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Journal of Mammalogy, 96(4):791–799, 2015 DOI:10.1093/jmammal/gyv084 Measuring variation in the frequency of group fission and fusion from continuous monitoring of group sizes Guillaume Body, Robert B. Weladji,* Øystein Holand, and Mauri Nieminen Department of Biology, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec H4B 1R6, Canada (GB, RBW) Department of Animal and Aquacultural Sciences, Norwegian University of Life Sciences, P.O. Box 5003, N-1432 Ås, Norway (ØH) Finnish Game and Fisheries Research Institute, Reindeer Research Station, Toivoniementie 246, FI-99910 Kaamanen, Finland (MN) * Correspondent: [email protected] The frequency of group fission and fusion is critical for understanding the evolution of sociality, as it measures group cohesiveness, which can confer a high fitness value on individuals who practice it. Variations in the frequency of fission and fusion can also reveal an individual’s response to environmental changes. As fission–fusion group dynamics influence the risk of disease transmission, measuring it accurately may have implications for population management. Here, we extend a method to assess the frequency of fission and fusion from continuous monitoring of group composition. The former method was based on the variation in group size (i.e., party/subgroup size) within which marked individuals were successively recorded. Upon validation of the method here, we propose a correction, using the coefficient of variation in group size instead of the variation in group size. We performed our study on an enclosed herd of reindeer, Rangifer tarandus. All reindeer were tracked using Global Positioning System (GPS) collars and followed by direct observations. GPS collars allowed us to obtain accurate measures of the frequency of fission and fusion that we used to validate our models. The resulting method best described the temporal variation in fission and fusion frequency without the confounding effect of the variation in group size. This method can easily be applied to other populations where some individuals can be recognized and their group size repeatedly recorded. It will also help in quantifying the 2nd dimension of the fission–fusion group dynamics: the variation in group size. Key words: caribou, fission–fusion, self-organization, sociality, ungulates © 2015 American Society of Mammalogists, www.mammalogy.org In animal societies, the fission and fusion of groups (also called parties or subgroups) is increasingly being recognized as an important component of social structure (Aureli et al. 2008; Sueur et al. 2011). Fission–fusion group dynamics (FFGD) were defined in a recent framework as the “extent of variation in spatial cohesion and individual membership in a group over time” (Aureli et al. 2008). According to the framework, the social systems of mammals can be compared along the gradient of variation and explained by ecological contexts. For example, Australian snubfin dolphins, Orcaella heinsohni, have large and cohesive groups with individuals maintaining strong bonds in contrast to Indo-Pacific humpback dolphins, Sousa chinensi. This difference in cohesiveness was then explained by a difference in food distribution as snubfin dolphins often feed on large and concentrated food resources, whereas humpback dolphins feed on patchy and irregular distribution of fish (Parra et al. 2011). Increased group cohesiveness may have evolved due to its fitness advantages. Female feral horses indeed have a higher reproductive success when they form strong social bonds, which is independent of their kinship (Cameron et al. 2009). The extent of group cohesiveness has also been linked with the evolution of cognitive abilities (the social brain hypothesis), as primate species displaying high levels of FFGD had higher success on inhibition tasks (Amici et al. 2008). Accordingly, the study of group living is shifting from a cost/benefit approach of the optimal group size toward the study of group cohesiveness (self-organization theory—Couzin and Laidre 2009) and individual decisions (Conradt et al. 2009). The most complex social systems are described by hierarchical organization of social structures. Indeed, elephants Loxodonta africana (Wittemyer et al. 2009), giraffes Giraffa camelopardalis (Bercovitch and Berry 2012; VanderWaal et al. 2014), and some primates (reviewed in Grueter et al. 2012) form multilevel societies where cohesive groups temporarily join each other and form subcommunities, which themselves could be grouped into communities. Another level of complexity corresponds to the formation of subgroups, as seen in giant noctule bats, Nyctalus lasiopterus (Popa-Lisseanu et al. 2008), 791 792 JOURNAL OF MAMMALOGY and spider monkeys, Ateles geoffroyi (Ramos-Fernández and Morales 2014), where a social group may split into unstable subgroups, which are often merging and splitting together. Subgroups vary in size and composition, while the overall group is stable in size, composition, and territory occupied. Although highly complex social systems have been the main focus of FFGD studies, simpler systems also display FFGD. Kangaroos, Macropus sp. (Caughley 1964), and many ungulates (e.g., chamois Rupicapra pyrenaica—Pépin and Gerard 2008; bison Bison bison—Fortin et al. 2009) have highly unstable and fluid groups. Groups often merge, split, and their composition may vary randomly (Gerard et al. 2002). By contrast, solitary and territorial species often form very stable groups, and some species previously classified as solitary are now being described by FFGD (e.g., Raccoon Procyon lotor—Prange et al. 2011). Therefore, any mammalian species (see Silk et al. 2014 for a review in birds) can be described as displaying a certain degree of FFGD (Couzin and Laidre 2009). The degree of FFGD can be summarized using 3 axes: the variation in spatial cohesion, the variation in group size, and the variation in group composition (Aureli et al. 2008). Differences in the degree of FFGD are expected among species (e.g., Parra et al. 2011), among populations (e.g., Kelley et al. 2011), and even within population through time (e.g., Body et al. 2015). These differences would reflect differences not only in social structure but also in population density, food distribution, or predation risk. Accurately estimating the degree of FFGD may prove to be valuable for a number of theoretical studies and management practices, such as the understanding of information spread (Couzin and Laidre 2009), the evolution of cognitive abilities (Shultz and Dunbar 2006; Pérez-Barbería et al. 2007), or the management of disease spread (Proffitt et al. 2012). However, the methods used to quantify these 3 dimensions of the FFGD are far from established, and much effort is needed to develop these methods. The aim of this article is to improve an existing method to assess the temporal variation in the frequency of fission and fusion, which directly influences the variation in group size. Indeed, from an individual viewpoint, the more frequent fission and fusion events are (i.e., groups are unstable), the more variable group sizes will be. This method has been proposed by Gower et al. (2009) on elk Cervus elaphus and then used on pronghorn Antilopacra americana (White et al. 2012) but has not yet been evaluated. Gower’s method is based on the variation in group size within which one particular individual is found during a particular interval of time. Results of these 2 studies showed a strong correlation between this measure of the rate of fission and fusion and the average group size. Such a variance–mean correlation is a common pattern as it corresponds to issues of heteroscedasticity (Crawley 2007). As a result, this method merits further reflection to ensure that the variation of the measured variable (variation in group size) actually reflects a variation in the rate of fission and fusion and not a variation in average group size, which would have a different biological interpretation. For instance, an increase of the average group size could be the consequence of either an increase in population density or an increase of predation risk. However, we expect increasing population density to increase the rate of fission and fusion and increasing predation risk to decrease the rate of fission and fusion. Consequently, our study has 3 objectives: 1) to improve Gower’s method by using the coefficient of variation instead of the group size variation, 2) to validate the modified-Gower’s method using an accurate record of the frequency of group fission and fusion in a controlled herd, and 3) to perform a sensitivity analysis on the model parameters to provide best practice advices when using the modified-Gower’s method. To achieve these objectives, we analyzed the temporal variation in the rate of group fission and fusion reported in reindeer Rangifer tarandus (Body et al. 2015) during one breeding season. During the rut, reindeer form harems that include 1 dominant male, with or without subdominant males, and females. Harems are not restrained in space and are highly unstable, although female interharem movements and harem fission are less frequent during the peak rut due to the herding activity of dominant males (Body et al. 2014a). From direct and indirect observations of group size, we predicted the variation in the rate of fission and fusion. To validate the tested methods, we compared their predictions with the temporal variation in the actual frequency of group fission and fusion. Materials and Methods Study herd and GPS methods.—In 2011, we studied a herd of 45 adult reindeer (34 females and 11 males), free to move in a 13.4-km2 fenced enclosure (the Sinioivi enclosure of the Kutuharju Field Reindeer Research Station, Finland), during 39 days in the breeding season (8 September–18 October). We equipped all individuals except 1 female with GPS collars, which synchronously recorded their positions every 15 min (hereafter “recording”). GPS collars weighed about 700 g (< 2% of reindeer body mass) and had an average positioning error of 13.8 m (Body et al. 2015). On each recording, we assessed aggregations of individuals based on a chain rule (Croft et al. 2008) stating that 2 neighbors were in the same aggregation if they were closer than 89 m (Body et al. 2015). These aggregations are named “group” based on Lent’s (1965) definition: “an aggregation of individuals separated by some distance from other aggregations, showing coordination of activities, such as travelling together or resting and feeding together.” This definition, therefore, matches the definition of “subgroup” or “party” used in other research domains (e.g., in primatology, see Aureli et al. 2008). Because male movement may be different from female movement (Ramos-Fernández and Morales 2014), particularly during the mating season, and to match previous research (Body et al. 2015), we focused our study only on females: the smallest group being a group of 1 female. We identified fission from fusion by comparing group composition among successive recordings. In the case of fission, females from one group at time t were recorded in different groups at time t + 1, and for fusion, females from different groups at time t were recorded together at time t + BODY ET AL.—MEASURING FISSION–FUSION FREQUENCY793 1. Every day from 21 September to 17 October, we also located groups and recorded their composition in the field. We recorded identities of females that comprised the groups using unique collar identification numbers (L’Italien et al. 2012). Handling of animals and data collection were done in agreement with the Animal Ethics and Care certificate provided by Concordia University (AREC-2010-WELA and AREC-2011-WELA) and by the Finnish National Advisory Board on Research Ethics. Research on live animals also followed American Society of Mammalogists guidelines (Sikes et al. 2011). Gower’s method and the modified-Gower’s method.— Gower et al. (2009) calculated their measure of the rate of fission and fusion from the group size variation as: “the absolute difference between a given group size and the mean group size for [the period in focus].” The corresponding statistical method used in their study (Gower et al. 2009; White et al. 2012) was a linear mixed model with the individual identity as the random term, the measure of the group size variation during the studied interval as the response variable, and time intervals, among others related to their specific questions, as explanatory variables. The methods we implemented were not exactly the method described in these articles: we adapted the Gower’s method (see next paragraph) to better suit our data set and some biology principles. These adaptations do not correspond to the main change we implemented as they should not influence the outcome of Gower’s method. Then, we modified the response variable used in Gower’s method (see 2nd paragraph below) to create the “modified-Gower’s method” and we compared the Gower’s and the modifiedGower’s methods. For our purpose, we adapted Gower’s method in 3 ways. First, to enhance the temporal variability, we calculated group size variation on sliding windows instead of time intervals. These windows are referred to as “the period in focus.” The influence of the chosen window length was then evaluated in a sensitivity analysis (see below). Second, when applying the method, we only used the group sizes related to each individual’s standpoint, and not to the population data set, which means that each individual had its own average group size for the period in focus. This approach matches individual exposure to group size more closely than using the average group size at the population level (Vander Wal et al. 2013) and is in line with the individual-based observation of group size (i.e., the typical group size—Jarman 1974). Gower’s method already takes into account this aspect by including the individual identity as a random term. The advantage of our version is the separation of interindividual variations in average group size from the interindividual variations in the rate of group fission and fusion. This is accomplished by controlling interindividual variations in average group size prior to analyses and by isolating interindividual differences in group size variation in the random term. Third, we used time as a continuous explanatory variable, instead of a categorical variable, and consequently, we used a generalized additive mixed model (GAMM) with the individual identity as random factor (i.e., intercept only), instead of a linear mixed model, as it provided a better fit of the temporal variability. These adaptations applied to both Gower’s method and the modified-Gower’s method. The modified-Gower’s method only differed from Gower’s method on the response variable. Instead of the group size variation, we calculated the coefficient of variation (i.e., CV = SD / X ) using the same data (i.e., the group sizes recorded per individual during the sliding window). The main advantage of using the coefficient of variation is its independence from the average group size, in contrast to the variation in group size as previously calculated by Gower et al. (2009). This modification would, therefore, allow us to differentiate a variation in the rate of group fission and fusion from a variation in average group size. We compared the actual variation in the frequency of fission and fusion (see paragraph below) to the variation of the predictions obtained from both methods when applied to testing data sets (see below). Validation analysis and comparison of the Gower’s and the modified-Gower’s methods.—We applied the Gower’s and the modified-Gower’s methods on 2 different data sets to evaluate their performance in predicting the frequency of fission and fusion. Data set A included GPS-based group size data and thus was very precise. In contrast, data set B included group size data from daily field observation and thus was more realistic (Fig. 1). Each data set was formed by 3 sub–data sets: 1) the actual variation in average group size, 2) the actual variation in frequency of fission and fusion, and the testing sub–data set based on 3) the observed group compositions (Fig. 1). We applied the Gower’s method and the modified-Gower’s method on the observed group composition sub–data sets to obtain 2 sets of predictions in temporal variations of the fission and fusion frequency, for each data set (Fig. 1). Then, we compared these 2 predictions with the actual variation in average group size and with the actual variation in fission and fusion frequency, for each data set A and B separately (Fig. 1). Data sets A and B differed in data frequencies; window lengths and periods analyzed but have similar number of data per window. The actual variations of the average group size and the actual variations of the frequency of fission and fusion were based on the GPS data as it provided the most accurate data. Average group sizes were calculated for each individual using the GPS data at the frequency given by the data set (A or B). The frequencies of fission and fusion were evaluated from the 15-min GPS data set as previously described (see “Study Herd and GPS Methods”), but frequencies of fission and fusion were reported per hour, i.e., it corresponded to the number of these events occurring during the period in focus divided by the window length in hour. We fitted a GAMM on each of these 4 sub–data sets, with the time as an explanatory variable and the individual’s identity as a random variable. The smoothing parameter of the GAMM depended on the data set. Data set A is entirely based on GPS data on a 4-h frequency and it uses a window length of 12 h. We used the entire study period for this data set (39 days) and each GAMM used a smoothing parameter (k) equal to 8 (in agreement with previous analyses—Body et al. 2014a). We used a 4-h frequency to increase the independence among successive group compositions and to 794 JOURNAL OF MAMMALOGY Fig. 1.—Organization of data sets A and B as described in “Materials and Methods.” Actual variation in group size and in the frequency of fission and fusion are obtained from GPS data; the testing sub–data sets are the individual-based group size obtained from the GPS data (data set A) or from field observations (data set B). The frequency of fission and fusion is estimated by the Gower’s method and the modified-Gower’s (m-G’s) method from each testing data set. Predictions from each model corresponding to the actual and estimated temporal variations in the frequency of fission and fusion are compared, on each data set, based on 3 criteria. Continuous-line boxes correspond to actual data or their derived predictions, while dashed-line boxes are estimations of the frequency of fission and fusion. accelerate analyses, as higher data frequency did not improve the results (see “Results”). We used a window length of 12 h to match the number of data per window of data set B. The testing sub–data set of data set B is based on daily field observation of group composition (see “Study Herd and GPS Methods”). Occasionally individual females were recorded multiple times within a day. We retained these records in data set B as they are rare and thus not expected to influence the analysis. To match the data frequency of the testing sub–data set, we used a data frequency of 1 day for the actual variation in average group size and in the frequency of fission and fusion. We used a 3-day window length because it provided the largest number of data (i.e., windows that included at least 2 records of group size observed in the field, for each individual) for the shortest window length. A shorter window length provided a better fit of the data (see result of the sensitivity analysis) and best matched the temporal variations. As the period analyzed was 27 days long, the smoothing parameters of the GAMMs were reduced to k = 6 (proportional to the period analyzed). To facilitate the application of the modified-Gower’s method, we provide the R script (Supporting Information S1) and the testing sub–data set of data set B (i.e., the group compositions based on field observation; Supporting Information S2) for an example of application. For data sets A and B, we compared the predictions obtained from the Gower’s and the modified-Gower’s method with the actual variations in average group size and in frequency of fission and fusion based on 3 criteria: a visual observation of temporal variation patterns, the Pearson correlation coefficient, and the sum of square errors (SSEs). As predicted values from the Gower’s and modified-Gower’s methods do not have the same units as the fitted values for the average group size and for the fission and fusion frequencies, we scaled (centered and reduced) each prediction and fitted value before calculating the SSE, which is used as a goodness-of-fit measure. Sensitivity analysis.—We performed the modified-Gower’s method on a range of data frequencies and window lengths based on the GPS data. We rarefied data frequencies to 2, 4, 8, 12, 24, 36, and 48 h and enlarged window lengths to 4, 8, 12, 24, 48, 72, 96, 120, 144, 168, 192, 216, and 240 h (i.e., from 4 h to 10 days), with all possible combinations having at least 2 records per window. To compare results based on these different arrangements, each fitted model was then used to predict a single time frame of 4-h intervals covering the whole study period. Then, we compared these predictions with the fitted values of the frequency of fission and fusion obtained on the data set A based on the Pearson correlation and the SSE (with scaled values). BODY ET AL.—MEASURING FISSION–FUSION FREQUENCY795 We ran a generalized linear model (GLM) to assess the effect of data frequency and window length on the Pearson correlation coefficient as well as on the SSE. We included in the models the interaction between the data frequency and the window length to verify whether the number of data per window was a good criterion to improve the Pearson correlation and to reduce the SSE. We fitted a logistic model for the Pearson correlation model, as it is limited by the interval [0,1] and a linear model (i.e., with a Gaussian error structure) for the SSE model. Using a data frequency of 4 h and a window length of 12 h (as for data set A), we performed a sensitivity analysis on the number of females followed. We used 33 (i.e., the maximal), 30, 25, 20, 15, 10, 5, and 2 randomly chosen females, with 10 different subsamples for each subsample size (except for 33 females that was the full data set). Again, we calculated the Pearson correlation and the SSE (with scaled values) between the predictions made by these models and the actual variation in frequency of fission and fusion. Then, we performed a GLM to explain the variation of the Pearson correlation coefficients (logistic model) and the variation of the SSE (Gaussian error structure) by the number of females followed. All analyses were performed with R 3.0.3 (R Core Team 2014). Results Methods comparison and validation.—Slightly different results were obtained from the GPS records (data set A; Fig. 2) and from the records based on direct observation of the group compositions (data set B; Fig. 3). In both cases, the actual average group size increased until the peak rut week, then decreased (Figs. 2a and 3a). The actual frequency of fission and fusion showed the opposite pattern, decreasing until the beginning of the peak rut, and then increasing (Figs. 2b and 3b). Results from data set A support the prediction that the modified-Gower’s method better fits the temporal variation of the frequency of fission and fusion than the Gower’s method. Predictions from the modified-Gower’s method were more strongly correlated with the actual variation of the frequency of fission and fusion and had a lower SSE than predictions from the Gower’s method (Gower’s method: r = 0.43, P < 0.001, SSE = 262, Fig. 2c; modified-Gower’s method: r = 0.94, P < 0.001, SSE = 27, Fig. 2d). In addition, predictions from the modified-Gower’s method were negatively correlated to the actual variation in average group size (r = −0.69, P < 0.001), which matched the negative correlation between the actual variation in frequency of fission and fusion and the actual variation in average group size (r = −0.63, P < 0.001). Predictions from Gower’s method were positively correlated to the average group size (r = 0.25, P < 0.001). Visual observation also revealed a high similarity between the temporal variation predicted by the modified-Gower’s method (Fig. 2d) and the temporal variation in frequency of fission and fusion (Fig. 2b), while the temporal variation predicted by the Gower’s method was different (Fig. 2c). Results from data set B are less clear. Gower’s method provided a stronger correlation between its predictions and the Fig. 2.—Data set A: Temporal variability of a) the actual average group size (individual-based) and b) the actual frequency of fission and fusion (FF). Predictions of the temporal variation in the frequency of fission and fusion are presented according to c) the Gower’s method and d) the modified-Gower’s method. The temporal variations are presented with their 95% CI. The gray vertical bar represents the peak rut week. actual variation in frequency of fission and fusion and a lower SSE than the modified-Gower’s method (Gower’s method: r = 0.84, P < 0.001, SSE = 8.3; modified-Gower’s method: r = 0.73, P < 0.001, SSE = 13.5). As for the data set A, Gower’s method predictions were positively, but not significantly, correlated with the average group size (r = 0.29, P = 0.14), while the predictions from the modified-Gower’s method were negatively correlated with the average group size (r = −0.53, P = 0.005). By comparison, the actual variation in average group size and in frequency in fission and fusion were not significantly correlated in the data set B (r = −0.10, P = 0.61). Visual observations of the temporal variations showed a better fit when predicted by the modified-Gower’s method than by the Gower’s method: the modified-Gower’s method predictions decreased then increased (Fig. 3d), which correctly matched the main temporal variation in frequency of fission and fusion (Fig. 3b), while the predictions from the Gower’s method only display a straight increase (Fig. 3c). Sensitivity analysis.—None of the parameters used in the modified-Gower’s method statistically influenced the Pearson correlation between the predicted and the actual frequency of fission and fusion (data frequency: P = 0.11, window length: P = 0.09, interaction: P = 0.18, and number of females followed: P = 0.07). However, visual inspection of the graphs suggested that we obtained the highest Pearson correlation for high data frequency and short window length (Figs. 4a and 4b) 796 JOURNAL OF MAMMALOGY and that decreasing the number of data per window increased the Pearson correlation (Fig. 4c). Increasing the number of females followed gradually increased the Pearson correlation and reduced its variability among subsamples (Fig. 4d). Model parameters that best fit the variation in the frequency of fission and fusion, and thus minimized the SSE, were similar to those that maximized the Pearson correlation and were statistically significant. The SSE decreased when the data were more frequent for short window length (P < 0.001; Fig. 4e). Decreasing the window length for frequent data reduced the SSE (P < 0.001; Fig. 4f), and decreasing the number of data per window reduced the SSE (P < 0.001; Fig. 4g). Increasing the number of females followed gradually decreased the SSE and reduced its variability among subsamples (P < 0.001; Fig. 4h). Discussion Fig. 3.—Data set B: Temporal variability of a) the actual average group size (individual-based) and b) the actual frequency of fission and fusion (FF). Predictions of the temporal variation in the frequency of fission and fusion are presented according to c) the Gower’s method and d) the modified-Gower’s method. The temporal variations are presented with their 95% CI. The gray vertical bar represents the peak rut week. In this article, we suggested and evaluated a method to assess the temporal variation in group size experienced by individuals. Furthermore, we showed that the suggested method accurately described the variation in the frequency of fission and fusion. As we illustrated here with the difference between the Gower’s method and the modified-Gower’s method, to avoid erroneous interpretations, it is critical that methods used to quantify patterns be evaluated and validated, prior to testing process. The main change from the Gower’s method to the modifiedGower’s method is the mathematical independence between the predicted variation in the frequency of fission and fusion and the variation in average group size. The relative variations of these 2 parameters of the population could help us to Fig. 4.—Variation of a–d) the Pearson correlation and e–h) the sum of square error (SSE) between the actual and predicted frequency of fission and fusion according to the variation in a and e) the data frequency, b and f) the window length, c and g) the number of data per window, and d and h) the number of females followed. Points are actual Pearson correlation coefficients or SSE and lines are the predictions from the models. The darkness of points and lines are correlated in panels a and e to the window length (lightest: window length = 4 h; darkest: window length = 240 h) and in panels b, c, f, and g to the data frequency (lightest: data every 48 h; darkest: data every 2 h). BODY ET AL.—MEASURING FISSION–FUSION FREQUENCY797 distinguish at least 2 situations. While being just an example, this distinction requires further validation. Increasing population density increases the number of groups and the average group size (Caughley 1964; Beauchamp 2011), and increasing the density of groups increases the frequency of fusion (Pépin and Gerard 2008), while increasing group size increases group fission probability (Body et al. 2015). Therefore, when both the frequency of fission and fusion and the average group size are increasing, it may represent an increase in population density (situation 1). In contrast, opposite trends in these variables, i.e., an increase in the average group size combined with a decrease in the frequency of fission and fusion, likely represent changes in individual’s behavior (situation 2), such as the increase of group cohesiveness in our study population (Body et al. 2015). Indeed, decreasing fission probability increased the average group size and reduced the opportunities for group fusion, as fewer groups were present in the environment (Body et al. 2015). The opposite scenarios could also happen: a decrease of group cohesiveness could increase the frequency of both fission and fusion, while decreasing the average group size and a reduction in density could decrease the average group size and the frequency of fission and fusion. The latter scenario may explain the decrease in elk average group size in open habitat in response to fine-scale predation risk (Creel and Winnie 2005). The elk population density in open habitat is likely to decrease in the presence of wolves, as elk antipredator behavior consists of finding refuge under cover of wooded areas (Creel et al. 2005). Analyzing the frequency of fission and fusion in this population independent of average group size may strengthen this explanation. As shown by our study, Gower’s method led to erroneous conclusions, as the estimated variation in the frequency of fission and fusion was also influenced by the variation in average group size. Therefore, we advise that previous studies using Gower’s method (Gower et al. 2009; White et al. 2012) be updated with the modified-Gower’s method to verify their findings. We recommend using the shortest possible sliding window, given that windows should include at least 2 records per individual. We also showed that it was not necessary to increase the frequency of data to improve the fit of statistics. However, the frequency of data should match the temporal scale of the biological question. In our system, we were looking at the temporal variation in the rate of fission and fusion during the breeding season. Therefore, daily records were more appropriate than, for instance, weekly records, as it would have provided too few data. Researchers should also invest more time obtaining regular records of a subsample of females rather than obtaining irregular records of a large number of females. If the number of females followed is sufficiently large, a valuable addition would be to bootstrap the analysis and obtain confidence intervals or to study variations among individuals in their experience of group size variations. The modified-Gower’s method performed well on daily fieldbased group composition records. It can, therefore, be applied to studied natural populations where group size records include some recognizable individuals, as the modified-Gower’s method is based on individual variation in group size. Populations studied using radiocollared individuals are perfect candidates for the modified-Gower’s method, as it is relatively easy to repeatedly and regularly observe the group size in which these individuals are present. The modified-Gower method can also be applied to populations studied through transect counts where some individuals are tagged, although observation of each individual may be less regular. The variation in group size due to fission and fusion is a pattern representing group stability, not a process. Processes underlying change in this pattern could be assessed following self-organization principles (Couzin and Krause 2003) and should be studied at the individual or group level. The influence of both external variables (e.g., population density and landscape—Pays et al. 2012) and internal variables (e.g., fear of predation, sociality, and mating tactics—see Sueur et al. 2011) should be investigated by analyzing fission or fusion probability of groups or individuals. For instance, the decrease in frequency of fission and fusion during the rut of reindeer was first linked to an increase of group cohesiveness (Body et al. 2015), which was then explained by males herding females back into the group (Body et al. 2014a). In spider monkeys, individuals were splitting less often and were merging more often when the group sex-ratio was biased toward their own gender. Therefore, this process at the individual level can explain the observed pattern of sexual segregation in this species (Ramos-Fernández and Morales 2014). Fission–fusion group dynamics is often understood as an adaptive behavior with respect to variation in food distribution. This interpretation suggests that fission–fusion societies evolved from stable groups to better adjust the level of intragroup feeding competition. An alternative explanation would be that group cohesiveness evolved from highly unstable groups. Null models assuming independence of movement among individuals indeed produce a high degree of FFGD due to similar space use (Ramos-Fernández et al. 2006; Holdo and Roach 2013). Group cohesion is also increased by the presence of uninformed individuals who will copy the behaviors of their neighbors (Conradt et al. 2009; Pillot et al. 2011). The fitness benefit provided by social integration (kinship recognition, friendship, or stable dominance hierarchies—Cameron et al. 2009) may then increase the structuring of societies into cliques (stable units) and the evolution of multilevel societies. At any point in this evolution of cohesiveness, the evolution of territoriality also decreases FFGD as it prevents individuals (or social groups) from merging with each other. Group cohesiveness is, therefore, a good alternative measure to group size in assessing species social complexity (Shultz and Dunbar 2006), and measuring it in species that are not highly social is fundamental to better understand the early steps of the evolution of sociality. The modified-Gower’s method provides one way to quantify the 2nd dimension of FFGD. We identified 2 other promising methods to assess this dimension depending on the kind of data available: the method described in Fortin et al. (2009) should be applied on GPS data, while the mathematical 798 JOURNAL OF MAMMALOGY model described in Juanico (2009) could provide a solution when group sizes are recorded but individuals are not identified. For a complete assessment of the framework proposed by Aureli et al. (2008), the 2 other axes should be evaluated as well. The 1st dimension, the variation in spatial cohesion, could be measured by interindividual distances (e.g., Aureli et al. 2012; King et al. 2012; Ramos-Fernández and Morales 2014), although we personally prefer to use the distance to the nearest neighbor. It is a more efficient metric for defining groups (Croft et al. 2008; Body et al. 2014a), an easier metric to measure in field conditions, and it is theoretically independent of group size. The 3rd dimension, the variation in group composition, could be measured by metrics from social network analysis (Croft et al. 2008; Ramos-Fernández and Morales 2014). Simultaneously quantifying the 3 axes of FFGD will be a great challenge for the coming years, and we hope that the modified-Gower’s method will be a good starting point. Acknowledgments We thank J. Siitari and H. Törmänen of the Finnish Game and Fisheries Research Station for the management of GPS collars data and individuals’ data and M. Tervonen of the Finnish Reindeer Herder’s Association for the management of reindeers in Finland. We thank S. Engelhardt, N. Melnycky, and H. Gjøstein who helped with data collection and 2 anonymous referees for valuable comments on an early version of this manuscript as well as Q. Gray and E. Anderson for language correction. We also acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada to RBW, the Research Council of Norway to ØH, and the Quebec Centre for Biodiversity Sciences to GB. Supporting Information The Supporting Information documents are linked to this manuscript and are available at Journal of Mammalogy online (jmammal.oxfordjournals.org). The materials consist of data provided by the author that are published to benefit the reader. The posted materials are not copyedited. 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