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Nonparametric Statistics
aka, distribution-free statistics makes no assumption about the underlying
distribution, other than that it is continuousthe data can be non-quantitative, rank order,
etc. Competitors of the t- and F- procedures we
used in chapters 11 and 12.generally less efficient, require larger sample
sizes for the same confidence level and power
Some Commonly Used Statistical Tests
Normal theory based test
Corresponding nonparametric test
Purpose of test
t test for independent
samples
Mann-Whitney U test; Wilcoxon rank-sum test
Compares two independent samples
Paired t testWilcoxon matched pairs
signed-rank testExamines a set of differences
Pearson correlation coefficient
Spearman rank correlation coefficient
Assesses the linear association between two variables.
One way analysis of variance (F test)
Kruskal-Wallis analysis of variance by ranks
Compares three or more groups
Two way analysis of variance
Friedman Two way analysis of variance
Compares groups classified by two different factors
Source: Gerard E. Dallal, Ph.D., Nonparametric Statistics. http://www.jerrydallal.com/LHSP/npar.htm
ETM 620 - 09U2
Test of the median: the Sign Test Tests hypotheses about the median of a
continuous distribution, i.e.,
Recall that the median is that value for which
Therefore, the sign test looks at the number of values above (R+) and below (R-) the hypothesized median. When the null hypothesis is true, R = min(R+, R-) follows the binomial distribution with sample size n and p = 0.5, i.e.
ETM 620 - 09U3
H0 : ˜ ˜ 0H1 : ˜ ˜ 0
P(X ˜ 0) P(X ˜ 0) 0.5
min
0min )5.0()5.0()(
R
r
rnr
r
nRRP
An example: Recall the example comparing two methods for
testing shear strength in steel girders. Suppose we are interested in testing whether or not the actual median of the Karlsruhe method is 1.2, that is …
given the data as shown on pg 293 and in the Excel data file.
Note the difference between the algorithm given in the textbook (as done in Excel) and the results from Minitab …
ETM 620 - 09U4
2.1~:
2.1~:
1
0
H
H
The Sign Test for paired samples Same as for single samples, but the null
hypothesis is that the median difference = 0, i.e.
Example, paired comparison of example 11-17 ignoring the normality assumption …
Calculate P-value as the probability that number of data points is less than or equal to the minimum R value given a binomial distribution with p = 0.5, i.e.
ETM 620 - 09U5
0~:
0~:
1
0
D
D
H
H
min
0min )5.0()5.0()(
R
r
rnr
r
nRRP
Determining β Recall that β is the probability of a Type II
error, i.e.
This is highly dependent on the shape of the underlying distributionsee, for example, the example on pg. 491 of
your textbook
ETM 620 - 09U6
)|Pr( 0 aXx
Wilcoxon signed rank test Sign test only focuses on whether the data are
above or below the presumed median, ignoring the magnitude
If we assume a symmetrical continuous distribution, we can use the Wilcoxon signed rank test Similar to the sign test, but now we order the
differences from the mean in order of magnitude and add the ranks together.
Let’s do this once on Excel and once on Minitab. (Note the differences!)
ETM 620 - 09U7
Large sample approximation Given n >20, then it can be shown that R is
approximately normally distributed with
and a test of H0: µ = µ0 can be based on the statistic
24)12)(1(
4)1(
2
nnn
nn
R
R
24/)12)(1(
4/)1(0
nnn
nnRZ
Comparing 2 means: Wilcoxon rank sum Order all data from lowest to highest, keeping
up with which data point belongs to which groupFor example, see example 16-5, pg 500
Then, R1=sum(rank order for sample 1) and R2=sum(rank order for sample 2)From table IX, obtain R*
α for n1 and n2 at α of 0.01 and 0.05
Alternatively, using Mann-Whitney on Minitab …
ETM 620 - 09U9
Large sample approximation Given n1 and n2 >8, then it can be shown that
R1 is approximately normally distributed with
and a test of H0: µ1 = µ2 can be based on the statistic
12)1(
2)1(
21212
211
1
1
nnnn
nnn
R
R
1
110
R
RRZ
Analysis of Variance: the Kruskal-Wallis Test
Expands the rank-sum method to more than one factor level
Use Minitab to perform the statistical analysis …
Look at example 16-6, pg. 503
ETM 620 - 09U11
Other nonparametric tests … Mood’s Median Test
similar to Kruskal-Wallis, more robust against outliers but less robust when samples are from different distributions
Friedman Testtest of the randomized block design
(nonparametric equivalent to the two-way ANOVA)
Runs testchecks for data runs (> expected number of
observations above or below the median)
ETM 620 - 09U12