Likelihood based observability analysis and confidence intervals for predictions of dynamic models

ABSTRACT: BACKGROUND: Predicting a system's behavior based on a mathematical model is a primary task in Systems Biology. If the model parameters are estimated from experimental data, the parameter uncertainty has to be translated into confidence intervals for model predictions. For dynamic models of biochemical networks, the nonlinearity in combination with the large number of parameters hampers the calculation of prediction confidence intervals and renders classical approaches as hardly feasible. RESULTS: In this article reliable confidence intervals are calculated based on the prediction profile likelihood. Such prediction confidence intervals of the dynamic states can be utilized for a data-based observability analysis. The method is also applicable if there are non-identifiable parameters yielding to some insufficiently specified modelpredictions that can be interpreted as non-observability. Moreover, a validation profile likelihood is introduced that should be applied when noisy validation experiments are to be interpreted. CONCLUSIONS: The presented methodology allows the propagation of uncertainty from experimental to model pre-dictions. Although presented in the context of ordinary differential equations, the concept is general and also applicable to other types of models. Matlab code which can be used as a template to implement the method is provided at http://www.fdmold.uni-freiburg.de/~ckreutz/PPL .     

Approximate confidence intervals for risk prediction in genetic counseling.

In current genetic counseling practice, a single risk estimate is often quoted to a family rather than a range of risks. Such point estimates are predicated on knowing basic parameters like recombination fractions exactly when, in fact, there may be considerable uncertainty about them. Using the large sample theory of statistics, it is possible to derive approximate risk intervals that incorporate known statistical imprecision. The necessary theory will be briefly discussed and illustrated by an application to family counseling for Duchenne muscular dystrophy in the presence of two flanking markers. Some of the problems of the theory will be mentioned. These include lack of adequate sample size to justify the conclusions of large sample theory, pronounced nonlinearity in the risk function, and failure to take into proper account genetic interference. Except in trivial cases, sophisticated computer software is needed to carry out the computations of risk intervals.     

On confidence intervals for genotype relative risks and attributable risks from case parent trio designs for candidate-gene studies

Scherag et al. [Hum Hered 2002;54:210-217] recently proposed point estimates and asymptotic as well as exact confidence intervals for genotype relative risks (GRRs) and the attributable risk (AR) in case parent trio designs using single nucleotide polymorphism (SNP) data. The aim of this study was the investigation of coverage probabilities and bias in estimates if the marker locus is not identical to the disease locus. Using a variety of parameter constellations, including marker allele frequencies identical to and different from the SNP at the disease locus, we performed an analytical study to quantify the bias and a Monte-Carlo simulation study for quantifying both bias and coverage probabilities. No bias was observed if marker and trait locus coincided. Two parameters had a strong impact on coverage probabilities of confidence intervals and bias in point estimates if they did not coincide: the linkage disequilibrium (LD) parameter delta and the allele frequency at the marker SNP. If marker allele frequencies were different from the allele frequencies at the functional SNP, substantial biases occurred. Further, if delta between the marker and the disease locus was lower than the maximum possible delta, estimates were also biased. In general, biases were towards the null hypothesis for both GRRs and AR. If one GRR was not increased, as e.g. in a recessive genetic model, biases away from the null could be observed. If both GRRs were in identical directions and if both were substantially larger than 1, the bias always was towards the null. When applying point estimates and confidence intervals for GRRs and AR in candidate gene studies, great care is needed. Effect estimates are substantially biased towards the null if either the allele frequencies at the marker SNP and the true disease locus are different or if the LD between the marker SNP and the disease locus is not at its maximum. A bias away from the null occurs only in uncommon study situations; it is small and can therefore be ignored for applications.     

The distribution of genetic parameter estimates and confidence intervals from small disconnected diallels

The distributions of genetic variance components and their ratios (heritability and type-B genetic correlation) from 105 pairs of six-parent disconnected half-diallels of a breeding population of loblolly pine (Pinus taeda L.) were examined. A series of simulations based on these estimates were carried out to study the coverage accuracy of confidence intervals based on the usual t-method and several other alternative methods. Genetic variance estimates fluctuated greatly from one experiment to another. Both general combining ability variance (²_g) and specific combining ability variance (²_s) had a large positive skewness. For ²_g and ²_s, a skewness-adjusted t-method proposed by Boos and Hughes-Oliver (Am Stat 54:121–128, 2000) provided better upper endpoint confidence intervals than t-intervals, whereas they were similar for the lower endpoint. Bootstrap BCa-intervals (Efron and Tibshirani, An introduction to the bootstrap. Chapman & Hall, London 436 p, 1993) and Halls transformation methods (Zhou and Gao, Am Stat 54:100–104, 2000) had poor coverages. Coverage accuracy of Fiellers interval endpoint(J R Stat Soc Ser B 16:175–185, 1954) and t-interval endpoint were similar for both h² and r_B for sample sizes n10, but for n=30 the Fiellers method is much better.     

Nonparametric confidence intervals for the one- and two-sample problems

Confidence intervals for the mean of one sample and the difference in means of two independent samples based on the ordinary-t statistic suffer deficiencies when samples come from skewed families. In this article we evaluate several existing techniques and propose new methods to improve coverage accuracy. The methods examined include the ordinary-t, the bootstrap-t, the biased-corrected acceleration and three new intervals based on transformation of the t-statistic. Our study shows that our new transformation intervals and the bootstrap-t intervals give best coverage accuracy for a variety of skewed distributions, and that our new transformation intervals have shorter interval lengths.     

Uncertainty in population growth rates: determining confidence intervals from point estimates of parameters

Background

Demographic models are widely used in conservation and management, and their parameterisation often relies on data collected for other purposes. When underlying data lack clear indications of associated uncertainty, modellers often fail to account for that uncertainty in model outputs, such as estimates of population growth.

Methodology/Principal Findings

We applied a likelihood approach to infer uncertainty retrospectively from point estimates of vital rates. Combining this with resampling techniques and projection modelling, we show that confidence intervals for population growth estimates are easy to derive. We used similar techniques to examine the effects of sample size on uncertainty. Our approach is illustrated using data on the red fox, Vulpes vulpes, a predator of ecological and cultural importance, and the most widespread extant terrestrial mammal. We show that uncertainty surrounding estimated population growth rates can be high, even for relatively well-studied populations. Halving that uncertainty typically requires a quadrupling of sampling effort.

Conclusions/Significance

Our results compel caution when comparing demographic trends between populations without accounting for uncertainty. Our methods will be widely applicable to demographic studies of many species.     

A comparison of reflected versus test-based confidence intervals for the median survival time, based on censored data

The small-sample performance of some recently proposed nonparametric methods of constructing confidence intervals for the median survival time, based on randomly right-censored data, is compared with that of two new methods. Most of these methods are equivalent for large samples. All proposed intervals are either 'test-based' or 'reflected' intervals, in the sense defined in the paper. Coverage probabilities for the interval estimates were obtained by exact calculation for uncensored data, and by stimulation for three life distributions and four censoring patterns. In the range of situations studied, 'test-based' methods often have less than nominal coverage, while the coverage of the new 'reflected' confidence intervals is closer to nominal (although somewhat conservative), and these intervals are easy to compute.