Normalization in the fitting of data by iterative methods. Application to tracer kinetics and enzyme kinetics

1. The normalization of biochemical data to weight them appropriately for parameter estimation is considered, with reference particularly to data from tracer kinetics and enzyme kinetics. If the data are in replicate, it is recommended that the sum of squared deviations for each experimental variable at each time or concentration point is divided by the local variance at that point. 2. If there is only one observation for each variable at each sampling point, normalization may still be required if the observations cover more than one order of magnitude, but there is no absolute criterion for judging the effect of the weighting that is produced. The goodness of fit that is produced by minimizing the weighted sum of squares of deviations must be judged subjectively. It is suggested that the goodness of fit may be regarded as satisfactory if the data points are distributed uniformly on either side of the fitted curve. A chi-square test may be used to decide whether the distribution is abnormal. The proportion of the residual variance associated with points on one or other side of the fitted curve may also be taken into account, because this gives an indication of the sensitivity of the residual variance to movement of the curve away from particular data points. These criteria for judging the effect of weighting are only valid if the model equation may reasonably be expected to apply to all the data points. 3. On this basis, normalizing by dividing the deviation for each data point by the experimental observation or by the equivalent value calculated by the model equation may both be shown to produce a consistent bias for numerically small observations, the former biasing the curve towards the smallest observations, the latter tending to produce a curve that is above the numerically smaller data points. It was found that dividing each deviation by the mean of observed and calculated variable appropriate to it produces a weighting that is fairly free from bias as judged by the criteria mentioned above. This normalization factor was tested on published data from both tracer kinetics and enzyme kinetics.     

Using the EM algorithm to weight data sets of unknown precision when modelling fish stocks

Stocks of commercial fish are often modelled using sampling data of various types, of unknown precision, and from various sources assumed independent. We want each set to contribute to estimates of the parameters in relation to its precision and goodness of fit with the model. Iterative re-weighting of the sets is proposed for linear models until the weight of each set is found to be proportional to (relative weighting) or equal to (absolute weighting) the set-specific residual invariances resulting from a generalised least squares fit. Formulae for the residual variances are put forward involving fractional allocation of degrees of freedom depending on the numbers of independent observations in each set, the numbers of sets contributing to the estimate of each parameter, and the number of weights estimated. To illustrate the procedure, numbers of the 1984 year-class of North Sea cod (a) landed commercially each year, and (b) caught per unit of trawling time by an annual groundfish survey are modelled as a function of age to estimate total mortality, Z, relative catching power of the two fishing methods, and relative precision of the two sets of observations as indices of stock abundance. It was found that the survey abundance indices displayed residual variance about 29 times higher than that of the annual landings.     

Deregressing estimated breeding values and weighting information for genomic regression analyses

Background

Genomic prediction of breeding values involves a so-called training analysis that predicts the influence of small genomic regions by regression of observed information on marker genotypes for a given population of individuals. Available observations may take the form of individual phenotypes, repeated observations, records on close family members such as progeny, estimated breeding values (EBV) or their deregressed counterparts from genetic evaluations. The literature indicates that researchers are inconsistent in their approach to using EBV or deregressed data, and as to using the appropriate methods for weighting some data sources to account for heterogeneous variance.

Methods

A logical approach to using information for genomic prediction is introduced, which demonstrates the appropriate weights for analyzing observations with heterogeneous variance and explains the need for and the manner in which EBV should have parent average effects removed, be deregressed and weighted.

Results

An appropriate deregression for genomic regression analyses is EBV/r² where EBV excludes parent information and r² is the reliability of that EBV. The appropriate weights for deregressed breeding values are neither the reliability nor the prediction error variance, two alternatives that have been used in published studies, but the ratio (1 - h²)/[(c + (1 - r²)/r²)h²] where c > 0 is the fraction of genetic variance not explained by markers.

Conclusions

Phenotypic information on some individuals and deregressed data on others can be combined in genomic analyses using appropriate weighting.     

Physiological Performance and Stream Microhabitat use by the Centrarchids Lepomis megalotis and Lepomis macrochirus

Two centrarchids, Lepomis megalotis and L. macrochirus, were compared in laboratory studies of prey capture success, swimming endurance, morphology, hydrodynamic drag, and thermal tolerance, as well as field observations of focal point velocity and depth. For both species, capture of planktonic prey declined as current velocity increased, but L. megalotis was more efficient than L. macrochirus at higher current velocities. Capture of floating prey was not influenced by current velocity, but L. megalotis was more efficient overall at all velocities. Of the two species, L. megalotis was significantly more streamlined, had relatively lower hydrodynamic drag, and had higher swimming endurance in current. Both species had significantly higher critical thermal maxima (CTMax) in summer than in winter, but variance in CTMax was greater for L. megalotis than for L. macrochirus in both summer and winter. Differences between L. megalotis and L. macrochirus in performance and morphology may have direct influence on their relative abundances in small streams. Field observations showed L. megalotis was more common than L. macrochirus in faster, shallower microhabitats.     

Synonymous and nonsynonymous substitutions in mammalian genes and the nearly neutral theory

The nearly neutral theory of molecular evolution predicts larger generation-time effects for synonymous than for nonsynonymous substitutions. This prediction is tested using the sequences of 49 single-copy genes by calculating the average and variance of synonymous and nonsynonymous substitutions in mammalian star phylogenies (rodentia, artiodactyla, and primates). The average pattern of the 49 genes supports the prediction of the nearly neutral theory, with some notable exceptions.The nearly neutral theory also predicts that the variance of the evolutionary rate is larger than the value predicted by the completely neutral theory. This prediction is tested by examining the dispersion index (ratio of the variance to the mean), which is positively correlated with the average substitution number. After weighting by the lineage effects, this correlation almost disappears for nonsynonymous substitutions, but not quite so for synonymous substitutions. After weighting, the dispersion indices of both synonymous and nonsynonymous substitutions still exceed values expected under the simple Poisson process. The results indicate that both the systematic bias in evolutionary rate among the lineages and the episodic type of rate variation are contributing to the large variance. The former is more significant to synonymous substitutions than to nonsynonymous substitutions. Isochore evolution may be similar to synonymous substitutions. The rate and pattern found here are consistent with the nearly neutral theory, such that the relative contributions of drift and selection differ between the two types of substitutions. The results are also consistent with Gillespie's episodic selection theory.     

Using the biomasses of soil nematode taxa as weighting factors for assessing soil food web conditions

We propose the use of nematode generic biomasses as weighting factors for calculation of nematode community indices. Three data sets were used to calculate the indices using guild-based weighting (i.e., fixed weighting of nematode guilds) and genus-based weighting (i.e., weighting based on the nematode generic biomasses). The genus-based weighting factors were quadratically correlated with guild-based weighting factors, but the genus-based weighting factors were highly variable within each nematode guild, indicating that important information was likely missing when guild-based weighting was used. Although variation patterns of in the indices in response to management practices and land use were often similar for the two weighting systems, they sometimes differed substantially, and the specific index values frequently differed depending on which weighting system was used. In addition, the absolute values of the indices were frequently found to be different between the two weighting systems. Based on the comparison of indices from the two systems, we found that the genus-based system was complementary rather than superior to the guild-based system. It was suggested that both weighting systems should be used for the calculation of the nematode community indices in a study in order to better distinguish the treatment effects.     

Optimal cue integration in ants

In situations with redundant or competing sensory information, humans have been shown to perform cue integration, weighting different cues according to their certainty in a quantifiably optimal manner. Ants have been shown to merge the directional information available from their path integration (PI) and visual memory, but as yet it is not clear that they do so in a way that reflects the relative certainty of the cues. In this study, we manipulate the variance of the PI home vector by allowing ants (Cataglyphis velox) to run different distances and testing their directional choice when the PI vector direction is put in competition with visual memory. Ants show progressively stronger weighting of their PI direction as PI length increases. The weighting is quantitatively predicted by modelling the expected directional variance of home vectors of different lengths and assuming optimal cue integration. However, a subsequent experiment suggests ants may not actually compute an internal estimate of the PI certainty, but are using the PI home vector length as a proxy.     

Possibilities and limitations of weighted averaging

A statistical analysis of a weighted averaging procedure for the estimation of small signals buried in noise (Hoke et al. 1984a) is given. The weighting factor used by this method is in inverse proportion to the variance estimated for the noise. It is shown that, compred to conventional averaging, weighted averaging can improve the signal-to-noise ratio to a high extent if the variance of the noise changes as a function of time. On the other hand, uncritical application of the method involves the danger that the signal amplitude is underestimated. How serious this effect is depends on the number of degrees of freedom available for the estimation of the weighting factor. The effect can be neglected, if this number is sufficiently increased by means of an appropriate preprocessing.     

An assessment of the individual and collective effects of variants on height using twins and a developmentally informative study design

In a sample of 3,187 twins and 3,294 of their parents, we sought to investigate association of both individual variants and a genotype-based height score involving 176 of the 180 common genetic variants with adult height identified recently by the GIANT consortium. First, longitudinal observations on height spanning pre-adolescence through adulthood in the twin sample allowed us to investigate the separate effects of the previously identified SNPs on pre-pubertal height and pubertal growth spurt. We show that the effect of SNPs identified by the GIANT consortium is primarily on prepubertal height. Only one SNP, rs7759938 in LIN28B, approached a significant association with pubertal growth. Second, we show how using the twin data to control statistically for environmental variance can provide insight into the ultimate magnitude of SNP effects and consequently the genetic architecture of a phenotype. Specifically, we computed a genetic score by weighting SNPs according to their effects as assessed via meta-analysis. This weighted score accounted for 9.2% of the phenotypic variance in height, but 14.3% of the corresponding genetic variance. Longitudinal samples will be needed to understand the developmental context of common genetic variants identified through GWAS, while genetically informative designs will be helpful in accurately characterizing the extent to which these variants account for genetic, and not just phenotypic, variance.     

An Ensemble Weighting Approach for Dendroclimatology: Drought Reconstructions for the Northeastern Tibetan Plateau

Traditional detrending methods assign equal mean value to all tree-ring series for chronology developments, despite that the mean annual growth changes in different time periods. We find that the strength of a tree-ring model can be improved by giving more weights to tree-ring series that have a stronger climate signal and less weight to series that have a weaker signal. We thus present an ensemble weighting method to mitigate these potential biases and to more accurately extract the climate signals in dendroclimatology studies. This new method has been used to develop the first annual precipitation reconstruction (previous August to current July) at the Songmingyan Mountain and to recalculate the tree-ring chronology from Shenge site in Dulan area in northeastern Tibetan Plateau (TP), a marginal area of Asian summer monsoon. The ensemble weighting method explains 31.7% of instrumental variance for the reconstructions at Songmingyan Mountain and 57.3% of the instrumental variance in the Dulan area, which are higher than those developed using traditional methods. We focus on the newly introduced reconstruction at Songmingyan Mountain, which showsextremely dry (wet) epochs from 1862–1874, 1914–1933 and 1991–1999 (1882–1905). These dry/wet epochs were also found in the marginal areas of summer monsoon and the Indian subcontinent, indicating the linkages between regional hydroclimate changes and the Indian summer monsoon.     

Statistical error in isothermal titration calorimetry: variance function estimation from generalized least squares

The method of generalized least squares (GLS) is used to assess the variance function for isothermal titration calorimetry (ITC) data collected for the 1:1 complexation of Ba(2+) with 18-crown-6 ether. In the GLS method, the least squares (LS) residuals from the data fit are themselves fitted to a variance function, with iterative adjustment of the weighting function in the data analysis to produce consistency. The data are treated in a pooled fashion, providing 321 fitted residuals from 35 data sets in the final analysis. Heteroscedasticity (nonconstant variance) is clearly indicated. Data error terms proportional to q(i) and q(i)/v are well defined statistically, where q(i) is the heat from the ith injection of titrant and v is the injected volume. The statistical significance of the variance function parameters is confirmed through Monte Carlo calculations that mimic the actual data set. For the data in question, which fall mostly in the range of q(i)=100-2000 microcal, the contributions to the data variance from the terms in q(i)(2) typically exceed the background constant term for q(i)>300 microcal and v<10 microl. Conversely, this means that in reactions with q(i) much less than this, heteroscedasticity is not a significant problem. Accordingly, in such cases the standard unweighted fitting procedures provide reliable results for the key parameters, K and DeltaH(degrees) and their statistical errors. These results also support an important earlier finding: in most ITC work on 1:1 binding processes, the optimal number of injections is 7-10, which is a factor of 3 smaller than the current norm. For high-q reactions, where weighting is needed for optimal LS analysis, tips are given for using the weighting option in the commercial software commonly employed to process ITC data.     

Memory and the efficient use of information

We consider the problem of how an animal's memory should be designed in order to cope with a stochastic and changing environment. In particular we consider the problem of forming the best estimate of an unknown and possibly changing environmental parameter. Under the simple model we consider, the effect of an observation is to update this estimate using a linear operator. Two models of a changing environment are analysed. For each model we show how estimates change as a function of time elapsed and observations taken. The effect of a regular sequence of observations is also considered, and it is shown that an exponential weighting of past observations is a sufficient statistic on which to base decisions. The weighting factors are different in the two model environments considered, but each is shown to be a function of the rate at which the environment is changing.     

Diversity, distribution and larval habitats of North Swedish blackflies (Diptera: Simuliidae)

1. Blackfly species richness and community structure were analysed at fifty-six sites in northern Sweden in two seasons. The sites were situated in a wide range of streams and rivers from small springbrooks, bog streams and lake-outlet streams to medium-sized forest rivers and large rivers draining montane regions.
2. Thirty-nine blackfly species were found, with between two and thirteen species per site. Neither species richness nor abundance could be related to the environmental variables measured.
3. An analysis of labral fan size of blackflies indicated a clear trend for the prevalence of larvae with small fans in large rivers and larvae with larger fan size in small streams. Similarly, fan size related to current velocities so that large fans were associated with slow current velocities and small fans with high velocities.
4. A strong relationship existed between species composition and habitat, as seen in ordination by non-metric multidimensional scaling. The relationship found between fan size and habitat size-related variables, such as channel width, depth, velocity and substratum particle size, along with longitude and altitude, in partial least squares regression analysis offered an explanation of the species composition–habitat relationship.
5. In addition to testing that distributions of blackfly larvae reflect morphological traits, we tested two general hypotheses pertaining to distribution patterns: (a) that blackfly communities show bimodal distributions; and (b) that their distributions are nested. Neither of these two hypotheses was supported by our observations. However, widespread blackfly species were locally more abundant than those found at relatively few sites, thus showing a positive abundance–occupancy relationship.     

Microtubule sliding in flagellar axonemes of Chlamydomonas mutants missing inner- or outer-arm dynein: velocity measurements on new types of mutants by an improved method.

To help understand the functional properties of inner and outer dynein arms in axonemal motility, sliding velocities of outer doublets were measured in disintegrating axonemes of Chlamydomonas mutants lacking either of the arms. Measurements under improved solution conditions yielded significantly higher sliding velocities than those observed in a previous study [Okagaki and Kamiya, 1986, J. Cell Biol. 103:1895-1902]. As in the previous study, it was found that the velocities in axonemes of wild type (wt) and a mutant (oda1) missing the outer arm differ greatly: 18.5 +/- 4.1 microns/sec for wt and 4.4 +/- 2.3 microns/sec for oda1 at 0.5 mM Mg-ATP. In contrast, axonemes of two types of mutants (ida2 and ida4) that lacked different sets of two inner-arm heavy chains displayed velocities almost identical with the wild-type velocity. Moreover, axonemes of a non-motile double mutant ida2 X ida4 underwent sliding disintegration at a similar high velocity, although less frequently than in axonemes of single mutants. These observations support the hypothesis that the inner and outer dynein arms in disintegrating axonemes drive microtubules at different speeds and it is the faster outer arm that determines the overall speed when both arms are present. The inner arm may be important for the initiation of sliding. The axoneme thus appears to be equipped with two (or more) types of motors with different intrinsic speeds.     

Variance-stabilizing transformations for two-color microarrays

MOTIVATION: Authors of several recent papers have independently introduced a family of transformations (the generalized-log family), which stabilizes the variance of microarray data up to the first order. However, for data from two-color arrays, tests for differential expression may require that the variance of the difference of transformed observations be constant, rather than that of the transformed observations themselves. RESULTS: We introduce a transformation within the generalized-log family which stabilizes, to the first order, the variance of the difference of transformed observations. We also introduce transformations from the 'started-log' and log-linear-hybrid families which provide good approximate variance stabilization of differences. Examples using control-control data show that any of these transformations may provide sufficient variance stabilization for practical applications, and all perform well compared to log ratios.     

Three-Dimensional Analysis of the Swimming Behavior of Daphnia magna Exposed to Nanosized Titanium Dioxide

Due to their surface characteristics, nanosized titanium dioxide particles (nTiO₂) tend to adhere to biological surfaces and we thus hypothesize that they may alter the swimming performance and behavior of motile aquatic organisms. However, no suitable approaches to address these impairments in swimming behavior as a result of nanoparticle exposure are available. Water fleas Daphnia magna exposed to 5 and 20 mg/L nTiO₂ (61 nm; polydispersity index: 0.157 in 17.46 mg/L stock suspension) for 96 h showed a significantly (p<0.05) reduced growth rate compared to a 1-mg/L treatment and the control. Using three-dimensional video observations of swimming trajectories, we observed a treatment-dependent swarming of D. magna in the center of the test vessels during the initial phase of the exposure period. Ensemble mean swimming velocities increased with increasing body length of D. magna, but were significantly reduced in comparison to the control in all treatments after 96 h of exposure. Spectral analysis of swimming velocities revealed that high-frequency variance, which we consider as a measure of swimming activity, was significantly reduced in the 5- and 20-mg/L treatments. The results highlight the potential of detailed swimming analysis of D. magna for the evaluation of sub-lethal mechanical stress mechanisms resulting from biological surface coating and thus for evaluating the effects of nanoparticles in the aquatic environment.     

Development of weighting factors for G20 countries—explore the difference in environmental awareness between developed and emerging countries

Purpose

Weighting is one of the steps involved in LCIA. This enables us to integrate various environmental impacts and facilitates the interpretation of environmental information. Many different weighting methodologies have already been proposed, and the results of many case studies with a single index have been published. LIME2 (Itsubo et al. Int J Life Cycle Assess 17(4):488-498, 2012) developed weighting factors for four different areas of protection that reflect environmental awareness among the Japanese public. This method has already been widely used in Japan, but difficulties exist universally using the Japanese weighting factors around the world. It is presumed that the weighting varies depending on economic, cultural, and social conditions, and there are still few cases in which weighting factors have been specifically invented or studied in consideration of variance in these elements. This study attempted to develop weighting factors applicable to the Group of Twenty (G20) countries with a view toward developing those that could be used in different countries. In the study, a survey was conducted with a uniform questionnaire in G20 countries to compare the weighting factors calculated for different countries, along with an investigation on development and utilization of global weighting factors.

Methods

A conjoint analysis was conducted to give a weighting between the four areas of protection defined by LIME: human health, social assets, biodiversity, and primary production. The analysis is suitable for measuring the value of each of the multiple attributes of the environment. This study conducted a questionnaire in all the G20 member states. The survey puts priority on making the questions understood by the respondents and minimizing bias, adopting interviews, visiting surveys, and surveys in venues in the 11 emerging countries. In the developed countries, Internet surveys were conducted after confirming that their results are statistically significant from the pretest results in these states. In both surveys, random sampling was performed to take 200–250 samples (households) in each of the emerging countries and 500–600 samples in each of the developed countries. The surveys collected a total of 6400 responses. Statistical values based on this model can be considered to reflect the variability between each individual’s environmental thoughts. The calculated results can then be used to compare the variety of environmental thoughts in developed and emerging countries.

Results and discussion

The study was able to obtain two different kinds of results: dimensionless weighting factors and economic indicators using the amount of willingness to pay. This paper solely presents the former. The weighting factors in the entire G20 community, in the group of developed countries (G8) and in the group of emerging countries (G20 states excluding the G8) and those in the individual G20 countries, were estimated. The calculated values were significant statistically at the 1 % level (all p values for the safeguard subject coefficients were less than 0.0001), with the exception of monetary attributes for several emerging countries. Converted into dimensionless values, so that the total sum for the four subjects equals 1, the weighting factor was the highest for human health in the entire G20 circles, at 0.34, followed by biodiversity at 0.29, and primary production at 0.23. The weighting for social assets was relatively poor, at 0.13. In the G8 developed states, the figures of biodiversity and primary production were relatively higher than those of the same two subjects in the full G20. Biodiversity had the highest value, at 0.34, and was followed by human health at 0.30. On the other hand, in emerging countries, the weighting of health impacts was particularly significant, at 0.44, whereas the three other subjects had almost equivalent weightings—biodiversity at 0.19, social assets at 0.18, and primary product at 0.18. The weighting factors by country and the variance of preference intensities by country showed minor differences among developed countries while they reflected considerable differences among emerging countries.

Conclusions

Accurate weighting factors representing the environmental attitudes of the world and national public are needed in order to conduct general purpose LCA. This study is the world’s first to conduct surveys with the use of the same questionnaire not only in developed countries but also in emerging countries, and to compare the findings. A total of 6400 responses were obtained via interviews and Internet surveys. The survey thus gained a statistically significant result on all the environmental attributes including the weighting factors for the G20 circles, G8 states, emerging countries exclusive of the G8 states, and individual countries in which surveys took place. The results have revealed a relatively minor difference in weighting factors and variation coefficients between the areas of protection in the developed countries whereas a considerable difference was observed between those subjects in emerging countries.

     

FREQUENCY DISTRIBUTIONS OF LENGTHS OF POSSIBLE NETWORKS FROM A DATA MATRIX

Abstract— It is common practice to attempt to find the minimum length tree (also known as the Wagner tree) for a given data matrix on a group of OTUs (taxa). However, little study has been made of the pattern of frequency distributions when the lengths of all possible networks (unrooted trees) are taken into consideration. A published real data matrix with eight OTUs was compared with randomly generated data, when the former showed a much larger variance and very marked skewness. A number of published data matrices with a larger number of OTUs were studied by random selection of 10240 out of the possible trees: these were compared with 32 randomly generated data sets with 13 OTUs, using the same program. An algorithm has been found for calculation of the expected mean, variance and skewness for random binary data with up to 13 OTUs, based on the number of characters representing each type of partition of the OTUs. The calculation requires listing of the possible topologies and their relative weighting, which are tabulated.     

Comparison of adjustment methods for stratified two‐sample tests in the context of ROC analysis

We compare several nonparametric and parametric weighting methods for the adjustment of the effect of strata. In particular, we focus on the adjustment methods in the context of receiver‐operating characteristic (ROC) analysis. Nonparametrically, rank‐based van Elteren's test and inverse‐variance (IV) weighting using the area under the ROC curve (AUC) are examined. Parametrically, the stratified t‐test and IV AUC weighted method are applied based on a binormal monotone transformation model. Stratum‐specific, pooled, and adjusted estimates are obtained. The pooled and adjusted AUCs are estimated. We illustrate and compare these weighting methods on a multi‐center diagnostic trial and through extensive Monte‐Carlo simulations.