Differences Between Conditional Proportions

Why Compare Conditional Proportions?

A two-way table can describe how the proportion with an outcome differs between two groups. For example, a survey might record whether a person uses public transit and whether that person lives in an urban or rural area. Comparing the overall number of transit users would not answer whether transit use is more common in one group: the groups may contain different numbers of people. Instead, calculate the proportion of each group that uses transit, then subtract those proportions.

The result is a descriptive measure of association between two categorical variables. It tells us the size and direction of the difference for the particular outcome and group ordering we chose. It does not, by itself, explain why the proportions differ.

Definition and Sign Convention

Consider a 2×2 table with two groups, $G_1$ and $G_0$, and two outcome categories: “Yes” and “No.” Let $a$ and $b$ be the counts in the Yes and No cells for $G_1$, and let $c$ and $d$ be the corresponding counts for $G_0$.

Group	Outcome: Yes	Outcome: No	Group total
$G_1$	$a$	$b$	$a+b$
$G_0$	$c$	$d$	$c+d$

Assume the table counts observations in exactly one of the two groups and exactly one of the two outcome categories. Each group total must be positive to calculate its conditional proportion. The proportion with a Yes outcome is calculated within each group, so the denominator is that group’s total.

Definition: The difference between the conditional proportions for the Yes outcome, comparing $G_1$ with $G_0$, is $$ \hat{p}_1-\hat{p}_0 = \frac{a}{a+b}-\frac{c}{c+d}. $$ Here, $\hat{p}_1$ is the sample proportion with a Yes outcome in $G_1$, and $\hat{p}_0$ is the sample proportion with a Yes outcome in $G_0$. The order is part of the definition: the proportion for $G_0$ is subtracted from the proportion for $G_1$.

A positive difference means the Yes outcome is more common in $G_1$ than in $G_0$; a negative difference means it is less common in $G_1$. A difference of zero means the two observed proportions are equal. The difference is measured in proportion units. Multiplying it by 100 expresses it in percentage points.

Worked Calculations

Worked Example: Comparing Transit Use in Two Areas

Suppose a hypothetical survey records transit use for 120 urban residents and 100 rural residents. Of the urban residents, 42 report using public transit; of the rural residents, 24 do. Compare the Yes proportions for urban residents and then rural residents.

Let $G_1$ be urban residents and $G_0$ be rural residents. The group totals, rather than the total number of survey respondents, are the denominators:

Urban proportion: $\hat{p}_1=42/120=0.350$, or 35.0%.

Rural proportion: $\hat{p}_0=24/100=0.240$, or 24.0%.

Subtract in the stated order:

\hat{p}_1-\hat{p}_0 = \frac{42}{120}-\frac{24}{100} = 0.350-0.240 = 0.110.

The transit-use proportion among the surveyed urban residents is 0.110, or 11.0 percentage points, higher than the proportion among the surveyed rural residents. This is a description of the hypothetical survey data; the difference alone does not establish why the proportions differ.

Worked Example: Interpreting a Negative Difference

In a hypothetical survey, 18 of 60 people in Group A report using a particular service, while 28 of 70 people in Group B report using it. Define the comparison as Group A minus Group B.

Group A proportion: $\hat{p}_1=18/60=0.300$, or 30.0%.

Group B proportion: $\hat{p}_0=28/70=0.400$, or 40.0%.

The signed difference is

\hat{p}_1-\hat{p}_0 = \frac{18}{60}-\frac{28}{70} = 0.300-0.400 = -0.100.

The negative sign shows that the service-use proportion in Group A is lower than the proportion in Group B. The difference is $-0.100$ proportion units, or $-10.0$ percentage points for Group A minus Group B. Equivalently, the observed proportion in Group A is 10.0 percentage points lower than in Group B. Stating the group order makes the sign interpretable.

Worked Example: Unequal Group Sizes and Percentage Points

Suppose a hypothetical poll records a Yes response from 16 of 40 respondents in Group C and from 15 of 60 respondents in Group D. Calculate the difference for Group C minus Group D.

The proportions use each group’s own total:

\hat{p}_C=\frac{16}{40}=0.400, \qquad \hat{p}_D=\frac{15}{60}=0.250.

Therefore,

\hat{p}_C-\hat{p}_D = 0.400-0.250 = 0.150.

The Yes proportion in Group C is 15.0 percentage points higher than in Group D. The raw counts, 16 and 15, are close, but the group sizes differ. Comparing counts alone would miss the difference in proportions.

The result is not a 15% increase. The percentage-point difference is 15.0 points. If one separately described the increase relative to Group D’s proportion, it would be $0.150/0.250=0.600$, or 60.0% relative to that baseline. That is a different comparison; the difference between proportions is reported in percentage points.

How the Direction Affects the Result

A signed difference is not just a size: it records which group’s proportion was placed first. If the same two proportions are subtracted in the reverse order, the result changes sign:

\hat{p}_0-\hat{p}_1 = -(\hat{p}_1-\hat{p}_0).

For example, a difference of 0.110 for urban minus rural is a difference of $-0.110$ for rural minus urban. Both calculations describe the same gap, but their signs answer differently ordered comparisons. State the order whenever reporting a signed difference.

The chosen outcome matters as well. In a two-category outcome, the proportion with a No outcome is one minus the proportion with a Yes outcome in each group. Consequently, the difference for No is the negative of the difference for Yes:

(1-\hat{p}_1)-(1-\hat{p}_0) = -(\hat{p}_1-\hat{p}_0).

Thus, a statement about the difference in Yes proportions should name Yes as the outcome being compared. The corresponding difference in No proportions has the opposite sign.

Connection to the Cross-Product Criterion

The difference also has a useful algebraic relationship to the cell counts in the 2×2 table. Combining the two fractions over a common denominator gives

\begin{aligned} \hat{p}_1-\hat{p}_0 &= \frac{a}{a+b}-\frac{c}{c+d}\\ &= \frac{a(c+d)-c(a+b)}{(a+b)(c+d)}\\ &= \frac{ad-bc}{(a+b)(c+d)}. \end{aligned}

The last step follows by expanding the numerator: $a(c+d)-c(a+b)=ac+ad-ac-bc=ad-bc$. Since both group totals are positive, the denominator is positive. Therefore the difference is zero exactly when $ad=bc$, and its sign is the sign of $ad-bc$. This is consistent with the cross-product criterion for a 2×2 table: equal conditional Yes proportions correspond to equal cross-products. The difference adds information about direction and expresses the size of the gap in proportion units.

Reporting the Difference Clearly

A complete description identifies the outcome, both groups, the subtraction order, and the units. For example: “The proportion reporting transit use was 11.0 percentage points higher among the surveyed urban residents than among the surveyed rural residents.” This wording gives the outcome, comparison groups, and direction without implying that the difference is caused by group membership.

When reporting the calculation, show or state each conditional proportion as well as the difference. Doing so makes the denominators visible and helps readers check whether the comparison answers the intended question.

Common Mistakes

Using the wrong denominator. Dividing each outcome count by the grand total gives joint proportions, not the proportion with the outcome within each group. Use $a+b$ for $G_1$ and $c+d$ for $G_0$.
Leaving the subtraction order unstated. A positive result does not mean much without knowing which group was subtracted from which. Name the comparison as $G_1$ minus $G_0$.
Calling a proportion difference a percent change. A difference of 0.11 is 11 percentage points. It is not automatically an 11% relative increase.
Comparing counts instead of proportions. Groups of different sizes can have similar counts but different outcome proportions. Calculate the conditional proportions before comparing.
Interpreting the difference as a cause. A descriptive difference shows how the observed proportions compare. It does not, by itself, establish why they differ.

Key takeaway: For a 2×2 table, calculate each group’s outcome proportion using that group’s total, then subtract in a clearly stated order. Interpret the signed result in context and report it in percentage points when using percentages.

Check Your Understanding

For each question, identify the denominator or interpretation that makes the comparison meaningful.

A table has 25 Yes and 75 No outcomes in Group A, and 18 Yes and 42 No outcomes in Group B. Calculate the Yes-proportion difference for Group A minus Group B and interpret its sign.
In a comparison defined as Group X minus Group Y, the difference in Yes proportions is $-0.08$. Which group has the higher observed Yes proportion, and by how many percentage points?
Why should the Yes count in each group be divided by that group’s total rather than by the grand total?
If the difference for a Yes outcome is 0.12 for Group A minus Group B, what is the difference for Group B minus Group A?
Explain why a difference of 0.10 proportion units is described as 10 percentage points, not automatically as a 10% increase.

Group	Outcome: Yes	Outcome: No	Group total
\(G_1\)	\(a\)	\(b\)	\(a+b\)
\(G_0\)	\(c\)	\(d\)	\(c+d\)