Question

Consider again the conditions of Exercise 5. Explain how to change the numbers in Table 10.47 in such a way that each row total and each column total remains unchanged, but the value of the χ2 test statistic is increased.

Short answer

An example for the change in observed values is:

Group/blood types	1	2	3	Total
A	24-10=14	43	69+10=79	136
B	6	24	47	77
AB	5+10=15	7	22-10=12	34
O	15	26	62	103
Total	50	100	200	350

The test statistic is 29.99.

Step-by-step solution

Given Information

The count of rows is 3 (r).

The count of columns is 4(c).

The significance level is 0.1\(\left( \alpha \right)\)

The given information is tabulated as follows,

Group/blood types	1	2	3	Total
A	24	43	69	136
B	6	24	47	77
AB	5	7	22	34
O	15	26	62	103
Total	50	100	200	350

Formulate the hypotheses

The hypothesis is formulated that the distribution of blood types is the same in all three groups as follows:

\({H_0}\): Distribution of blood types is the same in all three groups.

\({H_1}\): Distribution of blood types is not the same in all three groups.

State the formula for test statistics and the expected frequencies

The expected values are obtained using the following formula,

\({E_{ij}} = \frac{{{r_i} \times {c_j}}}{G}\)

Thus, the expected values are:

Group/ blood types	1	2	3	Total
A	19.43	38.86	77.71	136
B	11	22	44	77
AB	4.86	9.71	19.43	34
O	14.71	29.43	58.86	103
Total	50	100	200	350

The test statistic is formulated using the following formula;

\({\chi ^2} = \sum\limits_{i,j = 1}^n {\frac{{{{\left( {{N_{ij}} - {E_{ij}}} \right)}^2}}}{{{E_{ij}}}}} \)

It is required to change the observed values such that:

• Row and column total stay constant.
• Test statistic increases.

Obtain the new observed values

Expected value would remain same if the row and column table changes.

The value of the\({\chi ^2}\)test statistic will be increased when some values\({N_{i,j}}\)in the table deviate more strongly from their expected value\({E_{i,j}}\).

The test statistic\({\chi ^2}\)is increased in following steps:

• Choosing two distinct rows\(i,i'\) and choose two distinct columns\(j,j'\)such that,\(\begin{array}{l}{N_{i,j}} > {E_{i,j}}\\{N_{i',j'}} > {E_{i',j'}}\\{N_{i',j}} < {E_{i',j}}\\{N_{i,j'}} < {E_{i,j'}}\end{array}\).

• Increase\({N_{i,j}}\)and\({N_{i'j'}}\)by some constant c, and decrease\({N_{i'j}}\)and\({N_{i,j'}}\)by the constant c such that none of the counts become negative.

Example:

The observed values are changes as follows:

Group/blood types	1	2	3	Total
A	24-10=14	43	69+10=79	136
B	6	24	47	77
AB	5+10=15	7	22-10=12	34
O	15	26	62	103
Total	50	100	200	350

Obtain the new chi-square test statistic

The chi-square test statistic is obtained using the expected values computed using the new observed values are:

New expected values:

Group/ blood types	1	2	3	Total
A	19.43	38.86	77.71	136
B	11.00	22.00	44.00	77
AB	4.86	9.71	19.43	34
O	14.71	29.43	58.86	103
Total	50	100	200	350

New chi-square test statistic:

Group/ blood types	1	2	3	Total
A	1.5168	0.4417	0.0213	136
B	2.2727	0.1818	0.2045	77
AB	21.1807	0.7584	2.8403	34
O	0.0055	0.3994	0.1678	103
Total	50	100	200	350

Thus, the original chi-square test statistic is 6.829.

The new test statistic value is 29.99.

Therefore, the test statistic value is increased to 29.99 by maintaining the row and column totals.