Testing causal structure in the biogeography of avian extinctions on oceanic islands


*E-mail: karels@csun.edu


In their recent comment in this journal, T. M. Blackburn and colleagues called into question the use of standardized partial regression modelling (also called path analysis and structural equation modelling) when null expectations for regression coefficients are not zero. Here, we answer their critique by showing how randomization can be used to illuminate and interpret causal modelling in analyses that have non-zero expectations. Causal modelling is a powerful tool that can yield novel insights in biogeography when properly interpreted.

Blackburn et al. (2009) presented a critique of our causal modelling of avian extinctions on oceanic islands. We had suggested using causal modelling as an alternative to general linear mixed modelling (Blackburn et al., 2004) because the variables involved in considering these avian extinctions (namely, island area and isolation, avifauna size, number of introduced mammalian predators, number of avian extinctions) have a causal order that can refine and inform the analysis (Li, 1975). In fact, using the causal order of variables (for example, island area and isolation can influence the size of bird faunas which then influences the number of extinctions, but the reverse does not occur) allows for indirect influences to be quantified and provides a more accurate quantification of direct influences on the number of extinctions on islands. Like Blackburn et al. (2004), Karels et al. (2008) also found both area and number of introduced mammalian predators to be important. However, causal modelling showed that the effect of area on extinctions is indirect through size of the avifauna and number of introduced predators, in addition to direct and indirect effects of island isolation. Generalized linear mixed modelling (GLMM) used by Blackburn et al. (2009) does not, and cannot, reveal the orderly structure of complex systems such as when the effects of a factor (e.g. area) on the variable of interest (e.g. number of extinctions) is mediated through another factor (e.g. number of introduced predators). Blackburn et al. (2009) suggested that our causal model is not interpretable, and this conclusion could easily encourage one to ignore a valuable technique that provides important insights.

First, Blackburn et al. (2009) indicate that we examined the number of extinctions on islands, while their analyses are of extinction probability. This is true, but both their analyses and ours include all the variables of interest. They estimated the probability of extinction by dividing the number of species that have gone extinct on an island by the number of species in the original avifauna (Blackburn et al., 2004). We included both the size of the original avifauna and the number of species that have gone extinct in our path model (Fig. 1; Karels et al., 2008). One problem with using a ratio of two variables in an analysis is that other variables, potential causal factors in the environment in this case, can influence both the numerator and the denominator of the calculated proportion, but it may be difficult to ascertain where the influence lies. Path analysis can quantify such confounded causal processes separately (Escudero et al., 2000), and thus can produce a clearer view of influences on the two variables in the proportion. This beneficial property comes directly from the structure of the path diagram that models the relationships among variables.

Figure 1.

 Path model of extinction of island avifauna assuming a random extinction rate of 0.114 for each island’s avifauna. The number of extinct species for each island was determined as a random binomial variable where the number of original species represented the number of independent trials where a species was randomly determined as extinct with a probability of 0.114. This was performed 50 times for each island, thus creating 50 randomly determined sets of extinction values. The path model was run 50 times, changing only the data used for the number of extinctions. The data sets were consistent with the structure of the model (χ2 = 0.03, d.f. = 1, P = 0.87) with 62% of the variation in number of extinctions explained mostly by ‘No. of Avifauna’, as would be expected if extinction were random. Path coefficients are shown as the mean with ± 1.96 standard deviations of coefficients from 50 runs of the model. Endogenous paths and correlations do not vary between model runs since those relationships remain constant when only the variable ‘No. of Extinctions’ is randomized.

Blackburn et al. (2009) suggest that if the relationship between variables is as expected at random, then it is uninteresting. In fact, if extinctions of bird species on islands were as expected at random with respect to species richness it would be fairly easy and accurate to predict the number of extinctions from the size of the original bird fauna, a useful and interesting prediction to say the least. At random, one would expect to see a strong association of the size of the bird fauna and the number of extinctions. This would suggest that ‘a random draw of species from the islands’ provides an explanation for the number of species that have gone extinct. However, the size of the avifauna has a significantly poor association with the number of extinctions (Karels et al., 2008). This means that something is causing deviation from the random expectation, an interesting result that we suggest may be partly due to heavily impacted islands being large enough to support agriculture, which in turn may promote successful colonizations of introduced mammals (Duncan & Forsyth, 2006), and far enough from the mainland that isolation promotes more intense agriculture and deforestation (Didham et al., 2005) and dependence on indigenous avifauna (Steadman & Martin, 2003).

To us, quantification of potential influences on extinction is important. Blackburn et al. (2009) state that variation in extinctions among islands is obviously not random, because different islands have different extinction probabilities. They suggest examination of their Fig. 1A from Blackburn et al. (2004), which does not quantify a random expectation for either extinction numbers or probabilities. We feel that it is important to quantify effect sizes for the influences of possible factors that affect extinctions, so that the biological importance of influences can be compared and judged. For example, we showed that the degree of isolation of islands had a similar statistical influence on the number of extinctions as did the number of introduced predators (Karels et al., 2008). Island isolation was not supported as a significant influence on extinctions in Blackburn et al.’s (2004) original analysis. An hypothesis or even a null expectation that explained this influence would be interesting. But this question would not appear interesting without the quantitative analysis of Karels et al. (2008).

Blackburn et al. (2009) suggest that the influence of island isolation on the number of extinctions shown by Karels et al. (2008) may not mean anything, because they do not know how the null expectation that we incorporated might influence expectations of influences of other independent variables. However, in path analysis, path (standardized partial regression) coefficients are calculated with other influences in the model held statistically invariant (Li, 1975). Therefore, the strength of one independent variable is not influenced by other independent variables unless collinearity occurs (Petraitis et al., 1996); this was tested and rejected as a major influence within our analyses. Blackburn et al. (2009) also call into question our use of model assumptions of normality. Multivariate normality is the most critical assumption of path analysis (Kline, 2005); this possible source of bias is something that we also tested and rejected (Karels et al., 2008). Furthermore, models in which the data do not conform to multivariate normality are more likely to be rejected (Kline, 2005). The fit of the data to our model was very good (χ2 = 0.03, d.f. = 1, P = 0.88), which further supports the idea that violation of the assumption of normality was not an issue in our analysis.

It is possible to evaluate further the null expectation for the relationship of island isolation on number of extinctions of avifauna on oceanic islands, and in the process also to examine whether there are other null expectations that should be taken into account. To illustrate this point, we took our original model (Fig. 1 in Karels et al., 2008), which uses hypothesized causal relationships among variables that might explain the number of extinctions on islands (see the placement of variables in Fig. 1). Following similar protocols to those we used in Trevino et al. (2008), we used PopTools (Hood, 2000) to generate randomly expected numbers of extinct species for each of the islands and assuming a binomial distribution, using the numbers of species in the original avifaunas on each island as the number of binomial trials, with an average probability of extinction of 0.114 (determined from the data set). In other words, we performed random draws of the number of species to be counted as ‘extinct’ for each island not exceeding the number of original species (extant and extinct) on that island. We repeated this process 50 times to generate 50 data sets with random extinction, but keeping all other variables at their real values. We then computed all of the path coefficients and error terms in our path model for each of these 50 data sets using the structural equation program amos 16 (Arbuckle, 2007). From the 50 runs of the model, we calculated the mean and standard deviation for each path coefficient and residual error estimate (unexplained variance).

Not surprisingly, there was a strong and significant standardized path coefficient (p) for the influence of the size of the avifaunas on the simulated random number of extinctions (p = 0.70; Fig. 1) which was significantly greater (χ2 = 4.3, d.f. = 1, P < 0.01) than the coefficient we previously found with real numbers of extinctions. This shows that the null expectation for this relationship is strongly positive. The standardized path coefficient of 0.70 under random extinction is not significantly different from the unstandardized path of 0.88 that we reported previously in Karels et al. (2008), despite differences in how they were determined. The unstandardized expected random coefficient of 0.88 was used to set the random expectation in our path model in Karels et al. (2008). This corresponds to a standardized (path) coefficient of 0.65 that is within the 95% confidence limits (0.59–0.81) of the path coefficient of 0.70 shown in Fig. 1. Thus, the two methods of identifying null expectations for an association of size of the avifaunas and number of extinctions were consistent.

Our randomization analysis also shows that the direct influences of island area, island isolation and number of introduced predators are all not significantly different from zero, which is what we would expect if extinctions were mostly explained by the size of the original avifauna. This strengthens our acceptance of zero as a reasonable null expectation for a non-significant path coefficient. As Blackburn et al. (2009) suggested from our previous analysis (Karels et al., 2008), the relationship between the size of the avifaunas and the number of extinctions is lower than expected under random extinctions because extinction is non-random. Our path model in Karels et al. (2008), when compared with the null expectation in our analysis here (Fig. 1), supports this suggestion, but also shows that non-random extinction can partially be explained by the significantly greater than expected effects on extinction of island isolation and introduced predators. Area, while having no significant direct effect, still has strong indirect effects on the size of the avifaunas, which is not surprising, and strong effects on number of introduced predators, which is highly important when considering the number of extinctions. However, there are clearly unmeasured variables that also account for the non-random nature of extinction on oceanic islands, as our model in Karels et al. (2008) indicates that 70% of the variation in extinction remains unexplained.

Although our evaluation of null expectations has not altered our conclusion that island isolation has a slightly, but not significantly, stronger direct influence than the number of introduced mammalian predators on number of extinctions (Karels et al., 2008), we have found this to be a useful thought exercise. We suggest that doing this sort of analysis a priori would be a useful approach, both to identify possible null expectations of relationships that differ significantly from zero and to serve as a check on those direct relationships that can be tested against a null expectation of zero.

In conclusion, on average 62% of the variation in the number of extinctions on islands could be explained primarily by the size of the original avifauna if extinction were random. In our original analysis of the real data (Karels et al., 2008), all of the variables explained only 30% of the variation in the number of extinctions. The pattern of extinctions of island avifaunas has considerable non-random structure and this is truly ‘something of interest’ and importance that requires further study.

Editor: Lisa Manne