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Nonparametric Tests
IPS Chapter 15
© 2009 W.H. Freeman and Company
Objectives (IPS Chapter 15)
Nonparametric Tests
The Wilcoxon rank sum test
The Normal approximation for W
What hypotheses does Wilcoxon test?
The Wilcoxon signed rank test
The Normal approximation for W+
Dealing with ties
The Kruskal-Wallis test
Assumptions for inference
For the inference methods for means we have already studied, we
assumed that the variables have Normal distributions in the
population(s) from which we draw our data.
Robustness: some skewness was acceptable, especially if the sample
size was large.
What happens if plots suggest the data are clearly not Normal,
especially if the sample size is small?
Options for non-Normal data and small n1. Is lack of Normality due to outliers? If an outlier appears to be “real data,”
you have to leave it in, but if you have reason to think there is an error in
that data, you may be able to remove it.
2. Try transforming the data. For example use a logarithm for right-skewed
data.
3. Try another standard distribution. Other procedures can replace the t
procedures if data (especially right-skewed data) fits another distribution.
4. Use modern bootstrap methods and permutation tests. Heavy computing
avoids requiring Normality or any other specific form of sampling
distribution.
5. Use other nonparametric methods. Discussed in this chapter.
Ranks
Hypotheses for rank tests just replace the mean with the median.
For strongly skewed data, we prefer the median to the mean for describing the center of the data.
To rank observations, first arrange them in order from smallest to
largest. The rank of each observation is its position in this ordered list,
starting with rank 1 for the smallest observation.
Example: Weeds among the corn
Does the presence of small numbers of weeds reduce the yield of corn?
Lamb’s-quarter is a common weed in corn fields. A researcher planted corn at
the same rate in 8 small plots of ground, then weeded the corn rows by hand to
allow no weeds in 4 randomly selected plots and exactly 3 lamb’s–quarter plants
per meter of row in the other 4 plots.
Here are the yields of corn (bushels per acre) in each of the plots.
A Back-to-back stemplot shows non-Normality, possible outliers, and small sample sizes.
Example: Weeds among the corn
First rank all 8 observations together.
Arrange them in order from smallest to largest.
The shaded numbers are those with no weeds.
Note that 4 of the 5 highest yields are from the no weeds group.
The idea of rank tests is to look just at position in this list.
Working with ranks allows us to dispense with the numerical values of the data and the specific conditions on the shape of the distribution such as Normality.
Example: Weeds among the corn
If the presence of weeds reduces corn yields, we expect the ranks of the yields
from plots without weeds to be larger as a group than the ranks from plots with
weeds.
Compare the sums of the ranks from the two treatments.
If the weeds have no effect, we would expect the sum of the ranks in either group to be 18. Why?
Wilcoxon Rank Sum TestDraw an SRS of size n1 from one population and draw an independent SRS of
size n2 from a second population. There are N observations in all, where N = n1
+ n2. Rank all N observations. The sum W of the ranks for the first sample is
the Wilcoxon rank sum statistic. If the two populations have the same
continuous distribution, then W has mean
and standard deviation
The Wilcoxon rank sum test rejects the hypothesis that the two populations
have identical distributions when the rank sum W is far from its mean.
1( 1)
2W
n Nμ +=
1 2( 1)
12W
n n Nσ +=
Example: Weeds among the corn
In this study, we want to test the hypotheses
H0: No difference in distribution of yields.
Ha: Yields are systematically higher in weed-free plots.
The test statistic is the rank sum W = 23 for the weed-free plots.
Conditions for Wilcoxon test are met:
data come from a randomized comparative experiment.
yield of corn in bushels per acre has a continuous distribution.
Example: Weeds among the corn
N = 8, n1 (no weeds) = 4, and n2 (3 weeds per meter) = 4.
The sum of ranks for the weed-free plants has mean and standard deviation:
The observed rank sum W = 23 is only 1.4 standard deviations above the mean.
Software tells us that the P-value for P(W 23) is 0.1.
We cannot reject the null hypothesis.
We do not have enough evidence to say that yields are systematically
higher in weed-free plots.
A larger sample size might clarify the effect of weeds on corn yield.
.
1( 1) 4(9)18
2 2W
n Nμ += = =
1 2( 1) (4)(4)(9)3.464
12 12W
n n Nσ += = =
The Normal Approximation for W
To calculate the P-value for the rank sum Wilcoxon test, we need to
know the sampling distribution of W when the null hypothesis is true.
With or without software, P-values for the Wilcoxon test are often
based on the fact that the rank sum statistic W becomes
approximately Normal as the two sample sizes increase.
Test statistic: 1
1 2
( 1) / 2
( 1) /12W
W
W W n Nz
n n N
μσ− − +
= =+
Example: Weeds among the corn
We can improve this approximation by using the continuity correction. You
use this for a variable that takes only whole-number values, like W. Act as if
each whole number occupies the entire interval from 0.5 below the number to
0.5 above it.
Software tells us that the exact P-value for P(W 23) is 0.1.
23 181.44
3.464
value ( 1.44) 0.0749
W
W
Wz
P P Z
μσ− −
= = =
− = ≥ =
22.5 181.30
3.464
value ( 1.30) 0.0968
W
W
Wz
P P Z
μσ− −
= = =
− = ≥ =
Use software that gives the
exact P-value for the Wilcoxon
test rather than the Normal
approximation.
Here is the output from JMP; notice
that it only gives the normal
approximation for the Wilcoxon
Rank Sum test.
Using technologyOneway Analysis of yield By group
Quantiles Level Minimum 10% 25% Median 75% 90% Maximum no weeds
165 165 165.425 169.45 175.725 176.9 176.9
weeds 153.1 153.1 153.825 157.3 171.95 176.4 176.4 Wilcoxon / Kruskal- Wallis Tests (Rank Sums) Level Count Score Sum Score Mean (Mean-
Mean0)/Std0 no weeds 4 23.000 5.75000 1.299 weeds 4 13.000 3.25000 - 1.299 2- Sample Test, Normal Approximation
S Z Prob>|Z| 13 - 1.29904 0.1939
Small sample sizes. Refer to statistical tables for tests, rather than large- sample approximations.
Here's a place to find a small sample table:
http://www.socr.ucla.edu/Applets.dir/WilcoxonRankSumTable.htm
What hypotheses does Wilcoxon test?If we assume that our sample is Normally distributed, we can use
the two-sample t test for means.
When the distribution may not be Normal, we might restate the
hypotheses in terms of population medians rather than means.
The Wilcoxon rank sum test will test the hypotheses above only if an
additional condition is met: both populations must have
distributions of the same shape.
H0: μ1 = μ2
Ha: μ1 > μ2
H0: median1 = median2
Ha: median1 > median2
What hypotheses does Wilcoxon test?The same shape condition is too strict to be reasonable in practice.
A more useful statement of the hypotheses compares two
continuous distributions, whether or not they have the same
shape.
These hypotheses are considered “nonparametric” because they do
not include a parameter. They are just stated in words.
H0: the two distributions are the same
Ha: one has values that are systematically larger
Dealing with ties in rank testsUp until now, our data has had no two values exactly the same.
However, we often find observations tied at the same value.
The usual practice is to assign all tied values the average of the
ranks they occupy - these are sometimes called midranks.
In practice, software is required to use rank tests when the data contain
tied values.
ANOVA hypotheses:
Data should come from independent random samples, all Normally distributed with the
same standard deviation
Kruskal-Wallis hypotheses:
1. Data should come from independent random samples, the response has a continuous
(but not necessarily Normal) distribution.
2. Data should come from independent random samples, the response has a continuous
(but not necessarily Normal) distribution, and the samples come from population
distributions of the same shape (not necessarily Normal).
Comparing several samples: the Kruskal-Wallis test
0 0 1 3 9:
: not all four medians are equala
H M M M M
H
= = =
Example: Weeds among the corn
Lamb’s-quarter is a common weed in corn fields. A researcher planted corn at
the same rate in 16 small plots of ground, then randomly assigned the plots to 4
groups. He weeded the corn rows by hand to allow a fixed number of lamb’s-
quarter plants to grow in each meter of corn row. These numbers were 0, 1, 3,
and 9 in the four groups of plots. No other weeds were allowed to grow, and all
plots received identical treatment except for the weeds.
Here are the yields of corn (bushels per acre) in each of the plots.
Example: Weeds among the corn
Here are the summary statistics for the corn yield.
Can we safely use ANOVA? The standard deviations don’t pass the largest s < 2 (smallest s) test, and there were outliers in the original data that cannot be removed.
Can we use the median Kruskal-Wallis test? The different standard deviations suggest that the distributions do not all have the same shape.
Example: Weeds among the cornRank all 16 observations in order from smallest to largest.
Note the tied observations
Kruskal-Wallis test statistic
Example: Weeds among the corn
Kruskal-Wallis test statistic:
Using Table E with df = 3, the P-value is 0.10 < P < 0.15.
We do not reject the null hypothesis.
JMP output for the
Kruskal-Wallis test
and ANOVA.
Using technologyOneway Analysis of yield By group
Quantiles Level Minimum 10% 25% Median 75% 90% Maximum 0 weeds
165 165 165.425 169.45 175.725 176.9 176.9
1 weeds
157.3 157.3 158.25 163.65 166.575 166.7 166.7
3 weeds
153.1 153.1 153.825 157.3 171.95 176.4 176.4
9 weeds
142.4 142.4 147.4 162.55 162.775 162.8 162.8
Wilcoxon / Kruskal- Wallis Tests (Rank Sums) Level Count Score Sum Score Mean (Mean-
Mean0)/Std0 0 weeds 4 52.500 13.1250 2.184 1 weeds 4 33.500 8.3750 0.000 3 weeds 4 25.000 6.2500 - 1.032 9 weeds 4 25.000 6.2500 - 1.032 1- way Test, ChiSquare Approximation
ChiSquare DF Prob>ChiSq 5.5725 3 0.1344
Small sample sizes. Refer to statistical tables for tests, rather than large- sample approximations.
