What Does It Mean for Profiles to Be Parallel?
A profile records a group mean at each of several occasions. Two profiles are parallel when their vertical separation does not change across occasions. With several groups, the same condition must hold for every pair of groups. This is a statement about changes across occasions, not about whether the groups have the same mean level.
Write the mean profile matrix as \(M\), with \(g\) rows for groups and \(p\) columns for occasions. Its entry \(\mu_{ij}\) is the population mean for group \(i\) at occasion \(j\). The difference between groups \(i\) and \(k\) is constant across occasions precisely when \(\mu_{ij}-\mu_{kj}\) does not depend on \(j\).
The equivalence in the definition follows by subtracting group differences at two occasions. For any groups \(i,k\) and occasions \(j,\ell\), parallelism requires \(\mu_{ij}-\mu_{kj}=\mu_{i\ell}-\mu_{k\ell}\). Rearranging gives \((\mu_{ij}-\mu_{i\ell})=(\mu_{kj}-\mu_{k\ell})\): the occasion-to-occasion change is common across groups.
Constructing the Group-by-Occasion Contrasts
Let \(A\) be any full-row-rank group contrast matrix of dimension \((g-1)\times g\), so \(A\mathbf{1}_g=0\). Let \(B\) be any full-row-rank occasion contrast matrix of dimension \((p-1)\times p\), so \(B\mathbf{1}_p=0\). Here \(\mathbf{1}_g\) and \(\mathbf{1}_p\) are vectors of ones. The matrix \(A M B^{T}\) first contrasts group means and then contrasts those group differences over occasions.
The choice of full-rank contrast bases does not affect the null hypothesis. A different basis changes the coordinates used to express the contrasts, but not whether all contrasts are zero. For concrete calculations, adjacent differences are often convenient: the rows of \(B\) can be \((-1,1,0,\ldots,0)\), \((0,-1,1,\ldots,0)\), and so on.
Worked Example: Parallel Profiles with Different Group Levels
Suppose three groups have four occasion means: \(\mu_1=(10,12,14,16)\), \(\mu_2=(13,15,17,19)\), and \(\mu_3=(8,10,12,14)\). Use adjacent occasion differences and adjacent group contrasts.
Take \(A\) to have rows \((1,-1,0)\) and \((0,1,-1)\), and \(B\) to have rows \((-1,1,0,0)\), \((0,-1,1,0)\), and \((0,0,-1,1)\). The occasion differences for the three profiles are \(B\mu_1^T=(2,2,2)^T\), \(B\mu_2^T=(2,2,2)^T\), and \(B\mu_3^T=(2,2,2)^T\).
The first row of \(A M B^T\) subtracts the occasion differences of group 2 from those of group 1, giving \((0,0,0)\). Its second row subtracts those of group 3 from those of group 2, also giving \((0,0,0)\). Thus \(A M B^T=0\), and the profiles are parallel. Their levels are not equal: for example, group 2 is three units above group 1 at every occasion.
Why the Joint Contrast Null Is Exactly Parallelism
The following result makes the matrix hypothesis precise. It also shows why a single omnibus null is appropriate: it captures every group comparison across every independent occasion contrast.
Proof. Define the centering projections \(P_g=I_g-\mathbf{1}_g\mathbf{1}_g^T/g\) and \(P_p=I_p-\mathbf{1}_p\mathbf{1}_p^T/p\). The rows of \(A\) span the subspace orthogonal to \(\mathbf{1}_g\), as do the rows of \(P_g\); likewise the rows of \(B\) and \(P_p\) span the subspace orthogonal to \(\mathbf{1}_p\). Therefore \(A M B^T=0\) exactly when \(P_g M P_p=0\): both say that every doubly centered group-by-occasion contrast vanishes.
If \(P_g M P_p=0\), decompose \(M\) using \(I_g=P_g+(I_g-P_g)\) and \(I_p=P_p+(I_p-P_p)\): \(M=(I_g-P_g)M+P_gM(I_p-P_p)+P_gMP_p\). The last term is zero. Since \(I_g-P_g=\mathbf{1}_g\mathbf{1}_g^T/g\), the first term is \(\mathbf{1}_g a^T\), where \(a^T=\mathbf{1}_g^T M/g\). Since \(I_p-P_p=\mathbf{1}_p\mathbf{1}_p^T/p\), the second term is \(b\mathbf{1}_p^T\), where \(b=P_gM\mathbf{1}_p/p\). This proves the stated form. Conversely, if \(M=\mathbf{1}_g a^T+b\mathbf{1}_p^T\), then \(P_gMP_p=0\), because \(P_g\mathbf{1}_g=0\) and \(\mathbf{1}_p^TP_p=0\).
Finally, the difference between rows \(i\) and \(k\) of this form is \((b_i-b_k)\mathbf{1}_p^T\), constant across occasions. Conversely, if every pairwise row difference is constant across occasions, choosing one group as a reference expresses every row as that reference profile plus a group-specific constant vector, which has the displayed additive form. Thus the matrix null and profile parallelism are equivalent. \(\square\)
Worked Example: Detecting a Nonparallel Pair
Keep the same matrices \(A\) and \(B\), but let the profiles be \(\mu_1=(10,12,14,16)\), \(\mu_2=(13,15,17,19)\), and \(\mu_3=(8,10,13,14)\). Their adjacent occasion differences are \((2,2,2)\), \((2,2,2)\), and \((2,3,1)\), respectively.
The first row of \(A M B^T\), comparing groups 1 and 2, is \((2,2,2)-(2,2,2)=(0,0,0)\). The second row, comparing groups 2 and 3, is \((2,2,2)-(2,3,1)=(0,-1,1)\). Consequently, \(A M B^T\ne 0\). In particular, the group 2 minus group 3 difference is \((5,5,4,5)\), which is not constant across occasions. These profiles are not parallel.
Turning the Hypothesis into a Multivariate Test
For subject \(k\) in group \(i\), let \(Y_{ik}\) be the \(p\)-dimensional vector of repeated measurements and let \(E[Y_{ik}]=\mu_i\). Transform each subject’s response to \(Z_{ik}=B Y_{ik}\), a \(q\)-dimensional vector where \(q=p-1\) for a full occasion contrast basis. Its group mean is \(\theta_i=B\mu_i\). The parallelism null becomes equality of these transformed group means: \(H_0:A\Theta=0\), where \(\Theta\) has rows \(\theta_i^T\).
Thus, profile parallelism is a one-way multivariate group effect on the occasion-contrast responses. The responses are contrasts, not the original occasion levels; the tested effect concerns whether groups differ in their pattern of change. Under independent multivariate normal observations with a common positive-definite covariance matrix, this is the ordinary one-way MANOVA hypothesis. The transformed covariance is \(B\Sigma B^T\), which is positive definite when \(\Sigma\) is positive definite and \(B\) has full row rank.
For group sample sizes \(n_i\), let \(N=\sum_i n_i\), let \(\bar Z_i\) be the transformed sample mean in group \(i\), and let \(\bar Z\) be the overall transformed mean. The within-group and between-group sums-of-products matrices are
When \(E\) is nonsingular, Wilks’ lambda is \(\Lambda=|E|/|E+H|\). A small value indicates that the group means differ relative to within-group variation in at least some linear combination of occasion contrasts. The usual MANOVA criteria, including Wilks’ lambda, test the same joint null; their calibration and comparative behavior are part of the standard MANOVA framework.
Under the stated normal model and \(N-g\ge q\), \(E\) is positive definite with probability one: after fitting \(g\) group means, the residual space has dimension \(N-g\), and the residual vectors have a nonsingular covariance in the \(q\)-dimensional response space. Under \(H_0\), the standard one-way MANOVA decomposition gives independent Wishart matrices \(E\) and \(H\), with degrees of freedom \(N-g\) and \(g-1\), respectively, and common scale \(B\Sigma B^T\). This supplies the null distribution used to calibrate multivariate tests. The condition concerns residual degrees of freedom: for example, three groups with two subjects each have \(N-g=6-3=3\), sufficient for full rank when \(q=3\) under this model.
Worked Example: Exact F Test with Two Occasions
With two occasions, there is one independent occasion contrast, so \(q=1\). Consider three groups with observed within-subject differences \(d=Y_2-Y_1\): group 1 has \(1,2,3\); group 2 has \(3,4,5\); group 3 has \(0,1,2\). Parallelism is exactly equality of the three population mean differences.
The sample means are \(2,4,1\), and the grand mean is \((2+4+1)/3=7/3\). The between-group sum of squares is \(3[(2-7/3)^2+(4-7/3)^2+(1-7/3)^2]\) \(=3[(1+25+16)/9]=14\).
Within each group, the deviations from its mean are \(-1,0,1\), so each group contributes \(2\) to the within-group sum of squares. Therefore \(E=6\), with \(N-g=9-3=6\) residual degrees of freedom. The between-group degrees of freedom are \(g-1=2\), giving \(F=(14/2)/(6/6)=7\), with \(2\) and \(6\) degrees of freedom under the null.
This is the usual one-way ANOVA F test applied to the subject-level difference scores. It is an exact test of profile parallelism under independent normal observations with common variance for the differences. It does not test whether the groups have equal means at either occasion separately.
Why the Joint Test Matters
The parallelism hypothesis is joint: all independent group-by-occasion contrasts must vanish. Testing each occasion difference separately and then informally combining the results does not, by itself, provide a test with a specified joint Type I error rate. It can also miss a coordinated multivariate pattern: group differences may be modest in each contrast but jointly informative. A MANOVA criterion evaluates the contrast-response vectors together and accounts for their covariance.
A useful interpretation is therefore: the test asks whether group membership is associated with the vector of within-profile contrasts. Rejection indicates that at least one group comparison has a nonconstant difference across occasions. It does not identify which contrasts differ; follow-up estimates or planned contrasts are needed for that purpose.
Common Mistakes
- Equating parallelism with equal profiles. Parallel profiles may have different vertical offsets. The null tests equality of changes across occasions, not equality of all group means.
- Testing levels instead of changes. A group effect at one occasion is not itself evidence of nonparallelism. The relevant quantities are group differences after occasion contrasts.
- Using only one group comparison. With \(g>2\), checking one pair does not establish that every pair of profiles is parallel. A full-rank group contrast basis spans all group comparisons.
- Interpreting a nonsignificant result as proof of parallelism. Failure to reject means the data do not provide sufficient evidence against the joint null at the chosen level; it does not establish exact equality of changes.
Check Your Understanding
Use the definitions and matrix interpretation above to answer these questions.
- In words, what must remain constant across occasions for two group profiles to be parallel?
- Why does \(A M B^T=0\) not require the group profiles to have equal levels?
- For three occasions, what does an adjacent-difference contrast basis measure?
- With two occasions, why can profile parallelism be tested by comparing group means of the subject-level differences?
- What does rejection of the joint parallelism null establish, and what does it not identify by itself?