Umbrella test

The umbrella test according to Mack and Wolfe represents the generalization of the Jonkheere-Terpstra test . In contrast to this test, however, a monotonous trend is not assumed, but trends with a peak.

The null hypothesis H ₀ for the expected values ​​G of the groups reads: ${\displaystyle G_{1}\ =\ G_{2}\ =\ \dots \ =\ G_{c}}$

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

Calculation of the test variable

The test statistic MW for a number of groups with a peak at each measurement is: ${\displaystyle c}$ ${\displaystyle l}$ ${\displaystyle n}$

${\displaystyle MW\ =\ \sum _{r=1}^{l-1}\ \sum _{s=r+1}^{l}U_{rs}\ +\ \sum _{r=l}^{c-1}\ \sum _{s=r+1}^{c}U_{sr}}$

For the r-th and the s-th group , or is also defined as ${\displaystyle U_{rs}}$ ${\displaystyle U_{sr}}$ ${\displaystyle 1\leq r\ <s\leq c}$

${\displaystyle U_{rs}\ =\ \sum _{h=1}^{n_{r}}\ \sum _{i=1}^{n_{s}}\Psi (X_{rh}\ -\ X_{si})}$

and

${\displaystyle U_{sr}\ =\ n_{r}n_{s}\ -\ U_{rs}}$

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 if there is a biphasic trend between the groups. ${\displaystyle MW}$

Under general conditions, the test variable shows a normal distribution. ${\displaystyle MW}$

Checking the significance

The following formulas apply to the expected value and its variance , which ultimately result from adding the statistics of the Jonkheere-Terpstra test : ${\displaystyle \mu _{MW}}$ ${\displaystyle \sigma _{MW}}$

${\displaystyle \mu _{MW}\ =\ {\frac {m_{1}^{2}\ +\ m_{2}^{2}\ -\ \sum _{i=1}^{c}n_{i}^{2}\ -\ n_{l}^{2}}{4}}\ }$

and

${\displaystyle \ \sigma _{MW}\ =\ {\sqrt {\frac {2(m_{1}^{3}+m_{2}^{3})\ +\ 3(m_{1}^{2}+m_{2}^{2})\ -\ \sum _{i=1}^{c}n_{i}^{2}(2n_{i}+3)\ -\ n_{l}^{2}(2n_{l}+3)\ +\ 12n_{l}m_{1}m_{2}\ -\ 12n_{l}^{2}N}{72}}}}$

With

${\displaystyle N\ =\ \sum _{i=1}^{c}n_{i},\quad m_{1}\ =\ \sum _{i=1}^{l}n_{i}\quad {\text{und}}\ m_{2}\ =\ \sum _{i=l}^{c}n_{i}}$

The resulting variable has a standard normal distribution if the total number of all samples is greater than 12: ${\displaystyle Z}$

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

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

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

Individual evidence

1. ^ HB Mack, DA Wolfe: K-sample rank tests for umbrella alternatives. In: J. Amer. Extra Ass. , 76, 1981, pp. 175-181, doi: 10.1080 / 01621459.1981.10477625 , JSTOR 2287064
2. ^ 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. 327-333
3. ^ AR Jonkheere: A distribution-free K- sample test against ordered alternatives. In: Biometrika , 41, 1954, pp. 133-145, doi: 10.1093 / biomet / 41.1-2.133 , JSTOR 2333011