import Pkg
Pkg.activate(".")
using CSV
using Plots
using HypothesisTests
using DataFrames Activating project at `~/sandbox/dataalltheway/posts/011-01-non-parametric-hypothesis-tests-julia`Non-parametric hypothesis tests with examples in Julia
How To
Non-parametric Tests
Wilcoxon Test
Mann Whitney U Test
Kruskal-Wallis Test
Julia
A tutorial on non-parametric hypothesis tests with examples in Julia.
Dhruva Sambrani 
Update history
2022-11-30 First draft
Introduction
This article is an extension of Farmer. 2022. “Non-Parametric Hypothesis Tests with Examples in R.” November 18, 2022. Please check out the parent article for the theoretical background.
- Wilcoxon rank-sum (Mann-Whitney U test) (Section 3)
- Wilcoxon signed-rank test (Section 4)
- Kruskal-Wallis test (Section 5)
Import packages
Data import and cleanup
I have subsetted the data from 1928 onward and dropped any columns with all NAs or zeros. To do so for eachcol of data we first calculate whether all the elements are !ismissing && !=0 (! = not). Then pick all rows for those columns, while disallowmissing data.
temp_file = download("https://zenodo.org/record/7081360/files/1.%20Cement_emissions_data.csv")
data = CSV.read(temp_file, DataFrame)
dropmissing!(data, :Year)
filter!(:Year => >=(1928), data)
picked_cols_mask = eachcol(data) .|>
col -> all(x->(!ismissing(x) && x!=0), col)
data = disallowmissing(data[!, picked_cols_mask])94×24 DataFrame
69 rows omitted
| Row | Year | Argentina | Australia | Belgium | Brazil | Canada | Chile | China | Democratic Republic of the Congo | Denmark | Egypt | Finland | Italy | Japan | Mozambique | Norway | Peru | Portugal | Romania | Spain | Sweden | Turkey | USA | Global |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Int64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | |
| 1 | 1928 | 116.3 | 378.0 | 1505.0 | 43.61 | 868.6 | 54.57 | 39.78 | 21.81 | 385.2 | 43.8 | 138.1 | 1519.0 | 1897.0 | 7.27 | 158.3 | 25.42 | 36.34 | 163.6 | 763.2 | 233.2 | 29.07 | 15420.0 | 35616.0 |
| 2 | 1929 | 174.4 | 356.2 | 1606.0 | 47.25 | 963.0 | 72.62 | 76.51 | 29.07 | 396.1 | 87.22 | 138.1 | 1730.0 | 2112.0 | 10.9 | 159.1 | 25.43 | 43.61 | 156.3 | 901.3 | 284.0 | 32.71 | 15000.0 | 36873.0 |
| 3 | 1930 | 189.0 | 348.9 | 1508.0 | 43.58 | 926.7 | 79.88 | 73.45 | 32.71 | 385.2 | 148.5 | 101.6 | 1723.0 | 1853.0 | 10.9 | 160.2 | 10.9 | 47.29 | 196.3 | 908.4 | 304.6 | 29.07 | 14290.0 | 35561.0 |
| 4 | 1931 | 265.3 | 196.3 | 1218.0 | 83.59 | 799.5 | 50.88 | 97.93 | 21.81 | 250.8 | 119.9 | 79.95 | 1519.0 | 1788.0 | 10.9 | 109.7 | 14.54 | 47.25 | 98.14 | 806.8 | 257.9 | 50.83 | 10810.0 | 30931.0 |
| 5 | 1932 | 247.1 | 123.6 | 1039.0 | 72.69 | 363.4 | 54.51 | 79.57 | 7.27 | 203.6 | 119.9 | 76.23 | 1545.0 | 1843.0 | 10.9 | 117.3 | 10.9 | 58.15 | 105.4 | 705.1 | 241.0 | 54.46 | 6656.0 | 24721.0 |
| 6 | 1933 | 254.5 | 159.9 | 963.1 | 109.0 | 189.0 | 69.05 | 113.2 | 3.63 | 272.6 | 141.7 | 80.05 | 1756.0 | 2366.0 | 10.18 | 110.8 | 14.53 | 79.95 | 109.0 | 694.1 | 200.7 | 58.15 | 5587.0 | 23866.0 |
| 7 | 1934 | 279.8 | 207.2 | 937.6 | 159.9 | 272.6 | 101.7 | 97.93 | 3.63 | 381.6 | 145.4 | 112.7 | 2013.0 | 2304.0 | 7.27 | 123.6 | 21.82 | 90.94 | 156.3 | 672.3 | 287.1 | 83.59 | 6900.0 | 28542.0 |
| 8 | 1935 | 356.3 | 276.2 | 1087.0 | 181.6 | 272.6 | 141.6 | 156.1 | 3.63 | 374.3 | 189.0 | 134.3 | 2086.0 | 2904.0 | 7.27 | 130.8 | 29.07 | 105.4 | 189.0 | 668.7 | 367.1 | 65.42 | 6684.0 | 32090.0 |
| 9 | 1936 | 410.7 | 323.5 | 1163.0 | 239.7 | 388.9 | 123.5 | 428.5 | 3.63 | 392.4 | 167.2 | 163.5 | 1890.0 | 3082.0 | 7.27 | 149.0 | 36.34 | 119.9 | 185.3 | 297.9 | 392.5 | 69.05 | 9950.0 | 38763.0 |
| 10 | 1937 | 512.3 | 363.4 | 1486.0 | 283.5 | 483.4 | 156.3 | 437.7 | 3.77 | 334.4 | 163.5 | 203.4 | 2155.0 | 2984.0 | 7.27 | 159.9 | 39.96 | 127.1 | 225.3 | 189.0 | 432.5 | 105.4 | 10370.0 | 40829.0 |
| 11 | 1938 | 614.3 | 178.1 | 1508.0 | 305.5 | 432.5 | 181.7 | 9.18 | 7.5 | 316.2 | 185.3 | 236.2 | 2279.0 | 2729.0 | 11.99 | 163.6 | 50.86 | 130.8 | 221.7 | 294.4 | 490.6 | 130.9 | 9239.0 | 38551.0 |
| 12 | 1939 | 556.0 | 338.0 | 1261.0 | 345.5 | 450.7 | 167.2 | 223.4 | 17.44 | 345.3 | 185.0 | 279.6 | 2526.0 | 2508.0 | 14.54 | 192.6 | 58.18 | 145.4 | 261.7 | 588.8 | 585.1 | 141.7 | 10790.0 | 35687.0 |
| 13 | 1940 | 534.4 | 348.8 | 105.4 | 367.1 | 592.4 | 189.0 | 272.4 | 11.45 | 218.1 | 178.1 | 149.0 | 2373.0 | 2101.0 | 14.54 | 167.2 | 61.78 | 134.5 | 196.3 | 770.5 | 345.3 | 130.9 | 11550.0 | 31431.0 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| 83 | 2010 | 4178.0 | 3549.0 | 2582.0 | 21290.0 | 6005.0 | 1046.0 | 639600.0 | 189.0 | 672.2 | 20510.0 | 525.7 | 13280.0 | 24320.0 | 340.9 | 754.0 | 3338.0 | 3376.0 | 2778.0 | 11200.0 | 1324.0 | 29980.0 | 31450.0 | 1.2549e6 |
| 84 | 2011 | 4586.0 | 3496.0 | 2762.0 | 22840.0 | 6020.0 | 1080.0 | 708600.0 | 175.2 | 861.8 | 20100.0 | 557.8 | 12580.0 | 24980.0 | 373.3 | 749.0 | 3300.0 | 2813.0 | 3089.0 | 9523.0 | 1361.0 | 31450.0 | 32210.0 | 1.3498e6 |
| 85 | 2012 | 4184.0 | 3518.0 | 2643.0 | 25000.0 | 6532.0 | 1128.0 | 714800.0 | 157.9 | 871.1 | 20970.0 | 497.2 | 10070.0 | 25620.0 | 452.7 | 725.0 | 3731.0 | 2550.0 | 3150.0 | 8754.0 | 1479.0 | 31370.0 | 35270.0 | 1.3846e6 |
| 86 | 2013 | 4581.0 | 3294.0 | 2541.0 | 26650.0 | 5973.0 | 1086.0 | 748300.0 | 174.0 | 867.1 | 20210.0 | 481.8 | 8877.0 | 26810.0 | 505.6 | 731.0 | 4257.0 | 2814.0 | 2695.0 | 7642.0 | 1402.0 | 33910.0 | 36370.0 | 1.4441e6 |
| 87 | 2014 | 4336.0 | 3138.0 | 2643.0 | 26910.0 | 5912.0 | 1022.0 | 778600.0 | 127.9 | 887.3 | 20760.0 | 468.8 | 8339.0 | 26560.0 | 585.9 | 727.0 | 4590.0 | 3096.0 | 2944.0 | 8897.0 | 1399.0 | 34500.0 | 39440.0 | 1.4999e6 |
| 88 | 2015 | 4571.0 | 3076.0 | 2348.0 | 25080.0 | 6185.0 | 1033.0 | 722000.0 | 154.6 | 931.5 | 21650.0 | 462.1 | 8196.0 | 25940.0 | 614.2 | 672.0 | 4476.0 | 2921.0 | 3337.0 | 9216.0 | 1537.0 | 34440.0 | 39910.0 | 1.4444e6 |
| 89 | 2016 | 4029.0 | 2931.0 | 2436.0 | 22420.0 | 6114.0 | 1120.0 | 743000.0 | 98.04 | 1095.0 | 22820.0 | 553.2 | 7680.0 | 25970.0 | 947.8 | 684.0 | 4340.0 | 2297.0 | 3181.0 | 9414.0 | 1554.0 | 37530.0 | 39440.0 | 1.4876e6 |
| 90 | 2017 | 4362.0 | 3019.0 | 2291.0 | 19080.0 | 6827.0 | 865.9 | 758200.0 | 348.7 | 1194.0 | 21770.0 | 603.7 | 7711.0 | 26430.0 | 910.6 | 766.0 | 4291.0 | 2531.0 | 3310.0 | 9449.0 | 1484.0 | 39470.0 | 40320.0 | 1.5079e6 |
| 91 | 2018 | 4369.0 | 2942.0 | 2534.0 | 19340.0 | 6915.0 | 782.2 | 786700.0 | 406.1 | 1160.0 | 21000.0 | 601.7 | 7757.0 | 26180.0 | 930.0 | 730.0 | 4320.0 | 2251.0 | 3505.0 | 9667.0 | 1607.0 | 39410.0 | 38970.0 | 1.5692e6 |
| 92 | 2019 | 4141.0 | 3040.0 | 2819.0 | 19860.0 | 7125.0 | 825.3 | 826900.0 | 451.0 | 1129.0 | 19670.0 | 583.5 | 7912.0 | 25330.0 | 1011.0 | 722.0 | 4546.0 | 2225.0 | 3828.0 | 9064.0 | 1349.0 | 32350.0 | 40900.0 | 1.6175e6 |
| 93 | 2020 | 3508.0 | 2820.0 | 2634.0 | 22050.0 | 6625.0 | 825.3 | 858200.0 | 451.0 | 1227.0 | 18130.0 | 569.7 | 7059.0 | 24490.0 | 1011.0 | 725.0 | 4546.0 | 2310.0 | 3901.0 | 8192.0 | 1272.0 | 40810.0 | 40690.0 | 1.6375e6 |
| 94 | 2021 | 4671.0 | 2820.0 | 2634.0 | 23790.0 | 6625.0 | 825.3 | 853000.0 | 451.0 | 1227.0 | 16160.0 | 569.7 | 7059.0 | 23790.0 | 1011.0 | 701.3 | 4546.0 | 2310.0 | 3901.0 | 8609.0 | 1272.0 | 44390.0 | 41200.0 | 1.6729e6 |
plot(
data[!, :Year],
Array(log.(data[!, Not(:Year)])),
label=reshape(string.(propertynames(data)[2:end]), 1, :),
legend= :outerright,
size=(900, 400)
)
Wilcoxon rank-sum (Mann-Whitney U test)
mwut_results = MannWhitneyUTest(data[!, :USA], data[!, :Canada], )Approximate Mann-Whitney U test
-------------------------------
Population details:
parameter of interest: Location parameter (pseudomedian)
value under h_0: 0
point estimate: 28503.5
Test summary:
outcome with 95% confidence: reject h_0
two-sided p-value: <1e-30
Details:
number of observations in each group: [94, 94]
Mann-Whitney-U statistic: 8763.0
rank sums: [13228.0, 4538.0]
adjustment for ties: 24.0
normal approximation (μ, σ): (4345.0, 373.05)Right tailed test
pvalue(mwut_results, tail=:right)1.2040605143479147e-31Wilcoxon signed-rank test
dt = select(filter(:Year=> y-> y==2000 || y==2020, data), Not(:Year))
x = collect(dt[1, :])
y = collect(dt[2, :])
SignedRankTest(x, y)Exact Wilcoxon signed rank test
-------------------------------
Population details:
parameter of interest: Location parameter (pseudomedian)
value under h_0: 0
point estimate: 16.0
95% confidence interval: (-3848.0, 621.5)
Test summary:
outcome with 95% confidence: fail to reject h_0
two-sided p-value: 0.5803
Details:
number of observations: 23
Wilcoxon rank-sum statistic: 119.0
rank sums: [119.0, 157.0]
adjustment for ties: 0.0Kruskal-Wallis test
KruskalWallisTest(collect(eachcol(data[:, Not(:Year)]))...)Kruskal-Wallis rank sum test (chi-square approximation)
-------------------------------------------------------
Population details:
parameter of interest: Location parameters
value under h_0: "all equal"
point estimate: NaN
Test summary:
outcome with 95% confidence: reject h_0
one-sided p-value: <1e-99
Details:
number of observation in each group: [94, 94, 94, 94, 94, 94, 94, 94, 94, 94 … 94, 94, 94, 94, 94, 94, 94, 94, 94, 94]
χ²-statistic: 1253.58
rank sums: [100736.0, 98787.5, 1.11532e5, 1.21338e5, 1.21336e5, 55862.0, 138744.0, 20383.0, 71558.5, 1.03662e5 … 19940.0, 59493.0, 68485.5, 84067.0, 1.0239e5, 132068.0, 83910.0, 1.10562e5, 1.7974e5, 195500.0]
degrees of freedom: 22
adjustment for ties: 0.999999Citation
BibTeX citation:
@online{sambrani2022,
author = {Sambrani, Dhruva},
title = {Non-Parametric Hypothesis Tests with Examples in {Julia}},
date = {2022-11-30},
url = {https://dataalltheway.com/posts/011-01-non-parametric-hypothesis-tests-julia},
langid = {en}
}
For attribution, please cite this work as:
Sambrani, Dhruva. 2022. “Non-Parametric Hypothesis Tests with
Examples in Julia.” November 30, 2022. https://dataalltheway.com/posts/011-01-non-parametric-hypothesis-tests-julia.