Statistical Methods

Unit 7 - Chi-square (X²) distribution

7.6. Computation of in r x c contingency table
The r x c contingency table is an extension of 2 x 2 contingency table in which the data are classified into ‘r’ rows and ‘c’ columns (table 3). In this table the frequencies which occupy cells of the table are called ‘cell frequencies’ whereas row and column totals are called the ‘marginal frequencies;
r x c contingency table

A\B	B1	B2	...	Bj	...	Bc	Total
A1	O11	O12	...	O1j	...	O1c	A1
A2	O21	O22	...	O2j	...	O2c	A2
. . .	...	...	...	...	...	...	...
Ai	Oi1	Oi2	...	Oij	...	Oic	Ai
. . .	...	...	...	...	...	...	...
Ar	Or1	Or2	...	Orj	...	Orc	Ar
Total	(B1)	(B2)	..	Bj	..	BC	N

As the table consists of ‘r’ rows and ‘c’ columns, there will be (r x c) observed frequencies, one in each cell. Corresponding to each observed frequency, there is expected frequency, computed based on certain hypothesis. Under the null hypothesis of no relationship or of independence between the attributes, expected frequency of each cell is computed by multiplying totals of the row and column to which the cell belongs divided by the total number of observations. For instance, the expected frequency of the cell in 1st row and 2nd column is obtained by multiplying the 1st row total (A1) with the 2nd column total (B2) and then dividing by the total number of observations, ‘n’. After calculating the expected frequencies for each cell, is computed using the formula,

Which has (r-1) (c-1) degrees of freedom.

Example: In a fish tagging experiment, the length frequency of tagged fishes and recoveries were as under. Test whether the length distributions can be accepted as same?

	Length group (cm)
	10-20	20-30	30-40	40-50	50-60	Total
Fishes tagged	108	140	256	358	111	1000
Fishes recovered	9	15	28	40	8	100

Answer:
(i) Hypotheses
Ho : Length distribution of tagged and recovered fishes is the same.
Hi : Length distribution of tagged and recovered fishes is not the same.
(ii) Test statistics

	Length group (cm)
	10-20	20-30	30-40	40-50	50-60	Total
Fishes tagged	108	140	256	358	111	1000
Fishes recovered	9	15	28	40	8	100
Total	117	155	284	425	119	1100

To compute statistic the following computations are to be made;


= 1101.40 - 1100 = 1.40
(iii) Statistical decision
Table value of with 4 df at 5% level of significance is 9.488. As the calculated value of is less than table value of , the null hypothesis is not rejected.