Sampling distributions, biases, variances, and confidence intervals for genetic correlations

Genetic correlations are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of are not known, and accurate methods have not been described for estimating the precision of estimates of Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for MANOVA estimates of were normally distributed. REML and ML estimates were normally distributed for but skewed for and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, and r. Received: 6 July 1995 / Accepted: 8 March 1996     

Evaluation of the jackknife technique for fitting multiexponential functions to biochemical data

The jackknife technique was tested by fitting a two-exponential function to the time course of disappearance of radioactivity from the area of a wheat leaf that had been fed ¹⁴CO₂. The function was fitted by both unweighted and weighted least squares, first without and then with the jackknife. Weighting altered the estimates of the function's parameters, but jackknifing did not. Hence jackknifing did not remove any of the bias introduced by incorrect weighting. The confidence limits of the parameters calculated by jackknifing were greater than those estimated from the variance-covariance matrix of the regression, but similar to those derived from replicate experiments. The jackknife also allowed confidence limits for the rate constants and transit time of the underlying two-compartment model to be derived.     

Application of jackknife procedures to inter-experiment comparisons of parameter estimates for the Michaelis-Menten equation.

The jackknife procedure is introduced as a means of making comparisons among Michaelis-Menten parameter estimates for six different experimental conditions. In addition to providing a solution to the general inter-experimental comparison problem, the jackknife procedure will provide valid parameter estimates even when some of the assumptions usually required for statistical analysis are violated, e.g., the random errors are not normally distributed and the variances are not homogeneous. Other recent variations of the jackknife have also been introduced and briefly investigated: (i) the linear jackknife, which is more efficient computationally, and (ii) the weighted jackknife, which reduces the influence of design points (substrate concentrations) that have an excessive influence on the precision of parameter estimates.     

ldne: a program for estimating effective population size from data on linkage disequilibrium

ldne is a program with a Visual Basic interface that implements a recently developed bias correction for estimates of effective population size (N(e) ) based on linkage disequilibrium data. The program reads genotypic data in standard formats and can accommodate an arbitrary number of samples, individuals, loci, and alleles, as well as two mating systems: random and lifetime monogamy. ldne calculates separate estimates using different criteria for excluding rare alleles, which facilitates evaluation of data for highly polymorphic markers such as microsatellites. The program also introduces a jackknife method for obtaining confidence intervals that appears to perform better than parametric methods currently in use.     

Recruitment rates in forest plots: Gf estimates using growth rates and size distributions

1 In censuses of tree populations in permanent plots, short census intervals and small population size lead to uncertainty in the observed recruitment rate of a minimum size. Increasing the census interval, however, underestimates the rate because of unrecorded 'recruit and die' events.
2 We propose a new Gf procedure for estimation of recruitment rates. Recruitment rate per area is obtained by multiplication of the density in the smallest size class (f) and the average size growth rate in that class (G) divided by the width of the class. This procedure is valid when the size distribution of the population examined is continuous with size.
3 When tree size structure is negative‐exponentially distributed, as is often the case in natural rain forest populations, the Gf estimate of the recruitment rate for a given size class was least biased close to the midpoint size of this class.
4 Gf estimates agreed well with census estimates of recruitment rate from permanent plots in rain forests. A tendency for Gf estimates to be larger than census estimates disappeared when census estimates were corrected for mortality after recruitment.
5 The effects of plot size, census interval and variation in growth rate on estimates of recruitment rate were simulated using model populations. Small plot size caused substantially more among‐plot deviation for the census count of recruitment events than for the Gf estimate. The census recruitment rate also showed larger variation among plots for shorter intervals than the Gf estimate, which was independent of census interval. The Gf estimates were therefore more accurate than census counts in many situations. More than several tens of trees were needed in a size class to allow a reliable Gf estimates.     

A comparison of two methods for making statistical inferences on Nei's measure of genetic distance

The delta and jackknife methods can be used to estimate Nei's measure of genetic distance and calculate confidence intervals for this estimate. Computer stimulations were used to study the bias and variance of each estimator and the accuracy of the corresponding approximate 95% confidence intervals. The simulations were conducted using 3 sets of data and several sample sizes. The results showed: (1) the jackknife reduced bias; (2) in 8 out of 9 cases the variance and mean square error of the jackknife estimator were less; (3) a second order jackknife reduced the bias the most but suffered a corresponding increase in variance; (4) both the first order jackknife and delta methods yielded intervals whose confidence levels were approximately equal but less than 95%.     

Comparison of confidence intervals for adjusted attributable risk estimates under multinomial sampling

The epidemiologic concept of the adjusted attributable risk is a useful approach to quantitatively describe the importance of risk factors on the population level. It measures the proportional reduction in disease probability when a risk factor is eliminated from the population, accounting for effects of confounding and effect-modification by nuisance variables. The computation of asymptotic variance estimates for estimates of the adjusted attributable risk is often done by applying the delta method. Investigations on the delta method have shown, however, that the delta method generally tends to underestimate the standard error, leading to biased confidence intervals. We compare confidence intervals for the adjusted attributable risk derived by applying computer intensive methods like the bootstrap or jackknife to confidence intervals based on asymptotic variance estimates using an extensive Monte Carlo simulation and within a real data example from a cohort study in cardiovascular disease epidemiology. Our results show that confidence intervals based on bootstrap and jackknife methods outperform intervals based on asymptotic theory. Best variants of computer intensive confidence intervals are indicated for different situations.     

An expanded plastid DNA phylogeny of Orchidaceae and analysis of jackknife branch support strategy

An expanded plastid DNA phylogeny for Orchidaceae was generated from sequences of rbcL and matK for representatives of all five subfamilies. The data were analyzed using equally weighted parsimony, and branch support was assessed with jackknifing. The analysis supports recognition of five subfamilies with the following relationships: (Apostasioideae (Vanilloideae (Cypripedioideae (Orchidoideae (Epidendroideae))))). Support for many tribal-level groups within Epidendroideae is evident, but relationships among these groups remain uncertain, probably due to a rapid radiation in the subfamily that resulted in short branches along the spine of the tree. A series of experiments examined jackknife parameters and strategies to determine a reasonable balance between computational effort and results. We found that support values plateau rapidly with increased search effort. Tree bisection-reconnection swapping in a single search replicate per jackknife replicate and saving only two trees resulted in values that were close to those obtained in the most extensive searches. Although this approach uses considerably more computational effort than less extensive (or no) swapping, the results were also distinctly better. The effect of saving a maximal number of trees in each jackknife replicate can also be pronounced and is important for representing support accurately.     

The size distribution of human hematogenous metastases.

The development of primary cancers and their subsequent metastases occur through a complex sequence of discrete steps. A hypothesis is proposed here whereby the time available for the growth of metastases is normally distributed, presumably as a consequence of the summation of multiple independently distributed time intervals from each of the steps and of the Central Limit Theorem. For exponentially growing metastases, the corresponding size distribution would be lognormal; Gompertzian growth would imply a modified (Gompertz-normal) distribution, where larger metastases would occur less frequently as a consequence of a decreased growth rate. These two size distributions were evaluated against 18 human autopsy cases where precise size measurements had been collected from over 3900 macroscopic hematogenous organ metastases. The lognormal distribution provided an approximate agreement. Its main deficiency was a tendency to over-represent metastases greater than 10 mm diameter. The Gompertz-normal distribution provided more stringent agreement, correcting for this over-representation. These observations supported the hypothesis of normally distributed growth times, and qualified the utility of the lognormal and Gompertz-normal distributions for the size distribution of metastases.     

Comparing Bayesian estimates of genetic differentiation of molecular markers and quantitative traits: an application to Pinus sylvestris

Comparison of the level of differentiation at neutral molecular markers (estimated as F(ST) or G(ST)) with the level of differentiation at quantitative traits (estimated as Q(ST)) has become a standard tool for inferring that there is differential selection between populations. We estimated Q(ST) of timing of bud set from a latitudinal cline of Pinus sylvestris with a Bayesian hierarchical variance component method utilizing the information on the pre-estimated population structure from neutral molecular markers. Unfortunately, the between-family variances differed substantially between populations that resulted in a bimodal posterior of Q(ST) that could not be compared in any sensible way with the unimodal posterior of the microsatellite F(ST). In order to avoid publishing studies with flawed Q(ST) estimates, we recommend that future studies should present heritability estimates for each trait and population. Moreover, to detect variance heterogeneity in frequentist methods (ANOVA and REML), it is of essential importance to check also that the residuals are normally distributed and do not follow any systematically deviating trends.     

Estimating index of population trend by re-sampling techniques (jackknife and bootstrap) and its application to the life table study of rice leaf roller, Cnaphalocrocis medinalis (Lepidoptera: Pyralidae)

Taking a published natural population life table of rice leaf roller, Cnaphalocrocis medinalis (Lepidoptera: Pyralidae), as an example, we estimated the population trend index, I, via re‐sampling methods (jackknife and bootstrap), determined its statistical properties and illustrated the application of these methods in determining the control effectiveness of bio‐agents and chemical insecticides. Depending on the simulation outputs, the smoothed distribution pattern of the estimates of I by delete‐1 jackknife is visually distinguishable from the normal density, but the smoothed pattern produced by delete‐d jackknife, and logarithm‐transformed smoothed patterns produced by both empirical and parametric bootstraps, matched well the corresponding normal density. Thus, the estimates of I produced by delete‐1 jackknife were not used to determine the suppressive effect of wasps and insecticides. The 95% percent confidence intervals or the narrowest 95 percentiles and Z‐test criterion were employed to compare the effectiveness of Trichogramma japonicum Ashmead and insecticides (powder, 1.5% mevinphos + 3% alpha‐hexachloro cyclohexane) against the rice leaf roller based on the estimates of I produced by delete‐d jackknife and bootstrap techniques. At α= 0.05 level, there were statistical differences between wasp treatment and control, and between wasp and insecticide treatments, if the normality is ensured, or by the narrowest 95 percentiles. However, there is still no difference between insecticide treatment and control. By Z‐test criterion, wasp treatment is better than control and insecticide treatment with P‐value < 0.01. Insecticide treatment is similar to control with P‐value > 0.2 indicating that 95% confidence intervals procedure is more conservative. Although similar conclusions may be drawn by re‐sampling techniques, such as the delta method, about the suppressive effect of trichogramma and insecticides, the normality of the estimates can be checked and guaranteed, and the correlation among sequential life stages of rice leaf roller is also considered in the estimation. Judged by the P‐values from Z‐test, the delta method is more conservative.     

Power of quantitative trait locus mapping for polygenic binary traits using generalized and regression interval mapping in multi-family half-sib designs

A generalized interval mapping (GIM) method to map quantitative trait loci (QTL) for binary polygenic traits in a multi-family half-sib design is developed based on threshold theory and implemented using a Newton-Raphson algorithm. Statistical power and bias of QTL mapping for binary traits by GIM is compared with linear regression interval mapping (RIM) using simulation. Data on 20 paternal half-sib families were simulated with two genetic markers that bracketed an additive QTL. Data simulated and analysed were: (1) data on the underlying normally distributed liability (NDL) scale, (2) binary data created by truncating NDL data based on three thresholds yielding data sets with three different incidences, and (3) NDL data with polygenic and QTL effects reduced by a proportion equal to the ratio of the heritabilities on the binary versus NDL scale (reduced-NDL). Binary data were simulated with and without systematic environmental (herd) effects in an unbalanced design. GIM and RIM gave similar power to detect the QTL and similar estimates of QTL location, effects and variances. Presence of fixed effects caused differences in bias between RIM and GIM, where GIM showed smaller bias which was affected less by incidence. The original NDL data had higher power and lower bias in QTL parameter estimates than binary and reduced-NDL data. RIM for reduced-NDL and binary data gave similar power and estimates of QTL parameters, indicating that the impact of the binary nature of data on QTL analysis is equivalent to its impact on heritability.     

Parametric and jackknife confidence interval estimators for two-factor mating design genetic variance ratios

Summary Confidence interval estimators have not been defined for dominance to additive genetic variance () and average degree of dominance () for the nested, factorial, and backcross mating designs. The objective of this paper was to describe interval estimators for these parameters. Approximate F random variables were defined for expected mean square (EMS) ratios for linear models with one environmental effect. Approximate 1– parametric interval estimators were defined for and using these random variables. Random variables defined for linear models with no environmental effects are not approximately distributed as F random variables because common EMS are involved in the numerators and denominators of the EMS ratios. Delete-one jackknife (jackknife) interval estimators were defined for and for linear models with zero or one environmental effect(s); In transformed analysis of variance point estimates were used in pseudovalue estimators.Oregon Agricultural Experiment Station Technical Paper No. 8067     

Computer simulation of error rates of Poisson-based interval estimates of plankton abundance

The routine assignment of error rates (confidence intervals) to Poisson distribution estimates of plankton abundance should be rejected. In addition to the interval estimation procedure being pseudoreplicative, it is not robust to common violations of its assumptions. Because the spatial dispersion of organisms in sampling units from the counting chamber to the field is rarely random and because counting protocols are usually terminated by a count threshold having been equalled or exceeded, Poisson based estimates are usually derived from sampling non-Poisson distributions. Computer simulation was used to investigate the quantitative consequences of such estimates. The expected mean error rate of 95% confidence intervals is inflated from 5% to 15% as contagion increases, as the parametric variance-mean ratio increases from 1 to 2. Also, count threshold termination of the counting protocol effects both a biased estimate of the parametric mean (or total) and alters expected mean error rates, especially if the total count is low (< 100 organisms) and the mean density in the sampling unit is low.     

Growth estimation of mangrove cockle Anadara tuberculosa (Mollusca: Bivalvia): application and evaluation of length-based methods

Growth is one of the key processes in the dynamic of exploited resources, since it provides part of the information required for structured population models. Growth of mangrove cockle, Anadara tuberculosa was estimated through length-based methods (ELEFAN I y NSLCA) and using diverse shell length intervals (SLI). The variability of L(infinity), k and phi prime (phi') estimates and the effect of each sample were quantified by jackknife techniques. Results showed the same L(infinity) estimates from ELEFAN I and NSLCA across each SLI used, and all L(infinity) were within the expected range. On the contrary, k estimates differed between methods. Jackknife estimations uncovered the tendency of ELEFAN I to overestimate k with increases in SLI, and allowed the identification of differences in uncertainty (PE and CV) between both methods. The average values of phi' derived from NSCLA1.5 and length-age sources were similar and corresponded to ranges reported by other authors. Estimates of L(infinity), k and (phi' from NSCLA1.5 were 85.97 mm, 0.124/year and 2.953 with jackknife and 86.36mm de L(infinity), 0.110/year de k and 2.914 de phi' without jackknife, respectively. Based on the observed evidence and according to the biology of the species, NSCLA is suggested to be used with jackknife and a SLI of 1.5 mm as an ad hoc approach to estimate the growth parameters of mangrove cockle.     

A parametric model for estimating prevalence,incidence, and mean bout duration from point sampling

Point or instantaneous sampling refers to the scoring of presence or absence of behavior at the end of equally spaced intervals of time and is used to estimate prevalence. The literature cited demonstrates that point sampling does not adequately estimate frequency or mean bout duration. A parametric model is developed based on exponentially distributed times of behavior and intervening nonbehavior, thus enabling estimators of mean bout length and incidence. Variance estimators are provided and a method is suggested for designing sample situations which control the variance of the prevalence estimator. The paper concludes the theoretical investigation with a thorough Monte Carlo investigation and application to a “real-life” problem. The point sample estimators compare favorably with continuous observation under appropriate choice of sampling interval and under approximately exponential assumptions.     

Noise underlies switching behavior of the bacterial flagellum

We report the switching behavior of the full bacterial flagellum system that includes the filament and the motor in wild-type Escherichia coli cells. In sorting the motor behavior by the clockwise bias, we find that the distributions of the clockwise (CW) and counterclockwise (CCW) intervals are either exponential or nonexponential with long tails. At low bias, CW intervals are exponentially distributed and CCW intervals exhibit long tails. At intermediate CW bias (0.5) both CW and CCW intervals are mainly exponentially distributed. A simple model suggests that these two distinct switching behaviors are governed by the presence of signaling noise within the chemotaxis network. Low noise yields exponentially distributed intervals, whereas large noise yields nonexponential behavior with long tails. These drastically different motor statistics may play a role in optimizing bacterial behavior for a wide range of environmental conditions.     

Evaluation of jackknife and bootstrap for defining confidence intervals for pairwise agreement measures

Several research fields frequently deal with the analysis of diverse classification results of the same entities. This should imply an objective detection of overlaps and divergences between the formed clusters. The congruence between classifications can be quantified by clustering agreement measures, including pairwise agreement measures. Several measures have been proposed and the importance of obtaining confidence intervals for the point estimate in the comparison of these measures has been highlighted. A broad range of methods can be used for the estimation of confidence intervals. However, evidence is lacking about what are the appropriate methods for the calculation of confidence intervals for most clustering agreement measures. Here we evaluate the resampling techniques of bootstrap and jackknife for the calculation of the confidence intervals for clustering agreement measures. Contrary to what has been shown for some statistics, simulations showed that the jackknife performs better than the bootstrap at accurately estimating confidence intervals for pairwise agreement measures, especially when the agreement between partitions is low. The coverage of the jackknife confidence interval is robust to changes in cluster number and cluster size distribution.     

Comparing sire and dam estimates of heritability: jackknife and likelihood approaches

Three estimates of heritability are available from the half-sib pedigree design: the sire, dam and genotypic estimates. Because of its significantly smaller standard error, the genotypic estimate is preferred provided that there are no non-additive effects that inflate the estimate. I present two methods to test for such effects: these are a t-test of the paired sire and dam pseudovalues from the jackknife procedure and the likelihood ratio test from the animal model. Both methods are shown to be valid tests for significant dominance and/or maternal effects. SPLUS coding for the implementation of the jackknife method is provided. Unless sample sizes are very large, the power of the tests is low and hence caution is advised in the use of the genotypic estimate following a nonsignificant test. An approximate power analysis can be done using the data from the jackknife method but the estimated power is typically a substantial underestimate of the true power and its use is not recommended.     

Frigatebirds, Tropicbirds, and Ciconiida: Excesses of Confidence Probability

Because it is based on a significance test that takes the shape of the tree as given, the Rzhetsky/Nei Confidence Probability (CP) can attribute high "confidence" to groups with little or even literally no support. CP further overestimates confidence in that it takes no account of reliability of alignment, and it shows instability in that drastic changes in results can be produced by small changes in data. Instability can arise when alignment is uncertain, since different alignment strategies can lead to slightly different matrices. Parsimony jackknifing offers a more reliable and stable way of assessing support. To take ambiguities of alignment into account with parsimony jackknifing, we suggest "consensus" and "average" methods of summarizing jackknife results from several alignments. Reanalyzing 12S and 16S rRNA data on pelecaniform birds, we find that CP has overestimated support for the Ciconiida, for placing frigatebirds with condors, and for placing tropicbirds with cormorants.