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𝜍: ESTIMATES AND TESTS

Business Statistics

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

Correlation coefficient ▪ 𝑠

𝑌,𝑍 = 𝑡𝑌,𝑍 𝑡𝑌𝑡𝑍

Or written in full ▪ 𝑠

𝑌,𝑍 = σ𝑗=1

𝑜

𝑦𝑗− ҧ 𝑦 𝑧𝑗− ത 𝑧 σ𝑗=1

𝑜

𝑦𝑗− ҧ 𝑦 2 σ𝑗=1

𝑜

𝑧𝑗− ത 𝑧 2

Or using the sums-of-squares notation ▪ 𝑠

𝑌,𝑍 = 𝑇𝑇𝑌,𝑍 𝑇𝑇𝑌,𝑌 𝑇𝑇𝑍,𝑍

THE CORRELATION COEFFICIENT

Correlation coefficient ▪ for two related numerical variables (paired data: 𝑌, 𝑍 = 𝑦1, 𝑧1 , 𝑦2, 𝑧2 , … , 𝑦𝑜, 𝑧𝑜 ) ▪ −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

Some points to observe ▪ 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

because for correlation, the variables are standardized

Note: ▪ correlation implies no causality

▪ sources: tylervigen.com and forbes.com

THE CORRELATION COEFFICIENT

THE CORRELATION COEFFICIENT

Which figure has a larger correlation coefficient? EXERCISE 1

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 𝐼0: 𝜍 = 0 or 𝐼0: 𝜍 ≥ 0.3 etc.

▪ never 𝑠 = 0 or so!

TESTING THE CORRELATION COEFFICIENT

And the null distribution? ▪ It is known that

𝑆 1−𝑆2 / 𝑜−2 ~𝑢𝑜−2

Important limitation: ▪ The distribution of the test statistic

𝑆 1−𝑆2 / 𝑜−2 is only

for valid for 𝜍 = 0 (crucial in step 3)

▪ so we can only test 𝐼0: 𝜍 = 0 ▪ fortunately, that’s by far the most interesting hypothesis

TESTING THE CORRELATION COEFFICIENT

Test of 𝜍 = 𝜍0 ≠ 0: Google “Fisher transformation”; not in this course

▪ Step 1:

▪ 𝐼0: 𝜍 = 0; 𝐼1: 𝜍 ≠ 0; 𝛽 = 0.05

▪ Step 2:

▪ sample statistic: 𝑆; reject for “too small” and “too large” values

▪ Step 3:

▪ if 𝐼0 is true,

𝑆 1−𝑆2 / 𝑜−2 ~𝑢𝑜−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 𝐼0: 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 TESTING THE CORRELATION COEFFICIENT

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 (→ 𝑌𝑠 and 𝑍

𝑠)

▪ compute the (Pearson) correlation coefficient of 𝑌𝑠 and 𝑍

𝑠

𝑠

𝑇 𝑌, 𝑍 = 𝑠 𝑌𝑠, 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. 𝐼0: 𝜍𝑇 = 0 We can use similar but slightly different test as for 𝜍 ▪ i.e.

𝑆𝑇 1/ 𝑜−1 ~𝑂 0,1

▪ which requires 𝑜 ≥ 20, but not normality of 𝑌 and 𝑍

NON-LINEAR RELATIONSHIPS

21 May 2015, Q1k-l OLD EXAM QUESTION

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