How does the size of the cross-tabulation table impact the chi-square value.

necessaryh 2021-01-22 Answered
How does the size of the cross-tabulation table impact the chi-square value.
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oppturf
Answered 2021-01-23 Author has 94 answers

The chi-squared data is often expressed in a cross-tabulation, where each row represents a level or category of one categorical variable, and each column represents a level or category of the other categorical variable.
The value recorded in a cell corresponding to the ithrowandjth column is the value of the variable of interest for the ith category of the row variable and jth category of the column category.
Step 2
Explanation:
Suppose there are r rows, that is, there are r levels of the row variable, and c columns, that is, there are c levels of the column variable.
Since the grand total is a constant, and row total = column total = grand total, (r1) categories of the row variable and (c1) categories of the column variable can take values freely. As a result, row degrees of freedom =(r1), column degrees of freedom =(c1).
For the cross-tabulation, the total degrees of freedom, df=(r1)×(c1).
For a chi-squared test, there are two ways to conclude the hypothesis test- using the critical value, and using the p-value.
The critical value for the usual right-tailed test, at level of significance, α is:
The p-value for the usual right-tailed test, for the statistic value, x2 is:
Note that, the distribution of the chi-squared test statistic under the null hypothesis, which is the chi-squared distribution with df=(r1)×(c1) degrees of freedom, is used in both cases.
Clearly, the size of the cross-tabulation determines the degrees of freedom of the null chi-squared distribution, which, in turn, is the key point in establishing the rejection rule, and hence, forming the conclusion of the chi-squared test of significance.

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