Pearson (rho), Spearman (R), and Kendall (tau) correlation coefficients: See Section 8.5 in the text for Spearman R and Section 8.3 for Kendall tau NOTE: Bootstrap methods do quite well for Kendall`s tau (see below and Section 8.4), but not as well for Pearson`s rho and Spearman`s R Data: Amount of pre-packed engine lubricant and engine survival times: (Source of data unknown.) Lubricant Survival Lubricant Survival 1. 29 23 11. 84 429 2. 21 36 12. 37 489 3. 30 67 13. 94 504 4. 45 104 14. 56 925 5. 48 147 15. 67 980 6. 92 164 16. 79 1254 7. 74 247 17. 70 2961 8. 61 304 18. 93 5351 9. 99 355 19. 76 9915 10. 24 408 20. 72 11301 Initializing random-number generator at 123456 Number of permutations and/or bootstrap replications: 10000 Data sorted into one set of columns: 1 29 23 2 21 36 3 30 67 4 45 104 5 48 147 6 92 164 7 74 247 8 61 304 9 99 355 10 24 408 11 84 429 12 37 489 13 94 504 14 56 925 15 67 980 16 79 1254 17 70 2961 18 93 5351 19 76 9915 20 72 11301 Pearson rho for X vs Y and Student-t P-value: Rho=0.290556 T=1.2883 P=0.2140 (df=18) Permutation test for P-value of Pearson rho: (Permuting values of Y for fixed X.) Number greater-or-equal (absolute value) and number of permutations: 2249 10000 95% Conf.Int. for true permutation-P-value bracketed by estimate: 0.2167 0.2249 0.2331 (nsims=10000) Fisher z-transform 95% Conf.Int. for true rho bracketing rho: -0.1744 0.2906 0.6496 Bootstrap 95% Percentile Conf.Int. for rho bracketing Median: 0.0819 0.3086 0.5694 Bias: 0.4430 Bootstrap P-value for H_0:rho=0: P=0.0066 (2-sided) X vs log(Y): 1 29 3.135 2 21 3.584 3 30 4.205 4 45 4.644 5 48 4.990 6 92 5.100 7 74 5.509 8 61 5.717 9 99 5.872 10 24 6.011 11 84 6.061 12 37 6.192 13 94 6.223 14 56 6.830 15 67 6.888 16 79 7.134 17 70 7.993 18 93 8.585 19 76 9.202 20 72 9.333 Pearson rho for X vs log(Y) and Student-t P-value: Rho=0.546574 T=2.76915 P=0.0126 (df=18) Permutation test for P-value of Pearson rho: (Permuting values of log(Y) for fixed X.) Number greater-or-equal (absolute value) and number of permutations: 126 10000 95% Conf.Int. for true permutation-P-value bracketed by estimate: 0.0104 0.0126 0.0148 (nsims=10000) Fisher z-transform 95% Conf.Int. for true rho bracketing rho: 0.1372 0.5466 0.7965 Bootstrap 95% Percentile Conf.Int. for rho bracketing Median: 0.2109 0.5570 0.7900 Bias: 0.4711 Bootstrap P-value for H_0:rho=0: P=0.0040 (2-sided) Kendall K and Kendall tau (assuming no ties): K=72 tau=0.378947 Z=2.33599 P=0.0195 (2-sided) Permutation test for P-value of (absolute value of) Kendall tau: Nge and number of permutations: 197 10000 95% Conf.Int. for true two-sided P-value bracketed by estimate: 0.0170 0.0197 0.0224 Distribution-free confidence interval for tau (Section 8.3): 0.0897 0.3789 0.6682 Bootstrap 95% Percentile Conf.Int. for Kendall tau bracketing Median: 0.0526 0.3684 0.6263 Bias: 0.5286 Bootstrap P-value for H_0:tau=0: P=0.0260 (2-sided) Within-sample midranks for Spearman R: Lubricant, Lubricant midranks, Survival, Survival midranks, Tiegroups: 1 29 3.0 23 1.0 0 0 2 21 1.0 36 2.0 0 0 3 30 4.0 67 3.0 0 0 4 45 6.0 104 4.0 0 0 5 48 7.0 147 5.0 0 0 6 92 17.0 164 6.0 0 0 7 74 13.0 247 7.0 0 0 8 61 9.0 304 8.0 0 0 9 99 20.0 355 9.0 0 0 10 24 2.0 408 10.0 0 0 11 84 16.0 429 11.0 0 0 12 37 5.0 489 12.0 0 0 13 94 19.0 504 13.0 0 0 14 56 8.0 925 14.0 0 0 15 67 10.0 980 15.0 0 0 16 79 15.0 1254 16.0 0 0 17 70 11.0 2961 17.0 0 0 18 93 18.0 5351 18.0 0 0 19 76 14.0 9915 19.0 0 0 20 72 12.0 11301 20.0 0 0 Spearman R, large-sample Z, large-sample P-value: 0.5083 2.2155 0.0267 (2-sided) Permutation test for P-value of (absolute value of) Spearman R: Nge and number of permutations: 230 10000 95% Conf.Int. for true two-sided P-value bracketed by estimate: 0.0201 0.0230 0.0259 Large-sample 95% Conf.Int. for R bracketing R (Section 8.5): 0.0586 0.5083 0.9579 Bootstrap 95% Percentile Conf.Int. for R bracketing Median: 0.1466 0.5030 0.8940 Bias: 0.5129 Bootstrap P-value for H_0:R=0: P=0.0052 (2-sided)