 RNA log-odds covariance

RNA log-odds covariance describes correlated errors between transformed population estimates at different temperatures in an RNA melting log-odds plot. Shared RNA melting baselines can generate it even when individual detector readings are independent.

Let $f_i$ be unfolded fractions and $y_i=\ln [f_i/(1-f_i)]$. First-order propagation givesIn matrix form, $C_y\approx J{C}_f{J}^{\mathsf{T}}$ with diagonal $J_{ii}=1/[f_i(1-f_i)]$. This is the multivariate form of RNA melting log-odds uncertainty; the derivatives are evaluated at the estimated fractions.^[ref]

A concrete shared-input model uses $y_i=\ln (A_i-A_{F,i})-\ln (A_{U,i}-A_i)$. If all folded-state predictions contain the same uncertain offset $a$, thenThis component has one shared error direction. It is not equivalent to adding independent noise with variance $g_i^2{u}_a^2$ to every observation. Other baseline parameters contribute through RNA baseline prediction covariance.

For an original simplified example, nine observations each have independent log-odds standard uncertainty0.03 and the same shared-offset contribution 0.01. The standard uncertainty of their mean isIncorrectly treating the shared term as independent would give $\sqrt{(0.03^2+0.01^2)/9}\approx 0.0105$. Repetition reduces the independent component while leaving this common offset contribution intact.

A model fit can use the joint covariance matrix in generalized least squares, with an objective of the form $r^{\mathsf{T}}{C}_y^{-1}r$ when that inverse exists. Alternatively, fit the underlying absorbance and baseline parameters jointly. Shared-error components alone may be singular, and near population endpoints the logarithmic linearization may fail. A precise straight-line fit does not independently validate the two-state RNA melting model or its baseline assumptions.