[Simulation] Statistical Errors

in Projects & Research on R, Simulation, Markdown

연구 목적

모의실험의 목적은 표본크기와 분산 등의 변화에 따라 비모수적 검정과, 모수적 검정시 범하는 1종오류, 2종오류의 차이를 보고자 하는 것이다.

연구 방법

R을 통해 Normal distribution, Uniform distribution, Chi-squared distribution, Exponential distribution를 따르는 난수를 발생시켜 주어진 상황에 따라 표본을 뽑아 총 10,000번 모수적 검정인 t-test, 비모수적 검정인 Wilcoxon rank sum test를 시행하여 각 상황에 따른 차이들을 1종오류와 2종오류에서 확인하여 볼 것이다.

따라서 각 조건별로 Type I error와 Type II error를 구해서 테이블로 제시하고, 그래프로 변화하는 양상을 살펴볼 수 있도록 그래프로 제시할 것이다.

Type I error: 귀무가설이 참임에도 불구하고 귀무가설을 기각하는 것

Type II error: 귀무가설이 참이 아닌데 귀무가설을 받아들이는 것

즉, Type I error는 두 집단의 모평균이 같을 때, Type II error는 두 집단의 모평균이 다를 때의 상황을 조건으로 한다.

주어진 상황은 다음과 같다.

  • Type I error
    • 등분산, 표본크기가 같은 경우
    • 등분산, 표본크기가 다른 경우
    • 분산이 다르고, 표본크기는 같은 경우
  • Type II error
    • 등분산, 표본크기가 같은 경우
    • 등분산, 표본크기가 다른 경우
    • 분산이 다르고, 표본크기는 같은 경우
library(kableExtra)
library(exactRankTests)
library(plyr)
alpha.p <- 0.05
n.simulations <- 10000

Type I error

N=10000,등분산,표본크기가 같은 경우

\[Table 1\] Type I error, 등분산,표본크기가 같은 경우: Normal distribution
n5 n10 n25 n50 n75 n100
N(0,1) N(0,1) t-test 0.0485 0.0507 0.0505 0.0539 0.0484 0.0492
N(0,1) N(0,1) Wilcoxon 0.0333 0.0398 0.0478 0.0457 0.0495 0.0503
N(0,2) N(0,2) t-test 0.0516 0.0447 0.0483 0.0491 0.0528 0.0472
N(0,2) N(0,2) Wilcoxon 0.0325 0.0448 0.0470 0.0469 0.0514 0.0495
N(0,3) N(0,3) t-test 0.0464 0.0499 0.0484 0.0494 0.0523 0.0461
N(0,3) N(0,3) Wilcoxon 0.0331 0.0414 0.0501 0.0491 0.0505 0.0509
N(0,4) N(0,4) t-test 0.0473 0.0520 0.0512 0.0508 0.0500 0.0480
N(0,4) N(0,4) Wilcoxon 0.0322 0.0442 0.0488 0.0495 0.0506 0.0484
N(0,5) N(0,5) t-test 0.0521 0.0538 0.0493 0.0518 0.0468 0.0494
N(0,5) N(0,5) Wilcoxon 0.0316 0.0447 0.0517 0.0503 0.0500 0.0483
N(0,6) N(0,6) t-test 0.0493 0.0549 0.0498 0.0506 0.0562 0.0523
N(0,6) N(0,6) Wilcoxon 0.0331 0.0429 0.0498 0.0480 0.0494 0.0516

\[Table 2\] Type I error, 등분산,표본크기가 같은 경우: Uniform distribution
n5 n10 n25 n50 n75 n100
U(0,1) U(0,1) t-test 0.0563 0.0539 0.0517 0.0485 0.0540 0.0494
U(0,1) U(0,1) Wilcoxon 0.0311 0.0454 0.0509 0.0489 0.0509 0.0504
U(0,2) U(0,2) t-test 0.0561 0.0522 0.0490 0.0527 0.0516 0.0511
U(0,2) U(0,2) Wilcoxon 0.0285 0.0405 0.0525 0.0490 0.0504 0.0510
U(0,3) U(0,3) t-test 0.0555 0.0492 0.0516 0.0530 0.0508 0.0526
U(0,3) U(0,3) Wilcoxon 0.0347 0.0440 0.0475 0.0489 0.0541 0.0481
U(0,4) U(0,4) t-test 0.0494 0.0487 0.0499 0.0488 0.0530 0.0520
U(0,4) U(0,4) Wilcoxon 0.0294 0.0436 0.0461 0.0498 0.0528 0.0516
U(0,5) U(0,5) t-test 0.0554 0.0489 0.0526 0.0496 0.0512 0.0489
U(0,5) U(0,5) Wilcoxon 0.0319 0.0388 0.0487 0.0488 0.0492 0.0509
U(0,6) U(0,6) t-test 0.0500 0.0567 0.0488 0.0487 0.0469 0.0463
U(0,6) U(0,6) Wilcoxon 0.0291 0.0443 0.0500 0.0461 0.0505 0.0488

\[Table 3\] Type I error, 등분산,표본크기가 같은 경우: Chi-squared distribution
n5 n10 n25 n50 n75 n100
chisq(1) chisq(1) t-test 0.0303 0.0403 0.0463 0.0445 0.0463 0.0468
chisq(1) chisq(1) Wilcoxon 0.0313 0.0438 0.0493 0.0482 0.0508 0.0517
chisq(2) chisq(2) t-test 0.0399 0.0418 0.0469 0.0448 0.0458 0.0485
chisq(2) chisq(2) Wilcoxon 0.0306 0.0428 0.0474 0.0507 0.0492 0.0497
chisq(3) chisq(3) t-test 0.0408 0.0485 0.0447 0.0506 0.0510 0.0527
chisq(3) chisq(3) Wilcoxon 0.0310 0.0396 0.0488 0.0502 0.0516 0.0493
chisq(4) chisq(4) t-test 0.0475 0.0523 0.0446 0.0493 0.0482 0.0484
chisq(4) chisq(4) Wilcoxon 0.0312 0.0419 0.0492 0.0471 0.0482 0.0515
chisq(5) chisq(5) t-test 0.0462 0.0497 0.0460 0.0501 0.0508 0.0525
chisq(5) chisq(5) Wilcoxon 0.0315 0.0466 0.0497 0.0510 0.0505 0.0497
chisq(6) chisq(6) t-test 0.0448 0.0469 0.0478 0.0482 0.0568 0.0511
chisq(6) chisq(6) Wilcoxon 0.0303 0.0404 0.0532 0.0497 0.0472 0.0490

\[Table 4\] Type I error, 등분산,표본크기가 같은 경우: Exponential distribution
n5 n10 n25 n50 n75 n100
exp(1/1) exp(1/1) t-test 0.0370 0.0438 0.0514 0.0502 0.0504 0.0486
exp(1/1) exp(1/1) Wilcoxon 0.0293 0.0417 0.0462 0.0494 0.0525 0.0493
exp(1/2) exp(1/2) t-test 0.0395 0.0441 0.0504 0.0514 0.0508 0.0470
exp(1/2) exp(1/2) Wilcoxon 0.0313 0.0412 0.0483 0.0498 0.0519 0.0497
exp(1/3) exp(1/3) t-test 0.0412 0.0443 0.0503 0.0511 0.0453 0.0483
exp(1/3) exp(1/3) Wilcoxon 0.0321 0.0439 0.0461 0.0517 0.0522 0.0519
exp(1/4) exp(1/4) t-test 0.0397 0.0427 0.0466 0.0503 0.0535 0.0482
exp(1/4) exp(1/4) Wilcoxon 0.0293 0.0466 0.0512 0.0469 0.0509 0.0521
exp(1/5) exp(1/5) t-test 0.0369 0.0452 0.0458 0.0475 0.0506 0.0491
exp(1/5) exp(1/5) Wilcoxon 0.0328 0.0455 0.0501 0.0483 0.0516 0.0524
exp(1/6) exp(1/6) t-test 0.0377 0.0426 0.0473 0.0509 0.0512 0.0516
exp(1/6) exp(1/6) Wilcoxon 0.0287 0.0426 0.0519 0.0505 0.0507 0.0473

N=10000,등분산,표본크기가 다른 경우

\[Table 5\] Type I error, 등분산,표본크기가 다른 경우: Normal distribution
n15, n5 n30, n10 n45, n15 n75, n25 n105, n35
N(0,1) N(0,1) t-test 0.0495 0.0520 0.0479 0.0496 0.0535
N(0,1) N(0,1) Wilcoxon 0.0433 0.0528 0.0512 0.0477 0.0515
N(0,2) N(0,2) t-test 0.0480 0.0523 0.0506 0.0497 0.0494
N(0,2) N(0,2) Wilcoxon 0.0392 0.0528 0.0516 0.0467 0.0489
N(0,3) N(0,3) t-test 0.0493 0.0525 0.0490 0.0515 0.0471
N(0,3) N(0,3) Wilcoxon 0.0425 0.0450 0.0516 0.0494 0.0499
N(0,4) N(0,4) t-test 0.0487 0.0483 0.0538 0.0490 0.0496
N(0,4) N(0,4) Wilcoxon 0.0371 0.0479 0.0498 0.0503 0.0508
N(0,5) N(0,5) t-test 0.0488 0.0477 0.0483 0.0505 0.0487
N(0,5) N(0,5) Wilcoxon 0.0433 0.0520 0.0468 0.0505 0.0497
N(0,6) N(0,6) t-test 0.0509 0.0508 0.0484 0.0485 0.0494
N(0,6) N(0,6) Wilcoxon 0.0429 0.0468 0.0475 0.0490 0.0450

\[Table 6\] Type I error, 등분산,표본크기가 다른 경우: Uniform distribution
n15, n5 n30, n10 n45, n15 n75, n25 n105, n35
U(0,1) U(0,1) t-test 0.0511 0.0509 0.0463 0.0490 0.0488
U(0,1) U(0,1) Wilcoxon 0.0403 0.0511 0.0536 0.0521 0.0483
U(0,2) U(0,2) t-test 0.0544 0.0566 0.0511 0.0531 0.0497
U(0,2) U(0,2) Wilcoxon 0.0431 0.0493 0.0532 0.0516 0.0508
U(0,3) U(0,3) t-test 0.0498 0.0533 0.0521 0.0480 0.0471
U(0,3) U(0,3) Wilcoxon 0.0399 0.0493 0.0484 0.0495 0.0477
U(0,4) U(0,4) t-test 0.0511 0.0515 0.0472 0.0441 0.0528
U(0,4) U(0,4) Wilcoxon 0.0409 0.0519 0.0456 0.0503 0.0491
U(0,5) U(0,5) t-test 0.0499 0.0523 0.0506 0.0508 0.0487
U(0,5) U(0,5) Wilcoxon 0.0425 0.0527 0.0482 0.0456 0.0470
U(0,6) U(0,6) t-test 0.0561 0.0516 0.0471 0.0506 0.0491
U(0,6) U(0,6) Wilcoxon 0.0390 0.0499 0.0510 0.0494 0.0518

\[Table 7\] Type I error, 등분산,표본크기가 다른 경우: Chi-squared distribution
n15, n5 n30, n10 n45, n15 n75, n25 n105, n35
chisq(1) chisq(1) t-test 0.0451 0.0440 0.0460 0.0469 0.0508
chisq(1) chisq(1) Wilcoxon 0.0434 0.0482 0.0499 0.0478 0.0486
chisq(2) chisq(2) t-test 0.0404 0.0493 0.0457 0.0490 0.0445
chisq(2) chisq(2) Wilcoxon 0.0392 0.0505 0.0482 0.0494 0.0517
chisq(3) chisq(3) t-test 0.0465 0.0519 0.0504 0.0485 0.0504
chisq(3) chisq(3) Wilcoxon 0.0414 0.0557 0.0534 0.0508 0.0485
chisq(4) chisq(4) t-test 0.0430 0.0483 0.0510 0.0520 0.0492
chisq(4) chisq(4) Wilcoxon 0.0428 0.0476 0.0486 0.0500 0.0503
chisq(5) chisq(5) t-test 0.0484 0.0532 0.0478 0.0484 0.0461
chisq(5) chisq(5) Wilcoxon 0.0416 0.0478 0.0496 0.0493 0.0482
chisq(6) chisq(6) t-test 0.0448 0.0482 0.0494 0.0494 0.0501
chisq(6) chisq(6) Wilcoxon 0.0456 0.0543 0.0500 0.0490 0.0476

\[Table 8\] Type I error, 등분산,표본크기가 다른 경우: Exponential distribution
n15, n5 n30, n10 n45, n15 n75, n25 n105, n35
exp(1/1) exp(1/1) t-test 0.0416 0.0477 0.0447 0.0442 0.0432
exp(1/1) exp(1/1) Wilcoxon 0.0419 0.0507 0.0486 0.0480 0.0533
exp(1/2) exp(1/2) t-test 0.0494 0.0426 0.0487 0.0489 0.0499
exp(1/2) exp(1/2) Wilcoxon 0.0411 0.0509 0.0486 0.0485 0.0462
exp(1/3) exp(1/3) t-test 0.0468 0.0476 0.0434 0.0532 0.0479
exp(1/3) exp(1/3) Wilcoxon 0.0433 0.0519 0.0493 0.0545 0.0509
exp(1/4) exp(1/4) t-test 0.0446 0.0482 0.0461 0.0483 0.0484
exp(1/4) exp(1/4) Wilcoxon 0.0388 0.0456 0.0463 0.0487 0.0483
exp(1/5) exp(1/5) t-test 0.0462 0.0467 0.0468 0.0506 0.0520
exp(1/5) exp(1/5) Wilcoxon 0.0415 0.0497 0.0467 0.0498 0.0501
exp(1/6) exp(1/6) t-test 0.0426 0.0503 0.0492 0.0500 0.0498
exp(1/6) exp(1/6) Wilcoxon 0.0406 0.0480 0.0514 0.0520 0.0515

N=10000, 분산이 다르고, 표본크기는 같은 경우

\[Table 9\] Type I error, 분산이 다르고, 표본크기는 같은 경우: Normal distribution
n5 n10 n25 n50 n75 n100
N(0,1.5) N(0,1) t-test 0.0491 0.0505 0.0511 0.0513 0.0505 0.0516
N(0,1.5) N(0,1) Wilcoxon 0.0322 0.0417 0.0508 0.0450 0.0516 0.0498
N(0,2) N(0,2) t-test 0.0504 0.0517 0.0512 0.0488 0.0477 0.0502
N(0,2) N(0,2) Wilcoxon 0.0319 0.0437 0.0494 0.0484 0.0532 0.0521
N(0,2.5) N(0,3) t-test 0.0501 0.0480 0.0476 0.0515 0.0520 0.0469
N(0,2.5) N(0,3) Wilcoxon 0.0300 0.0436 0.0474 0.0497 0.0497 0.0460
N(0,3) N(0,4) t-test 0.0499 0.0487 0.0505 0.0490 0.0485 0.0545
N(0,3) N(0,4) Wilcoxon 0.0315 0.0425 0.0520 0.0451 0.0487 0.0485
N(0,3.5) N(0,5) t-test 0.0548 0.0542 0.0508 0.0519 0.0466 0.0478
N(0,3.5) N(0,5) Wilcoxon 0.0296 0.0470 0.0512 0.0513 0.0517 0.0472
N(0,4) N(0,6) t-test 0.0492 0.0502 0.0505 0.0509 0.0505 0.0462
N(0,4) N(0,6) Wilcoxon 0.0339 0.0425 0.0482 0.0571 0.0500 0.0503

\[Table 10\] Type I error, 분산이 다르고, 표본크기는 같은 경우: Uniform distribution
n5 n10 n25 n50 n75 n100
U(0,2.46)-0.73 U(0,1) t-test 0.0766 0.0602 0.0537 0.0524 0.0467 0.0522
U(0,2.46)-0.73 U(0,1) Wilcoxon 0.0509 0.0611 0.0696 0.0681 0.0701 0.0730
U(0,2.84)-0.42 U(0,2) t-test 0.0585 0.0525 0.0487 0.0504 0.0465 0.0512
U(0,2.84)-0.42 U(0,2) Wilcoxon 0.0328 0.0497 0.0527 0.0541 0.0544 0.0535
U(0,3.16)-0.08 U(0,3) t-test 0.0579 0.0528 0.0499 0.0495 0.0518 0.0544
U(0,3.16)-0.08 U(0,3) Wilcoxon 0.0298 0.0446 0.0473 0.0480 0.0526 0.0466
U(0,3.5)+0.25 U(0,4) t-test 0.0536 0.0541 0.0508 0.0494 0.0520 0.0482
U(0,3.5)+0.25 U(0,4) Wilcoxon 0.0326 0.0442 0.0488 0.0531 0.0495 0.0502
U(0,3.76)+0.62 U(0,5) t-test 0.0550 0.0523 0.0522 0.0485 0.0505 0.0525
U(0,3.76)+0.62 U(0,5) Wilcoxon 0.0341 0.0476 0.0500 0.0504 0.0509 0.0544
U(0,4.0)+1 U(0,6) t-test 0.0613 0.0567 0.0510 0.0501 0.0516 0.0516
U(0,4.0)+1 U(0,6) Wilcoxon 0.0395 0.0496 0.0521 0.0535 0.0569 0.0538

\[Table 11\] Type I error, 분산이 다르고, 표본크기는 같은 경우: Chi-squared distribution
n5 n10 n25 n50 n75 n100
chisq(1.5)-0.5 chisq(1) t-test 0.0423 0.0455 0.0442 0.0488 0.0531 0.0493
chisq(1.5)-0.5 chisq(1) Wilcoxon 0.0447 0.0713 0.1353 0.2049 0.2868 0.3530
chisq(2)-0 chisq(2) t-test 0.0374 0.0451 0.0474 0.0494 0.0493 0.0519
chisq(2)-0 chisq(2) Wilcoxon 0.0320 0.0435 0.0510 0.0484 0.0499 0.0466
chisq(2.5)+0.5 chisq(3) t-test 0.0417 0.0459 0.0476 0.0494 0.0466 0.0531
chisq(2.5)+0.5 chisq(3) Wilcoxon 0.0319 0.0452 0.0520 0.0601 0.0651 0.0642
chisq(3)+1 chisq(4) t-test 0.0459 0.0491 0.0520 0.0475 0.0536 0.0518
chisq(3)+1 chisq(4) Wilcoxon 0.0325 0.0422 0.0558 0.0590 0.0658 0.0705
chisq(3.5)+1.5 chisq(5) t-test 0.0463 0.0478 0.0463 0.0449 0.0497 0.0461
chisq(3.5)+1.5 chisq(5) Wilcoxon 0.0318 0.0475 0.0601 0.0628 0.0658 0.0747
chisq(4)+2 chisq(6) t-test 0.0440 0.0515 0.0508 0.0486 0.0462 0.0489
chisq(4)+2 chisq(6) Wilcoxon 0.0327 0.0453 0.0562 0.0596 0.0675 0.0711

\[Table 12\] Type I error, 분산이 다르고, 표본크기는 같은 경우: Exponential distribution
n5 n10 n25 n50 n75 n100
exp(1/1.5)-0.5 exp(1/1) t-test 0.0560 0.0534 0.0561 0.0511 0.0491 0.0525
exp(1/1.5)-0.5 exp(1/1) Wilcoxon 0.0460 0.0808 0.1411 0.2372 0.3157 0.4020
exp(1/2)-0 exp(1/2) t-test 0.0425 0.0446 0.0456 0.0502 0.0481 0.0453
exp(1/2)-0 exp(1/2) Wilcoxon 0.0318 0.0444 0.0493 0.0497 0.0490 0.0502
exp(1/2.5)+0.5 exp(1/3) t-test 0.0423 0.0512 0.0500 0.0512 0.0463 0.0481
exp(1/2.5)+0.5 exp(1/3) Wilcoxon 0.0357 0.0500 0.0782 0.1075 0.1250 0.1588
exp(1/3)+1 exp(1/4) t-test 0.0481 0.0491 0.0522 0.0503 0.0502 0.0519
exp(1/3)+1 exp(1/4) Wilcoxon 0.0406 0.0623 0.1011 0.1607 0.2247 0.2654
exp(1/3.5)+1.5 exp(1/5) t-test 0.0470 0.0509 0.0513 0.0496 0.0508 0.0499
exp(1/3.5)+1.5 exp(1/5) Wilcoxon 0.0434 0.0790 0.1274 0.1948 0.2842 0.3566
exp(1/4)+2 exp(1/6) t-test 0.0543 0.0538 0.0490 0.0551 0.0529 0.0478
exp(1/4)+2 exp(1/6) Wilcoxon 0.0478 0.0803 0.1413 0.2288 0.3243 0.4027

Type II error

N=10000,등분산,표본크기가 같은 경우

\[Table 13\] Type II error, 등분산,표본크기가 같은 경우: Normal distribution
n5 n10 n25 n50 n75 n100
N(0,1)+2 N(0,1) t-test 0.2015 0.0106 0.0000 0.0000 0.0000 0e+00
N(0,1)+2 N(0,1) Wilcoxon 0.3302 0.0184 0.0000 0.0000 0.0000 0e+00
N(0,2)+2 N(0,2) t-test 0.4979 0.1562 0.0012 0.0000 0.0000 0e+00
N(0,2)+2 N(0,2) Wilcoxon 0.6124 0.1946 0.0022 0.0000 0.0000 0e+00
N(0,3)+2 N(0,3) t-test 0.6311 0.3075 0.0207 0.0000 0.0000 0e+00
N(0,3)+2 N(0,3) Wilcoxon 0.7363 0.3640 0.0232 0.0000 0.0000 0e+00
N(0,4)+2 N(0,4) t-test 0.7117 0.4324 0.0674 0.0006 0.0000 0e+00
N(0,4)+2 N(0,4) Wilcoxon 0.7942 0.4862 0.0790 0.0022 0.0000 0e+00
N(0,5)+2 N(0,5) t-test 0.7675 0.5337 0.1320 0.0059 0.0002 0e+00
N(0,5)+2 N(0,5) Wilcoxon 0.8319 0.5752 0.1478 0.0099 0.0007 0e+00
N(0,6)+2 N(0,6) t-test 0.7932 0.5935 0.1912 0.0203 0.0011 1e-04
N(0,6)+2 N(0,6) Wilcoxon 0.8518 0.6353 0.2185 0.0253 0.0022 1e-04

\[Table 14\] Type II error, 등분산,표본크기가 같은 경우: Uniform distribution
n5 n10 n25 n50 n75 n100
U(0,1)+0.5 U(0,1) t-test 0.3608 0.0337 0.0000 0.0000 0.0000 0.0000
U(0,1)+0.5 U(0,1) Wilcoxon 0.4950 0.0861 0.0003 0.0000 0.0000 0.0000
U(0,2)+0.5 U(0,2) t-test 0.8003 0.5697 0.1489 0.0100 0.0002 0.0000
U(0,2)+0.5 U(0,2) Wilcoxon 0.8698 0.6298 0.2128 0.0232 0.0015 0.0001
U(0,3)+0.5 U(0,3) t-test 0.8769 0.7790 0.4942 0.1854 0.0605 0.0175
U(0,3)+0.5 U(0,3) Wilcoxon 0.9243 0.8113 0.5301 0.2365 0.0851 0.0336
U(0,4)+0.5 U(0,4) t-test 0.9116 0.8586 0.6865 0.4351 0.2514 0.1337
U(0,4)+0.5 U(0,4) Wilcoxon 0.9402 0.8790 0.7038 0.4739 0.2840 0.1782
U(0,5)+0.5 U(0,5) t-test 0.9258 0.8896 0.7850 0.5961 0.4447 0.3243
U(0,5)+0.5 U(0,5) Wilcoxon 0.9511 0.9050 0.7981 0.6280 0.4619 0.3566
U(0,6)+0.5 U(0,6) t-test 0.9326 0.9132 0.8319 0.7012 0.5866 0.4772
U(0,6)+0.5 U(0,6) Wilcoxon 0.9576 0.9220 0.8399 0.7185 0.6009 0.4969

\[Table 15\] Type II error, 등분산,표본크기가 같은 경우: Chi-squared distribution
n5 n10 n25 n50 n75 n100
chisq(1)+0.5 chisq(1) t-test 0.8868 0.8364 0.7319 0.5456 0.4094 0.2921
chisq(1)+0.5 chisq(1) Wilcoxon 0.8646 0.6730 0.3034 0.0587 0.0089 0.0014
chisq(2)+0.5 chisq(2) t-test 0.9380 0.9113 0.8486 0.7532 0.6582 0.5668
chisq(2)+0.5 chisq(2) Wilcoxon 0.9384 0.8819 0.7402 0.5223 0.3436 0.2203
chisq(3)+0.5 chisq(3) t-test 0.9441 0.9277 0.8911 0.8188 0.7542 0.6946
chisq(3)+0.5 chisq(3) Wilcoxon 0.9558 0.9238 0.8460 0.7337 0.6281 0.5287
chisq(4)+0.5 chisq(4) t-test 0.9458 0.9308 0.9085 0.8590 0.8033 0.7619
chisq(4)+0.5 chisq(4) Wilcoxon 0.9602 0.9379 0.8803 0.8055 0.7402 0.6787
chisq(5)+0.5 chisq(5) t-test 0.9461 0.9334 0.9079 0.8742 0.8436 0.7979
chisq(5)+0.5 chisq(5) Wilcoxon 0.9660 0.9447 0.9051 0.8568 0.8019 0.7467
chisq(6)+0.5 chisq(6) t-test 0.9506 0.9377 0.9211 0.8911 0.8522 0.8236
chisq(6)+0.5 chisq(6) Wilcoxon 0.9618 0.9405 0.9131 0.8748 0.8381 0.7908

\[Table 16\] Type II error, 등분산,표본크기가 같은 경우: Exponential distribution
n5 n10 n25 n50 n75 n100
exp(1/1)+0.5 exp(1/1) t-test 0.8642 0.7770 0.5512 0.2927 0.1401 0.0638
exp(1/1)+0.5 exp(1/1) Wilcoxon 0.8743 0.6954 0.3058 0.0665 0.0117 0.0021
exp(1/2)+0.5 exp(1/2) t-test 0.9409 0.9096 0.8456 0.7446 0.6516 0.5624
exp(1/2)+0.5 exp(1/2) Wilcoxon 0.9437 0.8855 0.7312 0.5162 0.3484 0.2123
exp(1/3)+0.5 exp(1/3) t-test 0.9447 0.9345 0.9088 0.8606 0.8135 0.7823
exp(1/3)+0.5 exp(1/3) Wilcoxon 0.9548 0.9178 0.8461 0.7418 0.6284 0.5342
exp(1/4)+0.5 exp(1/4) t-test 0.9544 0.9465 0.9310 0.9046 0.8758 0.8500
exp(1/4)+0.5 exp(1/4) Wilcoxon 0.9665 0.9368 0.8870 0.8317 0.7584 0.7010
exp(1/5)+0.5 exp(1/5) t-test 0.9577 0.9489 0.9361 0.9176 0.9008 0.8893
exp(1/5)+0.5 exp(1/5) Wilcoxon 0.9641 0.9407 0.9108 0.8701 0.8320 0.7911
exp(1/6)+0.5 exp(1/6) t-test 0.9589 0.9518 0.9417 0.9289 0.9177 0.9092
exp(1/6)+0.5 exp(1/6) Wilcoxon 0.9674 0.9442 0.9240 0.8984 0.8683 0.8323

N=10000,등분산,표본크기가 다른 경우

\[Table 17\] Type II error, 등분산,표본크기가 다른 경우: Normal distribution
n15, n5 n30, n10 n45, n15 n75, n25 n105, n35
N(0,1)+0.5 N(0,1) t-test 0.8508 0.7361 0.6235 0.4363 0.2799
N(0,1)+0.5 N(0,1) Wilcoxon 0.8768 0.7524 0.6395 0.4578 0.3028
N(0,2)+0.5 N(0,2) t-test 0.8970 0.8400 0.7811 0.6691 0.5641
N(0,2)+0.5 N(0,2) Wilcoxon 0.9177 0.8478 0.8001 0.6884 0.5782
N(0,3)+0.5 N(0,3) t-test 0.9151 0.8794 0.8458 0.7650 0.6798
N(0,3)+0.5 N(0,3) Wilcoxon 0.9350 0.8829 0.8465 0.7708 0.7005
N(0,4)+0.5 N(0,4) t-test 0.9282 0.8975 0.8688 0.8131 0.7520
N(0,4)+0.5 N(0,4) Wilcoxon 0.9388 0.9045 0.8796 0.8197 0.7654
N(0,5)+0.5 N(0,5) t-test 0.9287 0.9083 0.8870 0.8394 0.8012
N(0,5)+0.5 N(0,5) Wilcoxon 0.9375 0.9131 0.8940 0.8525 0.8024
N(0,6)+0.5 N(0,6) t-test 0.9368 0.9123 0.8990 0.8585 0.8202
N(0,6)+0.5 N(0,6) Wilcoxon 0.9406 0.9136 0.9034 0.8669 0.8293

\[Table 18\] Type II error, 등분산,표본크기가 다른 경우: Uniform distribution
n15, n5 n30, n10 n45, n15 n75, n25 n105, n35
U(0,1)+0.5 U(0,1) t-test 0.1122 0.0020 0.0000 0.0000 0.0000
U(0,1)+0.5 U(0,1) Wilcoxon 0.2095 0.0094 0.0010 0.0000 0.0000
U(0,2)+0.5 U(0,2) t-test 0.6526 0.3670 0.1797 0.0380 0.0056
U(0,2)+0.5 U(0,2) Wilcoxon 0.7314 0.4381 0.2480 0.0718 0.0180
U(0,3)+0.5 U(0,3) t-test 0.8226 0.6688 0.5283 0.3026 0.1582
U(0,3)+0.5 U(0,3) Wilcoxon 0.8564 0.7044 0.5690 0.3531 0.2135
U(0,4)+0.5 U(0,4) t-test 0.8849 0.7916 0.7074 0.5475 0.4125
U(0,4)+0.5 U(0,4) Wilcoxon 0.9007 0.8057 0.7343 0.5848 0.4520
U(0,5)+0.5 U(0,5) t-test 0.9064 0.8480 0.7971 0.6925 0.5740
U(0,5)+0.5 U(0,5) Wilcoxon 0.9214 0.8586 0.8140 0.7080 0.6104
U(0,6)+0.5 U(0,6) t-test 0.9197 0.8826 0.8410 0.7639 0.6926
U(0,6)+0.5 U(0,6) Wilcoxon 0.9374 0.8904 0.8504 0.7793 0.7126

\[Table 19\] Type II error, 등분산,표본크기가 다른 경우: Chi-squared distribution
n15, n5 n30, n10 n45, n15 n75, n25 n105, n35
chisq(1)+0.5 chisq(1) t-test 0.8867 0.8043 0.7338 0.6223 0.5186
chisq(1)+0.5 chisq(1) Wilcoxon 0.7085 0.5072 0.3777 0.1780 0.0760
chisq(2)+0.5 chisq(2) t-test 0.9359 0.8958 0.8626 0.7995 0.7381
chisq(2)+0.5 chisq(2) Wilcoxon 0.8880 0.8064 0.7295 0.6166 0.4921
chisq(3)+0.5 chisq(3) t-test 0.9384 0.9193 0.8975 0.8576 0.8175
chisq(3)+0.5 chisq(3) Wilcoxon 0.9209 0.8830 0.8452 0.7724 0.7100
chisq(4)+0.5 chisq(4) t-test 0.9467 0.9291 0.9101 0.8820 0.8493
chisq(4)+0.5 chisq(4) Wilcoxon 0.9340 0.9032 0.8839 0.8403 0.7907
chisq(5)+0.5 chisq(5) t-test 0.9453 0.9339 0.9200 0.8980 0.8744
chisq(5)+0.5 chisq(5) Wilcoxon 0.9407 0.9138 0.8979 0.8711 0.8419
chisq(6)+0.5 chisq(6) t-test 0.9438 0.9325 0.9226 0.9008 0.8895
chisq(6)+0.5 chisq(6) Wilcoxon 0.9461 0.9250 0.9089 0.8915 0.8662

\[Table 20\] Type II error, 등분산,표본크기가 다른 경우: Exponential distribution
n15, n5 n30, n10 n45, n15 n75, n25 n105, n35
exp(1/1)+0.5 exp(1/1) t-test 0.8286 0.6931 0.5763 0.3977 0.2686
exp(1/1)+0.5 exp(1/1) Wilcoxon 0.7374 0.5206 0.3608 0.1758 0.0774
exp(1/2)+0.5 exp(1/2) t-test 0.9362 0.8916 0.8636 0.8055 0.7425
exp(1/2)+0.5 exp(1/2) Wilcoxon 0.8838 0.7945 0.7440 0.6050 0.4936
exp(1/3)+0.5 exp(1/3) t-test 0.9497 0.9320 0.9152 0.8870 0.8624
exp(1/3)+0.5 exp(1/3) Wilcoxon 0.9189 0.8678 0.8470 0.7744 0.7108
exp(1/4)+0.5 exp(1/4) t-test 0.9547 0.9427 0.9331 0.9190 0.8999
exp(1/4)+0.5 exp(1/4) Wilcoxon 0.9344 0.8980 0.8801 0.8450 0.8093
exp(1/5)+0.5 exp(1/5) t-test 0.9548 0.9508 0.9440 0.9302 0.9156
exp(1/5)+0.5 exp(1/5) Wilcoxon 0.9369 0.9197 0.9050 0.8777 0.8498
exp(1/6)+0.5 exp(1/6) t-test 0.9611 0.9495 0.9471 0.9400 0.9302
exp(1/6)+0.5 exp(1/6) Wilcoxon 0.9431 0.9251 0.9183 0.9030 0.8831

N=10000, 분산이 다르고, 표본크기는 같은 경우

\[Table 21\] Type II error, 표본크기는 같고 분산이 다른 경우: Normal distribution
n5 n10 n25 n50 n75 n100
N(0,1.5)+0.5 N(0,1) t-test 0.9006 0.8423 0.6577 0.3989 0.2282 0.1231
N(0,1.5)+0.5 N(0,1) Wilcoxon 0.9306 0.8605 0.6763 0.4205 0.2453 0.1402
N(0,2)+0.5 N(0,2) t-test 0.9169 0.8804 0.7786 0.5890 0.4248 0.3022
N(0,2)+0.5 N(0,2) Wilcoxon 0.9459 0.8984 0.7763 0.5923 0.4439 0.3170
N(0,2.5)+0.5 N(0,3) t-test 0.9297 0.9021 0.8217 0.6812 0.5467 0.4337
N(0,2.5)+0.5 N(0,3) Wilcoxon 0.9539 0.9174 0.8262 0.6959 0.5685 0.4605
N(0,3)+0.5 N(0,4) t-test 0.9375 0.9092 0.8448 0.7347 0.6387 0.5322
N(0,3)+0.5 N(0,4) Wilcoxon 0.9583 0.9234 0.8540 0.7437 0.6372 0.5507
N(0,3.5)+0.5 N(0,5) t-test 0.9341 0.9244 0.8644 0.7795 0.6832 0.5923
N(0,3.5)+0.5 N(0,5) Wilcoxon 0.9571 0.9283 0.8693 0.7807 0.7052 0.6068
N(0,4)+0.5 N(0,6) t-test 0.9366 0.9203 0.8804 0.7956 0.7258 0.6452
N(0,4)+0.5 N(0,6) Wilcoxon 0.9579 0.9354 0.8838 0.8076 0.7348 0.6598

\[Table 22\] Type II error, 표본크기는 같고 분산이 다른 경우: Uniform distribution
n5 n10 n25 n50 n75 n100
U(0,2.46)-0.23 U(0,1) t-test 0.7758 0.5262 0.1076 0.0040 0.0000 0.0000
U(0,2.46)-0.23 U(0,1) Wilcoxon 0.8516 0.6667 0.3032 0.0580 0.0100 0.0015
U(0,2.84)+0.08 U(0,2) t-test 0.8456 0.6930 0.3240 0.0588 0.0086 0.0008
U(0,2.84)+0.08 U(0,2) Wilcoxon 0.9042 0.7583 0.4181 0.1270 0.0315 0.0075
U(0,3.16)+0.42 U(0,3) t-test 0.8842 0.7901 0.5092 0.2053 0.0696 0.0222
U(0,3.16)+0.42 U(0,3) Wilcoxon 0.9158 0.8218 0.5444 0.2534 0.1080 0.0398
U(0,3.5)+0.75 U(0,4) t-test 0.9099 0.8477 0.6414 0.3771 0.2031 0.1026
U(0,3.5)+0.75 U(0,4) Wilcoxon 0.9383 0.8652 0.6801 0.4230 0.2455 0.1414
U(0,3.76)+1.12 U(0,5) t-test 0.9145 0.8743 0.7321 0.5191 0.3401 0.2143
U(0,3.76)+1.12 U(0,5) Wilcoxon 0.9447 0.9020 0.7658 0.5935 0.4313 0.3026
U(0,4.0)+1.5 U(0,6) t-test 0.9189 0.8931 0.7835 0.6128 0.4633 0.3264
U(0,4.0)+1.5 U(0,6) Wilcoxon 0.9483 0.9110 0.8200 0.7049 0.5740 0.4602

\[Table 23\] Type II error, 표본크기는 같고 분산이 다른 경우: Chi-squared distribution
n5 n10 n25 n50 n75 n100
chisq(1.5)-0 chisq(1) t-test 0.9403 0.8981 0.7898 0.6339 0.4917 0.3848
chisq(1.5)-0 chisq(1) Wilcoxon 0.9358 0.8701 0.6940 0.4458 0.2590 0.1463
chisq(2)-0.5 chisq(2) t-test 0.9343 0.9064 0.8468 0.7507 0.6608 0.5693
chisq(2)-0.5 chisq(2) Wilcoxon 0.9415 0.8811 0.7332 0.5158 0.3376 0.2273
chisq(2.5)+1 chisq(3) t-test 0.9405 0.9219 0.8689 0.8106 0.7396 0.6641
chisq(2.5)+1 chisq(3) Wilcoxon 0.9507 0.9043 0.8024 0.6279 0.4908 0.3644
chisq(3)+1.5 chisq(4) t-test 0.9435 0.9326 0.8930 0.8401 0.7816 0.7298
chisq(3)+1.5 chisq(4) Wilcoxon 0.9545 0.9114 0.8323 0.6976 0.5786 0.4838
chisq(3.5)+2 chisq(5) t-test 0.9393 0.9285 0.9007 0.8572 0.8162 0.7677
chisq(3.5)+2 chisq(5) Wilcoxon 0.9605 0.9215 0.8507 0.7498 0.6492 0.5506
chisq(4)+2.5 chisq(6) t-test 0.9420 0.9319 0.9108 0.8688 0.8355 0.8000
chisq(4)+2.5 chisq(6) Wilcoxon 0.9574 0.9260 0.8688 0.7792 0.7024 0.6164

\[Table 24\] Type II error, 표본크기는 같고 분산이 다른 경우: Exponential distribution
n5 n10 n25 n50 n75 n100
exp(1/1.5)-0 exp(1/1) t-test 0.9382 0.8862 0.7413 0.4908 0.3148 0.1991
exp(1/1.5)-0 exp(1/1) Wilcoxon 0.9461 0.8961 0.7749 0.5913 0.4327 0.3118
exp(1/2)+0.5 exp(1/2) t-test 0.9339 0.9088 0.8505 0.7568 0.6554 0.5772
exp(1/2)+0.5 exp(1/2) Wilcoxon 0.9438 0.8818 0.7209 0.5087 0.3453 0.2245
exp(1/2.5)+1 exp(1/3) t-test 0.9310 0.9163 0.8828 0.8350 0.7930 0.7405
exp(1/2.5)+1 exp(1/3) Wilcoxon 0.9409 0.8806 0.7384 0.5198 0.3570 0.2376
exp(1/3)+1.5 exp(1/4) t-test 0.9278 0.9184 0.9011 0.8709 0.8429 0.8146
exp(1/3)+1.5 exp(1/4) Wilcoxon 0.9378 0.8802 0.7364 0.5396 0.3711 0.2493
exp(1/3.5)+2 exp(1/5) t-test 0.9320 0.9232 0.9102 0.8946 0.8826 0.8571
exp(1/3.5)+2 exp(1/5) Wilcoxon 0.9353 0.8781 0.7465 0.5492 0.3952 0.2633
exp(1/4)+2.5 exp(1/6) t-test 0.9310 0.9247 0.9172 0.9106 0.8850 0.8769
exp(1/4)+2.5 exp(1/6) Wilcoxon 0.9375 0.8867 0.7500 0.5567 0.4059 0.2726