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Association

Association. Predicting One Variable from Another. Correlation. Usually refers to Pearson’s r computed on two interval/ratio scale variables. It measures the degree to which variance in one variable is “explained” by a second variable

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Association

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  1. Association Predicting One Variable from Another

  2. Correlation • Usually refers to Pearson’s r computed on two interval/ratio scale variables. • It measures the degree to which variance in one variable is “explained” by a second variable • It measures the strength of a linear relationship between the variables

  3. Definition of r

  4. Properties of r • r is symmetrical and varies from -1 to +1 • 0 indicates no correlation or relationship • ±1 indicates a perfect correlation (knowledge of one variable makes it possible to predict the second one without any error).

  5. Properties of r2 • r2 is symmetrical and varies from 0 to 1 • r2 is the proportion of the variability in one variable that is “explained by” the other variable • cor.test(x, y, method=“pearson”) • cor(x, y, method=“pearson”)

  6. Spearman’s rho • For rank/ordinal data. • Pearson correlation computed on ranks • If Spearman coefficient is larger than Pearson, it may indicate a non-linear relationship • Ties make it difficult to compute p values

  7. Kendall’s tau • For rank/ordinal data • Evaluate pairs of observations (xi, yi) and (xj, yj) • Concordant – (xi > xj) and (yi > yj) OR (xi < xj) and (yi < yj) • Discordant – (xi > xj) and (yi < yj) OR (xi < xj) and (yi > yj)

  8. Kendall’s tau-a

  9. Kendall’s tau b • Divide by total number of pairs adjusted for all ties

  10. Kendall’s tau c • For grouped (tabled data) where the table is not square (rows ≠ columns)

  11. Nominal Measures • Measures based on Chi-Square: • Phi coefficient • Cramer’s V • Contingency coefficient • Odds ratio

  12. Phi and Cramer’s V • Phi ranges from 0 to 1 in a 2x2 table but can exceed 1 in larger tables. Cramer’s V adds a correction to keep the maximum value at 1 or less:

  13. Contingency Coefficient • Ranges from 0 to <1 depending on the number of rows and columns with 1 indicating a high relationship and 0 indicating no relationship

  14. Odds Ratio • For 2 x 2 tables it shows the relative odds between the two variables

  15. > Table <- xtabs(~Sex+Goods, data=EWG2) > Table Goods Sex Absent Present Female 38 28 Male 16 30 > ChiSq <- chisq.test(Table) > ChiSq Pearson's Chi-squared test with Yates' continuity correction data: Table X-squared = 4.7644, df = 1, p-value = 0.02905

  16. library(vcd) > assocstats(Table) X^2 df P(> X^2) Likelihood Ratio 5.7073 1 0.016894 Pearson 5.6404 1 0.017552 Phi-Coefficient : 0.224 Contingency Coeff.: 0.219 Cramer's V : 0.224 > cor(as.numeric(EWG2$Sex), as.numeric(EWG2$Goods), use="complete.obs") [1] 0.2244111 > oddsratio(Table, log=FALSE) [1] 2.544643

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