Parameter estimation toolbox

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Selecting the appropriate level of complexity for mechanistic models is a much less straightforward task. Occam’s razor still should be a guiding principle, but the objective is to devise a model with as few superfluous mechanisms as is possible, rather than aiming strictly to minimize the number of parameters involved. Objective statistical methods may provide some assistance with the number of parameters required to describe certain mechanisms adequately, but will rarely be able to assist with the number of mechanisms that should be included. The Pauling quote cautions that, even with relatively few parameters, almost any pattern can plausibly be represented, with little necessary connection to reality. Even relatively simple and abstract ecological models often have upwards of 10 parameters, sufficient to make the elephant ride a bicycle whilst playing a trumpet. Many models have over 100 parameters, and could draw the whole circus!

Apart from the possibility of including spurious processes with too many parameters, a very large number of parameters will make a model unworkable, as they will be impossible to estimate and it will be equally impossible to investigate the sensitivity of model solutions to these estimates.

There is an important distinction between parameters estimated extrinsically and those estimated from the same time series with which the model is being tested. If parameters are determined entirely extrinsically, there is no necessary limitation on the number of parameters, provided not too much post hoc reasoning has gone into the choice of mechanisms included in the model. However, estimating any more than a single parameter from the time series on which a mechanistic model is tested severely limits the ability to draw any useful conclusions about the fit of the model to those same data.

Selecting an appropriate statistical model

Most ecologists are familiar with the idea of using a hypothesis-testing approach to decide whether additional components should be included in a statistical model. For example, in a two-way analysis of variance you might examine the ratio of the interaction mean square (for two factors A and B) to the error mean square, and compare it with a tabulated F ratio. On this basis, you might decide whether it was necessary to include parameters for the interaction between A and B. The idea can easily be extended to generalized linear models (McCullagh & Nelder, 1989) by using a likelihood ratio test. These approaches are only valid, however, when comparing nested models. That is, a model is compared with another, which is the original model with added components. Likelihood ratios or F ratios cannot be used to compare models that are not nested. For example, in a mark–recapture study estimating survival, it is possible to use a likelihood ratio to compare a model in which survival is constant through time and with age with another in which survival

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is constant through time, but varies with age. The former model is nested within the latter. It is not possible, however, to use a likelihood ratio to compare the model in which survival is constant through time, but varies with age, with one in which survival varies through time, but not with age. The two models are not nested.

Recently the Akaike information criterion (AIC) (Akaike, 1985; Burnham & Anderson, 1992) has been suggested as a tool for comparing non-nested models. It is defined as

where p is the number of parameters fitted, and the log-likelihood (l ) is the value calculated using maximum likelihood estimates for each of the parameters fitted in that particular model. The model with the lowest AIC is selected as optimal for the given data. The AIC is derived from information theory, but heuristically can be understood as the likelihood component, representing the discrepancy between the model and the data, which will never increase as more parameters are added, plus a penalty for adding additional parameters.

Rescaling and redimensioning models

One of the most useful general tools available to minimize problems with parameter estimation is that of redimensioning or rescaling the model. This involves re-expressing the model so that variables are defined either relative to parameters, or possibly relative to other variables. The approach is most suitable for fairly abstract models developed for the purpose of understanding the qualitative behaviour of ecological systems. It is less likely to be useful for highly detailed tactical models, as they will usually be expressed in particular units that are relevant for the specific problem at hand.

There are several potential advantages of rescaling a model. First, rescaling may reduce the number of different parameters that need to be estimated. In some cases, it may be possible to define away a problem with a particularly intractable parameter by defining the entire system in units of that parameter. Second, the process of rescaling will often show that generic features of the model dynamics are determined by particular parameter combinations, especially ratios, rather than by particular parameters alone. The rescaling exercise in itself, then, produces useful information, and essentially performs an algebraic sensitivity analysis. The very simple example in Box 2.1 illustrates some of these points.

Changing the variables in a model is a similar, but slightly more complex process. If a model includes more than one variable, then it may be worthwhile to replace one or more original variables with ratios of one variable to

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Box 2.1 Rescaling the Ricker equation

The Ricker equation is one of the simplest ways of representing intraspecific competition in a difference equation. The main purpose of the model is to help in understanding the effect of overcompensation on population dynamics. Overcompensation occurs when, beyond a certain point, the more individuals there are in one generation, the fewer there are in the next. Consider a population of a univoltine species, with the population in generation t represented by Nt. The carrying capacity is K, and a parameter R describes both the rate of reproduction and the extent of overcompensation. The following equation then describes the model:

It is very easy to confirm that the model has an equilibrium at Nt = K, because the term in the brackets is then 0, and exp(0) is 1. The greater the value of R, the more humped is the relationship described by the equation.

Equation (1) has two parameters, but K can easily be removed by rescaling in the following way. Define a new variable Xt = Nt/K. This means that we are measuring population size in units of the carrying capacity or equilibrium population size. The original Nt = KXt. Substituting this into eqn (1) gives

This means that the dynamic behaviour of the original model depends only on the parameter R, and that any conclusions drawn about the behaviour of eqn (3) will apply directly to eqn (1). The general behaviour of the system can easily be explored numerically on a spreadsheet. As R, the extent of the overcompensation, increases, the model moves from a smooth approach to equilibrium through an oscillatory approach to sustained cycles and finally to chaos (see May, 1974a).

If we were interested in fitting this model to real data to see what sort of dynamic behaviour would be predicted, the rescaling tells us that only R needs to be estimated. Note, however, that slightly more complex functional forms do a much better job at representing real insect populations (see Bellows, 1981).

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another, or with sums or differences involving other variables. This may simplify the model structure or make its behaviour easier to interpret. It is essential that the new variables should have direct biological meaning. For example, in a model that originally has numbers of individuals of each of three genotypes as its variables, it may well be worth considering changing variables to a model with total population size as one variable, and frequency of two of the three genotypes as the other two. (Note that changing variables will not change the number of equations necessary to specify the model fully.) In a differential equation model, the standard product and ratio rules of calculus should be used to form the new equations.

It is difficult to make general rules to determine whether a change of variable is worthwhile. It is largely a matter of trying the exercise to see if the new model is simpler or more tractable than the original. An example where a change of variable is very useful is given in Box 2.2.

Parameter uncertainty and sensitivity analysis

It is extremely rare that any ecological parameter can be measured exactly. Using the methods discussed throughout this book, the best that can be done is to estimate the most likely value that a parameter may have, given a set of observed data. However, the observed data will also be consistent with the parameter taking a range of other values, with somewhat lower likelihood. It is necessary to see how the outcome of the model may vary over the range of plausible parameter values. This is sensitivity analysis.

Understanding such sensitivity is also critical to designing any program of parameter estimation. The outcome of any model is more sensitive to variations in some parameters than it is to variation in others. There is little value in expending large amounts of effort to obtain a precise estimate of a parameter if the behaviour of the model is relatively invariant to changes in the parameter. Often, the major goal of a modelling exercise may be to perform a sensitivity analysis to guide further research. With abstract models capable of being represented algebraically, sensitivity can often be determined by careful examination of the equations themselves (see Box 2.2).

With more complex models, an extensive numerical investigation is required. A good general overview of the possible approaches is provided by Swartzman and Kaluzny (1987, pp. 217–25). A common approach is to perturb each of the uncertain parameters in turn, recording the response of the model, whilst holding all other parameters constant at their most likely point estimates. Such one-at-a-time sensitivity analysis may produce very misleading results. In many cases, the model output may be almost invariant to perturbation of either of two parameters singly, but joint perturbation of both in a particular direction may produce a major change in the response.

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To take a particularly simple example, consider a simple spreadsheet model that was constructed to examine the economic viability of moving use of the Australian pastoral zone away from concentration on sheep grazing towards a high-value harvest of kangaroos for human consumption. At present, the majority of the kangaroo harvest is of low value, being used for skins and pet food. Using a scenario based on current practices, a relatively small proportion of the harvest is used for game meat (25% of the harvest is for meat, most being for skins, and only 5% of the meat is for human consumption), and the value of the game meat is low ($A0.50 per kilogram for game versus $A0.45 per kilogram for pet food). Increasing the proportion of the harvest used for human consumption has little effect on the total value of the harvest, because the price of the meat is low. Holding the proportion of the harvest taken for human consumption constant but increasing the price of the game meat also has little effect, because so little of the harvest is used for human consumption. However, increasing both together produces a substantially increased total value. This is a particularly obvious example, but is based on a real simulation (Hardman, 1996). In many other circumstances, similar parameter interactions will occur, but will be hidden by the complexity of the model. Parameter interactions may also be compensatory: changes in the model’s response due to alteration of one parameter may approximately cancel changes in response to alteration of another parameter.

A factorial model, in which combinations of the selected factors are varied systematically (Henderson-Sellers & Henderson-Sellers, 1993) is a more rigorous approach to sensitivity analysis than one-at-a-time parameter variations. The difficulty is that the number of parameter combinations rapidly becomes unworkable. If only two perturbations of each parameter are considered (‘high’ and ‘low’), the number of combinations of n parameters is 2n. One way to reduce this number is to use fractional factorial models, in which some multifactor interactions are ignored. For example, the full factorial design with six factors and two levels requires 64 runs, but a design with only 16 runs could potentially identify all the main effects and nine of the 15 two-factor interactions. For specific guidelines on constructing such designs, see HendersonSellers and Henderson-Sellers (1993).

An alternative approach is to select combinations of values of the parameters for use in sensitivity analysis at random (Swartzman & Kaluzny, 1987, p. 223), sampling each parameter from an assumed probability distribution. The approach will generate the sensitivity of the output to variation in each parameter, averaged across the sensitivity in the response to all the other variable parameters. This is in contrast to looking at each parameter in turn, holding all others at their average value. A difficulty of entirely random sampling is that, by chance, some areas of parameter space may be missed entirely. Random sampling of parameter space may also be misleading if some parameters are correlated.

Fig. 2.1 Latin Hypercube sampling. The diagram illustrates a sensitivity analysis for three parameters. Parameter P1 has a normal distribution, P2 has a uniform distribution and P3 has a triangular distribution. For six model runs, each distribution is divided into six segments of equal area (probability). For each model run, a value for each parameter

is selected at random from within one of these segments. Segments are sampled for successive runs without replacement, leading to the table of parameter values shown to the right of the figure. Note that not all possible parameter combinations are used (this would require 63 rather than 6 model runs). From Swartzman and Kaluzny (1987).

Latin hypercube sampling (Iman & Conover, 1980) is one solution to the problem that random sampling may miss areas of parameter space. Suppose that there are k parameters for which sensitivity is required to be analysed, and n simulation runs available for the sensitivity analysis. For each parameter, the range over which the sensitivity is to be examined is then divided into n intervals into which the parameter might fall with equal probability. If the hypothesized distribution of the parameter were uniform, these would be of equal width. More usually, a distribution such as a normal distribution or perhaps triangular distribution would be used, so that the intervals are wider at the boundaries of the distribution (Fig. 2.1). For each parameter, one of the n intervals is selected at random without replacement, and within that interval a value is selected at random. When this process is repeated for each parameter, the first parameter combination is formed. The whole process is repeated n times. As intervals are sampled without replacement, a complete coverage for

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each individual parameter is obtained, and the combinations of parameter values that are investigated are selected at random. This process seems almost baroque in its complexity, but can be programmed without too much difficulty, and is the default sampling procedure in the risk analysis package @Risk (Palisade Corporation, Newfield, New York).

Sensitivity analysis is particularly important for stochastic models, such as population viability analysis models, because little guidance is likely to be provided by algebraic analysis. However, the computational requirements may be very large indeed, given that 500 –1000 replications may be necessary to gain a good description of the probability distribution of outcomes for a single parameter combination. Statistical approaches such as logistic regression are helpful for interpreting such results, provided it is recognized that the significance levels are a function of the number of replicates used in the simulation. For an example of this approach, see McCarthy et al. (1995).

Statistical tools

Accuracy and precision

The ideal for any estimator (estimation procedure) is that it is unbiased and of minimum variance. ‘Minimum variance’ means that repeated estimates are as close as possible to each other. This is often called ‘precision’. An estimator is unbiased if its expected value (loosely, the average of many repeated estimates) is equal to the true value of the entity being estimated. Lack of bias, together with precision, is often termed ‘accuracy’ (Fig. 2.2). Ideally, an estimate should satisfy both criteria simultaneously, but on occasions, it is necessary to compromise one to achieve major gains in the other. All

Fig. 2.2 The target analogy for bias and precision. Estimating a parameter can be imagined as aiming for a target, in which the bull’s-eye is the value of the parameter itself. Each estimation is represented by one shot at the target. (a) An estimator (estimation procedure) that is unbiased, but relatively imprecise compared with (b). The estimates are clustered around the correct value, but there is considerable scatter. (b) An estimator that is more precise than (a), but is biased. The estimates are tightly clustered together, with little scatter, but they are not clustered around the correct parameter value.

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estimation problems are subject to a constraint of limited available effort, whether effort is measured in time or money. The design problem is therefore to allocate limited effort in such a way that the estimate is as accurate as possible.

Most biologists are familiar with the standard statistical methods of analysis of variance and regression. There is a plethora of books on these approaches for biologists, and the subject is covered well in almost all elementary statistics texts. I will not elaborate on the details of the methods here. There are, however, a couple of general points that are worth making.

First, statistical methods are usually taught to biologists with a heavy emphasis on hypothesis testing. Hypothesis testing using analysis of variance, regression, etc. uses a so-called parametric model, meaning that it proceeds by a process of estimating parameters in a statistical model. Hypothesis tests are essentially a process of determining whether particular parameters or groups of parameters could possibly be zero (or less commonly, some other specified value), given the observed data. This means that it is straightforward to turn any parametric statistical test around so that it estimates one or more parameters, rather than simply tests hypotheses. Usually, a parameter estimate is more useful than a straightforward hypothesis test. For example, Box 2.3 shows how a t test can be used to obtain an estimate for the size of the treatment effect, and shows how this approach provides substantially more information than does the hypothesis test alone.

Second, analysis of variance and regression are versions of an identical statistical model: the general linear model. All this means is that the parameters of the model are separated from each other by plus signs, and that the random variation associated with each observation (or possibly group of observations) is added on to the systematic (nonrandom) part. Each source of random variation or error is also assumed to have a normal distribution, although the method performs fairly well (it is ‘robust’) with many nonnormal error distributions. An important consequence of the assumption of additive normal error is that the amount of variability in ‘treatments’ should not be related to their predicted mean response. Linear models can be built with complex nested or crossed error structures (see Cochran & Cox, 1957; Sokal & Rohlf, 1995; Zar, 1999). The principal advantages of linear models should be familiar to most ecologists. They are very well understood and they can be solved algebraically, using almost all standard statistical packages. In many cases, conventional linear models will be perfectly adequate to estimate parameters for ecological models. Even if the ecological model appears nonlinear, or the errors are non-normal, a transformation of the original data may be able to be represented linearly. By a substantial margin, logarithmic transformations are the most common, but a variety of other functions can be used.

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Box 2.3 contd

a

b

c

d

e

f

g

h

Difference between treatment and control

Case b is similarly unequivocal. It is statistically significant, as the interval does not include 0, and is biologically important at this confidence level, as the error bars do not leave the region of biological importance.

Case c is statistically significant, but is biologically unimportant, as the error bars do not leave the hatched region.

The remaining cases are equivocal. In d and e, the result is statistically significant, but may or may not be biologically important. Cases f and g are not statistically significant, but do not preclude the possibility of a biologically important difference in one direction. Case h is similarly not statistically significant, but biologically important differences in either direction are not inconsistent with the result.

Logistic regression

Many parameter estimation problems in ecology involve estimating proportions. For example, estimating a survival rate is a problem of estimating the proportion of individuals in a particular class that survive from one time period to the next. Elementary statistical theory states that, if the probability of an event happening to any one individual is p, and there are n independent trials, the number of individuals x to which the event happens follows a binomial distribution. As this distribution is not a normal distribution, and its variance is related to the mean, standard linear methods may be inappropriate. A further problem is that a linear model can easily generate estimated probabilities that are either less than zero or greater than one.

Logistic regression is a statistical method that handles these problems. It

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Box 2.4 contd

provided the groups each contain a reasonable number of observations. The number of observations in each group that is ‘reasonable’ can be determined by the same rule of thumb used in conventional contingency table analysis: the expected number of both success and failures in at least 80% of groups should be 5 or greater.

If the residual deviance is greater than the upper 5% tail of a tabulated χ2 distribution with i − Np degrees of freedom, then there is evidence of overdispersion, also known as extra-binomial variation or heterogeneity. It may sometimes be reduced or removed by fitting an additional covariate, factor or interaction. Frequently, this will not be possible, but you may still wish to use the model, both to make inferences about which explanatory variables are important influences on the response, and to estimate parameters.

There are several ways in which overdispersion can be handled (see McCullagh & Nelder, 1989; Crawley, 1993, Collett, 1991, Wilson & Grenfell, 1997). The simplest solution is to assume that the binomial variance has been inflated by some constant scale factor φ. This can be estimated by:

φ = D0/(i − Np ),

where D0 is the residual deviance. It is probably slightly better to use Pearson’s X2 for the full model, in place of the residual deviance, but it is less convenient, and the improvement will be slight (Collett, 1991). φ can then be used to adjust changes in deviance to see if particular terms can be omitted from the model. For example, suppose you wished to determine whether a factor with k levels could be omitted from a full model with Np parameters, and a residual deviance D0, and that the residual deviance with the factor omitted was D1. Instead of comparing D0 − D1 with a χ2 distribution with k − 1 degrees of freedom, you would form the ratio

F = (D0 − D1)/(k − 1),

φ

and compare it with an F distribution with k − 1 and i − Np degrees of freedom. The parallels with a standard analysis of variance should be obvious.

Overdispersion has a very small effect on the parameter estimates themselves, but will cause their standard errors to be underestimated. An approximate solution is to multiply the standard errors produced in the standard analysis by φ (Collett, 1991).

Some packages (e.g. GLIM and PROC GENMOD in SAS) allow a scale parameter φ to be specified in a logistic regression. Running a standard analysis, estimating φ from the equation above, and then rerunning the analysis specifying that value of φ as the scale parameter will perform very similar adjustments to those just described (Crawley, 1993). More sophisticated ways of handling overdispersion are discussed by Collett (1991).

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Maximum likelihood estimation

Linear models are not adequate for all estimation problems in ecology. There are two problems, frequently linked. First, many ecological processes and therefore models are intrinsically nonlinear. This means that there is no transformation capable of producing a model with all parameters separated by plus signs. Second, data may have a non-normal error structure, which means that the data cannot satisfactorily be modelled as a systematic part plus a normally distributed random part. The result of these two problems is that the statistical theory upon which linear model theory is based is invalid, possibly causing biased parameter estimators, and more probably, incorrect measures of the precision of the resulting estimates. Logistic regression, described above, is a common example of a statistical model for which conventional least squares estimation cannot be used.

In such cases, a much more general approach, that of maximum likelihood, is often helpful. It is important to note that least squares estimation, applied to a linear model with normally distributed errors, generates maximum likelihood estimates for the model parameters. Maximum likelihood estimation thus encompasses linear least squares estimation. It is not an alternative method. There is an outstanding review of the general approach and philosophy in Edwards (1972), and a recent discussion from an ecological perspective is provided by Hilborn and Mangel (1997).

It is worth digressing a little to explain the basic approach, because it underlies most estimation problems discussed in this book. Most ecologists will probably treat the process as a ‘black box’, using one of a number of computer packages that can handle maximum likelihood estimation for common models. The basic idea is, however, relatively straightforward, and there will be occasions when it is necessary to compute a maximum likelihood estimate from first principles, rather than relying on a computer package.

The principle is best explained first with a specific example, rather than being tackled abstractly. The data in Table 2.1 come from an experiment investigating the mortality induced in the fish Poecilia latipinna by the protozoan parasite Ichthyophthirius multifiliis. The table shows, for a range of intensities of infection xi , the number yi of fish surviving five days after infection out of a total number mi that were in the infection class originally. The objective is to fit, using maximum likelihood, the logistic regression model

where pi is the probability of a fish surviving five days, given it is in infection class xi.

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Table 2.1 Survival of black mollies (Poecilia latipinna) infected with Ichthyophthirius multifiliis. The following data are from an experiment conducted by McCallum (1982b) (also summarized in McCallum, 1985). Black mollies were subjected to Ichthyophthirius infective stages at several densities. Two days after infection, the parasite burden established on each fish was assessed by counting the number of the ectoparasitic ciliates visible on the fish’s flank within a counting frame with an area of 0.16 cm2. The number of days the fish survived after infection, up to a maximum of 10 days, was then recorded. The data are abstracted from this information, and were obtained by grouping the fish into ranges of infection intensity, and then recording the proportion of the fish in each of these infection classes surviving to day 6 after infection. (A better, but more complex, way to analyse these

data is to model the survival time of each fish as a function of parasite burden. This approach is taken in Chapter 4)

Assuming that the survival of one fish does not affect the survival of another, the probability of a given number of fish in each infection class surviving should follow a binomial distribution:

The joint probability of observing each number of surviving fish, given known p values, is simply a series of equations of the form above multiplied together:

This equation could calculate the probability of any given outcome, provided the probabilities pi were known. What we actually have, however, is one set of observed outcomes yi, and our task is to estimate the probabilities pi.

The basic idea of maximum likelihood estimation is very simple and logical. The objective is to find the set of parameters that are the most likely of all possible sets to have generated the observed result. This will be the set for which the probability of detecting the observed outcome is as large as possible. In other words, we need to find values of α and β that maximize the value of

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–0.01

–0.02

β

–0.03

–0.04

Fig. 2.3 Contour plot of the likelihood surface for the black mollie survival data in Table 2.1. β is plotted on the vertical axis, and α on the horizontal axis. The contour interval is 2,

and hence the first contour encloses an approximate 95% confidence region for the joint distribution of the parameters. The profile likelihood confidence limits for β are shown as dashed lines.

achievable log-likelihood (obtained when the predicted and observed values are identical) and that attained by the model under consideration (McCullagh & Nelder, 1989, p. 118). The adequacy of the fitted model is then often assessed by comparing the residual deviance with a χ2 distribution with i − Np degrees of freedom, where i is the number of classes in the data, and Np is the number of fitted parameters. This is only appropriate, however, under restrictive conditions (see McCullagh & Nelder, 1989, p. 118; and Box 2.4). In the particular example considered above, the maximum achievable log-likelihood is that obtained by substituting the observed frequencies themselves into eqn (2.5) (Box 2.5).

A more reliable procedure to assess whether a model is adequate is to examine the change in deviance when additional parameters are added to the model. If the true value of the additional parameters is zero, the change in deviance should have a χ2 distribution with degrees of freedom equal to the number of additional parameters. This will be the case asymptotically (that is, when the sample size is sufficiently large). In practice, it means that you can determine whether additional parameters can be justified by comparing the change in deviance with the upper tail of a tabulated χ2 distribution (see Box 2.4).

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Box 2.5 contd

Step 3. Copy the linear predictor and log-likelihood terms down columns, and sum the log-likelihood terms in cell F10:

Step 4. Use Solver (Tools Solver) to maximize cell F10 by changing cells K2 and L2:

continued on p. 35

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Maximum likelihood and nonlinear least squares

The parameter estimates produced by ordinary least squares analysis of linear models are maximum likelihood estimates, provided that the error can be assumed to be additive, normally distributed and with a constant variance. In fact, it is possible to go considerably further than this. If you want to fit a nonlinear relationship f(x,β ) to a data set with a response y, predictor variables x and parameters β, then, provided the relationship can be assumed to be of the form

where ε are normally distributed errors with constant variance, a standard nonlinear least squares fitting procedure will produce maximum likelihood estimates for the parameters β (see, for example, Seber & Wild, 1989).

In many ecological problems, the assumption of additive normal error with constant variance in eqn (2.9) will not be reasonable, particularly if y is a measure of abundance. Taylor (1961) showed that the variance in animal abundance is usually related to mean abundance via a power relationship with an exponent close to 2. If Taylor’s power law applies, and the exponent is close to 2, the error may be approximately lognormal. This means that log( y) will be the appropriate dependent variable to use in the nonlinear least squares procedure, and the resulting parameters should be maximum likelihood estimators. Log transformations should be familiar to most readers from linear regression, where they are employed both to linearize relationships and to normalize errors. In linear regression, there may be a conflict between the most appropriate transformation to normalize errors and the most appropriate transformation to linearize the relationship. If nonlinear regression is used, this conflict does not arise, and you are free to select the transformation that does the best job of producing normally distributed residuals with approximately equal variance. Standard residual analysis should always follow a nonlinear regression, as it should follow any regression.

Selecting the most appropriate model from several alternatives is rather harder with nonlinear regression than it is for linear regression. This is partially because statistical theory is better developed for linear models, but also because the potential range of possible models is far greater, once nonlinear relationships are allowed. Some software may produce an R2 analogous to that from linear regressions. Although this statistic has some use, its interpretation is not straightforward in nonlinear regression (Ryan, 1997). Comparing nested models can done with some semblance of statistical rigour. (Recall that two models are nested if the simpler one can be obtained from the more complex one by fixing one or more parameters at particular values.)

Suppose we are comparing a full model, with f parameters estimated from