Worked Design of a Scientific Contrast Family

From a Scientific Question to a Contrast Family

A multivariate analysis can produce many intervals, but the intervals are useful only if they answer the intended scientific question. That question determines which mean differences matter, how each difference should be encoded, and what it means for the full set of results to cover successfully. The design matters too: a contrast that is scientifically meaningful may not be estimable if the study did not observe the groups needed to identify it.

This tutorial develops a practical design sequence: write the mean vector in a stated order, translate each requested comparison into a coefficient row, check the matrix’s rank and the design’s estimability, and state the joint coverage target. The goal is not to choose an interval procedure, but to define precisely what the procedure must cover.

For example, with means ordered as control, low dose, and high dose, the difference between high dose and control is \((0,-1,1)\mu\). The zero-sum condition makes this a comparison of means rather than a general linear combination that also depends on their common level. Always state the ordering: the same row has a different interpretation if the coordinates are reordered.

Worked Example: Comparisons With a Control

Rank and the Meaning of Redundancy

The rank of \(C\) is the dimension of the linear span of its rows: it counts the number of linearly independent contrast directions, not necessarily the number of rows. A dependent row is a linear combination of other rows, so it adds no new direction to that span. The number of requested interval statements and the rank therefore answer different questions. Rank is useful for describing the family’s linear structure, but it does not by itself determine a procedure or the simultaneous coverage event.

For instance, adding the contrast \(\mu_2-\mu_1\) to the three treatment-versus-control contrasts does not add a direction, because \((\mu_2-\mu_1)=(\mu_2-\mu_0)-(\mu_1-\mu_0)\). It may still be scientifically useful to report that comparison. If it is included, the family has four listed contrasts but rank three. Also, a basis for the row space is not automatically a replacement for the original family’s coordinatewise interval target: covering the basis contrasts does not, without further reasoning, assert coverage for every listed contrast.

Worked Example: A Two-Factor Scientific Question

Estimability Depends on the Design

A contrast matrix describes what the investigator wants to learn. It does not guarantee that the data can identify every row. In a fixed-design linear model, write \(E[Y]=X\beta\), where \(X\) is the design matrix and \(\beta\) is the parameter vector. A linear function \(a^\top\beta\) is estimable if some linear combination of the observations has that expectation.

Proof. If \(X^\top d=a\), then \(E[d^\top Y]=d^\top X\beta=a^\top\beta\), so \(a^\top\beta\) is estimable by definition. Conversely, if a linear combination \(d^\top Y\) has expectation \(a^\top\beta\) for every \(\beta\), then \(d^\top X\beta=a^\top\beta\) for every \(\beta\). Equality of these linear functions for every \(\beta\) requires \(X^\top d=a\). The vectors of the form \(X^\top d\) are exactly the row space of \(X\), proving the equivalence. Applying the criterion row by row proves the matrix statement. \(\square\)

When \(\beta\) is the vector of group means and \(X\) has one indicator column per group, this condition has a direct interpretation: a group mean cannot be recovered if its group has no observations. More generally, if cell means are represented as \(\mu=K\beta\), then a target \(c^\top\mu\) is estimable precisely when \(c^\top K\) lies in the row space of \(X\). This formulation separates the scientific definition of a target from the parameterization and design used to estimate it.

Worked Example: A Missing Group

State the Simultaneous Coverage Target

Once the rows of \(C\) have been chosen and their estimability checked, state what successful joint inference means. If \([L_j,U_j]\) is an interval for the \(j\)th contrast, the simultaneous coverage target at level \(1-\alpha\) is

where \(m\) is the number of listed rows. The event requires all specified contrasts to be covered together, not merely each contrast considered separately. This statement makes the family’s scientific scope explicit: if one requested comparison is omitted from \(C\), it is not part of this coverage claim. Conversely, adding a scientifically important comparison changes the stated target even if it is linearly dependent on existing rows.

Common Mistakes

• Using an ambiguous mean order. A row has meaning only relative to the stated order of coordinates. Write the group order before interpreting coefficients.
• Treating rank as an estimability check. Rank describes dependencies among rows of \(C\); estimability requires checking those rows against the row space of \(X\).
• Assuming a contrast is estimable because its coefficients sum to zero. Zero-sum coefficients make a group-mean comparison a contrast, but missing or confounded design information can still make it unestimable.
• Equating separate coverage with simultaneous coverage. Intervals that each have a stated marginal coverage level do not, just by that fact, guarantee that all intervals cover their targets together.