Estimability of Repeated-Measures Hypotheses

Why Estimability Comes Before Testing

A contrast can be scientifically meaningful but impossible to estimate from a particular design. For example, a researcher may ask whether the average response at a later occasion differs from the average at an earlier occasion. If some occasion-by-group cells were never observed, the data may not determine that overall difference without additional model assumptions. Before constructing a test statistic, first ask whether the hypothesis is estimable.

The answer is a property of the mean model and its design matrix. It is not decided by whether the contrast looks familiar, whether its coefficients sum to zero, or whether software can return a number after fitting the model. The relevant question is whether different parameter values that produce the same observable mean also produce the same value of the target hypothesis.

Definition: In the linear model \(E[Y]=X\beta\), where \(Y\) is an \(n\)-vector of responses, \(X\) is a known \(n\times p\) design matrix, and \(\beta\) is a \(p\)-vector of parameters, the linear function \(c'\beta\) is estimable if it has the same value for every pair of parameter vectors \(\beta_1,\beta_2\) satisfying \(X\beta_1=X\beta_2\). Equivalently, it is determined by the model mean \(X\beta\).

A hypothesis about occasion means can be written in this form. Let \(\mu=M\beta\) be the vector of occasion means specified by the model, where \(M\) is a known matrix. A scalar contrast is \(l'\mu=l'M\beta\); a collection of hypotheses has the form \(C\mu=CM\beta\). The rows of \(CM\) are the parameter coefficient vectors whose estimability must be checked.

The Row-Space Criterion

The row space of \(X\), written \(\operatorname{Row}(X)\), is the set of all linear combinations of its rows. It captures which parameter combinations are determined by the design. The null space, \(\operatorname{Null}(X)\), consists of vectors \(v\) with \(Xv=0\): moving from \(\beta\) to \(\beta+v\) changes the parameters but leaves the model mean unchanged.

Theorem: The function \(c'\beta\) is estimable if and only if \(c\in\operatorname{Row}(X)\). Equivalently, \(c'v=0\) for every \(v\in\operatorname{Null}(X)\). Thus \(C\mu=CM\beta\) is estimable component by component exactly when every row of \(CM\) belongs to \(\operatorname{Row}(X)\).

Proof. If \(c'\beta\) is estimable, then any two parameter vectors with the same mean give the same value of the function. For any \(v\in\operatorname{Null}(X)\), the vectors \(\beta\) and \(\beta+v\) have equal means, because \(X(\beta+v)=X\beta\). Therefore \(c'(\beta+v)=c'\beta\), which implies \(c'v=0\). So \(c\) is orthogonal to \(\operatorname{Null}(X)\). The fundamental theorem of linear algebra gives \(\operatorname{Null}(X)^\perp=\operatorname{Row}(X)\), hence \(c\in\operatorname{Row}(X)\).

Conversely, if \(c\in\operatorname{Row}(X)\), there is a vector \(a\) such that \(c=X'a\). If \(X\beta_1=X\beta_2\), then \(c'\beta_1=a'X\beta_1=a'X\beta_2=c'\beta_2\). Thus the function has the same value for all parameter vectors producing that mean and is estimable. Applying this argument to each row of \(CM\) proves the matrix statement. \(\square\)

The criterion also has an estimation interpretation. If \(c=X'a\), then \(a'Y\) is a linear unbiased estimator of \(c'\beta\), because \(E[a'Y]=a'X\beta=c'\beta\). Conversely, if a linear estimator \(a'Y\) is unbiased for every \(\beta\), then \(a'X\beta=c'\beta\) for every \(\beta\), so \(X'a=c\). Therefore a linear unbiased estimator exists exactly when the target is in the row space. No assumption of independent errors is needed for this mean-based criterion.

Worked Example: A Contrast With Complete Occasion Coverage

Suppose two subjects are measured at each of three occasions. Use a separate mean parameter for each occasion, \(\beta=(\mu_1,\mu_2,\mu_3)'\). Each observation at occasion \(j\) has design row \(e_j'\), where \(e_j\) is the \(j\)th coordinate vector. The design matrix is

X= \begin{pmatrix} 1&0&0\\ 1&0&0\\ 0&1&0\\ 0&1&0\\ 0&0&1\\ 0&0&1 \end{pmatrix}.

The contrast comparing occasion 1 with occasion 3 has coefficient vector \(c=(1,0,-1)'\). This vector is in the row space: it is the first row of \(X\) minus the fifth row. Hence \(\mu_1-\mu_3\) is estimable. In fact, the corresponding difference of sample means is

\bar Y_1-\bar Y_3 =\frac{Y_1+Y_2}{2}-\frac{Y_5+Y_6}{2}, \qquad E[\bar Y_1-\bar Y_3]=\mu_1-\mu_3.

The repeated observations affect the precision of this estimator, but not the row-space decision. Here all three occasion means are represented by observed design rows, so every linear contrast of them is estimable.

Redundant Parameterizations

A design matrix can have linearly dependent columns. In that case, the parameter vector is not uniquely determined by the model mean, but some functions of its components can still be estimable. The row-space criterion checks the target function directly; it does not require the individual coefficients to be identifiable.

Worked Example: A Difference Despite Redundant Coefficients

Suppose there are two occasions, with two observations at each. A redundant parameterization gives occasion means \(\mu_1=\beta_1+\beta_3\) and \(\mu_2=\beta_2+\beta_3\), so the design rows are \((1,0,1)\) for occasion 1 and \((0,1,1)\) for occasion 2:

X= \begin{pmatrix} 1&0&1\\ 1&0&1\\ 0&1&1\\ 0&1&1 \end{pmatrix}.

The target difference is \(\mu_1-\mu_2=\beta_1-\beta_2\), with coefficient vector \(c=(1,-1,0)'\). It is estimable because

$$ (1,-1,0)=(1,0,1)-(0,1,1), $$

so \(c\) is a linear combination of rows of \(X\). Its unbiased estimator is the difference between the two occasion averages.

By contrast, \(\beta_1\) alone has coefficient vector \((1,0,0)'\), which is not in the row space. To verify this, the nonzero vector \(v=(-1,-1,1)'\) satisfies \(Xv=0\), but \((1,0,0)v=-1\), not zero. Thus \(\beta_1\) changes along a parameter direction that leaves all fitted means unchanged. The occasion difference remains estimable because its coefficient vector is orthogonal to that direction.

Least-squares fitting does not remove this distinction. When \(X\) is rank deficient, there may be multiple coefficient vectors that minimize the residual sum of squares, but all give the same fitted vector. An estimable function has the same value at every least-squares solution.

Result: If \(\hat\beta\) is any least-squares solution, then every estimable \(c'\beta\) has the same value \(c'\hat\beta\) for all least-squares solutions.

Proof. Let \(\hat\beta_1\) and \(\hat\beta_2\) both minimize the sum of squared residuals. Their fitted vectors are equal: each is the orthogonal projection of \(Y\) onto the column space of \(X\), and an orthogonal projection onto a fixed subspace is unique. Therefore \(X\hat\beta_1=X\hat\beta_2\). By the definition of estimability, \(c'\hat\beta_1=c'\hat\beta_2\). This proves the result. \(\square\)

Missing Design Cells Can Block a Marginal Contrast

Estimability depends on the design and the mean model together. With a saturated group-by-occasion model, each group-by-occasion cell has its own mean parameter. An absent cell then contributes no design row to determine that cell mean. A different, more restrictive model might impose relationships among cell means and thereby change what is estimable; such relationships must be justified as model assumptions, not inferred from the row-space calculation alone.

Worked Example: A Missing Group-by-Occasion Cell

Consider two groups and two occasions, with cell means \(\beta_{11},\beta_{12},\beta_{21},\beta_{22}\). Suppose cells \((1,1)\), \((1,2)\), and \((2,1)\) are observed, but cell \((2,2)\) is entirely missing. In this order of parameters, the observed design rows are \(e_1',e_2',e_3'\); for example, with one observation per available cell,

X= \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 \end{pmatrix}.

Suppose the target is the difference in equally weighted group-average occasion means. The occasion means are \(\mu_1=(\beta_{11}+\beta_{21})/2\) and \(\mu_2=(\beta_{12}+\beta_{22})/2\), so

\mu_2-\mu_1 =\frac{1}{2}(-\beta_{11}+\beta_{12}-\beta_{21}+\beta_{22}).

Its coefficient vector is \((-1/2,1/2,-1/2,1/2)'\), which is not in the row space of \(X\): every row-space vector has fourth coordinate zero. Equivalently, \(v=(0,0,0,1)'\) is in \(\operatorname{Null}(X)\), but the target coefficient vector has nonzero inner product \(1/2\) with \(v\). The marginal occasion difference is therefore not estimable under this model.

A different target is estimable: the occasion difference within group 1, \(\beta_{12}-\beta_{11}\), has coefficient vector \((-1,1,0,0)'\), which is a linear combination of the first two rows of \(X\). The missing cell blocks the equally weighted marginal comparison, not every possible comparison across occasions.

A Practical Check and a Common Mistake

For a specified hypothesis, write down its coefficient vector in the model parameters before attempting a test. If the hypothesis concerns occasion means \(\mu=M\beta\), form \(CM\), not just \(C\): the design matrix is expressed in terms of \(\beta\), so the target must be expressed in those same parameters. Then check whether each row of \(CM\) is in \(\operatorname{Row}(X)\). Equivalently, check whether each is orthogonal to every vector in \(\operatorname{Null}(X)\).

Write the mean model.
Identify the parameter vector, the design matrix \(X\), and how the occasion means depend on the parameters.

Express the target in parameters.
For \(C\mu\) with \(\mu=M\beta\), calculate the coefficient matrix \(CM\).

Check the row space.
Test each target row for membership in \(\operatorname{Row}(X)\), or test orthogonality to \(\operatorname{Null}(X)\).

Interpret the conclusion under the stated model.
If the target is not estimable, explain which unobserved or redundant parameter direction prevents it from being determined.

Common mistake: It is tempting to conclude that a contrast is estimable because its occasion weights sum to zero. That condition only says the contrast removes a common level when the occasion means themselves are available; it does not guarantee that the design identifies every occasion mean in the contrast. In the missing-cell example, the marginal difference has weights that sum to zero, but it still depends on the unobserved parameter \(\beta_{22}\). The correction is to express the contrast in model parameters and apply the row-space test.

Key takeaway: Estimability is about whether the observed design determines the value of a hypothesis, not whether the hypothesis can be written down or a fitting routine returns a coefficient. For repeated-measures hypotheses, translate occasion contrasts into parameter coordinates and check their rows against the design matrix’s row space.

Check Your Understanding

For each question, use the mean model and design matrix specified rather than relying on the appearance of the contrast alone.

State the row-space criterion for a scalar function \(c'\beta\) to be estimable.
Why can a contrast of two occasion means be estimable even when the parameter coefficients used to represent those means are not individually estimable?
For a design matrix with rows \(e_1'\) and \(e_2'\) and three occasion-mean parameters, is \(\mu_1-\mu_2\) estimable? Is \(\mu_1-\mu_3\)? Explain using the row space.
In the two-group missing-cell example, identify a null-space vector that shows why the equally weighted occasion difference is not estimable.
Does a contrast’s weights summing to zero, by itself, establish estimability? Give the reason.