Profile Analysis as Structured Mean Hypotheses

Profile Analysis as Mean Structure

In a repeated-measures study, each group has a mean at each occasion. Profile analysis asks how these means vary across groups and occasions. The central issue is that several statements that sound similar describe different features: groups may have profiles with the same shape but different levels; their average levels may agree even if their shapes differ; or each group’s profile may be constant across occasions.

These are hypotheses about the mean structure, not yet procedures for testing it. Writing them as contrast equations makes their meanings precise and separates the group dimension from the occasion dimension.

Let \(\mathbf{1}_a\) and \(\mathbf{1}_p\) denote vectors of ones. Choose matrices \(A\) and \(B\) whose columns are bases for the group-contrast and occasion-contrast spaces, respectively. Thus \(A\) is \(a\times(a-1)\), \(B\) is \(p\times(p-1)\), both have full column rank, and

The columns need not be orthogonal or normalized. Their role is to represent every possible contrast among groups and every possible contrast among occasions. The zero constraints make each column sum to zero, so each contrast compares levels rather than measuring an overall mean.

Three Distinct Mean Hypotheses

The hypotheses below refer to equally weighted occasions. This is the natural definition when the occasions are treated symmetrically; a different scientific weighting would require defining a different overall level.

The equations differ in which dimensions they contrast. The parallelism equation contrasts groups and occasions simultaneously. Equal overall level contrasts groups after averaging over occasions. Constancy contrasts occasions separately within every group. In particular, equal overall level does not say that profiles are parallel, and parallelism does not say that group averages are equal.

Why the Contrast Equations Have These Meanings

The following equivalences justify the matrix formulations. They also show why the hypotheses do not depend on the particular bases chosen for \(A\) and \(B\).

Proof. Write \(m_g'\) for row \(g\) of \(M\). If \(A'MB=0\), then every group contrast applied to the rows gives a vector whose occasion contrasts are zero. In particular, for any two groups \(g,h\), the group difference is a contrast, so \((m_g-m_h)'B=0\). The columns of \(B\) span the subspace orthogonal to \(\mathbf{1}_p\). A vector orthogonal to that whole subspace must be a multiple of \(\mathbf{1}_p\). Hence \(m_g-m_h=c_{gh}\mathbf{1}_p'\) for some constant \(c_{gh}\): the difference is the same at every occasion.

Conversely, suppose every pairwise row difference is constant across occasions. Every group contrast is a linear combination of pairwise row differences, so its resulting occasion vector is constant across occasions. Multiplication by \(B\) removes a constant vector because \(\mathbf{1}_p'B=0\). Thus \(A'MB=0\). This establishes the equivalence. Since the argument uses only the subspaces spanned by the contrast matrices, changing their bases leaves the zero hypothesis unchanged. \(\square\)

A second way to see the structure is to separate the mean matrix into four components. Define the projection matrices \(P_a=\mathbf{1}_a\mathbf{1}_a'/a\), \(Q_a=I_a-P_a\), \(P_p=\mathbf{1}_p\mathbf{1}_p'/p\), and \(Q_p=I_p-P_p\). The identities \(P_a+Q_a=I_a\) and \(P_p+Q_p=I_p\) give the decomposition

The terms represent, respectively, the grand mean, group differences in average level, occasion differences shared across groups, and group-by-occasion interaction. These are matrix components, not four additional assumptions. In particular, parallelism says the interaction component is zero; equal overall level says the group-level component is zero; and constancy says both occasion-varying components are zero.

Proof. Under parallelism, every difference between two profiles is a constant across occasions, by the equivalence just proved. Under equal overall level, the average of each such difference across occasions is zero. The average of a constant vector is that same constant, so the constant difference must be zero. Every pair of profiles is therefore identical.

Conversely, if all profiles coincide, their pairwise differences are zero. They are constant across occasions, and their occasion averages are equal across groups. Thus both parallelism and equal overall level hold. \(\square\)

Worked Contrast Calculations

For three occasions, use the occasion-contrast basis

For two groups, take \(A=(1,-1)'\). If \(d\) is the first group’s profile minus the second group’s profile, then \(A'MB=d\), and \(dB=(d_1-d_3,\ d_2-d_3)\). Thus the two displayed contrasts compare the first and second occasions with the third. Both being zero means all three differences are equal.

Reading the Hypotheses Without Confusing Them

The profile hypotheses can be compared by asking which contrasts must vanish. A zero interaction contrast concerns whether group differences change with occasion. A zero group contrast of occasion averages concerns only group level. A zero within-group occasion contrast concerns whether there is any occasion variation in each row.

The implications are also useful. Constancy implies parallelism, because constant profiles have constant pairwise differences. It does not imply equal overall level. Parallelism together with equal overall level implies coincident profiles, as proved above. Equal overall level alone implies neither parallelism nor constancy.

Common Mistake and Practical Importance

A frequent mistake is to treat “the groups have the same average” as equivalent to “the profiles are parallel.” That reasoning uses one number per group and discards the occasions’ pattern. In the second worked example, both averages equal \(6\), but the occasion-by-occasion differences are \(-2,0,2\), so the profiles are not parallel. The correction is to use the contrast that matches the scientific claim: \(A'M\mathbf{1}_p/p\) for equality of average levels, and \(A'MB\) for parallelism.

This formulation matters because the same observed group means can support different scientific questions. A treatment may have a constant advantage over time, may have no average advantage while changing differently over time, or may produce no change over time within either group. Specifying the mean hypothesis before choosing a test makes clear which pattern is being studied. The hypotheses here define the target; their testing and calibration require separate treatment.