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

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oppturf · Accepted Answer

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 $i^{t h} r o w and j^{t h}$ column is the value of the variable of interest for the $i^{t h}$ category of the row variable and $j^{t h}$ 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, $(r - 1)$ categories of the row variable and $(c - 1)$ categories of the column variable can take values freely. As a result, row degrees of freedom $= (r - 1)$ , column degrees of freedom $= (c - 1) .$
For the cross-tabulation, the total degrees of freedom, $d f = (r - 1) \times (c - 1)$ .
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, $x^{2}$ is:
Note that, the distribution of the chi-squared test statistic under the null hypothesis, which is the chi-squared distribution with $d f = (r - 1) \times (c - 1)$ 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.

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

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