Worked Example: Balanced and Confounded Cohorts

Why the Cohort-by-Platform Table Matters

Suppose a molecular feature differs between two cohorts, but each cohort was measured on a different platform. The observed difference could reflect cohort biology, platform behavior, or both. Statistical adjustment cannot separate these explanations just by choosing a more elaborate formula: the observed cohort-by-platform pattern determines which contrasts the data can support.

This tutorial uses small hypothetical tables to distinguish a balanced design, a partially overlapping design, and a perfectly confounded design. The central question is whether the observed cohort and platform combinations connect the levels well enough to estimate cohort contrasts separately from platform contrasts.

Definition: For one molecular feature, an additive cohort-and-platform model is $$ y_i=\mu+\gamma_{c(i)}+\delta_{b(i)}+\varepsilon_i, $$ where $y_i$ is the measured value for sample $i$, $\mu$ is an overall intercept, $\gamma_c$ is the effect for cohort $c$, $\delta_b$ is the effect for platform $b$, and $\varepsilon_i$ is an error term. A common identification convention is $\sum_c\gamma_c=0$ and $\sum_b\delta_b=0$. These constraints define the reference location for the effects; they do not assert that cohort or platform effects are absent.

The model assumes that, for this feature, the cohort and platform contributions add: there is no cohort-by-platform interaction in the mean. For ordinary least-squares estimation, the errors are taken to have mean zero conditional on the design; independent errors with a common variance are a standard additional assumption for the usual standard errors. The identifiability conclusions below are about the design matrix, not about an error distribution. Even with non-normal errors, a contrast that is not identified by the design cannot be recovered by fitting this model.

A cohort-by-platform table records which cells have samples. A useful equivalent representation is a bipartite graph: one kind of vertex represents cohorts, the other represents platforms, and an edge joins a cohort to a platform when that cell is observed. The edge may have one or many samples. For identifiability, the key initial question is whether this graph is connected.

A Balanced, Fully Crossed Design

A balanced design has equal sample counts in the observed cells. In a fully crossed design, every cohort is measured on every platform. Balance is helpful because cohort and platform comparisons are not weighted unevenly by cell sample counts. More fundamentally, full crossing connects every cohort to every platform. Balance is not itself necessary for identifiability: a connected but unbalanced design can also identify the additive contrasts.

Worked Example: Balanced Two-Cohort, Two-Platform Table

Suppose a hypothetical study has four samples in each cell. The entries below are the cell means for one feature, measured in arbitrary normalized units.

	Platform P1	Platform P2
Cohort C1	10	12
Cohort C2	14	16

The grand mean of the four cell means is $(10+12+14+16)/4=13$. The cohort means are 11 for C1 and 15 for C2, so their deviations from the grand mean are $-2$ and $2$. The platform means are 12 for P1 and 14 for P2, giving deviations $-1$ and $1$. These effects satisfy both sum-to-zero constraints.

The fitted additive means are therefore $13-2-1=10$, $13-2+1=12$, $13+2-1=14$, and $13+2+1=16$, matching all four cell means. In this balanced example, the estimated C2-minus-C1 contrast is $15-11=4$ units, and the estimated P2-minus-P1 contrast is $14-12=2$ units.

The no-interaction pattern can also be checked by the difference-in-differences contrast: $10-12-14+16=0$. This agreement is consistent with the additive model for these cell means. With noisy data, a nonzero sample contrast would not by itself prove an interaction; its size must be judged relative to sampling variation.

Overlap and Identifiability

The graph view makes the role of overlap precise. If a cohort was measured on two platforms, those platforms are linked through that cohort. If a platform was used in two cohorts, those cohorts are linked through that platform. A chain of such overlaps can connect levels even when many cohort-platform cells are empty.

Key result: In the additive model, if the graph of observed cohort-platform cells is connected and every listed level has observations, cohort contrasts and platform contrasts are identifiable under the two sum-to-zero constraints. If the graph has disconnected components, contrasts between levels in different components are generally not separately identifiable from cohort and platform effects.

Here, “identifiable” means that no two distinct parameter values satisfying the stated constraints give the same fitted means for every observed sample, except where the contrast itself is unchanged. The intercept and effects without constraints have redundant parameterizations. This is not a defect in the fitted means; it is a question of whether the effects have been given a unique numerical representation.

Why a Connected Graph Is Enough

A short argument shows why the graph criterion works. Consider a change in parameters that leaves every fitted value unchanged. Write the changes as $\Delta\mu$, $\Delta\gamma_c$, and $\Delta\delta_b$. For every observed cohort-platform cell, the change in its fitted mean must be zero:

\Delta\mu+\Delta\gamma_c+\Delta\delta_b=0

On a connected graph, moving from one observed edge to another through a shared cohort or platform propagates the same cohort shift across all cohort vertices and the same platform shift across all platform vertices. Thus any parameter change leaving all fitted means fixed must have $\Delta\gamma_c=a$ for every cohort and $\Delta\delta_b=b$ for every platform, with $\Delta\mu=-a-b$. Without constraints, there are two independent redundant directions: a common shift of cohort effects offset by the intercept, and a common shift of platform effects offset by the intercept.

The two sum-to-zero constraints remove both directions. If there are $C$ cohorts and $B$ platforms, the changed sums are $Ca$ and $Bb$. Requiring both sums to remain zero gives $Ca=0$ and $Bb=0$; since $C$ and $B$ are positive, $a=b=0$, and then $\Delta\mu=0$. Therefore the constrained parameterization is unique. In particular, the cohort and platform contrasts are identifiable. The two constraints eliminate the two redundancies; they do not leave one remaining ambiguity.

Worked Example: Partial Overlap That Still Connects the Design

Consider three cohorts and three platforms, with hypothetical cell means in normalized units. Empty cells were not observed.

	Platform P1	Platform P2	Platform P3
Cohort C1	17	18	—
Cohort C2	—	20	21
Cohort C3	—	—	23

The observed graph is the path P1—C1—P2—C2—P3—C3. It is connected even though only five of the nine possible cells are observed. Suppose the additive model and sum-to-zero constraints hold. The following parameter values reproduce the table: $\mu=20$, cohort effects $(-2,0,2)$, and platform effects $(-1,0,1)$. For example, the C2-P3 fitted mean is $20+0+1=21$, and the C3-P3 fitted mean is $20+2+1=23$. The cohort effects sum to zero, as do the platform effects.

Overlap supplies the comparisons. At P2, the C2-minus-C1 cell-mean difference is $20-18=2$ units; under the additive model this equals $\gamma_{\mathrm{C2}}-\gamma_{\mathrm{C1}}$, because the platform effect cancels. At C2, the P3-minus-P2 difference is $21-20=1$ unit, which equals $\delta_{\mathrm{P3}}-\delta_{\mathrm{P2}}$. The path links the remaining levels as well. The sum constraints then set the location of the effects and yield the values above.

This example demonstrates identifiability, not a general recommendation to estimate effects by chaining raw differences. In real data, cell means can be noisy and cell counts unequal; fitting the specified model uses all observed samples. Also, the absence of unobserved cells means the data cannot directly check every possible cohort-by-platform interaction. The additive assumption matters.

Perfect Confounding: No Separate Attribution

Perfect confounding occurs when cohort membership determines platform, so the cohort and platform patterns cannot be disentangled. In graph terms, the observed table separates into disconnected pieces. A difference between pieces may be real, but its division into cohort and platform components is not learned from those observations alone.

Worked Example: Two Cohorts on Separate Platforms

Suppose a hypothetical study measures C1 only on P1 and C2 only on P2. The observed means are 10 and 16 normalized units.

	Platform P1	Platform P2
Cohort C1	10	—
Cohort C2	—	16

The graph has two separate edges, so there is no observed bridge between the cohorts or between the platforms. Under the sum-to-zero constraints, write the cohort effects as $(-a,a)$ and the platform effects as $(-b,b)$. The two fitted means require

\mu-a-b=10,\qquad \mu+a+b=16.

Adding these equations gives $2\mu=26$, so $\mu=13$. Subtracting the first equation from the second gives $2a+2b=6$, or $a+b=3$. The observed difference of 6 units is therefore the sum of the cohort contrast $2a$ and the platform contrast $2b$; it does not determine either contrast separately.

For instance, $a=3,\ b=0$ gives cohort effects $(-3,3)$ and no platform contrast. Alternatively, $a=0,\ b=3$ gives no cohort contrast and platform effects $(-3,3)$. Both choices, with $\mu=13$, reproduce the observed means exactly. The constraints have fixed the overall location but have not made the separate cohort and platform contrasts identifiable.

Diagnostics, Alternatives, and Limitations

Before fitting or interpreting an adjustment, make a table of observed cell counts and inspect its graph. Check that every level being compared has observations, identify disconnected components, and note cells supported by only one or a few samples. A connected graph is a necessary design check for separate additive contrasts, but it does not establish that the additive model is scientifically adequate or that the resulting estimates are precise.

A common mistake is to assume that software output proves the cohort and platform effects were separately learned. A program may return coefficients after imposing reference coding, sum constraints, or a generalized inverse. Those conventions can produce a particular numerical solution even when the data support multiple equivalent decompositions. The correction is to inspect the design and ask whether the contrast of interest is unchanged across all parameterizations that give the same fitted values. If not, it is not identifiable from the observed design.

Reference coding and sum-to-zero coding are alternative ways to represent effects. In a connected design, they yield different coefficient labels and intercepts, but properly translated cohort and platform contrasts agree. In a disconnected design, changing coding cannot create missing comparisons. A cell-means model is another useful representation: it estimates observed cell means directly, but those means alone do not separate cohort from platform contributions without the additive assumption and overlap structure.

If a design is disconnected or perfectly confounded, possible remedies must provide information rather than just change the parameterization. New bridge samples, reference materials, or technical measurements on more than one platform can connect parts of the design. These approaches have their own assumptions and do not automatically remove every source of bias. If no suitable additional information is available, report the observed combined differences and state that separate attribution is unsupported.

Key takeaway: A balanced, fully crossed table makes comparisons transparent. Partial overlap can still identify additive cohort and platform contrasts when the observed-cell graph is connected. Perfect confounding leaves some contrasts inseparable, regardless of the fitting software or coding convention.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

Three cohorts are each measured on P1 and P2, but not on P3. Is the observed graph connected across the observed cohort and platform levels? What does that imply for additive contrasts?
In a connected design with two cohorts and three platforms, how many independent redundant parameter directions are present before constraints, and what constraints are used here?
In a two-cohort, two-platform diagonal table with cell means 8 and 14, can the cohort contrast and platform contrast be estimated separately? What combined contrast is observed?
Does equal sample size in every observed cell guarantee that the cohort-by-platform effects are identifiable? Explain.
Can reference coding resolve a disconnected design? Explain why or why not.

Solutions. (1) Yes: the shared platforms link all three cohorts, and the two platforms are connected through each cohort. Under the additive model, cohort and platform contrasts are identifiable with the stated constraints. (2) There are two independent redundant directions: a common cohort shift and a common platform shift, each offset by the intercept. The constraints $\sum_c\gamma_c=0$ and $\sum_b\delta_b=0$ remove them in a connected design. (3) No. The observed difference is $14-8=6$ units, which is the sum of the cohort contrast and platform contrast, not a separate estimate of either. (4) No. Equal counts make the observed cells balanced, but if the graph has disconnected components, contrasts across those components remain confounded. (5) No. Reference coding selects a representation; it cannot supply the unobserved overlap needed to distinguish cohort from platform effects.