Testing Minimum Fold Changes with TREAT

Why Test a Minimum Effect?

A gene can have strong evidence of a nonzero expression difference while the estimated difference remains too small to matter scientifically. Conversely, a sizeable estimated difference may be too uncertain to establish that the true effect clears a meaningful threshold. TREAT addresses this distinction by testing whether the absolute log-fold change is larger than a prespecified minimum, not merely whether it differs from zero.

The threshold must be chosen before examining the gene-level results and expressed on the same scale as the contrast. For example, a minimum fold change of 1.5 corresponds to a log2-scale threshold of \(\log_2(1.5)\), approximately \(0.585\). An absolute log2 fold change of \(0.585\) means a ratio of 1.5 in one direction or \(1/1.5\) in the other.

Model, Contrast, and Interval Null

For gene \(g\), write the gene-wise linear model as \(Y_g=X\theta_g+\varepsilon_g\), where \(Y_g\) is the vector of expression measurements, \(X\) is the design matrix, and \(\theta_g\) is the coefficient vector. The errors are assumed to have mean zero and, conditional on the design, a common residual variance for that gene with an appropriate normal model for moderated t inference. The design and contrast must represent the intended comparison; full column rank makes all coefficients identifiable, but even if \(X\) is rank deficient, a particular contrast can be estimable if it lies in the row space of \(X\).

Let \(c\) be the contrast vector defining the comparison of interest. The scalar effect is \(\beta_g=c^\mathsf{T}\theta_g\), and its estimate is \(\hat{\beta}_g=c^\mathsf{T}\hat{\theta}_g\). For a two-group comparison with expression measured on a log2 scale, this contrast is a log2 fold change, with the sign determined by the order of the groups.

The test uses the moderated standard error and moderated degrees of freedom obtained from the linear-model variance analysis. Write the standard error of the contrast as \(SE_g\) and its moderated degrees of freedom as \(df_g\). The TREAT statistic is the estimated distance beyond the threshold, in standard-error units:

When \(|\hat{\beta}_g|\leq\tau\), this statistic is nonpositive: the point estimate itself has not cleared the minimum. When it is positive, the estimate is beyond the threshold, but its size relative to \(SE_g\) determines how compelling the evidence is. In the two-direction test, the p-value accounts for the possibility of an effect beyond either boundary. For a positive statistic \(t_{\mathrm{TREAT},g}\), it is \(2P(T_{df_g}\geq t_{\mathrm{TREAT},g})\); a value below zero does not provide evidence of threshold exceedance. In applying the two-tail calculation, the p-value is capped at 1.

Why the Statistic Measures Exceedance

Consider first a positive effect at the boundary of the null, \(\beta_g=\tau\). The estimated excess over the threshold is \(\hat{\beta}_g-\tau\), and dividing by its standard error gives the standardized distance from that boundary. For a negative boundary, \(\beta_g=-\tau\), the corresponding excess in the negative direction is \(-\hat{\beta}_g-\tau\). Taking the absolute estimate combines these two directions: both are represented by \(|\hat{\beta}_g|-\tau\).

This explains why the threshold is subtracted before standardizing. Testing \(\hat{\beta}_g/SE_g\) instead would ask whether the effect differs from zero, a different question. Under the moderated model, the reference distribution for the standardized contrast uses \(df_g\); TREAT evaluates the evidence relative to the boundary effects, rather than treating the zero-effect null as the relevant benchmark. Testing either of the two possible directions requires accounting for both directions in the p-value.

Worked Tests in Both Directions

Choosing the Threshold and Reading Results

A threshold is meaningful only in relation to the contrast scale and scientific question. If the contrast is a log2 fold change, \(\tau=1\) means the effect must exceed a two-fold change in either direction. A smaller threshold, such as \(\tau=\log_2(1.5)\approx0.585\), asks for evidence of at least a 1.5-fold change. A threshold chosen after inspecting results can make the reported inference misleading, because it no longer represents an independently specified minimum of interest.

TREAT and a zero-null moderated t-test answer different questions. A zero-null test evaluates \(H_0:\beta_g=0\); with a large sample or small standard error it can flag effects that are statistically distinguishable from zero but smaller than the chosen minimum. TREAT instead tests whether the effect exceeds that minimum. A confidence interval can also help describe effect uncertainty, but merely checking whether an interval for \(\beta_g\) includes \(\tau\) is not a substitute for the TREAT interval-null test: the test addresses both positive and negative thresholds and uses its own boundary-based statistic and calibration.

Assumptions, Diagnostics, and Failure Modes

TREAT inherits the gene-wise linear-model assumptions and the variance estimates used for moderated inference. The contrast must be estimable and its sign and scale must match the stated comparison. The model must adequately represent the design, and residual behavior should not be so severe that the normal-based moderated reference distribution is unreliable. Empirical Bayes moderation can stabilize gene-wise variance estimates, especially with few samples, but it does not repair a misspecified design or make an invalid contrast meaningful.

Before interpreting results, check that the threshold has the intended units, that the design encodes the groups or covariates correctly, and that the contrast has the intended direction. Examine residual and mean-variance diagnostics from the fitted models, and look for unusual genes or samples that could dominate estimates. A p-value just above the cutoff is not proof that an effect is below the threshold; a p-value just below it is not a measure of biological importance. The estimated effect and its uncertainty remain important context.

A common mistake is to compare \(|\hat{\beta}_g|\) with \(\tau\) and conclude that a gene passes because its point estimate is larger. In the second worked example, \(1.8>1\), but the TREAT p-value is \(0.06866\), so the test does not reject at 0.05. The correction is to standardize the excess \(|\hat{\beta}_g|-\tau\) by the moderated standard error and evaluate it against the moderated reference distribution.