Size & Shape

Size and Shape

If this is your bailiwick, definitely have a look at Claude's (2008) Morphometrics with R. Rather unfortunately, there is no R library from the book. But you can get the code from: http://www.springer.com/us/book/9780387777894 and go "Functions listed in the book." The R libraries "shapes" and "geomorph" also have geometric morphometry.

We will look at Kyra Stull's data for age, humerus, radius, femur, and tibia length. This is available from her web-page (and see "download data"), but if you mark and copy the yellow highlighted stuff below and paste to R then you will already have her data as an R object.

Stull=structure(list(age=c(4.890410959,2.619178082,9.087671233,8.273972603,7.950684932,5.745205479,8.252054795,6.068493151,
6.873972603,2.920547945,7.824657534,2.016438356,12.47945205,3.232876712,2.369863014,5.408219178,3.84109589,3.539726027,9.855,2.879452055,8.679452055,6.542465753,8.635616438,2.802739726,3.449315068,9.008219178,6.934246575,1.695890411,10.80547945,
4.978082192,3.567123288,10.92328767,8.082191781,2.282191781,10.47945205,4.515068493,2.942465753,4.797260274,3.076712329,7.602739726,6.665753425,6.5151,3.136986301,3.720547945,7.564383562,4.284931507,11.83835616,3.336986301,10.19178082,3.479452055,
4.9896,6.073972603,10.32328767,9.567123288,10.42191781,7.728767123,8.652054795,2.169863014,6.764383562,11.58082192,8.556164384,6.405479452,6.695890411,5.208219178,11.61917808,2.923287671,12.63835616,12.4,12.41369863,6.887671233,10.2411,12.40547945,
5.728767123,12.27123288,11.04931507,7.539726027,11.78356164,8.065753425,3.712328767,9.128767123,7.561643836,5.915068493,11.32054795,6.312328767,7.726027397,6.969863014,8.178082192,6.156164384,12.1260274,1.583561644,10.93150685,8.736986301,
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11,6.750684932,12.46027397,10.49589041,9.446575342,8.887671233,11.53150685,3.454794521,6.2,1.742465753,6.106849315,10.95068493,5.008219178,1.567123288,11.63561644,3.128767123,4.087671233,11.83013699,11.51232877,3.931506849,2.923287671,7.35890411,
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6.421917808,4.602739726,7.117808219),HML=c(168.73,136.84,222.37,210.94,197.3,170.54,204.88,183.7,185.61,143.59,197.71,120.27,246.43,154.59,134.78,169.57,168.88,151.29,226.66,135.84,237.57,187.62,206.47,142.32,140.06,214.45,206.7,117.35,276.81,166.08,139.79,248,209.01,150.38,236.44,165.62,140.9,174.1,142.03,207.08,195.14,185.62,140.88,161.5,217.76,167.17,289.4,157.18,231.54,164.86,157.56,188.36,236.8,224.53,235.27,209.32,220.19,117.8,206.22,
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229.77,212.59,145.62,228.71,160.53,175.69,192.95,148.85,228.13,167.22,142.27,201.29,130.19,168.8,152.81,230.28,195.41,163.18,207.47,142.46,231.94,237.51,125.49,163.22,236.99,264.78,198.78,258.46,252.3,164.31,195.72,169.5,175.55,143.2,215.32,204.32,203.08,227.96,176.77,215.52,241.83,159.99,159.17,136.21,216.55,257.14,179.64,233.98,185.13,250.7,233.93,215.71,193.07,224.73,195.95,185,115.83,177.93,252.11,188.06,139.22,253.29,150.95,162.23,254.69,
275.88,157.33,156.86,200.69,222.98,152.21,226.4,158.15,232.93,260.46,192.24,123.19,176.58,143.97,134.62,125.66,170.8,134.99,219.01,151.28,176.25,147.81,176.28,234.59,207.28,197.24,204.85,184.9,191.91,144.03,208.09,171.74,211.38,178.29,175.84,229.69,210.04,140.3,184.7,217.37,183.44,155.42,246.83,227.86,165.76,203.52,233.24,213.75,260,137.99,145.74,180.47,224.15,241.31,210.97,175,205,212.54,151.36,141.26,219.25,251.7,214.02,223.43,167.24,
230.05,238.52,203.13,209.45,133.74,115.67,193.56,99.72,244.9,249.52,148.04,204.63,167.44,114.13,194,187.04,213.67,209.04,189.09,158.26,198.63,193.76,161.67,180.52,249.55,221.61,215.9,209.42,208.16,145.26,186.61,231.28,219.96,143.17,139.7,203.28,212.37,233.2,219.37,215.5,131.62,162.03,227.07,194.38,127.11,214.74,198.7,169.44,204.08,199.67,158.27,155.24,184.95,217.99,203.06,235.66,209.15,123.25,223.86,157.52,158.01,225.87,221.1,171.52,235.45,
196.17,243.38,177.38,159.08,254.83,111.55,165,190.5,194.08,160.57,230.49,101.92,120.55,132.93,162.76,153,134.65,185.63,194.61,149.22,162.65,222.79,236.62,214.04,207.93,228.98,138.18,194.96,127.34,178.88,134.99,174.11,113.36,177.25,192.49,263.05,136.02,135.47,124.55,210.38,219.5,181.4,170.43,216.81,147.67,195.79,178.85,226.38,169.44,251.7,237.26,136.25,145.71,220.04,230.97,208.47,149.78,136.3,135.3,159.79,158.03,118.69,145.65,195.85,222.75,
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183.24,175.52,238.13,176.08,307.88,209.86,256.17,215.67,265.55,314.39,289.91,279.91,286.97,265.15,276.93,203.07,290.82,246.02,306.31,256.77,247.18,344.58,298.17,189.17,267.37,319.15,253.07,210.49,359.47,336.6,242.1,300.54,345.88,306.36,359.65,187.3,194.92,240.18,308.92,331.34,327.45,239.77,278.6,303,189.62,190.8,312.63,354.14,300.16,329.58,235.79,337.43,353.41,280.31,300.6,182.88,151.66,267.9,132.01,333.47,336.29,199.63,278.03,232.79,142.14,
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332.91,278.42,346.81,261.96,271.48,290.98,266.08,246.3,327.47,288.84,216.33,215.87,313.62,236.67,364.73,301.38,288.34,303.69,214.22,287.97),TML=c(188.78,160.93,281.61,245.65,218.97,198.64,240.45,209.63,234.8,162.44,236.54,127.62,271.62,171.06,150.75,179.56,194.46,165.55,280.22,150.52,285.05,209.82,239.41,149.09,171.13,242.68,233.73,130.65,311.72,194.76,166.46,306.83,258.49,167.02,303.79,185.76,159.95,196.07,155.04,246.47,234.62,213.01,161.85,
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173.1,253.29,212.03,255.02,223.69,246.74,224.04,174.09,172.38,273.83,244.02,257.27,235.69,125.33,234.6,140.75,241.85,205.81,225.62,218.43,298.26,325.26,241.05,291.17,201.29,147.72,241.29,204.41,234.15,303.7,169.38,204.98,239.73,195.08,153.53,267,233.07,179.52,314.39,259.39,272.05,190.36,158.08,286.73,186.31,188.6,252.32,206.72,289.98,216.2,239.45,279.94,292.38,277.92,261.92,206.56,206.71,300.12,237.5,143.75,120.97,287.66,259.06,167.37,
176.35,216.66,266.4,278.28,267.49,287.98,264.89,187.27,170.14,141.32,141.29,282.92,219.99,297.94,168.15,268.21,147.41,142.72,173.32,209.02,174.13,158.08,170.85,264.49,271.91,289.98,345.89,218.41,285.44,255.1,229.91,178.35,169.45,261.98,278.03,239.97,288.79,223.27,298.26,219.81,287.64,144.53,209.09,307.93,156.06,267.05,215.83,298.68,311.49,229.4,202.23,129.93,232.6,198.44,120.24,236.08,215.45,280.39,247.43,117.26,251.65,317.16,179.15,120.76,
291.17,263.82,274.81,207.24,297.3,226.13,328.64,221.96,145.32,131.55,203.14,164.23,271.23,150.71,274.08,241.09,185.54,310.4,274.3,249.42,164.6,270.54,180.29,193.84,230.54,161.76,280.12,180.49,157.93,241.14,139.8,192.15,164.35,253.78,220.94,188.54,249.26,153.98,289.72,280.3,141.22,182.03,277.56,310.49,233.92,284.72,312.48,194.16,230.69,188.23,199.59,154.21,235.69,252.92,230.82,280.4,197.9,265.26,281.83,177.28,187.03,156.57,258.4,322.57,
204.49,268.78,220.63,308.28,268.2,259.85,221.47,275.38,219.83,223.02,126.72,205.29,312.32,217.68,154.76,301.39,182.69,180.59,298.82,328.42,186.96,175.42,213.49,275.33,158.57,260.32,179.06,279.72,350.11,216.59,130.62,208.17,170.26,156.17,139.07,196.58,135.83,252.13,170.43,204.34,174.94,210.6,271.77,229.09,234.2,234.63,206.36,224.1,168.77,245.3,194.81,256.78,211.03,200.7,284.56,234.98,158.98,214.53,259.52,203.18,169.07,296.73,275.7,192.42,
251.79,299.81,244.91,318.72,150.83,163.1,206.03,253.27,276.88,280.17,193.54,229.93,260.03,157.49,158.45,255.64,293.45,246.75,263.2,199.96,279.72,294.32,227.8,239.09,155.48,125.84,202.46,105.86,287.04,287.22,165.39,228.98,198.62,117.12,230.14,220.68,244.83,250.4,214.55,175.24,223.82,224.75,186.3,208.97,297.6,274.07,258.69,246.56,238.02,168.46,210.86,267.9,259.29,161.84,150.36,259.12,251.99,288.63,261.77,259.75,149.86,185.95,262.34,223.62,
132.76,252.95,226.35,196.3,243.48,248.08,184.99,171.63,203.13,290.89,231.18,287.84,252.84,133.97,263.81,175.68,173.78,283.45,275.26,195.8,269.5,228.93,280.41,197.63,185.86,298.66,117.89,184.07,229.77,230.9,186.93,260.83,115.36,127.06,144.35,186.34,173.17,143,215.65,220.29,172.45,176.54,265.77,300.89,254.23,247,265.6,155.02,217.61,141.45,192.31,150.39,206.62,117.56,209.44,223.95,336.6,145.2,145.1,135.93,252.81,256.39,201.15,193.33,258.86,
166.16,207.44,205.77,258.77,200.16,315.97,296.65,150.33,168.82,246.42,272.02,255.24,168.91,160.08,152.16,190.64,167.51,128.97,171.54,248.93,261.42,199.11,151.95,133.21,157.42,294.38,261.13,275.97,261.3,154.42,272.9,292.27,236.33,172.93,228.96,271.82,247.5,333.87,247.53,221.68,313.81,173.39,181.86,252.55,163.51,125.42,227.26,171.51,220.02,235.98,235.79,268.55,222.24,251.43,256.53,258.11,264.02,280.71,159.23,265.91,227.54,270.41,132.7,212.66,
166.95,217.95,236.54,178.88,183.16,275.3,143.59,302.16,268.01,272,171.15,297.88,284.4,252.8,259.49,218.41,281.37,246.52,174.23,148.49,212.25,273.88,226.79,268.16,329.56,190.55,321.3,271.79,263.56,255.79,144.36,265.46,327.49,239.9,182.82,222.66,166.42,257.83,293.96,170.01,306.02,151.41,254.84,229.79,250.22,243.67,157.34,304.47,250.35,257.87,264.7,308.82,229.56,176.28,262.86,253.92,283.37,235.93,156.66,232.43,251.96,253.79,241.96,206.93,140.66,
216.52,232.37,193.57,181.78,247.73,291.81,188.83,161.81,291.65,271.52,243.83,169.52,155.76,184.95,327.13,200.46,282.73,262.07,175.15,300.58,263.67,150.45,213.29,260.39,193.87,312.24,152.34,255.18,265.29,280.84,193.1,170.94,230.02,275.68,244.9,168.56,275.61,239.13,267.59,268.63,158.25,244.9,130.52,230.16,195.35,294.4,215.04,160.9,145.6,201.29,152.37,222.28,279.73,233.71,263.49,146.35,272.36,239.44,293.28,176.98,228.75,230.77,257.59,215.27,
221.36,207.91,230.36,243.12,290.9,308.78,231.02,116.06,187.29,272.74,223.17,211.75,298.46,131.08,157.65,155.05,172.09,192.9,231.47,89.16,151.78,157.23,141.25,132.09,118.13,273.89,146.31,101.72,291.88,149.58,91.89,98.67,133.28,105.07,149.44,94.82,206.85,146,156.73,116,190.82,157.91,142.76,135.73,190.73,77.94,223.17,174.3,271.63,177.46,200.79,163.65,148.19,147.5,94.04,167.85,172.82,140.73,164.44,187.32,150.12,71.04,175.73,149.84,256.9,
98.36,150.99,147.62,91.69,191.14,172.75,233.8,119.73,317.75,109.2,71.54,117.52,278.51,151.92,79.3,92.18,124.79,204.04,105.51,77.33,124.89,97.95,68.64,278.51,79.22,117.61,266.72,222.92,253.83,219.23,257.48,265.22,343.39,253.03,185.83,277.37,227,291.63,200.15,222.68,231.18,213.14,201.08,270.01,243.74,175.22,180.18,270.69,190.59,305.07,248.47,233.31,247.64,175.7,243.58)),.Names=c("age","HML","RML","FML","TML"),row.names=c(1L,3L,4L,
5L,9L,10L,11L,12L,13L,14L,15L,16L,19L,22L,23L,24L,25L,26L,27L,29L,30L,31L,32L,33L,34L,40L,41L,42L,43L,45L,46L,50L,52L,54L,58L,62L,63L,64L,65L,73L,75L,78L,79L,80L,82L,85L,86L,87L,89L,90L,91L,92L,93L,94L,96L,97L,99L,101L,105L,107L,108L,109L,111L,112L,115L,118L,119L,121L,122L,123L,125L,126L,127L,131L,133L,137L,139L,140L,141L,142L,143L,145L,146L,147L,148L,150L,151L,152L,154L,155L,158L,159L,161L,163L,165L,167L,170L,172L,174L,
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458L,459L,461L,462L,464L,465L,466L,467L,468L,469L,471L,472L,475L,476L,478L,480L,482L,483L,485L,486L,489L,491L,492L,493L,494L,495L,499L,500L,502L,505L,506L,507L,509L,510L,512L,513L,515L,516L,517L,518L,520L,521L,525L,526L,528L,531L,534L,535L,536L,538L,541L,542L,544L,547L,548L,549L,550L,551L,553L,554L,555L,556L,557L,558L,559L,560L,561L,562L,563L,564L,567L,568L,569L,571L,572L,573L,574L,575L,576L,578L,579L,581L,582L,585L,586L,589L,591L,592L,
593L,594L,596L,598L,600L,603L,604L,607L,609L,610L,611L,613L,615L,616L,617L,618L,619L,620L,622L,623L,625L,626L,627L,628L,629L,630L,631L,632L,634L,635L,636L,638L,639L,641L,642L,643L,644L,648L,649L,652L,653L,654L,655L,656L,658L,659L,660L,661L,662L,664L,665L,666L,667L,668L,669L,673L,674L,677L,679L,684L,685L,688L,689L,691L,692L,693L,695L,696L,697L,698L,699L,701L,702L,705L,706L,707L,708L,711L,713L,716L,717L,720L,721L,726L,727L,728L,729L,731L,
733L,737L,741L,742L,746L,748L,749L,751L,752L,753L,754L,756L,759L,760L,761L,762L,764L,765L,766L,767L,768L,769L,770L,771L,772L,773L,775L,776L,777L,778L,779L,781L,782L,784L,786L,787L,790L,791L,792L,793L,798L,799L,800L,801L,802L,803L,805L,808L,810L,811L,812L,813L,815L,816L,817L,818L,820L,821L,823L,824L,825L,826L,827L,829L,830L,831L,832L,833L,834L,836L,837L,839L,842L,844L,845L,846L,847L,852L,854L,857L,858L,859L,860L,861L,862L,863L,864L,865L,
866L,868L,869L,870L,871L,873L,874L,875L,876L,877L,880L,881L,882L,883L,884L,885L,886L,887L,888L,890L,891L,892L,893L,894L,896L,897L,898L,900L,901L,902L,903L,904L,906L,908L,910L,911L,912L,913L,914L,917L,920L,921L,922L,924L,926L,928L,929L,930L,931L,934L,935L,936L,939L,940L,941L,943L,949L,951L,952L,953L,954L,955L,956L,960L,961L,962L,964L,965L,966L,967L,968L,969L,971L,974L,976L,980L,981L,982L,983L,984L,985L,987L,988L,989L,991L,993L,994L,995L,
996L,997L,998L,999L,1000L,1001L,1002L,1003L,1004L,1006L,1013L,1016L,1017L,1019L,1021L,1022L,1023L,1026L,1027L,1028L,1029L,1035L,1036L,1038L,1039L,1041L,1042L,1043L,1044L,1045L,1047L,1049L,1050L,1052L,1053L,1054L,1055L,1056L,1057L,1058L,1060L,1062L,1064L,1066L,1068L,1069L,1070L,1071L,1072L,1075L,1077L,1082L,1083L,1084L,1086L,1088L,1089L,1090L,1091L,1092L,1093L,1094L,1095L,1096L,1097L,1098L,1101L,1102L,1104L,1105L,1106L,1111L,1113L,
1115L,1116L,1119L,1121L,1128L,1129L,1131L,1133L,1136L,1137L,1138L,1142L,1143L,1144L,1145L,1149L,1150L,1151L,1152L,1154L,1155L,1158L,1160L,1162L,1163L,1165L,1166L,1168L,1170L,1172L,1173L,1175L,1183L,1184L,1186L,1189L,1190L,1192L,1193L,1195L,1196L,1197L,1202L,1203L,1205L,1206L,1207L,1209L,1212L,1213L,1214L,1215L,1219L,1220L,1223L,1224L,1226L,1227L,1229L,1231L,1232L,1234L,1235L,1237L,1238L,1239L,1241L,1243L,1245L,1249L,1250L,1254L,
1255L,1256L,1258L,1259L,1263L,1265L,1268L,1269L,1271L,1272L,1274L,1276L,1277L,1280L,1283L,1284L,1285L,1287L,1290L,1293L,1294L,1295L,1296L,1298L,1299L,1300L,1301L,1302L,1304L,1305L,1306L,1308L,1310L),class="data.frame")
# This is the end

To "see" the data:

pairs(Stull,cex=.2,upper.panel=NULL)

There is a matrix-y way to deal with size and shape using principal components, but we can use a simpler approach, which is Darroch and Mosimann size and shape variables. DM define individual size as the geometric mean (the p^th root of the product of the p variables) of the raw scale (non-log) measurements, and shape variables as the individual variables divided by their geometric mean within the case. So:

For individual i, size is: (HML*RML*FML*TML)^(1/4) and shape variables are the measurements divided by size. But in the log scale, log size is:

ln(size) = ln((HML*RML*FML*TML)^(1/4)) = (ln(HML)+ln(RML)+ln(FML)+ln(TML))/4, so the simple average of the log measurements. And any given log shape variable (e.g., for HML) is:

ln(HML/size) = ln(HML) – ln(size), so the log measurement minus the average log measurement within the case. And "off she goes" (boring harp version or the more exciting squeeze box version). The example below uses the long bone measurements on the "raw" (not log) scale.

DM = function (dat2=Stull)

{
labs = c('HML shape','RML shape','FML shape','TML shape')
N=NROW(dat2)
dat=dat2[,-1]
opar = par(no.readonly = T)
on.exit(par(opar))

# DM Size

size=apply(dat,1,prod)^(1/4)
plot(sqrt(dat2[,1]),size,xlab='Age on square root scale',ylab='DM Size',axes=FALSE)
box()
axis(2)
axis(1,at=sqrt(1:12),lab=1:12)

if (.Platform$OS.type != "windows" ) {windows=function() quartz()}
windows()

# DM Shape

shape = dat/size%o%rep(1,NCOL(dat))

Age = dat2[,1]

par(mfrow=c(1,2))

for(i in c(1,3)){
plot(Age,shape[,i],ylab=labs[i])
}

windows()
par(mfrow=c(1,2))
for(i in c(2,4)){
plot(Age,shape[,i],ylab=labs[i])
}
}

This will produce three plots, but R has the annoying habit of placing the three plots exactly over each other, so drag the two in the "forefront" slightly out of the way so that you can view all three plots. The plot of size against (square root of) age shows just what you would expect, that the kids get larger with age. The two plots of humerus and femur shape against age show also just what you would expect. The arm grows with negative allometry while the thigh grows with positive allometry. The same can be said for the forearm and the leg. And for the non-anatomists out there, the arm is just the humerus while the leg is the tibia and fibula. This differs from the colloquial use of "arm" and "leg" to mean the upper limb and lower limb.