Non Parametric Part 1 Adobe

NON PARAMETRIC TESTS

1

REVIEW

2

PARAMETRIC & NON PARAMETRIC

**Non parametric tests can be applied to Normal data but parametric tests

have greater power IF assumptions met.

Wilcoxon Rank Sum test

Wilcoxon Matched-Pair Rank test

3

Non-parametric methods have fewer assumptions than parametric tests. So useful when these assumptions not met.

Nonparametric procedures are sometimes less powerful than tests designed for use with a specific distr ibution I t is better to use a test that is powerful when we believe that our assumptions are approximately satisfied than a less powerful test with fewer assumptions.

Often used when sample size is small and dif ficult to tel l i f normally distr ibuted.

Appropriate for dealing with data that are measured on a nominal or

ordinal scale and whose distr ibution is unknown.

Nonparametric tests use the order/rank of the values rather than the actual values themselves.

NON PARAMETRIC TESTS

4

A Wilcoxon Rank Sum Test is most often used for three types of study design as an alternative for the independent t -test:

Determining if there are differences between two independent groups

Determining if there are differences between interventions

Determining if there are differences in change scores

t -test relies on several conditions: independent samples, normality, and equal variances. We can use the Wilcoxon

Rank Sum Test as an alternative when the distribution is not

normal.

5

Assumption #1: The dependent variable should be measured at the ordinal or continuous level . The independent

variable should consist of two categorical, independent

groups.

Assumption #2: The underlying distribution is not normal.

Assumption #3: The observations are chosen randomly and independent.

Assumption #4: The distribution of scores for both groups of the independent variable have the same shape (identical).

WILCOXON RANK SUM TEST

6


To compare the running time of the first grade boys (B) and girls

(G), the following information is collected.

Is there a difference between the running time of boys and

girls?

Time (in secs) 20 24 29 33 57 35 10 17 23 19 22 21

Sex G G G G G G B B B B B B

7

Hypotheses are sometimes defines in terms of population medians , but can also be expressed in words.

H0: = (The median running time is not different between boys and girls.)

There is no difference in the running time of boys and girls

(i.e. the sum of ranks for group 1 is no different than the sum of ranks for

group 2).

Ha: (The median running time is different between boys and girls.)

There is a difference in the running time of boys and girls

(i.e. the sum of ranks for group 1 is significantly different from the sum of

ranks for group 2).

The Wilcoxon Rank Sum Test requires that the two tested samples be similar in shape (boxplot/histogram).

8

Rank all 12 observations. Arrange them in order from

smallest to largest.

The rank of each observation is its position in this ordered

list, starting with rank 1 from

the smallest observation.

Assign all tied values the average of the ranks they

occupy (if any).

Sex Time Rank

1

2

3

4

5

6

7

8

9

10

11

12 9

Rank all 12 observations. Arrange them in order from

smallest to largest.

The rank of each observation is its position in this ordered

list, starting with rank 1 from

the smallest observation.

Assign all tied values the average of the ranks they

occupy.

Sex Time Rank

B 10 1

B 17 2

B 19 3

G 20 4

B 21 5

B 22 6

B 23 7

G 24 8

G 29 9

G 33 10

G 35 11

G 57 12 10

Sex Sum of ranks

Boys

Girls

Calculate the sum of ranks per group

The Wilcoxon W statistic is the smaller of these two numbers:

(obtained value).

Critical value?

= 0.05, two tailed test

1 = 6 , 2 = 6 (1 is always smaller of the two groups)

From table D.8, critical value =

11

Sex Sum of ranks

Boys 24

Girls 54 Calculate the sum of ranks per group

The Wilcoxon W statistic is the smaller of these two numbers: 24

(obtained value).

Critical value?

= 0.05, two tailed test 1 = 6 , 2 = 6 (1 is always smaller of the two groups) From table D.8, critical value = 28

Reject or Do not reject H0 ?

(The obtained value needs to be equal to or less than the critical value in

order to be statistically significant).

12

There is enough evidence to conclude that there is a difference in the running time of boys and girls (The ranks in the two groups differed) where 1 = 6, 2 = 6 = 24, = .015.

13

A study of early childhood education asked nursery school

pupils to retell a fairy tale that had been read to them earlier in

the week. There were 5 high-progress readers and 5 low -

progress readers. An expert listened to a recording of the

children and assigned a score for certain uses of languages.

High 0.55 0.57 0.72 0.70 0.84

Low 0.40 0.72 0.00 0.36 0.55

To what extent the scores of the high progress group are

significantly higher than those of the low progress group?


14

Can be used as an alternative to the paired -samples t -test (dependent t -test) when the population cannot be assumed to be normally distributed.

Assumption #1: The dependent variable should be measured at the ordinal or continuous level. The independent variable should consist of two categorical, "related groups" or "matched pairs.

Assumption #2: The data are paired and the differences come from the same population.

Assumption #3: Each pair is chosen randomly and independent .

Assumption #4: The data need not be normally distributed, but the distribution should be symmetric around the median.

WILCOXON MATCHED-PAIRS SIGNED RANK

TEST

15

A group of 10 students with chronic

anxiety receive sessions of cognitive

therapy. Quality of Life (QoL) scores

are measured before and after therapy.

Is there a dif ference on the QoL scores

after the cognitive therapy?


TEST

H0: = 0 (The median QoL scores is not different before and after the cognitive therapy.

Ha: 0 (The median QoL scores is different before and after the cognitive therapy).

QoL scores

Before After

6 9

5 12

3 9

4 9

2 3

1 1

3 2

8 12

6 9

12 10

16

Before After Difference, d Sign Absolute value Rank

6 9

5 12

3 9

4 9

2 3

1 1

3 2

8 12

6 9

12 10

WILCOXON MATCHED-PAIRS SIGNED RANKS

TEST

The Wilcoxon W+ statistic is the sum of the ranks of the positive differences: ??

The Wilcoxon W- statistic is the sum of the ranks of the negative differences: ??

17

Before After Difference, d Sign Absolute value Rank

6 9 -3 - 3 4.5

5 12 -7 - 7 9

3 9 -6 - 6 8

4 9 -5 - 5 7

2 3 -1 - 1 1.5

1 1 0

3 2 1 + 1 1.5

8 12 -4 - 4 6

6 9 -3 - 3 4.5

12 10 2 + 2 3

WILCOXON MATCHED-PAIRS SIGNED RANKS

TEST

The Wilcoxon W+ statistic is the sum of the ranks of the positive differences: 1.5 + 3 = 4.5 .

The Wilcoxon W- statistic is the sum of the ranks of the negative differences: 1.5 + 4.5 + 4.5 + 6 + 7

+ 8 + 9 = 40.5

Whichever of these sums is the smaller, is our value of W. So, W = 4.5 (obtained value)

18

Critical value?

= 0.05 , two tailed test

n= 10-1 = 9 (omitting 0 dif ference)

From table D.7, critical value = 5.0195

Reject or Do not reject H 0 ?

(The obtained value needs to be equal to or less than the

critical value in order to be statistically significant).

WILCOXON MATCHED-PAIRS SIGNED

RANK TEST

19


TEST

The distribution was symmetric around the median. The Wilcoxon matched-pairs

signed rank test determined that there was a statistically significant median

different before and after the cognitive therapy (W=4.5, p=.033). 20

A study of early childhood education asked nursery school pupils to retell a fairy tale that had been read to them earlier in the week. Each child told two stories. The rest had been read to them, and the second had been read but also illustrated with pictures. There were 5 low-progress readers. An expert listened to a recording of the children and assigned a score for certain uses of languages.

Are the story 2 scores significantly higher than the story 1 scores?


TEST

Child 1 2 3 4 5

Story 2 0.77 0.49 0.66 0.28 0.38

Story 1 0.40 0.72 0.00 0.36 0.55

21

generally considered the nonparametric alternative to the one -way ANOVA, which can be used when the data fail the assumptions of the one-way ANOVA.

Assumption #1: The dependent variable is measured at the continuous or ordinal level and the independent variable consists of two or more categorical , independent groups.

Assumption #2: You should have independence of observations, which means that there is no relationship between the observations in each group of the independent variable or between the groups themselves .

Assumption #3: The distribution of scores for each group of the independent variable have the same shape. (N.B., having the same shape also means having the same variability ).

KRUSKAL-WALLIS TEST

22

The Kruskal-Wallis hypothesis test:

H0: All k populations have the same distribution.

Ha: Not all k populations have the same distribution.

The Kruskal-Wallis test statistic:

where

n=sum of sample sizes

k=number of samples

R j=sum of ranks in the jth sample

n j=size of the jth sample

If each n j > 5, then H is approximately distributed as a 2 (Chi-

square distribution).

KRUSKAL-WALLIS TEST

23

Do dif ferent departments have dif ferent class sizes?

KRUSKAL-WALLIS TEST

Class size (Math) Class size (English) Class size (History)

23 55 30

41 60 40

54 72 18

78 45 34

66 70 44

HO: MedianM = MedianE = MedianH

(The three departments have the same median class size)

Ha: Not all of the three department medians are equal.

24

KRUSKAL-WALLIS TEST

Group Class size Rank

M 23

M 41

M 54

M 78

M 66

E 55

E 60

E 72

E 45

E 70

H 30

H 40

H 18

H 34

H 44

Sum of group rank

for M =?

Sum of group rank

for E =?

Sum of group rank

for H=?

25

KRUSKAL-WALLIS TEST


M 23 2

M 41 6

M 54 9

M 78 15

M 66 12

E 55 10

E 60 11

E 72 14

E 45 8

E 70 13

H 30 3

H 40 5

H 18 1

H 34 4

H 44 7

Sum of group rank

for M =?

Sum of group rank

for E =?

Sum of group rank

for H=?

26

KRUSKAL-WALLIS TEST


M 23 2

M 41 6

M 54 9

M 78 15

M 66 12

E 55 10

E 60 11

E 72 14

E 45 8

E 70 13

H 30 3

H 40 5

H 18 1

H 34 4

H 44 7

Sum of group rank

for M =44

Sum of group rank

for E =56

Sum of group rank

for H =20

=12

15(15+1)

442

5+

562

5+

202

5 3 15 + 1 = 6.72

27

H=6.72 (Obtained value)

Critical value=?

=0.05, two tailed test

df=k-1=2

From chi-square dist. Table A.4, the critical value = 5.992

Reject or Do not reject H 0 ? (H is statistically significant if it is

larger than the critical value of Chi-Square)

If the null hypothesis in the Kruskal-Wallis test is rejected, then

we may wish, in addition, compare each pair of populations to

determine which are dif ferent and which are the same.

KRUSKAL-WALLIS TEST

28

KRUSKAL-WALLIS TEST

KW

2

1,KW

C>D ifReject

11

12

)1()(C

:scomparison paired for thepoint critical The

. and population

from nsobservatio theof ranks theofmean theis and where

:statistic test comparison pairwise The

ji

k

ji

ji

nn

nn

ji

RR

RRD

Further Analysis (Pairwise Comparisons of Average Ranks)

29

KRUSKAL-WALLIS TEST

= 2

0.05,2

15(15 + 1)

12

1

5+

1

5

= 5.992 (20)(2

5)

= 47.936 = 6.925

Reject if D > CKW

=44

5= 8.8

=56

5= 11.2

=20

5= 4

, = 8.8 11.20 = 2.4 , = 11.20 4 = 7.2 , = 8.8 4 = 4.8

Pairwise Comparisons of Average Ranks

30

KRUSKAL-WALLIS TEST

Distributions of class size were similar for all groups, as assessed by visual inspection of a

boxplot. The medians of class size were statistically significantly different between the Math,

History and English departments , 2 (2) = 6.72, p = .035. The post hoc analysis revealed statistically significant differences in the class size between the History and English (p =.033)

departments but not between the other group combinations.

31

Documents

Non Parametric Part 1 Adobe