Why Separate Baseline Differences From Treatment-Period Changes?
An omics feature can differ between patients who later respond to immunotherapy and those who do not before treatment begins. It can also change during treatment, and that change may differ by response group. These are distinct patterns. A pretreatment contrast may identify a baseline response-associated marker, but it does not show that treatment altered the feature. A contrast measured only on treatment can combine the baseline difference with any subsequent divergence.
The key analysis therefore preserves the within-patient pairing of pretreatment and on-treatment samples. For each feature, it estimates the baseline group contrast separately from the change over time in each group. The difference between those two changes is the response-by-time contrast. It is an association between response group and the observed treatment-period change, not by itself a causal effect of immunotherapy.
A Paired Model and Its Interpretation
Let \(Y_{igt}\) denote the measured abundance of one feature for patient \(i\), response group \(g\), and time \(t\). Code \(G_i=1\) for responders and \(G_i=0\) for nonresponders, and code \(T_t=1\) for on-treatment and \(T_t=0\) for pretreatment. A linear mixed model for a continuous, suitably normalized feature is
Here \(b_i\) is a patient-specific random intercept, and \(\varepsilon_{igt}\) is the residual. In this coding, \(\beta_0\) is the nonresponder pretreatment mean, \(\beta_G\) is the pretreatment responder-minus-nonresponder contrast, \(\beta_T\) is the nonresponder mean change, and \(\beta_{GT}\) is the difference in changes. The responder mean change is \(\beta_T+\beta_{GT}\). Thus, a feature can have a nonzero \(\beta_G\) but a near-zero \(\beta_{GT}\): it distinguishes groups at baseline without showing a group-specific change over time.
The model relies on a design and measurement scale that make the comparisons meaningful. The same feature should be measured on a comparable scale at both times, and the patient identifiers must correctly link each pair. Patients should be independent of one another; the within-patient dependence must be represented by the random effect or by an equivalent paired analysis. For a Gaussian model, residual behavior should be sufficiently well approximated for the intended standard errors and tests. The model also presumes that missing measurements do not create an unaddressed selection problem. These assumptions should be considered feature by feature, rather than taken for granted because a model converges.
The interaction interpretation follows directly from the four group-time means. The nonresponder change is \(\mu_{N1}-\mu_{N0}\); the responder change is \(\mu_{R1}-\mu_{R0}\). Subtracting the first from the second gives
This identity shows why comparing only on-treatment group means is inadequate: that comparison is \(\beta_G+\beta_{GT}\), not \(\beta_{GT}\) alone. The baseline contrast remains part of the on-treatment difference.
Worked Example: Decomposing Four Group-Time Means
Suppose a hypothetical normalized protein abundance, measured in log2 units, has these group means for paired patients:
| Response group | Pretreatment mean | On-treatment mean | Mean change |
|---|---|---|---|
| Responders | 5.00 | 6.50 | 1.50 |
| Nonresponders | 4.50 | 4.75 | 0.25 |
The pretreatment contrast is \(5.00-4.50=0.50\) log2 units. The on-treatment contrast is \(6.50-4.75=1.75\) log2 units. The difference in changes is
Equivalently, the on-treatment contrast minus the baseline contrast is \(1.75-0.50=1.25\) log2 units. The groups already differed before treatment, and their mean difference became larger over the observed period. These means alone do not establish whether that change is statistically distinguishable from sampling variation, nor whether immunotherapy caused it.
Calculating a Difference in Changes
When each patient has both measurements, first form the within-patient change for the feature. A simple group comparison of these changes estimates the interaction directly. For independent response groups, an unpooled standard error for the difference in mean changes is
where \(D\) is the on-treatment minus pretreatment measurement, and \(s_R\) and \(s_N\) are the sample standard deviations of changes in the two groups. This is a Welch standard error; it does not require equal variances. Its use still assumes independent patients across groups and a change scale appropriate to the measurement and analysis.
Worked Example: Estimating a Feature’s Group Difference in Change
For a hypothetical RNA feature, suppose the four responder changes, in normalized log2 units, are \(1,2,3,2\), and the four nonresponder changes are \(0,1,0,1\). Each listed value is already a within-patient on-treatment-minus-pretreatment change.
The responder mean change is \(\bar{D}_R=(1+2+3+2)/4=2.00\). Deviations from that mean are \(-1,0,1,0\), so the sample variance is \(s_R^2=(1+0+1+0)/(4-1)=2/3\). The nonresponder mean change is \(\bar{D}_N=(0+1+0+1)/4=0.50\); the squared deviations sum to \(1\), giving \(s_N^2=1/3\).
The estimated difference in mean changes is \(2.00-0.50=1.50\) log2 units. Its standard error is
A Welch comparison would therefore have \(t=1.50/0.50=3.00\). Its degrees of freedom are approximately
This small hypothetical dataset gives an estimated difference in change of 1.50 log2 units with standard error 0.50. The uncertainty is substantial with only four patients per group; in an actual omics analysis, the feature-wise result would also need to be interpreted in the context of testing many features and the study design.
Feature-Wise Testing and Multiplicity
An omics assay may yield thousands of feature-wise interaction tests. Testing each at level \(\alpha=0.05\) without adjustment can produce many small p-values by chance. A common approach is to control the false discovery rate (FDR), the expected proportion of false rejections among the rejected hypotheses, under the procedure’s assumptions. The Benjamini–Hochberg procedure orders the \(m\) p-values from smallest to largest and compares the \(j\)th ordered value with \(j\alpha/m\). Equivalently, adjusted values are computed and compared with \(\alpha\). Dependence among molecular features matters: the standard procedure is justified under independence and certain forms of positive dependence; more conservative alternatives can be needed under more general dependence.
The hypothesis family should match the scientific question. If the question is which features show response-associated changes, test \(H_0:\beta_{GT}=0\) feature by feature and adjust across that family. Baseline contrasts \(\beta_G\) answer a different question and should not be described as interaction findings. The direction and size of estimated effects remain important: an adjusted p-value is not a measure of biological importance.
Worked Example: Adjusting Five Interaction Tests
Suppose a hypothetical analysis tests interaction coefficients for five preselected features and obtains sorted p-values \(0.004, 0.012, 0.030, 0.200,\) and \(0.600\). For Benjamini–Hochberg adjustment with \(m=5\), the raw adjusted quantities are \(mp_{(j)}/j\):
| Rank \(j\) | p-value \(p_{(j)}\) | \(5p_{(j)}/j\) | Adjusted value after monotonicity |
|---|---|---|---|
| 1 | 0.004 | 0.020 | 0.020 |
| 2 | 0.012 | 0.030 | 0.030 |
| 3 | 0.030 | 0.050 | 0.050 |
| 4 | 0.200 | 0.250 | 0.250 |
| 5 | 0.600 | 0.600 | 0.600 |
The adjusted values are already nondecreasing, so the monotonicity step does not alter them. At FDR level \(0.05\), the first three tests meet the threshold; the final two do not. This conclusion concerns this defined family of five interaction tests, not every possible feature or analysis choice.
Diagnostics, Alternatives, and Limits
Useful checks begin with the data structure. Confirm that pretreatment and on-treatment samples are correctly paired, that response labels and sampling times are accurate, and that each feature is on a comparable scale across times. Inspect within-patient changes and residuals for extreme values, skew, or variance patterns that conflict with the model. Check whether technical batches or sample-processing differences align with time or response group; if so, the estimated contrasts may reflect measurement structure as well as biology. These checks can flag problems, but cannot prove that all sources of bias have been removed.
A paired change analysis is a direct and transparent alternative to a mixed model when there are exactly two time points and complete pairs. A mixed model is useful when the paired structure needs explicit representation or the design has more complexity. An unpaired comparison of on-treatment values alone is generally not an adequate substitute because it discards pairing and mixes baseline and change contrasts. A baseline-only comparison can be appropriate when the question concerns pretreatment associations, but it cannot answer whether groups changed differently during treatment.
Several limitations remain even with a well-specified model. Response groups are usually observed rather than randomized, so differences may reflect clinical characteristics, treatment exposure, sampling, or other factors. On-treatment samples can be missing for reasons related to health or response, and complete-pair analyses may then represent a selected subset. A measured bulk-tumor feature may also change because cellular composition changes rather than because expression changes within a cell type. Finally, association with response does not establish predictive utility in new patients; independent validation is needed before a feature is treated as a reliable marker.
- Mistake: “The on-treatment responder mean is higher, so immunotherapy increased this feature more in responders.”
- Correction: Compare within-patient changes between groups. The on-treatment contrast includes any baseline contrast, and the observed interaction is not automatically causal.
- Mistake: Calling a pretreatment group difference a treatment-response change.
- Correction: Label baseline contrast \(\beta_G\) and response-by-time contrast \(\beta_{GT}\) separately, and report their units and uncertainty.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- If responders have a baseline mean of 7 and nonresponders a baseline mean of 6, what is the baseline contrast, and does it establish a different treatment-period change?
- Responders change by 0.8 units and nonresponders by 0.3 units. Calculate the difference in changes and state its interpretation.
- Why is an on-treatment group contrast not generally the same as the response-by-time interaction?
- What does an FDR-adjusted value of 0.04 mean when tests are assessed at FDR level 0.05?
Solutions. (1) The baseline contrast is \(7-6=1\) unit in favor of responders; it says nothing by itself about whether changes differ. (2) The difference in changes is \(0.8-0.3=0.5\) units, an estimated response-group difference in observed treatment-period change. (3) The on-treatment contrast equals the baseline contrast plus the difference in changes, so it includes both components. (4) The test meets the 0.05 FDR decision threshold under the chosen procedure and assumptions; it does not mean there is a 96% probability that the finding is true.