Transcript Nonparametric Statistics

Nonparametric Statistics
 aka, distribution-free statistics
 makes no assumption about the underlying distribution, other than
that it is continuous
 the 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 test
Wilcoxon matched pairs
signed-rank test
Examines 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
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Test of the median: the Sign Test
 Tests hypotheses about the median of a continuous distribution,
i.e.,
˜  
˜0
H 0 : 
˜  
˜0
H1 : 
 Recall that the median is that value for which

˜ 0 )  P(X  
˜ 0 )  0.5
P(X  
 Therefore, the sign test looks at the number of values above (R+)

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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.
R min  n 
P( R  R min )    ( 0.5) r ( 0.5) n  r
r 0  r 
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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 …
~  1.2
H0 : 
~  1.2
H1 : 
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 …
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The Sign Test for paired samples
 Same as for single samples, but the null hypothesis is that the
median difference = 0, i.e.
~ 0
H :
0
D
~ 0
H1 : 
D
 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.
R min  n 
P( R  R min )    ( 0.5) r ( 0.5) n  r
r 0  r 
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Determining β
 Recall that β is the probability of a Type II error, i.e.
  Pr(x  X a |  0 )
 This is highly dependent on the shape of the underlying
distribution
 see, for example, the example on pg. 491 of your textbook
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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!)
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Large sample approximation
 Given n >20, then it can be shown that R is approximately
normally distributed with
n( n  1)
R 
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n( n  1)(2n  1)
2
R 
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and a test of H0: µ = µ0 can be based on the statistic
Z0 
R  n( n  1)/ 4
n( n  1)(2n  1)/24
Comparing 2 means: Wilcoxon rank sum
 Order all data from lowest to highest, keeping up with which data
point belongs to which group
 For 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 …
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Large sample approximation
 Given n1 and n2 >8, then it can be shown that R1 is approximately
normally distributed with
n1 ( n1  n2  1)
 R1 
2
n1n2 ( n1  n2  1)
2
 R1 
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and a test of H0: µ1 = µ2 can be based on the statistic
Z0 
R1   R1
 R1
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
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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 Test
 test of the randomized block design (nonparametric equivalent to
the two-way ANOVA)
 Runs test
 checks for data runs (> expected number of observations above or
below the median)
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