R. I. C. Chris Francis

Testing for mean and variance differences with samples from distributions that may be non-normal with unequal variances

The problem of comparing several samples to decide whether the means and/or variances are significantly different is considered. It is shown that with very non-normal distributions even a very robust test to compare the means has poor properties when the distributions have different variances, and therefore a new testing scheme is proposed. This starts by using an exact randomization test for any significant difference (in means or variances) between the samples. If a non-significant result is obtained then testing stops. Otherwise, an approximate randomization test for mean differences (but allowing for variance differences) is carried out, together with a bootstrap procedure to assess whether this test is reliable. A randomization version of Levenes test is also carried out for differences in variation between samples. The five possible conclusions are then that (i) there is no evidence of any differences, (ii) evidence for mean differences only, (iii) evidence for variance differences only, (iv) evidence for mean and variance differences, or (v) evidence for some indeterminate differences. A simulation experiment to assess the properties of the proposed scheme is described. From this it is concluded that the scheme is useful as a robust, conservative method for comparing samples in cases where they may be from very non-normal distributions.

Use of otolith weight in length-mediated estimation of proportions at age

Each year almost a million fish are aged from otoliths, primarily to estimate proportions at age for use in stock assessments. The preparation and reading of otoliths is time-consuming and thus expensive. Two techniques have been proposed to reduce costs. The first is length-mediated estimation, in which the length distribution from a large sample of fish is converted to an age distribution, using information (usually in the form of an age-length key) from a smaller sample containing length and age data. The second is to infer age from otolith weight (and/or other otolith measurements). These two cost-saving ideas are combined in a new method, length-mediated mixture analysis. It requires three samples - one with lengths only, one with lengths and otolith measurements, and one with lengths, otolith measurements and ages - and estimation is by maximum likelihood. The use of this method, which can be thought of as a generalisation of three established methods of age inference, is illustrated in two simulation experiments in a cost-benefit framework.

Theory & Methods: Analysis of Variance by Randomization when Variances are Unequal

If there are significant factor and interaction effects with analysis of variance using ran-domization inference, they can be detected by tests that compare the F-statistics for the real data with the distributions of these statistics obtained by randomly allocating either the original observations or the residuals to the various factor combinations. Such tests involve the assumption that the effect of factors or interactions is to shift the observations for a factor combination by a fixed amount, without changing the amount of variation at that combination. In reality the expected amount of variation at each factor combination, as measured by the variance, may not be constant, which may upset the properties of the tests for the effects of factors and interactions. This paper discusses several possible methods for adjusting the randomization procedure to allow for this type of problem, including generalizations of methods that have been proposed for comparing the means of several samples when there is unequal variance but no factor structure. A simulation study shows that the best of the methods examined is one for which the randomized sets of data are designed to approximate the distributions of F-statistics when unequal variance is present.

Links between climate and recruitment of New Zealand hoki (Macruronus novaezelandiae) now unclear

Abstract We investigated the relationship between climate variation and year‐class strengths for hoki (Macruronus novaezelandiae) in New Zealand waters. Our analyses extended those of a previous study, by using an additional 6 years’ data and considering some additional predictands (total year‐class strength and proportion migrating) relating to an alternative stock‐structure hypothesis, and also updated them, by using revised versions of some predictors and year‐class strengths. Predictors considered were based on the Southern Oscillation Index, weather patterns, sea‐surface temperatures, wind speeds, and modelled mixed‐layer depths and nutrient concentrations. In contrast to the earlier analyses we found little or no predictive power for either year‐class strength or proportion migrating. The substantial correlations found in the earlier study were greatly reduced. Such reversals are not uncommon in the climate‐recruitment literature.

Comment on “Differences in predicted catch composition between two widely used catch equation formulations”Appears in Can. J. Fish. Aquat. Sci. 66: 126–132.

Branch (2009. Can. J. Fish. Aquat. Sci. 66: 126–132) described the two most common catch equation formulations in stock assessment models: the continuous (Baranov) one, which represents fishing mortality as an instantaneous rate, F, and the discrete one, in which it is represented as an exploitation rate, u. He claimed that the continuous formulation is preferable at high fishing mortality where a fish could encounter multiple sets of gear within a year. This claim is wrong for two reasons. First, it is based on the false supposition that the discrete catch equations require the assumption that fish encounter at most only one set of fishing gear in a year. Second, it is not possible to determine, for a specific stock assessment, whether one formulation is preferable to another solely on the basis of information about the fishery. The appropriate way to make this decision is to see which fits the data better.

Optimum design for catch sampling of eels

Data from the first two seasons of a catch-sampling programme for New Zealand freshwater eels (Anguilla australis and A. diefenbachii) are described. These are used in two simulation experiments to provide information to optimize future sampling. Results are presented in the form of equations that predict the precision of estimates (of species composition, mean size, and mean age at two different sizes) as a function of various survey design variables. Precision typically depends more on the number of landings (catches) that are sampled than the number of eels sampled per landing. Also, the precision obtainable from a given design varies substantially from fishery to fishery. For example, if 50 eels were measured and 10 otoliths collected from each of five landings, estimated standard errors varied by a factor of 9 for species composition, 19 for mean weight, and 5 for mean age at the minimum legal size, depending on which fishery was sampled. Results for mean age estimates are more restricted and less certain (than those for species composition and mean size) because age data were fewer. Three further optimization issues are discussed: sampling costs, the importance of ‘minor’ species, and the pool of fishers from which samples are collected.