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Lecture Statistical Comparisons Stevens - Spring 2018 1 of 5

CEE 6660 -- Statistical Comparisons I. Comparing alternatives

A. Dancing around statistical comparisons using min and max

B. Statistical comparison of alternatives

1. Needs a. mean or median for each alternative b. standard deviation or rankings

2. Procedures a. t test for normally distributed data

1) and if t > specific value (t/2,), the alternatives are different

2) 1, 2 are the mean values for the defender and challenger, respectively

3) and

4) t/2, is the critical value from the t distribution (similar to normal) for  = n1 + n2 - 2 degrees of freedom for a required probability, p = 1 - (often  = 0.05 for 95% certainty, but it’s your choice)

b. rank sum test (aka Wilcoxon or Mann-Whitney) for non-parametric (or any, really) data 1) combine the data and sort 2) assign ranks (1 to (n1+n2)) 3) separate data and ranks by group 4) sum the ranks for each

5) calculate and

6) look up the smaller of U1 and U2 in significance tables (plot) 3. Examples

a. data (annual worth of two alternatives) 1) alt. 1 - (552, 563, 489, 484, 440, 512, 355, 530, 480, 480) 2) alt. 2 - (513, 538, 552, 555, 476, 547, 497, 571, 485, 457)

b. t-test for normally distributed (use a probability plot to help decide) 1) 1 = 488.5, 2 = 519.1 2) s1 = 59.8, s2 = 38.9

3)

4)

5) with  = 10 + 10 - 2 = 18

6) t1crit,,= 1.734 > t and there is no compelling evidence that alternative 2 is better than alternative 1

1. Row 18, Column 5 in t-distribution table (below)

t 2 1– s2 1– ----------------=

s2 1– spooled 1 n1 ----- 1

n2 -----+ 

 = spooled n1 1– s12 n2 1– s22+

n1 n2 2–+ ------------------------------------------------------=

U1 R1 n1 n1 1+ 

2 ------------------------–= U2 R2

n2 n2 1+  2

------------------------–=

spooled 10 1– 59.82 10 1– 38.92+

10 10 2–+ ------------------------------------------------------------------------ 2544.6 50.4= = =

s2 1– spooled 1 n1 ----- 1

n2 -----+ 

  50.4 1 10 ------ 1

10 ------+ 

  22.54= = =

t 519.1 488.5– 22.54

--------------------------------- 1.36= =

Lecture Statistical Comparisons Stevens - Spring 2018 2 of 5

c. Wilcoxon test (for data that are not normally distributed

1) combine and rank data

2) Separate and sum ranks

3) Look up in table for n1 = 10 and n2 = 10 for  = 0.05: Rlower = 82, Rupper =128 and since our range is within this one, the probability that they are different is < 95% = 90.3%. See Wilcoxon Rank Sum Table for R values

4) Find U for lower R1

5) See Table A5.07 for the U values

x 552 563 489 484 440 512 355 530 480 480 513 538 552 555 476 547 497 571 485 457 id x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 Rank 16.5 19.0 9.0 7.0 2.0 11.0 1.0 13.0 5.5 5.5 12.0 14.0 16.5 18.0 4.0 15.0 10.0 20.0 8.0 3.0

Table 1. Student’s t distribution

 Tail area probability, 

0.40 0.25 0.10 0.05 0.025 0.01 0.005 0.001 1 0.3249 1.0000 3.078 6.314 12.706 31.821 63.657 318.309 2 0.2887 0.8165 1.886 2.920 4.303 6.965 9.925 22.327 3 0.2767 0.7649 1.638 2.353 3.182 4.541 5.841 10.215 4 0.2707 0.7407 1.533 2.132 2.776 3.747 4.604 7.173 5 0.2672 0.7267 1.476 2.015 2.571 3.365 4.032 5.893 6 0.2648 0.7176 1.440 1.943 2.447 3.143 3.707 5.208 7 0.2632 0.7111 1.415 1.895 2.365 2.998 3.499 4.785 8 0.2619 0.7064 1.397 1.860 2.306 2.896 3.355 4.501 9 0.2610 0.7027 1.383 1.833 2.262 2.821 3.250 4.297

10 0.2602 0.6998 1.372 1.812 2.228 2.764 3.169 4.144 11 0.2596 0.6974 1.363 1.796 2.201 2.718 3.106 4.025 12 0.2590 0.6955 1.356 1.782 2.179 2.681 3.055 3.930 13 0.2586 0.6938 1.350 1.771 2.160 2.650 3.012 3.852 14 0.2582 0.6924 1.345 1.761 2.145 2.624 2.977 3.787 15 0.2579 0.6912 1.341 1.753 2.131 2.602 2.947 3.733 16 0.2576 0.6901 1.337 1.746 2.120 2.583 2.921 3.686 17 0.2573 0.6892 1.333 1.740 2.110 2.567 2.898 3.646 18 0.2571 0.6884 1.330 1.734 2.101 2.552 2.878 3.610 19 0.2569 0.6876 1.328 1.729 2.093 2.539 2.861 3.579 20 0.2567 0.6870 1.325 1.725 2.086 2.528 2.845 3.552 21 0.2566 0.6864 1.323 1.721 2.080 2.518 2.831 3.527 22 0.2564 0.6858 1.321 1.717 2.074 2.508 2.819 3.505 23 0.2563 0.6853 1.319 1.714 2.069 2.500 2.807 3.485 24 0.2562 0.6848 1.318 1.711 2.064 2.492 2.797 3.467 25 0.2561 0.6844 1.316 1.708 2.060 2.485 2.787 3.450 26 0.2560 0.6840 1.315 1.706 2.056 2.479 2.779 3.435 27 0.2559 0.6837 1.314 1.703 2.052 2.473 2.771 3.421 28 0.2558 0.6834 1.313 1.701 2.048 2.467 2.763 3.408 29 0.2557 0.6830 1.311 1.699 2.045 2.462 2.756 3.396 30 0.2556 0.6828 1.310 1.697 2.042 2.457 2.750 3.385 40 0.2550 0.6807 1.303 1.684 2.021 2.423 2.704 3.307 60 0.2545 0.6786 1.296 1.671 2.000 2.390 2.660 3.232

120 0.2539 0.6765 1.289 1.658 1.980 2.358 2.617 3.160 Inf 0.2533 0.6745 1.282 1.645 1.960 2.326 2.576 3.090

R1 16.5 19 9 7 2 11 1 13 5.5 5.5+ + + + + + + + +  89.5= = R2 12 14 16.5 18 4 15 10 20 8 3+ + + + + + + + +  120.5= =

U1 R1 n1 n1 1+ 

2 ------------------------– 89.5 10 10 1+ 

2 --------------------------– 34.5= = =

U2 R2 n2 n2 1+ 

2 ------------------------– 120.5 10 10 1+ 

2 --------------------------– 65.5= = =

Lecture Statistical Comparisons Stevens - Spring 2018 3 of 5

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Table A5.07: Critical Values for the Wilcoxon/Mann-Whitney Test (U Nondirectional =.05 (Directional =.025)

n2 n1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1 - - - - - - - - - - - - - - - - - - - 2 - - - - - - - 0 0 0 0 1 1 1 1 1 2 2 2 3 - - - - 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 4 - - - 0 1 2 3 4 4 5 6 7 8 9 10 11 11 12 13 5 - - 0 1 2 3 5 6 7 8 9 11 12 13 14 15 17 18 19 6 - - 1 2 3 5 6 8 10 11 13 14 16 17 19 21 22 24 25 7 - - 1 3 5 6 8 10 12 14 16 18 20 22 24 26 28 30 32 8 - 0 2 4 6 8 10 13 15 17 19 22 24 26 29 31 34 36 38 9 - 0 2 4 7 10 12 15 17 21 23 26 28 31 34 37 39 42 45

10 - 0 3 5 8 11 14 17 20 23 26 29 33 36 39 42 45 48 52 11 - 0 3 6 9 13 16 19 23 26 30 33 37 40 44 47 51 55 58 12 - 1 4 7 11 14 18 22 26 29 33 37 41 45 49 53 57 61 65 13 - 1 4 8 12 16 20 24 28 33 37 41 45 50 54 59 63 67 72 14 - 1 5 9 13 17 22 26 31 36 40 45 50 55 59 64 67 74 78 15 - 1 5 10 14 19 24 29 34 39 44 49 54 59 64 70 75 80 85 16 - 1 6 11 15 21 26 31 37 42 47 53 59 64 70 75 81 86 92 17 - 2 6 11 17 22 28 34 39 45 51 57 63 67 75 81 87 93 99 1 18 - 2 7 12 18 24 30 36 42 48 55 61 67 74 80 86 93 99 106 1 19 - 2 7 13 19 25 32 38 45 52 58 65 72 78 85 92 99 106 113 1 20 - 2 8 14 20 27 34 41 48 55 62 69 76 83 90 98 105 112 119 1

Nondirectional =.01 (Directional =.005) n2

n1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 - - - - - - - - - - - - - - - - - - - 2 - - - - - - - - - - - - - - - - - - 0 3 - - - - - - - - 0 0 0 1 1 1 2 2 2 2 3 4 - - - 0 0 1 1 2 2 3 3 4 5 5 6 6 7 5 - - - 0 1 1 2 3 4 5 6 7 7 8 9 10 11 12 6 - - - 0 1 2 3 4 5 6 7 9 10 11 12 13 15 16 17 7 - - - 0 1 3 4 6 7 9 10 12 13 15 16 18 19 21 22 8 - - - 1 2 4 6 7 9 11 13 15 17 18 20 22 24 26 28 9 - - 0 1 3 5 7 9 11 13 16 18 20 22 24 27 29 31 33

10 - - 0 2 4 6 9 11 13 16 18 21 24 26 29 31 34 37 39 11 - - 0 2 5 7 10 13 16 18 21 24 27 30 33 36 39 42 45 12 - - 1 3 6 9 12 15 18 21 24 27 31 34 37 41 44 47 51 13 - - 1 3 7 10 13 17 20 24 27 31 34 38 42 45 49 53 56 14 - - 1 4 7 11 15 18 22 26 30 34 38 42 46 50 54 58 63 15 - - 2 5 8 12 16 20 24 29 33 37 42 46 51 55 60 64 69 16 - - 2 5 9 13 18 22 27 31 36 41 45 50 55 60 65 70 74 17 - - 2 6 10 15 19 24 29 34 39 44 49 54 60 65 70 75 81 18 - - 2 6 11 16 21 26 31 37 42 47 53 58 64 70 75 81 87 19 - 0 3 7 12 17 22 28 33 39 45 51 56 63 69 74 81 87 93 20 - 0 3 8 13 18 24 30 36 42 46 54 60 67 73 79 86 92 99 1

Uobt is the lesser of the two calculated test statistics (U1 & U2). If Uobt Ucrit, reject H0. Dashes (-) indicate that the sample size is too small to reject the Null Hypothesis at the chosen level.

If n > 20 this table cannot be used. A p can be computed for Uobt, using the normal distribution approximatio

12 1)n(nnn

2 nn-U

z 2121

21 obt

U

Lecture Statistical Comparisons Stevens - Spring 2018 5 of 5