2. Key Terms
• Power of a test refers to the probability of rejecting a false
null hypothesis (or detect a relationship when it exists)
• Power Efficiency the power of the test relative to that of its
most powerful alternative. For example, if the power
efficiency of a certain nonparametric test for difference of
means with sample size 10 is 0.9, it means that if interval
scale and the normality assumptions can be made (more
powerful), we can use the t-test with a sample size of 9 to
achieve the same power.
3. Choice of nonparametric test
• It depends on the level of measurement obtained (nominal, ordinal, or
interval), the power of the test, whether samples are related or
independent, number of samples, availability of software support (e.g.
SPSS)
• Related samples are usually referred to match-pair (using randomization)
samples or before-after samples.
• Other cases are usually treated as independent samples. For instance, in
a survey using random sampling, we have a sub-sample of males and a
sub-sample of females. They can be considered as independent samples
as they are all randomly selected.
4. Non-parametric Tests
Sign Test paired data
2 independent
Mann-Whitney U Test
samples
> 2 independent
Kruskal-Wallis Test
samples
5. Sign Test
• Used for paired data
– Can be ordinal or continuous
• Very simple and easy to interpret
• Makes no assumptions about distribution of
the data
• Not very powerful
6. Sign Test: null hypothesis
• The null hypothesis for the sign test is
H0: the median difference is zero
• To evaluate H0 we only need to know the signs
of the differences
– If half the differences are positive and half are
negative, then the median = 0 (H0 is true).
– If the signs are more unbalanced, then that is
evidence against H0.
7. child
Rating
before
Rating
after change sign Example:
Body image data
1 1 5 4 +
2 1 4 3 +
3 3 1 -2 -
4 2 3 1 + • The sign test looks at the
5 4 4 0 0 signs of the differences
6 1 4 3 +
7 3 5 2 +
– 15 children felt better
8 1 5 4 + about their teeth (+
9 1 4 3 + difference in ratings)
10 4 4 0 0
11 1 1 0 0
– 1 child felt worse (- diff.)
12 1 4 3 + – 4 children felt the same
13 1 4 3 + (difference = 0)
14 2 4 2 +
15 1 4 3 + • Looks like good evidence
16 2 5 3 +
17 1 4 3 +
18 1 5 4 +
19 4 4 0 0
20 3 5 2 +
8. Mann-Whitney U test
• The Mann-Whitney U test, also called the rank sum
test, is a non-parametric test that compares two
independent (unmatched) groups.
• This means that either the data are at the ordinal
level or data are at the interval/ratio level but not
normally distributed.
• The test statistic is the U statistic. This is the test that
you use if you cannot fulfill the assumptions of the t-
test.
9. Mann-Whitney U test
• Assumption of normality or equality of variance is
not met.
• Like many non-parametric tests, uses the ranks of
the data rather than their raw values to calculate the
statistic.
• Since this test does not make a distribution
assumption, it is not as powerful as the t-test.
10. Mann-Whitney U test
The hypotheses for the comparison of two
independent groups are:
• Ho: The two samples come from identical
populations / the sum of the ranks is similar
• Ha: The two samples come from different
populations / the sum of the ranks is different
11. Procedure for Mann-Whitney U-Test
1. Choose Mann-Whitney Test
2. Hypotheses Null and hypotheses Alternative
3. Assign ranks to all the scores in the
experiment.
4. Compute the sum of the ranks for each
group.
5. Compute the two version of the Mann-
Whitney U. Fist compute U1 for Group 1
using the formula:
U1 = (n1)(n2) + n1 (n1 + 1) - Σ R1
2
12. Procedure for Mann-Whitney U-Test
Next compute U2 for Group 2 using the formula:
U2 = (n1)(n2) + n2 (n2 + 1) - Σ R2
2
4. Determine the Mann_Whitney Uobt.
5. Find the critical value
6.Compare Uobt to Ucritt .
7. Intrepret, and make decision.
8. Draw conclusion
13.
14.
15. Solution:
• H0: There is no difference in the test scores for
the two type of classes.
• H1: There is a difference in test score for the
two type of classes (claim)
19. Solution:
To check your computation of U:
U1 + U2 = n1.n2 197 +37 = (18)(13)
234 = 234
It checks out, and because U is the smaller of U1 and
U2, U = 37
Critical value: Using n1 = 18 and n2 = 13, at a = 0.05,
the critical value is 67.
Reject null hypothesis , since Uobt less than Ucrit
There is a difference between the two classes on the
algebra readiness test.
20. The Kruskal-Wallis H Test
• The Kruskal-Wallis H Test is a nonparametric
procedure that can be used to compare more
than two populations in a completely
randomized design.
• All n = n1+n2+…+nk measurements are jointly
ranked (i.e.treat as one large sample).
• We use the sums of the ranks of the k samples
to compare the distributions.
21. The Kruskal-Wallis H Test
• Rank the total measurements in all k samples from 1
to n. Tied observations are assigned average of the
ranks they would have gotten if not tied.
• Calculate Ti = rank sum for the ith sample i = 1, 2,
…,k
• And the test statistic
12 Ti 2
H= ∑ − 3(n + 1)
n(n + 1) ni
22. The Kruskal-Wallis H Test
H0: the k distributions are identical versus
Ha: at least one distribution is different
Test statistic: Kruskal-Wallis H
When H0 is true, the test statistic H has an
approximate chi-square distribution with df = k-1.
Use a right-tailed rejection region or p-value based
on the Chi-square distribution.
23. Spearman’s Rank Correlation
• Spearman's Rank Correlation is a technique
used to test the direction and strength of the
relationship between two variables.
• In other words, its a device to show whether
any one set of numbers has an effect on
another set of numbers
• It uses the statistic Rs which falls between -1
and +1
24. Procedure for using
Spearman's Rank Correlation
• State the null hypothesis i.e. "There is no
relationship between the two sets of data."
• Rank both sets of data from the highest to the
lowest.
• Make sure to check for tied ranks.
• Subtract the two sets of ranks to get the difference
d.
• Square the values of d.
• Add the squared values of d to get Sigma d2.
25. Procedure for using
Spearman's Rank Correlation
• Use the formula Rs = 1-(6Ʃd2/n3-n) where n is the number of
ranks you have.
• If the Rs value...
– ... is -1, there is a perfect negative correlation.
– ...falls between -1 and -0.5, there is a strong negative correlation.
– ...falls between -0.5 and 0, there is a weak negative correlation.
– ... is 0, there is no correlation
– ...falls between 0 and 0.5, there is a weak positive correlation.
– ...falls between 0.5 and 1, there is a strong positive correlation
– ...is 1, there is a perfect positive correlation
between the 2 sets of data. If the Rs value is 0, state that null
hypothesis is accepted. Otherwise, say it is rejected.
26. Run Test for Randomness
• Run test is used for examining whether or not a set
of observations constitutes a random sample from
an infinite population
• A run is defined as a series of increasing values or a
series of decreasing values.
• For example, the males and females in a line can
have patterns such as M F M F M F M F and M M M
M F F F F, which have 8 and 2 runs, respectively
27. Run Test for Randomness
• Hypothesis: To test the run test of
randomness, first set up the null and
alternative hypothesis.
• In run test of randomness, null hypothesis
assumes that the distribution of the sample is
random. The alternative hypothesis will be the
opposite of the null hypothesis.
28. Run Test for Randomness
• The second step is the calculation of the mean and
variance.
• Where N= Total number of observations =N1+N2
N1=Number of + symbols
N2=Number of – symbols
R= number of runs
• If Rc (lower) <= R<= Rc (Upper), accept Ho. Otherwise
reject Ho
29. Cox-Stuart Test for Trend
• This test is useful for detecting positively or
negatively sloping gradual trends in a sequence of
independents on a single random variable
• One of the three alternative hypotheses are possible
– An upward or downward trend exists
– An upward trend exists or
– A downward trend exists
• If the null hypothesis is accepted, the result
indicates that the measurements within the
ordered sequence are identically distributed
30. Kolmogorov-Smirnov Test
• In situations where there is unequal number of
observations in two samples, K-S test is appropriate
• This test is used to test whether there is any
significant difference between two treatments A and
B
• The test Hypothesis is
– Ho: No difference in the effect of treatments A and B
– H1: There is some difference in the effect of treatments A
and B
