Permutation TestsHal Whitehead

BIOL4062/5062

• Introduction to permutation tests

• Exact and randomized permutation tests

• Permutation tests using standard statistics

• Mantel tests

• ANOSIM

Permutation TestsAllow hypotheses to be tested when:

• Distributional properties of test statistic under null hypothesis are not known– e.g. measures of genetic distance

• Distributional properties of test statistic under null hypothesis are complex

• Assumptions about data necessary for standard tests or measure of uncertainty (e.g. normality) are not met

• Good for small data sets

Permutation Tests

• Useful when hypotheses can be phrased in terms of order or allocation of data points:

• e.g. When dogs meet, larger dog barks for longer

• e.g. Social relationships are stronger within same sex pairs

Exact and Random Permutation Tests

• Data => Real Test Statistic

• Either:

• Compute statistic for all possible permutations of data (“Exact test”)

• Or:

• Compute statistic for, say, 1,000 random permutations (“Random test”)

Permutation Tests

• Exact test

• Compare real test statistic with distribution of values of all other possible test statistics

• Random test

• Compare real test statistic with distribution of values of random test statistics

Permutation Tests If:

• real statistic is greater than or equal to 3/128 possible statistics (exact test):– reject null hypothesis that allocation or ordering of units does not

affect statistic:• P=0.023 (1-tailed test)

• P=0.046 (2-tailed test)

• real statistic is greater than or equal to 12/1000 random statistics (random test):– reject null hypothesis that allocation or ordering of units does not

affect statistic:• P=0.012 (1-tailed test)

• P=0.024 (2-tailed test)

Example: dogs• Null hypothesis:

– Longer barking unrelated to size ordering

• Alternative hypothesis:– Larger dog barks longer

• Data– 7 dogs: A > B > C > D > E > F > G

Pair of dogs Who barks longer?AB A, A, B, A

AF F, A, A

BD B, D, D

CF C, C, C

EG G

EF F

Test statistic:No. times larger dog barks longer

Y= 9

Example: dogs– 7 dogs: A > B > C > D > E > F > G

Pair of dogs Who barks longer?AB A, A, B, A

AF F, A, A

BD B, D, D

CF C, C, C

EG G

EF F

Test statistic:No. times larger dog barks longer

Y= 9

RANDOM: G > B > A > C > F > D > EPair of dogs Who barks longer?

AB A, A, B, AAF F, A, ABD B, D, DCF C, C, CEG GEF F

Random statistic:No. times larger dog barks longer

Y= 8

Example: dogs

• No. times larger dog barks longer: Y= 9

• In 5040 exact permutations:– Y>9 1635 times– do not reject null hypothesis (P=0.324)

• In 1000 random permutations:– Y>9 332 times– do not reject null hypothesis (P=0.332)

Can use permutation tests with normal test statistics when assumptions are not valid

Example: contingency table with small sample sizes

A B C D E F

I 0 0 1 2 2 0

II 0 0 4 3 0 0

III 0 0 0 1 0 1

IV 3 1 2 0 1 0

G=25.18 df=15 P=0.047But expected numbers are too small for valid G-test

Random permutation (totals same)

1 0 2 1 1 0

1 0 3 1 2 0

0 0 1 0 0 1

1 1 1 4 1 0

G(r)=15.82

For 10,000 random permutations: G>G(r) in 304; P=0.0304

Comparing Association Matrices:Mantel Test

• May help with problems of independence

• 2 association matrices, indexed by same units:– Evolution: genetic similarity and environmental

similarity between populations

– Behaviour: gender similarity (1/0) and association index between individuals

– Population genetics: genetic similarity and geographic distance between populations

Comparing Association Matrices:Mantel Test

• Matrices can be 0:1's

• Matrix correlation coefficient: similarity between the two association matrices

• Mantel test tests the null hypothesis that there is no relationship between the associations shown on the two matrices

Mantel TestsGiven two symmetric association matrices:

a11 a12 a13 .... a1k b11 b12

b13 .... b1k

a21 a22 a23 .... a2k b21 b22

b23 .... b2k

a31 a32 a33 .... a3k b31 b32

b33 .... b3k

... ...

ak1 ak2 ak3 .... akk bk1 bk2

bk3 .... bkk

Matrix correlation coefficient (r) is the correlation between:{a21, a31, a32, ... , ak1, ak2, ak3,.... , akk-1}, and{b21, b31,b32, ... , bk1, bk2, bk3,.... , bkk-1}

[Cannot be tested using standard methods because of lackof independence]

r=1 : maximal positive relationshipr=0 : no relationshipr=-1 : maximal negative relationship

Partial Mantel Tests

• Are X and Y related, controlling for V?

• Among populations of an organism– Is genetic similarity related to morphological

similarity controlling for geographical distance?

Mantel Tests Mantel test uses statistic:

k k

Z = Σ Σ aij . bij

i=1 j=1

Z can be transformed into a variable W, approximately normal (0 mean and s.d.

1) under the null hypothesis (r=0)

Somewhat dubious at small k

Mantel Tests

• Better to:– randomly permute the individuals in one matrix

many times

– each time calculate Z (Zm’s)

• Compare real Z with Zm’s

• If Z>97.5% of the Zm’s, or Z<97.5% of Zm’s, then the null hypothesis that r=0 is rejected– there is a relationship between variables

Mantel test: exampleDo bottlenose whales associate with their kin?

• 14 whales• Microsatellite-based

estimate of kin relatedness versus association index:– Matrix correlation r =

-0.09– Mantel test P = 0.83 (1,000

perms)

• They do not seem to preferentially associate with their kin

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Association index: propn of time together

Kin

ship

Mantel Test: ExampleCoda repertoire of sperm whales

Repertoire similarity R1 R2 R3 R4 R5 R6 R7 R8

R1 1 0.62 0.34 0.03 -0.06 0 0.13 -0.06

R2 0.62 1 0.71 0.11 -0.18 0.13 0.11 0.13

R3 0.34 0.71 1 0.15 -0.07 0.37 0.13 0.02

R4 0.03 0.11 0.15 1 0.52 0.44 0.71 0.71

R5 -0.06 -0.18 -0.07 0.52 1 0.32 0.62 0.52

R6 0 0.13 0.37 0.44 0.32 1 0.33 0.34

R7 0.13 0.11 0.13 0.71 0.62 0.33 1 0.67

R8 -0.06 0.13 0.02 0.71 0.52 0.34 0.67 1

Groups:

Group similarity R1 R2 R3 R4 R5 R6 R7 R8R1 1 1 1 0 0 0 0 0 R2 1 1 1 0 0 0 0 0 R3 1 1 1 0 0 0 0 0 R4 0 0 0 1 1 1 0 0 R5 0 0 0 1 1 1 0 0 R6 0 0 0 1 1 1 0 0 R7 0 0 0 0 0 0 1 1 R8 0 0 0 0 0 0 1 1

Mantel test:Group vs RepertoireP=0.00Groups seem to havedistinct repertoires

Mantel Test: ExampleCoda repertoire of sperm whales

Repertoire similarity R1 R2 R3 R4 R5 R6 R7 R8

R1 1 0.62 0.34 0.03 -0.06 0 0.13 -0.06

R2 0.62 1 0.71 0.11 -0.18 0.13 0.11 0.13

R3 0.34 0.71 1 0.15 -0.07 0.37 0.13 0.02

R4 0.03 0.11 0.15 1 0.52 0.44 0.71 0.71

R5 -0.06 -0.18 -0.07 0.52 1 0.32 0.62 0.52

R6 0 0.13 0.37 0.44 0.32 1 0.33 0.34

R7 0.13 0.11 0.13 0.71 0.62 0.33 1 0.67

R8 -0.06 0.13 0.02 0.71 0.52 0.34 0.67 1

Groups:

Group similarity R1 R2 R3 R4 R5 R6 R7 R8R1 1 1 1 0 0 0 0 0 R2 1 1 1 0 0 0 0 0 R3 1 1 1 0 0 0 0 0 R4 0 0 0 1 1 1 0 0 R5 0 0 0 1 1 1 0 0 R6 0 0 0 1 1 1 0 0 R7 0 0 0 0 0 0 1 1 R8 0 0 0 0 0 0 1 1

Partial Mantel test:Group vs Repertoirecontrolling for clanP=0.69Groups do not seem to have distinct repertoireswithin clans

Clans:

Clan similarity 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1

ANOSIMAnalysis of Similarities

(“R test”)• Version of ANOVA for similarity of

dissimilarity matrices– Similarity/dissimilarity matrix with units

grouped

• Closely related to Mantel test– In which one matrix indicates group

membership

• Programme PRIMER

Dissimilarity matrixwith groups of units

A B C D E

A 0

B 0.2 0

C 0.4 0.7 0

D 0.6 0.5 0.1 0

E 0.7 0.8 0.3 0.6 0

A B C D E

A 0

B 2 0

C 4 8.5 0

D 6.5 5 1 0

E 8.5 10 3 6.5 0

A B C D E

A 0

B 0.2 0

C 0.4 0.7 0

D 0.6 0.5 0.1 0

E 0.7 0.8 0.3 0.6 0

Ranks

A B C D E

A 0

B 2 0

C 4 8.5 0

D 6.5 5 1 0

E 8.5 10 3 6.5 0

Ranks

• Mean rank within groups rW= 3.125

• Mean rank between groups rB = 7.083

• ANOSIM statistic R = (rB – rW)/[n(n-1)/4]

– = 0.791

ANOSIM statistic

• ANOSIM statistic R = (rB – rW)/[n(n-1)/4]

• -1 < R < 1

• R = 0 if high and low ranks perfectly mixed between versus within groups

• R = 1 or -1 for maximal differences between groups

• But is R statistically different from 0?

Testing ANOSIM statistic

• Permute group assignations many times, and calculate R*’s

• Compare with real R

ANOSIM

• Can be done with more than 2 groups

• More complex designs– Two-way– Nested designs

• Can be done without ranking– Then absolute value of R has less meaning– Almost same as Mantel test

Issues with Permutation Tests

• Results of permutation test strictly refer to only the data set not the wider population– unless sampled at random

• How many permutations?– Depends on test and p-value– Tradeoff between accuracy and computer time– Usually 100-10,000 permutations

Permutation Tests• Allow hypotheses to be tested when:

– Distributional properties unknown

– Distributional properties of test statistic complex

– Usual assumptions not met (need independence)

• Good for small data sets• Can check analytically-based tests• Mantel tests compare two or more association matrices

– may help deal with independence issues

• ANOSIM (or Mantel tests) can do ANOVA-like analyses of similarity or dissimilarity matrices