The Cross-Product Criterion for a 2×2 Table

Why Compare Opposite-Corner Products?

A two-way table organizes counts for two categorical variables. When each variable has two categories, the table has four interior counts. Independence means that, within the table, knowing the category of one variable does not change the distribution of the other. The cross-product criterion turns that idea into a compact check: multiply the counts in one pair of opposite corners, multiply the other pair, and compare the results.

This criterion is useful because it expresses independence directly in terms of the four counts. It does not require calculating a separate proportion for every cell. To use it correctly, however, the cells must be placed and labeled consistently, and it is important to distinguish a property of the observed table from a conclusion about a larger population.

Set Up the Table and the Criterion

Suppose each individual is classified by one of two row categories and one of two column categories. Let \(a,b,c,d\) be the four counts, with row categories in the rows and column categories in the columns:

	Column 1	Column 2	Total
Row 1	\(a\)	\(b\)	\(a+b\)
Row 2	\(c\)	\(d\)	\(c+d\)
Total	\(a+c\)	\(b+d\)	\(N\)

Here \(N=a+b+c+d\) is the total number of individuals. Assume the counts come from a complete classification: each individual contributes to exactly one cell, and the two categories of each variable are mutually exclusive and exhaustive. Also assume \(N>0\). These are counts, so \(a,b,c,d\) are nonnegative integers.

Definition: A two-by-two table is independent when the joint proportion in every cell equals the product of the corresponding row proportion and column proportion. For example, the proportion in the upper-left cell is \(a/N\); independence requires it to equal \(((a+b)/N)((a+c)/N)\). In a two-by-two table, independence is equivalent to the cross-product criterion \(ad=bc\): the products of the two opposite-corner pairs are equal.

The products \(ad\) and \(bc\) use opposite corners of the table. The criterion compares those products, not the sums of the rows or columns. Its value is that it gives a simple exact test of independence for a two-by-two table of counts.

Why the Criterion Is Equivalent to Independence

Start with the definition of independence for the upper-left cell. Its joint proportion is \(a/N\), the row 1 marginal proportion is \((a+b)/N\), and the column 1 marginal proportion is \((a+c)/N\). Thus independence requires

\frac{a}{N}=\frac{a+b}{N}\frac{a+c}{N}.

Since \(N>0\), multiply both sides by \(N^2\), then expand and use \(N=a+b+c+d\):

$$ aN=(a+b)(a+c) $$ $$ a(a+b+c+d)=a^2+ab+ac+bc $$ $$ a^2+ab+ac+ad=a^2+ab+ac+bc $$ $$ ad=bc. $$

This proves that independence requires equal cross-products. To prove the converse, suppose \(ad=bc\). The difference between the two sides of the upper-left-cell independence equation after clearing denominators is

$$ aN-(a+b)(a+c)=ad-bc=0. $$

So the upper-left cell has the required product-of-marginals proportion. The other cells do as well: the corresponding differences for the upper-right, lower-left, and lower-right cells simplify, respectively, to \(bc-ad\), \(bc-ad\), and \(ad-bc\). Each is zero when \(ad=bc\). Therefore all four joint proportions equal their corresponding products of marginal proportions, which is independence. This establishes the equivalence in both directions.

The argument also covers zero counts. For example, if an entire row has count zero, both cross-products are zero; the row has marginal proportion zero, so its two joint proportions are zero as independence requires. No division by an individual cell count is needed.

Checking Independence in a Table

For an observed table, compute the two diagonal products using the four cell counts. If they are equal, the table’s joint proportions satisfy the definition of independence. If they differ, the variables are not independent in that observed table. The check is exact; it is not a test of whether a population relationship is statistically significant.

Worked Example: Equal Cross-Products

In a hypothetical set of 80 appointment records, suppose the counts are arranged by reminder status and attendance as follows:

	Attended	Did not attend	Total
Received reminder	12	18	30
No reminder	20	30	50
Total	32	48	80

The table is a complete two-by-two classification: each record falls in one reminder category and one attendance category. The cross-products are

ad=12(30)=360,\qquad bc=18(20)=360.

Since \(ad=bc\), the table satisfies the cross-product criterion. Thus reminder status and attendance are independent in these 80 records, in the precise descriptive sense that the table’s joint proportions equal the products of the corresponding marginal proportions. This result by itself does not establish that reminder status and attendance are independent in a broader population.

Worked Example: Unequal Cross-Products

Consider a second hypothetical table showing completion of a practice activity by two groups:

	Completed	Did not complete	Total
Group A	16	24	40
Group B	18	42	60
Total	34	66	100

Each of the 100 hypothetical records contributes to exactly one cell. Compute the two cross-products:

ad=16(42)=672,\qquad bc=24(18)=432.

The products are unequal: \(672\ne432\). Therefore this table is not independent. The conclusion concerns the arrangement of these counts; the criterion does not, on its own, measure uncertainty or determine whether a population-level relationship exists.

Using the Criterion to Find a Count

The same relationship can be used in reverse when one cell is unknown. This requires an explicit assumption: if the table is to be independent, its unknown count must make the cross-products equal. Solving that equation gives the count compatible with independence. It does not prove independence from incomplete information; it tells us what the count would need to be.

Worked Example: Solving for a Missing Count

Suppose a hypothetical two-by-two table has counts \(x\), 24, 15, and 30, arranged as

	Column 1	Column 2
Row 1	\(x\)	24
Row 2	15	30

Assume for this calculation that the table is independent. The cross-product criterion gives

$$ x(30)=24(15). $$

Calculate the known product and divide by 30:

30x=360,\qquad x=\frac{360}{30}=12.

The count that makes this table independent is \(x=12\). Checking the result gives \(12(30)=360\) and \(24(15)=360\), as required. All four counts are nonnegative integers, so the value is compatible with a count table. If \(x\) were an observed count rather than an unknown one, it would need to equal 12 for this table to be independent.

What the Criterion Does—and Does Not—Say

The cross-product criterion is a compact description of the association pattern in a two-by-two table. Equal products mean independence in the table; unequal products mean the table does not satisfy independence. Because it uses only the four counts, the criterion is unchanged in substance if the categories are reordered: swapping the two rows or the two columns exchanges which diagonal product is called \(ad\) and which is called \(bc\), but equality remains equality.

When the counts are collected from a sample, this check describes that sample’s table exactly. It does not account for sampling variability, and it is not a significance test for a population parameter. Such an inference requires additional methods and assumptions. Likewise, a table association alone does not show that one variable caused the other.

Common Mistakes

Comparing the wrong quantities. Independence is checked by multiplying opposite-corner counts, \(ad\) and \(bc\), not by comparing the two row totals or the two column totals.
Using the rule without fixing the layout. First identify which cell is \(a\), \(b\), \(c\), and \(d\). With a consistent two-by-two arrangement, the two diagonals are \(ad\) and \(bc\).
Treating a matching product as a population conclusion. The criterion establishes independence in the table being examined. If the counts are a sample, it does not show that the population variables are independent.
Forgetting the assumption in a missing-count problem. A value found by setting \(ad=bc\) is the value required under an independence assumption; it is not automatically the actual count.

Key takeaway: For a two-by-two table with counts \(a,b,c,d\) placed in row order, independence is equivalent to \(ad=bc\). Check the opposite-corner products carefully, and interpret the result as a statement about the table unless an appropriate inferential method is used.

Check Your Understanding

Try these questions using the cross-product criterion.

In the layout shown above, which two products must be equal for the table to be independent?
A table has \(a=9\), \(b=15\), \(c=12\), and \(d=20\). Calculate both cross-products and state whether the table is independent.
Explain why the cross-product criterion can be used even if one cell count is zero.
If three cells are 8, 12, and 18, what value must the fourth cell have for the table to satisfy independence? State which cell you treat as unknown.
A sample table has unequal cross-products. What can you conclude about independence in that table, and what can you not conclude about the population from this check alone?