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The individual estimates in a generalized linear model can also be tested to see whether they are significantly different from zero. This just involves comparing the estimate divided by its estimated standard error,

z = b/SÊ(b)

with critical values for the standard normal distribution. Thus if the absolute value of z exceeds 1.96, then the estimate is significantly different from zero at about the 5% level.

One complication that can occur with some generalized linear models is overdispersion, with the differences between the observed and fitted values of the dependent variable being much larger than what is expected from the assumed error distribution. There are ways of handling this should it arise. For more information on this and other aspects of these models, see McCullagh and Nelder (1989) or Dobson (2001).

Example 3.3:  Dolphin Bycatch in Trawl Fisheries

The accidental capture of marine mammals and birds in commercial fishing operations is of considerable concern in many fisheries around the world. This example concerns one such situation, which is of catches of the common dolphin (Delphinus delphis) and the bottlenose dolphin (Tursiops truncatus) in the Taranaki Bight trawl fishery for jack mackerel (Trachurus declivis, T. novaezelandiae, and T. murphyi) off the west coast of New Zealand.

The New Zealand Ministry of Fisheries puts official observers on a sample of fishing vessels to monitor dolphin bycatch, and Table 3.12 shows a summary of the data collected by these observers for the six fishing seasons 1989/90 to 1994/95, as originally published by Baird (1996, Table 3). The table shows the number of observed trawls and the number of dolphins accidentally killed, categorized by up to eight conditions for each fishing year. These are the fishing area (the northern or southern Taranaki Bight), the gear type (bottom or midwater), and the time (day or night). Excluding five cases where there were no observed trawls, this gives 43 observations on the bycatch under different conditions, in different years. Some results from fitting a generalized linear model to these data are also shown in the last two columns of the table.

The data in Table 3.12 will be used here to examine how the amount of bycatch varies with the factors recorded (year, area, gear type, and time of day). Because the dependent variable (the number of dolphins killed) is a count, it is reasonable to try fitting the data using a log-linear model with Poisson errors. A simple model of that type for the ith count is

where Ti is the number of tows involved; α(fi) depends on the fishing year fi when the observation was collected in such a way that α(fi) = α(1) for observations in 1989/90, α(fi) = α(2) for observations in 1990/91, and so on up to α(fi) = α(6) for observations in 1994/95; Xi1 is 0 for North Taranaki and 1 for South Taranaki; Xi2 is 0 for bottom trawls and 1 for

3.7  Chapter Summary

•Statistical models describe observations on variables in terms of parameters of distributions and the nature of the random variation involved.

•The properties of discrete random variables are briefly described, including definitions of the mean and the variance. The hyper­ geometric, binomial, and Poisson distributions are described.

•The properties of continuous random variables are briefly described, including definitions of a probability density function, the mean, and the variance. The exponential, normal, and lognormal distributions are described.

•The theory of linear regression is summarized as a way of relating the values of a variable Y to the corresponding values of some other

variables X1, X2, …, Xp. The coefficient of multiple determination, R2, is defined. Tests for the significance of relationships and the use of analysis of variance with regression are described.

•The use of multiple regression is illustrated using an example where chlorophyll-a concentrations are predicted from phosphorus and nitrogen concentrations in lakes.

•The difference between factors and variables is described. The models for one-, two-, and three-factor analysis of variance are defined, together with their associated analysis-of-variance tables.

•A two-factor example on the survival time of fish is used to illustrate analysis of variance, where the two factors are the type of fish and the treatment before the fish were kept in a mixture of toxic metals.

•The structure of data for a repeated-measures analysis of variance is defined.

•The use of multiple testing methods for comparing means after analysis of variance is briefly discussed, as is the fact that these methods are not approved of by all statisticians. The use of contrasts is also discussed briefly.

•The structure of generalized linear models is defined. Tests for goodness of fit, and for the existence of effects based on the comparison of deviances with critical values of the chi-squared distribution, are described, as are tests for the significance of the coefficients of individual X variables.

•The use of a generalized linear model is illustrated by an example where the number of dolphins accidentally killed during commercial fishing operations is related to the year of fishing, the type of fishing gear used, and the time of day of fishing. A log-linear model, with the number of dolphins killed assumed to have a Poisson distribution, is found to give a good fit to the data.

92 Statistics for Environmental Science and Management, Second Edition

Exercises

Exercise 3.1

At the start of the chapter the hypergeometric, binomial, Poisson, normal, and lognormal distributions are described, with examples of where each might be used. Give one further environmental example where each distribution would likely be appropriate for data.

Exercise 3.2

The data in Table 3.15 data come from a study by Green (1973). These data show four soil variables (X1 to X4) and one vegetation variable (Y), where the variables are X1 = % soil with constant lime enrichment, X2 = % meadow soil with calcium groundwater, X3 = % soil with coral bedrock under conditions of constant lime enrichment, X4 = % alluvial and organic soils adjacent to rivers and saline organic soil at the coast, and Y = % deciduous seasonal broadleaf forest. The sample units for this study were 2.5 × 2.5-km plots in the Corozal District of Belize in Central America. Use multiple regression to relate Y to the X variables. Plot Y against each of the X variables and determine whether the relationships, if any, appear to be linear. If they are linear, then a regression of the form

Y = β0 + β1X1 + β2X2 + β3X3 + β4X4 + e

can be fitted. Otherwise, modify this equation as appropriate. For example, you might decide that the relationship with one of the variables does not seem to be linear, so you might decide to relate Y to both X and X2. Fit the equation that you have selected using any statistical package that is available to you, or in a spreadsheet. Produce residual plots to test assumptions. Remove any X variables from the equation if they do not appear to be related to Y. Clearly state your conclusions from this analysis about the extent that the amount of deciduous seasonal broadleaf forest can be predicted by the soil characteristics.

Exercise 3.3

Table 1.1 shows data from Norwegian lakes on SO4 concentrations (mg/L) as well as other variables. Carry out a two-factor analysis of variance on the SO4 data and report your conclusions concerning (a) whether there was any change from year to year and (b) whether the assumptions of the analysis of variance model are reasonable. For (b), residual plots should be examined. You can, for example, plot the residuals against latitudes and longitudes. An important question concerns whether the residuals from the fitted model show correlation in time or space. If they do, then a simple two-factor analysis of variance is not appropriate for the data. Note that this is a two-factor analysis of variance with some missing data for some lakes in some years. This can be handled by some statistical packages under what may be referred to as a general linear model