Transcript a. Performing the Test

16
Chapter
Nonparametric Tests
Why Use Nonparametric Tests?
One-Sample Runs Test
Wilcox on Signed-Rank Test
Mann-Whitney Test
Kruskal-Wallis Test for
Independent Samples
Friedman Test for Related
Samples
Spearman Rank Correlation Test
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
Why Use Nonparametric Tests?
Parametric Tests
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Parametric hypothesis tests require the
estimation of one or more unknown
parameters (e.g., population mean or
variance).
Often, unrealistic assumptions are made
about the normality of the underlying
population.
Large sample sizes are often required to
invoke the Central Limit Theorem.
Why Use Nonparametric Tests?
Nonparametric Tests
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Nonparametric or distribution-free tests
- usually focus on the sign or rank of the
data rather than the exact numerical value.
- do not specify the shape of the parent
population.
- can often be used in smaller samples.
- can be used for ordinal data.
Why Use Nonparametric Tests?
Advantages and Disadvantages of
Nonparametric Tests
Table 16.1
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Why Use Nonparametric Tests?
Some Common Nonparametric Tests
Figure 16.1
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One-Sample Runs Test
Wald-Wolfowitz Runs Test
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The one-sample runs test (Wald-Wolfowitz
test) detects non-randomness.
Ask – Is each observation in a sequence of
binary events independent of its predecessor?
A nonrandom pattern suggests that the
observations are not independent.
The hypotheses are
H0: Events follow a random pattern
H1: Events do not follow a random pattern
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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To test the hypothesis, first count the number
of outcomes of each type.
n1 = number of outcomes of the first type
n2 = number of outcomes of the second type
n = total sample size = n1 + n2
A run is a series of consecutive outcomes of
the same type, surrounded by a sequence of
outcomes of the other type.
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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For example, consider the following series
representing 44 defective (D) or acceptable (A)
computer chips:
DAAAAAAADDDDAAAAAAAADDAAAAAAAADDD
DAAAAAAAAAA
• The grouped sequences are:
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A run can be a single outcome if it is preceded
and followed by outcomes of the other type.
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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There are 8 runs (R = 8).
n1 = number of defective chips (D) = 11
n2 = number of acceptable chips (A) = 33
n = total sample size = n1 + n2 = 11 + 33 = 44
The hypotheses are:
H0: Defects follow a random sequence
H1: Defects follow a nonrandom sequence
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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When n1 > 10 and n2 > 10, then the number of
runs R may be assumed to be normally
distributed with mean mR and standard
deviation sR.
calc
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One-Sample Runs Test
Wald-Wolfowitz Runs Test
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The test statistic is:
calc
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For a given level of significance a, find the
critical value za for a two-tailed test.
Reject the hypothesis of a random pattern if
z < -za or if z > +za .
One-Sample Runs Test
Wald-Wolfowitz Runs Test
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Decision rule for large-sample runs tests:
Figure 16.2
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Wilcox on Signed-Rank Test
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The Wilcox on signed-rank test compares a
single sample median with a benchmark
using only ranks of the data instead of the
original observations.
It is used to compare paired observations.
Advantages are
- freedom from the normality assumption,
- robustness to outliers
- applicability to ordinal data.
The population should be roughly symmetric.
Wilcox on Signed-Rank Test
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To compare the sample median (M) with a
benchmark median (M0), the hypotheses are:
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When evaluating the difference between
paired observations, use the median
difference (Md) and zero as the benchmark.
Wilcox on Signed-Rank Test
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Calculate the difference between the paired
observations.
Rank the differences from smallest to
largest by absolute value.
Add the ranks of the positive differences to
obtain the rank sum W.
Wilcox on Signed-Rank Test
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For small samples, a special table is
required to obtain critical values.
For large samples (n > 20), the test statistic
is approximately normal.
calc
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Use Excel or Appendix C to get a p-value.
Reject H0 if p-value < a.
Mann-Whitney Test
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The Mann-Whitney test is a nonparametric
test that compares two populations.
It does not assume normality.
It is a test for the equality of medians,
assuming
- the populations differ only in centrality,
- equal variances
The hypotheses are
H0: M1 = M2 (no difference in medians)
H1: M1 ≠ M2 (medians differ)
Mann-Whitney Test
Performing the Test
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Step 1: Sort the combined samples from
lowest to highest.
Step 2: Assign a rank to each value.
If values are tied, the average of the ranks
is assigned to each.
Step 3: The ranks are summed for each
column (e.g., T1, T2).
Step 4: The sum of the ranks T1 + T2 must be
equal to n(n + 1)/2, where n = n1 + n2.
Mann-Whitney Test
Performing the Test
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Step 5: Calculate the mean rank sums T1
and T2.
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Step 6: For large samples (n1 < 10, n2 > 10),
use a z test.
calc
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Step 7: For a given a, reject H0 if
z < -za or z > +za
Kruskal-Wallis Test
for Independent Samples
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The Kruskal-Wallis (K-W) test compares c
independent medians, assuming the
populations differ only in centrality.
The K-W test is a generalization of the
Mann-Whitney test and is analogous to a
one-factor ANOVA (completely randomized
model).
Groups can be of different sizes if each
group has 5 or more observations.
Populations must be of similar shape but
normality is not a requirement.
Kruskal-Wallis Test
for Independent Samples
Performing the Test
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First, combine the
samples and assign
a rank to each
observation in each
group. For example:
When a tie occurs,
each observation is
assigned the
average of the ranks.
Table 16.7
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Kruskal-Wallis Test
for Independent Samples
Performing the Test
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Next, arrange
the data by
groups and sum
the ranks to
obtain the Tj’s.
Remember,
STj = n(n+1)/2.
Table 16.8
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Kruskal-Wallis Test
for Independent Samples
Performing the Test
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The hypotheses to be tested are:
H0: All c population medians are the same
H1: Not all the population medians are the same
For a completely randomized design with c
groups, the tests statistic is
calc
where n = n1 + n2 + … + nc
nj = number of observations in group j
Tj = sum of ranks for group j
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Kruskal-Wallis Test
for Independent Samples
Performing the Test
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The H test statistic follows a chi-square
distribution with n = c – 1 degrees of
freedom.
This is a right-tailed test, so reject H0 if H >
c2a or if p-value < a.
Friedman Test for Related
Samples
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The Friedman test determines if c treatments
have the same central tendency (medians)
when there is a second factor with r levels and
the populations are assumed to be the same
except for centrality.
This test is analogous to a two-factor ANOVA
without replication (randomized block design)
with one observation per cell.
The groups must be of the same size.
Treatments should be randomly assigned
within blocks.
Data should be at least interval scale.
Friedman Test for Related
Samples
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In addition to the c treatment levels that define
the columns, the Friedman test also specifies r
block factor levels to define each row of the
observation matrix.
The hypotheses to be tested are:
H0: All c populations have the same median
H1: Not all the populations have the same median
Unlike the Kruskal-Wallis test, the Friedman
ranks are computed within each block rather
than within a pooled sample.
Friedman Test for Related
Samples
Performing the Test
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First, assign a rank to each observation within
each row. For example, within each Trial:
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When a tie occurs, each observation is
assigned the average of the ranks.
Friedman Test for Related
Samples
Performing the Test
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Compute the test statistic:
calc
where r = the number of blocks (rows)
c = the number of treatments (columns)
Tj = the sum of ranks for treatment j
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Friedman Test for Related
Samples
Performing the Test
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The Friedman test statistic F, follows a chisquare distribution with n = c – 1 degrees of
freedom.
Reject H0 if F > c2a or if p-value < a.
Spearman Rank Correlation Test
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Spearman’s rank correlation coefficient
(Spearman’s rho) is an overall
nonparametric test that measures the
strength of the association (if any) between
two variables.
This method does not assume interval
measurement.
The sample rank correlation coefficient rs
ranges from -1 < rs < +1.
Spearman Rank Correlation Test
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The sign of rs indicates whether the relationship
is
direct – ranks tend to vary in the same
direction, or
inverse – ranks tend to vary in opposite
directions
The magnitude of rs indicated the degree of
relationship. If
rs is near 0 – there is little or no agreement
between rankings
rs is near +1 – there is strong direct agreement
rs is near -1 – there is strong inverse
agreement
Spearman Rank Correlation Test
Performing the Test
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First, rank
each
variable. For
example,
If more than
one value is
the same,
assign the
average of
the ranks.
Table 16.11
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Spearman Rank Correlation Test
Performing the Test
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The sums of ranks within each column must
always be n(n+1)/2.
Next, compute the difference in ranks di for
each observation.
The rank differences should sum to zero.
Spearman Rank Correlation Test
Performing the Test
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Calculate the sample rank correlation
coefficient rs.
where di = difference in ranks for case i
n = sample size
For a right-tailed test, the hypotheses to be
tested are
H0: True rank correlation is zero (rs < 0)
H1: True rank correlation is positive (rs > 0)
Spearman Rank Correlation Test
Performing the Test
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If n is large (at least 20 observations), then rs
may be assumed to follow the Student’s t
distribution with degrees of freedom n = n - 1
calc
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Reject H0 if t > ta or if p-value < a.
Correlation versus Causation
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Caution: correlation does not prove
causation.
Correlations may prove to be “significant”
even when there is no causal relation
between the two variables.
However, causation is not ruled out.
Multiple causes may be present.
Applied Statistics in
Business & Economics
End of Chapter 16
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