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Sigmoid curves, non-linear double-reciprocal plots and allosterism.

ABSTRACT: 1. The theory of plane curves was applied to the graphical methods used in enzyme kinetics and a mathematical analysis of the possible graph shapes is given. 2. The belief that allosterism can be inferred from steady-state data alone is subjected to criticism and the mathematical significance of sigmoid curves and non-linear double-reciprocal plots is explored. 3. It is suggested that the usual methods of interpreting steady-state kinetic data are often based on over-restrictive assumptions which prevent maximum utilization of the available data. 4. Methods for obtaining the degree of the rate equation from graph shapes obtained directly from initial-rate measurements and from replots of asymptotic behaviour as chi approach the level 0 and chi approach the level infinity are discussed. 5. Detailed proofs of the theorems given in the text have been deposited as Supplementary Publication SUP 50049 (10 pages) at the British Library (Lending Division), Boston Spa, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem. J. (1975), 145, 5.

SUBMITTER: Bardsley WG

PROVIDER: S-EPMC1165625 | biostudies-other | 1975 Aug

REPOSITORIES: biostudies-other

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Project description:BackgroundMeta-analyses of studies evaluating survival (time-to-event) outcomes are a powerful technique to assess the strength of evidence for a given disease or treatment. However, these studies rely on the adequate reporting of summary statistics in the source articles to facilitate further analysis. Unfortunately, many studies, especially within the field of prognostic research do not report such statistics, making secondary analyses challenging. Consequently, methods have been developed to infer missing statistics from the commonly published Kaplan-Meier (KM) plots but are liable to error especially when the published number at risk is not included.MethodsWe therefore developed a method using non-linear optimisation (nlopt) that only requires the KM plot and the commonly published P value to better estimate the underlying censoring pattern. We use this information to then calculate the natural logarithm of the hazard ratio (ln (HR)) and its variance (var) ln (HR), statistics important for meta-analyses.ResultsWe compared this method to the Parmar method which also does not require the number at risk to be published. In a validation set consisting of 13 KM studies, a statistically significant improvement in calculating ln (HR) when using an exact P value was obtained (mean absolute error 0.014 vs 0.077, P?=?0.003). Thus, when the true HR has a value of 1.5, inference of the HR using the proposed method would set limits between 1.49/1.52, an improvement of the 1.39/1.62 limits obtained using the Parmar method. We also used Monte Carlo simulations to establish recommendations for the number and positioning of points required for the method.ConclusionThe proposed non-linear optimisation method is an improvement on the existing method when only a KM plot and P value are included and as such will enhance the accuracy of meta-analyses performed for studies analysing time-to-event outcomes. The nlopt source code is available, as is a simple-to-use web implementation of the method.

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Sigmoid curves, non-linear double-reciprocal plots and allosterism.

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