Why Adjusted Means Need a Target
In a factorial design, an adjusted mean summarizes a factor level after averaging across levels of another factor. The averaging weights matter: they specify which mix of the other factor’s levels the summary represents. If one combination was never observed, an adjusted mean may depend on a cell mean that the data cannot identify. The result is not merely a missing calculation. It is a question about what the data can determine and what must be supplied by assumptions.
This tutorial considers a two-factor design. The response is quantitative, factor A has levels \(a=1,\ldots,A\), and factor B has levels \(b=1,\ldots,B\). A cell is one combination of factor levels. Some cells may have observations and others may be empty. An empty cell means there are no observations for that combination; it does not mean the response there is zero.
An adjusted mean for level \(a\) of factor A can be defined using weights \(w_b\) across levels of B:
For an average representing a mix of B levels, the weights are usually nonnegative as well as summing to one. Using the same weights for each A level makes the resulting comparison refer to a common mix of B. Different choices of weights define different targets, even when the observed data are unchanged.
A Two-Factor Table With One Empty Cell
Consider two levels of each factor. Write the four cell means as \(\mu_{11},\mu_{12},\mu_{21},\mu_{22}\). Suppose the cell \((2,2)\) is empty and the other three cells are observed. Their hypothetical cell-mean estimates are 10, 14, and 12 response units, respectively. The value of \(\mu_{22}\) is unknown from these data.
| B level 1 | B level 2 | |
|---|---|---|
| A level 1 | 10 observed | 14 observed |
| A level 2 | 12 observed | Empty |
The numbers in observed cells can be used directly in weighted averages. But any target that assigns positive weight to \(\mu_{22}\) also depends on a mean for which the design provides no observations.
Worked Example: A Weighted Comparison That Depends on the Empty Cell
Use a common weighting scheme that gives B level 1 weight 0.75 and B level 2 weight 0.25. Find the two adjusted means and their difference, taking the difference as A level 2 minus A level 1.
For A level 1, both cells are observed:
For A level 2, the weighted mean includes the empty cell:
Consequently, the comparison is
The first adjusted mean is estimable, but the second is not: it depends on \(\mu_{22}\) with a nonzero coefficient. Their difference is also non-estimable for the same reason. For illustration, if \(\mu_{22}=8\), the difference would be \(-2+0.25(8)=0\) response units. If instead \(\mu_{22}=20\), it would be \(-2+0.25(20)=3\) response units. These are conditional calculations, not estimates of the comparison: the observed cells do not tell us which value of \(\mu_{22}\) is appropriate.
The standard errors and normality assumptions of classical ANOVA do not solve this problem. They concern uncertainty for quantities the model can estimate; they cannot provide information about a cell with no observations unless the model imposes a restriction that supplies it.
Why the Empty-Cell Coefficient Decides Estimability
The reason can be shown directly. Collect the four cell means in the vector \(\boldsymbol{\mu}=(\mu_{11},\mu_{12},\mu_{21},\mu_{22})^\mathsf{T}\). The observed design contains information about the first three entries but none about the fourth. For any number \(t\), replacing \(\mu_{22}\) by \(\mu_{22}+t\) leaves all observed cell means unchanged.
A linear target \(L=c_{11}\mu_{11}+c_{12}\mu_{12}+c_{21}\mu_{21}+c_{22}\mu_{22}\) changes under that replacement by \(c_{22}t\). If \(c_{22}\ne 0\), different values of the unobserved cell mean produce different target values while fitting the observed cell means identically. Therefore the target is not determined by the data. If \(c_{22}=0\), the target contains only observed cell means, so it is determined by them.
This argument is a simple way to check a proposed adjusted mean or contrast in this example: write it as a linear combination of cell means and inspect the coefficient of the empty cell. A nonzero coefficient means that the target requires information the observed design does not contain.
Worked Example: Changing the Weights Changes the Comparison
Return to the same table. Let \(w\) be the weight assigned to B level 2, so B level 1 receives weight \(1-w\). Use the same weights for both A levels and compute the comparison symbolically.
The adjusted means are
Subtracting gives
When \(w=0\), the comparison is \(2\) response units: it compares the observed cell means at B level 1, \(12-10=2\). The empty cell has coefficient zero, so this target is estimable. When \(w=0.25\), the comparison becomes \(-2+0.25\mu_{22}\), as in the previous example, and is not estimable. With equal weights, \(w=0.5\), it is \(-6+0.5\mu_{22}\), also not estimable.
Thus, the weights do more than change the numerical scale of a summary. They determine the population mix being compared and can determine whether the target uses the empty cell. Setting its weight to zero avoids that cell, but it also changes the question: the comparison now concerns B level 1 alone, not an average across both B levels.
Model-Based Extrapolation Is an Added Assumption
A model can sometimes assign a value to the empty cell by relating it to observed cells. For example, the no-interaction or additive model assumes that the effect of changing A does not depend on the level of B. In a two-factor cell-means description, this restriction implies
The right-hand side uses only observed cells. It yields a value for the empty cell only because the additive restriction rules out other possible values. This is model-based extrapolation, not a direct estimate from observations in cell \((2,2)\).
Worked Example: Filling an Empty Cell Under Additivity
Use the observed cell means 10, 14, and 12, and suppose the additive model is appropriate. Calculate the model-implied empty-cell mean and the adjusted comparison using \(w=0.25\).
Under additivity, the difference between B levels is the same at both A levels. At A level 1 that difference is \(14-10=4\) response units. The model therefore assigns A level 2, B level 2 the value
Substituting this model-implied value into the weighted means gives
The resulting comparison is \(13-11=2\) response units. This number follows from the observed means and the no-interaction restriction. It is not the unrestricted comparison: without that restriction, the comparison remains \(-2+0.25\mu_{22}\).
The observed table alone cannot check whether the missing cell follows the additive pattern, because there are no observations in that cell. Replication within observed cells could help estimate within-cell variation, but it would not reveal whether the unobserved combination has a different mean. The extrapolated value is therefore conditional on the structural assumption, and conclusions should say so.
Common Mistake and a Practical Check
A common mistake is to treat a software-reported adjusted mean as if it were automatically determined by the data. For example, a reader might say, “The average uses the observed cells, so the missing cell does not matter.” The correction is to write the target as a weighted sum of cell means and check the coefficient assigned to every empty cell. If that coefficient is nonzero, the target requires a value for the empty cell, whether that value is displayed, suppressed, or obtained under a model restriction.
A second mistake is to compare adjusted means without specifying their weights. Equal weights, weights reflecting a particular population mix, and weights that omit an empty level define different targets. The weights should be chosen to match the question, not selected merely because they make a calculation possible. If the scientifically intended target is non-estimable under the design, report that limitation or state the additional assumption used to obtain a model-based result.
State the adjusted mean or comparison and make the weights explicit.
Combine terms and identify which cell means have nonzero coefficients.
If a nonzero coefficient remains on an empty cell, the target is not identified by the observed cell means alone.
If a model supplies the missing cell value, describe the assumption and interpret the result as conditional on it.
This distinction matters because an adjusted mean is not just a computational summary. It represents a specified comparison across factor levels. The design determines which such comparisons are supported directly; the model determines whether a broader comparison can be obtained through extrapolation.
Check Your Understanding
For each question, identify the target and explain what the empty cell does to it.
- In the two-factor table above, what is the comparison of A level 2 with A level 1 when all weight is placed on B level 1? Is it estimable?
- Using \(w=0.5\), write the adjusted comparison as a function of \(\mu_{22}\). Explain why the observed table does not determine its value.
- Why does an empty cell not mean that its mean response is zero?
- Under the additive assumption, calculate the implied empty-cell mean if the observed means are 7, 11, and 9 in cells \((1,1),(1,2),(2,1)\). State what assumption makes the calculation possible.
- Explain why choosing weights that avoid an empty cell may change the scientific question being answered.