Nonparametric Statistical Methods Presented by Guo Cheng, Ning
Liu, Faiza Khan, Zhenyu Zhang, Du Huang, Christopher Porcaro,
Hongtao Zhao, Wei Huang 1 Slide 2 Introduction Slide 3 Definition
Nonparametric methods 1: rank-based methods are used when we have
no idea about the population distribution from which the data is
sampled. Used for small sample sizes. Used when the data are
measured on an ordinal scale and only their ranks are meaningful. 3
Slide 4 Outline 1. Sign Test 2. Wilcoxon Signed Rank Test 3.
Inferences for Two Independent Samples 4. Inferences for Several
Independent Samples 5. Friedman Test 6. Spearmans Rank Correlation
7. Kendalls Rank Correlation Coefficient 4 Slide 5 1.Sign Test 5
Slide 6 Parameter of interest: Median Median is used as a parameter
because it is a better measure of data as compared to the mean for
skewed distributions. 6 Slide 7 Hypothesis test H 0 : = 0 vs H a :
> 0 where 0 is a specified value and is unknown median 7 Slide 8
Testing Procedure Step 1: Given a random sample x 1, x 2, , x n
from a population with unknown median , count the number of x i s
that exceed 0. Denote them by s +. s - = n - s + Step 2: Reject H 0
if s + is large or s - is small. 8 Slide 9 How to reject H 0 ? To
determine how large s + must be in order to reject H 0, we need to
find out the distribution of the corresponding random variable S +.
X i : random variable corresponding to the observed values x i S -
: random variable corresponding to s - 9 Slide 10 Distribution of S
+ and S - 10 Slide 11 Calculating P-value 11 Slide 12 Rejection
criteria 12 Slide 13 Large sample z-test 13 Slide 14 Confidence
Interval 14 Slide 15 Example 15 Slide 16 SAS code 16 DATA
themostat; INPUT temp; datalines; 202.2 203.4 ; PROC UNIVARIATE
DATA=themostat loccount mu0=200; VAR temp; RUN; Slide 17 SAS Output
Basic Statistical Measures Location Variability Mean 201.7700 Std
Deviation 2.41019 Median 201.7500 Variance 5.80900 Mode. Range
8.30000 Interquartile Range 2.90000 Tests for Location: Mu0=200
Test -Statistic- -----p Value------ Student's t t 2.322323 Pr >
|t| 0.0453 Sign M 3 Pr >= |M| 0.1094 Signed Rank S 19.5 Pr >=
|S| 0.048 17 Slide 18 2. Wilcoxon signed rank test 18 Slide 19
Inventor Frank Wilcoxon (2 September 1892 in County Cork, Ireland
18 November 1965, Tallahassee, Florida, USA) was a chemist and
statistician, known for development of several statistical tests.
19 Slide 20 What is it used for? Two related samples Matched
samples Repeated measurements on a single sample Slide 21
Hypothesis 21 Slide 22 Testing procedure 22 Slide 23 Example 23
Slide 24 SAS codes 24 DATA thermo; INPUT temp; datalines; 202.2
203.4 ; PROC UNIVARIATE DATA=thermo loccount mu0=200; TITLE
"Wilcoxon signed rank test the thermostat"; VAR temp; RUN; Slide 25
SAS outputs (selected results) 25 8 Basic Statistical Measures
Location Variability Mean 201.7700 Std Deviation 2.41019 Median
201.7500 Variance 5.80900 Mode. Range 8.30000 Interquartile Range
2.90000 Tests for Location: Mu0=200 Test -Statistic- -----p
Value------ Student's t t 2.322323 Pr > |t| 0.0453 Sign M 3 Pr
>= |M| 0.1094 Signed Rank S 19.5 Pr >= |S| 0.048 Slide 26
Large sample approximation 26 Slide 27 Derive E(x) & Var(x) 27
Slide 28 Rejection region: 28 Slide 29 3. Inferences for Two
Independent Samples 29 Slide 30 Hypothesis Slide 31 Definition 31
Slide 32 Definition 32 Slide 33 Wilcoxon sum rank test 33 Slide 34
Mann-Whitney-U test 34 Slide 35 Between two tests 35 Slide 36
Advantages 36 Slide 37 For large samples 37 Slide 38 For large
samples 38 Slide 39 Treatment of ties 39 Slide 40 Example To test
if the grades of two classes which have the same teacher are the
same, we randomly pick 7 students from Class A and 9 from Class B,
their scores are as follows A: 8.50 9.48 8.65 8.16 8.83 7.76 8.63
B: 8.27 8.20 8.25 8.14 9.00 8.10 7.20 8.32 7.70 40 Slide 41 Example
7.207.707.768.108.148.168.208.25 BBABBABB 12345678
8.278.328.508.638.658.839.009.48 BBAAAABA 910111213141516 41 Slide
42 Example 42 Slide 43 Example 43 Slide 44 SAS code Data exam;
Input group $ score @@; Datalines; A 8.50 A 9.48 A 8.65 A 8.16 A
8.83 A 7.76 A 8.63 B 8.27 B 8.20 B 8.25 B 8.14 B 9.00 B 8.10 B 7.20
B 8.32 B 7.70 ; 44 Slide 45 SAS code Proc npar1way data=exam
wilcoxon; Var score; Class group; Exact wilcoxon; Run; 45 Slide 46
Output Wilcoxon Scores (Rank Sums) for Variable score Classified by
Variable group groupNSum of Scores Expected Under H0 Std Dev Under
H0 Mean Score A775.059.509.44722210.714286
B961.076.509.4472226.777778 46 Slide 47 Output Wilcoxon Two-Sample
Test Statistic (S)75.0000 Normal Approximation Z1.5878 One-Sided Pr
> Z0.0562 Two-Sided Pr > |Z|0.1123 t Approximation One-Sided
Pr > Z0.0666 Two-Sided Pr > |Z|0.1332 Exact Test One-Sided Pr
>= S0.0571 Two-Sided Pr >= |S - Mean|0.1142 Z includes a
continuity correction of 0.5. 47 Slide 48 Output 48 Slide 49 4.
Inferences for Several Independent Samples 49 Slide 50 Introduction
We know that if our data is normally distributed and that the
population standard deviations are equal, we can test for a
difference among several populations by using the One-way ANOVA F
test. 50 Slide 51 When to use Kruskal-Wallis test? But what happens
when our data is not normal? This is when we use the nonparametric
Kruskal-Wallis test to compare more than two populations as long as
our data comes from a continuous distribution. The notion of the kw
rank test is to rank all the data from each group together and then
apply one-way ANOVA to the ranks rather than to the original data.
51 Slide 52 Kruskal-Wallis Test (kw Test) A non-parametric method
for testing whether samples originate from the same distribution.
Used for comparing more than two samples that are independent. 52
Slide 53 Kruskal-Wallis Test: History William Henry Kruskal October
10 th, 1919 April 21 st, 2005 Obtained Bachelors and Masters degree
in Mathematics at Harvard University and received his Ph. D. from
Columbia University in 1955. Wilson Allen Wallis November 5 th,1912
October 12 th, 1998 Undergraduate work at the University of
Minnesota and Graduate work at the University of Chicago in 1933.
53 Slide 54 Kruskal-Wallis Test: Steps 1. Create Hypothesis: Null
Hypothesis (H o ): The samples from populations are identical
Alternative Hypothesis (H a ): At least one sample is different 54
Slide 55 Kruskal-Wallis Test: Steps 2. Rank all the data. The
lowest number gets the lowest rank and so on. Tied data gets the
average of the ranks they would have obtained if they werent tied.
3. All the ranks of the different samples are added together. Label
these sums L 1, L 2, L 3, and L 4. 55 Slide 56 Kruskal-Wallis Test:
Steps 4. Find Test Statistic: n = total number of observations in
all samples L i = total rank of each sample kw = test statistic 5.
Reject H o if H is greater than the chi-square table value. 56
Slide 57 Kruskal-Wallis Test: Example An experiment was done to
compare four different ways of teaching a concept to a class of
students. In this experiment, 28 tenth grade classes were randomly
assigned to the four methods (7 classes per method). A 45 question
test was given to each class. The average test scores of the
classes are given in the following table. Apply the Kruskal- Wallis
test to the test scores data set. 57 Slide 58 Kruskal-Wallis Test:
Example Given Data Ranks of Data values 58 Slide 59 Kruskal-Wallis
Test: Example 59 Slide 60 Kruskal-Wallis Test: Example 60 Slide 61
SAS Input data test; input methodname $ scores; cards; case 14.59
case 23.44 case 25.43 case 18.15 Case 20.82 Case 14.06 Case 14.26
Formula 20.27 Formula 26.84 Formula 14.71 Formula 22.34 Formula
19.49 Formula 24.92 Formula 20.20 Equation 27.82 Equation 24.92
Equation 28.68 Equation 23.32 Equaiton 32.85 Equation 33.90
Equation 23.42 Unitary 33.16 Unitary 26.93 Unitary 30.43 Unitary
36.43 Unitary 37.04 Unitary 29.76 Unitary 33.88 ; proc npar1way
data=test wilcoxon; class methodname; var scores; run; 61 Slide 62
SAS Output Wilcoxon Scores (Rank Sums) for Variable scores
Classified by Variable methodname Sum of Expected Std Dev Mean
methodname N Scores Under H0 Under H0 Score case 7 49.00 101.50
18.845498 7.000000 formula 7 66.50 101.50 18.845498 9.500000
equation 7 125.50 101.50 18.845498 17.928571 unitary 7 165.00
101.50 18.845498 23.571429 Average scores were used for ties.
Kruskal-Wallis Test Chi-Square 18.1390 DF 3 Pr > Chi-Square
0.0004 62 Slide 63 4. Friedman Test 63 Slide 64 Introduction A
distribution-free rank-based test for comparing the treatments is
known as the Friedman test, named after the Nobel Laureate
economist Milton Friedman who proposed it. The Friedman Test is a
version of the repeated-Measures ANOVA that can be performed on
ordinal(ranked) data. 64 Slide 65 Steps in the Friedman test 65
Slide 66 Steps in the Friedman test 66 Slide 67 Example Now we have
8 treatments separated in 3 blocks, = 0.025 67 Slide 68 Define Null
and Alternative Hypothesis H 0 : There is no difference between 8
treatments H a : There exists difference between 8 treatments 68
Slide 69 Rank Sum 69 Slide 70 Friedman Test 70 Slide 71 Conclusion
71 Slide 72 5. Spearmans Rank Correlation Coefficient 72 Slide 73
Introduction From Pearson to Spearman Spearmans Rank Correlation
Coefficient Large-Sample Approximation Hypothesis Test Examples 73
Slide 74 From Pearson to Spearman Pearsons Measure only the degree
of linear association Based on the assumption of bivariate normally
of two variables Spearmans Take in account only the ranks Measure
the degree of monotone association Inferences on the rank
correlation coefficients are distribution-free 74 Slide 75 From
Pearson to Spearman 75 Slide 76 From Pearson to Spearman Charles
Edward Spearman As a psychologist General factor of intelligence
the nature and causes of variations in human As a statistician Rank
correlation two-way analysis Charles Edward Spearman (10 Sept. 1863
17 Sept. 1945) Correlation coefficient 76 Slide 77 Spearmans Rank
Correlation Coefficient 77 Slide 78 Spearmans Rank Correlation
Coefficient 78 Slide 79 Large sample approximation 79 Slide 80
Hypothesis testing 80 Slide 81 Example Table 5.1 Wine Consumption
and Heart Disease Deaths Country Australia2.5211Netherlands1.8167
Austria3.9167New Zealand1.9266 Belgium2.9131Norway0.8227
Canada2.4191Spain6.586 Denmark2.9220Sweden1.6207
Finland0.8297Switzerland5.8115 France9.171U.K.1.3285
Iceland0.8211U.S.1.2199 Ireland0.7300W. Germany2.7172 Italy7.9107
81 Slide 82 Example 82 Slide 83 Example Table 5.2 Ranks of Wine
Consumption and Heart Disease Deaths 11112.5-1.51186.51.5
2156.58.512916-7.0 313.55-8.513315-12.0 41091.01417215.0
513.514-0.515711-4.0 6318-15.016 412.0 719118.0176 -11.0
8312.5-9.518510-5.0 9119-18.0191284.0 1018315.0 83 Slide 84 Example
84 Slide 85 Example 85 Slide 86 6. Kendalls Rank Correlation
Coefficient 86 Slide 87 Kendalls Tau It is a coefficient use to
measure the association between two pairs of ranked data. Named
after British statistician Maurice Kendall who developed it in
1938. Ranges from -1.0 to 1.0 Tau-a (with no ties) and Tau-b (with
ties) 87 Slide 88 Formula for Tau-a 88 Slide 89 Concordant and
Discordant 89 Slide 90 Example 1 Kendalls tau-a Raw data for 11
students in 2 exams: Exam 1Exam 2 85 9895 9080 8375 5770 6365 7773
9993 8079 9688 6974 90 Slide 91 Ranks of exam results Exam1 xExam 2
ycd 1 291 2 190 3 380 4 561 5 460 6 741 7 640 8 921 9 820 101101
10C=50D=5 91 Slide 92 Calculation for 92 Slide 93 Steps for
calculating 1.Sort data x in ascending order, pair y ranks with x
2.Count c and d for each y 3.Sum C and D 4.Use formula to calculate
93 Slide 94 Formula for tau-b(with ties) 94 Slide 95 Example 2
Kendalls tau-b Wine Consumption and heart disease deaths data
iCountryxiyicd 1Ireland0.7300018 2Iceland0.8211311 2Norway0.8227213
4Finland0.8297015 5U.S.1.219959 6U.K1.3285013 7Sweden1.620739
8Netherlands1.816755 9N. Z1.9266010 Canada2.419127
11Australia2.521117 12Germany2.717216 13Belgium2.913124
14Denmark2.922005 15Austria3.916704 16Switzerland5.811503
17Spain6.58611 18Italy7.910701 19France9.17100 C=25D=141 95 Slide
96 Calculation for tau-b 96 Slide 97 Hypothesis Test for 97 Slide
98 Hypothesis test results 98 Slide 99 Hypothesis test results 99
Slide 100 100 Slide 101 Example 1 extension Exam1 xExam 2 y
Kendall's Spearman r s 129111 219011 338000 456111 546011 674111
764011 892111 982011 10110111 1011 C=50D=5 101 Slide 102 102 Slide
103 103 Slide 104 SAS Code Data exams; Input exam1 exam2;
Datalines; 85 98 95 ; Run; Proc corr data=exams kendall; Var exam1
exam2; Run; 104 Slide 105 SAS output 105 Slide 106 7. Conclusion
106 Slide 107 Summary Nonparametric tests are very useful when we
dont know anything about the distributions. Especially when the
distribution is not normal, we cant use T-test, then we have to
study the nonparametric methods. Median is a better measurement of
central tendency for non-normal population. Sample can be ordinal
and sample size is usually small. 107 Slide 108 Summary In summary,
we have briefly introduced some most common methods in our
presentation including: Sign test Wilcoxon rank sum test and signed
rank test Kruskal-Wallis Test Friedman Test Spearmans Rank
Correlation Kendalls Rank Correlation Coefficient 108 Slide 109
Questions Q1Q2 109 Slide 110 The End. Thank You ! 110
