Optimal species distinction by discriminant analysis: comparing established methods of character selection with a combination procedure using ant morphometrics as a case study
2006; Wiley; Volume: 45; Issue: 1 Linguagem: Inglês
10.1111/j.1439-0469.2006.00372.x
ISSN1439-0469
AutoresKarl Moder, Birgit C. Schlick‐Steiner, Florian M. Steiner, Sylvia Cremer, Erhard Christian, Bernhard Seifert,
Tópico(s)Animal Behavior and Reproduction
ResumoJournal of Zoological Systematics and Evolutionary ResearchVolume 45, Issue 1 p. 82-87 Free Access Optimal species distinction by discriminant analysis: comparing established methods of character selection with a combination procedure using ant morphometrics as a case study K. Moder, K. Moder Institute of Mathematics and Applied Statistics, Department of Spatial-, Landscape-, and Infrastructure-Sciences, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria These authors contributed equally to this work.Search for more papers by this authorB. C. Schlick-Steiner, B. C. Schlick-Steiner Institute of Forest Entomology, Forest Pathology and Forest Protection, Department of Forest and Soil Sciences, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria Institute of Zoology, Department of Integrative Biology and Biodiversity Research, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria These authors contributed equally to this work.Search for more papers by this authorF. M. Steiner, F. M. Steiner Institute of Forest Entomology, Forest Pathology and Forest Protection, Department of Forest and Soil Sciences, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria Institute of Zoology, Department of Integrative Biology and Biodiversity Research, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria These authors contributed equally to this work.Search for more papers by this authorS. Cremer, S. Cremer Institute of Biology, Department of Population Biology, University of Copenhagen, Copenhagen East, DenmarkSearch for more papers by this authorE. Christian, E. Christian Institute of Zoology, Department of Integrative Biology and Biodiversity Research, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, AustriaSearch for more papers by this authorB. Seifert, B. Seifert State Museum of Natural History Görlitz, Görlitz, GermanySearch for more papers by this author K. Moder, K. Moder Institute of Mathematics and Applied Statistics, Department of Spatial-, Landscape-, and Infrastructure-Sciences, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria These authors contributed equally to this work.Search for more papers by this authorB. C. Schlick-Steiner, B. C. Schlick-Steiner Institute of Forest Entomology, Forest Pathology and Forest Protection, Department of Forest and Soil Sciences, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria Institute of Zoology, Department of Integrative Biology and Biodiversity Research, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria These authors contributed equally to this work.Search for more papers by this authorF. M. Steiner, F. M. Steiner Institute of Forest Entomology, Forest Pathology and Forest Protection, Department of Forest and Soil Sciences, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria Institute of Zoology, Department of Integrative Biology and Biodiversity Research, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, Austria These authors contributed equally to this work.Search for more papers by this authorS. Cremer, S. Cremer Institute of Biology, Department of Population Biology, University of Copenhagen, Copenhagen East, DenmarkSearch for more papers by this authorE. Christian, E. Christian Institute of Zoology, Department of Integrative Biology and Biodiversity Research, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Vienna, AustriaSearch for more papers by this authorB. Seifert, B. Seifert State Museum of Natural History Görlitz, Görlitz, GermanySearch for more papers by this author First published: 01 November 2006 https://doi.org/10.1111/j.1439-0469.2006.00372.xCitations: 10 Authors' addresses: Karl Moder, Institute of Mathematics and Applied Statistics, Department of Spatial-, Landscape-, and Infrastructure-Sciences, University of Natural Resources and Applied Life Sciences Vienna, BOKU, Gregor-Mendel-Str. 33, A-1180 Vienna, Austria. E-mail: [email protected]; Birgit C. Schlick-Steiner (for correspondence), Florian M. Steiner, Erhard Christian, Institute of Forest Entomology, Forest Pathology and Forest Protection, Department of Forest and Soil Sciences; and Institute of Zoology, Department of Integrative Biology and Biodiversity Research, BOKU, University of Natural Resources and Applied Life Sciences Vienna, Gregor-Mendel-Str. 33, A-1180 Vienna, Austria. E-mail: [email protected], [email protected]; Sylvia Cremer, Institute of Biology, Department of Population Biology, University of Copenhagen, Universitetsparken 15, 2100 Copenhagen East, Denmark. E-mail: [email protected]; Bernhard Seifert, State Museum of Natural History Görlitz, PSF 300154, D-02826 Germany. E-mail: [email protected] AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract We compare the performances of established means of character selection for discriminant analysis in species distinction with a combination procedure for finding the optimal character combination (minimum classification error, minimum number of required characters), using morphometric data sets from the ant genera Cardiocondyla, Lasius and Tetramorium. The established methods are empirical character selection as well as forward selection, backward elimination and stepwise selection of discriminant analysis. The combination procedure is clearly superior to the established methods of character selection, and is widely applicable. Introduction Increasing knowledge of the dimensions of biodiversity challenges systematic biologists by the difficult task of species characterization and discrimination (Wilson 2003). This is especially so with cryptic diversity, i.e. highly similar species that have formerly been regarded as one nominal species (Knowlton 1993; Saez and Lozano 2005). The need to correctly address these species in taxonomy, as well as in determination routine, entails more rigorous and precise analysis (Seifert 2002). In the early days of systematic biology, only qualitative and simple numerical characterizations were available (e.g. Linnaeus 1758); in recent decades, however, quantitative character assessment proved to be necessary (Rohlf and Marcus 1993). Data from a wide range of disciplines and methods such as morphometrics (Moraes et al. 2004), pigmentation quantification (Hollocher et al. 2000), semiochemistry (Anyanwu et al. 2000), acoustics (Kingston et al. 2001) and molecular genetics (Desrosiers et al. 1999) are now analysed statistically. Discriminant analysis (DA), introduced by Fisher (1936), is a well-established procedure for species distinction (e.g. Bühler 1964; Flury and Riedwyl 1983; Lamprecht 1999; Seifert 2002). On the basis of the discriminant function, a DA classifies observations according to predefined classes (Jobson 1992; SAS Institute 2004) – species in our case. For taxonomic and determination routine purposes, DA is used to determine which characters best predict an individual's species membership. The performance of a discriminant function can be evaluated by estimating the classification error, i.e. the portion of samples classified into wrong classes (SAS Institute 2004). The classification error varies with the number and the discriminatory power (with respect to the underlying hypothesis) of characters (cf. SAS Institute 2004). In the following, we use the term 'optimal discrimination' for discrimination with minimum classification error (and the fewest characters, if more than one combination of characters produces the minimum classification error). Up until recently, it was common practice to select characters for DA empirically, e.g. by using the classification error of a promising character combination as feedback for improving the combination, or by fixing a certain cut-off of correlation coefficients (e.g. Seifert 2002; Csősz and Markó 2004). Increasingly, however, statistical packages have facilitated character selection with three well-defined methods: forward selection (FW), backward elimination (BW) and stepwise selection (SW) (Klecka 1980). As far as we are aware, the three methods have never been properly comparatively evaluated, although some authors discuss advantages and disadvantages of one or the other method (e.g. Flury and Riedwyl 1983, for FW and BW; Krishnaiah and Kanal 1990, for FW, BW and SW). A comprehensive comparison of the performances of the three methods is beyond the scope of our paper. From the biologist's perspective, it seems more urgent to learn how the established selection methods perform compared with the optimal discrimination in the above sense. In this paper, we thus present a novel method to find the optimal combination of characters for DA and compare the performance to established procedures by morphometric data sets from morphologically most similar species groups of ants. This includes a reanalysis of DAs empirically selected by Seifert (2003) for a series of species pairs of Cardiocondyla and a comparison with forward selection, backward elimination and stepwise selection as implemented in stepdisc of SAS 9.1 for two cryptic species of Lasius and a species group of nine most similar Tetramorium ants. Materials and Methods We programmed the novel combination procedure as a macro in SAS 9.1 (SAS Institute 2004). It runs all DAs with all possible combinations of a given set of characters. Given a number m of characters, the number M of possible character combinations is described by Eqn 1: (1) where i is the actually selected number of characters. The method starts with DAs including the minimum number of characters, i.e. 1, and ends with DAs including the maximum number of characters, i.e. m. For any number of characters, repeatedly one character is exchanged until all characters have been included. After each run the classification error for the included characters is saved. When all runs are performed, the character set that produces the lowest classification error is known for any number of characters. Comparison of these lowest classification errors for all numbers of characters allows the optimal discrimination to be identified. The program stepdisc in SAS 9.1 selects characters for DA based on the probabilities derived from covariance analysis, with the new variable as covariate (SAS Institute 2004). There are three different methods of character selection (SAS Institute 2004): (1) Forward selection, FW, begins with no character in the model. At each step, stepdisc enters the character that contributes most to the discriminatory power of the model as measured by Wilks' lambda (see below for definition), the likelihood ratio criterion. When none of the unselected characters meets the entry criterion, the FW process stops. (2) Backward elimination, BW, begins with all characters in the model except those that are linearly dependent on previous characters. At each step, the character that contributes least to the discriminatory power of the model as measured by Wilks' lambda is removed. When all remaining characters meet the criterion to stay in the model, the BW process stops. (3) Stepwise selection, SW, begins with no characters in the model. At each step, the model is examined. If the character in the model that contributes least to the discriminatory power of the model as measured by Wilks' lambda fails to meet the criterion to stay, then that character is removed. Otherwise, the character not in the model that contributes most to the discriminatory power of the model is entered. When all characters in the model meet the criterion to stay and none of the other characters meet the criterion to enter, the SW process stops. SW can be expected to obtain slightly better results than FW and BW because more models are evaluated, which enhances the chance to find a model with a smaller classification error. We calculated Wilks' lambda for character combinations derived by FW, BW, SW and the combination procedure to characterize the discriminatory power of the combinations (Backhaus et al. 2003). Wilks' lambda is a direct measure of the proportion of variance in each combination of characters that is unaccounted for by the model. The larger the proportion of the variance accounted for by the combination of characters, the smaller the value of Wilks' lambda. The data sets (Table 1) were composed of 14–25 morphometric characters from a total of 1282 workers from 497 nest samples belonging to 23 species of the ant genera Cardiocondyla (from Seifert 2003), Lasius (B. Seifert, unpublished data) and Tetramorium (Steiner et al. 2006; Schlick-Steiner et al. in press). Our data met the general demands of DA (Backhaus et al. 2003): all characters were continuous, no zero-values were included, no character was constant in any of the species, variances were homogeneous within species, and data for all characters were normally distributed within species. The definitions of classes for DA, i.e. the putative species identities, were derived by qualitative morphological analysis of workers and gynes in Cardiocondyla (Seifert 2003), by gas chromatography–mass spectrometry of cuticular hydrocarbons in Lasius (S. Cremer et al., unpublished data) and by phylogenetic analysis of mitochondrial DNA in Tetramorium (Steiner et al. 2006; Schlick-Steiner et al. in press). DAs were performed on single individuals in Cardiocondyla and Tetramorium and on nest means of three to five individuals in Lasius. Table 1. Size of morphometric data sets of the ant genera Cardiocondyla, Lasius and Tetramorium Genus Species w n m Cardiocondyla bicoronata Seifert, 2003 29 16 18 kagutsushi Terayama, 1999 117 57 18 koshewnikovi Ruzsky, 1902 35 16 18 mauritanica Forel, 1890 138 68 18 minutior Forel, 1899 78 41 18 nigra Forel, 1905 46 21 18 shuckardi Forel, 1891 14 9 18 stambuloffi Forel, 1892 52 23 18 tjibodana Karavajev, 1935 31 17 18 venustula Wheeler, 1908 26 13 18 sp. l 15 9 18 sp. m 15 7 18 Lasius sp. 2 126 32 14 sp. 3 87 23 14 Tetramorium caespitum (Linnaeus, 1758) 54 18 25 hungaricum Röszler, 1935 48 16 25 impurum (Foerster, 1850) 48 16 25 tsushimae Emery, 1925 51 17 25 sp. A 48 16 25 sp. B 71 15 25 sp. C 48 12 25 sp. D 36 12 25 sp. E 69 23 25 w, number of workers; n, number of nests; m, number of morphometric characters. Results In most cases, established approaches to selecting characters for DA did not produce the optimal character combination (Fig. 1). The empirical approach of Seifert (2003) yielded an optimal discrimination for only one of six species pairs of Cardiocondyla (Cardiocondyla shuckardi versus Cardiocondyla venustula; Fig. 1a); for two other species pairs (Cardiocondyla koshnewikovi versus Cardiocondyla stambuloffi and Cardiocondyla sp. l versus Cardiocondyla sp. m), a classification error of 0.0 was found by both our combination procedure and the empirical approach, but Seifert (2003) needed a higher number of characters. For the remaining three couples, Seifert (2003) did not detect a character combination with a classification error as low as in the optimal discrimination. In the most intricate species pair, the difference between the performances was most pronounced: 0.083 classification error for 11 characters by the combination procedure versus 0.164 for 18 characters by the empirical approach. Figure 1Open in figure viewerPowerPoint Performance of DA based on combinations of morphometric characters of ants selected by the novel combination procedure (filled circles) and by established methods, i.e. empirically (open diamonds) by Seifert (2003; a) and by forward selection (open triangles), backward elimination (open pentagons) and stepwise selection (open squares) implemented in stepdisc of SAS 9.1 (b, c). (a) Cardiocondyla, reanalysis of data from Seifert (2003), each DA designed to discriminate between two species (a, Cardiocondyla bicolor versus Cardiocondyla nigra; b, Cardiocondyla kagutsushi versus Cardiocondyla mauritanica; c, Cardiocondyla koshewnikovi versus Cardiocondyla stambuloffi; d, Cardiocondyla sp. l versus Cardiocondyla sp. m; e, C. minutior versus C. tjibodana; f, Cardiocondyla shuckardi versus Cardiocondyla venustula), for the combination procedure only optimal character combination shown; (b) Lasius sp. 2 versus sp. 3; (c) Tetramorium, discrimination of nine species, Tetramorium caespitum, Tetramorium hungaricum, Tetramorium impurum, Tetramorium tsushimae, Tetramorium sp. A–E) (combinations of one to four characters not shown). Ant drawings from Hölldobler and Wilson (1990): Cardiocondyla and Lasius after Smith (1947), Tetramorium after R.W. Taylor, unpublished data (F. Nanninga artist) In Lasius no distinctive combination was found by BW among two characters, and by FW, BW and SW among seven characters (Fig. 1b, Table 2), because the significance level of two or more characters was almost equal, leading to the selection of all or none of these characters. For one and 14 characters the classification error was identical in FW, BW, SW and the combination procedure, for three characters it was identical in BW, SW and the combination procedure; for all other numbers of characters, FW, BW and SW performed worse than the combination procedure. Overall, FW, BW and SW performed very similarly; the character combination with the smallest error among the three methods was achieved for nine combinations by FW, and for 10 combinations by BW and SW (only those numbers of characters included, for which all three methods achieved a result; n = 10). The best discrimination found by any of the three established selection methods included three characters and resulted in a classification error of 0.078 (BW and SW); the optimal discrimination, however, included five characters with a classification error of 0.046. In our combination procedure, the lowest error for a given number of characters was stationary if more than five and increased if >11 characters were selected. Values of Wilks' lambda for all methods decreased with increasing numbers of characters in the combinations (Table 2). Values of FW, BW and SW – with one exception (three characters, FW) – were equal to or smaller than values of the combination procedure. Table 2. Values of Wilks' lambda for combinations of morphometric characters of two Lasius species (Lasius sp. 2 versus sp. 3; total number of characters 14) and nine Tetramorium species (Tetramorium caespitum, Tetramorium hungaricum, Tetramorium impurum, Tetramorium tsushimae, Tetramorium sp. A–E; total number of characters 25), selected by the novel combination procedure (Comb) and by established methods, i.e. by forward selection (FW), backward elimination (BW) and stepwise selection (SW) implemented in stepdisc of SAS 9.1 Characters Lasius Tetramorium FW BW SW Comb FW BW SW Comb 1 0.5277 0.5277 0.5277 0.5277 0.5764 0.6386 2 0.4302 0.4302 0.4565 0.2814 0.2688 3 0.3818 0.3651 0.3651 0.3651 0.1235 0.1506 0.1235 0.1391 4 0.3463 0.3463 0.3463 0.3484 0.0775 0.0911 0.0775 0.1022 5 0.3338 0.3338 0.3338 0.3427 0.0647 6 0.3255 0.3255 0.3255 0.3427 0.0422 0.0638 7 0.3388 0.0307 0.0307 0.0442 8 0.3188 0.3188 0.3188 0.3382 0.0237 0.0230 0.0237 0.0396 9 0.3174 0.3174 0.3174 0.3382 0.0197 0.0186 0.0197 0.0305 10 0.3167 0.3167 0.3167 0.3373 0.0165 0.0155 0.0165 0.0167 11 0.3160 0.3160 0.3160 0.3344 0.0141 0.0159 12 0.3157 0.3157 0.3157 0.3195 0.0121 0.0111 0.0121 0.0152 13 0.3157 0.3157 0.3157 0.3172 0.0098 0.0098 0.0123 14 0.3157 0.3157 0.3157 0.3157 0.0093 0.0087 0.0087 0.0113 15 0.0082 0.0078 0.0078 0.0111 16 0.0074 0.0071 0.0090 17 0.0067 0.0065 0.0065 0.0084 18 0.0061 0.0059 0.0059 0.0082 19 0.0056 0.0055 0.0072 20 0.0052 0.0051 0.0051 0.0062 21 0.0049 0.0048 0.0048 0.0060 22 0.0046 0.0046 0.0046 0.0048 23 0.0043 0.0043 0.0043 0.0048 24 0.0041 0.0041 0.0041 0.0043 25 0.0040 0.0040 0.0040 0.0040 Values for optimal character combinations are given in bold letters. Missing values indicate failure in finding a distinctive combination for the respective numbers of characters. For the same reasons as in Lasius, the character selection methods applied by stepdisc found no distinctive combinations for several numbers of characters in Tetramorium (FW: 1, 2, 5, 6, 11, 13; BW: 5, 7, 11, 16, 19; and SW: 1, 2, 5, 6; Fig. 1c, Table 2). None of the established methods FW, BW or SW ever reached the minimum classification error. In this genus, the three methods also performed very similarly. Absolute differences between the classification errors were very small; the character combination with the smallest error among the three methods was achieved for seven combinations by FW, for 10 by BW and for 12 by SW (only those numbers of characters included, for which all three methods achieved a result; n = 15). The best combination found by any of the established methods required all 25 characters (classification error 0.155; FW, BW and SW), the optimal discrimination 21 characters (classification error 0.152). Like in Lasius, in our combination procedure the classification error increased slightly if more than 21 characters were selected. Values of Wilks' lambda for all methods decreased with increasing numbers of characters of the combinations (Table 2). Values of FW, BW and SW – with two exceptions (two and three characters, BW) – were equal to or smaller than values of the combination procedure. Discussion Our study shows that established methods of selecting characters for DA frequently lead to suboptimal results: to a higher classification error or to a higher number of characters required. This applies to DAs with two or more predefined classes. In the case of empirical character selection, the reason for suboptimal performance is clearly the high number of possible combinations that should – but cannot – be completely tested, even more so when extremely similar species necessitate many characters for discrimination. When only a few characters ensure reliable discrimination, however, experienced workers can find the optimal character combination even with the empirical approach (Fig. 1a). The methods forward selection, backward elimination and stepwise selection always performed very similarly (but in the Tetramorium example the expected slight advantage of SW came into effect) and always distinctly worse than the combination procedure. This is in line with general modelling methodology: exhaustive search is the only method guaranteed to find the optimal subset for an arbitrarily complex problem (Dwinnel 1998). The reason why FW, BW and SW never detected the optimal character combination in our study may be that selection processes based on analyses of variances do not sufficiently consider the dependencies among characters. The insufficiency of the three methods is illustrated by comparing their classification errors (Fig. 1b,c) and their Wilks' lambda values (the measure of their discriminatory power in the character selection process, Table 2): according to the latter, their performance is practically always better than or equal to that of the combination procedure, despite the clearly worse performance as measured by the classification error. Moreover, for character combinations selected by the combination procedure – like for those selected by FW, BW and SW – Wilks' lambda decreases with an increasing number of characters. In the combination procedure performance, this means that adding characters beyond the optimal discrimination (five characters in Lasius, 21 in Tetramorium) lowers Wilks' lambda but not the classification error. In other words, as already suspected by SAS Institute (2004), Wilks' lambda is no ideal measure of discriminatory power. This finding is relevant to all those who apply FW, BW and SW, because these methods select characters based on analyses of variances (independent of the package used). More characters do not necessarily mean better discrimination, as became evident in Lasius and Tetramorium. Although this appears counter-intuitive at first glance, it is well explicable. In DA, the ratio of variances between and variances within groups is maximised (e.g. Backhaus et al. 2003). This, however, is not necessarily identical with minimizing the classification error. If character sets contain not only characters that support the hypothesis underlying the classification, but also non-discriminating, neutral characters, or even characters supporting an alternative hypothesis, then inclusion of characters other than the supporting ones into the model will lead to stationarity or increase the classification error. Our Lasius example (Fig. 1b) impressively illustrates this: the characters included in combinations of one to five characters support the underlying hypothesis, those added in combinations of six to 11 characters are neutral, but those added in combinations of 12–14 characters would rather support an alternative classification. In conclusion, paring down the number of characters for a specific species distinction is not only desirable for practical data assessment reasons, but also to arrive at more accurate models – a phenomenon likewise known from practical experience in other fields of modelling (Dwinnel 1998). Our method is novel in as much as a similar combination procedure has never before been programmed. However, already Bühler (1964) proposed to calculate DAs for all possible character combinations (referring to a total number of only four characters, though), accompanied by the warning that this approach is 'very time-consuming' [translated]. The advent of powerful computers has shifted our understanding of 'time-consumption' to a new dimension, but the computational complexity of our combination procedure remains an – inevitable – disadvantage. In Eqn 1, the time problem is an exponential one, making it, in terms of computer science, NP-hard. The computation time depends on the number of characters, on the number of species and the number of individuals (because every run involves computing a subsequent DA for the chosen character combination), and on the power of the computer. The computation time needed for the complete search for the optimal discrimination in our Tetramorium example (Fig. 1c; nine species, 473 individuals, 25 characters; resulting in a total of 33 554 431 runs) may serve as illustration: c. 10 days using an AMD Athlon 64 × 2 Dual Core Processor 4800+. What about conceivable cases in which considerably more characters only minimally decrease the classification error? No general decision rule can be offered here. The user must balance 'optimal discrimination' with an excess of working time, giving top priority either to (1) the minimum error rate (possibly with a predefined cut-off, e.g. 0.05, cf. Wiens and Servedio 2000), or to (2) reduced effort. The optimal character combination as derived by our combination procedure can be used for purposes of classification routine, i.e. for classificatory discriminant analyses (SAS Institute 2004), provided that (1) the 'primary' data, i.e. the data used in the combination procedure, were indeed representative; (2) the 'secondary' data, i.e. the data to be classified subsequently, had not been included in the primary data; and (3) all secondary data came from species characterised by the primary data, i.e. a careful preidentification of individuals has been conducted. The combination procedure presented here is likewise applicable to other classification tasks such as sexual dimorphism (Ostrand et al. 2001), caste differences (Eickwort 1969), habitat stratification within populations (Prinzing et al. 2004), hybrid zone studies (Buckley et al. 2003) and, finally, discrimination tasks outside biology. 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