NONPARAMETRIC TESTS
Why NP Test?
When the sample distribution is unknown.
When the population distribution is abnormal i.e., data involves too
many variables.
Make minimal assumptions about the underlying distribution of the data.
Broad categories of NP Tests
The NP tests can be grouped into three Broad categories based on how the
data are organized:
A one-sample test - analyzes one field.
A test for related samples - compares two or more fields for the same set of cases.
An independent-samples test - analyzes one field that is grouped by categories of another field.
Various NP tests
There are a number of nonparametric tests. The important one are;
Chi-Square Test
Tests the hypothesis that the observed frequencies do not differ from
their expected values.
Example;
A large hospital schedules discharge support staff assuming that
patients leave the hospital at a fairly constant rate throughout the week.
However, because of increasing complaints of staff shortages, the hospital
administration wants to determine whether the number of discharges varies by
the day oftheweek. C:\ProgramFiles\SPSSInc\PASWStatistics18\Samples\English\dischargedata.sav
Data
Hypothesis
Use Chi-Square Test to test the assumption that patients leave the
hospital at a constant rate.
Computations
Test Results
Discussion
chi-square statistic equals 29.389. This is computed by squaring the
residual for each day, dividing by its expected value, and summing across all
days.
Number of expected values that can vary before the rest are completely
determined.
For a one-sample chi-square test, df is equal to the number of rows
minus 1.
Asymp. Sig. is the estimated probability of obtaining a chi-square value
greater than or equal to 29.389 if patients are discharged evenly across the
week.
The low significance value (.000) suggests that the average rate of
patient discharges really does differ by day of the week.
One-Sample Kolmogorov-Smirnov
The One-Sample Kolmogorov-Smirnov
procedure is used to test the null hypothesis that a sample comes from a
particular distribution. (Diagnostic test).
Computational Procedure
It involves finding the largest
difference (in absolute value) between two cumulative distribution functions
(CDFs)--one computed directly from the data; the other, from mathematical
theory.
Example
An insurance analyst wants to
model the number of automobile accidents per driver. She has randomly sampled
data on drivers in a certain region. She wants to test to confirm that the number
of accidents (X) follows a Poisson distribution.
Results and discussion
The Poisson distribution is
indexed by only one parameter--the mean. This sample of drivers averaged about
1.72 accidents over the past five years.
The next three rows fall under the
general category Most Extreme Differences. The differences referred to are the
largest positive and negative points of divergence between the empirical and
theoretical CDFs.
The first difference value,
labeled Absolute, is the absolute value of the larger of the two difference
values .
This value will be required to
calculate the test statistic.
The Positive difference is the
point at which the empirical CDF exceeds the theoretical CDF by the greatest
amount.
At the opposite end of the
continuum, the Negative difference is the point at which the theoretical CDF
exceeds the empirical CDF by the greatest amount.
The Z test statistic is the
product of the square root of the sample size and the largest absolute difference
between the empirical and theoretical CDFs.
Unlike much statistical testing, a
significant result here is bad news. The probability of the Z statistic is
below 0.05, meaning that the Poisson distribution with a parameter of 1.72 is
not a good fit for the number of accidents within the past five years in this
sample of drivers.
Generally, a significant
Kolmogorov-Smirnov test means one of two things--either the theoretical
distribution is not appropriate, or an incorrect parameter was used to generate
that distribution.
Looking at the previous results,
it is hard for the analyst to believe that the Poisson distribution is not the
appropriate one to use for modeling automobile accidents.
Poisson is often used to model
rare events and, fortunately, automobile accidents are relatively rare.
The analyst wonders if gender may
be confounding the test. The total sample average assumes that males and
females have equal numbers of accidents, but this is probably not true. She
will split the sample by gender, using each gender's average as the Poisson
parameter in separate tests.
The analyst wonders if gender may
be confounding the test. The total sample average assumes that males and
females have equal numbers of accidents, but this is probably not true.
She will split the sample by
gender, using each gender's average as the Poisson parameter in separate tests.
The statistics table provides
evidence that a single Poisson parameter for both genders may not be correct.
Males in this sample averaged
about two accidents over the past five years, while females tended to have
fewer accidents.
When assessing goodness of fit,
remember that a statistically significant Z statistic means that the chosen
distribution does not fit the data well.
Unlike the previous test, however,
we see a much better fit when splitting the file by gender.
Increasing the Poisson parameter
from 1.72 to 1.98 clearly provides a better fit to the accident data for men.
Similarly, decreasing the Poisson
parameter from 1.72 to 1.47 provides a better fit to the accident data for
women
Summary
Using the One-Sample
Kolmogorov-Smirnov Test procedure, we found that, overall, the number of
automobile accidents per driver do not follow a Poisson distribution.
However, once we split the file on
gender, the distributions of accidents for males and females can individually
be considered Poisson.
Conclusion
These results demonstrate that the
one-sample Kolmogorov-Smirnov test requires not only that we choose the
appropriate distribution but the appropriate parameter(s) for it as well.
If we want to compare the
distributions of two variables, the two-sample Kolmogorov-Smirnov test in the
Two-Independent-Samples
Tests procedure is to be used.
The Runs Test Procedure
Many statistical tests assume that
the observations in a sample are independent; in other words, that the order in
which the data were collected is irrelevant.
If the order does matter, then the
sample is not random, and we cannot draw accurate conclusions about the
population from which the sample was drawn.
Therefore, it is prudent to check
the data for a violation of this important assumption.
We can use the Runs Test procedure
to test whether the order of values of a variable is random.
The procedure first classifies
each value of the variable as falling above or below a cut point and then tests
to ensure that there is no order to the resulting sequence.
The cut point is based either on a
measure of central tendency ( mean, median, or mode) or a custom value.
We can obtain descriptive
statistics and/or quartiles of the test variable.
Example
An e-commerce firm enlisted beta
testers to browse and then rate their new Web site. Ratings were recorded as
soon as each tester finished browsing. The team is concerned that ratings may
be related to the amount of time spent browsing.
The ratings are collected in the
file siteratings.sav. Test the
hypothesis that time spent in browsing is correlated with site rating.
Nonparametric Tests for Two Independent Samples
The nonparametric tests for two
independent samples are useful for determining whether or not the values of a
particular variable differ between two groups.
This is especially useful when the
assumptions of the t test are not met.
When we want to test for
differences between two groups, the independent-samples t test comes naturally
to mind.
However, despite its simplicity,
power, and robustness, the independent-samples t test is invalid when certain
critical assumptions are not met.
These assumptions center around
the parameters of the test variable (in this case, the mean and variance) and
the distribution of the variable itself.
Most important, the t test assumes
that the sample mean is a valid measure of center. While the mean is valid when
the distance between all scale values is equal, it's a problem when our test
variable is ordinal because in ordinal scales the distances between the values
are arbitrary.
Furthermore, because the variance
is calculated using squared distances from the mean, it too is invalid if those
distances are arbitrary.
Finally, even if the mean is a
valid measure of center, the distribution of the test variable may be so
non-normal that it makes us suspicious of any test that assumes normality.
If any of these circumstances is
true for our analysis, we should consider using the nonparametric procedures
designed to test for the significance of the difference between two groups.
They are called nonparametric because they
make no assumptions about the parameters of a distribution, nor do they assume
that any particular distribution is being used.
Two popular nonparametric tests of
location (or central tendency)--the Mann-Whitney and Wilcoxon tests--and a test
of location and shape--the two-sample Kolmogorov-Smirnov test.
From the above two tests,
Mann-Whitney and Wilcoxon tests is commonly used test.
Mann-Whitney and Wilcoxon tests
We can use the Mann-Whitney and
Wilcoxon statistics to test the null hypothesis that two independent samples
come from the same population.
Their advantage over the
independent-samples t test is that Mann-Whitney and Wilcoxon do not assume
normality and can be used to test ordinal variables.
Physicians randomly assigned
female stroke patients to receive only physical therapy or physical therapy
combined with emotional therapy. Three months after the treatments, the
Mann-Whitney test is used to compare each group's ability to perform common
activities of daily life.
Data File
The results are in the file
adl.sav. Test to determine
whether the
two groups' abilities differ.
The U statistic is simple (but tedious) to calculate. For each case in
group 1, the number of cases in group 2 with higher ranks is counted. Tied
ranks count as 1/2. This process is repeated for group 2. The Mann-Whitney U
statistic displayed in the table is the smaller of these two values.
References
Siegel, S., and N. J. Castellan.
1988. Nonparametric statistics for the behavioral sciences. New York:
McGraw-Hill, Inc..
Conover, W. J. 1980. Practical
Nonparametric Statistics, 2nd ed. New York: John Wiley and Sons.
Daniel, W. W. 1995. Biostatistics,
6th ed. New York: John Wiley and Sons.
Norusis, M. 2004. SPSS 13.0
Guide to Data Analysis. Upper Saddle-River, N.J.: Prentice Hall, Inc..
Norusis, M. 2004. SPSS 13.0
Statistical Procedures Companion. Upper Saddle-River, N.J.: Prentice Hall,
Inc..
