Nonparametrics.Zip (a compressed version of nonparamtrics) Tom Hettmansperger Department of Statistics, Penn State University References: 1. Higgins (2004) Intro to Modern Nonpar Stat 2. Hollander and Wolfe (1999) Nonpar Stat Methods 3. Arnold Notes 4. Johnson, Morrell, and Schick (1992) Two- Sample Nonparametric Estimation and Confidence Intervals Under Truncation, Biometrics, 48, 1043-1056. 5. Website: http://www.stat.wmich.edu/slab/RGLM/
Transcript
Nonparametrics.Zip(a compressed version of nonparamtrics)
Tom HettmanspergerDepartment of Statistics, Penn State University
References:1. Higgins (2004) Intro to Modern Nonpar Stat
2. Hollander and Wolfe (1999) Nonpar Stat Methods
3. Arnold Notes
4. Johnson, Morrell, and Schick (1992) Two-Sample Nonparametric Estimation and Confidence Intervals Under Truncation, Biometrics, 48, 1043-1056.
5. Website: http://www.stat.wmich.edu/slab/RGLM/
Single Sample Methods
• Robust Data Summaries
• Graphical Displays
• Inference: Confidence Intervals and Hypothesis Tests
The median is a robust measure of location. It is not affected by outliers.It is efficient when the population has heavier tails than a normal population.
The sign test is also robust and insensitive to outliers. It is efficient when the tails are heavier than those of a normal population.
Similarly for the confidence interval.
In addition, the test and the confidence interval are distribution free anddo not depend on the shape of the underlying population to determinecritical values or confidence coefficients.
They are only 64% efficient relative to the mean and t-test when the population is normal.
If the population is symmetric then the Wilcoxon Signed Rank statistic can be used, and it is robust against outliers and 95% efficient relative to the t-test.
Two-Sample Methods
Two-Sample Comparisons
85% CI-Boxplots
Mann-Whitney-Wilcoxon Rank Sum Statistic
•Estimate of difference in locations•Test of difference in locations•Confidence Interval for difference in locations
Levene’s Rank Statistic for differences in scaleor variance.
Mann-Whitney-Wilcoxon Statistic: The sign statistic on the pairwise differences.
Unlike the sign test (64% efficiency for normal population, the MWW testhas 95.5% efficiency for a normal population. And it is robust againstoutliers in either sample.
Compute |Y – Med(Y)| and |X – Med(X)|, called absolute deviations.
Apply MWW to the absolute deviations. (Rank the absolute deviations)
The test rejects equal spreads in the two populations when differencein average ranks of the absolute deviations is too large.
Idea: After we have centered the data, then if the null hypothesisof no difference in spreads is true, all permutations of the combined dataare roughly equally likely. (Permutation Principle)
So randomly select a large set of the permutations say B permutations. Assign the first n to the Y sample and the remaining m to the X sample and compute MMW on the absolute deviations.
The approximate p-value is #MMW > original MMW divided by B.
Difference of rank mean abso devs 51.9793
levenerk
Frequency
524530150-15-30-45
120
100
80
60
40
20
0
Mean 0.1644StDev 16.22N 1000
Histogram of levenerkNormal
So we easily reject the null hypothesis of no difference in spreads and conclude that the two populations have significantly different spreads.
k-Sample Methods
One Sample Methods
Two Sample Methods
NGC-4382NGC-4494NGC-3379M-81M-31
28
27
26
25
24
23
22
21
20
85% CI-Boxplot Planetray Nebula Luminosities
Variable Mean StDev Median .75IQR Skew KurtosisMessier 31 22.685 0.969 23.028 1.069 -0.67 -0.67
All one-sample and two-sample methods can be applied one at a timeor two at a time. Plots, summaries, inferences.
We begin k-sample methods by asking if the location differences betweenthe NGC nebulae are statistically significant.
We will briefly discuss issues of truncation.
KWforondistributisamplingeapproximat
asFreedomofDegreeskchisquareauseGenerally
NRn
NRn
NRn
NN
RRN
nnRR
N
nnRR
N
nn
NNKW
constructRandRRwith
datacombinedofrankswithsizesampletotalNGiven
samplesseveraltoMWWExtending
)21(
})2
1()
2
1()
2
1({
)1(
12
})()()({)1(
12
:,,
233
222
211
232
32231
31221
21
321
Kruskal-Wallis Test on NGC
sub N Median Ave Rank Z1 45 26.23 29.6 -9.392 101 26.66 104.5 0.363 59 26.97 156.4 8.19Overall 205 103.0
KW = 116.70 DF = 2 P = 0.000
This test can be followed by multiple comparisons.
For example, if we assign a family error rateof .09, then we would conduct 3 MWW tests, eachat a level of .03. (Bonferroni)
What to do about truncation.
1. See a statistician
2. Read the Johnson, Morrell, and Schick reference. and thensee a statistician.
Here is the problem: Suppose we want to estimate the difference in locationsbetween two populations: F(x) and G(y) = F(y – d).
But (with right truncation at a) the observations come from
ayforandayfordaF
dyFyG
axforandaxforaF
xFxF
a
a
1)(
)()(
1)(
)()(
Suppose d > 0 and so we want to shift the X-sample to the right toward the truncation point. As we shift the Xs, some will pass the truncation point andwill be eliminated from the data set. This changes the sample sizes and requires adjustment when computing the corresponding MWW to see ifit is equal to its expectation. See the reference for details.
What more can we do?
1. Multiple regression
2. Analysis of designed experiments (AOV)
3. Analysis of covariance
4. Multivariate analysis
These analyses can be carried out using the website:
http://www.stat.wmich.edu/slab/RGLM/
Professor Lundquist, in a seminar on compulsive thinkers, illustrates his brainstapling technique.
