Question

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

Chi-square tests
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asked 2021-01-22
How does the size of the cross-tabulation table impact the chi-square value.

Answers (1)

2021-01-23

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 \(\displaystyle{i}^{{{t}{h}}}{r}{o}{w}{\quad\text{and}\quad}{j}^{{{t}{h}}}\) column is the value of the variable of interest for the \(\displaystyle{i}^{{{t}{h}}}\) category of the row variable and \(\displaystyle{j}^{{{t}{h}}}\) 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, \(\displaystyle{d}{f}={\left({r}–{1}\right)}\times{\left({c}–{1}\right)}\).
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, \(\displaystyle\alpha\) is:
The p-value for the usual right-tailed test, for the statistic value, \(\displaystyle{x}^{{2}}\) is:
Note that, the distribution of the chi-squared test statistic under the null hypothesis, which is the chi-squared distribution with \(\displaystyle{d}{f}={\left({r}–{1}\right)}\times{\left({c}–{1}\right)}\) 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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