Two-Way Tables and Association · Tutorial 213 of 996

Association Patterns in Ordered Categories

Use row proportions to describe how categorical outcomes change across ordered groups, and decide whether the pattern is consistently increasing, decreasing, or non-monotone.

Beginner 9 min read

What You'll Learn

Identify when one variable’s categories have a meaningful order.
Calculate and compare the same conditional cell proportion across ordered groups.
Distinguish increasing, decreasing, and non-monotone sequences of proportions.
Describe patterns using percentage points and the correct denominator.
Recognize why increasing one outcome proportion does not mean every other outcome proportion also increases.
Explain what a descriptive pattern does and does not establish.

Why Order Matters in a Two-Way Table

Some categorical variables have a meaningful order. Examples include education level, frequency categories such as “never” through “often,” and stages such as mild, moderate, and severe. When one variable is ordered, a two-way table can show more than whether group distributions differ: it can show how a particular outcome proportion changes as we move through the ordered categories.

The key is to compare the same conditional proportion in each group. If the group categories are ordered from lower to higher, list the proportion of interest in that order. A sequence that keeps rising is increasing; one that keeps falling is decreasing. If it rises in some places and falls in others, it is non-monotone. These are descriptions of the observed table, not conclusions about what caused the pattern.

Definition: Suppose the categories of $X$ have a meaningful order, $i=1,\ldots,k$, and $Y$ has categories $j=1,\ldots,m$. Let $n_{ij}$ be the count in row $i$, outcome category $j$, and let $n_i$ be the total count in row $i$. The observed conditional cell proportion is $$ \hat{p}_{ij}=\frac{n_{ij}}{n_i}. $$ For a fixed outcome category $j$, its sequence across the ordered $X$ categories is increasing if it never decreases, decreasing if it never increases, and non-monotone if it changes direction. A constant sequence is both nondecreasing and nonincreasing, but shows no change.

The denominator $n_i$ is the total for the particular ordered group, so $\hat{p}_{ij}$ describes the share of that group in outcome category $j$. This is a row conditional proportion when $X$ is arranged in the rows. The counts $n_{ij}$ alone are not enough to compare groups fairly if their row totals differ.

Reading a Pattern Correctly

To describe a pattern, first state the order of the groups and the outcome category being tracked. Then compare its proportions in sequence. If the values are $0.25, 0.40, 0.55,$ and $0.70$, the proportion increases at every step. If they are $0.25, 0.45, 0.38,$ and $0.60$, the sequence is non-monotone: it rises, falls, and rises again.

A pattern can be monotone without changing by the same amount at each step. For example, $0.20, 0.30, 0.55, 0.60$ is increasing, although the successive increases are not equal. Nor does an ordered category scale guarantee equal spacing. The change from “never” to “sometimes” need not represent the same amount as the change from “sometimes” to “often.” Describe the order and the proportions; do not assume a constant rate of change.

Proportions can be reported as decimals or percentages. When comparing percentages, a change from 30% to 42% is an increase of 12 percentage points. It is not a 12% increase: relative to 30%, the increase is $12/30=40\%$. For a table pattern, percentage-point differences are usually the clearest way to describe the step from one group to the next.

Worked Example: A Consistently Increasing Proportion

In a hypothetical skills exercise, participants are grouped by the number of practice sessions completed. The categories are ordered from fewer to more sessions. The table records whether each participant answered a particular question correctly.

Practice sessions	Correct	Incorrect	Row total	Correct proportion
0	12	28	40	12/40 = 30%
1	21	29	50	21/50 = 42%
2	32	32	64	32/64 = 50%
3 or more	39	21	60	39/60 = 65%

For each row, divide the number correct by that row’s total. For example, among participants with two sessions, $$ \hat{p}_{\text{correct}}=\frac{32}{64}=0.50=50\%. $$ The ordered sequence is $30\%, 42\%, 50\%, 65\%$. Each value is greater than or equal to the one before it, so the observed proportion answering correctly is consistently increasing across these practice categories. The step changes are $42\%-30\%=12$, $50\%-42\%=8$, and $65\%-50\%=15$ percentage points. The increases are not equal, so the sequence is increasing but not at a constant step size.

The table describes an association between practice category and the observed proportion correct. It does not by itself establish that completing more practice sessions caused higher performance.

Non-Monotone Patterns and Multiple Outcomes

A sequence can show a clear difference between its first and last values and still be non-monotone. The intermediate values matter. For example, if a proportion changes from 25% to 45%, then to 38%, and finally to 60%, the final value is above the initial value, but the pattern is not consistently increasing because it falls from 45% to 38%.

When $Y$ has more than two categories, examine each outcome category separately. Different categories can have different patterns. Within any one row, the conditional proportions across all outcome categories add to 1, because the outcome categories account for every observation in that row. Consequently, if one outcome category’s proportion rises, at least one other category’s proportion must fall somewhere in that row distribution. It does not follow that every other category must fall at every step.

Worked Example: Increasing, Decreasing, and Non-Monotone Sequences

A hypothetical service survey records the preferred support channel for respondents in three ordered groups of technology familiarity. Each respondent selects one channel.

Familiarity	Printed guide	Phone	Web	Row total
Novice	25	15	10	50
Intermediate	24	24	32	80
Advanced	7	14	49	70

Calculate the conditional proportions using each familiarity group’s row total. For novices, the proportions are $25/50=50\%$ for printed guides, $15/50=30\%$ for phone, and $10/50=20\%$ for web. For the intermediate group, they are $24/80=30\%$, $24/80=30\%$, and $32/80=40\%$. For the advanced group, they are $7/70=10\%$, $14/70=20\%$, and $49/70=70\%$.

The web proportions, $20\%, 40\%, 70\%$, are increasing. The printed-guide proportions, $50\%, 30\%, 10\%$, are decreasing. The phone proportions, $30\%, 30\%, 20\%$, are nonincreasing: they stay the same at first and then fall. They are not strictly decreasing because the first two values are equal. These descriptions apply to the observed groups in this table; they do not say why respondents prefer different channels.

Why the Row Denominator Matters

Comparing counts instead of row proportions can reverse the impression a table gives. A larger group may have a larger count in an outcome category even when that category makes up a smaller share of the group. The pattern of interest is usually the distribution within each ordered group, so calculate each proportion with that group’s total as its denominator.

Worked Example: Counts Can Hide a Decreasing Proportion

In a hypothetical questionnaire, two ordered age groups are asked whether they prefer an online appointment. The younger group contains 40 people, of whom 24 prefer online appointments. The older group contains 100 people, of whom 50 prefer them.

The counts are 24 and 50, so the older group has more respondents preferring an online appointment. But the within-group proportions are $$ \frac{24}{40}=0.60=60\%, \qquad \frac{50}{100}=0.50=50\%. $$ Across these two ordered groups, the observed proportion preferring an online appointment decreases by 10 percentage points. Comparing the counts alone would miss this pattern because the older group has many more respondents overall.

This comparison is descriptive. It does not imply that age causes the preference or that the pattern continues beyond the two groups shown.

A Short Justification: Why Row Proportions Sum to One

Within row $i$, every observation belongs to exactly one of the $m$ outcome categories. Therefore, the sum of the cell counts in the row is its total: $$ \sum_{j=1}^{m} n_{ij}=n_i. $$ Dividing both sides by $n_i$, provided $n_i>0$, gives $$ \sum_{j=1}^{m}\hat{p}_{ij} = \sum_{j=1}^{m}\frac{n_{ij}}{n_i} = \frac{\sum_{j=1}^{m}n_{ij}}{n_i} = \frac{n_i}{n_i} = 1. $$

Thus, each row’s conditional proportions form a complete distribution. In a two-outcome table, if one proportion increases from one group to the next, the other must decrease by the same number of percentage points: the two proportions always add to 100%. With three or more outcomes, a rise in one proportion means the others’ combined proportion falls by the same amount, but individual categories among those others can move in different directions.

Assumptions and Limits of the Description

This is a descriptive comparison of a table, not an inferential procedure. It requires no sampling-distribution model or significance test. To make the proportions meaningful, however, the group categories must be clearly ordered for the question at hand, the outcome categories must be defined consistently across groups, and each row total used as a denominator must be positive. If categories do not have a meaningful order, arranging them in a sequence and calling a pattern increasing or decreasing would be arbitrary.

A monotone pattern is not necessarily strong, important, or causal. A small increase at each step is still increasing, while a large rise followed by a small fall is non-monotone. State the actual proportions and changes, not just a label. Also keep the scope of the conclusion tied to the table: an observed pattern across the categories shown does not establish what happens outside them.

Common Mistakes

Comparing counts rather than conditional proportions. A larger group can have more cases in a cell but a smaller share in that category. Divide each cell count by its own row total before comparing across ordered groups.
Calling a sequence increasing because its last value is highest. Check every step. If the values rise, fall, and rise, the sequence is non-monotone even if its final value exceeds its initial value.
Treating an ordered scale as equally spaced. “Low,” “medium,” and “high” have an order, but the gaps between them are not automatically equal. Describe the sequence without assuming equal increments.
Confusing percentage points with percent change. Going from 30% to 42% is an increase of 12 percentage points. A relative percent increase would use 30% as the baseline and is a different calculation.
Claiming that one group characteristic caused the pattern. A two-way table describes how proportions differ across groups. The table alone does not establish a causal explanation.

Key takeaway: To describe association across ordered categories, choose an outcome category, calculate its within-group proportion for every ordered group, and inspect the full sequence. State whether it increases, decreases, stays constant, or changes direction, and support the description with the proportions and their units.

Check Your Understanding

Use the definitions and examples above to answer the following questions.

An outcome proportion across four ordered groups is 18%, 32%, 32%, and 47%. Is it increasing, strictly increasing, or non-monotone? Explain.
Across ordered categories, a proportion is 40%, 55%, 51%, and 62%. Why is “consistently increasing” not an accurate description, even though the last value is the largest?
A cell count rises from 15 to 24 across two groups, while the row totals rise from 30 to 80. Calculate both conditional proportions and describe the change.
In a three-outcome table, one outcome proportion rises by 8 percentage points between two groups. What must happen to the combined proportion of the other two outcomes? Must each one fall?
Explain why the labels “mild,” “moderate,” and “severe” can be ordered without assuming that the step from mild to moderate equals the step from moderate to severe.

Back to Tutorials