1. Regression and Correlation
GTECH 201
Lecture 18

2. ANOVA Analysis of Variance
Continuation from matched-pair difference of means tests; but now for 3+ cases
We still check whether samples come from one or more distinct populations
Variance is a descriptive parameter
ANOVA compares group means and looks whether they differ sufficiently to reject H0

3. ANOVA H0 and HA

4. ANOVA Test Statistic
MSB = between-group mean squares
MSW = within-group mean squares
Between-group variability is calculated in three steps:
Calculate overall mean as weighted average of sample means
Calculate between-group sum of squares
Calculate between-group mean squares (MSB)

5. Between-group Variability
Total or overall mean
Between-group sum of squares
Between-group mean squares

6. Within-group Variability
Within-group sum of squares
Within-group mean squares

7. Kruskal-Wallis Test Nonparametric equivalent of ANOVA
Extension of Wilcoxon rank sum W test to 3+ cases
Average rank is Ri / ni
Then the Kruskal-Wallis H test statistic is
With N =n1 + n2 + … +nk = total number of observations, and
Ri = sum of ranks in sample i

8. ANOVA Example House prices by neighborhood in ,000 dollars A B C D
196

9. ANOVA Example, continued
Sample statistics
n X s A B C D
Total
Now fill in the six steps of the ANOVA calculation

10. The Six Steps

11. Correlation Co-relatedness between 2+ variables
As the values of one variable go up, those of the other change proportionally
Two step approach:
Graphically - scatterplot
Numerically – correlation coefficients

12. Is There a Correlation?

13. Scatterplots
Exploratory analysis

14. Pearson’s Correlation Index
Based on concept of covariance
= covariation between X and Y
= deviation of X from its mean
= deviation of Y from its mean
Pearson’s correlation coefficient

15. Sample and Population r is the sample correlation coefficient
Applying the t distribution, we can infer the correlation for the whole population
Test statistic for Pearson’s r

16. Correlation Example
Lake effect snow

17. Spearman’s Rank Correlation
Non-parametric alternative to Pearson
Logic similar to Kruskal and Wilcoxon
Spearman’s rank correlation coefficient

18. Regression
In correlation we observe degrees of association but no causal or functional relationship
In regression analysis, we distinguish an independent from a dependent variable
Many forms of functional relationships
bivariate
linear
multivariate
non-linear (curvi-linear)

19. Graphical Representation
In correlation analysis either variable could be depicted on either axis
In regression analysis, the independent variable is always on the X axis
Bivariate relationship is described by a best-fitting line through the scatterplot

20. Least-Square Regression
Objective: minimize

21. Regression Equation
Y = a + bX

22. Strength of Relationship
How much is explained by the regression equation?

23. Coefficient of Determination
Total variation of Y (all the bucket water)
Large ‘Y’ = dependent variable
Small ‘y’ = deviation of each value of Y from its mean
e = explained; u = unexplained

24. Explained Variation
Ratio of square of covariation between X and Y to the variation in X
where Sxy = covariation between X and Y
Sx2 = total variation of X
Coefficient of determination

25. Error Analysis
r 2 tells us what percentage of the variation is accounted for by the independent variable
This then allows us to infer the standard error of our estimate which tells us, on average, how far off our prediction would be in measurement units