Permutation test

From Rosetta Code
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Task
Permutation test
You are encouraged to solve this task according to the task description, using any language you may know.

A new medical treatment was tested on a population of volunteers, with each volunteer randomly assigned either to a group of treatment subjects, or to a group of control subjects.

Members of the treatment group were given the treatment, and members of the control group were given a placebo. The effect of the treatment or placebo on each volunteer was measured and reported in this table.

Table of experimental results
Treatment group Control group
85 68
88 41
75 10
66 49
25 16
29 65
83 32
39 92
97 28
98

Write a program that performs a permutation test to judge whether the treatment had a significantly stronger effect than the placebo.

  • Do this by considering every possible alternative assignment from the same pool of volunteers to a treatment group of size and a control group of size (i.e., the same group sizes used in the actual experiment but with the group members chosen differently), while assuming that each volunteer's effect remains constant regardless.
  • Note that the number of alternatives will be the binomial coefficient .
  • Compute the mean effect for each group and the difference in means between the groups in every case by subtracting the mean of the control group from the mean of the treatment group.
  • Report the percentage of alternative groupings for which the difference in means is less or equal to the actual experimentally observed difference in means, and the percentage for which it is greater.
  • Note that they should sum to 100%.

Extremely dissimilar values are evidence of an effect not entirely due to chance, but your program need not draw any conclusions.

You may assume the experimental data are known at compile time if that's easier than loading them at run time. Test your solution on the data given above.

Ada[edit]

with Ada.Text_IO; with Iterate_Subsets;
 
procedure Permutation_Test is
 
type Group_Type is array(Positive range <>) of Positive;
 
Treat_Group: constant Group_Type := (85, 88, 75, 66, 25, 29, 83, 39, 97);
Ctrl_Group: constant Group_Type := (68, 41, 10, 49, 16, 65, 32, 92, 28, 98);
 
package Iter is new Iterate_Subsets(Treat_Group'Length, Ctrl_Group'Length);
 
Full_Group: constant Group_Type(1 .. Iter.All_Elements)
 := Treat_Group & Ctrl_Group;
 
function Mean(S: Iter.Subset) return Float is
Sum: Natural := 0;
begin
for I in S'Range loop
Sum := Sum + Full_Group(S(I));
end loop;
return Float(Sum)/Float(S'Length);
end Mean;
 
package FIO is new Ada.Text_IO.Float_IO(Float);
 
T_Avg: Float := Mean(Iter.First);
S_Avg: Float;
S: Iter.Subset := Iter.First;
Equal: Positive := 1; -- Mean(Iter'First) = Mean(Iter'First)
Higher: Natural  := 0;
Lower: Natural  := 0;
 
begin -- Permutation_Test;
-- first, count the subsets with a higher, an equal or a lower mean
loop
Iter.Next(S);
S_Avg := Mean(S);
if S_Avg = T_Avg then
Equal := Equal + 1;
elsif S_Avg >= T_Avg then
Higher := Higher + 1;
else
Lower := Lower + 1;
end if;
exit when Iter.Last(S);
end loop;
 
-- second, output the results
declare
use Ada.Text_IO;
Sum: Float := Float(Higher + Equal + Lower);
begin
Put("Less or Equal: ");
FIO.Put(100.0*Float(Lower+Equal) / Sum, Fore=>3, Aft=>1, Exp=>0);
Put(Integer'Image(Lower+Equal));
New_Line;
Put("More: ");
FIO.Put(100.0*Float(Higher) / Sum, Fore=>3, Aft=>1, Exp=>0);
Put(Integer'Image(Higher));
New_Line;
end;
end Permutation_Test;

This solution uses an auxiliary package Iterate_Subsets. Here is the Spec:

generic
Subset_Size, More_Elements: Positive;
package Iterate_Subsets is
 
All_Elements: Positive := Subset_Size + More_Elements;
subtype Index is Integer range 1 .. All_Elements;
type Subset is array (1..Subset_Size) of Index;
 
-- iterate over all subsets of size Subset_Size
-- from the set {1, 2, ..., All_Element}
 
function First return Subset;
procedure Next(S: in out Subset);
function Last(S: Subset) return Boolean;
 
end Iterate_Subsets;
 

And here is the implementation:

package body Iterate_Subsets is
 
function First return Subset is
S: Subset;
begin
for I in S'Range loop
S(I) := I;
end loop;
return S;
end First;
 
procedure Next(S: in out Subset) is
I: Natural := S'Last;
begin
if S(I) < Index'Last then
S(I) := S(I) + 1;
else
while S(I-1)+1 = S(I) loop
I := I - 1;
end loop;
S(I-1) := S(I-1) + 1;
for J in I .. S'Last loop
S(J) := S(J-1) + 1;
end loop;
end if;
return;
end Next;
 
function Last(S: Subset) return Boolean is
begin
return S(S'First) = Index'Last-S'Length+1;
end Last;
 
end Iterate_Subsets;
Output:
Less or Equal:  87.2 80551
More:           12.8 11827


BBC BASIC[edit]

      ntreated% = 9
nplacebo% = 10
DIM results%(ntreated% + nplacebo% - 1)
results%() = 85, 88, 75, 66, 25, 29, 83, 39, 97, \ REM treated group
\ 68, 41, 10, 49, 16, 65, 32, 92, 28, 98 : REM placebo group
 
greater% = 0
FOR comb% = 0 TO 2^(ntreated%+nplacebo%)-1
IF FNnbits(comb%) = ntreated% THEN
tsum% = 0 : psum% = 0
FOR b% = 0 TO ntreated%+nplacebo%-1
IF comb% AND 2^b% THEN
tsum% += results%(b%)
ELSE
psum% += results%(b%)
ENDIF
NEXT
meandiff = tsum%/ntreated% - psum%/nplacebo%
IF comb% = 2^ntreated% - 1 THEN
actual = meandiff
ELSE
greater% -= meandiff > actual
groups% += 1
ENDIF
ENDIF
NEXT
 
percent = 100 * greater%/groups%
PRINT "Percentage groupings <= actual experiment: "; 100 - percent
PRINT "Percentage groupings > actual experiment: "; percent
END
 
DEF FNnbits(N%)
N% -= N% >>> 1 AND &55555555
N% = (N% AND &33333333) + (N% >>> 2 AND &33333333)
N% = (N% + (N% >>> 4)) AND &0F0F0F0F
N% += N% >>> 8 : N% += N% >>> 16
= N% AND &7F
Output:
Percentage groupings <= actual experiment: 87.1970296
Percentage groupings >  actual experiment: 12.8029704

C[edit]

#include <stdio.h>
 
int data[] = { 85, 88, 75, 66, 25, 29, 83, 39, 97,
68, 41, 10, 49, 16, 65, 32, 92, 28, 98 };
 
int pick(int at, int remain, int accu, int treat)
{
if (!remain) return (accu > treat) ? 1 : 0;
 
return pick(at - 1, remain - 1, accu + data[at - 1], treat) +
( at > remain ? pick(at - 1, remain, accu, treat) : 0 );
}
 
int main()
{
int treat = 0, i;
int le, gt;
double total = 1;
for (i = 0; i < 9; i++) treat += data[i];
for (i = 19; i > 10; i--) total *= i;
for (i = 9; i > 0; i--) total /= i;
 
gt = pick(19, 9, 0, treat);
le = total - gt;
 
printf("<= : %f%%  %d\n > : %f%%  %d\n",
100 * le / total, le, 100 * gt / total, gt);
return 0;
}
Output:
<= : 87.197168%  80551
> : 12.802832% 11827

C++[edit]

This is a translaion of C

#include<iostream>
#include<vector>
#include<numeric>
#include<functional>
 
class
{
public:
int64_t operator()(int n, int k){ return partial_factorial(n, k) / factorial(n - k);}
private:
int64_t partial_factorial(int from, int to) { return from == to ? 1 : from * partial_factorial(from - 1, to); }
int64_t factorial(int n) { return n == 0 ? 1 : n * factorial(n - 1);}
}combinations;
 
int main()
{
static constexpr int treatment = 9;
const std::vector<int> data{ 85, 88, 75, 66, 25, 29, 83, 39, 97,
68, 41, 10, 49, 16, 65, 32, 92, 28, 98 };
 
int treated = std::accumulate(data.begin(), data.begin() + treatment, 0);
 
std::function<int (int, int, int)> pick;
pick = [&](int n, int from, int accumulated)
{
if(n == 0)
return accumulated > treated ? 1 : 0;
else
return pick(n - 1, from - 1, accumulated + data[from - 1]) +
(from > n ? pick(n, from - 1, accumulated) : 0);
};
 
int total = combinations(data.size(), treatment);
int greater = pick(treatment, data.size(), 0);
int lesser = total - greater;
 
std::cout << "<= : " << 100.0 * lesser / total << "% " << lesser << std::endl
<< " > : " << 100.0 * greater / total << "% " << greater << std::endl;
}
Output:
<= : 87.197168%  80551
> : 12.802832% 11827

Common Lisp[edit]

(defun perm-test (s1 s2)
(let ((more 0) (leq 0)
(all-data (append s1 s2))
(thresh (apply #'+ s1)))
(labels
((recur (data sum need avail)
(cond ((zerop need) (if (>= sum thresh)
(incf more)
(incf leq)))
((>= avail need)
(recur (cdr data) sum need (1- avail))
(recur (cdr data) (+ sum (car data)) (1- need) (1- avail))))))
 
(recur all-data 0 (length s1) (length all-data))
(cons more leq))))
 
(let* ((a (perm-test '(68 41 10 49 16 65 32 92 28 98)
'(85 88 75 66 25 29 83 39 97)))
(x (car a))
(y (cdr a))
(s (+ x y)))
(format t "<=: ~a ~6f%~% >: ~a ~6f%~%"
x (* 100e0 (/ x s))
y (* 100e0 (/ y s))))
output
<=: 80551 87.197%
>: 11827 12.803%

D[edit]

import std.stdio, std.algorithm, std.array, combinations3;
 
auto permutationTest(T)(in T[] a, in T[] b) pure nothrow @safe {
immutable tObs = a.sum;
auto combs = combinations!false(a ~ b, a.length);
immutable under = combs.count!(perm => perm.sum <= tObs);
return under * 100.0 / combs.length;
}
 
void main() {
immutable treatmentGroup = [85, 88, 75, 66, 25, 29, 83, 39, 97];
immutable controlGroup = [68, 41, 10, 49, 16, 65, 32, 92, 28, 98];
immutable under = permutationTest(treatmentGroup, controlGroup);
writefln("Under =%6.2f%%\nOver =%6.2f%%", under, 100.0 - under);
}
Output:
Under = 87.20%
Over  = 12.80%

Alternative version:

Translation of: C
void main() @safe {
import std.stdio, std.algorithm, std.range;
 
immutable treatment = [85, 88, 75, 66, 25, 29, 83, 39, 97];
immutable control = [68, 41, 10, 49, 16, 65, 32, 92, 28, 98];
immutable both = treatment ~ control;
immutable sTreat = treatment.sum;
 
T pick(T)(in size_t at, in size_t remain, in T accu) pure nothrow @safe @nogc {
if (remain == 0)
return accu > sTreat;
 
return pick(at - 1, remain - 1, accu + both[at - 1]) +
(at > remain ? pick(at - 1, remain, accu) : 0);
}
 
alias mul = reduce!q{a * b};
immutable t = mul(1.0, iota(both.length, treatment.length + 1, -1))
.reduce!q{a / b}(iota(treatment.length, 0, -1));
immutable gt = pick(both.length, treatment.length, 0);
immutable le = cast(int)(t - gt);
writefln(" > : %2.2f%%  %d", 100.0 * gt / t, gt);
writefln("<= : %2.2f%%  %d", 100.0 * le / t, le);
}
Output:
 > : 12.80%  11827
<= : 87.20%  80551

Elixir[edit]

Translation of: Ruby
defmodule Permutation do
def statistic(ab, a) do
sumab = Enum.sum(ab)
suma = Enum.sum(a)
suma / length(a) - (sumab - suma) / (length(ab) - length(a))
end
 
def test(a, b) do
ab = a ++ b
tobs = statistic(ab, a)
{under, count} = Enum.reduce(comb(ab, length(a)), {0,0}, fn perm, {under, count} ->
if statistic(ab, perm) <= tobs, do: {under+1, count+1},
else: {under , count+1}
end)
under * 100.0 / count
end
 
defp comb(_, 0), do: [[]]
defp comb([], _), do: []
defp comb([h|t], m) do
(for l <- comb(t, m-1), do: [h|l]) ++ comb(t, m)
end
end
 
treatmentGroup = [85, 88, 75, 66, 25, 29, 83, 39, 97]
controlGroup = [68, 41, 10, 49, 16, 65, 32, 92, 28, 98]
under = Permutation.test(treatmentGroup, controlGroup)
:io.fwrite "under = ~.2f%, over = ~.2f%~n", [under, 100-under]
Output:
under = 87.20%, over = 12.80%

GAP[edit]

a := [85, 88, 75, 66, 25, 29, 83, 39, 97];
b := [68, 41, 10, 49, 16, 65, 32, 92, 28, 98];
 
# Compute a decimal approximation of a rational
Approx := function(x, d)
local neg, a, b, n, m, s;
if x < 0 then
x := -x;
neg := true;
else
neg := false;
fi;
a := NumeratorRat(x);
b := DenominatorRat(x);
n := QuoInt(a, b);
a := RemInt(a, b);
m := 10^d;
s := "";
if neg then
Append(s, "-");
fi;
Append(s, String(n));
n := Size(s) + 1;
Append(s, String(m + QuoInt(a*m, b)));
s[n] := '.';
return s;
end;
 
PermTest := function(a, b)
local c, d, p, q, u, v, m, n, k, diff, all;
p := Size(a);
q := Size(b);
v := Concatenation(a, b);
n := p + q;
m := Binomial(n, p);
diff := Sum(a)/p - Sum(b)/q;
all := [1 .. n];
k := 0;
for u in Combinations(all, p) do
c := List(u, i -> v[i]);
d := List(Difference(all, u), i -> v[i]);
if Sum(c)/p - Sum(d)/q > diff then
k := k + 1;
fi;
od;
return [Approx((1 - k/m)*100, 3), Approx(k/m*100, 3)];
end;
 
# in order, % less or greater than original diff
PermTest(a, b);
[ "87.197", "12.802" ]

Go[edit]

A version doing all math in integers until computing final percentages.

package main
 
import "fmt"
 
var tr = []int{85, 88, 75, 66, 25, 29, 83, 39, 97}
var ct = []int{68, 41, 10, 49, 16, 65, 32, 92, 28, 98}
 
func main() {
// collect all results in a single list
all := make([]int, len(tr)+len(ct))
copy(all, tr)
copy(all[len(tr):], ct)
 
// compute sum of all data, useful as intermediate result
var sumAll int
for _, r := range all {
sumAll += r
}
 
// closure for computing scaled difference.
// compute results scaled by len(tr)*len(ct).
// this allows all math to be done in integers.
sd := func(trc []int) int {
var sumTr int
for _, x := range trc {
sumTr += all[x]
}
return sumTr*len(ct) - (sumAll-sumTr)*len(tr)
}
 
// compute observed difference, as an intermediate result
a := make([]int, len(tr))
for i, _ := range a {
a[i] = i
}
sdObs := sd(a)
 
// iterate over all combinations. for each, compute (scaled)
// difference and tally whether leq or gt observed difference.
var nLe, nGt int
comb(len(all), len(tr), func(c []int) {
if sd(c) > sdObs {
nGt++
} else {
nLe++
}
})
 
// print results as percentage
pc := 100 / float64(nLe+nGt)
fmt.Printf("differences <= observed: %f%%\n", float64(nLe)*pc)
fmt.Printf("differences > observed: %f%%\n", float64(nGt)*pc)
}
 
// combination generator, copied from combination task
func comb(n, m int, emit func([]int)) {
s := make([]int, m)
last := m - 1
var rc func(int, int)
rc = func(i, next int) {
for j := next; j < n; j++ {
s[i] = j
if i == last {
emit(s)
} else {
rc(i+1, j+1)
}
}
return
}
rc(0, 0)
}
Output:
differences <= observed: 87.197168%
differences  > observed: 12.802832%

Haskell[edit]

binomial n m = (f !! n) `div` (f !! m) `div` (f !! (n - m))
where f = scanl (*) 1 [1..]
 
permtest treat ctrl = (fromIntegral less) / (fromIntegral total) * 100
where
total = binomial (length avail) (length treat)
less = combos (sum treat) (length treat) avail
avail = ctrl ++ treat
combos total n a@(x:xs)
| total < 0 = binomial (length a) n
| n == 0 = 0
| n > length a = 0
| n == length a = fromEnum (total < sum a)
| otherwise = combos (total - x) (n - 1) xs
+ combos total n xs
 
main = let r = permtest
[85, 88, 75, 66, 25, 29, 83, 39, 97]
[68, 41, 10, 49, 16, 65, 32, 92, 28, 98]
in do putStr "> : "; print r
putStr "<=: "; print $ 100 - r
Output:
> : 12.80283184307952
<=: 87.19716815692048

Somewhat faster, this goes from top down:

binomial n m = (f !! n) `div` (f !! m) `div` (f !! (n - m))
where f = scanl (*) 1 [1..]
 
perms treat ctrl = (less,total) where
total = binomial (length ctrl + length treat) (length treat)
less = length $ filter (<= sum treat)
$ sums (treat ++ ctrl) (length treat)
sums x n
| l < n || n < 0 = []
| n == 0 = [0]
| l == n = [sum x]
| otherwise = [a + b | i <- [0..n], a <- sums left i, b <- sums right (n - i)]
where (l, l1) = (length x, l `div` 2)
(left, right) = splitAt l1 x
 
main = print $ (lt, 100 - lt) where
(a, b) = perms [85, 88, 75, 66, 25, 29, 83, 39, 97]
[68, 41, 10, 49, 16, 65, 32, 92, 28, 98]
lt = (fromIntegral a) / (fromIntegral b) * 100

In cases where the sample data are a large number of relatively small positive integers, counting number of partial sums is a lot faster:

combs maxsum len x = foldl f [(0,0,1)] x where
f a n = merge a (map (addNum n) $ filter (\(l,_,_) -> l < len) a)
addNum n (a,s,c)
-- anything larger than maxsum is as good as infinity
| s + n > maxsum = (a+1, maxsum + 1, c)
| otherwise = (a+1, s+n, c)
 
merge a [] = a
merge [] a = a
merge a@((a1,a2,a3):as) b@((b1,b2,b3):bs)
| a1 == b1 && a2 == b2 = (a1,a2,a3+b3):merge as bs
| a1 < b1 || (a1 == b1 && a2 < b2) = (a1,a2,a3):merge as b
| otherwise = (b1,b2,b3):merge a bs
 
permtest a b = (lt, ge) where
lt = sum $ map (\(a,b,c) -> if a == la && b < sa then c else 0)
$ combs sa la (a++b)
ge = (binomial (la + lb) la) - lt
(sa, la, lb) = (sum a, length a, length b)
 
binomial n m = (f !! n) `div` (f !! m) `div` (f !! (n - m))
where f = scanl (*) 1 [1..]
 
-- how many combinations are less than current sum
main = print$ permtest [85, 88, 75, 66, 25, 29, 83, 39, 97]
[68, 41, 10, 49, 16, 65, 32, 92, 28, 98]

J[edit]

require'stats'
trmt=: 0.85 0.88 0.75 0.66 0.25 0.29 0.83 0.39 0.97
ctrl=: 0.68 0.41 0.1 0.49 0.16 0.65 0.32 0.92 0.28 0.98
difm=: -&mean
result=: trmt difm ctrl
all=: trmt(#@[ ({. difm }.) |:@([ (comb [email protected],"1 [email protected]])&# ,) { ,) ctrl
smoutput 'under: ','%',~":100*mean all <: result
smoutput 'over: ','%',~":100*mean all > result

Result:

under: 87.1972%
over: 12.8028%

jq[edit]

Works with: jq version 1.4

Part 1: Combinations

# combination(r) generates a stream of combinations of r items from the input array.
def combination(r):
if r > length or r < 0 then empty
elif r == length then .
else ( [.[0]] + (.[1:]|combination(r-1))),
( .[1:]|combination(r))
end;
 

Part 2: Permutation Test

# a and b should be arrays:
def permutationTest(a; b):
 
def normalize(a;b): # mainly to avoid having to compute $sumab
(a|add) as $sa
| (b|add) as $sb
| (($sa + $sb)/((a|length) + (b|length))) as $avg
| [(a | map(.-$avg)), (b | map(.-$avg))];
 
# avg(a) - avg(b) (assuming ab==a+b and avg(ab) is 0)
def statistic(ab; a):
(a | add) as $suma
# (ab|add) should be 0, by normalization
| ($suma / (a|length)) +
($suma / ((ab|length) - (a|length)));
 
normalize(a;b)
| (a + b) as $ab # pooled observations
| .[0] as $a | .[1] as $b
| statistic($ab; $a) as $t_observed # observed difference in means
| reduce ($ab|combination($a|length)) as $perm # for each combination...
([0,0]; # state: [under,count]
if statistic($ab; $perm) <= $t_observed then .[0] += 1 else . end
| .[1] += 1 )
| .[0] * 100.0 / .[1] # under/count
;

Example:

def treatmentGroup: [85, 88, 75, 66, 25, 29, 83, 39, 97];
def controlGroup: [68, 41, 10, 49, 16, 65, 32, 92, 28, 98];
 
permutationTest(treatmentGroup; controlGroup) as $under
| "% under=\($under)", "% over=\(100 - $under)"
Output:
$ jq -n -r -f permutation_test.jq
% under=87.14304271579813
% over=12.856957284201869

Julia[edit]

Works with: Julia version 0.6

The primary function for this solution is permutation_test, which relies on Julia's combinations (from Combinatorics module) function to provide all of the possible study arm assignments. bifurcate splits the pooled results into "treatment" and "control" groups according to the indices provided by combinations.

Functions

using Combinatorics
 
meandiff(a::Vector{T}, b::Vector{T}) where T <: Real = mean(a) - mean(b)
 
function bifurcate(a::AbstractVector, sel::Vector{T}) where T <: Integer
x = a[sel]
asel = trues(length(a))
asel[sel] = false
y = a[asel]
return x, y
end
 
function permutation_test(treated::Vector{T}, control::Vector{T}) where T <: Real
effect0 = meandiff(treated, control)
pool = vcat(treated, control)
tlen = length(treated)
plen = length(pool)
better = worse = 0
for subset in combinations(1:plen, tlen)
t, c = bifurcate(pool, subset)
if effect0 < meandiff(t, c)
better += 1
else
worse += 1
end
end
return better, worse
end

Main

const treated = [85, 88, 75, 66, 25, 29, 83, 39, 97]
const control = [68, 41, 10, 49, 16, 65, 32, 92, 28, 98]
 
(better, worse) = permutation_test(treated, control)
 
tot = better + worse
 
println("Permutation test using the following data:")
println("Treated: ", treated)
println("Control: ", control)
println("\nThere are $tot different permuted groups of these data.")
@printf("%8d, %5.2f%% showed better than actual results.\n", better, 100 * better / tot)
print(@sprintf("%8d, %5.2f%% showed equalivalent or worse results.", worse, 100 * worse / tot))
Output:
Permutation test using the following data:
Treated:  [85, 88, 75, 66, 25, 29, 83, 39, 97]
Control:  [68, 41, 10, 49, 16, 65, 32, 92, 28, 98]

There are 92378 different permuted groups of these data.
   11827, 12.80% showed better than actual results.
   80551, 87.20% showed equalivalent or worse results.

Kotlin[edit]

Translation of: C
// version 1.1.2
 
val data = intArrayOf(
85, 88, 75, 66, 25, 29, 83, 39, 97,
68, 41, 10, 49, 16, 65, 32, 92, 28, 98
)
 
fun pick(at: Int, remain: Int, accu: Int, treat: Int): Int {
if (remain == 0) return if (accu > treat) 1 else 0
return pick(at - 1, remain - 1, accu + data[at - 1], treat) +
if (at > remain) pick(at - 1, remain, accu, treat) else 0
}
 
fun main(args: Array<String>) {
var treat = 0
var total = 1.0
for (i in 0..8) treat += data[i]
for (i in 19 downTo 11) total *= i
for (i in 9 downTo 1) total /= i
val gt = pick(19, 9, 0, treat)
val le = (total - gt).toInt()
System.out.printf("<= : %f%%  %d\n", 100.0 * le / total, le)
System.out.printf(" > : %f%%  %d\n", 100.0 * gt / total, gt)
}
Output:
<= : 87.197168%  80551
 > : 12.802832%  11827

Mathematica[edit]

"<=: " <> ToString[#1] <> " " <> ToString[100. #1/#2] <> "%\n >: " <> 
ToString[#2 - #1] <> " " <> ToString[100. (1 - #1/#2)] <> "%" &[
Count[Total /@ Subsets[Join[#1, #2], {[email protected]#1}],
n_ /; n <= [email protected]#1],
Binomial[[email protected]#1 + [email protected]#2, [email protected]#1]] &[{85, 88, 75, 66, 25,
29, 83, 39, 97}, {68, 41, 10, 49, 16, 65, 32, 92, 28, 98}]
Output:
<=: 80551 87.1972%
 >: 11827 12.8028%

Perl[edit]

#!/usr/bin/perl
use warnings;
use strict;
 
use List::Util qw{ sum };
 
 
sub means {
my @groups = @_;
return map sum(@$_) / @$_, @groups;
}
 
 
sub following {
my $pattern = shift;
my $orig_count = grep $_, @$pattern;
my $count;
do {
my $i = $#{$pattern};
until (0 > $i) {
$pattern->[$i] = $pattern->[$i] ? 0 : 1;
last if $pattern->[$i];
--$i;
}
$count = grep $_, @$pattern;
} until $count == $orig_count or not $count;
undef @$pattern unless $count;
}
 
 
my @groups;
my $i = 0;
while (<DATA>) {
chomp;
$i++, next if /^$/;
push @{ $groups[$i] }, $_;
}
 
my @orig_means = means(@groups);
my $orig_cmp = $orig_means[0] - $orig_means[1];
 
my $pattern = [ (0) x @{ $groups[0] },
(1) x @{ $groups[1] }
];
 
my @cmp = (0) x 3;
while (@$pattern) {
my @perms = map { my $g = $_;
[ (@{ $groups[0] }, @{ $groups[1] } ) [ grep $pattern->[$_] == $g, 0 .. $#{$pattern} ] ];
} 0, 1;
my @means = means(@perms);
$cmp[ ($means[0] - $means[1]) <=> $orig_cmp ]++;
} continue {
following($pattern);
}
my $all = sum(@cmp);
my $length = length $all;
for (0, -1, 1) {
printf "%-7s %${length}d %6.3f%%\n",
(qw(equal greater less))[$_], $cmp[$_], 100 * $cmp[$_] / $all;
}
 
 
__DATA__
85
88
75
66
25
29
83
39
97
 
68
41
10
49
16
65
32
92
28
98
Output:
equal     313  0.339%
less    80238 86.858%
greater 11827 12.803%

Perl 6[edit]

Works with: rakudo version 2015-09-30
sub stats ( @test, @all ) {
(([+] @test) / [email protected] ) - ([+] flat @all, (@test X* -1)) / (@all - @test)
}
 
 
my int @treated = <85 88 75 66 25 29 83 39 97>;
my int @control = <68 41 10 49 16 65 32 92 28 98>;
my int @all = flat @treated, @control;
 
my $base = stats( @treated, @all );
 
my @trials = 0, 0, 0;
 
@trials[ 1 + ( stats( $_, @all ) <=> $base ) ]++ for @all.combinations(+@treated);
 
say 'Counts: <, =, > ', @trials;
say 'Less than  : %', 100 * @trials[0] / [+] @trials;
say 'Equal to  : %', 100 * @trials[1] / [+] @trials;
say 'Greater than : %', 100 * @trials[2] / [+] @trials;
say 'Less or Equal: %', 100 * ( [+] @trials[0,1] ) / [+] @trials;
Output:
Counts: <, =, > 80238 313 11827
Less than    : %86.858343
Equal to     : %0.338825
Greater than : %12.802832
Less or Equal: %87.197168

Phix[edit]

Translation of: C
constant data = {85, 88, 75, 66, 25, 29, 83, 39, 97,
68, 41, 10, 49, 16, 65, 32, 92, 28, 98 }
 
function pick(int at, int remain, int accu, int treat)
if remain=0 then return iff(accu>treat?1:0) end if
return pick(at-1, remain-1, accu+data[at], treat) +
iff(at>remain?pick(at-1, remain, accu, treat):0)
end function
 
int treat = 0, le, gt
atom total = 1;
for i=1 to 9 do treat += data[i] end for
for i=19 to 11 by -1 do total *= i end for
for i=9 to 1 by -1 do total /= i end for
 
gt = pick(19, 9, 0, treat)
le = total - gt;
 
printf(1,"<= : %f%%  %d\n > : %f%%  %d\n",
{100*le/total, le, 100*gt/total, gt})
Output:
<= : 87.197168%  80551
 > : 12.802832%  11827

PicoLisp[edit]

(load "@lib/simul.l")  # For 'subsets'
 
(scl 2)
 
(de _stat (A)
(let (LenA (length A) SumA (apply + A))
(-
(*/ SumA LenA)
(*/ (- SumAB SumA) (- LenAB LenA)) ) ) )
 
(de permutationTest (A B)
(let
(AB (append A B)
SumAB (apply + AB)
LenAB (length AB)
Tobs (_stat A)
Count 0 )
(*/
(sum
'((Perm)
(inc 'Count)
(and (>= Tobs (_stat Perm)) 1) )
(subsets (length A) AB) )
100.0
Count ) ) )
 
(setq
*TreatmentGroup (0.85 0.88 0.75 0.66 0.25 0.29 0.83 0.39 0.97)
*ControlGroup (0.68 0.41 0.10 0.49 0.16 0.65 0.32 0.92 0.28 0.98) )
 
(let N (permutationTest *TreatmentGroup *ControlGroup)
(prinl "under = " (round N) "%, over = " (round (- 100.0 N)) "%") )
Output:
under = 87.85%, over = 12.15%

PureBasic[edit]

Given a treatment group with [n=9] and a control group with [m=10]. The numbers [x] from [1] to [1<<(n+m)] exhaust the possible states.

Any bit-String of Length [n+m] containing [n=9] "1's" is a Valid bit String, as tested by: IsValidBitString(x,n+m,n).

Then we can use these bits to Select whether a particular index For our array should be assigned to: the treatment group or the control group

 
 
Define.f meanTreated,meanControl,diffInMeans
Define.f actualmeanTreated,actualmeanControl,actualdiffInMeans
 
Dim poolA(19)
 
poolA(1) =85 ; first 9 the treated
poolA(2) =88
poolA(3) =75
poolA(4) =66
poolA(5) =25
poolA(6) =29
poolA(7) =83
poolA(8) =39
poolA(9) =97
 
poolA(10) =68 ; last 10 the control
poolA(11) =41
poolA(12) =10
poolA(13) =49
poolA(14) =16
poolA(15) =65
poolA(16) =32
poolA(17) =92
poolA(18) =28
poolA(19) =98
 
Procedure.i IsValidBitString(x,pool,treated)
Protected c,i
For i=1 to pool
mask=1<<(i-1)
If mask&x:c+1:EndIf
Next
If c=treated :ProcedureReturn x
Else  :ProcedureReturn 0
EndIf
EndProcedure
 
treated=9
control=10
 
pool =treated+control
 
; actual Experimentally observed difference in means
 
For i=1 to Treated
sumTreated+poolA(i)
Next
For i=Treated+1 to Treated+Control
sumControl+poolA(i)
Next
 
actualmeanTreated=sumTreated /Treated
actualmeanControl=sumControl /Control
actualdiffInMeans=actualmeanTreated-actualmeanControl
 
; exhaust the possibilites
For x=1 to 1<<pool
 
; Valid? i.e. are there 9 "1's" ?
If IsValidBitString(x,pool,treated)
TotalComBinations+1:sumTreated=0:sumControl=0
 
; separate the groups
For i=pool to 1 Step -1
mask=1<<(i-1):idx=pool-i+1
If mask&x
sumTreated+poolA(idx)
Else
sumControl+poolA(idx)
EndIf
Next
 
meanTreated=sumTreated /Treated
meanControl=sumControl /Control
diffInMeans=meanTreated-meanControl
; gather the statistics
If (diffInMeans)<=(actualdiffInMeans)
diffLessOrEqual+1
Else
diffGreater+1
EndIf
 
EndIf
Next
; show our results
; cw(StrF(100*diffLessOrEqual/TotalComBinations,2)+" "+Str(diffLessOrEqual))
; cw(StrF(100*diffGreater /TotalComBinations,2)+" "+Str(diffGreater))
 
Debug StrF(100*diffLessOrEqual/TotalComBinations,2)+" "+Str(diffLessOrEqual)
Debug StrF(100*diffGreater /TotalComBinations,2)+" "+Str(diffGreater)
 
Output:
87.20 80551
12.80 11827


Python[edit]

Translation of: Tcl
from itertools import combinations as comb
 
def statistic(ab, a):
sumab, suma = sum(ab), sum(a)
return ( suma / len(a) -
(sumab -suma) / (len(ab) - len(a)) )
 
def permutationTest(a, b):
ab = a + b
Tobs = statistic(ab, a)
under = 0
for count, perm in enumerate(comb(ab, len(a)), 1):
if statistic(ab, perm) <= Tobs:
under += 1
return under * 100. / count
 
treatmentGroup = [85, 88, 75, 66, 25, 29, 83, 39, 97]
controlGroup = [68, 41, 10, 49, 16, 65, 32, 92, 28, 98]
under = permutationTest(treatmentGroup, controlGroup)
print("under=%.2f%%, over=%.2f%%" % (under, 100. - under))
Output:
under=89.11%, over=10.89%

The above solution does a different thing than the other solutions. I'm not really sure why. If you want to do the same thing as the other solutions, then this is the solution:

from itertools import combinations as comb
 
def permutationTest(a, b):
ab = a + b
Tobs = sum(a)
under = 0
for count, perm in enumerate(comb(ab, len(a)), 1):
if sum(perm) <= Tobs:
under += 1
return under * 100. / count
 
treatmentGroup = [85, 88, 75, 66, 25, 29, 83, 39, 97]
controlGroup = [68, 41, 10, 49, 16, 65, 32, 92, 28, 98]
under = permutationTest(treatmentGroup, controlGroup)
print("under=%.2f%%, over=%.2f%%" % (under, 100. - under))
Output:
under=87.20%, over=12.80%

R[edit]

permutation.test <- function(treatment, control) {
perms <- combinations(length(treatment)+length(control),
length(treatment),
c(treatment, control),
set=FALSE)
p <- mean(rowMeans(perms) <= mean(treatment))
c(under=p, over=(1-p))
}
> permutation.test(c(85, 88, 75, 66, 25, 29, 83, 39, 97),
+ c(68, 41, 10, 49, 16, 65, 32, 92, 28, 98))
under over
0.8719717 0.1280283


Racket[edit]

Translation of: Common Lisp
#lang racket/base
 
(define-syntax-rule (inc! x)
(set! x (add1 x)))
 
(define (permutation-test control-gr treatment-gr)
(let ([both-gr (append control-gr treatment-gr)]
[threshold (apply + control-gr)]
[more 0]
[leq 0])
(let loop ([data both-gr] [sum 0] [needed (length control-gr)] [available (length both-gr)])
(cond [(zero? needed) (if (>= sum threshold)
(inc! more)
(inc! leq))]
[(>= available needed) (loop (cdr data) sum needed (sub1 available))
(loop (cdr data) (+ sum (car data)) (sub1 needed) (sub1 available))]
[else (void)]))
(values more leq)))
 
(let-values ([(more leq) (permutation-test '(68 41 10 49 16 65 32 92 28 98)
'(85 88 75 66 25 29 83 39 97))])
(let ([sum (+ more leq)])
(printf "<=: ~a ~a%~n>: ~a ~a%~n"
more (real->decimal-string (* 100. (/ more sum)) 2)
leq (real->decimal-string (* 100. (/ leq sum)) 2))))
 
Output:
<=: 80551 87.20%
>:  11827 12.80%

REXX[edit]

This REXX program is modeled after the   C   version, with some generalizations and optimization added.

/*REXX program  performs a    permutation test   on     N + M   subjects  (volunteers): */
/* ↑ ↑ */
/* │ │ */
/* │ └─────control population. */
/* └────────treatment population. */
n=9
data= 85 88 75 66 25 29 83 39 97 68 41 10 49 16 65 32 92 28 98
w=words(data); m=w-n
say 'w=' w
say 'volunteer population given treatment:' right(n, length(w) )
say ' control population given a placebo:' right(m, length(w) )
say
say 'treatment population efficacy % (percentages):' subword(data, 1, n)
say ' control population placebo  % (percentages):' subword(data, n+1 )
say
do v= 0 for w  ; #.v=word(data, v+1) ; end
treat=0; do i= 0 to n-1  ; treat=treat + #.i  ; end
tot=1; do j= w to m+1 by -1 ; tot=tot * j  ; end
do k=w%2 to 1 by -1 ; tot=tot / k  ; end
 
GT=picker(n+m, n, 0) /*compute the GT value from PICKER func*/
LE=tot - GT /* " " LE " via subtraction.*/
say "<= " format(100 * LE / tot, ,3)'%' LE /*display number with 3 decimal places.*/
say " > " format(100 * GT / tot, ,3)'%' GT /* " " " " " " */
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
picker: procedure expose #. treat; parse arg it,rest,eff /*get args.*/
if rest==0 then return eff > treat /*Zero ? */
if it>rest then q=picker(it-1, rest, eff) /*recurse. */
else q=0
itP=it - 1 /*temp var.*/
return picker(itP, rest - 1, eff+#.itP) + q /*recurse. */
output   when using the default input:
volunteer population given treatment:  9
 control  population given a placebo: 10

treatment population efficacy % (percentages): 85 88 75 66 25 29 83 39 97
 control  population placebo  % (percentages): 68 41 10 49 16 65 32 92 28 98

<=  87.197% 80551
 >  12.803% 11827

Ruby[edit]

Translation of: Python
def statistic(ab, a)
sumab, suma = ab.inject(:+).to_f, a.inject(:+).to_f
suma / a.size - (sumab - suma) / (ab.size - a.size)
end
 
def permutationTest(a, b)
ab = a + b
tobs = statistic(ab, a)
under = count = 0
ab.combination(a.size) do |perm|
under += 1 if statistic(ab, perm) <= tobs
count += 1
end
under * 100.0 / count
end
 
treatmentGroup = [85, 88, 75, 66, 25, 29, 83, 39, 97]
controlGroup = [68, 41, 10, 49, 16, 65, 32, 92, 28, 98]
under = permutationTest(treatmentGroup, controlGroup)
puts "under=%.2f%%, over=%.2f%%" % [under, 100 - under]
Output:
under=87.20%, over=12.80%

Seed7[edit]

$ include "seed7_05.s7i";
include "float.s7i";
 
const array integer: treatmentGroup is [] (85, 88, 75, 66, 25, 29, 83, 39, 97);
const array integer: controlGroup is [] (68, 41, 10, 49, 16, 65, 32, 92, 28, 98);
const array integer: both is treatmentGroup & controlGroup;
 
const func integer: pick (in integer: at, in integer: remain, in integer: accu, in integer: treat) is func
result
var integer: picked is 0;
begin
if remain = 0 then
picked := ord(accu > treat);
else
picked := pick(at - 1, remain - 1, accu + both[at], treat);
if at > remain then
picked +:= pick(at - 1, remain, accu, treat);
end if;
end if;
end func;
 
const proc: main is func
local
var integer: experimentalResult is 0;
var integer: treat is 0;
var integer: total is 1;
var integer: le is 0;
var integer: gt is 0;
var integer: i is 0;
begin
for experimentalResult range treatmentGroup do
treat +:= experimentalResult;
end for;
total := 19 ! 10; # Binomial coefficient
gt := pick(19, 9, 0, treat);
le := total - gt;
writeln("<= : " <& 100.0 * flt(le) / flt(total) digits 6 <& "% " <& le);
writeln(" > : " <& 100.0 * flt(gt) / flt(total) digits 6 <& "% " <& gt);
end func;
Output:
<= : 87.197168%  80551
 > : 12.802832%  11827

Sidef[edit]

Translation of: Ruby
func statistic(ab, a) {
var(sumab, suma) = (ab.sum, a.sum)
suma/a.size - ((sumab-suma) / (ab.size-a.size))
}
 
func permutationTest(a, b) {
var ab = (a + b)
var tobs = statistic(ab, a)
var under = (var count = 0)
ab.combinations(a.len, {|*perm|
statistic(ab, perm) <= tobs && (under += 1)
count += 1
})
under * 100 / count
}
 
var treatmentGroup = [85, 88, 75, 66, 25, 29, 83, 39, 97]
var controlGroup = [68, 41, 10, 49, 16, 65, 32, 92, 28, 98]
var under = permutationTest(treatmentGroup, controlGroup)
say ("under=%.2f%%, over=%.2f%%" % (under, 100 - under))
Output:
under=87.20%, over=12.80%

Tcl[edit]

package require Tcl 8.5
 
# Difference of means; note that the first list must be the concatenation of
# the two lists (because this is cheaper to work with).
proc statistic {AB A} {
set sumAB [tcl::mathop::+ {*}$AB]
set sumA [tcl::mathop::+ {*}$A]
expr {
$sumA / double([llength $A]) -
($sumAB - $sumA) / double([llength $AB] - [llength $A])
}
}
 
# Selects all k-sized combinations from a list.
proc selectCombinationsFrom {k l} {
if {$k == 0} {return {}} elseif {$k == [llength $l]} {return [list $l]}
set all {}
set n [expr {[llength $l] - [incr k -1]}]
for {set i 0} {$i < $n} {} {
set first [lindex $l $i]
incr i
if {$k == 0} {
lappend all $first
} else {
foreach s [selectCombinationsFrom $k [lrange $l $i end]] {
lappend all [list $first {*}$s]
}
}
}
return $all
}
 
# Compute the permutation test value and its complement.
proc permutationTest {A B} {
set whole [concat $A $B]
set Tobs [statistic $whole $A]
set undercount 0
set overcount 0
set count 0
foreach perm [selectCombinationsFrom [llength $A] $whole] {
set t [statistic $whole $perm]
incr count
if {$t <= $Tobs} {incr undercount} else {incr overcount}
}
set count [tcl::mathfunc::double $count]
list [expr {$overcount / $count}] [expr {$undercount / $count}]
}

Demonstration code:

set treatmentGroup {0.85 0.88 0.75 0.66 0.25 0.29 0.83 0.39 0.97}
set controlGroup {0.68 0.41 0.10 0.49 0.16 0.65 0.32 0.92 0.28 0.98}
lassign [permutationTest $treatmentGroup $controlGroup] over under
puts [format "under=%.2f%%, over=%.2f%%" [expr {$under*100}] [expr {$over*100}]]
Output:
under=86.90%, over=13.10%

Ursala[edit]

#import std
#import nat
#import flo
 
treatment_group = <85,88,75,66,25,29,83,39,97>
control_group = <68,41,10,49,16,65,32,92,28,98>
 
f = # returns the fractions of alternative mean differences above and below the actual
 
float~*; -+
vid^~G(plus,~&)+ (not [email protected])*|@htX; ~~ float+ length,
minus*+ mean^~*C/~& ^DrlrjXS(~&l,choices)^/-- [email protected]+-
 
#show+
 
t = --* *-'%'@lrNCC printf/$'%0.2f' times/$100. f(treatment_group,control_group)
Output:
12.80%
87.20%

zkl[edit]

A solution that is not going to scale gracefully at all.

Translation of: D
fcn permutationTest(a,b){
ab  := a.extend(b);
tObs  := a.sum(0);
combs := Utils.Helpers.pickNFrom(a.len(),ab); // 92,378
under := combs.reduce('wrap(sum,perm){ sum+(perm.sum(0) <= tObs) },0);
100.0 * under / combs.len();
}
 
treatmentGroup := T(85, 88, 75, 66, 25, 29, 83, 39, 97);
controlGroup  := T(68, 41, 10, 49, 16, 65, 32, 92, 28, 98);
under  := permutationTest(treatmentGroup, controlGroup);
println("Under =%6.2f%%\nOver =%6.2f%%".fmt(under, 100.0 - under));
Output:
Under = 87.20%
Over  = 12.80%