Pertemuan 4: Uji McNemar, Cox Stuart, Wilcoxon

Statistika Nonparametrik

Offline di Departemen Matematika
Published

November 4, 2024

Tabel Guide Uji Nonparametrik

Uji McNemar

  1. Anggota klub pendakian gunung telah lama berdebat tentang mana yang lebih sulit di antara 2 jalur panjat tebing. Berharap untuk menyelesaikan argumen satu anggota memeriksa buku catatan klub. Catatan ini berisi apakah tiap anggota berhasil suatu pendakian. Log menunjukkan bahwa 108 anggota telah mencoba kedua pendakian dengan hasil yang dirangkum dalam Tabel. Apakah ada bukti bahwa satu pendakian lebih sulit dari lainnya?

climb=matrix(c(73, 14, 9, 12), byrow=TRUE, nrow=2, ncol=2)
climb
     [,1] [,2]
[1,]   73   14
[2,]    9   12

Uji Hipotesis :

\(H_0: \text{Tidak terdapat perbedaan signifikan}\)

\(H_1: \text{Tidak demikian}\)

library(stats)

#H0: Tidak terdapat perbedaan signifikan
mcnemar.test(climb) #Package stats

    McNemar's Chi-squared test with continuity correction

data:  climb
McNemar's chi-squared = 0.69565, df = 1, p-value = 0.4042

Apa maksudnya Continuity Correction?

Telusuri sumber berikut: Continuity Correction on Wikipedia

Kenapa digunakan pada fungsi mcnemar.test()?

Telusuri sumber berikut: Continuity Correction for Pearson and McNemar’s Chi Square Test

mcnemar.test(climb, correct=FALSE)

    McNemar's Chi-squared test

data:  climb
McNemar's chi-squared = 1.087, df = 1, p-value = 0.2971

Lebih baik jika kita menggunakan mcnemar.exact() dari library exact2x2

library(exact2x2)
Warning: package 'exact2x2' was built under R version 4.4.1
Loading required package: exactci
Warning: package 'exactci' was built under R version 4.4.1
Loading required package: ssanv
Loading required package: testthat
mcnemar.exact(climb) #Package exact2x2

    Exact McNemar test (with central confidence intervals)

data:  climb
b = 14, c = 9, p-value = 0.4049
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
 0.6271247 4.0741737
sample estimates:
odds ratio 
  1.555556
Dokumentasi Fungsi mcnemar.test()

https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/mcnemar.test

atau run ?mcnemar.test setelah import library stats

Latihan Soal Uji McNemar

  1. One hundred thirty-five citizens were selected at random and were asked to state their opinion regarding U.S. foreign policy. Forty-three were opposed to the U.S. foreign policy. After several weeks, during which they received an informative newsletter, they were again asked their opinion; 37 were opposed, and 30 of the 37 were persons who originally were not opposed to the U.S. foreign policy. Is the change in numbers of people opposed to the U.S. foreign policy significant?
Not Opposed Opposed
Not Opposed 72 30
Opposed 36 7

Uji Cox Stuart

  1. Departemen Perdagangan AS menerbitkan perkiraan yang diperoleh dari sampel independen setiap tahun dari jarak tempuh tahunan rata-rata yang dicakup oleh berbagai kelas kendaraan di Amerika Serikat. Angka-angka untuk mobil dan truk (dalam ribuan mil) diberikan di bawah ini untuk setiap tahun 1970-1983. Apakah ada bukti tren monoton dalam kedua kasus?
Cars 9.8 9.9 10.0 9.8 9.2 9.4 9.5 9.6 9.8 9.3 8.9 8.7 9.2 9.3
Trucks 11.5 11.5 12.2 11.5 10.9 10.6 11.1 11.1 11.0 10.8 11.4 12.3 11.2 11.2 
cars=c(9.8, 9.9, 10, 9.8, 9.2, 9.4, 9.5, 
          9.6, 9.8, 9.3, 8.9, 8.7, 9.2, 9.3)

trucks=c(11.5, 11.5, 12.2, 11.5, 10.9, 10.6, 11.1, 
            11.1, 11, 10.8, 11.4, 12.3, 11.2, 11.2)

Uji Hipotesis :

\(H_0: \text{Kasus Monoton}\)

\(H_1: \text{Tidak demikian}\)

library(randtests)

cox.stuart.test(cars, alternative="two.sided") #Package randtests

    Cox Stuart test

data:  cars
statistic = 0, n = 7, p-value = 0.01563
alternative hypothesis: non randomness
cox.stuart.test(trucks, alternative="two.sided")

    Cox Stuart test

data:  trucks
statistic = 3, n = 7, p-value = 1
alternative hypothesis: non randomness

Uji apakah cars memiliki trend naik/turun?

Uji Hipotesis :

\(H_0: \text{Kasus Monoton Tidak Naik}\)

\(H_1: \text{Terdapat Tren Naik}\)

cox.stuart.test(cars, alternative="right.sided") 

    Cox Stuart test

data:  cars
statistic = 0, n = 7, p-value = 1
alternative hypothesis: increasing trend

Uji Hipotesis :

\(H_0: \text{Kasus Monoton Tidak Turun}\)

\(H_1: \text{Terdapat Tren Turun}\)

cox.stuart.test(cars, alternative="left.sided") 

    Cox Stuart test

data:  cars
statistic = 0, n = 7, p-value = 0.007813
alternative hypothesis: decreasing trend

Latihan Soal Uji Cox Stuart

  1. For each of the last 34 years a small Midwestern college recorded the average heights of male freshmen. The averages were 68.3, 68.6, 68.4, 68.1, 68.4, 68.2, 68.7, 68.9, 69.0, 68.8, 69.0, 68.6, 69.2, 68.9, 68.6, 68.6, 68.8, 69.2, 68.8, 68.7, 69.5, 68.7, 68.8, 69.4, 69.3, 69.3, 69.5, 69.5, 69.0, 69.2, 69.2, 69.1, 69.9. Do these averages indicate an increasing trend in height?

Uji Wilcoxon

  1. Misalkan Set 1 adalah 𝑋, dan Set 2 adalah 𝑌.
Set I 1 1 1 1 1 3 3 5 5 7 7
Set II 1 2 2 4 4 4 4 5 5 5 7

Apakah di antara kelompok terdapat perbedaan yang signifikan?

x=c(1,1,1,1,1,3,3,5,5,7,7)
y=c(1,2,2,4,4,4,4,5,5,5,7)

Uji Hipotesis :

\(H_0: \text{Tidak terdapat perbedaan yang signfikan antara X dan Y (Ekspektasi X = Ekspektasi Y)}\)

\(H_1: \text{Tidak demikian}\)

Library stats memiliki function wilcox.test() yang dapat digunakan untuk melakukan uji Wilcoxon. Tetapi tidak bisa menghitung exact p-value pada data dengan ties.

wilcox.test(x, y, alternative='two.sided') #Package stats
Warning in wilcox.test.default(x, y, alternative = "two.sided"): cannot compute
exact p-value with ties

    Wilcoxon rank sum test with continuity correction

data:  x and y
W = 46.5, p-value = 0.3656
alternative hypothesis: true location shift is not equal to 0

Sebagai alternatif, gunakan fungsi wilcoxsign_test() dari library coin

library(coin)
Warning: package 'coin' was built under R version 4.4.1
Loading required package: survival

Attaching package: 'coin'
The following object is masked from 'package:testthat':

    expectation
wilcoxsign_test(y~x, distribution="exact", zero.method="Wilcoxon")

    Exact Wilcoxon Signed-Rank Test

data:  y by x (pos, neg) 
     stratified by block
Z = 1.5521, p-value = 0.1875
alternative hypothesis: true mu is not equal to 0
wilcoxsign_test(y~x, distribution="approximate", zero.method="Wilcoxon")

    Approximative Wilcoxon Signed-Rank Test

data:  y by x (pos, neg) 
     stratified by block
Z = 1.5521, p-value = 0.1932
alternative hypothesis: true mu is not equal to 0

Latihan Soal Uji Wilcoxon

  1. A random sample consisting of 20 people who drove automobiles was selected to see if alcohol affected reaction time. Each driver’s reaction time was measured in a laboratory before and after drinking a specified amount of a beverage containing alcohol. The reaction times in seconds were as follows

Does alcohol affect reaction time?