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

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

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.