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# spearman.jl | ||
# Spearman's rank correlation test in Julia | ||
# | ||
# Copyright (C) 2016 Diego Javier Zea | ||
# | ||
# Permission is hereby granted, free of charge, to any person obtaining | ||
# a copy of this software and associated documentation files (the | ||
# "Software"), to deal in the Software without restriction, including | ||
# without limitation the rights to use, copy, modify, merge, publish, | ||
# distribute, sublicense, and/or sell copies of the Software, and to | ||
# permit persons to whom the Software is furnished to do so, subject to | ||
# the following conditions: | ||
# | ||
# The above copyright notice and this permission notice shall be | ||
# included in all copies or substantial portions of the Software. | ||
# | ||
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, | ||
# EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF | ||
# MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND | ||
# NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE | ||
# LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION | ||
# OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION | ||
# WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. | ||
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export CorrelationTest, SpearmanCorrelationTest | ||
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abstract CorrelationTest <: HypothesisTest | ||
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"Sum squared difference of ranks (midranks for ties)" | ||
spearman_S(xrank, yrank) = sumabs2(xrank .- yrank) | ||
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immutable SpearmanCorrelationTest <: CorrelationTest | ||
# Tied ranking for x and y | ||
xrank::Vector{Float64} | ||
yrank::Vector{Float64} | ||
# Adjustment for ties | ||
xtiesadj::Float64 | ||
ytiesadj::Float64 | ||
# Sum squared difference of ranks | ||
S::Float64 | ||
# Number of points | ||
n::Int | ||
# Spearman's ρ | ||
ρ::Float64 | ||
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function SpearmanCorrelationTest(x, y) | ||
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n = length(x) | ||
(n != length(y)) && throw(ErrorException("x and y must have the same length")) | ||
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xrank, xtiesadj = HypothesisTests.tiedrank_adj(x) | ||
yrank, ytiesadj = HypothesisTests.tiedrank_adj(y) | ||
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S = spearman_S(xrank, yrank) | ||
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ρ = corspearman(x, y) | ||
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new(xrank, yrank, xtiesadj, ytiesadj, S, n, ρ) | ||
end | ||
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end | ||
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testname(::SpearmanCorrelationTest) = "Spearman's rank correlation test" | ||
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# parameter of interest: name, value under h0, point estimate | ||
population_param_of_interest(x::SpearmanCorrelationTest) = ("Spearman's ρ", 0.0, x.ρ) | ||
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function show_params(io::IO, x::SpearmanCorrelationTest, ident) | ||
println(io, ident, "Number of points: ", x.n) | ||
println(io, ident, "Spearman's ρ: ", x.ρ) | ||
println(io, ident, "S (Sum squared difference of ranks): ", x.S) | ||
println(io, ident, "adjustment for ties in x: ", x.xtiesadj) | ||
println(io, ident, "adjustment for ties in y: ", x.ytiesadj) | ||
end | ||
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function P_from_null_S_values(S_null, x::SpearmanCorrelationTest, tail) | ||
S_null_mean = mean(S_null) | ||
# S is approximately normally distributed | ||
# S and ρ are inversely proportional | ||
S_null[:] = S_null .- S_null_mean # center | ||
S_centered = x.S - S_null_mean # center | ||
if tail == :both | ||
modS = abs(S_centered) | ||
mean(S_null .<= -modS) + mean(S_null .>= modS) | ||
elseif tail == :right | ||
mean(S_null .<= S_centered) | ||
elseif tail == :left | ||
mean(S_null .>= S_centered) | ||
else | ||
throw(ArgumentError("tail=$(tail) is invalid")) | ||
end | ||
end | ||
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function spearman_P_exact(x::SpearmanCorrelationTest, tail) | ||
S_null = Float64[ spearman_S(perm, x.yrank) for perm in permutations(x.xrank) ] | ||
P_from_null_S_values(S_null, x, tail) | ||
end | ||
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function spearman_P_sampling(x::SpearmanCorrelationTest, tail) | ||
# 360000 samples gives an se(P)=0.0005 for P < 0.1 | ||
X = copy(x.xrank) | ||
S_null = Float64[ spearman_S(shuffle!(X), x.yrank) for sample in 1:360000 ] | ||
P_from_null_S_values(S_null, x, tail) | ||
end | ||
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# Use estimated mean and std for the S null distribution as in: | ||
# | ||
# Press WH, Teukolsky SA, Vetterling WT, Flannery BP. | ||
# Numerical recipes in C. | ||
# Cambridge: Cambridge university press; 1996. | ||
# Use estimated mean and std for the S null distribution as in: | ||
# | ||
# Press WH, Teukolsky SA, Vetterling WT, Flannery BP. | ||
# Numerical recipes in C. | ||
# Cambridge: Cambridge university press; 1996. | ||
function spearman_P_estimated(x::SpearmanCorrelationTest, tail) | ||
a = (x.n^3 - x.n) | ||
# Numerical Recipes (14.6.6) | ||
S_null_mean = (a/6.) - (x.xtiesadj/12.) - (x.ytiesadj/12.) | ||
# Numerical Recipes (14.6.7) | ||
S_null_std = sqrt((((x.n-1)*(x.n^2)*((x.n+1)^2))/36) * (1-(x.xtiesadj/a)) * (1-(x.ytiesadj/a))) | ||
zscore = (x.S - S_null_mean)/S_null_std | ||
# S is approximately normally distributed | ||
# S and ρ are inversely proportional | ||
if tail == :both | ||
cdf(Normal(), -abs(zscore)) + ccdf(Normal(), abs(zscore)) | ||
elseif tail == :right | ||
cdf(Normal(), zscore) | ||
elseif tail == :left | ||
ccdf(Normal(), zscore) | ||
else | ||
throw(ArgumentError("tail=$(tail) is invalid")) | ||
end | ||
end | ||
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# Using T test for n > 10 as in: | ||
# | ||
# McDonald JH. | ||
# Handbook of biological statistics. | ||
# Baltimore, MD: Sparky House Publishing; 2009 Aug. | ||
function spearman_P_ttest(x::SpearmanCorrelationTest, tail) | ||
ρ2 = x.ρ^2 | ||
df = x.n-2 | ||
t = sqrt((df*ρ2)/(1-ρ2)) | ||
if tail == :both | ||
cdf(TDist(df), -t) + ccdf(Normal(), t) | ||
elseif tail == :right | ||
ccdf(TDist(df), t) | ||
elseif tail == :left | ||
cdf(TDist(df), t) | ||
else | ||
throw(ArgumentError("tail=$(tail) is invalid")) | ||
end | ||
end | ||
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function pvalue(x::SpearmanCorrelationTest; tail=:both, method=:estimated) | ||
if x.n <= 10 | ||
# Exact P value using permutations | ||
return( spearman_P_exact(x, tail) ) | ||
end | ||
if method == :sampling | ||
return( spearman_P_sampling(x, tail) ) | ||
elseif method == :exact | ||
return( spearman_P_exact(x, tail) ) | ||
elseif method == :estimated | ||
return( spearman_P_estimated(x, tail) ) | ||
elseif method == :ttest | ||
return( spearman_P_ttest(x, tail) ) | ||
else | ||
throw(ArgumentError("method=$(method) is invalid")) | ||
end | ||
end | ||
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using HypothesisTests, Base.Test | ||
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# Test Exact P value: n <= 10 | ||
let x = [ 44.4, 45.9, 41.9, 53.3, 44.7, 44.1 ], | ||
y = [ 2.6, 3.1, 2.5, 5.0, 3.6, 4.0 ] | ||
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corr = HypothesisTests.SpearmanCorrelationTest(x, y) | ||
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# R values | ||
@test_approx_eq corr.ρ 0.6 | ||
@test_approx_eq_eps HypothesisTests.pvalue(corr) 0.2417 0.0001 | ||
@test_approx_eq_eps HypothesisTests.pvalue(corr, tail=:right) 0.1208 0.0001 | ||
@test_approx_eq_eps HypothesisTests.pvalue(corr, tail=:left) 0.9125 0.0001 | ||
end | ||
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show(IOBuffer(), | ||
HypothesisTests.SpearmanCorrelationTest([ 44.4, 45.9, 41.9, 53.3, 44.7, 44.1 ], | ||
[ 2.6, 3.1, 2.5, 5.0, 3.6, 4.0 ]) | ||
) | ||
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let x = collect(1:11), | ||
y = [6,5,4,3,2,1,7,11,10,9,8] | ||
# https://stat.ethz.ch/pipermail/r-devel/2009-February/052112.html | ||
# correct P value 0.03044548 | ||
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corr = HypothesisTests.SpearmanCorrelationTest(x, y) | ||
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@test_approx_eq_eps HypothesisTests.pvalue(corr, tail=:right, method=:exact) 0.03044548 1e-8 | ||
@test_approx_eq_eps HypothesisTests.pvalue(corr, tail=:right, method=:sampling) 0.030 1e-3 | ||
@test_approx_eq_eps HypothesisTests.pvalue(corr, tail=:right, method=:estimated) 0.030 1e-3 | ||
@test_approx_eq_eps HypothesisTests.pvalue(corr, tail=:right, method=:ttest) 0.03 1e-2 | ||
end | ||
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let x = collect(1:10), | ||
y = [5,4,3,2,1,6,10,9,8,7] | ||
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# R's pspearman: 0.05443067 is the exact value | ||
corr = HypothesisTests.SpearmanCorrelationTest(x, y) | ||
@test_approx_eq_eps HypothesisTests.pvalue(corr) 0.05443067 1e-8 | ||
end | ||
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