Transcript Lecture 8 - The Department of Statistics and Applied Probability, NUS

Non-parametric Tests
With histograms like these, there really isn’t a need to
perform the Shapiro-Wilk tests!
Data exploration and Statistical analysis
1. Data checking, identifying problems and characteristics
2. Understanding chance and uncertainty
3. How will the data for one attribute behave, in a
theoretical framework?
4. Theoretical framework assumes complete information,
need to address uncertainties in real data
5. Testing your beliefs, do the data support what you think
is true?
6. What happens when the assumptions of the theoretical
framework are not valid
Data
Data exploration,
categorical / numerical
outcomes
Model each outcome with
a theoretical distribution
Estimation of parameters,
quantifying uncertainty
Hypothesis testing
Parametric tests
(t-tests, ANOVA,
test of proportions)
Non-parametric tests
(Wilcoxon, KruskalWallis, rank test)
Non-parametric tests
Used when:
- assumptions on the distributions of the data are clearly not
valid;
- a small fraction of the data are considered outliers (either tail),
but are not removed as they are experimentally valid (related to
first reason above actually)
- occasionally when sample sizes are small and it becomes
meaningless looking at histograms or even using the ShapiroWilk tests
Non-parametric equivalent
For most parametric tests, there are non-parametric
equivalents:
Conceptual difference of non-parametric
tests
Parametric tests
Uses the actual observed values of the outcomes in calculating
the test statistics.
Non-parametric tests
Converts the observed values to ranks, and uses these ranks
for comparisons (to calculate the test statistics).
Test for one-sample
One-sample t-test compares the mean of the sample to the hypothesized
mean value under the null hypothesis.
For the non-parametric test: One-sample Wilcoxon Signed Rank test
It compares the median of the sample to the hypothesized median value
under the null hypothesis.
Null hypothesis
Alternative hypothesis
: Median = some value, m
: Median  m (two-sided hypothesis)
Median > m (one-sided hypothesis)
Median < m (one-sided hypothesis)
Sign test
Simple idea: Count the number of observations > m, out of a total number of
observations N. If the null hypothesis is true, then on average, we expect
about N / 2 observations to be greater than m, and about N / 2 observations
to be less than m.
So suppose x = number of observations > m. We can actually calculate the
Binomial probability of having at least x observations out of N, when the
probability that any observation will be > m is 50%.
Let X ~ Binomial(N, 0.5)
P(observation > m) = 0.5
P(at least x observations out of N > m) = Binomial probability of P(X  x)
However, this ignores the magnitude of the data, or the
distance of each value from m.
Wilcoxon Signed rank test
1. Calculate the difference between each value with m.
2. Note the sign of the difference, whether it is +ve or –ve difference.
3. Drop the sign, and rank the unsigned differences, from smallest
(assigned a rank of 1) to largest (assigned a rank of N).
4. Restore the sign to the assigned ranks.
5. Sum up the positive ranks (W+), and also the negative ranks (W-,
defined without the negative sign).
6. Conceptually, if the median of the data is genuinely closed to m, then
W+ will be similar to W-.
7. If W+ >> W-, then this implies that there is evidence that the median of
the data is > m.
8. If W- >> W+, then this implies that there is evidence that the median of
the data is < m.
Test for two independent samples
Mann-Whitney U test (a.k.a. Wilcoxon rank-sum test)
Again comparing the sum of ranks from two collection of data.
Null hypothesis
Alternative hypothesis
: Distributions of both groups are the same*
: Distributions of both groups are different
* Formally speaking: Probability of (a randomly chosen observation from
one group is > than a randomly chosen observation from the second group)
= 0.5.
Mann-Whitney U test
Conceptually
- Combine all the observations from the two groups into a single collection;
- Assign ranks to these observations, from smallest as rank 1 to largest as
rank N (where N represents the total number of observations when
combining both groups)
- Return the observations to the two groups they were originally from
- Sum up the ranks of the observations in each of the two groups.
- Conceptually expect the average ranks from both groups to be similar.
The details of the test are actually more complex, and the student is strongly
encouraged to find out more from the recommended textbooks (or online).
Test for paired samples
Sign test
Same as the sign test encountered during the 1-sample test. However, here
we are comparing the differences between the paired observations (which
eventually yields only one set of outcome, and thus a “1-sample” test)
Null hypothesis
Alternative hypothesis
: Probability that the difference is greater than 0 = 0.5
: Probability that the difference is greater than 0  0.5
Probability that the difference is greater than 0 > 0.5
Probability that the difference is greater than 0 < 0.5
Caveat: As before, the sign test only looks at the sign of the difference, but not
the magnitude of the difference. So there is actually additional information that
can be used.
Remember! Statistics is about understanding and minimizing
uncertainty, while trying to maximizing information! (or to
make use of as much data as possible)
Test for paired samples
Wilcoxon signed rank test
Again similar as the procedure introduced for 1-sample testing. But the focus
here is on the differences between the paired observations.
Null hypothesis
Alternative hypothesis
: Median of differences = 0
: Median of differences  0
Median of differences > 0
Median of differences < 0
Test for  2 independent samples
Kruskal-Wallis test
Compares the medians of all the groups to see whether they are equal.
Null hypothesis
Alternative hypothesis
: Medians of all the groups are identical
: At least one group has a different median
Compare this with the hypotheses for ANOVA:
Null hypothesis
: Means of all the groups are identical
Alternative hypothesis : At least one group has a different mean
Test statistic
Ri = sum of ranks in group i
ni = number of observations in group i
Which follows a chi-square distribution with k – 1 degrees of freedom.
Kruskal-Wallis test
As with ANOVA:
- Tests a global hypothesis of no difference between any of the
groups
- Need to identify which groups are different in the event of a
significant p-value (post-hoc tests of every possible pairwise
comparisons with Mann-Whitney U test)
- Post-hoc tests incur problem of multiple testing, standard
Bonferroni correction required.
Non-parametric equivalent
For most parametric tests, there are non-parametric
equivalents:
Non-parametric versus parametric
Question: If non-parametric tests are robust to issues
pertaining to outliers, sample sizes and distributional
assumptions, then why are they not the default tests to use?
Answer:
Parametric tests use the actual values for the comparisons,
whereas non-parametric tests use only the ranks.
This means the magnitude of the differences between the
observations are not used, and a difference of 1 or a 100 may
be reduced to just a difference in rank of 1.
This actually reduces the power of the non-parametric test,
relative to the parametric equivalent.
Non-parametric tests in SPSS
Consider the mathematics.xls dataset again.
1. It is traditionally believed that male students tend to outperform female
students in mathematics. Based on the marks before the start of the trial, is
there any evidence in support of this hypothesis.
2. Is there any evidence that consuming omega 3 improves the performance
in the mathematics exam?
3. Is there any difference in the marks before the trial between the three
schools? If there is, which school exhibited the best performance?
4. Is there any difference in the omega 3 consumption between male and
female students?
Let’s approach all these questions from
the non-parametric perspective!
1. It is traditionally believed that male students tend to outperform female students in
mathematics. Based on the marks before the start of the trial, is there any
evidence in support of this hypothesis.
Test of two-independent samples
Mann-Whitney U test
H0: Distributions of both groups are the same
H1: Mean ranks for females < mean ranks for
males
2. Is there any evidence that consuming omega 3 improves the performance in the
mathematics exam?
Test of two-related samples
Sign test
H0: Probability that the difference is greater than 0 = 0.5
H1: Probability that the difference is greater than 0 > 0.5
Wilcoxon Signed Rank Test
H0: Median of differences = 0
H1: Median of differences > 0
Again to derive one tailed p-value, we need
to half the p-value.
3. Is there any difference in the marks before the trial between the three schools? If
there is, which school exhibited the best performance?
Test of K-independent samples
Kruskal-Wallis test
Remember the need to perform
separate 2-independent samples
tests to identify the schools that are
different – should the Kruskal Wallis
test yields a significant result.
H0: Medians of all the groups are identical
H1: At least one group has a different median
4. Is there any difference in the omega 3 consumption between male and female
students?
Test of 2-independent samples
Mann-Whitney U test
Based on the outcome of this analysis, there
is no evidence to suggest that there is a
difference in omega 3 consumption between
male and female students.
Students should be able to
• understand the difference between a parametric and nonparametric test
• know when a parametric test should be used and when a nonparametric test should be used instead
• know the relative advantages and disadvantages of a nonparametric test
• know which non-parametric test should be used under the
specific scenario
• perform the appropriate analyses in SPSS and RExcel