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Measurement error revisited: its importance for the analysis of size and shape of birds

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Perktaş U., Gosler A. G. 2010. Measurement error revisited: its importance for the analysis of size and shape of birds. Abstract. Measurement error in morphological characters is an important issue for many ornithological studies (e.g. ecomorphology,
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  Measurement error revisited: its importance for the analysis of size andshape of birds Utku P ERKTA Ş 1 & Andrew G. G OSLER 2 1 Department of Biology (Zoology Section), Faculty of Science, Hacettepe University, Beytepe, 06800, Ankara, TURKEY, e-mail: perktas@hacettepe.edu.tr 2 Edward Grey Institute of Field Ornithology, Department of Zoology, University of Oxford, OX1 3PS, UK  Perkta ş  U., Gosler A. G. 2010. Measurement error revisited: its importance for the analysis of size and shape of birds.Acta Ornithol. 45: 161–172. DOI10.3161/000164510X551309 Abstract. Measurement error in morphological characters is an important issue for many ornithological studies (e.g.ecomorphology, quantitative studies of heritability, studies of systematic and geographic variation). The variation inexternal morphological characters, such as wing and tarsus length, is usually evaluated using multivariate statisticalmethods such as principal component analysis (PCA). These are often considered better than univariate statistical meth-ods for explaining size and shape variation in bird populations because they reduce the ‘dimensionality’ of the data —the size of individual measures (wing etc.) are assumed to contain a component reflecting a general character ‘size’.However, the effect of measurement error on principal components has not been formally assessed with respect to suchdata. Here we report three examples in order to assess the importance of measurement error for analyses within and between bird populations. The effect of measurement error on PCA is also discussed in relation to the importance of levels of error in shape components.Our results indicate that, in relation to size (PC1), principal component scores are affected less by measurementerror if the covariance matrix is used rather than the correlation matrix. However, the effects of relative measurementerror were substantially greater in subsequent axes, which represent shape variation rather than size, than they werein the size axis (PC1). Measurement error may, therefore, be a more important issue for shape axes than for the size axisand this problem may be exacerbated further if very few characters are used in the PCA. Our results also indicate thatPCA is especially sensitive to issues relating to sample size. We recommend that if reducing the measurement error insize and shape measures is not possible, and the sample size is small ( ≤ 30), principal component scores should bederived using the covariance matrix, as these are more likely to give more robust results. Key words: measurement error, size, shape, principal component analysis, Common Chaffinch, Fringilla coelebs , methods,museum studiesReceived — Febr. 2010, accepted — Oct. 2010ACTA ORNITHOLOGICAVol. 45 (2010) No. 2 INTRODUCTIONThe measurement error associated with vari-ous morphological characters has a bearing onmany ornithological studies, and although thesecharacters are used in studies of ecologicalprocesses and ecomorphology (e.g. Van Valen1965, Johnson 1966, Gosler 1987, Keast & Saunders1991, Mulvihill & Chandler 1991, Kaboli et al.2007), age and sex differences (e.g. Arenas & Senar2004), quantitative studies of heritability (e.g.Gosler & Harper 2000, Merilä & Sheldon 2000,Akesson et al. 2007), evolutionary studies of fluc-tuating asymmetry (Merilä & Björklund 1995) and evolutionary ecological studies in birds(Hromada et al. 2003, Tryjanowski & Š imek 2005,Tryjanowski et al. 2007), the importance of themeasurement error has generally been ignored.This deficiency may be particularly important insystematic studies of taxonomic relationshipsusing morphology, and in studies of geographicvariation in birds (Martin 1991, Slotow &Goodfriend 1996, Telleria & Carbonell 1999, Ponset al. 2004, Soobramoney et al. 2005, Dmitrenok etal. 2007). Since systematic inferences have also been made using morphological characters, thesemay be wrong if measurement error was not eval-uated correctly. For example, the likelihood of aType II statistical error (failure to detect a realeffect) increases as the measurement error associ-ated with a trait increases (Bailey & Byrnes 1990,Lougheed et al. 1991, Yezerinac et al. 1992).  Nevertheless, good examples do exist of wheremeasurement error has been taken into account inrelation to geographic variation in birds (e.g.Grant 1979, Zink 1986).To assess the scale of this problem, we consid-ered it in a morphological study of the CommonChaffinch Fringilla coelebs in which we calculatedthe relative measurement error of several externalmorphological characters in museum specimensusing repeated measurements and Model IIAnalysis of Variance. We also considered the ef -fects of measurement error on the first three com-ponents derived from a principal componentanalysis (hereafter PCA) because PCA is widelyrecognized for its value in evaluating overall bodysize (Rising & Somers 1989). As a general ap -proach, the first principal component, PC1, from aseries of morphological characters is interpretedas an allometric size variable, so that by definition,the ‘shape’ dimension interpreted from theremaining principal components lacks any infor-mation related to allometry (Bookstein 1989). Thefirst two principal components have been usedwidely in morphological studies of birds, andespecially in studies of geographic variation, toreduce the dimensionality of data and to reachrobust conclusions (Martin 1991, Martin &Pithoccelli 1991, Wiklund 1996, Rising 2001, Rojas-Soto 2003). To assess the effect of measurementerror, we used PCA in three ‘examples’ based onwithin- and between-population datasets (geo-graphic variation perspective) of Common Chaf -finches. Although this kind of measurement error eval-uation has been made before (Bailey & Byrnes1990, Lougheed et al. 1991, Yezerinac et al. 1992),we re-evaluated it to clarify its effect on size, andespecially shape, evaluation within a PCA. Wetried to find an appropriate evaluation process forPCA, assuming that measurement error isinevitable, and some suggestions are made as tohow to estimate the size of birds by using multi-variate statistical techniques, which are affectedless by measurement error than are conventionalanalyses.METHODSOur study is based on data from maleCommon Chaffinch skin specimens and skull sam -ples in the collections of the American Museum of Natural History, the British Museum of NaturalHistory and The Zoological Research Museum 162 U. Perkta ş & A. G. Gosler Alexander Koenig in Bonn. Eight external charac-ters (bill length, nostril, bill depth, bill width, wing length, 3th primary length, tail length and tar-sometatarsus length) from skin specimens andfour skull characters (Fig. 1) were measured.Measurements were taken by the first author(U.P.) to avoid interpersonal variability.It is important to make a distinction betweenthe variation that exists among individuals withinand between populations, which arises throughdifferential effects of age, sex, genotype, nutritionetc., and measurement error due for example toslight variation in measurement technique between specimens or observer (obviated in thisstudy as all measurements were made by U.P.).While the former is what we wish to study, the latter is inevitable, and if too great may mask theformer. Across the distribution range of theCommon Chaffinch, different subspecies have been recognized on the basis of plumage and body-size (Cramp & Perrins 1994). We assessedvariation in a number of external morphologicalcharacters in relation to this described subspecificvariation. In the first example, a subspecies, Fringilla coelebs coelebs , collected in Bonn andstored in The Zoological Research MuseumAlexander Koenig, was assessed to examine theeffect of measurement error on size variationwithin a population. In the second example, twosubspecies, Fringilla coelebs spodiogenys from Fig. 1. Measurements on skull specimens: 1 — Skull length, 2 — Pre-orbital length, 3 — Post-orbital length, and 4 — Orbitallength.  Tunisia (n = 15) and Fringilla coelebs coelebs fromBonn (n = 30), were evaluated together to exam-ine the effect of measurement error on size andshape differences between populations (i.e. a geo-graphic variation perspective). All measurementdetails follow Svensson (1992). Measurementswere taken using digital calipers (± 0.1 mm) for bill characters and tarsometatarsus; and a stoppedrule (wing-rule) for other measurements (i.e. wing length, 3th primary length, tail length; estimatedto ± 0.1 mm). In the third example, digital imagesof skull samples, which came from the differentparts of the Western Palearctic distribution rangeof the Common Chaffinch (n = 19), were evaluat-ed by using TpsDig software (Rohlf 2003) to deter-mine the effect of measurement error on size andshape, derived from direct, and thus more robust,skull measurements. Scale setting was made foreach skull in TpsDig to standardize measure-ments. We measured each character three timeson each skin specimen, and twice on each skullspecimen. Measurements were repeated after onedataset was collected on all specimens. Statistical Analysis Measurement error was calculated as the per-centage of within-individual variance in the totalvariance (Lougheed et al. 1991):ME%= × 100where ME% is measurement error, s 2within is with-in-individual variance and s 2among  is the varianceamong individuals.Model II Analysis of Variance (ANOVA) wasused to determine the within- and among-birdcomponents of variance (s 2within and s 2among  )(Bailey & Byrnes 1990), after which ME% was cal-culated using the equation presented above. AKolmogorov-Smirnov test was used to test fornormality of the data. Coefficient of variation (V*)for each character was calculated using correctionfor small sample size: V*= V (1 + 1/4n), where nis sample size and V is uncorrected coefficient of variation (Sokal & Rohlf 1995: 58). Grubbs' methodfor assessing outliers was used (Barnett & Lewis1994). A nested Analysis of Variance (Sokal &Rohlf 1995) was designed to determine both thevariance components and the importance of measurement error in Example 2. All characters,some of which had high measurement error (e.g.usually > 15%), were then evaluated together in aPCA, and principal component (PC) scores werederived. In the second step, those characters thathad high measurement error were excluded fromthe PCA, and PC scores were derived again with-out those characters. PCA was performed forthree examples using both the correlation matrixand covariance matrix. The srcinal models of PCA-based size and shape employed log-trans-formed data and the variance-covariance matrix(e.g. Jolicoeur 1963, Mosimann 1970). However,srcinal (i.e. untransformed) data have frequently been used in ornithological studies because birdsdisplay determinate growth (see e.g. Rising 2001).For this reason, only srcinal data were used inthe present studies for the PCAs based on correla-tion and covariance matrices.Two different test procedures, one parametric(i.e. Pearson’s product moment correlations) andone non-parametric (i.e. Spearman’s rank correla-tions), were used with the correlation matrix. Theexistence of significant correlation coefficients between variables, which were used in the corre-lation matrix based on the parametric test proce-dure, was tested with Barlett’s Test. To performthe PCA, XLStat 2009 was used. XLStat gives thePC scores as unstandardized values, as presentedin the standard description of PCA in Legendre &Legendre (1998).Finally, ANOVA was used to evaluate the effectof measurement error on PC scores in the threedifferent data sets.RESULTSMeasurement error (ME %) varied from 4.11%(bill length) to 38.58% (bill width) in the samples(Table 1). Across the eight measurements (Table 1),there was a positive and significant correlation between the percent measurement error of  Fringilla coelebs coelebs and Fringilla coelebs spodio- genys (r = 0.976, p < 0.001). That is, the characterswith the greatest and least measurement error inone subspecies were similarly more or less error-prone in the other subspecies, indicating that theerror relates to the measurement technique ratherthan the subspecies.ME% for data obtained from digital images of skull samples varied from 0.13% (skull length) to5.26% (orbital length) across the samples (Table 2).The relationship between coefficients of varia-tion and ME% was not significant in either of thetwo datasets (r = 0.084, p = 0.422; r = -0.071, p =0.433). Some characters, especially bill width andtarsus, had relatively high measurement error, buthad small coefficients of variation. We suspect, The importance of measurement error 163 s 2 within s 2 among  + s 2 within  of the within-population variation in the corre-lation matrix based on the parametric test proce-dure, the effect of measurement error for PC1 was 6.7% with all characters and 35.6% withoutcharacters that had high measurement error. ForPC2 in the same matrix based on the same proce-dure, it was 23.5% with all characters and 39.6%without characters that had high measurementerror (i.e. not as expected — see below). For PC3,also in the same matrix based on the same proce-dure, it was 38.7% with all characters and 23.9%without characters that had high measurementerror (Fig. 2). 164 U. Perkta ş & A. G. Gosler therefore, that having well defined measurementlandmarks is important for the measurement of museum skin specimens, and shrinkage, whichvaries in degree with age of the specimen (Jenni & Winkler 1989), might also be important in museum skin specimens because the relative posi-tion of measurement landmarks for the charactermight vary between specimens. Within-population study of  F. c. coelebs (Example 1) When PCA was used to assess the effect of measurement error on the first (PC1), second(PC2) and third principal component axes (PC3) Table 1. Descriptive statistics and percentage measurement error for the measurements of the Common Chaffinch Fringilla coelebs spodiogenys (n = 15) and Fringilla coelebs coelebs (n = 30) males. Geographic variation — males of both subspecies evaluatedtogether (n = 45). Measurements were given as millimeters (mm). V* — corrected coefficient of variation. Variable Mean SD Min Max V* ME% Fringilla coelebs spodiogenys Bill length 16.19 0.57 14.67 16.80 3.59 4.11Nostril 9.56 0.39 8.90 10.33 4.19 11.59Bill depth 7.88 0.37 7.27 8.50 4.75 6.27Bill width 6.79 0.21 6.50 7.07 3.10 38.58Wing length 89.6 2.76 84.67 93.67 3.13 4.993 th primary length 65.83 2.27 61.33 70.00 3.51 22.80Tail length 71.91 1.95 69.67 75.67 2.77 9.30Tarsus length 18.32 0.36 17.83 19.10 1.99 7.49 Fringilla coelebs coelebs Bill length 15.73 0.35 15.10 16.30 2.26 6.28Nostril 9.65 0.37 9.00 10.40 3.84 6.54Bill depth 7.20 0.26 6.70 7.80 3.67 7.34Bill width 6.52 0.19 6.10 6.90 2.93 33.41Wing length 89.19 1.82 85.00 91.70 2.05 7.163 th primary length 65.71 2.18 60.30 69.30 3.34 17.71Tail length 67.56 2.62 62.00 72.30 3.91 8.00Tarsus length 17.82 0.4 16.40 18.40 2.25 8.49Geographic variation Bill length 15.97 0.50 14.67 16.80 3.13 4.31Nostril 9.62 0.37 8.90 10.40 3.85 7.57Bill depth 7.43 0.44 6.70 8.50 5.92 6.87Bill width 6.61 0.23 6.10 7.07 3.48 37.35Wing length 89.31 2.19 84.67 93.67 2.45 6.203 th primary length 65.75 2.13 60.30 70.00 3.24 20.21Tail length 69.99 3.14 62.00 75.67 4.49 6.35Tarsus length 17.96 0.46 16.40 19.10 2.56 5.56 Table 2. Descriptive statistics and percentage measurement error for the measurements of male Common Chaffinch skulls (n = 19). Measurements were given as millimeters (mm). V* — corrected coefficient of variation. Variable Mean SD Min Max V* ME%Skull length 32.82 1.72 29.67 35.73 5.52 0.13Pre-orbital length 9.34 0.43 8.45 10.17 4.84 1.47Post-orbital length 15.28 0.55 14.43 16.38 3.79 4.76Orbital length 10.32 0.53 9.03 11.32 5.36 5.26  The importance of measurement error 165 The fact that measurement error increasedwhen high-error characters were omitted wasunexpected, but it occurred because this reducedvariable set essentially had only one reliable com-ponent, so that all remaining variance expressedin the other components was error.Using the covariance matrix, the effect of measurement error on PC1 was more or less thesame as when PC1 was derived using the correla-tion matrix. On PC2, however, it was notincreased by the removal of high measurement-error characters (Fig. 3). Between-population study of  F. c. spodiogenys and  F. c. coelebs (Example 2) In the between-population variation analysis,the measurement error was unimportant for onlya few characters, which had low measurementerror as determined from the nested ANOVAresults. Between populations and individualsvariation from variance components were moreimportant than the measurement error for allcharacters (Table 3). Because there is a significantcorrelation between the percentage error of totalvariance in the nested ANOVA and the measure-ment error of each character (Pearson productmoment: r = 0.987, p < 0.001); if measurementerror is high, it may affect the variation.No serious difference for PC1 was foundregarding the effect of measurement error be -tween the data sets with or without high measure-ment-error characters. In terms of the effect of theerror, however, PC3 was more sensitive than PC1and PC2 in the covariance matrix. Hence, thecovariance matrix calculated without high meas-urement-error characters provided a better evalu-ation process for the shape axes (Fig. 4). As inExample 1, we found little difference in measure-ment error between the full and restricteddatasets for PC1 and PC2 when using the covari-ance matrix. The effect of measurement error onPC1 was less than 5% for all combinations (Fig. 5). The analysis of skull samples (Example 3) The analysis of skull samples showed lessmeasurement error than did Examples 1 or 2,whether in relation to PC1 especially and irrespec-tive of whether the covariance or correlation Fig. 2. The effect of measurement error (% ME) on PC1, PC2and PC3 derived from correlation matrix based on non-para-metric test procedure (i.e. Spearman’s rank correlations) andparametric test procedure (i.e. Pearson’s product moment cor-relations) in Example 1. A — eight characters included: allcharacters had high or low measurement error, B — five char-acters included: only the characters that had low measurementerror.Fig. 3. The effect of measurement error (% ME) on PC1, PC2 and PC3 derived from covariance matrix in Example 1. A — eight characters included: all characters had high or lowmeasurement error, B — five characters included: only thecharacters that had low measurement error.Fig. 4. The effect of measurement error (% ME) on PC1, PC2and PC3 derived from correlation matrix based on non-para-metric test procedure (i.e. Spearman’s rank correlations) andparametric test procedure (i.e. Pearson’s product moment cor-relations) in Example 2. A — eight characters included: allcharacters had high or low measurement error, B — five char-acters included: only the characters that had low measurementerror. 5 10 15 20 25 30 35 40 455 10 15 20 25 30 35 40 455 10 15 20 25 30 35 40 455 10 15 20 25 30 35 40 45 %ME %ME SPEARMAN’S RANK PEARSON PRODUCT MOMENT %ME %ME PC3PC2PC1 AB PC3PC2PC1PC3PC2PC1PC3PC2PC15 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45 %ME %ME SPEARMAN’S RANK PEARSON PRODUCT MOMENT PC3PC2PC1PC3PC2PC15 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45 %ME %ME PC3PC2PC1PC3PC2PC1 AB 5 10 15 20 25 30 35 40 45 %ME PC3PC2PC15 10 15 20 25 30 35 40 45 %ME PC3PC2PC1 A B
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