Correction to: ‘Imputation of body mass’ (2021) by Turvey et al.

We previously regressed the survival of non-volant Caribbean Holocene mammals as a function of body mass and island characteristics [1]; body mass was lacking for one species, and island covariates were missing for several islands. We believed covariate values had been imputed, but we were wrong as we will explain in this correction. Several approaches have been developed to impute missing covariates in hierarchical Bayesian modelling and all involve specifying a sub-model to impute the missing covariate based on, for example, geography [2], or another covariate [3]. In the hierarchical Bayesian approach that we previously used [1], integrated nested Laplacian approximations (INLAs), the posterior predictive distribution can be used to impute values of the response variable [3] (in our case, binary survival or extinction in the logit scale). But imputing covariates requires specifying appropriate sub-models based on sources of information that relate to the covariate. We conflated one approach—the posterior prediction of the response—with covariate imputation. Consequently, the covariate values we attempted to impute were incorrect [1]. As the results from incorrectly imputed covariates were reported, those could be wrong too. We therefore applied the correct method of imputation for the critical covariate, body mass, to our data. After undertaking those analyses, all key conclusions of our paper hold [1].

There were missing covariates for both body mass and island characteristics, but we lack a sound basis to impute island values. Of 118 islands, 87 lacked dates of first human arrival, 86 lacked hurricane metrics and 78 lacked human impact values. These missing covariates were not randomly distributed and instead clustered, with 68 islands lacking all 3 covariates. We therefore focused imputation on a phylogenetic model of body mass as a normally distributed variable,

$$\begin{array}{r}{y}_i\sim \mathrm{G}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{a}\mathrm{n}\left(\mu ,{\sigma }^2\right),\end{array}$$

in which $\mu$ is the mean, and σ² is the variance. Given body mass observations per species y_i

$$\begin{array}{r} y_i= {\beta }_0+{\beta }_1+{\beta }_2+ {\epsilon }_i,\end{array}$$

with the global intercept given by ${\beta }_0$, and non-phylogenetic species-specific intercepts given by

$$\begin{array}{r}{\beta }_1 \sim \mathrm{G}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{a}\mathrm{n}\left(0, {\sigma }_0^2{I}_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{s}}\right),\end{array}$$

the phylogenetic variance–covariance $V_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{s}}$ also generates species-specific effects modelled by

$$\begin{array}{r}{\beta }_2 \sim \mathrm{G}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{a}\mathrm{n}\left(0, {\sigma }_0^2{V}_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{s}}\right).\end{array}$$

This last set of species-specific intercepts accounts for the correlated structure of errors and non-independence in observations through the phylogeny. We implemented these models with a single phylogeny from [4] and using the R package phyr v1.1.2 [5]. Following [3], we then generated 100 approximated posterior samples from the fitted model using the inla.posterior.sample routine in INLA [6]. These provide a frequency distribution of body mass imputed for the one species that lacked this value: Hexolobodontinae sp. nov., summarized by a low mode (1.29 kg), intermediate median (2.47 kg) and a high mean (5.3 kg), indicating a long tail in the posterior sample.

As in Rayfield, Cooke [7], we then modelled the probability of survival for each species-by-island observation by looping each imputed body mass value across the sample of 100 phylogenies from [4] such that each of 100 models comprised a unique imputed body mass and phylogeny combination. These results thus propagate the error of the imputed body mass to the extinction models. We used custom scripts to summarize those new results, shown in figure 1.

Figure 1.

Posterior estimates and high-probability intervals for sample-wide model coefficients including all non-collinear covariates and a single imputed body mass.

Compared to published results [1], all significant covariates are confirmed (figure 1). Volcano presence, however, was originally a negative covariate of survival probability (figure 2), but now this relationship is both stronger and statistically supported. Nearly identical results were obtained in analyses without the imputed body mass value (not shown).

Figure 2.

Survival probability as a function of body mass for islands at the most recent quartile (recent), median (intermediate) and top quartile (ancient) of first human arrival, with all other covariates corresponding to their sample means. Bottom ticks show jittered species body masses, with imputed body mass values (in order: mode, median, mean) in red.

To determine how the relationship between body mass and extinction probability had changed, we replotted predicted estimates from the phylogenetic regression (including the new values imputed for Hexolobodontinae sp. nov.). Results for these analyses were nearly identical to our original results [1], and patterns of intermediate body mass survival and interactions with first human arrivals were the same.

Results from our re-analyses, including a phylogenetic imputation sub-model for body mass, are consistent with the findings first reported in our paper [1]. As the incorrect imputations were part of our code, we have updated the GitHub repository to reflect these changes [8].

Data accessibility

Data are available on GitHub [8].