Decomposing Profile Hypotheses with Contrast Matrices

Three Questions About a Set of Profiles

Suppose \(a\) groups are measured at the same \(p\) occasions, and let \(M\) be the \(a \times p\) matrix of population means. Profile analysis separates three questions: do the groups have parallel profiles, do they have equal overall levels, and is the average profile flat across occasions? These are different questions, so each requires its own contrast.

The key is to distinguish comparisons across groups from comparisons across occasions. A group contrast removes a common group component; an occasion contrast removes a common occasion component. Combining them produces an interaction contrast, which is the part relevant to parallelism.

Definition: A group contrast matrix \(C\) is a full-row-rank matrix with \(a\) columns satisfying \(C\mathbf{1}_a=0\). An occasion contrast matrix \(A\) is a full-row-rank matrix with \(p\) columns satisfying \(A\mathbf{1}_p=0\). Here \(\mathbf{1}_k\) is the \(k\)-vector of ones. For complete contrast bases, \(C\) has \(a-1\) rows and \(A\) has \(p-1\) rows. The group-by-occasion interaction contrasts are \(CMA^{\mathsf T}\).

The choice of basis is flexible: rows may be scaled or replaced by other independent contrasts. A hypothesis that all group contrasts vanish does not depend on the particular full-rank basis used. For clarity, the examples use simple difference contrasts.

Writing the Three Hypotheses

Let \(C\) compare groups and \(A\) compare occasions. Parallelism asks whether each group difference is constant across occasions. Equality of profile levels asks whether group means, averaged over occasions, are equal. Flatness asks whether occasion means, averaged over groups, are equal.

Three profile hypotheses:

Parallelism: \(CMA^{\mathsf T}=0\).
Equal profile levels: \(CM\mathbf{1}_p=0\).
Flatness: \(\mathbf{1}_a^{\mathsf T}MA^{\mathsf T}=0\).

Dividing the last two expressions by \(p\) or \(a\), respectively, gives the corresponding contrasts of averages and does not change whether they equal zero.

The level hypothesis is not the claim that every group has the same mean at every occasion. It compares each group’s average across occasions. Likewise, flatness compares occasion averages across groups; it does not require every group’s own profile to be flat. Those stronger claims involve more than the respective marginal contrast.

Worked Example: Parallel Profiles With Different Levels

Consider three group profiles measured at four occasions:

M= \begin{pmatrix} 10&12&14&16\\ 13&15&17&19\\ 8&10&12&14 \end{pmatrix}.

Use group contrasts \(C=\begin{pmatrix}-1&1&0\\-1&0&1\end{pmatrix}\) and occasion adjacent-difference contrasts \(A=\begin{pmatrix}-1&1&0&0\\0&-1&1&0\\0&0&-1&1\end{pmatrix}\). First compare profiles:

CM= \begin{pmatrix} 3&3&3&3\\ -2&-2&-2&-2 \end{pmatrix}, \qquad CMA^{\mathsf T}= \begin{pmatrix} 0&0&0\\ 0&0&0 \end{pmatrix}.

Both group differences are constant across occasions, so the profiles are parallel. But their averages are \(13\), \(16\), and \(11\). Equivalently,

CM\mathbf{1}_4= \begin{pmatrix}12\\-8\end{pmatrix} \ne 0,

so equal profile levels are not supported by these means. The occasion averages across groups are \(31/3,37/3,43/3,49/3\); their adjacent differences are \(2,2,2\). Thus

\mathbf{1}_3^{\mathsf T}MA^{\mathsf T} = \begin{pmatrix}6&6&6\end{pmatrix} \ne 0.

The average profile is not flat. This example makes the separation visible: parallelism holds even though the group levels differ and the profiles change across occasions.

Why the Contrast Hypotheses Are Separate

The distinctions can be made precise by decomposing \(M\). Define the averaging projections \(P_a=\mathbf{1}_a\mathbf{1}_a^{\mathsf T}/a\) and \(P_p=\mathbf{1}_p\mathbf{1}_p^{\mathsf T}/p\), and the contrast projections \(Q_a=I_a-P_a\) and \(Q_p=I_p-P_p\). Then

$$ M=P_aMP_p+Q_aMP_p+P_aMQ_p+Q_aMQ_p. $$

The four terms are the grand mean component, group-level component, occasion component, and group-by-occasion interaction component, respectively.

Result: The three zero hypotheses correspond to three distinct components of the mean matrix: equal levels means \(Q_aMP_p=0\); flatness means \(P_aMQ_p=0\); parallelism means \(Q_aMQ_p=0\).

Proof. Since \(P_a\) and \(Q_a\) are complementary orthogonal projections, \(P_a+Q_a=I_a\); similarly \(P_p+Q_p=I_p\). Expanding \((P_a+Q_a)M(P_p+Q_p)\) gives the displayed four-term decomposition. For levels, \(Q_aMP_p=0\) holds exactly when \(Q_aM\mathbf{1}_p=0\), since \(P_p=\mathbf{1}_p\mathbf{1}_p^{\mathsf T}/p\). The columns of \(C^{\mathsf T}\) span the group contrast space, which is the column space of \(Q_a\). Therefore \(CM\mathbf{1}_p=0\) is equivalent to \(Q_aM\mathbf{1}_p=0\), and hence to \(Q_aMP_p=0\). The same argument on the occasion side shows that \(\mathbf{1}_a^{\mathsf T}MA^{\mathsf T}=0\) is equivalent to \(P_aMQ_p=0\). Finally, the row spaces of \(C\) and \(A\) span the group and occasion contrast spaces. Thus \(CMA^{\mathsf T}=0\) exactly when the interaction component \(Q_aMQ_p\) is zero. Each hypothesis sets a different component to zero. \(\square\)

The components are also orthogonal under the Frobenius inner product \(\langle U,V\rangle=\operatorname{tr}(U^{\mathsf T}V)\). For example, the group-level and interaction components have orthogonal occasion directions because \(P_pQ_p=0\). The other pairs are orthogonal by the same projection identities. Consequently, the three hypotheses do not collapse into one another: setting any one of these components to zero leaves the other two unconstrained.

Worked Example: Equal Marginal Means Without Parallelism

Take two groups and two occasions with

M= \begin{pmatrix} 1&3\\ 3&1 \end{pmatrix}, \qquad C=A=\begin{pmatrix}1&-1\end{pmatrix}.

Each group average is \(2\), and each occasion average is also \(2\). Directly, the level contrast is

CM\mathbf{1}_2 = \begin{pmatrix}1&-1\end{pmatrix} \begin{pmatrix}4\\4\end{pmatrix} =0,

and the flatness contrast is

\mathbf{1}_2^{\mathsf T}MA^{\mathsf T} = \begin{pmatrix}4&4\end{pmatrix} \begin{pmatrix}1\\-1\end{pmatrix} =0.

However, the interaction contrast is

CMA^{\mathsf T} = \begin{pmatrix}1&-1\end{pmatrix} \begin{pmatrix}-2&2\end{pmatrix} =-4.

Thus the marginal means are equal, but the group profiles are not parallel: one rises while the other falls. Equality of levels and flatness cannot substitute for a test of parallelism.

From Mean Contrasts to Test Conclusions

In an analysis, the entries of \(M\) are estimated from data, and each matrix hypothesis is tested separately using the relevant multivariate model. A contrast matrix specifies the hypothesis, not the uncertainty of its estimate. A test decision therefore also depends on sample sizes and the estimated covariance structure; a table of sample means alone cannot supply a p-value.

Worked Example: Interpreting Three Separate Test Results

Suppose three independent groups are measured at three occasions. Their estimated mean matrix is

\widehat M= \begin{pmatrix} 20&22&24\\ 23&25&27\\ 18&21&24 \end{pmatrix}.

Use \(C=\begin{pmatrix}-1&1&0\\-1&0&1\end{pmatrix}\) and \(A=\begin{pmatrix}-1&1&0\\0&-1&1\end{pmatrix}\). The estimated interaction contrasts are

C\widehat M= \begin{pmatrix} 3&3&3\\ -2&-1&0 \end{pmatrix}, \qquad C\widehat M A^{\mathsf T} = \begin{pmatrix} 0&0\\ 1&1 \end{pmatrix}.

The second group difference changes across occasions, so the estimated profiles are not exactly parallel. The estimated level contrasts are \(C\widehat M\mathbf{1}_3=(9,-3)^{\mathsf T}\), or average differences \((3,-1)^{\mathsf T}\). The occasion averages are \(61/3,68/3,75/3\), so the average profile also changes across occasions.

Now suppose the prespecified multivariate tests, using the study’s estimated covariance and an appropriate model, report p-values \(0.018\) for parallelism, \(0.041\) for equal levels, and \(0.006\) for flatness. At a significance level of \(0.05\), reject all three null hypotheses. The conclusions are: evidence of nonparallel profiles, evidence that average group levels differ, and evidence that the average profile is not flat. Each conclusion belongs to its own test; none is inferred merely from either of the other two.

Common Pitfall: Treating a Profile as One Hypothesis

A tempting but incorrect argument is: “The profiles are not parallel, so the groups must have different levels and the average profile must be nonflat.” This does not follow. The two-by-two example above has a nonzero interaction contrast while both marginal contrasts are zero. Conversely, the first example has parallel profiles while both marginal hypotheses fail at the level of its means.

A related mistake is to interpret a nonsignificant test as proof that the corresponding component is exactly zero. A test may fail to reject because the estimate is small, uncertainty is large, or the study has limited information. Report the hypothesis tested and the test result separately from a descriptive statement about the observed profiles.

Key takeaway: Use \(CMA^{\mathsf T}\) for parallelism, \(CM\mathbf{1}_p\) for equality of group levels, and \(\mathbf{1}_a^{\mathsf T}MA^{\mathsf T}\) for flatness. They isolate interaction, group, and occasion components, respectively; analyze and interpret them as distinct hypotheses.

Check Your Understanding

For each question, identify the relevant contrast or explain the distinction.

What matrix expression tests whether group differences are constant across occasions?
Why does \(CM\mathbf{1}_p=0\) not assert that the groups have equal means at every occasion?
In the two-group, two-occasion example, which two hypotheses hold even though the interaction contrast is nonzero?
Which component in the four-part decomposition is set to zero by the flatness hypothesis?
Why can a sample mean matrix not, by itself, determine the p-value for a multivariate contrast test?