1. Relationship between two variables e.g, as education , what does income do? Scatterplot Bivariate Methods

2. Correlation

3. Linear Correlation Source: Earickson, RJ, and Harlin, JM. 1994. Geographic Measurement and Quantitative Analysis. USA: Macmillan College Publishing Co., p. 209.

5. Wet – May 29/30Avg. – June 26/28Dry – August 22 Pond Branch -PG 11.25m DEM Glyndon – LIDAR0.5m DEM 11x11 R 2 =0.71 R 2 =0.29 R 2 =0.79 R 2 =0.24 R 2 =0.79 R 2 =0.10

6. Theta-TVDI Scatterplots

7. API-TVDI Scatterplot

8. Covariance: Interpreting Scatterplots General sense (direction and strength) Subjective judgment More objective approach Extent to which variables Y and X vary together Covariance

9. Covariance Formulae Cov [X, Y] =  (x i - x)(y i - y) i=1 i=n 1 n - 1

10. Covariance Example

11. 1 2 3 4 5

12. How Does Covariance Work? X and Y are positively related x i > x  y i > y x i < x  y i < y X and Y are negatively related x i > x  y i < y x i y __

13. Interpreting Covariances Direction & magnitude Cov[X,Y] > 0  positive Cov[X, Y] < 0  negative abs(Cov[X, Y]) ↑  strength ↑ Magnitude ~ units

14. Covariance  Correlation Magnitude ~ units Multiple pairs of variables  not comparable Standardized covariance Compare one such measure to another

15. Pearson’s product-moment correlation coefficient Cov [X, Y] sXsYsXsY r = r  (x i - x)(y i - y) i=1 i=n (n - 1) s X s Y = ZxZyZxZy r  i=1 i=n (n - 1) =

16. Pearson’s Correlation Coefficient r [–1, +1] abs(r) ↑  strength ↑ r cannot be interpreted proportionally ranges for interpreting r values 0 - 0.2Negligible 0.2 - 0.4Weak 0.4 - 0.6Moderate 0.6 - 0.8Strong 0.8 - 1.0Very strong

17. Example X = TVDI, Y = Soil Moisture Cov[X, Y] = -0.017063 S X = 0.170, S Y = 0.115 r ?

18. Pearson’s r - Assumptions 1.interval or ratio 2.Selected randomly 3.Linear 4.Joint bivariate normal distribution

19. Interpreting Correlation Coefficients Correlation is not the same as causation! Correlation suggests an association between variables 1.Both X and Y are influenced by Z

20. Interpreting Correlation Coefficients 2.Causative chain (i.e. X  A  B  Y) e.g. rainfall  soil moisture  ground water  runoff 3.Mutual relationship e.g., income & social status 4. Spurious relationship e.g., Temperature (different units) 5. A true causal relationship (X  Y)

21. Interpreting Correlation Coefficients 6.A result of chance e.g., your annual income vs. annual population of the world

22. Interpreting Correlation Coefficients 7. Outliers (Source: Fang et al., 2001, Science, p1723a)

23. Interpreting Correlation Coefficients Lack of independence –Social data –Geographic data –Spatial autocorrelation

24. A Significance Test for r An estimator r    = 0 ? t-test

25. A Significance Test for r t test = r SE r = r 1 - r 2 n - 2 = r 1 - r 2 df = n - 2

26. A Significance Test for r H 0 :  = 0 H A :   0 t test = rn - 2 1 - r 2