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Why nonparametric methods What test to use ? Rank Tests
Parametric and non-parametric statistical methodsfor the life sciences - Session I
Liesbeth Bruckers Geert Molenberghs
Interuniversity Institute for Biostatistics and statisticalBioinformatics (I-Biostat)
Universiteit Hasselt
June 7, 2011
June 6, 2011Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests
Table of contents
1 Why nonparametric methodsIntroductory exampleNonparametric test of hypotheses
2 What test to use ?Two independent samplesMore then two independent samplesTwo dependent samplesMore then two dependent samplesOrdered hypotheses
3 Rank TestsWilcoxon Rank Sum TestKruskal-Wallis TestFriedmann StatisticSign TestJonckheere-Terpstra Test
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Introductory example Nonparametric test of hypotheses
Why nonparametric methods ?
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Introductory example Nonparametric test of hypotheses
Introductory Example
The paper Hypertension in Terminal Renal Failure, ObservationsPre and Post Bilateral Nephrectomy (J. Chronic Diseases (1973):471-501) gave blood pressure readings for five terminal renalpatients before and 2 months after surgery (removal of kidney).
Patient 1 2 3 4 5Before surgery 107 102 95 106 112After surgery 87 97 101 113 80
Question: Does the mean blood pressure before surgery exceed the
mean blood pressure two months after surgery ?
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Introductory example Nonparametric test of hypotheses
Classical Approach
Paired t-test:
Patient 1 2 3 4 5Before surgery 107 102 95 106 112After surgery 87 97 101 113 80Difference Di 20 5 -6 -7 32
Hypotheses: H0 : µd = 0 versus H1 : µd > 0
µd : mean difference in blood pressure
Test-Statistic : t = D√1
n(n−1)
∑(Di−D)2
follows a t distribution with n − 1 d.f.
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Introductory example Nonparametric test of hypotheses
Assumptions
The statistic follows a t-distribution if the differences arenormally distributed ⇒ t-test = parametric method
Observations are made independent: selection of a patientdoes not influence chance of any other patient for inclusion
(Two sample t test): populations must have same variances
Variables must be measured in an interval scale, to interpretthe results
These assumptions are often not tested, but accepted.
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Introductory example Nonparametric test of hypotheses
Normal probability plot
Normality is questionable !
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Introductory example Nonparametric test of hypotheses
Nonparametric Test of Hypotheses
Follow same general procedure as parametric tests:
State null and alternative hypothesis
Calculate the value of the appropriate test statistic (choicebased on the design of the study)
Decision rule: either reject or accept depending on themagnitude of the statistic
PH0 (T ≥ c) = ??Exact distributionApproximation for the exact distribution
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Two independent samples More then two independent samples Two dependent samples More then two dependent samples Ordered hypotheses
When to use what test
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Two independent samples More then two independent samples Two dependent samples More then two dependent samples Ordered hypotheses
What test to use ?
Choice of appropriate test statistic depends on the design of thestudy:
number of groups ?
independent of dependent samples ?
ordered alternative hypothesis ?
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Two independent samples More then two independent samples Two dependent samples More then two dependent samples Ordered hypotheses
Two Independent Samples
Permeability constants of the human chorioamnion (a placentalmembrane) for at term (x) and between 12 to 26 weeks gestationalage (y) pregnancies are given in the table below. Investigate thealternative of interest that the permeability of the humanchorioamnion for a term pregnancy is greater than for a 12 to 26weeks of gestational age pregnancy.
X (at term) 0.83 1.89 1.04 1.45 1.38 1.91 1.64 1.46Y (12-26weeks) 1.15 0.88 0.90 0.74 1.21
Statistical Methods:
t-test
Wilcoxon Rank Sum Test
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Two independent samples More then two independent samples Two dependent samples More then two dependent samples Ordered hypotheses
More Than Two Independent Samples
Protoporphyrin levels were determined for three groups of people -a control group of normal workers, a group of alcoholics withsideroblasts in their bone marrow, and a group of alcoholicswithout sideroblasts. The data is shown below. Does the datasuggest that normal workers and alcoholics with and withoutsideroblasts differ with respect to protoporphyrin level ?
Group Protoporphyrin level (mg)Normal 22 27 47 30 38 78 28 58 72 56Alcoholics with sideroblasts 78 172 286 82 453 513 174 915 84 153Alcoholics without sideroblasts 37 28 38 45 47 29 34 20 68 12
Statistical Methods:
ANOVA
Kruskal-Wallis Test
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Two independent samples More then two independent samples Two dependent samples More then two dependent samples Ordered hypotheses
Two Dependent Samples
Twelve adult males were put on liquid diet in a weight-reducingplan. Weights were recorded before and after the diet. The dataare shown in the table below.
Subject 1 2 3 4 5 6 7 8 9 10 11 12Before 186 171 177 168 191 172 177 191 170 171 188 187After 188 177 176 169 196 172 165 190 165 180 181 172
Statistical Methods:
Paired t-test
Sign test; Signed-rank test
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Two independent samples More then two independent samples Two dependent samples More then two dependent samples Ordered hypotheses
Randomized Blocked Design
Effect of Hypnosis:
Emotions of fear, happiness, depression and calmness wererequested (in random order) from 8 subject during hypnosis
Response: skin potential (in millivolts)
Subject 1 2 3 4 5 6 7 8
Fear 23.1 57.6 10.5 23.6 11.9 54.6 21.0 20.3Happiness 22.7 53.2 9.7 19.6 13.8 47.1 13.6 23.6Depression 22.5 53.7 10.8 21.1 13.7 39.2 13.7 16.3Calmness 22.6 53.1 8.3 21.6 13.3 37.0 14.8 14.8
Statistical Methods:
Mixed Models
Friedmann test
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Two independent samples More then two independent samples Two dependent samples More then two dependent samples Ordered hypotheses
Ordered Treatments
Patients were treated with a drug a four dose levels (100mg,200mg, 300mg and 400mg) and then monitored for toxicity.
Drug ToxicityDose Mild Moderate Severe Drug Death100mg 100 1 0 0200mg 18 1 1 0300mg 50 1 1 0400mg 50 1 1 1
Statistical Methods:
Regression
Jonckheere-Terpstra Test
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Wilcoxon Rank Sum Test
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Wilxocon Rank Sum Test
Detailed Example:
Data : GAF scores
Control 25 10 35Treatment 36 26 40
Does treatment improve the functioning ?
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Parametric Approach: t-test
t = X̄1−X̄0SX1−X0
, where SX1−X0=
√s21n1
+s20n0
t test: means of two normally distributed populations areequal
H0 : µ1 = µ0
H1 : µ1 6= µ0 (one sided test H1 : µ1 ≥ µ0
equal sample sizes
two distributions have the same variance
X̄1 = 34.00, X̄0 = 23.33, SX1 = 7.21,SX0 = 12.58
t = 1.27
PH0(t ≥ 1.27) = 0.1358
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Wilxocon Rank Sum Test
Detailed Example:
Control 25 10 35Treatment 36 26 40
Order data: Position of patients on treatment as comparedwith position of patients in control arm ?
Ranks
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Treatment is effective if treated patients rank sufficientlyhigh in the combined ranking of all patients
Test statistic such that:
treatment ranks are high ⇔ value test statistic is hightreatment ranks are low ⇔ value test statistic is low
WS = S1 + S2 + . . .+ Sn (n=3, number of patients in treatment arm)
Ranks
Control 2 1 4(25) (10) (35)
Treatment 5 3 6(36) (26) (40)
WS = 5+3+6 =14
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Reject null hypothesis when WS is sufficiently large : WS ≥ c
PH0(WS ≥ c) = α (alpha=0.05)
Distribution of WS under H0 ?
Suppose no treatment effect (H0)
rank is solely determined by patients health statusrank is independent of receiving treatment or placebo“rank is assigned to patient before randomisation”
Random selection of patients for treatment ⇒ randomselection of 3 ranks out of 6
Randomisation divides ranks (1,2,...6) into two groups !
Number of possible combinations :(Nn
)= N!
n!(N−n)!
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
All posibilities: (each as a probability of 1/20 under H0)
treatment ranks (4,5,6) (3,5,6) (3,4,6) (3,4,5) (2,5,6)ws 15 14 13 12 13treatment ranks (2,4,6) (2,4,5) (2,3,6) (2,3,5) (2,3,4)w 12 11 11 10 9treatment ranks (1,5,6) (1,4,6) (1,4,5) (1,3,6) (1,3,5)ws 12 11 10 10 9treatment ranks (1,3,4) (1,2,6) (1,2,5) (1,2,4) (1,2,3)ws 8 9 8 7 6
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Distribution of WS under the null hypothesis:
w 6 7 8 9 10 11 12 13 14 15
PH0(Ws = w) 1
201
202
203
203
203
203
202
201
201
20
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
PHO(WS ≥ 14) = 0.1
Do not reject H0.
Conclusion: Treatment does not increase the GAF scores.
Power of this study ???
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Large Sample Size-case
(Nn
)increases rapidly with N and n(
2010
)= 184756(
126
)= 924
Asymptotic Null Distribution: Central Limit Theorem
Sum T of large number of independent random variables isapproximately normally distributed.
P
(T − E (T )√
Var(T )≤ a
)≈ Φ(a)
where Φ(a) is the area to the left of a under a standard normal curve
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
If both n and m are sufficiently large:
WS ≈ N(E (WS);√Var(WS))
E (WS) = 12n(N + 1)
Var(WS) = 112nm(N + 1)
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Kruskal-Wallis Test
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Kruskal- Wallis test
Example: Kruskal- Wallis test:
The following data represent corn yields per acre from threedifferent fields where different farming methods were used.
Method 1 Method 2 Method 3
92 94 10191 90 10084 81 9389 102
Question: is the yields different for the 4 methods ?
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Parametric Approach One-way ANOVA
Statistical test of whether or not the means of several groupsare all equal
Assumptions:
Independence of casesThe distributions of the residuals are normal : εi ∼ (0, σ2).Homoscedasticity
F = variance between groupsvariance within groups = MSTR
MSE
Statistic follows a F distribution with s − 1, n − s d.f.
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Small F:
Large F:
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
One-Way ANOVA results
X̄1 = 89, X̄2 = 88.33, X̄3 = 99
σ1 = 3.56, σ2 = 6.65, σ3 = 4.08
MSTR= 135.03 , MSE = 22.08
F= 6.11
PH0(F ≥ 6.11) = 0.0245
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Ranks:
Method 1 Method 2 Method 3
6 8 105 4 91 2 73 11
Ri .: 3.75 4.666 6.75
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Hypothesis :H0: No difference between the treatmentsH1: Any difference between the treatments
If treatments do not differ widely (H0):
Ri. are close to each otherRi. close to R..
If treatments do differ (H1):
Ri. differ substantialRi. not close to R..
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Evaluate the null hypothesis by investigating:
K =12
N(N + 1)
s∑i=1
ni (Ri . − R..)2
PH0(K ≥ c) = ?
Exact distribution of K under H0 :
ranks are determined before assignment to treatmentrandom assignment → all possibilities same chance of beingobserved
Number of possible combinations: multinomial coefficient :( 114,3,4
)=(11
4
)(73
)(44
)= 11550( N
n1,n2,...,ns
)=(Nn1
)(N−n1n2
). . .(N−n1−...−ns−1
ns
)Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
A few possible configurations:
Method 1 Method 2 Method 3 K
(1,2,3,4) (5,6,7) (8,9,10,11) 8.91(1,2,3,5) (4,6,7) (8,9,10,11) 8.32(1,2,3,6) (4,5,6) (8,9,10,11) 7.84(1,2,3,7) (4,5,6) (8,9,10,11) 7,48
. . .(1,3,5,6) (2,4,8) (7,9,10,11) 6.16
. . .
Each configuration has a probability of 111550 to happen.
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Exact Distribution of K :
PH0(K ≥ 6.16) = 0.0306
Conclusion: Reject H0: there is a difference between thefarming methods
Large sample size approximation ” χ2 distribution with s − 1d.f.
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Friedmann Test
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Friedmann Statistic
Setting 1: complete randomization:Kruskal-Wallis test p-value =0.8611Treatment effect is blurred by the variability between subjects
Setting 2: randomisation within age groups:p-value 0.0411Conclusion reject H0
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Procedure
Divide subjects in homogeneous subgroups (BLOCKS)
Compare subjects within the blocks w.r.t. treatment effects
(Generalisation of the paired comparison design)
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Example
DataAge-group
treatment 20-30 y 30-40 y 40-50 y 50-60 yA 19 21 43 46B 17 20 37 44C 23 22 39 42
Rank subjects within a block:Age-group
treatment 20-30 y 30-40 y 40-50 y 50-60 yA 2 2 3 3B 1 1 1 2C 3 3 2 1
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Mean of ranks for:
treatment A = RA.=104 = 2.5
treatment B = RB.=64 = 1.5
treatment C = RC .=94 = 2.25
If these mean ranks are different → reject H0
If these mean ranks are close → accept H0
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Measure for closseness of the mean ranks:if the Ri . are all close to each other
↓then they are close to the overall mean R..
and(Ri . − R..)
2 will be close to zero
Friedman Statistic
Q =12N
s(s + 1)
s∑i=1
(Ri . − R..)2
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
PH0(Q ≥ c) =?
Exact distribution of Q under H0:
A few possible configurations:Age-group Q
Treatment 20-30 y 30-40 y 40-50 y 50-60 yA 1 1 1 1 8B 2 2 2 2C 3 3 3 3A 3 3 3 3 8B 2 2 2 2C 1 1 1 1A 1 3 1 3 0B 2 2 2 2C 3 1 3 1. . .A 2 2 3 3 3.5B 1 1 1 2C 3 3 2 1
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Exact Distribution of Q:
Q Pr—————————————-.0000000 .694444444444444E-01
.5000000 .277777777777778
1.500000 .222222222222222
2.000000 .157407407407407
3.500000 .148148148148148
4.500000 .555555555555555E-01
6.000000 .277777777777778E-01
6.500000 .370370370370370E-01
8.000000 .462962962962963E-02
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Number of possibilities for the rank combinations:
age-group 20- 30 year: 3! = 6age-groups are independent
↓total number of possible combinations: (3!)4 = 1296
Under the null these are all equally likely : 11296
(s!)N , s=] treatment groups, N = ] of blocks
PH0(Q ≥ 3.5) = 0.2731
Do not reject H0
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Sign Test
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Sign Test
Special case of Friedmann test: blocks of size 2
subjects matched on e.g. age, gender, ...twinstwo eyes (hands) of a personsubject serves as own control: e.g. blood pressure before and after treatment
Example: Pain scores for lower back pain, before and afterhaving acupuncture
Pain score Pain score Sign Pain score Pain score SignPatient Before After Patient Before After1 5 6 - 8 7 6 +2 6 7 - 9 6 5 +3 7 6 + 10 5 7 -4 9 4 + 11 8 6 +5 6 7 - 12 8 4 +6 5 4 + 13 7 3 +7 4 8 - 14 8 5 +
15 6 7 -
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
9 pairs out 15 where treatment comes out ahead (reduction in
pain scores)
Sign Test: SN = 9
PH0(SN ≥ 9) =???
Exact Distribution of SN under H0 is binomial
N trials, N = number of ‘pairs’Success probability: 1
2
PH0(SN = a) =
(N
a
)1
2N
PH0(SN ≥ 9) = ((15
9
)+(15
10
)+ . . .+
(1515
)) 1
215 = 0.31
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Jonckheere-Terpstra Test
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Jonckheere-Terpstra Test
To be used when the H1 is ordered.
Ordinal data for the responses and an ordering in thetreatment/groups.
Example:
Data:
Three diets for ratsResponse: growthH1: Growth rate decreases from A to C : A ≥ B ≥ C
A 133 139 149 160 184B 111 125 143 148 157C 99 114 116 127 146
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Parametric Approach : Regression
Models the relationship between a dependent and independentvariable
yi = β0 + β1xi + εiAssumptions
εi ∼ N(0, σ2), εi are independenthomoscedasticityxi is measured without error
Doctoral School Medicine
Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
β0 = 169, p-value = < 0.0001
β1 = −16, p-value = 0.0133
R-square = 0.3866
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Jonckheere-Terpstra Test
Based on Mann-Whitney statistics for two treatments
Comparing the treatment groups two by twoif WBA is large: growth A > growth B : (WBA= 18
if WBC is large: growth B > growth C : (WBC = 18
if WCA is large: growth A > growth C : (WBA= 23
JT Statistic: W =∑
i<j Wij
Reject H0 when W is sufficiently large
W = 59
PH0(W ≥ c) = 0.0120
Compare with the result of a Kruskal-Wallis Test: p-value =0. 072
The distribution of W follows a normal distribution for largesamples
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Why nonparametric methods What test to use ? Rank Tests Wilcoxon Rank Sum Test Kruskal-Wallis Test Friedmann Statistic Sign Test Jonckheere-Terpstra Test
Parametric versus nonparametric tests
Parametric tests:
Assumptions about the distribution in the population
Conditions are often not tested
Test depends on the validity of the assumptions
Most powerful test if all assumptions are met
Nonparametric tests:
Fewer assumptions about the distribution in the population
In case of small sample sizes often the only alternative (unless the
nature of the population distribution is known exactly)
Less sensitive for measurement error (uses ranks)
Can be used for data which are inherently in ranks, even fordata measured in a nominal scale
Easier to learn
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