O R I G I NA L A RT I C L E doi:10.1111/evo.12435

JOINED AT THE HIP: LINKED CHARACTERS AND THE PROBLEM OF MISSING DATA IN STUDIES OF DISPARITY Andrew J. Smith,1,2,∗ Michael V. Rosario,1,3,∗ Thomas P. Eiting,1 and Elizabeth R. Dumont4 1

Graduate Program in Organismic and Evolutionary Biology, University of Massachusetts Amherst, Amherst,

Massachusetts 01003 2

E-mail: [email protected]

3

Department of Biology, Duke University, Durham, North Carolina 27708

4

Biology Department, University of Massachusetts Amherst, Amherst, Massachusetts 01003

Received September 11, 2013 Accepted April 13, 2014 Paleontological investigations into morphological diversity, or disparity, are often confronted with large amounts of missing data. We illustrate how missing discrete data affect disparity using a novel simulation for removing data based on parameters from published datasets that contain both extinct and extant taxa. We develop an algorithm that assesses the distribution of missing characters in extinct taxa, and simulates data loss by applying that distribution to extant taxa. We term this technique “linkage.” We compare differences in disparity metrics and ordination spaces produced by linkage and random character removal. When we incorporated linkage among characters, disparity metrics declined and ordination spaces shrank at a slower rate with increasing missing data, indicating that correlations among characters govern the sensitivity of disparity analysis. We also present and test a new disparity method that uses the linkage algorithm to correct for the bias caused by missing data. We equalized proportions of missing data among time bins before calculating disparity, and found that estimates of disparity changed when missing data were taken into account. By removing the bias of missing data, we can gain new insights into the morphological evolution of organisms and highlight the detrimental effects of missing data on disparity analysis. KEY WORDS:

Diversity, morphology, simulation, taphonomy.

Morphological diversity, or disparity, refers to the range of variation in form within a group of organisms. Disparity is a common measure of biodiversity in paleontological studies because it is relatively robust to uneven sampling of the fossil record and errors in species counts (Horner and Goodwin 2009; Bapst et al. 2012; Butler et al. 2012). Disparity analyses based on discrete characters can be used to evaluate changes in patterns of morphological evolution through time (Hughes et al. 2013). Disparity analysis has provided key insights into evolutionary radiation (Ruta et al. 2006; Prentice et al. 2011), evolutionary responses to mass ex∗ These

authors contributed equally to this work.

Archived in on-line supplementary information and Dryad.

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tinction (Friedman 2009; Thorne et al. 2011), and the evolution of morphological novelties (Anderson et al. 2011). One problem when including fossils in disparity analyses is that, except in cases of exceptional preservation (e.g., Lagerst¨atte deposits), few specimens are anatomically complete (Wiens 2003). This is problematic because missing data reduce the number of pairwise comparisons that can be made between taxa. Fewer pairwise comparisons can change the position of taxa in morphospace, and potentially decrease disparity relative to a complete dataset. The fragmentary nature of most fossil specimens impedes the ability to distinguish evolutionary patterns of morphological change from artificial patterns caused by missing data. Understanding how missing data affect analyses of disparity

C 2014 The Society for the Study of Evolution. 2014 The Author(s). Evolution  Evolution 68-8: 2386–2400

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is essential to critically assessing morphological evolution through time. The problem of missing data has been addressed using three main approaches: removing taxa from analyses if they possess excessive amounts of missing data (e.g., Wills et al. 1994; Brusatte et al. 2008b; Benson et al. 2012), replacing missing data with a weighted average value (Wills 2001), or performing some type of phylogenetic correction (Brusatte et al. 2011; Butler et al. 2012). Discarding taxa is common, especially if certain specimens are very incomplete. However, this comes at the cost of removing organisms that may exhibit novel morphologies, which could affect the range of disparity predicted for a clade. Replacing missing data with a weighted mean value can lead to spurious disparity outputs when dealing with very incomplete datasets. In this case, the first axis of a morphospace can reflect the amount of character data a taxon possesses, with those at the negative end being datapoor, and those taxa at the positive end being data-rich (Butler et al. 2012). The final approach involves the use of a phylogeny to replace missing data with an estimated state, or generate additional “ancestor” taxa from an internal node (Brusatte et al. 2011; Butler et al. 2012). Reconstructing missing data reduces the scatter of taxa in morphospace, causing them to form multiple tight clusters with less overlap between clades (Butler et al. 2012). Despite these studies, to our knowledge only one study has investigated whether and how missing data affect disparity analyses. Ciampaglio et al. (2001) investigated how several common disparity metrics responded to randomly removing increasing amounts of data. They found that removing up to 25% of the characters from a simulated dataset, and 30% of the characters from a fossil crinoid dataset, had little effect on disparity metrics. However, many character matrices in disparity studies are more than 25% incomplete (e.g., Ciampaglio et al. 2001; Brusatte et al. 2008b; Prentice et al. 2011; Thorne et al. 2011; Butler et al. 2012: 31%, 37%, 44%, 37%, 52%, respectively). We do not know how disparity analyses respond when levels of missing data surpass 25%. Also, the Ciampaglio et al. (2001) study simulated taxa with a maximum of 10 characters that were generated using a random walk procedure. Disparity studies typically contain >50 characters, so analyses using small character matrices may not reveal patterns that typify larger datasets. When considering how to study missing data, it is important to acknowledge that the fossilization of anatomical characters is unlikely to be random (Sansom et al. 2010). The probability of fossil preservation differs among organisms, geographical locations, and intervals of geological time (Benton et al. 2000; Eiting and Gunnell 2009; Wagner and Marcot 2013). These factors interact with taphonomic processes to influence which anatomical regions are preserved and which are lost (Moore 2012; Smith 2012). For example, vertebrate fossils from the Big Horn basin of North America are often disarticulated and incomplete because of

microbial action and scavenging by predators (Bown et al. 1994). The probability of an anatomical structure being preserved during fossilization also differs depending on its physical properties and size (Briggs 1995; Behrensmeyer et al. 2000; Sansom et al. 2010). For example, mammal teeth are often preserved due to the hardness of the enamel (MacFadden et al. 2010), whereas the small and fragile phalanges of most tetrapods are rarely preserved (Shipman 1981). Similarly, the presence or absence of characters that are in close anatomical proximity to one another or fused together are expected be correlated (Behrensmeyer et al. 2000; Grupe 2007). For example, there is likely to be a strong association between the presence or absence of dental and mandibular characters, as teeth sit in alveoli within the mandible. Conversely, we might expect dental and limb characters to be preserved together less often owing to the larger physical distance between them. Taphonomic studies have verified that suites of characters tend to be found together (Grupe 2007; Sansom et al. 2010, 2011). Suites of covarying characters are likely to contribute less to overall disparity than the same number of characters that are randomly distributed throughout the body. This is because characters that covary together are likely to be functionally and developmentally related, and evolving at similar rates (Clarke and Middleton 2008). Therefore, the covariation of missing data in a fossil may be a critical factor to consider in studies of morphological disparity. In this study, we use the term “linkage” to describe the correlation in presence or absence of morphological characters. This is analogous to the way that “linkage” is used to describe the higher likelihood of two alleles being inherited together when they are closely situated on a chromosome. Here, we examine two key questions: (1) how does the linkage among missing characters affect disparity analyses; and (2) how can we lessen the impact of missing data on disparity analyses? To address the first question we use 10 discrete character matrices from cladistic studies of a variety of vertebrate and invertebrate organisms. Each matrix contains a combination of fossil and extant taxa. We developed an algorithm that extracts the covariation of character presence/absence, or linkage, based on the fossil subset of each matrix. The algorithm uses this linkage structure to simulate data loss among extant taxa, iteratively removing larger proportions of data. The inclusion of presence/absence data from matrices of extinct taxa allow us to introduce taphonomic patterns into the character removal process without having to make a priori assumptions about how missing data are distributed. We compare the relative positions of taxa in morphospace and the resulting disparity analyses using the linkage algorithm to the results of a morphospace and disparity analyses based on a null model that randomly removes characters from each taxon. We predict that sequential removal of data will cause a shrinking of morphospace due to an increase in morphological similarity among taxa. We also predict a decline in all disparity metrics

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relative to complete datasets, given that removing characters results in less available morphology to contribute to variation. Finally, we predict that using the linkage algorithm to remove characters will produce a slower decline in disparity compared to the random algorithm. By removing characters from particular regions of the character matrix, the linkage algorithm will permit more pairwise comparisons between characters while keeping functionally and developmentally associated characters together. Hence, the linkage algorithm can retain more information about the amount of disparity a clade contains. To address the question of how to lessen the impact of missing data in disparity analyses, we present an algorithm that calculates a correction factor. The script applies the linkage algorithm to groups of taxa that are to be compared (in our examples, taxa from different temporal periods), and equalizes the mean proportion of missing data among them. Specifically, it sequentially deletes characters from data-rich groups until their mean proportions of missing data match that of the most data-poor group. In this way our method removes the bias brought about by comparing disparity among groups that contain different amounts of missing data. We demonstrate how this technique operates by applying it to the fossil component of the mammal dataset (Luo et al. 2007), which encompasses mammal evolution during the Mesozoic, and by re-analyzing the pterosaur dataset of Prentice et al. (2011). We assess how missing data may alter our inferences of mammalian and pterosaur disparity through time by comparing results based on a dataset that was corrected for missing data and the original, uncorrected matrix. A difference in disparity between corrected and uncorrected datasets would illustrate that missing data can bias disparity analyses. By removing the bias imposed by missing data, we illustrate the impact that missing data can have on disparity studies.

Materials and Methods DATA REMOVAL ALGORITHMS

To evaluate the impacts of missing data on disparity analyses, we developed algorithms to simulate random and linked character removal from complete datasets. We use the random and linkage algorithms in all of our data analysis, and then we employ the linkage algorithm as part of a correction factor for missing data. There are two steps involved in removing characters. The first is specifying the average number of characters to be removed across all taxa. The second step involves selecting which characters to remove in successive iterations.

bution to approximate a realistic pattern of missing data during simulation. The mean and the variance of a Poisson distribution are defined by the parameter λ. The user assigns a value to λ that matches the number of characters they wish to remove from the dataset. The resulting Poisson distribution contains a mean that is centered on the number of characters the user wishes to remove. The algorithm then randomly selects an integer value from within the newly generated distribution, which denotes how many characters will be lost from the first taxon in the dataset. A second random draw from the same Poisson distribution determines the number of characters to be removed from the second taxon. The process proceeds in the order that the taxa are arranged in the dataset. Random draws from the Poisson distribution continue until all taxa have been assigned a value that determines the number of characters each taxon will lose. At this point, the mean number of characters removed from the entire dataset is approximately equal to the Poisson distribution parameter λ. Because the Poisson distribution has no upper bound, the algorithm contains a cut-off point that is five characters under the total number of characters. This makes it impossible to draw a number that exceeds the total number of characters for that taxon. The second step in the process of removing characters from a complete dataset was determining which character to remove during each successive iteration. This process proceeded differently depending on whether we assumed that character removal was random or linked. Note that the algorithms do not distinguish between inapplicable characters (e.g., a lack of hind-limb characters in crown cetaceans) and missing characters. If a character state cannot be determined, then it is treated as missing. RANDOM REMOVAL ALGORITHM

The random removal algorithm simulates character removal from the extant dataset without any information from the extinct dataset (Fig. 1A). The simulations of character loss from this algorithm are equivalent to approaches used in Ciampaglio et al. 2001. To simulate random character removal, we first select how many characters we wish to remove from each taxon (n) from the total number of characters in each taxon (nc ), as discussed above. To determine which of the characters to remove from the extant dataset, we assigned relative weights to every character using a probability-loss vector (Ploss ). For the random loss algorithm, we represent the condition in which each character is equally probable of being lost by setting Ploss to a vector of 1’s. Once the random algorithm has selected a character from a taxon and removed it, that character can no longer be selected during the remainder of the simulation for that specific taxon (Fig. 1B).

SPECIFYING NUMBER OF CHARACTERS TO REMOVE

As missing data are rarely uniform within a dataset (i.e., some taxa contain more missing data than others), we used a Poisson distri-

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Ploss = {P1 , P2 , P3 , . . . , Pnc } = {1, 1, 1, . . . , 1}.

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Removing characters from the extant dataset using the linkage and random algorithms. Each wedge of a pie chart represents a different character that is associated with a region on the skeleton below the chart. (A) The first character to be removed is determined in different ways depending on the algorithm selected. The random algorithm selects the first character randomly. The linkage algorithm

Figure 1.

selects the first character by weighting the probability of each character by the number of times that character is absent in the extinct dataset. The more often a character is absent, the more likely it is to be selected first. (B) The random algorithm can select any character to be removed from the extant dataset. In this example, following the removal of character 2, the cranium, the algorithm next selects character 8, which is associated with the tail. The algorithm then randomly selects character 4, the ribs, to be lost next. (C) The linkage algorithm selects the first character to be removed based on how often characters are absent in the extinct dataset. Character 8 is removed first as the probability of selection was the highest compared to the other characters. Once character 8, the tail, is removed, the probabilities of selecting remaining characters update depending on which characters are often absent alongside the tail in the extinct dataset. This results in a lower probability that characters 1 (teeth), 2 (skull), and 3 (mandible) would be removed, and a higher probability that characters 6 (hind limbs) and 7 (pelvis) would be lost. In this example, the linkage algorithm selects character 6, the hind limbs, to be removed even though character 7 had a higher probability of being selected next. This highlights the fact that the linkage algorithm does retain an element of stochasticity. Therefore, removing data via the linkage algorithm should be more representative of the probabilistic aspect of the fossil record (Wiens 2003; Sansom et al. 2010, 2011). Once the characters update, the final character to be removed is number 7, the pelvis.

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LINKAGE ALGORITHM

The linkage algorithm is able to use information from an extinct dataset to plan how to remove characters from an extant dataset. This requires two steps: (1) Determine which character is to be removed first from the extant dataset (Fig. 1A). (2) Incorporate covariation (linkage) in character presence/absence based on the fossil dataset into subsequent character removals (Fig. 1C). The first step is achieved by calculating the number of times a character is present in the extinct dataset divided by the number of taxa. This will provide different probability-loss weights to the characters in our dataset. In essence, this means that the more frequently a character is missing, the higher the probability that it will be removed first. We therefore assign Ploss for a dataset with n characters by iterating through each character variable and dividing the number of taxa with that character missing, by the total number of taxa (nt ). This generates a Ploss of probabilities (Pi ) of removal for each character. Ploss = {P1 , P2 , P3 , . . . , Pnc }, where nt −

nt  n=1

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charin

charin = the presence of the ith character in the nth taxon. The second step is achieved by calculating the strength of correlation in presence/absence between two characters in the extinct dataset. Those characters that are often absent together are likely to be removed sequentially. This information can be stored in a covariation matrix (CM) that we can then use to update Ploss as characters are removed during our simulation. To calculate the covariation between two characters, we count the number of within-taxon matches in either presence or absence for all taxa, and divide this count by nt . This symmetric matrix contains information about how the presence or absence of different characters covaries with others. ⎡ ⎤ c(1,1) c(1,2) c(1,3) . . . c(1, j) ⎢ ⎥ ⎢ c(2,1) c(2,2) c(2,3) . . . c(2, j) ⎥ ⎢ ⎥ C M(i, j) = ⎢ · ·· ·· · · ·· ⎥ ⎢ · ⎥ ⎣ · · · · · ⎦ c(i,1) c(i,2) c(i,3) . . . c(i, j) where nt 

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We adapt this to our notation to create the following equation: Pi =

C M(i,C L[nl ]) Pi ∩ P (C L [n l ]) = , P (C L [n l ]) P (C L [n l ])

where nl represents the number of characters that have currently been lost. The algorithm continues to remove characters and update Ploss until the desired number of characters are lost for a single taxon. At this point, Ploss and CM can be reset and characters can be removed from another taxon.

nt

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After calculating Ploss and CM, the algorithm uses these data to create a vector of characters (CL) to be removed from a taxon in the extant dataset. The character is selected by using a weighted sampler with Ploss as the vector of probability weights for each character. The sampler selects the first character to be lost (CL[1]), and then updates Ploss by adjusting the probability weights of each character to reflect the information in the covariation (linkage) matrix. The adjustment of the probability weights is based on conditional probability which states:

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1i f charin = char nj . 0i f charin = char nj

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Principal Coordinate Ordination and Disparity Analysis ORDINATION SPACES

To understand how ordination spaces responded to increasing missing data, we tracked how the relative position of taxa changed with each simulation of data removal. We then directly compared the relative difference of the location of taxa in ordination space between the linkage and random algorithms. This was achieved by first performing a principal coordinate analysis (PCO) on the complete dataset, and then calculating the Euclidean distance between each of the taxa. In this application, PCO reduces the number of variables to a series of axes derived from combinations of variables that explain variation among taxa; the first axis represents the largest contribution to disparity, and each subsequent axis contributes less. We then simulated character removal (as described above) 100 times for each percentage of character loss, calculated PCO on these new datasets, and then calculated the Euclidean distance between taxa. Next, we subtracted each of the simulated Euclidean distance matrices from the original Euclidean distance matrix and squared the result, then took the square root. By summing each cell in the matrix, we return a value that represents the distance each taxon moved from its original position in PCO space. A value of zero would represent no change in the position of taxa in PCO morphospace. A mean and 95% confidence interval is then generated from the 100 distance simulations at each percentage of character loss. A good PCO

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ordination is one that retains relationships among taxa as data are removed. As data are removed, the algorithm that preserves the most information will produce values closer to zero. To visually demonstrate how the relative positions of taxa change in morphospace, we plotted all the taxa from a complete dataset using the first two PCO axes. We then removed characters from our datasets at three intervals, 25%, 50%, and 75%, using both the random and the linage algorithm, and plotted the resulting distributions of taxa. DISPARITY ANALYSIS

We calculated disparity based on a well-established protocol (Wills et al. 1994; Thorne et al. 2011; Brusatte et al. 2012). (1) Calculate a distance matrix from a taxon-by-character matrix. As character data were discrete in all datasets, we used the Manhattan method to calculate taxon–taxon distances. All characters were treated as unordered and equally weighted. (2) Calculate PCO scores for that distance matrix. (3) Bootstrap subsets of the PCO matrix 1000 times. As we removed characters to simulate increasing amounts of missing data, the total number of PCO axes with values of zero increased. By using a constant number of PCO axes for each dataset, all simulations ranging from 0% to 75% missing data contained PCO scores >0 (Table S1). We generated 100 unique taxon-by-character matrices for each percentage of character removal between 0% and 75%. We subjected each character matrix to Manhattan distance calculation and PCO ordination. We then bootstrap re-sampled each PCO matrix 1000 times with replacement to calculate a mean disparity value and 95% confidence intervals for each simulated character matrix (in the main body of the text we show sum of ranges; other metrics are presented in the Supporting Information). We then calculated a final mean from all the 100 disparity means, as well as 95% confidence intervals, to represent each percentage of missing data. The process of generating 100 unique datasets for ordination and disparity analysis was repeated for each percentage from 0% to 75% missing data. We calculated four common disparity metrics for each increment of character removal: sum of ranges, product of ranges, sum of variances, and the product of variances (e.g., Wills et al. 1994; Brusatte et al. 2008a,b, 2011). All product metrics were scaled to the kth root, where k is number of PCO axes. These metrics capture different aspects of disparity and reveal different sensitivities to missing data (Wills et al. 1994; Ciampaglio et al. 2001). In the main body of the text, we present results from a single metric, sum of ranges. Results for other metrics are similar and are discussed in the Supporting Information. Sum of ranges estimates the overall spread of morphological variation within a clade. It is calculated by first subtracting the value of the maximum PCO axis from the minimum for each of the included PCO axes. Then we take the sum of all of these range values across

all axes in the PCO matrix. At 0% missing data, the disparity of each clade represents the best estimate of disparity for that group. As characters are removed, any change in disparity value can be attributed to missing data.

Simulating Character Removal in Extant Datasets DATA REQUIREMENTS

The random and linkage algorithms require a taxon-by-character matrix that can be divided into extinct and extant components. The extinct and extant subsets are coded for identical morphological characters. The extinct component of the dataset consists of a binary matrix, with 0 representing absence of a character, and 1 representing presence. When using the linkage algorithm, the extinct matrix must include some missing data, otherwise the algorithm cannot extract any correlation in character absence or presence from the dataset. The extant component remains unaltered.

DATA MATRICES

We investigated the effects of missing data on disparity analysis using discrete taxon-by-character matrices that included both fossil and extant species. We used matrices from 10 published studies, which included datasets from vertebrates and invertebrates (Table S1). These datasets varied in the relative number of extant and extinct taxa and in the number of characters present. Gaps (characters that are not present in all species) and polymorphic states (different states in different species) were coded as missing. We applied the random and linkage character removal algorithms to the 10 extant datasets. As data were removed, character matrices were saved for subsequent ordination and disparity calculation at 1% increments, from 0% to 75% average missing data (i.e., the average of missing data within species across the entire matrix). We repeated the process of simulating character removal 100 times in each extant dataset at every 1% increment. Because both our algorithms were based on random draws, each character matrix had a slightly different distribution of missing data among taxa but they contained the same numbers of pairwise comparisons. Due to space constraints, we only discuss datasets of mammals (Luo et al. 2007), birds (Smith 2010), and arthropods (Rota-Stabelli et al. 2011) here; results from analyses of the remaining seven datasets can be found in the Supporting Information. We focus on these datasets to demonstrate the behavior of our algorithm under different numbers of characters and taxa. The mammal matrix contained 76 fossil mammals and mammaliforms from across the Mesozoic and Cenozoic, and 24

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extant placental and marsupial mammals (Luo et al. 2007). All mammal taxa were coded for 445 discrete morphological characters. The bird matrix contained eight fossil Pelecaniformes from the Late Cretaceous and Cenozoic, and 51 extant Pelecaniformes (Smith 2010). All Pelecaniformes were coded for 464 discrete morphological characters. The arthropod character matrix contained 16 fossil species spanning the Cambrian to Silurian, and 64 extant species from the subphyla Chelicerata, Myriapoda, Hexapoda, and Crustacea (Rota-Stabelli et al. 2011). All arthropod species were scored for 395 discrete morphological characters.

Correcting for Missing Data Using the Linkage Algorithm We developed a correction factor that tries to account for different amounts of missing data among time bins by using the linkage algorithm to remove data from the more complete time bins until they contain as much missing data as the most incomplete bin. This process can be thought of as analogous to rarefaction for character data (Foote 1992). The correction factor algorithm selects characters to be removed from taxa within a group (in our case a time bin) and retains a user-defined percentage of characters for each taxon. This percentage cap avoids removing all the characters from a single taxon. For illustrative purposes, we compared patterns of disparity across adjacent time bins from datasets with and without correction for missing data. We first applied the missing data correction to the fossil component of the mammal dataset (Luo et al. 2007). This dataset was a good candidate for this analysis because it contained a large number of characters and extinct taxa from across the Mesozoic. The covariation structure of the entire dataset was calculated and then partitioned into four Mesozoic time bins: Triassic and Early Jurassic, Middle and Late Jurassic, Early Cretaceous, and Late Cretaceous. The four time bins contained different amounts of missing data. We calculated the mean amount of missing data in each bin (Table S2), and used the CM to remove characters from the data-rich bins until they contained the same amount of missing data as the most data-poor bin. We excluded three taxa from the dataset (Tritylodontidae, Cimolodonta, and Plagiaulacida) because they were coded at a higher taxonomic level than the other taxa. We also excluded Cenozoic taxa because there were too few taxa, although we did use these Cenozoic taxa in the calculation of the CM. The final analysis included 73 genera. We also re-analyzed the pterosaur disparity dataset of Prentice et al. (2011). Fifty-three Pterosaur taxa from the suborders Rhamphorhynchoidea and Pterodactyloidea were divided into four time bins: Triassic, Jurassic, Early Cretaceous, and Late

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Cetaceous. Each bin spanned 50 million years, and the Early Cretaceous time bin contained the largest amount of missing data (45%; Table S2). To calculate a mean set of PCO scores for mammals and pterosaurs, we ran the linkage algorithm 100 times on the fossil dataset. Following Manhattan distance calculation, the data matrix was subjected to PCO calculation, which returns a matrix of species (rows) and PCO scores (columns). We generated a mean PCO matrix by averaging each taxon by PCO score cell across all 100 matrices. After separating each taxon’s PCO scores into time bins (taxa can occupy multiple bins), we then calculated rarefied disparity profiles for each bin by bootstrap resampling taxa with replacement 1000 times using a minimum of two taxa through the total number of taxa in the dataset. Rarefaction curves are used here to standardize for sample size (Foote 1992). We used paired sample t-tests to determine whether disparity significantly differed between successive geological stages (Anderson et al. 2011; Zelditch et al. 2012, p. 222). Using the statistical differences between time bins in the uncorrected datasets as a baseline, we then performed the same analysis on the corrected datasets to see if the pattern of differences among bins changed. Significance values were corrected for multiple comparisons using the Holm–Bonferroni method (Holm 1979). SIMULATIONS AND STATISTICS

All simulations, statistics, and disparity analyses were performed in R version 3.0.1 (R Core Team 2013) using the packages “vegan” for distance calculations (version 2.0–7; Oksanen et al. 2013), “reshape” to manipulate each distance matrix for disparity calculation (version 0.8.4; Wickham 2007) and “abind” to splice missing data corrected matrices back together (version 1.4–0; Plate and Heiberger 2011). All data, scripts, and instructional documentation developed for this study can be found in the Supporting Information and on Dryad.

Results The linkage algorithm had a lower impact on disparity metrics and ordination spaces than random data removal. When data were removed via linkage, the results of the disparity analyses were more similar to the complete datasets than when characters were removed randomly. Similarly, the relative position of taxa within a morphospace was more stable under the linkage removal algorithm compared to random character removal.

Linkage Algorithm Performance The linkage algorithm removes characters based on their probability of preservation and the strength with which they are associated

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Figure 2. Algorithm performance. (A) Order that characters are removed (y-axis) from two different taxa in the Luo et al. (2007) dataset following the linkage algorithm. (B) Order that characters are removed (y-axis) from two different taxa using the random algorithm. The

general anatomical positions of the characters are arranged along the x-axis with respect to the anatomical regions jaw-dental, cranial, and postcranial.

with other characters. Figure 2 shows how characters are removed during application of the linkage and random algorithms. The linkage algorithm causes most of the characters to be lost within the anatomical region where the first character is removed. When the algorithm deviates away from an anatomical region the character

responsible is often rarely preserved in an extinct organism. The two examples presented in Figure 2A demonstrate how the linkage algorithm can begin by removing data from two different regions. This pattern of character removal can be compared to the random algorithm, where character removal is more stochastic (Fig. 2B).

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in ordination space under linkage (A) and random models (B). Gray points and solid lines—original PCO space (no missing data); black squares and dashed lines—25% missing data; black triangles with dashed and dotted lines—50% missing data; black diamonds with dotted lines—75% missing data. (C) The relative movement of taxa away from their original position in PCO space following data removal under the linkage and random algorithms. Changes in sum of ranges disparity following data removal under the linkage and random algorithms are shown in D. Columns contain plots for mammals (left), arthropods (center), and birds (right). In C and D, output based on the linkage algorithm is shown in dark gray, whereas output from the random algorithm is shown in light gray. In D, solid lines indicate mean values from 0 to 75% missing characters and dashed lines represent 95% confidence intervals.

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Empirical Examples PCO ORDINATIONS

The overall size of the morphospace produced by the PCO analysis declined as missing data increased (Fig. 3A and B). Taxa began by residing in clusters defined by their morphological similarity. As missing data increased taxa became more scattered in morphospace, and moved toward values of zero on both PCO axes. However, the movement of taxa toward zero required the removal of more characters when the linkage algorithm (Fig. 3A) was applied compared to the random algorithm (Fig. 3B). A similar pattern is present across all of the datasets we tested (Fig. S1). Following data removal simulation, the linkage algorithm produces a more faithful representation of the original PCO space compared to the random algorithm (Fig. 3C). All 10 datasets exhibit a similar pattern (Fig. S2), whereby the position of taxa in morphospace calculated from the random algorithm is more erroneous than that of the linkage algorithm. The difference between the linkage and random algorithm is noticeable once 10–20% of the data is missing. All disparity analyses that utilize an ordination method (e.g., Wills et al. 1994; Sidlauskas 2008; Hughes et al. 2013) are susceptible to a reduction in morphospace volume as missing data increase. Compared to the random algorithm, the linkage algorithm was able to lessen the rate at which information was lost from the disparity analysis by selecting suites of characters to be lost together.

DISPARITY ANALYSIS

Disparity declined with increasing missing data in all of the datasets (Figs. 3D and S3–S6). However, the rate of decline differed depending on whether the random or linkage algorithm was applied. The random algorithm consistently caused all estimates of disparity to decline faster relative to the linkage algorithm, leading to much lower values of disparity at 75% missing data. In the mammal dataset, disparity exhibited a steady decline across both algorithms. The linkage algorithm caused disparity to decline very slowly until 35% of the characters had been removed and then increase slightly from 35% to 75%. In contrast, the random algorithm produced a steady and more rapid decline in disparity across the entire range of missing data. Given any amount of missing data, the random algorithm consistently returned lower disparity values than the linkage algorithm. At 30% missing data, the 95% confidence intervals no longer overlapped between the linkage and random algorithms. Differences in disparity metrics between the linkage and random algorithms were less distinct in the arthropod example, although the overall pattern was similar. Disparity calculated under the linkage algorithm declined more slowly than when characters

were lost randomly. Initially, both the linkage and random algorithms exhibited the same rate of decline. The rate of decline in disparity under the linkage algorithm deviated from the random pattern at 35% missing data; and then leveled out; by 40% missing data, the 95% confidence intervals between the random and linkage algorithms no longer overlapped. Like the mammal and arthropod examples, birds underwent a slower decline in disparity when data were removed via linkage compared to when data were removed randomly. The random and linkage algorithms initially reduced the disparity predicted for birds at a similar rate, but the 95% confidence intervals of the two algorithms diverged at 25% missing data.

Correction Factor The general pattern of mammalian disparity through the Mesozoic changed when we applied the correction factor to equalize the amount of missing data across time bins (Fig. 4A and B). Prior to correcting for missing data, there were statistical differences in disparity between each pair of adjacent time bins. Initially, mean disparity significantly decreased between the Triassic + Early Jurassic and Middle + Late Jurassic bins (P < 0.05). Disparity then increased significantly in the Early Cretaceous (P < 0.01), before declining again in the Late Cretaceous (P < 0.05; Fig. 4A). After correcting for missing data, all statistical differences between adjacent time bins disappeared (Fig. 4B). Trends in mammalian disparity also appeared to differ between corrected and uncorrected datasets depending on which disparity metric was used (not product of variances however; Figs. S7–S10; Table S3). The general disparity trends between pairs of adjacent time bins of did not change following the correction for missing data in pterosaurs, but the magnitude of differences in disparity increased in two of the three comparisons (Fig. 4C and D). Trends in pterosaur disparity remained similar between corrected and uncorrected datasets regardless of disparity metric (Figs. S7–S9 and S11; Table S3).

Discussion Using the linkage algorithm, we demonstrated how missing data can influence ordination spaces and the resulting disparity analysis. The linkage algorithm was better able to retain the original distances among taxa in an ordination space as character data were removed, relative to the algorithm that removed characters randomly (Fig. 3A–C). By better approximating the true ordination space after data were lost, the linkage algorithm caused disparity to decline at a slower rate than when characters were removed randomly (Fig. 3D). Finally, when the linkage algorithm was used to correct for differences in missing data between time bins, trends in disparity changed compared to when no correction

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Figure 4. Comparing missing data corrected and uncorrected sum of ranges disparity trends. Disparity through time without correcting for missing data (mammals on left [A], pterosaurs on right [C]). Percentages refer to the amount of missing data present in each time bin. Disparity through time after the missing data correction was applied (mammals on left [B], pterosaurs on right [D]). The results of statistical comparisons between each successive time bin are displayed for each dataset. NS = nonsignificant; ∗P < 0.05; ∗∗P < 0.01; ∗∗∗

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was applied (Fig. 4A–D), although the magnitude and direction of these changes varied between datasets. By better approximating the loss of data during taphonomic processes, the linkage algorithm produces a more faithful representation of original disparity than does the random character removal algorithm. It does this because of two key properties. First, it accounts for the fact that regions that are functionally and developmentally associated are lost/preserved together. For exam-

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ple, correlations within an anatomical region (e.g., the hand) are likely to be stronger than correlations among more disparate regions (e.g., the fore and hind limbs). The linkage algorithm tends to remove suites of correlated characters from within anatomical regions, and suites of characters drawn from the same anatomical region likely possess less morphological variation than characters drawn from multiple regions (Erwin 2007; Goswami and Polly 2010). The second key property of the linkage algorithm is that

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more pairwise comparisons can be made for each character in the data matrix than when characters are removed randomly. This occurs because the random algorithm can remove any character from any taxa, whereas the linkage algorithm will more often remove similar sets of characters among taxa (Fig. 2). Taken together, these two properties of the linkage algorithm allow it to produce ordination spaces and disparity outputs that are closer to those derived from of complete datasets. Essentially, the linkage algorithm preserves disparity, which is quickly lost when characters are removed using the random algorithm. The order of character removal during a simulation influences the results of disparity analyses. This is especially true when some anatomical regions possess more disparity than others (Rodrigues et al. 2011). For example, morphological disparity is generally greater in the hind limbs of pterosaurs than in their forelimbs, which are more constrained by the demands of flight (Dyke et al. 2006; Rodrigues et al. 2011, p. 250). If the linkage algorithm removes hind limbs first then disparity will be seen to decline faster compared to if forelimbs were removed first. Because the linkage algorithm runs multiple simulations for each percentage of character loss, the final output will represent the mean condition (i.e., the order of limb removal can switch from one simulation to the next). In other words, differences in regional disparity that are caused by the order of character loss explains variation around the mean disparity values. Disparity calculations are only possible when taxa have characters in common. Incalculable pairwise distances can occur when, for example, one taxon is known only from a skull and another is known only from a limb. Depending on how missing data are distributed in a matrix, it is possible that the linkage algorithm could remove all characters from a given region, producing further incalculable pairwise distances. On the other hand, linkage algorithm can also alleviate this problem. For example, if limb characters are poorly represented in a matrix, then these characters will likely be removed first. In this case, those characters that often survive the taphonomic process are more likely to be retained during the data loss simulation. When calculating disparity from the remaining data, the results will reflect the disparity of those retained regions.

Correcting for Missing Data in Disparity Analysis We demonstrated the utility of the linkage-based correction factor using two fossil datasets from mammals and pterosaurs. If correcting for missing data changes trends in disparity that may signal that results based on the uncorrected data matrix are biased. Without correcting for missing data in the mammal dataset, there were significant differences in disparity between adjacent time bins

(Fig. 4A). All of those differences disappeared when we corrected for missing data using the linkage algorithm (Fig. 4B). From this perspective, the mammal dataset was not robust to missing data, and uncorrected disparity values may have led to inaccurate conclusions about how morphological disparity changed through the Mesozoic. This is in contrast to the Prentice et al. (2011) dataset, in which general trends in disparity did not change after correction for missing data (Fig. 4C and D). With the bias of missing data alleviated, the differences in disparity between each time bin were even greater than originally thought. Given the differences in how each dataset responded to missing data, it appears that some analyses are more robust to missing data than others. We advocate reporting the results of both corrected and uncorrected analyses of disparity to provide insight into the potential impact of missing data on the analysis. The correction factor method we propose provides another method for controlling the impact of missing data on disparity analysis. The phylogenetic methods of Butler et al. (2012) and Brusatte et al. (2011) are two recently developed techniques that correct for missing data using an altogether different approach: by using a well-resolved, bifurcating phylogeny to reconstruct ancestral character states (Butler et al. 2012) or ancestral taxa (Brusatte et al. 2011) using parsimony. Reconstructing ancestral character states using parsimony necessarily reduces the scatter of taxa in morphospace, causing them to form multiple tight clusters with less overlap between clades (Butler et al. 2012). Reconstructing character states for ancestors at each internal node adds hypothesized taxa to the dataset that were not present in the fossil record (Brusatte et al. 2011). These ancestors were often morphologically distinct and increased the overall range of morphospace that a group occupied. However, missing data can render the phylogentic position of some taxa unstable, leading to poorly resolved trees (Wilkinson 2003; Sansom and Wills 2013). In addition, as missing data increase, the accuracy of character reconstruction decreases due to a reduction in phylogeny-informative characters (Wiens 2003). This is especially problematic when missing data are concentrated in a few taxa (Prevosti and Chemisquy 2010), or when there are few taxa in the character matrix (Wiens and Tiu 2012). Parsimony can also produce misleading results when rates of evolution are rapid, and when the probabilities of gaining or losing a character state are unequal (Cunningham et al. 1998). Reconstruction of missing character states for terminal taxa is also bounded by known character states; a character state could not be reconstructed as “3” if only 0, 1, and 2 are known to exist. This will limit the amount of morphological diversity and novelty, while increasing morphological similarity among taxa. Deciding which technique to use depends on the quality of the data and the evolutionary questions being addressed. Determining whether the linkage algorithm is the correct choice for missing data correction depends on the distribution of missing data in the

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dataset and the characters that will make-up the final disparity result. If the distribution and amount of missing data that needs to be removed causes incalculable pairwise distances between taxa to arise, then it cannot be used unless problematic taxa are removed. Therefore, the linkage correction factor method is most beneficial whenever there are unequal amounts of missing data among time bins and no incalculable distances will arise among taxa. This ensures the bias caused by missing data is alleviated and pairwise distances can still be calculated. This study highlights the fact that more effort should be focused on better understanding how missing data can affect disparity analysis, and developing methods to overcome the problem of missing data. The 10 disparity datasets we analyzed here exhibited different levels of sensitivity to the steady removal of data. Some of the differences observed among the datasets could reflect differences in their intrinsic properties. The number of taxa, number of morphological characters, how the matrix was coded, and the strength of the CM can all effect how individual groups respond to correcting for missing data. Our study used cladistic character matrices with discrete data; no studies have investigated how disparity calculated from continuous variables is affected by missing data. This is an important area for further analysis as continuous variables may be more informative regarding functional and ecological changes through time (Friedman 2010; Anderson et al. 2011; Anderson and Friedman 2012; Foth et al. 2012).

Conclusions Disparity analysis is greatly affected by missing data. Simulation studies often assess the effects of missing data on disparity using algorithms that remove characters randomly. Here, we presented algorithms that simulate character loss based on the probabilistic nature of the taphonomic process and the fact that suites of characters are often linked and either gained or lost together. We used cladistic datasets to examine the behavior of the algorithm then applied the algorithm to correct for bias in estimates of disparity due to differences in the amount of missing data between time bins. By using the linkage among characters to equalize the proportion of missing data across time bins, we showed that trends in disparity can be different between corrected and uncorrected analyses. The linkage algorithm is a useful simulation tool for studies of missing data and can provide a great deal of insight into how disparity or phylogenetic analyses respond to missing data.

ACKNOWLEDGMENTS The authors wish to thank the Behavior and Morphology group at UMass, Amherst for comments and discussion on early versions of the manuscript. We also wish to acknowledge P. S. L Anderson and M. Friedman for comments and providing R scripts to run statistical analyses. This manuscript

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was very much improved by the comments and insights of two anonymous reviewers. MVR was supported by a DOE CSGF number DEFG02-97ER25308. The authors declare no conflict of interest. DATA ARCHIVING The doi for our data is 10.5061/dryad.d380g. LITERATURE CITED Anderson, P. S. L., and M. Friedman. 2012. Using cladistic characters to predict functional variety: experiments using early gnathostomes. J. Vertebr. Paleontol. 32:1254–1270. Anderson, P. S. L., M. Friedman, M. D. Brazeau, and E. J. Rayfield. 2011. Initial radiation of jaws demonstrated stability despite faunal and environmental change. Nature 476:206–209. Bapst, D. W., P. C. Bullock, M. J. Melchin, H. D. Sheets, and C. E. Mitchell. 2012. Graptoloid diversity and disparity became decoupled during the Ordovician mass extinction. Proc. Natl. Acad. Sci. USA 109:3428–3433. Behrensmeyer, A. K., S. M. Kidwell, and R. A. Gastaldo. 2000. Taphonomy and paleobiology. Paleobiology 26(sp4):103–147. Benson, R. B. J., M. Evans, and P. S. Druckenmiller. 2012. High diversity, low disparity and small body size in plesiosaurs (Reptilia, Sauropterygia) from the Triassic-Jurassic boundary. PLoS ONE 7:e31838. Benton, M. J., M. A. Wills, and R. Hitchin. 2000. Quality of the fossil record through time. Nature 403:534–537. Bown, T. M., K. D. Rose, E. L. Simons, and S. L. Wing. 1994. Distribution and stratigraphic correlation of Upper Paleocene and Lower Eocene fossil mammal and plant localities of the Fort Union, Willwood, and Tatman Formations, Southern Bighorn Basin, Wyoming. Geol. Surv. Prof. Pap. 1540:1–103. Briggs, D. E. G. 1995. Experimental taphonomy. Palaios 10:539–550. Brusatte, S. L., M. J. Benton, M. Ruta, and G. T. Lloyd. 2008a. Superiority, competition, and opportunism in the evolutionary radiation of dinosaurs. Science 321:1485–1488. ––––––. 2008b. The first 50Myr of dinosaur evolution: macroevolutionary pattern and morphological disparity. Biol. Lett. 4:733–736. Brusatte, S. L., S. Montanari, H. Yi, and M. A. Norell. 2011. Phylogenetic corrections for morphological disparity analysis: new methodology and case studies. Paleobiology 37:1–22. Brusatte, S. L., R. J. Butler, A. Prieto-M´arquez, and M. A. Norell. 2012. Dinosaur morphological diversity and the end-Cretaceous extinction. Nat. Commun. 3:804. Butler, R. J., S. L. Brusatte, B. Andres, and R. B. J. Benson. 2012. How do geological sampling biases affect studies of morphological evolution in deep time? A case study of pterosaur (Reptilia: Archosauria) disparity. Evolution 66:147–162. Ciampaglio, C. N., M. Kemp, and D. W. Mcshea. 2001. Detecting changes in morphospace occupation patterns in the fossil record: characterization and analysis of measures of disparity. Paleobiology 27:695– 715. Clarke, J. A, and K. M. Middleton. 2008. Mosaicism, modules, and the evolution of birds: results from a Bayesian approach to the study of morphological evolution using discrete character data. Syst. Biol. 57:185– 201. Cunningham, C. W., K. E. Omland, and T. H. Oakley. 1998. Reconstructing ancestral character states: a critical reappraisal. Trends Ecol. Evol. 13:361–366. Dyke, G. J., R. L. Nudds, and J. M. V Rayner. 2006. Limb disparity and wing shape in pterosaurs. J. Evol. Biol. 19:1339–1342. Eiting, T. P., and G. F. Gunnell. 2009. Global completeness of the bat fossil record. J. Mamm. Evol. 16:151–173.

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Supporting Information Additional Supporting Information may be found in the online version of this article at the publisher’s website: Table S1. Paper reference information, number of soft characters, number of characters removed at each percentage, and the number of PCO axes used for each dataset. Table S2. Total amount of missing data contained within each time bin. Table S3. Statistical significance (Bonferroni–Holm corrected) between adjacent time bins for missing data uncorrected and corrected analyses. Figure S1. Ordination space changes following varying levels of data removal for all tested datasets. Figure S2. The relative movement of taxa away from their original position in PCO space following data removal under the linkage (black) and random (red) algorithms. Figure S3. Changes in sum of ranges disparity following data removal under the linkage (black) and random (red) algorithms. Figure S4. Changes in sum of variances disparity following data removal under the linkage (black) and random (red) algorithms. Figure S5. Changes in product of ranges disparity following data removal under the linkage (black) and random (red) algorithms. Figure S6. Changes in product of variances disparity following data removal under the linkage (black) and random (red) algorithms. Figure S7. Comparing missing data corrected and uncorrected sum of variances disparity trends. Figure S8. Comparing missing data corrected and uncorrected product of ranges disparity trends. Figure S9. Comparing missing data corrected and uncorrected product of variances disparity trends. Figure S10. Rarefaction profiles for all disparity metrics tested, calculated for four time bin subsets. Figure S11. Rarefaction profiles for all disparity metrics tested, calculated for four time bin subsets.

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