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Bivariate Methods: Correlation and Covariance in Statistical Relationships

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26 diapositives · 441 Views · Oct 18, 2012

Bivariate methods to analyze the relationship between two variables, such as education and income. Key concepts include scatterplots, correlation, and covariance, providing insights into how these statistical measures relate to one another. The interpretation of correlation coefficients, such as Pearson's r, is also discussed, emphasizing that correlation does not imply causation. Examples illustrate how covariance and correlation can vary in strength and direction, aiding in the comprehension of the complexities of variable relationships.

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Bivariate Methods: Correlation and Covariance in Statistical Relationships
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  1. Bivariate Methods • Relationship between two variables • e.g, as education ­, what does income do? • Scatterplot

  2. Correlation

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

  4. Wet – May 29/30 Avg. – June 26/28 Dry – August 22 Pond Branch - PG 11.25m DEM R2=0.79 R2=0.79 R2=0.71 Glyndon – LIDAR 0.5m DEM 11x11 R2=0.24 R2=0.10 R2=0.29

  5. Theta-TVDI Scatterplots

  6. API-TVDI Scatterplot

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

  8. i=n S 1 (xi - x)(yi - y) Cov [X, Y] = n - 1 i=1 Covariance Formulae

  9. Covariance Example

  10. 2 3 4 5 1 Covariance Example

  11. How Does Covariance Work? • X and Y are positively related • xi > x yi > y • xi < x yi < y • X and Y are negatively related • xi > x yi < y • xi < x  yi > y __ __ __ __ __ __ __ __

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

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

  14. i=n S (xi - x)(yi - y) i=1 (n - 1) sXsY r = r = i=n ZxZy S Cov [X, Y] r = sXsY i=1 (n - 1) Pearson’s product-moment correlation coefficient

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

  16. Example • X = TVDI, Y = Soil Moisture • Cov[X, Y] = -0.017063 • SX = 0.170, SY = 0.115 • r ?

  17. Pearson’s r - Assumptions • interval or ratio • Selected randomly • Linear • Joint bivariate normal distribution

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

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

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

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

  22. Interpreting Correlation Coefficients • Lack of independence • Social data • Geographic data • Spatial autocorrelation

  23. A Significance Test for r • An estimator • r  r • r = 0 ? • t-test

  24. r r r n - 2 ttest = = = SEr 1 - r2 1 - r2 n - 2 A Significance Test for r df = n - 2

  25. r n - 2 ttest = 1 - r2 A Significance Test for r • H0: r = 0 • HA: r¹ 0