= RNA melting log-odds uncertainty
{c}
<RNA melting log-odds uncertainty> quantifies uncertainty in $y=\ln[f_U/(1-f_U)]$, the transformed population used in an <RNA melting log-odds plot>. It depends on the unfolded fraction $f_U$ as well as its uncertainty. The transformation becomes particularly sensitive near 0 and 1.
For a sufficiently concentrated uncertainty distribution within $0<f_U<1$, first-order <uncertainty propagation> gives
$$
\frac{dy}{df_U}=\frac1{f_U(1-f_U)},\qquad
u_y\approx\frac{u_f}{f_U(1-f_U)}.
$$
With an original stipulated $u_f=0.010$, the <standard uncertainty> is 0.040 at $f_U=0.50$ but approximately 0.510 at $f_U=0.02$. Equal population uncertainties do not justify equal statistical treatment of transformed observations.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§5, sensitivity-coefficient method; derivative here follows from the stated logarithmic transformation.]{ref}
Nonlinearity also creates asymmetry. At $f_U=0.02$, $y\approx-3.892$. Changing the fraction to 0.01 gives $y\approx-4.595$, while changing it to 0.03 gives $y\approx-3.476$: equal changes in fraction produce unequal transformed changes. These are deterministic perturbations, not asserted confidence limits. If the uncertainty materially reaches 0 or 1, the linear approximation needs reconsideration; taking the logarithm of those endpoints is undefined.
The underlying <RNA melting baselines> can induce shared errors across temperatures. For an optical mixture with $A_F<A<A_U$,
$$
y=\ln(A-A_F)-\ln(A_U-A),
$$
so
$$
\frac{\partial y}{\partial A}=\frac1{A-A_F}+\frac1{A_U-A},\qquad
\frac{\partial y}{\partial A_F}=-\frac1{A-A_F},\qquad
\frac{\partial y}{\partial A_U}=-\frac1{A_U-A}.
$$
These derivatives propagate the joint uncertainty of the measured <absorbance> and fitted baselines. Reusing baseline parameters for many observations creates <RNA log-odds covariance>; merely assigning separate error bars does not remove it.
A fit can model the original <absorbance> observations and their <RNA melting baselines>, or handle the transformed data with an appropriate joint error model. Optical curves are fitted quantities in primary melting analyses, rather than direct counts of unfolded molecules.<Thermodynamic characterization of naturally occurring RNA tetraloops by Justin P. Sheehy and others>[Materials and Methods, optical curves and thermodynamic fitting.]{ref} Precision in a fitted straight line cannot establish the correctness of the <two-state RNA melting model> used to interpret it.
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