= RNA log-odds covariance
{c}
<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 gives
$$
\operatorname{Cov}(y_i,y_j)\approx
\frac{\operatorname{Cov}(f_i,f_j)}{f_i(1-f_i)f_j(1-f_j)}.
$$
In matrix form, $C_y\approx JC_fJ^{\mathsf T}$ 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.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§5.2, covariance propagation through sensitivity coefficients.]{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 $a$, then
$$
g_i=\frac{\partial y_i}{\partial a}=-\frac1{A_i-A_{F,i}},\qquad
C_{ij}^{(a)}\approx g_ig_ju_a^2.
$$
This component has one shared error direction. It is not equivalent to adding independent noise with variance $g_i^2u_a^2$ 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 uncertainty>0.03 and the same shared-offset contribution 0.01. The <standard uncertainty> of their mean is
$$
\sqrt{0.03^2/9+0.01^2}\approx0.0141.
$$
Incorrectly treating the shared term as independent would give $\sqrt{(0.03^2+0.01^2)/9}\approx0.0105$. 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.
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