Supplementary Figures

Supplementary Figure 1. Model order vs. mean square error. Model mean square error (MSE) calculated from the highest weighted model, or model-averaged coefficients from all equally weighted top models (∆QICu<2), for each model order (one to 15 covariates) included in the model selection procedure.


Supplementary
. Timing of population extirpations. Cumulative distribution plots show the timespan over which native vertebrate population extirpations took place, for all populations with associated information about the known or estimated year of extirpation (volant birds, n=190; non-volant birds, n=7; bats, n=20; non-volant mammals, n=45; amphibians, n=32; reptiles, n=46). A single point is shown in cases where multiple populations of a native species group were declared extirpated in the same year; this is consistent with the GEE model formulation, which grouped the data by island. Dotted lines show the 25 th , 50 th , and 75 th percentiles for the number of years to extirpation for each native species group.

Supplementary Tables
Supplementary Table 1. Summary data for all islands in the dataset (n=1,024). Min

Native and non-native island species dataset
Island breeding records for threatened and extinct terrestrial vertebrates in the Threatened Island Biodiversity (TIB) Database were linked to geographic island locations using the Global Islands Database (GID) 1 , a spatial dataset of more than 180,000 islands larger than 1 km 2 . When an island's identity and/or location were unclear, we used Google Earth 2 , geographic and political information found in the literature, and expert communications to obtain and validate island information. For islands smaller than 1 km 2 , we used this information to manually add additional breeding islands to the spatial dataset.
We populated the TIB Database using information from peer-reviewed literature, reports, and expert communications to determine whether a species currently persists on or has been extirpated from each of its historic breeding islands (see [12] for details). We excluded cases where a species' historic or current breeding status on an island was unknown or where records were deemed unreliable. We defined extirpation to include cases in which: (1) a species has not been found during surveys on one of its historic breeding islands since 1990, or (2) species experts or the IUCN Red List account define a species as being "potentially extirpated" from an island. We classified these "likely" extirpations as extirpation rather than persistence because: (1) there is reasonable evidence that these species have been extirpated from an island, and (2) if they do persist there, the populations are presumably at such low densities that the probability of long-term persistence is low 3 .
The above assumption almost certainly leads to the misclassification of some extant populations as extirpated, as it is more difficult to confirm a species' absence on an island than its presence.
These misclassifications could potentially bias the results of our analysis. Therefore, we reran the analysis with all "potentially extirpated" populations classified as "present." The original dataset (as presented in the main text) contained 651 extirpations out of a total 2,656 native populations (24.5% of populations extirpated). 144 of the populations we defined as extirpated were in the "potentially extirpated" category. When we reclassified these populations as extant, the modified dataset had 507 extirpations (19% of populations extirpated). We built a model using this modified dataset and the same model selection procedure presented in the main text. The resulting model was very similar to the one presented in the main text: it included the same main effects, with the addition of average annual temperature. Most of the interactions were also the same, with the exception that the class/volancy*felid interaction term was replaced by class/volancy*mustelid/mongoose. We consider these minor changes, and based on this and the reasoning above, we conclude that our assumption is the appropriate choice for this study.
We compiled data on two biological traits of native threatened species that could influence their probability of extinction in general and their vulnerability to introduced mammals in particular: flight ability and body mass. All bats were classified as volant, and all other mammals as nonvolant. We classified birds as volant or non-volant using BirdLife International's species factsheets 4 . Many other intrinsic traits of native species, such as home range size, ground vs.
arboreal nesting, and migration patterns likely also influence the vulnerability of insular species to invasive mammals, but a lack of sufficient data on these variables for all species in our data precluded us from investigating them. We deemed body mass and flight ability to be fundamentally important intrinsic factors because numerous studies have shown them to be primary determinants of species extinction risk on islands [5][6][7][8][9] . To ensure that the inclusion of volancy in addition to taxonomic class did not greatly affect our results, we also ran the analysis with just the four native taxonomic groups (amphibians, reptiles, birds, and mammals). The amphibians and reptiles, the standard metric for body size is snout-vent length (SVL), and body mass data were not available for most species. Thus, for species lacking information on body mass, we used SVL as well as total animal length (TL) when this was provided. We used allometric equations developed for each taxon to convert the minimum and maximum SVL or TL to body mass estimates 10 , and then took the arithmetic mean of the minimum and maximum mass estimates to obtain a single body mass estimate for each species. When information on the body mass (mammals and birds) or SVL (amphibians and reptiles) of a particular species was unavailable, we used either: (1) the body mass or SVL of a similarly sized congener, when such information was available, or (2) the mean value of the body masses or SVLs of many similarly sized congeners.
Body mass data were unavailable for some native species that are extinct or are very rare and have never been weighed. We excluded these species from the analysis because the model selection procedure can only accommodate complete rows of data. Of a total 1,257 native threatened or extinct species in the original dataset, mass data were unavailable for 53 species (4%). Despite this being a relatively small proportion of the data, it is possible that a taxonomic bias in the omitted species could influence the results. In particular, most of the species for which mass data were unavailable were extinct birds and mammals. Only two amphibian and two reptile species lacked mass data, because for these taxa we calculated body mass from animal length, which can be obtained from preserved specimens and are thus available even for most extinct species. We ran the full model selection procedure on the subset of data for which body mass data were available, and threatened species body mass was not selected in any of the top models, indicating that omitting it from the analysis did not affect the results.
We recorded the presence or absence of all non-native mammals on islands for which we could find information 11,12 . We identified ~160 invasive mammals that had become established on islands and classified them into 12 groups based on a combination of feeding guild, taxonomy, body size, and other traits (Table 1 and Supplementary Table 2). Our goal was to achieve a balance between creating as few groups as possible to maximize the representation of each group across islands (i.e. sample size within strata) and as many groups as needed to distinguish between impacts of different mammal types. For example, we grouped primates separately from other omnivores such as rodents, raccoons, and possums due to their primarily arboreal nature and unique behavior. We divided carnivores into three groups-felids, canids, and mustelids/mongooses-because we predicted they would have different impacts on native species based on life history, behavior (e.g. solitary or pack hunters), and home range size.
Although mongooses are most closely related to civets and genets (family Herpestidae), which we grouped with other medium-sized omnivores based on diet, we grouped mongooses with mustelids because the common island invasive species in both these groups are largely or entirely carnivorous. We separated rodents from the other omnivore groups due to their generally smaller body sizes, their high densities and population growth rates on islands, and their broad omnivorous diets. In addition, rodents are the most common target of invasive mammal eradication efforts 13 , so their unique impacts are particularly important to understand from a conservation perspective. We classified rats and mice separately to tease apart potentially different impacts of these groups due to differences in body size and behavior, and because rats are targeted for eradication more frequently than mice. The two most ubiquitous rat species on islands, Rattus rattus and R. norvegicus, differ in climbing ability (strong and weak climbers, respectively), and may therefore affect arboreal and terrestrial island species differently.
However, the particular species of rat present on many of the islands in our dataset is unknown, making it impossible to distinguish their individual impacts.

Island attributes dataset
For most islands, we obtained data on island climate and elevation from the publically available BioClim Dataset (resolution 30 seconds) 14 . We obtained a single estimate for each climatic variable per island by calculating the arithmetic mean of the center points of all raster grid cells located within the boundary of an island polygon. A disadvantage of this approach is that more climatic information is lost for larger islands than for small ones, particularly for islands that span large latitudinal gradients, are topographically complex, or have distinct wet and dry regions. However, our distribution data for native and introduced species were at the scale of whole islands, thus the island was the geographic unit of our analysis and we used island-level values for each geographic and climatic variable.
Some islands in the TIB Database had no overlapping raster grid cells in the BioClim climate or elevation datasets. For islands located less than 100 km from the nearest continent or island, we used climate values from the nearest grid cell in the BioClim raster. There were no islands with climate data within 100 km of the Bounty Islands (New Zealand), Pitcairn and Henderson Islands (UK Overseas Territories), or the islands of French Polynesia, so we obtained climate data for these groups from outside data sources [15][16][17] . For most islands lacking GIS elevation data, we obtained maximum elevation values from the literature [18][19][20][21][22][23][24] and from Google Earth 2 . We were unable to find elevation data for some small islands using any of these sources, and we estimated the maximum elevations of these islands using linear interpolation. Specifically, for each island lacking elevation data, we identified all the islands in our dataset that are located in the same archipelago and share the same geologic history (and therefore likely have similar topographic characteristics) as that island. We used the area-elevation relationship of the other islands in the archipelago to interpolate the island's maximum elevation based on its area.
There were nine islands in the dataset for which we were unable to find or reasonably estimate temperature, precipitation, or temperature seasonality. We excluded these islands because the model selection procedure can only analyze complete rows of data. However, these omitted islands represent a very small proportion of the total dataset and therefore are unlikely to influence the results of our analysis. Each of these nine islands contained one threatened vertebrate species; thus, we excluded nine native populations on nine islands, out of a total 2,657 native populations on 1,024 islands in the full dataset. The small proportions of omitted data due to missing climate information (<0.5% of native populations and <1% of islands) is unlikely to influence the results of the analysis beyond having minute effects on model coefficients.
Island isolation is a primary determinant of species immigration rates and equilibrium species richness 25 . However, we did not include isolation as a covariate in our model for several reasons.
First, island biogeography theory makes no predictions about the effect of isolation on short-term extinction risk, but rather focuses on long-term equilibrium species richness due to intrinsic extinction and immigration rates. Second, the only other global study that measured correlates of extinction risk specifically for threatened insular species found no relationship between isolation and either endemism rates or the severity of impacts from invasive mammals 26 . Finally, island isolation is unlikely to be related to invasion pressure because most mammal introductions resulted from intentional or unintentional human actions. Rats and mice, for example, are common stowaways on ships, and were introduced to extremely remote islands via landings and shipwrecks throughout the exploration and whaling periods of the last 500 years and earlier 27,28 .
Therefore, we did not consider isolation as a driver of short-term population extinction risk for the species and islands in our dataset.
The ecology of some very large islands in our dataset may, in some ways, resemble that of continents more than that of small islands; if this is true, including these islands in our analysis could obscure patterns that are unique to smaller islands. We kept even the largest islands (e.g. New Guinea, Borneo, Madagascar) in our analysis for three main reasons. First, like many smaller islands, even these very large islands contain unique radiations of taxa that evolved in conditions different from those found on continents, often including a lack or paucity of native mammals or terrestrial predators. Thus, we hypothesized that native species on large islands would show similar patterns of extirpation and persistence, with respect to our model covariates, to those on smaller islands. Second, the only other global analysis of threats to insular species from invasive mammals also included many of the world's largest islands 26  Nevertheless, to ensure that including the large islands did not strongly influence our results, we ran the analysis with the nine largest islands in the dataset excluded (New Zealand's North and South Islands, Java, Sulawesi, Honshu, Sumatra, Madagascar, Borneo, and New Guinea). We used the same model selection procedure as described in the main text, and the resulting model contained 11 main effects and 9 interaction terms. In addition to the main effects and interaction terms present in the final model presented in the main text, this model contained three additional main effects (area-elevation residuals, introduced canids, and temperature) and four interaction terms: temperature*mustelid/mongoose, class/volancy*mustelid/mongoose, class/volancy*canid, class/volancy*temperature. We speculate that the inclusion of more terms and interactions in this model is due to a smaller effect of island area, which allows some of the other variables and interactions to have greater importance in the model. However, interpreting a model of this complexity is not feasible because the individual and combined roles of each term and interaction cannot be untangled. An analysis focusing on a smaller subset of islands -based on size or any other geographic characteristic -would be an interesting follow-up to our study, to better understand the nuances of different variables' impacts and interactions at smaller spatial scales. For this global-scale study, however, we considered the very large islands to be important components of our analysis and results.
Extirpation patterns may differ across archipelagos, island regions, or ocean basins due to intrinsic factors (e.g. species assemblages in a region may share a more recent taxonomic history and may therefore be similarly vulnerable to anthropogenic threats) and/or external factors (e.g. different history of colonization by humans or invasive mammals). However, we did not include geographic region as a factor in our model because it is not directly relevant to the questions our study was intended to address. We chose a set of abiotic predictor variables and interactions for which we had specific hypotheses about their effects on extirpation patterns (Tables 1 and 2), and geographic region would not provide further ecological or mechanistic insights into the factors driving extirpations. For example, our hypothesis that insular non-volant birds are vulnerable to predation by invasive rats or cats, or that native omnivores would be negatively affected by competition with invasive omnivores, applies to islands in all regions of the globe.
These impacts may be mediated by island topography or climate, which we included in the analysis with relevant hypotheses, but are not likely mediated by geographic region. In addition, defining boundaries between archipelagos or island regions would be tricky and could be prone to bias or arbitrary delineation of regions. For example, the islands of the Pacific Ocean could be divided into any number of groups based on distance to the nearest continent, geologic or biological affinity with other islands or archipelagos, latitude or longitude, or larger-scale ocean basin. Thus, any choice of island regions could bias the results of our analysis and would be unlikely to advance our goal of understanding extirpation patterns and invasive mammal impacts at a global scale.
The TIB Database includes information on human habitation for most islands. Precise population data are not available for many islands, so each island is classified into one of six human population categories using whatever information is available. Islands are classified as uninhabited or as having permanent human populations numbering 1-10, 11-100, 101-1,000, 1001-10,000, or >10,000. For many islands, we were unable to find data specific enough to classify them into any of these categories. However, in all cases we were able to asses whether humans likely maintain permanent settlements on islands by conducting Web searches (e.g. many islands are tourist destinations with hotels and other infrastructure) and searching Google Earth images for buildings, roads, agriculture, or other signs of human habitation. Based on this information, we supplemented the information obtained from the TIB Database and classified all islands in the dataset as either inhabited or uninhabited.

Extirpation probability analysis
We constructed logistic models to investigate global patterns of population extirpation and persistence for threatened island vertebrates. The binary response variable was the extirpation or persistence of a native species on one island. Many islands contain multiple native threatened species, each with its own record of persistence or extirpation, while others have only one threatened species. Similarly, some native species occur (or occurred historically) on many islands, while others are (or were) single-island endemics.
We were unable to document the timing of extirpation events with enough precision to include time as a factor in our analysis, or to use our model to predict the timing of future extirpations.
About half the extirpation records in our data had associated timing information, which typically consisted of the last year a species was documented on an island and the year the species was declared or presumed extirpated. For populations with sufficient data, we calculated the timespan over which extirpations occurred (i.e. the number of years between these two events), which ranged from <5 years to ~500 years (all species included in the IUCN Red List persisted until after 1500 AD, thus the maximum time to extirpation was 510 years) ( Supplementary Fig. 5).
This wide range of values, along with the lack of temporal data for half of the extirpations and the geographic and taxonomic bias inherent in historic survey data, prevented us from incorporating time in the analysis.