The effect of intraspecific sample size on type I and type II error rates in comparative studies

Comparative studies have increased greatly in number in recent years due to advances in statistical and phylogenetic methodologies. For these studies, a trade-off often exists between the number of species that can be included in any given study and the number of individuals examined per species. Here, we describe a simple simulation study examining the effect of intraspecific sample size on statistical error in comparative studies. We find that ignoring measurement error has no effect on type I error of nonphylogenetic analyses, but can lead to increased type I error under some circumstances when using independent contrasts. We suggest using ANOVA to evaluate the relative amounts of within- and between-species variation when considering a phylogenetic comparative study. If within-species variance is particularly large and intraspecific sample sizes small, then either larger sample sizes or comparative methods that account for measurement error are necessary.     

Testing the stability of transfer functions

In dendroclimatology, testing the stability of transfer functions using cross-calibration verification (CCV) statistics is a common procedure. However, the frequently used statistics reduction of error (RE) and coefficient of efficiency (CE) merely assess the skill of reconstruction for the validation period, which does not necessarily reflect possibly instable regression parameters. Furthermore, the frequently used rigorous threshold of zero which sharply distinguishes between valid and invalid transfer functions is prone to an underestimation of instability. To overcome these drawbacks, we here introduce a new approach – the Bootstrapped Transfer Function Stability test (BTFS). BTFS relies on bootstrapped estimates of the change of model parameters (intercept, slope, and r²) between calibration and verification period as well as the bootstrapped significance of corresponding models. A comparison of BTFS, CCV and a bootstrapped CCV approach (BCCV) applied to 42,000 pseudo-proxy datasets with known properties revealed that BTFS responded more sensitively to instability compared to CCV and BCCV. BTFS performance was significantly affected by sample size (length of calibration period) and noise (explained variance between predictor and predictand). Nevertheless, BTFS performed superior with respect to the detection of instable transfer functions in comparison to CCV.     

Chemical composition and rumen degradability of three corn hybrids treated with insecticides against the European corn borer (Ostrinia nubilalis)

This experiment determined the chemical composition, rumen degradability (aNDF in stalks and starch in kernels) and in vitro gas production of kernels from three corn hybrids treated (TT) or not treated (control, CTR) with insecticides against the European corn borer (ECB, Ostrinia nubilalis). Two whole-plant silage hybrids belonging to the FAO rating 600 and 700 maturity class (S600 and S700, respectively) and one selected for grain production (G600, FAO rating 600, Dekalb-Monsanto Agricoltura S.p.A., Lodi, Italy) were sown in two main plots (TT and CTR) of an experimental field. Two subsequent treatments of pyrethroids (25 and 1.2 g/ha of cyfluthrin and deltamethrin, respectively) were applied to the TT plots. The insecticide treatment reduced the number of damaged plants (4.5 broken plants/plot versus 0.3 broken plants/plot, P<0.01) and increased the total grain yield by 11% (13.8 t/ha versus 12.4 t/ha), while hybrids did not differ. ECB larvae which bored into the stalk tunnels modified the chemical composition of stalks and kernels. In stalks, total sugars content (i.e. glucose, fructose, sucrose) was about twice that in TT versus CTR plants (123 g/kg versus 60 g/kg DM, P<0.01), while aNDF content was higher in CTR stalks (765 versus 702 g/kg DM, P<0.01). DM degradability after 48 h of incubation of stalks was higher in TT than in CTR, both in vitro (0.360 versus 0.298, P<0.01) and in situ (0.370 versus 0.298, P<0.05), while there were no differences in aNDF degradability. Kernels from TT plots contained less DM (615 g/kg versus 651 g/kg, P<0.01) and more CP (84 g/kg and 78 g/kg DM, P<0.05) than those from CTR plots, while in situ rumen starch disappearance and in vitro gas production were similar. Corn hybrid selected for yield of grain (G600) differed from S600 and S700 due to a higher (P<0.01) content of aNDF, ADF and lignin(sa) in the stalks, and a higher starch content (696 g/kg versus 674 and 671 g/kg DM, P<0.01) and CP (87 g/kg versus 77 and 76 g/kg DM, P<0.05) in grain. The G600 hybrid produced stalks with a lower (P<0.01) aNDF rumen degradability than the S600 and S700.     

Spatial Uncertainty and Ecological Models

Applied ecological models that are used to understand and manage natural systems often rely on spatial data as input. Spatial uncertainty in these data can propagate into model predictions. Uncertainty analysis, sensitivity analysis, error analysis, error budget analysis, spatial decision analysis, and hypothesis testing using neutral models are all techniques designed to explore the relationship between variation in model inputs and variation in model predictions. Although similar methods can be used to answer them, these approaches address different questions. These approaches differ in (a) whether the focus is forward or backward (forward to evaluate the magnitude of variation in model predictions propagated or backward to rank input parameters by their influence); (b) whether the question involves model robustness to large variations in spatial pattern or to small deviations from a reference map; and (c) whether processes that generate input uncertainty (for example, cartographic error) are of interest. In this commentary, we propose a taxonomy of approaches, all of which clarify the relationship between spatial uncertainty and the predictions of ecological models. We describe existing techniques and indicate a few areas where research is needed.     

Errors-in-variables in joint population pharmacokinetic/pharmacodynamic modeling

Pharmacokinetic (PK) models describe the relationship between the administered dose and the concentration of drug (and/or metabolite) in the blood as a function of time. Pharmacodynamic (PD) models describe the relationship between the concentration in the blood (or the dose) and the biologic response. Population PK/PD studies aim to determine the sources of variability in the observed concentrations/responses across groups of individuals. In this article, we consider the joint modeling of PK/PD data. The natural approach is to specify a joint model in which the concentration and response data are simultaneously modeled. Unfortunately, this approach may not be optimal if, due to sparsity of concentration data, an overly simple PK model is specified. As an alternative, we propose an errors-in-variables approach in which the observed-concentration data are assumed to be measured with error without reference to a specific PK model. We give an example of an analysis of PK/PD data obtained following administration of an anticoagulant drug. The study was originally carried out in order to make dosage recommendations. The prior for the distribution of the true concentrations, which may incorporate an individual's covariate information, is derived as a predictive distribution from an earlier study. The errors-in-variables approach is compared with the joint modeling approach and more naive methods in which the observed concentrations, or the separately modeled concentrations, are substituted into the response model. Throughout, a Bayesian approach is taken with implementation via Markov chain Monte Carlo methods.     

Modification of sample size in group sequential clinical trials

In group sequential clinical trials, sample size reestimation can be a complicated issue when it allows for change of sample size to be influenced by an observed sample path. Our simulation studies show that increasing sample size based on an interim estimate of the treatment difference can substantially inflate the probability of type I error in most practical situations. A new group sequential test procedure is developed by modifying the weights used in the traditional repeated significance two-sample mean test. The new test has the type I error probability preserved at the target level and can provide a substantial gain in power with the increase of sample size. Generalization of the new procedure is discussed.     

Approximate standard errors in semiparametric models

We consider semiparametric models with p regressor terms and q smooth terms. We obtain an explicit expression for the estimate of the regression coefficients given by the back-fitting algorithm. The calculation of the standard errors of these estimates based on this expression is a considerable computational exercise. We present an alternative, approximate method of calculation that is less demanding. With smoothing splines, the method is exact, while with loess, it gives good estimates of standard errors. We assess the adequacy of our approximation and of another approximation with the help of two examples.     

Codon Usage Decreases the Error Minimization Within the Genetic Code

The genetic code is not random but instead is organized in such a way that single nucleotide substitutions are more likely to result in changes between similar amino acids. This fidelity, or error minimization, has been proposed to be an adaptation within the genetic code. Many models have been proposed to measure this adaptation within the genetic code. However, we find that none of these consider codon usage differences between species. Furthermore, use of different indices of amino acid physicochemical characteristics leads to different estimations of this adaptation within the code. In this study, we try to establish a more accurate model to address this problem. In our model, a weighting scheme is established for mistranslation biases of the three different codon positions, transition/transversion biases, and codon usage. Different indices of amino acids physicochemical characteristics are also considered. In contrast to pervious work, our results show that the natural genetic code is not fully optimized for error minimization. The genetic code, therefore, is not the most optimized one for error minimization, but one that balances between flexibility and fidelity for different species.     

How much variance is explained by ecologists? Additional perspectives

Oecologia - A recent meta-analysis of meta-analyses by Møller and Jennions (2002, Oecologia 132:492–500) suggested that ecologists using statistical models are explaining between 2.5%...