▪ sources: tylervigen.com and forbes.com

THE CORRELATION COEFFICIENT

EXERCISE 1 Which figure has a larger correlation coefficient?

TESTING THE CORRELATION COEFFICIENT Can we do a hypothesis test on the correlation coefficient? First acknoweldge: ▪ 𝑠 is the correlation coefficient of the two samples ▪ 𝜍 is the correlation coefficient of the bivariate population ▪ so the null hypothesis would be 𝐼 0 : 𝜍 = 0 or 𝐼 0 : 𝜍 ≥ 0.3 etc. ▪ never 𝑠 = 0 or so!

TESTING THE CORRELATION COEFFICIENT And the null distribution? 𝑆 ▪ It is known that 1−𝑆 2 / 𝑜−2 ~𝑢 𝑜−2 Important limitation: 𝑆 ▪ The distribution of the test statistic 1−𝑆 2 / 𝑜−2 is only for valid for 𝜍 = 0 (crucial in step 3) ▪ so we can only test 𝐼 0 : 𝜍 = 0 ▪ fortunately, that’s by far the most interesting hypothesis Test of 𝜍 = 𝜍 0 ≠ 0 : Google “Fisher transformation”; not in this course

TESTING THE CORRELATION COEFFICIENT ▪ Step 1: ▪ 𝐼 0 : 𝜍 = 0 ; 𝐼 1 : 𝜍 ≠ 0 ; 𝛽 = 0.05 ▪ Step 2: ▪ sample statistic: 𝑆 ; reject for “too small” and “too large” values ▪ Step 3: 𝑆 ▪ if 𝐼 0 is true, 1−𝑆 2 / 𝑜−2 ~𝑢 𝑜−2 ▪ normally distributed populations needed ▪ Step 4: ▪ as usual (insert 𝑠 for calculated value of 𝑆 ) ▪ Step 5: ▪ as usual

TESTING THE CORRELATION COEFFICIENT What is the meaning of rejecting 𝐼 0 : 𝜍 = 0 ? Conclude: there is a significant linear correlation between 𝑌 and 𝑍 ▪ meaning: the correlation is not 0 ▪ do not conclude: 𝑌 causes 𝑍 (or 𝑍 causes 𝑌 ) ▪ do not conclude: 𝑌 has a large influence on 𝑍 (or the other way around) There is an important difference between a correlation coefficient and a regression coefficient ▪ we will come back to this soon

NON-LINEAR RELATIONSHIPS What to do in case of a non-linear relation? ▪ Two suggestions: ▪ transform, e.g. log 𝑌 vs log 𝑍 ▪ use ranked data

NON-LINEAR RELATIONSHIPS Suggestion 1: log transformation ▪ life expectancy vs. GDP/capita ▪ → life expectancy vs. log(GDP/capita) ▪ 𝑠 = 0.646 ▪ → 0.774

NON-LINEAR RELATIONSHIPS ▪ Note: zero linear correlation does not exclude strong non- linear relation ▪ e.g., quadratic

NON-LINEAR RELATIONSHIPS Solution 2: with ranked data ▪ replace data ( 𝑌 and 𝑍 ) by ranks (→ 𝑌 𝑠 and 𝑍 𝑠 ) ▪ compute the (Pearson) correlation coefficient of 𝑌 𝑠 and 𝑍 𝑠 𝑠 𝑇 𝑌, 𝑍 = 𝑠 𝑌 𝑠 , 𝑍 𝑠 ▪ This is the rank correlation coefficient 𝑠 𝑇 ▪ Also Spearman correlation coefficient ▪ after Charles Spearman, 1863-1945

NON-LINEAR RELATIONSHIPS Of course many properties for 𝑠 also hold for 𝑠 𝑇 : ▪ −1 ≤ 𝑠 𝑇 ≤ 1 ▪ If 𝑠 𝑇 > 0 increasing (decreasing) 𝑦 -values tend to be accompanied by increasing (decreasing) 𝑧 -values ▪ If 𝑠 𝑇 < 0 increasing (decreasing) 𝑦 -values tend to be accompanied by decreasing (increasing) 𝑧 -values

NON-LINEAR RELATIONSHIPS Example ▪ life expectancy vs. GDP/capita ▪ or expectancy vs. log GDP/capita ▪ 𝑠 𝑇 = 0.828 ▪ for both (obviously!)

NON-LINEAR RELATIONSHIPS Can we also test hypotheses for the rank correlation coefficient? ▪ i.e. 𝐼 0 : 𝜍 𝑇 = 0 We can use similar but slightly different test as for 𝜍 𝑆 𝑇 ▪ i.e. 1/ 𝑜−1 ~𝑂 0,1 ▪ which requires 𝑜 ≥ 20 , but not normality of 𝑌 and 𝑍

OLD EXAM QUESTION 21 May 2015, Q1k-l

FURTHER STUDY Doane & Seward 5/E 12.1, 16.7 Tutorial exercises week 5 hypothesis test