Permutation test: Difference between revisions
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Revision as of 21:44, 19 October 2018
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.
| 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
<lang Ada>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;</lang>
This solution uses an auxiliary package Iterate_Subsets. Here is the Spec: <lang Ada>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; </lang>
And here is the implementation:
<lang Ada>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;</lang>
- Output:
Less or Equal: 87.2 80551 More: 12.8 11827
BBC BASIC
<lang bbcbasic> 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</lang>
- Output:
Percentage groupings <= actual experiment: 87.1970296 Percentage groupings > actual experiment: 12.8029704
C
<lang C>#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;
}</lang> Output:<lang><= : 87.197168% 80551
> : 12.802832% 11827</lang>
C++
This is a translaion of C <lang cpp>#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;
}</lang> Output:<lang><= : 87.197168% 80551
> : 12.802832% 11827</lang>
Common Lisp
<lang lisp>(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))))</lang>output<lang><=: 80551 87.197%
>: 11827 12.803%</lang>
D
<lang d>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);
}</lang>
- Output:
Under = 87.20% Over = 12.80%
Alternative version:
<lang d>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);
}</lang>
- Output:
> : 12.80% 11827 <= : 87.20% 80551
Elixir
<lang elixir>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]</lang>
- Output:
under = 87.20%, over = 12.80%
GAP
<lang gap>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" ]</lang>
Go
A version doing all math in integers until computing final percentages. <lang go>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)
}</lang>
- Output:
differences <= observed: 87.197168% differences > observed: 12.802832%
Haskell
<lang haskell>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</lang>
- Output:
> : 12.80283184307952 <=: 87.19716815692048
Somewhat faster, this goes from top down: <lang haskell>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</lang>
In cases where the sample data are a large number of relatively small positive integers, counting number of partial sums is a lot faster: <lang haskell>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]</lang>
J
<lang j>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 ~.@,"1 i.@])&# ,) { ,) ctrl smoutput 'under: ','%',~":100*mean all <: result smoutput 'over: ','%',~":100*mean all > result</lang>
Result: <lang>under: 87.1972% over: 12.8028%</lang>
Java
<lang Java>public class PermutationTest {
private static final int[] data = new int[]{
85, 88, 75, 66, 25, 29, 83, 39, 97,
68, 41, 10, 49, 16, 65, 32, 92, 28, 98
};
private static int pick(int at, int remain, int accu, int treat) {
if (remain == 0) 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);
}
public static void main(String[] args) {
int treat = 0;
double total = 1.0;
for (int i = 0; i <= 8; ++i) {
treat += data[i];
}
for (int i = 19; i >= 11; --i) {
total *= i;
}
for (int i = 9; i >= 1; --i) {
total /= i;
}
int gt = pick(19, 9, 0, treat);
int le = (int) (total - gt);
System.out.printf("<= : %f%% %d\n", 100.0 * le / total, le);
System.out.printf(" > : %f%% %d\n", 100.0 * gt / total, gt);
}
}</lang>
- Output:
<= : 87.197168% 80551 > : 12.802832% 11827
jq
Part 1: Combinations <lang jq># 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;
</lang> Part 2: Permutation Test <lang jq># 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
- </lang>
Example: <lang jq>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)"</lang>
- Output:
$ jq -n -r -f permutation_test.jq % under=87.14304271579813 % over=12.856957284201869
Julia
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 <lang julia>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</lang>
Main <lang julia>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 $tot different permuted groups of these data.") @printf("%8d, %5.2f%% showed better than actual results.\n", better, 100 * better / tot) print(@sprintf("%8d, %5.2f%% showed equalivalent or worse results.", worse, 100 * worse / tot))</lang>
- Output:
Permutation test using the following data: Treated: [85, 88, 75, 66, 25, 29, 83, 39, 97] Control: [68, 41, 10, 49, 16, 65, 32, 92, 28, 98] There are 92378 different permuted groups of these data. 11827, 12.80% showed better than actual results. 80551, 87.20% showed equalivalent or worse results.
Kotlin
<lang scala>// version 1.1.2
val data = intArrayOf(
85, 88, 75, 66, 25, 29, 83, 39, 97, 68, 41, 10, 49, 16, 65, 32, 92, 28, 98
)
fun pick(at: Int, remain: Int, accu: Int, treat: Int): Int {
if (remain == 0) return if (accu > treat) 1 else 0
return pick(at - 1, remain - 1, accu + data[at - 1], treat) +
if (at > remain) pick(at - 1, remain, accu, treat) else 0
}
fun main(args: Array<String>) {
var treat = 0
var total = 1.0
for (i in 0..8) treat += data[i]
for (i in 19 downTo 11) total *= i
for (i in 9 downTo 1) total /= i
val gt = pick(19, 9, 0, treat)
val le = (total - gt).toInt()
System.out.printf("<= : %f%% %d\n", 100.0 * le / total, le)
System.out.printf(" > : %f%% %d\n", 100.0 * gt / total, gt)
}</lang>
- Output:
<= : 87.197168% 80551 > : 12.802832% 11827
M2000 Interpreter
<lang M2000 Interpreter> Module Checkit {
Global data(), treat=0
data()=(85, 88, 75, 66, 25, 29, 83, 39, 97,68, 41, 10, 49, 16, 65, 32, 92, 28, 98)
Function pick(at, remain, accu) {
If remain Else =If(accu>treat->1,0):Exit
=pick(at-1,remain-1,accu+data(at-1))+If(at>remain->pick(at-1, remain, accu),0)
}
total=1
For i=0 to 8 {treat+=data(i)}
For i=19 to 11 {total*=i}
For i=9 to 1 {total/=i}
gt=pick(19,9,0)
le=total-gt
Print Format$("<= : {0:1}% {1}", 100*le/total, le)
Print Format$(" > : {0:1}% {1}", 100*gt/total, gt)
} Checkit </lang>
- Output:
<= : 87.2% 80551 > : 12.8% 11827
Slower version, using a lambda function with a series of inner lambda functions to return each combination at a time.
<lang M2000 Interpreter> Module CheckThis {
Function CombinationsStep (a, nn) {
c1=lambda (&f, &a) ->{=car(a) : a=cdr(a) : f=len(a)=0}
m=len(a)
c=c1
n=m-nn+1
p=2
while m>n {
c1=lambda c2=c,n=p, z=(,) (&f, &m) ->{if len(z)=0 then z=cdr(m)
=cons(car(m),c2(&f, &z)):if f then z=(,) : m=cdr(m) : f=len(m)+len(z)<n
}
c=c1
p++
m--
}
=lambda c, a (&f) ->c(&f, &a)
}
treated=(85, 88, 75, 66, 25, 29, 83, 39, 97)
placebo=(68, 41, 10, 49, 16, 65, 32, 92, 28, 98)
treat=0
m=each(treated): while m {treat+=array(m)}
total=1
for i=len(placebo)+1 to len(placebo) +len(treated):total*=i:next i
for i=len(placebo)-1 to 1: total/=i:next i
d=total div 10**int(log(total))
k=false
StepA=CombinationsStep(cons(treated, placebo),len(treated))
counter=0
gt=0
While not k {
comb=StepA(&k)
accu=0
m=each(comb)
while m {accu+=array(m)}
gt+=if(accu>treat->1,0)
counter++
if counter mod d=0 then Print over str$(counter/total," #0.0%"): Refresh 1000
}
print over str$(counter/total," #0.0%")
print
lt=total-gt
print Format$("less or equal={0:1}%, greater={1:1}%, total={2}",lt/total*100, gt/total*100, total)
} CheckThis </lang>
Mathematica
<lang mathematica>"<=: " <> ToString[#1] <> " " <> ToString[100. #1/#2] <> "%\n >: " <>
ToString[#2 - #1] <> " " <> ToString[100. (1 - #1/#2)] <> "%" &[
Count[Total /@ Subsets[Join[#1, #2], {Length@#1}],
n_ /; n <= Total@#1],
Binomial[Length@#1 + Length@#2, Length@#1]] &[{85, 88, 75, 66, 25,
29, 83, 39, 97}, {68, 41, 10, 49, 16, 65, 32, 92, 28, 98}]</lang>
- Output:
<=: 80551 87.1972% >: 11827 12.8028%
Perl
<lang perl>#!/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 () {
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</lang>
- Output:
equal 313 0.339% less 80238 86.858% greater 11827 12.803%
Perl 6
The use of .race to allow concurrent calculations means that multiple 'workers' will be updating @trials simultaneously. To avoid race conditions, the ⚛++ operator is used, which guarantees safe updates without the use of locks. That is turn requires declaring that array as being composed of atomicint.
<lang perl6>sub stats ( @test, @all ) {
([+] @test / +@test) - ([+] 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 atomicint @trials[3] = 0, 0, 0;
@all.combinations(+@treated).race.map: { @trials[ 1 + ( stats( $_, @all ) <=> $base ) ]⚛++ }
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;</lang>
- Output:
Counts: <, =, > 80238 313 11827 Less than : %86.858343 Equal to : %0.338825 Greater than : %12.802832 Less or Equal: %87.197168
Phix
<lang Phix>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})</lang>
- Output:
<= : 87.197168% 80551 > : 12.802832% 11827
PicoLisp
<lang PicoLisp>(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)) "%") )</lang>
- Output:
under = 87.85%, over = 12.15%
PureBasic
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
<lang PureBasic>
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) </lang>
- Output:
87.20 80551 12.80 11827
Python
<lang python>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))</lang>
- 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: <lang python>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))</lang>
- Output:
under=87.20%, over=12.80%
R
<lang r>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))
}</lang>
<lang r>> 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 </lang>
Racket
<lang Racket>#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))))
</lang>
- Output:
<=: 80551 87.20% >: 11827 12.80%
REXX
This REXX program is modeled after the C version, with some generalizations and optimization added. <lang rexx>/*REXX program performs a permutation test on N + M subjects (volunteers): */
/* ↑ ↑ */
/* │ │ */
/* │ └─────control population. */
/* └────────treatment population. */
n= 9 /*define the number of the control pop.*/ 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 /*w: volunteers + control population*/ L= length(w) say '# of volunteers & control population: ' w say 'volunteer population given treatment: ' right(n, L) say ' control population given a placebo: ' right(m, L) 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 the arguments.*/
if rest==0 then return eff > treat /*is REST = to zero?*/
if it>rest then q= picker(it-1, rest, eff) /*maybe recurse. */
else q= 0
itP= it - 1 /*set temporary var.*/
return picker(itP, rest - 1, eff + #.itP) + q /*recurse. */</lang>
- output when using the default input:
# of volunteers & control population: 19 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
<lang ruby>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]</lang>
- Output:
under=87.20%, over=12.80%
Scala
Imperative version (Ugly, side effects)
- Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).
<lang Scala>object PermutationTest extends App {
private val data = Array(85, 88, 75, 66, 25, 29, 83, 39, 97, 68, 41, 10, 49, 16, 65, 32, 92, 28, 98) private var (total, treat) = (1.0, 0)
private def pick(at: Int, remain: Int, accu: Int, treat: Int): Int = {
if (remain == 0) return if (accu > treat) 1 else 0
pick(at - 1, remain - 1, accu + data(at - 1), treat) +
(if (at > remain) pick(at - 1, remain, accu, treat) else 0)
}
for (i <- 0 to 8) treat += data(i) for (j <- 19 to 11 by -1) total *= j for (g <- 9 to 1 by -1) total /= g
private val gt = pick(19, 9, 0, treat) private val le = (total - gt).toInt
println(f"<= : ${100.0 * le / total}%f%% ${le}%d")
println(f" > : ${100.0 * gt / total}%f%% ${gt}%d")
}</lang>
Seed7
<lang seed7>$ 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;</lang>
- Output:
<= : 87.197168% 80551 > : 12.802832% 11827
Sidef
<lang 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))</lang>
- Output:
under=87.20%, over=12.80%
Tcl
<lang tcl>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}]
}</lang> Demonstration code: <lang tcl>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 {$over*100}]]</lang>
- Output:
under=86.90%, over=13.10%
Ursala
<lang Ursala>#import std
- import nat
- import flo
treatment_group = <85,88,75,66,25,29,83,39,97> control_group = <68,41,10,49,16,65,32,92,28,98>
f = # returns the fractions of alternative mean differences above and below the actual
float~*; -+
vid^~G(plus,~&)+ (not fleq@rlX)*|@htX; ~~ float+ length, minus*+ mean^~*C/~& ^DrlrjXS(~&l,choices)^/-- length@l+-
- show+
t = --* *-'%'@lrNCC printf/$'%0.2f' times/$100. f(treatment_group,control_group)</lang>
- Output:
12.80% 87.20%
zkl
A solution that is not going to scale gracefully at all.
<lang zkl>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));</lang>
- Output:
Under = 87.20% Over = 12.80%