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Last updated on Fri May 21 11:52:19 IST 2010.

k-sample inference

Kruskal-Wallis Test

Example: Suppose that we want to compare among the performance of k fertilisers on growth of plants. For this we take k groups of plants, with ni many plants in group i, and apply fertiliser i on the i-th group. Let the j-th plant in the i-th group experience a growth of amount Xij. We assume that all the Xij's are mutually independent and 

   X1j's are iid F1 for j=1,...,n1
   X2j's are iid F2 for j=1,...,n2
   ...
   Xkj's are iid Fk for j=1,...,nk,
where the Fi's are unknown continuous distributions. We want to test
H0: All Fi's are the same Vs. H1: not H0.

In the Kruskal-Wallis test we first pool all the k samples. Let Rij denote the rank of Xij in the pooled sample. Define
Ri = ∑ j Rij.

Exercise 11.1: Find E(Ri) under H0.

In the KW test we use the test statistic
H = 12/(n(n+1) ∑ i [Ri-ni(n+1)/2]2/ni,
where n = ∑ ni.

Here is the motivation behind this test. By the last exercise we see that H is a linear combination of the squared deviations of Ri's from their respective null means. Thus a large value of R signals potential departure from H0. The constants are chosen appropriately so that the following asymptotic distribution holds:

Theorem If all the ni's are "moderately large" then under the null hypothesis
H is asymptotically χ2(k-1).

Proof: Not to be done in this course. Note that we have not even stated the theorem fully rigourously, leaving the term "moderately large" undefined.

Exercise 11.2: Argue that H is distribution-free under H0. No asymptotics here.

Exercise 11.3: Show that

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© Arnab Chakraborty (2010)