Additive and Multiplicative Models of Technical Batch Effects

Why Distinguish Shifts From Scale Changes?

A technical batch effect is a systematic difference in measurements associated with how, when, or where samples were processed, rather than with the biological signal of interest. In a multi-omics feature matrix, such differences can affect one feature differently from another. A constant offset and a proportional change are not interchangeable: they imply different relationships between the observed value and the underlying signal, and can have different consequences when features are compared or integrated.

Let \(x_{ij}\) be the measured value for sample \(i\) and feature \(j\), and let \(b(i)\) denote the batch in which sample \(i\) was processed. Let \(\theta_{ij}\) denote the sample's underlying signal for feature \(j\) in the absence of the batch effect. This signal is generally not observed directly. The models below describe how batch may alter it; they do not, by themselves, establish that an observed batch difference is technical rather than biological.

Definition: In an additive batch-effect model, \(x_{ij}=\theta_{ij}+\alpha_{b(i)j}+\varepsilon_{ij}\), where \(\alpha_{kj}\) is the additive effect of batch \(k\) on feature \(j\), and \(\varepsilon_{ij}\) is residual variation with conditional mean zero. In a multiplicative batch-effect model, \(x_{ij}=\gamma_{b(i)j}\theta_{ij}+\varepsilon_{ij}\), where \(\gamma_{kj}>0\) is the dimensionless scale factor for batch \(k\) and feature \(j\), and again the residual has conditional mean zero. The first model adds an amount; the second multiplies the underlying signal.

These are conditional mean models: for example, the additive model asserts \(E(x_{ij}\mid\theta_{ij},b(i)=k)=\theta_{ij}+\alpha_{kj}\). The residuals need not have the same variance in every batch, but any analysis that assumes a common variance requires that additional assumption. The multiplicative formulation describes a scale change in the conditional mean; it does not automatically mean that the residual variance is multiplied by \(\gamma_{kj}^2\). That variance structure would need to be specified separately.

Both effects are commonly feature-specific. A shared \(\alpha_k\) or \(\gamma_k\) across all features is a more restrictive model, appropriate only when a common effect across those features is scientifically and technically defensible. In the additive model, \(\alpha_{kj}\) has the same units as \(x_{ij}\). In the multiplicative model, \(\gamma_{kj}\) has no units. A value \(\gamma_{kj}=1\) means no multiplicative change; \(\gamma_{kj}>1\) indicates an increase for positive underlying signals, while \(0<\gamma_{kj}<1\) indicates a decrease.

Parameter Meaning and Identifiability

The parameter meanings become clear by comparing measurements for the same underlying signal in two batches. For an additive model, the difference between the batch-specific conditional means is the difference in offsets, regardless of the signal level. For a multiplicative model, the difference depends on the signal level, while the ratio of the means is the ratio of the scale factors when the underlying signal is nonzero.

\begin{aligned} \text{Additive:}\quad &E(x\mid \theta,k=H)-E(x\mid \theta,k=L)=\alpha_{Hj}-\alpha_{Lj},\\ \text{Multiplicative:}\quad &E(x\mid \theta,k=H)-E(x\mid \theta,k=L) =(\gamma_{Hj}-\gamma_{Lj})\theta,\\ &\frac{E(x\mid \theta,k=H)}{E(x\mid \theta,k=L)} =\frac{\gamma_{Hj}}{\gamma_{Lj}},\qquad \theta\ne 0. \end{aligned}

An additive difference is constant across signal levels. A multiplicative difference grows or shrinks with the signal. In the special comparison of a signal \(m\) before and after a single scale factor \(\gamma\), the change is \((\gamma-1)m\), and the relative change is \(\gamma-1\) when \(m\ne0\). In particular, for \(0<m_L<m_H\) and \(\gamma\ne1\), the magnitudes of the changes at the two signal levels are \(|\gamma-1|m_L\) and \(|\gamma-1|m_H\). Their ratio is \(m_H/m_L\). This is a statement about magnitudes under those conditions, not a claim that the signed change always has that ratio.

Neither model's parameters are automatically identifiable from measurements alone. For instance, in the additive model, adding a constant to every \(\theta_{ij}\) and subtracting it from every \(\alpha_{kj}\) leaves the fitted means unchanged. A reference-batch convention, such as \(\alpha_{1j}=0\), or a sum-to-zero constraint, such as \(\sum_k\alpha_{kj}=0\), fixes this bookkeeping ambiguity. Similarly, one can set \(\gamma_{1j}=1\) for a reference batch. Such constraints define parameterization; they do not create the experimental information needed to separate batch from biology.

Worked Models and Calculations

Worked Example: Interpreting an Additive Offset

Suppose the same stable control material is measured twice in each of two batches for one feature, in arbitrary intensity units. The hypothetical readings are 12.2 and 11.8 in batch A, and 9.9 and 10.1 in batch B. Treat the control as having the same underlying signal in both batches, and use batch B as the reference, so \(\alpha_{Bj}=0\).

The batch means are \(\bar{x}_{Aj}=(12.2+11.8)/2=12.0\) and \(\bar{x}_{Bj}=(9.9+10.1)/2=10.0\). Under the additive model for this control, the estimated offset for batch A relative to B is \(12.0-10.0=2.0\) intensity units. Thus the model describes batch A's control mean as 2.0 units higher, not as 2.0 percent higher. The within-batch deviations, \(\pm0.2\) and \(\pm0.1\) units, are residual variation; they do not alter the estimated difference in these means.

This interpretation depends on the control having the same underlying signal in both batches. If the control material changed between runs, the difference could not be attributed solely to batch from these measurements.

Worked Example: Interpreting a Multiplicative Effect at Two Levels

Consider a hypothetical feature measured at two positive underlying levels, \(m_L=4\) and \(m_H=6\) units. Suppose one batch applies the same scale factor \(\gamma=1.5\), with no residual error for this calculation.

At the lower level, the observed mean is \(1.5(4)=6\) units, a change of \(6-4=2\) units. At the higher level, it is \(1.5(6)=9\) units, a change of \(9-6=3\) units. The relative changes are \(2/4=0.5\) and \(3/6=0.5\), both 50%. The change magnitudes have ratio \(3/2=1.5\), which equals \(m_H/m_L=6/4\). This agrees with the general result because \(0<m_L<m_H\) and \(\gamma\ne1\).

The scale factor is dimensionless, whereas the changes of 2 and 3 are measured in feature units. A constant additive offset of 2 units would match the lower-level change but not the higher-level change, illustrating why the two models encode different patterns.

Worked Example: Batch and Biology Completely Confounded

Suppose a hypothetical study measures a feature in 10 controls, all processed in batch A, and 10 treated samples, all processed in batch B. The control mean is 10 units and the treated mean is 15 units. Consider the additive mean model \(E(x)=\mu+\beta T+\alpha_B I(B)\), where \(T=1\) for treated samples and \(I(B)=1\) for batch B; the control reference values are \(T=0\) and \(I(B)=0\).

For controls, the model gives \(10=\mu\), so \(\mu=10\). For treated samples, it gives \(15=10+\beta+\alpha_B\), and therefore \(\beta+\alpha_B=5\). The data determine this sum, but not its two components. For example, \(\beta=5,\alpha_B=0\) and \(\beta=0,\alpha_B=5\) both reproduce the observed means exactly. Infinitely many pairs with \(\beta+\alpha_B=5\) fit as well.

Consequently, these data cannot establish how much of the five-unit difference is treatment-related and how much is batch-related. Replication within each of the two groups does not resolve the problem because treatment and batch never vary independently. A design that places both treatment groups in both batches, or suitable shared controls, is needed to distinguish their effects under the model.

Log Scales, Assumptions, and Model Checks

For positive measurements, a multiplicative mean relationship has a useful log-scale representation. If the error-free relation is \(x=\gamma\theta\), with \(x>0\), \(\theta>0\), and \(\gamma>0\), then \(\log x=\log\theta+\log\gamma\). Thus a multiplicative factor on the original scale becomes an additive offset on the log scale. This exact identity applies to the multiplicative signal relation. With residual error, the distribution and conditional mean on the log scale depend on the error model; in general, \(E(\log X)\) is not simply \(\log E(X)\). A log transformation therefore does not make arbitrary error assumptions equivalent.

An additive offset on the original scale does not generally become a constant offset on the log scale: \(\log(\theta+\alpha)-\log(\theta)\) depends on \(\theta\), when both logarithms are defined. Likewise, multiplicative effects are unsuitable for describing negative underlying signals with positive scale factors as a simple proportional increase or decrease. Earlier material on log-ratio transforms and zeros and detection limits is relevant when deciding whether a log-scale interpretation is meaningful; values at or below zero cannot be logged without an explicit modeling decision.

The basic models assume that the chosen feature and measurement scale are meaningful and that the model's stated conditional mean pattern is a reasonable approximation. Interpreting a batch parameter as technical also requires design or external information supporting separation of batch from biology. Useful checks include plotting measurements or residuals against fitted signal separately by batch, comparing control materials across batches, and checking whether differences look approximately constant or proportional across the range of signal. A residual-versus-fitted pattern that changes systematically by batch, or markedly different spread from that assumed, is evidence that the chosen mean or error model may be inadequate. Such checks can reveal problems but cannot cure complete confounding.

Model-selection guide: An additive model is a natural description when a batch difference is approximately constant in feature units across the relevant signal range. A multiplicative model is more appropriate when batch differences are approximately constant as ratios or percentages. A log-additive model describes multiplicative effects on the original positive scale when its error and measurement assumptions are also plausible. None is universally preferable; the scale, control data, residual patterns, and experimental design matter.

Model	Batch contrast at signal \(\theta\)	Parameter interpretation	Useful evidence
Additive	Constant difference \(\alpha_{Hj}-\alpha_{Lj}\)	Offset in the feature's units	Shared controls show similar absolute differences at different levels
Multiplicative	Difference proportional to \(\theta\)	Dimensionless scale factor \(\gamma_{Hj}/\gamma_{Lj}\)	Shared controls show similar ratios across positive levels
Log-additive	Constant difference on the log scale	Log ratio on the original scale	Positive measurements and a defensible log-scale model

Common Pitfalls and Practical Limits

Calling every batch difference a technical effect. A difference in observed means can also reflect different biological composition across batches. Without overlap, controls, or other design information, the cause may not be identifiable.
Confusing a factor with an absolute change. A factor such as 1.2 means a 20% increase relative to a positive baseline, not an increase of 1.2 measurement units.
Choosing a model from one signal level. At one level, an offset and a factor can produce the same observed difference. Measurements across relevant levels or shared controls help distinguish constant shifts from proportional changes.
Assuming that logging solves every problem. Logging changes the scale and requires positive values; it does not establish that batch is separable from biology or guarantee a suitable residual distribution.
Ignoring feature-specific behavior. A single batch parameter shared across a whole omics block can conceal features with different offsets, scale factors, or noise patterns.

The models specify patterns that can be used to reason about technical effects; they are not, on their own, instructions for removing those effects. In particular, a model cannot recover a unique biological-versus-batch decomposition when the study design provides no information to distinguish the two. Even with an identifiable design, incorrect assumptions about additive versus proportional behavior can leave systematic differences unexplained or obscure meaningful signal.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In an additive model, what units does \(\alpha_{kj}\) have, and what does it mean to set the reference-batch effect to zero?
A multiplicative factor of 0.8 is applied to positive signals of 5 and 10 units. Calculate the observed values, absolute changes, and relative changes.
Why can treatment and batch effects not be separated when every control is in one batch and every treated sample is in another?
What relationship becomes additive after taking logarithms, and what positivity condition is needed for that transformation?

Solutions.

\(\alpha_{kj}\) is measured in the same units as feature \(j\). Setting the reference-batch effect to zero defines that batch's offset as the comparison baseline; other additive effects are interpreted relative to it.
The observed values are \(0.8(5)=4\) and \(0.8(10)=8\) units. The changes are \(4-5=-1\) and \(8-10=-2\) units, and each relative change is \(-1/5=-0.2\), or \(-20\%\), and \(-2/10=-0.2\), or \(-20\%\). The larger signal has the larger absolute change magnitude, but the same relative change.
The model's observed group contrast is the sum of the treatment contrast and the batch contrast. If treatment status and batch always occur together, the data provide no independent variation to identify the two components separately.
A multiplicative relation \(x=\gamma\theta\) becomes \(\log x=\log\gamma+\log\theta\). Both \(x\) and \(\theta\) must be positive for these logarithms to be defined; modeling residuals on the log scale requires additional assumptions.