Jonckheere-Terpstra test

The Jonckheere-Terpstra test is a parameter-free statistical test that, similar to the Kruskal-Wallis test , uses an analysis of variance to compare whether different independent samples (groups) differ with regard to an ordinally scaled variable . The difference to the Kruskal-Wallis test is that it tests for the presence of a trend between the groups.

The null hypothesis H ₀ is that all sample values ​​were drawn from populations with an identical distribution: ${\displaystyle G_{i}}$ ${\displaystyle G_{1}=G_{2}=\dots =G_{c}}$

The following applies as an alternative hypothesis H _A , where at least one strict inequality applies. ${\displaystyle G_{1}\leq G_{2}\leq \dots \leq G_{c}}$

calculation

The test statistic for a number of groups , each with measurements: ${\displaystyle J}$ ${\displaystyle c}$ ${\displaystyle G_{i}}$ ${\displaystyle n_{i}}$

${\displaystyle J=\sum _{i<j}^{c}U_{ij}=\sum _{i=1}^{c-1}\sum _{j=i+1}^{c}U_{ij}}$

Where is defined as ${\displaystyle U_{ij}}$

${\displaystyle U_{ij}=\sum _{s=1}^{n_{i}}\sum _{t=1}^{n_{j}}\Psi (X_{jt}-X_{is})}$

With

${\displaystyle \Psi (u)={\begin{cases}1&{\text{wenn }}u>0\\0&{\text{wenn }}u\leq 0\end{cases}}\ }$ or in the case of ties (same measured values) ${\displaystyle \ \Psi (u)={\begin{cases}1&{\text{wenn }}u>0\\1/2&{\text{wenn }}u=0\\0&{\text{wenn }}u<0\end{cases}}}$

The calculated test variable increases when there is a trend between the groups. ${\displaystyle J}$

Under general conditions, the test variable is approximately normally distributed. The following formulas apply for the expected value and the variance : ${\displaystyle J}$ ${\displaystyle \mu _{J}}$ ${\displaystyle \sigma _{J}}$

${\displaystyle \mu _{J}={\frac {N^{2}-\sum _{i=1}^{c}n_{i}^{2}}{4}}\ }$

and

${\displaystyle \ \sigma _{J}={\sqrt {\frac {N^{2}(2N+3)-\sum _{i=1}^{c}n_{i}^{2}(2n_{i}+3)}{72}}}}$

The variable obtained from this through standardization is approximately standard normal distributed if the total number of all sample values ​​is greater than 12: ${\displaystyle Z}$ ${\displaystyle N}$

${\displaystyle Z={\frac {J-\mu _{J}}{\sigma _{J}}}}$

In other words: in the case of a one-sided test at the 5% level ( error of type 1 ) the test is significant if

${\displaystyle J>\mu _{J}+1{,}645\,\sigma _{J}}$.

generalization

In addition to a monotonous trend, models can also be edited in which an initial upward trend changes into a downward trend at a certain point. This is then the generalization of the Jonckheere-Terpsta test, the umbrella test according to Mack and Wolfe.

literature

• AR Jonckheere: A distribution-free k-sample test against ordered alternatives . In: Biometrica . 41, 1954, pp. 133-45. doi: 10.1093 / biomet / 41.1-2.133 JSTOR 2333011 .
• AR Jonckheere: A test of significance for the relation between m rankings and k ranked categories . In: British Journal of Statistical Psychology . 7, 1954, pp. 93-100. doi : 10.1111 / j.2044-8317.1954.tb00148.x .
• TJ Terpstra: The asymptotic normality and consistency of Kendall's test against trend, when ties are present in one ranking . In: Indagationes Mathematicae . 14, 1952, pp. 327-33.
• WJ Conover: Practical Nonparametric Statistics , 3rd Edition, John Wiley & Sons, New York 1999, ISBN 0-471-16068-7 , p. 5.4.

Individual evidence

1. ^ HB Mack, Wolfe, DA: K-sample rank tests for umbrella alternatives . In: J. Amer. Extra Ass. . 76, 1981, pp. 175-81. doi: 10.1080 / 01621459.1981.10477625 JSTOR 2287064 .