Quantifying uncertainty in parameter estimates for stochastic models of collective cell spreading using approximate Bayesian computation

Wound healing and tumour growth involve collective cell spreading, which is driven by individual motility and proliferation events within a population of cells. Mathematical models are often used to interpret experimental data and to estimate the parameters so that predictions can be made. Existing methods for parameter estimation typically assume that these parameters are constants and often ignore any uncertainty in the estimated values. We use approximate Bayesian computation (ABC) to estimate the cell diffusivity, D, and the cell proliferation rate, λ, from a discrete model of collective cell spreading, and we quantify the uncertainty associated with these estimates using Bayesian inference. We use a detailed experimental data set describing the collective cell spreading of 3T3 fibroblast cells. The ABC analysis is conducted for different combinations of initial cell densities and experimental times in two separate scenarios: (i) where collective cell spreading is driven by cell motility alone, and (ii) where collective cell spreading is driven by combined cell motility and cell proliferation. We find that D can be estimated precisely, with a small coefficient of variation (CV) of 2–6%. Our results indicate that D appears to depend on the experimental time, which is a feature that has been previously overlooked. Assuming that the values of D are the same in both experimental scenarios, we use the information about D from the first experimental scenario to obtain reasonably precise estimates of λ, with a CV between 4 and 12%. Our estimates of D and λ are consistent with previously reported values; however, our method is based on a straightforward measurement of the position of the leading edge whereas previous approaches have involved expensive cell counting techniques. Additional insights gained using a fully Bayesian approach justify the computational cost, especially since it allows us to accommodate information from different experiments in a principled way.

Commentary on the article by Swanenburg et al. ‘Validity and reliability of a German version of the Neck Disability Index (NDI-G)’

We commend Swanenburg et al. (2013) on translation, development, and clinimetric analysis of the NDI-G. However, the dual-factor structure with factor analysis and the high level of internal consistency (IC) highlighted in their discussion were not emphasized in the abstract or conclusion. These points may imply some inconsistencies with the final conclusions since determination of stable point estimates with the study's small sample are exceedingly difficult.

Heritability of working memory brain activation

Although key to understanding individual variation in task-related brain activation, the genetic contribution to these individual differences remains largely unknown. Here we report voxel-by-voxel genetic model fitting in a large sample of 319 healthy, young adult, human identical and fraternal twins (mean ± SD age, 23.6 ±1.8 years) who performed an n-back working memory task during functional magnetic resonance imaging (fMRI) at a high magnetic field (4 tesla). Patterns of task-related brain response (BOLD signal difference of 2-back minus 0-back) were significantly heritable, with the highest estimates (40 - 65%) in the inferior, middle, and superior frontal gyri, left supplementary motor area, precentral and postcentral gyri, middle cingulate cortex, superior medial gyrus, angular gyrus, superior parietal lobule, including precuneus, and superior occipital gyri. Furthermore, high test-retest reliability for a subsample of 40 twins indicates that nongenetic variance in the fMRI brain response is largely due to unique environmental influences rather than measurement error. Individual variations in activation of the working memory network are therefore significantly influenced by genetic factors. By establishing the heritability of cognitive brain function in a large sample that affords good statistical power, and using voxel-by-voxel analyses, this study provides the necessary evidence for task-related brain activation to be considered as an endophenotype for psychiatric or neurological disorders, and represents a substantial new contribution to the field of neuroimaging genetics. These genetic brain maps should facilitate discovery of gene variants influencing cognitive brain function through genome-wide association studies, potentially opening up new avenues in the treatment of brain disorders.

Accounting for individual variability in the von Bertalanffy growth model

Estimation of von Bertalanffy growth parameters has received considerable attention in fisheries research. Since Sainsbury (1980, Can. J. Fish. Aquat. Sci. 37: 241-247) much of this research effort has centered on accounting for individual variability in the growth parameters. In this paper we demonstrate that, in analysis of tagging data, Sainsbury's method and its derivatives do not, in general, satisfactorily account for individual variability in growth, leading to inconsistent parameter estimates (the bias does not tend to zero as sample size increases to infinity). The bias arises because these methods do not use appropriate conditional expectations as a basis for estimation. This bias is found to be similar to that of the Fabens method. Such methods would be appropriate only under the assumption that the individual growth parameters that generate the growth increment were independent of the growth parameters that generated the initial length. However, such an assumption would be unrealistic. The results are derived analytically, and illustrated with a simulation study. Until techniques that take full account of the appropriate conditioning have been developed, the effect of individual variability on growth has yet to be fully understood.

Maximum likelihood estimation of mortality and growth with individual variability from multiple length-frequency data

We consider estimation of mortality rates and growth parameters from length-frequency data of a fish stock and derive the underlying length distribution of the population and the catch when there is individual variability in the von Bertalanffy growth parameter L-infinity. The model is flexible enough to accommodate 1) any recruitment pattern as a function of both time and length, 2) length-specific selectivity, and 3) varying fishing effort over time. The maximum likelihood method gives consistent estimates, provided the underlying distribution for individual variation in growth is correctly specified. Simulation results indicate that our method is reasonably robust to violations in the assumptions. The method is applied to tiger prawn data (Penaeus semisulcatus) to obtain estimates of natural and fishing mortality.

Does the size of subsamples taken from multispecies trawl catches affect estimates of catch composition and abundance?

Although subsampling is a common method for describing the composition of large and diverse trawl catches, the accuracy of these techniques is often unknown. We determined the sampling errors generated from estimating the percentage of the total number of species recorded in catches, as well as the abundance of each species, at each increase in the proportion of the sorted catch. We completely partitioned twenty prawn trawl catches from tropical northern Australia into subsamples of about 10 kg each. All subsamples were then sorted, and species numbers recorded. Catch weights ranged from 71 to 445 kg, and the number of fish species in trawls ranged from 60 to 138, and invertebrate species from 18 to 63. Almost 70% of the species recorded in catches were "rare" in subsamples (less than one individual per 10 kg subsample or less than one in every 389 individuals). A matrix was used to show the increase in the total number of species that were recorded in each catch as the percentage of the sorted catch increased. Simulation modelling showed that sorting small subsamples (about 10% of catch weights) identified about 50% of the total number of species caught in a trawl. Larger subsamples (50% of catch weight on average) identified about 80% of the total species caught in a trawl. The accuracy of estimating the abundance of each species also increased with increasing subsample size. For the "rare" species, sampling error was around 80% after sorting 10% of catch weight and was just less than 50% after 40% of catch weight had been sorted. For the "abundant" species (five or more individuals per 10 kg subsample or five or more in every 389 individuals), sampling error was around 25% after sorting 10% of catch weight, but was reduced to around 10% after 40% of catch weight had been sorted.

Effect of individual variability on estimation of population parameters from length-frequency data

We consider estimation of mortality rates and growth parameters from length-frequency data of a fish stock when there is individual variability in the von Bertalanffy growth parameter L-infinity and investigate the possible bias in the estimates when the individual variability is ignored. Three methods are examined: (i) the regression method based on the Beverton and Holt's (1956, Rapp. P.V. Reun. Cons. Int. Explor. Mer, 140: 67-83) equation; (ii) the moment method of Powell (1979, Rapp. PV. Reun. Int. Explor. Mer, 175: 167-169); and (iii) a generalization of Powell's method that estimates the individual variability to be incorporated into the estimation. It is found that the biases in the estimates from the existing methods are, in general, substantial, even when individual variability in growth is small and recruitment is uniform, and the generalized method performs better in terms of bias but is subject to a larger variation. There is a need to develop robust and flexible methods to deal with individual variability in the analysis of length-frequency data.

An improved Fabens method for estimation of growth parameters in the vonBertalanffy model with individual asymptotes

In the analysis of tagging data, it has been found that the least-squares method, based on the increment function known as the Fabens method, produces biased estimates because individual variability in growth is not allowed for. This paper modifies the Fabens method to account for individual variability in the length asymptote. Significance tests using t-statistics or log-likelihood ratio statistics may be applied to show the level of individual variability. Simulation results indicate that the modified method reduces the biases in the estimates to negligible proportions. Tagging data from tiger prawns (Penaeus esculentus and Penaeus semisulcatus) and rock lobster (Panulirus ornatus) are analysed as an illustration.

The reliability of supply of feed grains in the northern region

Reliability of supply of feed grain has become a high priority issue for industry in the northern region. Expansion by major intensive livestock and industrial users of grain, combined with high inter-annual variability in seasonal conditions, has generated concern in the industry about reliability of supply. This paper reports on a modelling study undertaken to analyse the reliability of supply of feed grain in the northern region. Feed grain demand was calculated for major industries (cattle feedlots, pigs, poultry, dairy) based on their current size and rate of grain usage. Current demand was estimated to be 2.8Mt. With the development of new industrial users (ethanol) and by projecting the current growth rate of the various intensive livestock industries, it was estimated that demand would grow to 3.6Mt in three years time. Feed grain supply was estimated using shire scale yield prediction models for wheat and sorghum that had been calibrated against recent ABS production data. Other crops that contribute to a lesser extent to the total feed grain pool (barley, maize) were included by considering their production relative to the major winter and summer grains, with estimates based on available production records. This modelling approach allowed simulation of a 101-year time series of yield that showed the extent of the impact of inter-annual climate variability on yield levels. Production estimates were developed from this yield time series by including planted crop area. Area planted data were obtained from ABS and ABARE records. Total production amounts were adjusted to allow for any export and end uses that were not feed grain (flour, malt etc). The median feed grain supply for an average area planted was about 3.1Mt, but this varied greatly from year to year depending on seasonal conditions and area planted. These estimates indicated that supply would not meet current demand in about 30% of years if a median area crop were planted. Two thirds of the years with a supply shortfall were El Nino years. This proportion of years was halved (i.e. 15%) if the area planted increased to that associated with the best 10% of years. Should demand grow as projected in this study, there would be few years where it could be met if a median crop area was planted. With area planted similar to the best 10% of years, there would still be a shortfall in nearly 50% of all years (and 80% of El Nino years). The implications of these results on supply/demand and risk management and investment in research and development are briefly discussed.

A new data source for fisheries resource assessment: genetic estimates of the effective number of spawners. Final Report to the Fisheries Research and Development Corporation.

The development of innovative methods of stock assessment is a priority for State and Commonwealth fisheries agencies. It is driven by the need to facilitate sustainable exploitation of naturally occurring fisheries resources for the current and future economic, social and environmental well being of Australia. This project was initiated in this context and took advantage of considerable recent achievements in genomics that are shaping our comprehension of the DNA of humans and animals. The basic idea behind this project was that genetic estimates of effective population size, which can be made from empirical measurements of genetic drift, were equivalent to estimates of the successful number of spawners that is an important parameter in process of fisheries stock assessment. The broad objectives of this study were to 1. Critically evaluate a variety of mathematical methods of calculating effective spawner numbers (Ne) by a. conducting comprehensive computer simulations, and by b. analysis of empirical data collected from the Moreton Bay population of tiger prawns (P. esculentus). 2. Lay the groundwork for the application of the technology in the northern prawn fishery (NPF). 3. Produce software for the calculation of Ne, and to make it widely available. The project pulled together a range of mathematical models for estimating current effective population size from diverse sources. Some of them had been recently implemented with the latest statistical methods (eg. Bayesian framework Berthier, Beaumont et al. 2002), while others had lower profiles (eg. Pudovkin, Zaykin et al. 1996; Rousset and Raymond 1995). Computer code and later software with a user-friendly interface (NeEstimator) was produced to implement the methods. This was used as a basis for simulation experiments to evaluate the performance of the methods with an individual-based model of a prawn population. Following the guidelines suggested by computer simulations, the tiger prawn population in Moreton Bay (south-east Queensland) was sampled for genetic analysis with eight microsatellite loci in three successive spring spawning seasons in 2001, 2002 and 2003. As predicted by the simulations, the estimates had non-infinite upper confidence limits, which is a major achievement for the application of the method to a naturally-occurring, short generation, highly fecund invertebrate species. The genetic estimate of the number of successful spawners was around 1000 individuals in two consecutive years. This contrasts with about 500,000 prawns participating in spawning. It is not possible to distinguish successful from non-successful spawners so we suggest a high level of protection for the entire spawning population. We interpret the difference in numbers between successful and non-successful spawners as a large variation in the number of offspring per family that survive – a large number of families have no surviving offspring, while a few have a large number. We explored various ways in which Ne can be useful in fisheries management. It can be a surrogate for spawning population size, assuming the ratio between Ne and spawning population size has been previously calculated for that species. Alternatively, it can be a surrogate for recruitment, again assuming that the ratio between Ne and recruitment has been previously determined. The number of species that can be analysed in this way, however, is likely to be small because of species-specific life history requirements that need to be satisfied for accuracy. The most universal approach would be to integrate Ne with spawning stock-recruitment models, so that these models are more accurate when applied to fisheries populations. A pathway to achieve this was established in this project, which we predict will significantly improve fisheries sustainability in the future. Regardless of the success of integrating Ne into spawning stock-recruitment models, Ne could be used as a fisheries monitoring tool. Declines in spawning stock size or increases in natural or harvest mortality would be reflected by a decline in Ne. This would be good for data-poor fisheries and provides fishery independent information, however, we suggest a species-by-species approach. Some species may be too numerous or experiencing too much migration for the method to work. During the project two important theoretical studies of the simultaneous estimation of effective population size and migration were published (Vitalis and Couvet 2001b; Wang and Whitlock 2003). These methods, combined with collection of preliminary genetic data from the tiger prawn population in southern Gulf of Carpentaria population and a computer simulation study that evaluated the effect of differing reproductive strategies on genetic estimates, suggest that this technology could make an important contribution to the stock assessment process in the northern prawn fishery (NPF). Advances in the genomics world are rapid and already a cheaper, more reliable substitute for microsatellite loci in this technology is available. Digital data from single nucleotide polymorphisms (SNPs) are likely to super cede ‘analogue’ microsatellite data, making it cheaper and easier to apply the method to species with large population sizes.

Using genomic data to make indirect (and unauthorized) estimates of disease risk

The number of genetic factors associated with common human traits and disease is increasing rapidly, and the general public is utilizing affordable, direct-to-consumer genetic tests. The results of these tests are often in the public domain. A combination of factors has increased the potential for the indirect estimation of an individual's risk for a particular trait. Here we explain the basic principals underlying risk estimation which allowed us to test the ability to make an indirect risk estimation from genetic data by imputing Dr. James Watson's redacted apolipoprotein E gene (APOE) information. The principles underlying risk prediction from genetic data have been well known and applied for many decades, however, the recent increase in genomic knowledge, and advances in mathematical and statistical techniques and computational power, make it relatively easy to make an accurate but indirect estimation of risk. There is a current hazard for indirect risk estimation that is relevant not only to the subject but also to individuals related to the subject; this risk will likely increase as more detailed genomic data and better computational tools become available.

Uncertainty in forest simulators and forest planning systems

The forest simulator is a computerized model for predicting forest growth and future development as well as effects of forest harvests and treatments. The forest planning system is a decision support tool, usually including a forest simulator and an optimisation model, for finding the optimal forest management actions. The information produced by forest simulators and forest planning systems is used for various analytical purposes and in support of decision making. However, the quality and reliability of this information can often be questioned. Natural variation in forest growth and estimation errors in forest inventory, among other things, cause uncertainty in predictions of forest growth and development. This uncertainty stemming from different sources has various undesirable effects. In many cases outcomes of decisions based on uncertain information are something else than desired. The objective of this thesis was to study various sources of uncertainty and their effects in forest simulators and forest planning systems. The study focused on three notable sources of uncertainty: errors in forest growth predictions, errors in forest inventory data, and stochastic fluctuation of timber assortment prices. Effects of uncertainty were studied using two types of forest growth models, individual tree-level models and stand-level models, and with various error simulation methods. New method for simulating more realistic forest inventory errors was introduced and tested. Also, three notable sources of uncertainty were combined and their joint effects on stand-level net present value estimates were simulated. According to the results, the various sources of uncertainty can have distinct effects in different forest growth simulators. The new forest inventory error simulation method proved to produce more realistic errors. The analysis on the joint effects of various sources of uncertainty provided interesting knowledge about uncertainty in forest simulators.

Prediction of breeding values for average fruit weight in mango using a multivariate individual mixed model

Mango is an important horticultural fruit crop and breeding is a key strategy to improve ongoing sustainability. Knowledge of breeding values of potential parents is important for maximising progress from breeding. This study successfully employed a mixed linear model methods incorporating a pedigree to predict breeding values for average fruit weight from highly unbalanced data for genotypes planted over three field trials and assessed over several harvest seasons. Average fruit weight was found to be under strong additive genetic control. There was high correlation between hybrids propagated as seedlings and hybrids propagated as scions grafted onto rootstocks. Estimates of additive genetic correlation among trials ranged from 0.69 to 0.88 with correlations among harvest seasons within trials greater than 0.96. These results suggest that progress from selection for broad adaptation can be achieved, particularly as no repeatable environmental factor that could be used to predict G x E could be identified. Predicted breeding values for 35 known cultivars are presented for use in ongoing breeding programs.

An evaluation of length-frequency sampling procedures and subsequent data analysis for purse seine-caught yellowfin tuna in the Eastern Pacific Ocean

ENGLISH: A two-stage sampling design is used to estimate the variances of the numbers of yellowfin in different age groups caught in the eastern Pacific Ocean. For purse seiners, the primary sampling unit (n) is a brine well containing fish from a month-area stratum; the number of fish lengths (m) measured from each well are the secondary units. The fish cannot be selected at random from the wells because of practical limitations. The effects of different sampling methods and other factors on the reliability and precision of statistics derived from the length-frequency data were therefore examined. Modifications are recommended where necessary. Lengths of fish measured during the unloading of six test wells revealed two forms of inherent size stratification: 1) short-term disruptions of existing pattern of sizes, and 2) transition zones between long-term trends in sizes. To some degree, all wells exhibited cyclic changes in mean size and variance during unloading. In half of the wells, it was observed that size selection by the unloaders induced a change in mean size. As a result of stratification, the sequence of sizes removed from all wells was non-random, regardless of whether a well contained fish from a single set or from more than one set. The number of modal sizes in a well was not related to the number of sets. In an additional well composed of fish from several sets, an experiment on vertical mixing indicated that a representative sample of the contents may be restricted to the bottom half of the well. The contents of the test wells were used to generate 25 simulated wells and to compare the results of three sampling methods applied to them. The methods were: (1) random sampling (also used as a standard), (2) protracted sampling, in which the selection process was extended over a large portion of a well, and (3) measuring fish consecutively during removal from the well. Repeated sampling by each method and different combinations indicated that, because the principal source of size variation occurred among primary units, increasing n was the most effective way to reduce the variance estimates of both the age-group sizes and the total number of fish in the landings. Protracted sampling largely circumvented the effects of size stratification, and its performance was essentially comparable to that of random sampling. Sampling by this method is recommended. Consecutive-fish sampling produced more biased estimates with greater variances. Analysis of the 1988 length-frequency samples indicated that, for age groups that appear most frequently in the catch, a minimum sampling frequency of one primary unit in six for each month-area stratum would reduce the coefficients of variation (CV) of their size estimates to approximately 10 percent or less. Additional stratification of samples by set type, rather than month-area alone, further reduced the CV's of scarce age groups, such as the recruits, and potentially improved their accuracy. The CV's of recruitment estimates for completely-fished cohorts during the 198184 period were in the vicinity of 3 to 8 percent. Recruitment estimates and their variances were also relatively insensitive to changes in the individual quarterly catches and variances, respectively, of which they were composed. SPANISH: Se usa un diseño de muestreo de dos etapas para estimar las varianzas de los números de aletas amari11as en distintos grupos de edad capturados en el Océano Pacifico oriental. Para barcos cerqueros, la unidad primaria de muestreo (n) es una bodega de salmuera que contenía peces de un estrato de mes-área; el numero de ta11as de peces (m) medidas de cada bodega es la unidad secundaria. Limitaciones de carácter practico impiden la selección aleatoria de peces de las bodegas. Por 10 tanto, fueron examinados los efectos de distintos métodos de muestreo y otros factores sobre la confiabilidad y precisión de las estadísticas derivadas de los datos de frecuencia de ta11a. Se recomiendan modificaciones donde sean necesarias. Las ta11as de peces medidas durante la descarga de seis bodegas de prueba revelaron dos formas de estratificación inherente por ta11a: 1) perturbaciones a corto plazo en la pauta de ta11as existente, y 2) zonas de transición entre las tendencias a largo plazo en las ta11as. En cierto grado, todas las bodegas mostraron cambios cíclicos en ta11a media y varianza durante la descarga. En la mitad de las bodegas, se observo que selección por ta11a por los descargadores indujo un cambio en la ta11a media. Como resultado de la estratificación, la secuencia de ta11as sacadas de todas las bodegas no fue aleatoria, sin considerar si una bodega contenía peces de un solo lance 0 de mas de uno. El numero de ta11as modales en una bodega no estaba relacionado al numero de lances. En una bodega adicional compuesta de peces de varios lances, un experimento de mezcla vertical indico que una muestra representativa del contenido podría estar limitada a la mitad inferior de la bodega. Se uso el contenido de las bodegas de prueba para generar 25 bodegas simuladas y comparar los resultados de tres métodos de muestreo aplicados a estas. Los métodos fueron: (1) muestreo aleatorio (usado también como norma), (2) muestreo extendido, en el cual el proceso de selección fue extendido sobre una porción grande de una bodega, y (3) medición consecutiva de peces durante la descarga de la bodega. EI muestreo repetido con cada método y distintas combinaciones de n y m indico que, puesto que la fuente principal de variación de ta11a ocurría entre las unidades primarias, aumentar n fue la manera mas eficaz de reducir las estimaciones de la varianza de las ta11as de los grupos de edad y el numero total de peces en los desembarcos. El muestreo extendido evito mayormente los efectos de la estratificación por ta11a, y su desempeño fue esencialmente comparable a aquel del muestreo aleatorio. Se recomienda muestrear con este método. El muestreo de peces consecutivos produjo estimaciones mas sesgadas con mayores varianzas. Un análisis de las muestras de frecuencia de ta11a de 1988 indico que, para los grupos de edad que aparecen con mayor frecuencia en la captura, una frecuencia de muestreo minima de una unidad primaria de cada seis para cada estrato de mes-área reduciría los coeficientes de variación (CV) de las estimaciones de ta11a correspondientes a aproximadamente 10% 0 menos. Una estratificación adicional de las muestras por tipo de lance, y no solamente mes-área, redujo aun mas los CV de los grupos de edad escasos, tales como los reclutas, y mejoró potencialmente su precisión. Los CV de las estimaciones del reclutamiento para las cohortes completamente pescadas durante 1981-1984 fueron alrededor de 3-8%. Las estimaciones del reclutamiento y sus varianzas fueron también relativamente insensibles a cambios en las capturas de trimestres individuales y las varianzas, respectivamente, de las cuales fueron derivadas. (PDF contains 70 pages)

The Voting Rights Act, Shelby County, and Redistricting: Improving Estimates of Racially Polarized Voting in a Multiple-Election Context

The Supreme Court’s decision in Shelby County has severely limited the power of the Voting Rights Act. I argue that Congressional attempts to pass a new coverage formula are unlikely to gain the necessary Republican support. Instead, I propose a new strategy that takes a “carrot and stick” approach. As the stick, I suggest amending Section 3 to eliminate the need to prove that discrimination was intentional. For the carrot, I envision a competitive grant program similar to the highly successful Race to the Top education grants. I argue that this plan could pass the currently divided Congress.

Without Congressional action, Section 2 is more important than ever before. A successful Section 2 suit requires evidence that voting in the jurisdiction is racially polarized. Accurately and objectively assessing the level of polarization has been and continues to be a challenge for experts. Existing ecological inference methods require estimating polarization levels in individual elections. This is a problem because the Courts want to see a history of polarization across elections.

I propose a new 2-step method to estimate racially polarized voting in a multi-election context. The procedure builds upon the Rosen, Jiang, King, and Tanner (2001) multinomial-Dirichlet model. After obtaining election-specific estimates, I suggest regressing those results on election-specific variables, namely candidate quality, incumbency, and ethnicity of the minority candidate of choice. This allows researchers to estimate the baseline level of support for candidates of choice and test whether the ethnicity of the candidates affected how voters cast their ballots.