Methods for Time Series Analysis - Hydrologic Time Series Analysis: Theory and Practice - page 56

4.1.7 Bartlett Test

Techniques for comparing means of normally distributed populations generally
assume that the populations have the same variance. Before using ANOVA, it
should be confirmed whether this assumption of homogeneity of variance is
reasonable. The Bartlett test is widely used for equal variances. The step-by-
step procedure for applying this test to a hydrologic time series x t ( t = 1, 2, …,
n ) is given below.
Step 1: Fragment the entire series into ' K ' subseries with n i size of each i th
series ( i = 1, 2, …, K ).
Step 2: Setup the null hypothesis that the variances of all subseries are equal
and alternative hypothesis of unequal variances.
Step 3: Compute the sample variance ( s i 2 ) of each subseries as:

2

n

xx
n

ij i

Ç

s 2 =

(20)

1

i

j 1

Step 4: Compute the overall variance as (Kanji, 2001):

K

Ç

2

n 1 s

i i

i 1

s 2 = (21)

K

Ç

n 1

i

i 1

Step 5: Calculate the Bartlett test-statistic, which is defined as (Kanji, 2001):

K K

Ë Û

2.30259

Ç Ç

2 2

n 1log s n 1log s

B = Ì Ü (22) i i i

C

Ì Ü

Í Ý

i 1 i 1

where ' C ' is a bias correction factor and is mathematically expressed as
follows:

K

Î Þ

1 1 1

Ç

1

C = Ñ Ñ (23)

3 K 1 n 1

K

Ï ß

i

i 1

Ç

n 1

Ñ Ñ

i

Ð à

Step 6:
Case A: If n i > 6, ' B ' will approximate to a F 2 -distribution with ' K -1' degrees
of freedom. Critical values of the test-statistic can be obtained from
the standard tables of F 2 -distribution.
Case B: If n i 6, the test-statistic becomes BC = M , for which the critical
values can be obtained from Kanji (2001).
In both the cases, the null hypothesis of equal variances is rejected if the
computed value of the test-statistic is greater than its critical values.

i 1