JONCKHEERE TERPSTRA

JONCKHEERE TERPSTRA

Analysis Command

Perform a Jonckheere Terpstra test that k samples come from identical populations.

Analysis of Variance (ANOVA) is a data analysis technique for examining the significance of the factors (= independent variables) in a multi-factor model. The one factor model can be thought of as a generalization of the two sample t-test. That is, the two sample t-test is a test of the hypothesis that two population means are equal. The one factor ANOVA tests the hypothesis that k population means are equal.

The Kruskal-Wallis test is a nonparametric alternative to the one factor ANOVA. The Jonckheere Terpstra test is another nonparametric alernative. The distinction is that the Jonckheere-Terpstra test alternative is used when there is an expected rank order of the factor level effects. That is, given that there are k levels for the factor it tests the hypothesis

\( H_{0} = F_{1}(x) = F_{2}(x) = \ldots = F_{k}(x) \)
\( H_{a} = F_{1}(x) \ge F_{2}(x) \ge \ldots \ge F_{k}(x) \)

Given k samples \( X_1, X_2, \dots , X_k \) of size \( n_1, n_2, \ldots , n_k \), the Jonckheere-Terpa test statistic is defined as

\( S = \sum_{i = 1}^{k-1}{\sum_{j=i+1}^{k}{p_{ij}}} - \sum_{i = 1}^{k-1}{\sum_{j=i+1}^{k}{n_{i} n_{j}}} \)

where \( p_{ij} \) is sum of the number of times \( X_i < X_j \).

Dataplot computes the critical values for this test in two ways.

1. The first method uses a permutation test (see Higgins). That is, the response values are permuted NITER times and the test statistic is computed for each permutation. This forms the reference distribution The p-value is based on the number of permutations that have Jonckheere-Terpstra value greater than value computed from the original data.

2. The second method is a large sample normal approximation. Compute the adjusted statistic
   \( S_{adj} = \frac{S} {\sqrt{ \frac{1}{18}(n^{2}(2n + 3) - \sum_{i=1}^{k}{n_{i}^2(2n_{i} + 3)} }} \)

   This adjusted statistic is compared to a standard normal distribution. The p-value is then \( 1 - \phi^{-1}(S_{adj}) \) where \( \phi^{-1} \) is the cumulative distribution function of the standard normal distribution.

The Jonckheere-Terpstra test is an upper tailed test.

JONCKHEERE TERPSTRA <y> <x>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is the response (= dependent) variable;
            <x> is the factor (= independent) variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

JONCKHEERE TERPSTRA Y X
JONCKHEERE TERPSTRA Y X SUBSET X = 1 TO 4

The group-id variable should be coded from the smallest to largest anticpated ordering.

The data does not need to be pre-sorted either within level or between levels.

By default, Dataplot generates 4,000 permutations. To change this, enter the command
SET PERMUTATION TEST SAMPLE SIZE <value>

If <value> is less than 100, it will be set to 100. If <value> is greater than 100,000, it will be set to 100,000.

Permutation tests assume the observations are independent. However, no distributional assumptions are made about the response variable.

This routine uses a random permutation algorithm suggested by Knuth. Specifically, it adapts the RANDPERM routine of Knoble.

The following parameters are saved after the k sample permutation test is performed.

STATVAL	-	value of the test statistic
STATCDF	-	CDF of the test statistic
PVALUE	-	p-value of the test statistic
P80	-	80% upper critical value
P90	-	90% upper critical value
P95	-	95% upper critical value
P975	-	97.5% upper critical value
P99	-	99% upper critical value
P995	-	99.5% upper critical value
P999	-	99.9% upper critical value
P20	-	20% upper critical value
P10	-	10% upper critical value
P05	-	5% upper critical value
P025	-	2.5% upper critical value
P01	-	1% upper critical value
P005	-	0.5% upper critical value
P001	-	0.1% upper critical value

The computed values of the statistic for each permutation are written to the file dpst1f.dat. The use of this file is demonstrated in the program examples below.

The full permutations are written to the file dpst2f.dat.

None

JONCKHEERE TERPSTRA TEST

ONE WAY ANOVA	=	Perform a one factor analysis of variance.
KRUSKAL WALLIS TEST	=	Perform a Kruskal-Wallis nonparameteric one factor analysis.
MEDIAN POLISH	=	Carries out a robust ANOVA.
T TEST	=	Carries out a t test.
MANN WHITNEY TEST	=	Perform a Mann-Whitney test.
BOX PLOT	=	Generate a box plot.

Jonckheere (1954), "Distribution-Free k-Sample Test Against Ordered Alternatives," Biometrika, Vol. 41, No. 1/2, pp. 133-145.

Higgins (2004), "Introduction to Modern Nonparameteric Statistics," Duxbury Press, chapter 3, p. 101.

Knuth (1998), "The Art of Computer Programming: Volume 2 Seminumerical Algorithms, Third Edition", Section 3.4.2, Addison-Wesley.

Knoble RANDPERM algorithm downloaded from: "http://coding.derkeiler.com/Archive/Fortran/comp.lang.fortran/ 2006-03/msg00748.html"

One Factor Analysis

2024/03

 
set random number generator fibbonacci congruential
seed 26101
.
.           Read the data
.
read jonck.dat x y
.
.           Perform the Jonckheere-Terpstra test
.
jonckheere terpstra test y x

The following output is generated

             Jonckheere-Terpstra Permutation Test
             (Critical Values Determined from Permutation Test)
  
 Response Variable:  Y
 Group-ID Variable:  X
  
 H0: F1(x) = F2(x) = ... = Fk(x)
 Ha: F1(x) >= F2(x) >= ... >= Fk(x)
  
 Test:
 Number of Permutation Samples:                     4000
 Statistic Value:                                46.0000
 Test CDF Value:                                  0.9885
 Test P-Value:                                    0.0115
  
  
             Conclusions (Upper 1-Tailed Test)
  
 ------------------------------------------------------------
                                                         Null
    Significance           Test       Critical     Hypothesis
           Level      Statistic    Region (>=)     Conclusion
 ------------------------------------------------------------
           80.0%        46.0000        18.0000         REJECT
           90.0%        46.0000        28.0000         REJECT
           95.0%        46.0000        34.0000         REJECT
           99.0%        46.0000        48.0000         ACCEPT
  
  
             Jonckheere-Terpstra Test
             (Critical Values Determined from Normal Approximation)
  
 Response Variable:  Y
 Group-ID Variable:  X
  
 H0: F1(x) = F2(x) = ... = Fk(x)
 Ha: F1(x) >= F2(x) >= ... >= Fk(x)
  
 Test:
 Number of Permutation Samples:                     4000
 Statistic Value:                                46.0000
 Adjusted Statistic Value:                        2.1479
 Test CDF Value:                                  0.9841
 Test P-Value:                                    0.0159
  
  
             Conclusions (Upper 1-Tailed Test)
  
 ------------------------------------------------------------
                                                         Null
    Significance  Adjusted Test       Critical     Hypothesis
           Level      Statistic    Region (>=)     Conclusion
 ------------------------------------------------------------
           80.0%         2.1479         0.8416         REJECT
           90.0%         2.1479         1.2816         REJECT
           95.0%         2.1479         1.6449         REJECT
           99.0%         2.1479         2.3263         ACCEPT

.
.           Plot the results
.
title offset 7
title case asis
label case asis
y1label Count
x1label Jonckheere-Terpstra Permutations
 
let statval = round(statval,3)
let p95  = round(p95,3)
let pval = round(pvalue,3)
let statcdf = round(statcdf,3)
.
x2label color red
x2label JT of Original Sample: ^statval
x3label color blue
x3label 95 Percentile: ^P95
. xlimits -1.0 1.0
let niter = 4000
skip 1
read dpst1f.dat z
title Histogram of Jonckheere-Terpstra Test for ^niter Permutationscr() ...
      (Pvalue: ^pval, CDF: ^statcdf)
.
histogram z
.
line color red
line dash
drawdsds statval 20 statval 90
line color blue
line dash
drawdsds p95 20 p95 90
    

 
set random number generator fibbonacci congruential
seed 391631
.
.           Read the data
.
skip 25
read con326.dat y x
.
.           Perform the Jonckheere-Terpstra test
.
jonckheere terpstra test y x

             Jonckheere-Terpstra Permutation Test
             (Critical Values Determined from Permutation Test)
  
 Response Variable:  Y
 Group-ID Variable:  X
  
 H0: F1(x) = F2(x) = ... = Fk(x)
 Ha: F1(x) >= F2(x) >= ... >= Fk(x)
  
 Test:
 Number of Permutation Samples:                     4000
 Statistic Value:                               204.0000
 Test CDF Value:                                  0.9993
 Test P-Value:                                    0.0008
  
  
             Conclusions (Upper 1-Tailed Test)
  
 ------------------------------------------------------------
                                                         Null
    Significance           Test       Critical     Hypothesis
           Level      Statistic    Region (>=)     Conclusion
 ------------------------------------------------------------
           80.0%       204.0000        56.0000         REJECT
           90.0%       204.0000        82.0000         REJECT
           95.0%       204.0000       106.0000         REJECT
           99.0%       204.0000       148.0000         REJECT
  
  
             Jonckheere-Terpstra Test
             (Critical Values Determined from Normal Approximation)
  
 Response Variable:  Y
 Group-ID Variable:  X
  
 H0: F1(x) = F2(x) = ... = Fk(x)
 Ha: F1(x) >= F2(x) >= ... >= Fk(x)
  
 Test:
 Number of Permutation Samples:                     4000
 Statistic Value:                               204.0000
 Adjusted Statistic Value:                        3.4004
 Test CDF Value:                                  0.9997
 Test P-Value:                                    0.0003
  
  
             Conclusions (Upper 1-Tailed Test)
  
 ------------------------------------------------------------
                                                         Null
    Significance  Adjusted Test       Critical     Hypothesis
           Level      Statistic    Region (>=)     Conclusion
 ------------------------------------------------------------
           80.0%         3.4004         0.8416         REJECT
           90.0%         3.4004         1.2816         REJECT
           95.0%         3.4004         1.6449         REJECT
           99.0%         3.4004         2.3263         REJECT

.
.           Plot the results
.
title offset 7
title case asis
label case asis
y1label Count
x1label Jonckheere-Terpstra Permutations
let statval = round(statval,3)
let p95 = round(p95,3)
let pval = round(pvalue,3)
let statcdf = round(statcdf,3)
.
x2label color red
x2label JT of Original Sample: ^statval
x3label color blue
x3label 95 Percentile: ^P95
let niter = 4000
skip 1
read dpst1f.dat z
title Histogram of Jonckheere-Terpstra Test for ^niter Permutationscr() ...
      (Pvalue: ^pval, CDF: ^statcdf)
.
histogram z
.
line color red
line dash
drawdsds statval 20 statval 90
line color blue
line dash
drawdsds p95 20 p95 90