Must all cells have the same sample size?

No. Equal cell sizes are sufficient, but proportional patterns can contain unequal counts.

Does proportionality mean no interaction?

No. Interactions concern outcome patterns, not merely sample-size allocation.

Can an initially proportional study become nonproportional?

Yes. Uneven exclusions or missing observations can change the cell-count pattern used in analysis.

AI Glossary / Statistics and probability / Intermediate

Proportionality

Definition

Proportionality, in the supplied factorial-analysis-of-variance context, is a condition on cell sample sizes: the relative distribution across levels of one factor is the same at every level of another factor. In a two-factor table, proportional cell counts satisfy n_ij = n_i+ n_+j/N. Equal cell sizes are a special case. This entry concerns sample allocation, not the broader mathematical idea that one quantity is a constant multiple of another.

What it means

A factorial design combines the levels of two or more factors. Each combination forms a cell containing observations. When counts are proportional, differences in the total number of observations at one factor's levels do not change the composition of the other factor across those levels.

This helps separate main-effect information in the design. When counts are not proportional, factor membership can be associated in the observed sample, making some comparisons sensitive to adjustment and weighting. The condition describes counts, not outcome means. Proportional counts do not imply equal responses, no interaction, or satisfaction of the other assumptions used in ANOVA.

How it works

Arrange the sample sizes in a table indexed by factor levels. Calculate row totals, column totals, and the overall total. For a two-factor table with the relevant cells represented, compare each observed count with its row total multiplied by its column total and divided by N.

Alternatively, compare the within-row column proportions. If the same proportions recur in every row, the two-factor allocation is proportional. With positive cells, equivalent ratio checks can be convenient. For designs involving more factors, specify the appropriate factorization or balance condition rather than assuming that one pairwise check settles the full design.

Use the actual analysis sample, since missing outcomes or exclusions can disrupt an originally proportional allocation. If counts are nonproportional, define the scientific comparison and use a model and contrasts that match it. Different sums-of-squares conventions may test different hypotheses. Do not solve the problem by deleting valid observations merely to obtain a visually tidy table.

Three practical examples

A two-factor usability study checks whether experience-level proportions are identical across interface conditions.
An analyst reviews whether missing outcomes have disrupted the intended allocation in an AI-training experiment.
A quality study distinguishes unequal but proportional cell sizes from an allocation in which factor levels are associated.

Think of it this way

Think of two baskets with different total numbers of fruit but the same fruit mixture. One basket can be twice as large while preserving the same proportions of apples, pears, and oranges. In a factorial table, proportionality concerns that repeated mixture of factor levels, not equal totals for every basket.

Worked example

Consider an original two-by-three sample-size table. For factor A's first level, the counts across B's three levels are 10, 20, and 30. For A's second level, they are 20, 40, and 60.

The row totals are 60 and 120, the column totals are 30, 60, and 90, and N = 180. Within each row, the B proportions are 1/6, 1/3, and 1/2. The allocation is proportional despite its unequal cell sizes.

For example, the formula predicts the second row's middle count as 120 × 60/180 = 40, matching the table. It predicts the first row's last count as 60 × 90/180 = 30.

Now remove ten observations only from the second row's middle cell. That row becomes 20, 30, 60, with proportions 2/11, 3/11, and 6/11. They no longer match the first row. The final analysis table is nonproportional. Nothing in either count table specifies the cell means or rules out an interaction in the outcomes.

Illustrative case study

Fictional case study: Cedar Training compares two course formats across three experience groups. The planned allocation preserves the same experience mix for each format, though one format has more participants overall.

After collection, the team finds missing outcome measurements concentrated in one combination. It reviews the resulting cell counts and defines adjusted comparisons appropriate to the completed data and study objective. This fictional case illustrates why the realized analysis sample matters. It does not imply that statistical adjustment necessarily removes the consequences of selective missingness.

Common misconceptions and mistakes

Unequal cell sizes are not automatically nonproportional. The worked example has six cells of unequal size in a repeated ratio pattern. Conversely, equal row totals alone do not establish proportionality; the within-row distribution can still differ.

Proportional allocation is not evidence that the factors have no effect or no interaction. It concerns the explanatory-factor structure, while effects concern response differences. It also does not establish random assignment, independent errors, or equal residual variances. Finally, the term balanced is used differently across texts, sometimes requiring equal cell counts. State the actual count condition instead of relying on that label alone.

Business use

Checking proportionality helps business analysts understand whether raw factor comparisons reflect similar compositions of other factor levels. It can make experimental reports clearer and reveal when missing data or operational constraints changed the intended design.

For AI experiments, keep planned and realized allocation tables alongside outcome summaries. Document exclusions and the population over which marginal effects are averaged. If the allocation is not proportional, explain the chosen adjustment and weighting. A software-generated ANOVA table should not substitute for specifying which business comparison the analysis is intended to answer.

Check your understanding

A two-by-two table has rows (8, 12) and (16, 24). Is it proportional?

Answer: Yes. Each row has proportions 0.40 and 0.60, and the second row is twice the first. This conclusion concerns allocation only and says nothing about the response means.

Related terms

Factorial design
Main effect
Analysis of variance
Missing data
cell sample size
balanced design
unbalanced design
interaction
orthogonality
marginal mean

Learn next

Construct factorial count tables and compare equal, proportional, and nonproportional allocations. Then study marginal means, contrasts, and sums-of-squares hypotheses. Learn how missing data and interactions affect interpretation before relying on default ANOVA output.

Sources

Howell's glossary identifies proportionality as a factorial-ANOVA sample-size condition: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html

Brunner's University of Toronto factorial-ANOVA teaching material discusses proportional cell counts and factor relationships: https://utstat.toronto.edu/~brunner/oldclass/441s24/lectures/441s24FactorialAnova.pdf

The count tables, checks, and fictional case are original. The narrow source-specific meaning is stated explicitly; it is not a general glossary entry for every use of proportionality.

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