Independence in a Two-Way Table

Why Independence Is About Distributions

A two-way table organizes counts for two categorical variables. It can show, for example, whether a response category is distributed similarly across groups. The key question is not whether the groups have the same number of observations, but whether knowing an individual's category for one variable changes the distribution of the other variable.

If the conditional distribution of one variable is the same in every group defined by the other variable, the table shows no association between the variables. If those conditional distributions differ, the variables are associated in the table. Independence gives a precise way to describe the matching case.

Definition: Two categorical variables \(X\) and \(Y\) are independent if, for every category \(x_i\) with positive probability, the conditional distribution of \(Y\) given \(X=x_i\) matches the marginal distribution of \(Y\). Equivalently, the conditional distribution of \(X\) given each category of \(Y\) matches the marginal distribution of \(X\). In a two-way table, an observed table has exact empirical independence when these corresponding conditional and marginal distributions match exactly.

A marginal distribution describes one variable across the whole table, using its row or column totals. A conditional distribution describes one variable within a specified category of the other variable, using that row or column total as the denominator. Independence is about comparing these distributions, not comparing the raw cell counts.

The definition refers to distributions, so every category must be considered. For example, if there are three response categories, it is not enough to compare just one of them; the proportions for all three must match the corresponding marginal proportions. In a sample table, matching is exact only if the proportions are exactly equal, not merely close after rounding.

Checking Independence in a Table

Suppose rows represent categories of \(X\), and columns represent categories of \(Y\). To check for empirical independence, compare each row's conditional distribution of \(Y\) with the marginal distribution of \(Y\). You can also compare each column's conditional distribution of \(X\) with the marginal distribution of \(X\). When the variables are independent, the conditional distributions match their corresponding margins.

For a table of counts, let \(n_{ij}\) be the count in row \(i\), column \(j\); let \(n_{i+}\) be the total for row \(i\), \(n_{+j}\) the total for column \(j\), and \(n\) the grand total. The conditional proportion for column \(j\) within row \(i\) is \(n_{ij}/n_{i+}\). The marginal proportion for column \(j\) is \(n_{+j}/n\). These proportions, rather than the counts \(n_{ij}\) alone, are what you compare.

Worked Example: Matching Conditional and Marginal Distributions

In a hypothetical survey, 20 students in one program and 30 in another report whether they prefer option P or option Q. The table shows the counts.

Program	Option P	Option Q	Total
Program A	12	8	20
Program B	18	12	30
Total	30	20	50

Within Program A, the conditional distribution is \(12/20=0.60\) for P and \(8/20=0.40\) for Q. Within Program B, it is \(18/30=0.60\) for P and \(12/30=0.40\) for Q. The marginal distribution across all students is \(30/50=0.60\) for P and \(20/50=0.40\) for Q. Both row conditional distributions match the margin exactly.

The column conditionals match as well. Among students preferring P, \(12/30=0.40\) are in Program A and \(18/30=0.60\) are in Program B. Across all students, \(20/50=0.40\) are in Program A and \(30/50=0.60\) are in Program B. The distribution of program is the same within the P group as it is overall. The same check for Q gives \(8/20=0.40\) in Program A and \(12/20=0.60\) in Program B, again matching the marginal distribution.

Thus, this observed table has exact empirical independence: preference proportions match across programs, and program proportions match across preference categories. This describes the table; it does not by itself establish that preference and program are independent in a wider population.

Why the Two Directions Agree

Independence is a symmetric relationship: either variable can be used to define the groups, and the other variable's conditional distribution will match its margin. The following calculation shows this directly for a finite table.

Suppose every row has the same conditional proportion \(q_j\) in column \(j\). That is, \(n_{ij}/n_{i+}=q_j\) for every row \(i\). Then \(n_{ij}=n_{i+}q_j\). Summing those counts down column \(j\), the marginal proportion for that column is

\frac{n_{+j}}{n} = \frac{\sum_i n_{ij}}{n} = \frac{\sum_i n_{i+}q_j}{n} = q_j\frac{\sum_i n_{i+}}{n} = q_j.

The last equality holds because the row totals sum to the grand total: \(\sum_i n_{i+}=n\). Now consider the conditional distribution of the row category among observations in column \(j\), assuming that column has a positive total. For each row \(i\),

\frac{n_{ij}}{n_{+j}} = \frac{n_{i+}q_j}{nq_j} = \frac{n_{i+}}{n}.

This is the marginal proportion for row \(i\). So when all row distributions match the column margin, the column distributions also match the row margin. The same reasoning applies to probability distributions, with probabilities in place of relative frequencies.

When Conditional Distributions Differ

If a conditional distribution differs from its corresponding marginal distribution, the table shows association. The difference may be large or small, and it should be described in context. In particular, the statement concerns a relationship between the variables; it does not establish that one variable causes the other.

Worked Example: A Visible Difference Between Groups

In a hypothetical poll, 30 people in Group A and 30 in Group B answer whether they support a proposal. The counts are:

Group	Support	Do not support	Total
Group A	18	12	30
Group B	10	20	30
Total	28	32	60

The support proportion in Group A is \(18/30=0.600\), or 60.0%. In Group B it is \(10/30=0.333\), or 33.3%. The marginal support proportion is \(28/60=0.467\), or 46.7%. The conditional support proportions are not equal to each other or to the marginal proportion. The corresponding proportions for “Do not support” also differ: \(12/30=0.400\) in Group A, \(20/30=0.667\) in Group B, and \(32/60=0.533\) overall.

This observed table shows an association between group and response: support is more common in Group A than in Group B. The table alone does not explain why the groups differ, and it does not show that group membership caused the responses.

Counts Can Differ While Proportions Match

The number of observations in each row or column affects raw counts. As a result, independent variables do not generally produce equal cell counts. A larger group can have more observations in every category while having the same conditional distribution as a smaller group.

Worked Example: Unequal Counts, Matching Distributions

In a hypothetical table about a preferred travel mode, 20 people live near a campus and 80 live farther away. Their responses are:

Residence	Walk	Other mode	Total
Near	8	12	20
Far	32	48	80
Total	40	60	100

The conditional distribution for people living near is \(8/20=0.40\) for Walk and \(12/20=0.60\) for Other mode. For people living farther away, it is \(32/80=0.40\) for Walk and \(48/80=0.60\) for Other mode. The marginal distribution is \(40/100=0.40\) for Walk and \(60/100=0.60\) for Other mode. The distributions match exactly.

The counts in the two rows are quite different: 8 people in the near group walk, compared with 32 in the far group. Comparing those counts alone might suggest a difference. But the groups have different sizes, and their walking proportions are both 40.0%. In this table, residence and preferred mode are empirically independent.

Exact Independence and Small Differences

Real sample tables need not match their margins exactly, even when the variables are independent in the population. Sampling variation can make the observed conditional proportions differ. Conversely, a close match in a sample is not proof of independence in the population. The table provides a descriptive comparison; a conclusion about a population requires an appropriate inferential analysis.

Worked Example: A Small Sample Difference

In a hypothetical sample, 50 people in Group A and 80 in Group B answer a yes-or-no question. The observed counts are:

Group	Yes	No	Total
Group A	31	19	50
Group B	49	31	80
Total	80	50	130

The conditional Yes proportions are \(31/50=0.620\) in Group A and \(49/80=0.6125\) in Group B. The marginal Yes proportion is \(80/130\approx0.6154\). Thus the observed conditional proportions differ from one another by \(0.620-0.6125=0.0075\), or 0.75 percentage points. Neither exactly matches the marginal proportion.

The observed table therefore does not have exact empirical independence. The differences are small, but “small” is a descriptive judgment, not a rule that proves independence. The table alone does not determine whether the population variables are independent; it only shows that the sample proportions are close.

Common Mistakes

Comparing counts instead of proportions. “Group B has more Yes responses, so the variables are associated” ignores that Group B may also be larger. Compare conditional proportions using each group's total.
Comparing a conditional proportion with the wrong margin. A row conditional distribution of column categories should be compared with the column marginal distribution, not the row margin.
Checking only one category. Independence requires the distributions to match, meaning all category proportions must agree, subject to rounding when displayed.
Treating approximate equality as proof. Close sample proportions describe a close sample pattern; they do not establish population independence.
Interpreting association as causation. Different conditional distributions show a pattern in the table, not a causal explanation for that pattern.

Key takeaway: In a two-way table, independence means that conditioning on a category of one variable does not change the distribution of the other variable. Check conditional proportions against the appropriate marginal proportions, not raw counts.

Check Your Understanding

Use conditional and marginal distributions, rather than raw counts alone, to answer these questions.

A table has two rows. The proportion in the first column is 0.30 in each row, and the marginal proportion in that column is 0.30. What does this say about the observed distributions?
Why might two independent groups have very different counts in the same response category?
A row conditional distribution is compared with a row margin. Explain why this is not the appropriate comparison for checking independence.
In a sample, one group's Yes proportion is 0.48 and another's is 0.50, while the marginal Yes proportion is 0.49. What can be said descriptively, and what cannot be concluded about the population from this comparison alone?
What does it mean, in context, for two categorical variables to be associated in an observed two-way table?