Biostatistics, statistical software VII. Non-parametric tests: Wilcoxon’s signed rank test, Mann-Whitney U-test, Kruskal- Wallis test, Spearman’ rank correlation

Biostatistics, statistical software VII.

Non-parametric tests: Wilcoxon’s signed rank test, Mann-Whitney U-test, Kruskal-Wallis test, Spearman’ rank correlation.

Krisztina Boda PhD

Department of Medical Informatics, University of Szeged

INTERREG 2Krisztina Boda

Parametric tests

Parameter: a parameter is a number characterizing an aspect of a population (such as the mean of some variable for the population), or that characterizes a theoretical distribution shape.

Usually, population parameters cannot be known exactly; in many cases we make assumptions about them.


Parameters of the normal distribution: , Parameter of the binomial distribution: n, p Parameter of the Poisson distribution:


Normal distributions N(, )

Probability Density Function

y=normal(x;1;1)

-3 -2 -1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

Probability Density Function

y=normal(x;0;2)

-3 -2 -1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

N(0,1) N(1,1)

N(0,2)

, : parameters (a parameter is a number that describes the distribution)


Binomial distributions

1. Each trial results in one of two possible, mutually exclusive outcome. (success, failure)

2. The probability of a success, p, remains constant from trial to trial 3. The trials are independent. We are interested in being able to compute the probability of k successes in n

trials. The binomial distribution is useful for describing distributions of binomial events,

such as the number of males and females in a random sample of companies, or the number of defective components in samples of 20 units taken from a production process. The binomial distribution is defined as:

p is the probability that the respective event will occur q is equal to 1-p n is the maximum number of independent trials.

P P X kn

kp q k nkk n k

( ) , , ,...,0 1

n

k

n

k n kn n

!

!( )!, ! ...1 2


Example

Suppose that it is known that 30% of a certain population are immune to some disease. If a random sample of size n=10 is selected from this population, what is the probability that it will contain exactly k=4 immune persons?

0.2001210.1176490081.0210

7.03.0!6!4!10

7.03.04

10)4( 6464

XP


Number of SuccessProbabilty distribution Distribution function

Probabilty of "success"

0 0.028247525 0.028247525 0.31 0.121060821 0.1493083462 0.233474441 0.3827827863 0.266827932 0.6496107184 0.200120949 0.8497316675 0.102919345 0.9526510136 0.036756909 0.9894079227 0.009001692 0.9984096148 0.001446701 0.9998563149 0.000137781 0.999994095

10 5.9049E-06 1Összesen 1

Binomial distribution n=10, p can be changfed, k=0,1,…,10

Probabilty distribution

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11

Distribution function

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11


Poisson distribution

The Poisson distribution is also sometimes referred to as the distribution of rare events. Examples of Poisson distributed variables are number of accidents per person, number of sweepstakes won per person, or the number of catastrophic defects found in a production process.

If n tends to infinity, but at the same time np= is kept constant the binomial distribution approaches a fixed distribution

lim lim ( )!n

kn

k n kk

Pn

kp q f k

ke


Example. In a certain disease the number of new occurrences in a month is 3 in average. Assuming that the number of new occurrences follows a Poisson distribution, what is the probability that Nobody becomes ill (0.0498) There are exactly 2 new occurrences (0.224)

Number of events Probability Distribution functionAverage number of

events

0 0.049787068 0.049787068 31 0.149361205 0.1991482732 0.224041808 0.4231900813 0.224041808 0.6472318894 0.168031356 0.8152632455 0.100818813 0.9160820586 0.050409407 0.9664914657 0.021604031 0.9880954968 0.008101512 0.9961970089 0.002700504 0.998897512

10 0.000810151 0.999707663Total 0.999707663

Probability

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11

Distribution function

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11


Parametric tests

The null hypothesis contains a parameter of a distribution. The assumptions of the tests are that the samples are drawn from a normally distributed population.

One sample t-test: H0: =c, Two sample t-test: H0: 1=2,assumptions:

1=2


Nonparametric tests

We do not need to make specific assumptions about the distribution of data.

They can be used when The distribution is not normal The shape of the distribution is not evident Data are measured on an ordinal scale (low-

normal-high, passed – acceptable – good – very good)


Ranking data

Nonparametric tests can't use the estimations of population parameters. They use ranks instead. Instead of the original sample data we have to use its rank.

To show the ranking procedure suppose we have the following sample of measurements: 199, 126, 81, 68, 112, 112.

Sort the data in ascending order: 68, 81,112,112,126,199 Give ranks from 1 to n: 1, 2, 3, 4, 5, 6 Cases 5 and 6 are equal, they are assigned a rank of 3.5,

the average rank of 3 and 4. We say that case 5 and 6 are tied.

Ranks corrected for ties: 1, 2, 3.5, 3.5, 5, 6


Result of ranking data

Case Data Rank Ranks corrected for ties

4 68 1 1

3 81 2 2

5 112 3 3.5

6 112 4 3.5

2 126 5 5

1 199 6 6

The sum of all ranks must be

Using this formula we can check our computations. Now the sum of ranks is 21, and 6(7)/2=21.

rn n

ii

n

( )1

21


Nonparametric tests for paired data(nonparametric alternatives of paired t-test)

Sign test Wilcoxon’s matched pairs test Null hypothesis: the paired samples are

drawn from the same population


The sign test Example: 13 students

were measured in reading speed and comprehension at a course ending and after 1 month. Suppose we have reason to believe that the two distributions of reading scores are not normal.

Number of positive signs: 6

Number of negative signs: 5

Cases with no change are omitted

Student Score Score Difference Sign at course after

ending 1 month 1 50 52 -2 -2 48 51 -3 -3 46 46 04 50 49 1 +5 62 50 2 +6 80 70 10 +7 23 21 2 +8 30 33 -3 -9 45 46 -1 -10 53 53 011 49 48 1 +12 51 48 3 +13 46 48 -2 -


Table of the sign test

The table contains the acceptance region for given sample size and


Decision based on table

If the distributions of the two variables are the same (If the null hypothesis is true), the numbers of positive and negative differences should be similar.

The null hypothesis is accepted if both numbers lie in the interval given it table for the sign test

Number of positive signs: 6 Number of negative signs: 5 For n=11 and =0.05, this interval is 1-10. As both 5 and 6 lies in the interval 1-10, we accept the

null hypothesis at 5% level.


The Wilcoxon signed rank test Example: 13 students

were measured in reading speed and comprehension at a course ending and after 1 month. Suppose we have reason to believe that the two distributions of reading scores are not normal.

Sum of ranks belonging to positive signs: R+=40.5

Sum of ranks belonging to negative signs: R-=25.5

Cases with no change are omitted

Student Score Score Difference Rank at course after ignoring

ending 1 month signs1 50 52 -2 5.52 48 51 -3 93 46 46 04 50 49 1 25 62 50 2 5.56 80 70 10 117 23 21 2 8 30 33 -3 99 45 46 -1 210 53 53 011 49 48 1 212 51 48 3 913 46 48 -2 5.5


Table of the Wilcoxon signed rank test

The table contains the acceptance region for given sample size and



If the distributions of the two variables are the same (If the null hypothesis is true), the sum of positive and negative ranks should be similar.

The null hypothesis is accepted if both numbers lie in the interval given it table for the test

Sum of ranks belonging to positive signs: R+=40.5 Sum of ranks belonging to negative signs: R-=25.5 For n=11 and =0.05, this interval is 10-56. As both rank sums are in this interval, we do not reject the null

hypothesis and claim that the difference is not significant at 5% level.


The case of large samples

When the sample size is large, we can count the mean and standard deviation of the ranks and use the normal distribution to get the p-value. Computer packages use this normal approximation also in case of small sample size

)1,0(~)24/)12)(1((

4/)1(

1

2

1 Nnnn

nnR

R

RRz

n

ii

n

ii


Nonparametric test for data in independent groups(nonparametric alternatives of two sample t-test)

Mann-Whitney U test

Null hypothesis: the samples are drawn from the same population


Hypothetical example

The change of body weight are compared in two groups: patients having a special diet and control patients.

Null hypothesis: the diet is not effective, data are drawn from the same population.

The original data are ranked and the sum of ranks in each group is computed. If the null hypothesis is true, the sum of ranks in the two groups are similar.


Patient Change in body weight (kg)

Group Rank Rank corrected for ties

1. -1 Diet 3 3 2. 5 Diet 16 16.5 3. 3 Diet 12 13 4. 10 Diet 21 21 5. 6 Diet 18 19 6. 4 Diet 15 15 7. 0 Diet 4 5.5 8. 1 Diet 8 9 9. 6 Diet 19 19

10. 6 Diet 20 19 Sum of ranks, R1 140

11. 2 Control 11 11 12. 0 Control 5 5.5 13. 1 Control 9 9 14. 0 Control 6 5.5 15. 3 Control 13 13 16. 1 Control 10 9 17. 5 Control 17 16.5 18. 0 Control 7 5.5 19 -2 Control 1 1.5 20. -2 Control 2 1.5 21. 3 Control 14 13

Sum of ranks R2 91


Table of the Mann-Whitney U test



If the distributions of the two variables are the same (If the null hypothesis is true), the sum of ranks in the two groups should be similar.

The test statistic T is the sum of the ranks in the smaller group. The null hypothesis is accepted T lies in the interval given it table for the test

Sum of ranks in the first group (n=10): R1=140 Sum of ranks in the second group (n=11): R2=91 The test statistic T is the sum of the ranks in the smaller group. T=140. For n1=10 and n2=11 and =0.05, this interval is 81-139. As T lies outside of this interval, we reject the null hypothesis and

claim that the difference is significant at 5% level.


An alternative test statistic The statistic U (due to Mann Whitney) is the

number of all possible pairs of observations comprising one from each sample, say xi and yi , for which xi<yi. This if the sample sizes are n1 and n2, the U/n1n2 is the proportion of all such pairs, and so is also the estimated probability that a new observation from the first population will be less than a new observation sampled from the second population.

TnnnnU )1(2

11121


The case of large samples

When the sample size is large, T test statistic T has an approximately Normal distribution And we can calculate the test statistic z according to the following formula: (ns and nL are the sample sizes in the smaller and larger group respectively).

Computer packages use this normal approximation also in case of small sample size

)1,0(~

12

)1(

2/)1(N

nnnn

nnnRz

LsLs

Lsss


Comparing several independent groups: the Kruskal-Wallis test

It is also called nonparametric one-way ANOVA It tests whether k independent samples that are defined

by a grouping variable are from the same population. This test assumes that there is no a priori ordering of the

k populations from which the samples are drawn. As a result, it gives one p-value. If the null hypothesis is rejected, further tests are

required to make pairwise comparisons. These pairwise comparisons are generally not available in standard statistical packages. Pairwise comparisons can be performed by Mann Whitney U tests and p-values can be corrected by Bonferroni correction.


Comparison of several related samples: the Friedman test

The Friedman test is the nonparametric equivalent of a one-sample repeated measures design or a two-way analysis of variance with one observation per cell.

Friedman tests the null hypothesis that k related variables come from the same population. For each case, the k variables are ranked from 1 to k. The test statistic is based on these ranks.

As a result, it gives one p-value. If the null hypothesis is rejected, further tests are required to make

pairwise comparisons. These pairwise comparisons are generally not available in standard statistical packages. Pairwise comparisons can be performed by Wilxocon signed rank tests and p-values can be corrected by Bonferroni correction.


Review questions and exercises

Problems to be solved by hand-calculations ..\Handouts\Problems hand VII.doc

Solutions ..\Handouts\Problems hand VII solutions.doc

Problems to be solved using computer - none


Useful WEB pages

http://www-stat.stanford.edu/~naras/jsm http://www.ruf.rice.edu/~lane/rvls.html http://my.execpc.com/~helberg/statistics.html http://www.math.csusb.edu/faculty/stanton/m26

2/index.html

Documents

Biostatistics, statistical software VII. Non-parametric tests: Wilcoxon’s signed rank test, Mann-Whitney U-test, Kruskal- Wallis test, Spearman’ rank correlation