Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 54

is firstly divided into ' K ' subseries of equal or unequal sample size. The total
variance of the samples is then calculated by the following expression (Kanji,
2001):

K

Ç

2

n 1 s

i i

i 1

s 2 =

(11)

nK

where s i = variance of the i th sample, and n = total sample size. Now the test-
static or limit is computed as (Kanji, 2001):

qs
n

W = (12)

12
t

where q is the Studentized range. The critical values for q can be obtained at
degrees of freedom (Q) from the standard table available in textbooks on
statistics (e.g., Sachs, 1972; Kanji, 2001). The degree of freedom (Q) can be
computed as:

K

È Ø

Ç

nK

Q (13)

É Ù

i

Ê Ú i 1

Here, n t is expressed as:

K

n t = (14)

11 1

È Ø

¹¹¹

Ê Ú

nn n

1 2 K

If the limit ( W ) exceeds by the absolute difference between any two
sample means, it suggests that the corresponding population means differ
significantly.

4.1.5 Link-Wallace Test
The Link-Wallace test is employed for the purpose similar to the Tukey test;
however it has the limitation that the sample size of all populations must be
equal. It is a parametric test based on the assumption that the ' K ' populations
are normally distributed with equal variances. This test can be used to examine
the homogeneity of any hydrologic time series, x t ( t = 1, 2, …, n ) after
dividing the entire series into ' K ' subseries of equal sample size n k . The test-
statistic ( K L ) is defined as (Kanji, 2001):

nwx ()

k

K L = (15)

K

Ç

wx ()

i

i 1