ppular fr medicinal use and which families are the least ppular by native peple in these regins? T answer these questins, we cnsider regressin mdeling

Transcription

1 Statistical Analysis f Ppularity f Medicinal Plants Lianfen Qian and Sujin Wang Flrida Atlantic University and Texas A&M University Abstract The similarity amng six regins n the distributin f the medicinal species is analyzed. A cmparative statistical analysis is cnducted using three mdeling methds. The mst standard f which has been emplyed in an earlier study by Merman et al. (1999). The mst apprpriate mdel amng the three fr analyzing such data is chsen accrding t the empirical results f the data analyses. The least and mst utilized medicinal families are identied fr all six regins, including a new data set cllected fr the regin f Shuar. It is als cncluded that there is distinct medicinal use f plant species in the six dierent regins. Keywrds and phrases: Heterscedasticity, medicinal usage, plant selectin, regressin mdeling, variable transfrmatin. 1 Intrductin It is a ppular research area t identify plant species used as medicine. Recent references included Berlin and Berlin (1996), Jhns (1990), Merman (1998) and Merman et al. (1999). A few studies such as Merman (1991, 1996), Phillips and Gentry (1993), Merman et al. (1999) shwed that sme plants are mre ppularly selected fr medicinal use than thers. Merman et al. (1999) cnducted a cmparative analysis f data fr ve regins using regressin mdeling, residual and crrelatin analyses. Five mst and least ppular medicinal ras were identied. Recently, a new data set cllected fr Shuar is kindly prvided by Dr. Brad Bennett f Flrida Internatinal University. This data set has ttally 116 families f species with the ttal number f 823 species, as given in the appendix. In this paper, we study the similarity amng the six regins n the distributin f the medicinal species. In each f the six regins, there are 100 r mre families f plants. Fr each family, there are tw variables cllected: the ttal number f families f species and the number f families f medicinal species. Let Y ij and x ij be the number f medicinal species and the ttal number f species in the ith regin and the jth family, respectively. With these data sets, several questins are t be addressed: (1) Is there any similarity amng the regins n using medicinal species? (2) Hw d the x's relate t the Y 's? (3) Which families are the mst 1

2 ppular fr medicinal use and which families are the least ppular by native peple in these regins? T answer these questins, we cnsider regressin mdeling fr the tw variables. Merman et al. (1999, p. 59) pint ut why it is mre reasnable t use regressin mdeling fr Y n x than t use nly the ranks f families in terms f the number f medicinal species (Y ) r ranks f the prprtin f medicinal species per family (Y=x). The mdel cnsidered by Merman et al. (1999) is as fllws: Y ij = i + i x ij + ij ; j = 1; :::; n i ; i = 1; :::; k; (1.1) where ij N(0; 2 ij). The standard linear regressin analysis assumes that 2 ij = 2 stays cnstant fr all i; j. Since if there is n species in a family, naturally there is n medicinal species, it appears t be mre reasnable t cnsider a smewhat dierent mdel frm (1.1) in this paper as fllws: Y ij = i x ij + ij ; j = 1; :::; n i ; i = 1; :::; k: (1.2) One main bjective in ur analyses is t cnduct the statistical test f the hypthesis H 0 : i = 0 ; i = 1; :::; k: (1.3) where 0 is unknwn, i.e., t test the equal mdel fr all k regins. Because f the unknwn 0, it is nt straightfrward t deal with. Fr this purpse, we rewrite (1.3) as Y ij = ( x ij + ij ; Thus the crrespnding null hypthesis becmes i = k x ij + i x ij + ij ; i = 1; :::; k 1: (1.4) H 0 : i = 0; i = 1; :::; k 1: (1.5) Let Y, X and E be the stack clumn vectrs fr Y ij, x ij and ij. Dente x i = (x i1 ; :::; x ini ) 0 ; O i = (0; :::; 0) 0 in R n i ; i = 1; :::; k 1 and X i = (O 0 1; O 0 2; :::; O 0 i 1; x 0 i; O 0 i+1; :::; O 0 k 1) 0 ; a vectr in R n 1++n k. Then mdel (1.4) can be rewritten as Y = X + k 1 X i=1 i X i + E: (1.6) Merman et al. (1999) cmpared the data fr ve regins: Nrth America (NA), Krea (KO), Chiapas Highlands (CH), Kashmir (KA), Ecuadr (EC). Fr each regin, the data cntains the ttal number f species and the number f medicinal species per family. Our 2

3 study will include the new data set fr Shuar (SH) prvided by Dr. Bennett by analyzing all the data fr the six regins tgether. There are n NA = 255; n KO = 136; n CH = 144; n KA = 100; n EC = 118 and n SH = 116 families having species in regins NA, KO, CH, KA, EC and SH, respectively. There are ttally 273 dierent families in the six regins. We cnducted the sequential analysis using mdel (1.6). First we analyze the data fr the six regins indexed (EC, NA, KO, SH, CH, KA)=(1,2,3,4,5,6) using the rdinary linear regressin fr the riginal data. The results are shwn in Tables 1 and 2. Insert Tables 1 and 2 here Table 1 shws that the regressin relatinship is statistically signicant with p value= Table 2 shws that all cecients i 's (i = 1; :::; 5) are signicantly dierent frm zer with p values less than r equal t.001. That is, the pattern f medicinal use in KA is signicantly dierent frm all ther ve regins. Figure 1 illustrates a plt f the estimated ij 's against ^Y ij after tting an rdinary least squares mdel (1.6) t the cmbined riginal data fr all six regins. Frm Figure 1, we bserve that Asteraceae in CH and NA are having the highest psitive residuals indicating the mst ppular families fr medicinal use in all regins. While Paceae in NA and CH, Cyperaceae in NA and Orchidaceae in CH are the least ppular ras in the six regins. Nte that bth the mst ppular and least ppular ras in the six regins are lcated in NA and CH. This shws the typical pattern f the tw regins. We have als cmputed the hat matrix and the DFIT t detect the high leverage pints and inuential pints, respectively (see Belsley, Kuh and Welsch, 1980). Asteraceae and Paeceae frm NA have high leverage because they have the tw largest ttal number f species (2688, 1505) that prduce the large diagnal elements f the hat matrix. There are many inuential pints, amng them Asteraceae and Fabaceae frm CH, Paceae frm NA are extremely inuential because f having large prprtins (37.5%, 21.7% and 30%, respectively) f medicinal use. Insert Figure 1 here Figure 1 indicates that larger residuals fr the larger predicted values expect a few in- uential and/r leverage pints. This is because the respnse is actually cunts (number f medicinal species) whse distributin seems t have similarity t the Pissn distributin, where the variance is identical t the mean. This kind f relatinship between the mean and variance suggests t use the square rt transfrmatin fr the respnse and the predictr. The new estimated cecients f this methd fr the square rt transfrmed data are shwn in Table 3. Detailed numerical results using this mdel as well as thers are prvided in the next sectin. 3

4 2 Stabilizing transfrmatin As Sen and Srivastava (1990, pp ) pint ut, when heterscedasticity ccurs we may take ne f the tw types f actins t make the ij apprximately equal. One cnsists f transfrming the riginal data when the variance f Y ij depends n its mean; the ther invlves weighting the regressin. Frm the discussin abve, here we take the Freeman and Tukey (1950) square rt transfrmatin fr bth Y ij and x ij. Figure 2 shws the square rt transfrmed data fr all six regins. It is seen frm Figure 2 that the transfrmatin stabilizes the errr structure f the data fr all six regins. Thus, we mdel the data with the fllwing mdel: Y 1=2 ij = ( x 1=2 ij + ij ; i = 6; x 1=2 ij + i x 1=2 ij + ij ; i = 1; :::; 5: Insert Figure 2 and Table 3 here As befre, the mdel is signicant with p value= The square rt transfrmatin has little eect n the verall t f the mdel and the estimated regressin parameters, see Table 3. Figure 3 shws the plt f the residuals against the tted values fr the transfrmed data which suggests a much better mdel t than the residual plt fr the riginal data in Figure 1. The dierences between the results f statistical inferences using the riginal and transfrmed data are nt large, but the later represents mre clsely the nature f the underlying distributin f the respnse variable. The square rt transfrmatin stabilizes the errr variability s that the resultant statistical inferences are mre reliable. Insert Figure 3 here Based n Tables 2 and 3, we can rst eliminate KA frm the plled square rt transfrmed data. Then we rerganize the transfrmed data fr the remaining ve regins and run the regressin fr the ve regins indexed by (NA, KO, SH, CH, EC)=(1,2,3,4,5). The results shw that the regressin is statistically signicant with p value=0.000 and all cecients ( i ; i = 1; :::; 4) are signicant which indicates that the pattern f medicinal use in EC is signicant frm the rest fur regins (NA, KO, SH, CH). This analysis can be cntinued further sequentially fr the rest fur regins with labeling (KO, SH, CH, NA)=(1,2,3,4). The results indicate that the pattern in NA is signicant frm ther three regins. Then further analysis can be perfrmed fr the rest three regins: (SH, CH, KO)=(1,2,3). The cnclusin here is that they are all similar in pattern. Therefre, we cnclude that the regins shw fur patterns n the medicinal use f species. They are KA, (SH, KO, CH), NA, EC. Amng the grup (SH, KO, CH), the similarity between SH and KO is strnger than between CH and KO. Amng the fur patterns f the six regins, EC has the lwest usage f medicinal 4

5 species, fllwed by that f NA, then the grup f three regins (SH, KO, CH), then the highest in KA. 3 Cmparative analysis In rder t cmpare with the results f Merman, et al. (1999), we nw turn t statistical analyses fr each regin separately. Similar t what we have experienced with the cmbined data abve, it wuld be useful t cnsider the fllwing tw issues. First, instead f using the simple regressin with intercept as was dne in Merman, et al. (1999), regressin thrugh the rigin is used. This has the intuitive explanatin that if there is n species f a family in a regin, f curse there is n medicinal species. Secnd, there exists strng heterscedasticity in the randm errrs within each regin. Thus, we cnduct the analyses fr bth the riginal and the square rt transfrmed data fr each regin using linear regressin thrugh the rigin. The measure f gdness-f-t R 2 is als reprted. Thus, fr each regin, we use the mdel Y i = x i + i ; i = 1; :::; n; (3.1) where Y i is the number (r its square rt) f medicinal species and x i is the ttal number (r its square rt) f species in the ith family, n is the ttal number f families, i is the randm errr and x i is the mean regressin functin fr the ith family. Insert Table 4 here The regressin functin fr the regressin thrugh the rigin is ^Y i = ^x i ; i = 1; :::; n; where ^ = P n i=1 x iy i = P n i=1 x2 i is the rdinary least squared estimatr f. Dene the residual by e i = Y i ^Y i ; i = 1; : : : ; n. The number n f families varies frm regin t regin. Table 4 shws that the prprtin f the variability in Y explained by x, R 2 = 1 P n i=1 e2 i =P n Y 2 i=1 i, is higher fr the square rt transfrmed data than the riginal data fr all six regins. Apart frm the intuitive explanatins give abve, this numerical bservatin indicates that mdel (3.1) seems t be indeed a better alternative t Merman's mdel. We further nte that using the square rt transfrmed data appears t give a much better mdel t. Insert Tables 5 and 6 here The ve mst ppular and least ppular families are listed in Tables 5 and 6 fr the riginal and transfrmed data. In Tables 5 and 6, the a, b and c represent the rankings 5

6 f the mst and least ppular plants, i.e., with the largest psitive and negative residuals respectively fr the three mdeling methds: the methd f Merman et al., regressin thrugh the rigin fr the riginal data and fr the transfrmed data. We rst discuss methds a and b. Fr the ve mst ppular families, sme f their rankings are changed slightly. Fr example, the rankings f (a; b) = (4; 5) fr Rsaceae, while that f (a; b) = (5; 4) fr Ranunculace in NA. This means that methd a ranks Rsaceae the furth ppular ra, while methd b ranks it the fth. It is similar fr ther tw pairs: Frm (2; 3) t (3; 2) fr Asteraceae and Lamiaceae in KO, and (6; 5) fr Apiaceae, and (2; 3) t (3; 2) fr Piperaceae and Malvaceae in SH. Fr the ve least ppular families, the rst three rankings are identical fr all regins and nly a few rankings f furth and fth are changed. Using the square rt transfrmatin and then regressin thrugh the rigin (methd c) gives smewhat dierent classicatin. Fr the ve mst ppular families, it requires 14 ut f 20 families withut verlap t list the tp 5 in 4 f the 6 regins (all but Ecuadr and Shuar), and 20 ut 30 families withut verlap t list the tp 5 fr all six regins. This shws sme dierences in the detailed rankings f the families amng the regins fr utilizing medicinal plants. The tp ve families in NA have rankings f three methds (a; b; c)=(1; 1; 2), (2; 2; 4), (3; 3; 1), (5; 4; 3) and (6; 6; 5) fr Asteraceae, Apiaceae, Eriaceae, Ranunculace and Salicaceae, respectively. In KO, the rankings are (6; 6; 1), (1; 1; 2), (8; 7; 3), (3; 2; 4) and (11; 9; 5) fr Araliaceae, Liliaceae, Areceae, Lamiaceae and Rutaceae, respectively. One can see that there is little dierence amng the methds fr the rankings f the ppularity. Methd c shws that Araliaceae is the mst ppular family in KO, and s n. Since there are 136 families f species in KO, the change f the rankings is relatively small. Please refer t Tables 5 and 6 fr details fr the ther regins and fr the ve least ppular families. It is wrth pinting ut that ne family, Orchidaceae in SH, is nt identied as a tp family fr medicinal use, but its ranking is raised signicantly by methd c. Table 6 shws that it is ranked 51 ut f 255 in NA and 50 ut f 116 in SH frm the bttm. This cntrasts the rankings f 10 and 5 frm the bttm in NA and SH by bth methds a and b. Since this family is ne f the largest f plants as is pinted ut in Merman et al. (1999), ur results appear t be mre reasnable fr this family. On the ther hand this family is still pretty much ignred by native peples in NA, KO, KA and CH. Similar families are Cyperaceae and Paceae. Cyperaceae is preferred by SH with rankings (5; 7; 9) frm the tp, but ignred in NA, KO, KA, CH and EC. Paceae is f middle use fr medicinal purpses with rankings frm the bttm (67; 68; 68) ut f 118 families in EC and (17; 14; 59) ut f 116 families in SH, but it is quite ignred by native peples in NA, KO, KA and CH. In cnclusin, in this paper we have cnsidered three dierent mdels t analyze the medicinal plant data, arguing fr the methd f regressin thrugh the rigin fr the square rt transfrmed data (methd c) as the mst apprpriate apprach amng the three fr 6

7 such analyses. The resultant ndings and cmparisns derived frm the data analyses pint t sme interesting patterns and phenmena regarding medicinal use f plants in the six regins that the data have cvered. research area. All this is pen fr interpretatin f experts in this References [1] Berlin, E. & Berlin, B. (1996) Medicinal Ethnbilgy f the Highland Maya f Chiapas, Mexic: The Gastrintestinal Disease. Princetn University Press, Princetn. [2] Belsley, D. A. & Kuh, E. & Welsch, R. E. (1980) Regressin Diagnstics, Jhn Wiley & Sns, New Yrk. [3] Jhns, T. (1990) With Bitter Herbs They Shall Eat It: Chemical Eclgy and the Origins f Human Diet and Medicine. University f Arizna Press, Tucsn. [4] Freeman, M. F. & Tukey, J. W. (1950) Transfrmatins related t the angular and the square rt. Annal f Mathematical Statistics, 21, pp [5] Merman, D. E. (1998) Native American Ethnbtany. Timber Press, Prtland, OR. [6] Merman, D. E. (1991) The medicinal ra f native Nrth America: An analysis. Jurnal f Ethnpharmaclgy, 31, pp [7] Merman, D. E. (1996) An analysis f the fd plants and drug plants f native Nrth America. Jurnal f Ethnpharmaclgy, 52, pp [8] Merman, D. E. & Pembertn, R.W. & Kiefer, D. & Berlin, B. (1999) A cmparative analysis f ve medicinal ras. Jurnal f Ethnbilgy, 19, pp [9] Phillips, O. & Gentry, A. H. (1993) The useful plants f Tambpata, Peru: I. Statistical hypthesis tests with a new quantitative technique. Ecnmic Btany, 47, pp [10] Sen, A. K. & Srivastava, M. (1990) Regressin Analysis: Thery, Methds, and Applicatins. Springer-Verlag, New Yrk. 7

8 List f Captins Tables: Table 1: ANOVA fr (1.6) fr the cmbined data Table 2: P-values f testing H 0 in (1.5) fr the cmbined data Table 3: P-values f testing H 0 in (1.5) fr the transfrmed cmbined data Table 4: Slpe parameter estimates with R-squared values Table 5: Five mst ppular families f medicinal species (empty cell means n family fund in the regin) Table 6: Five least ppular families f medicinal species (empty cell means n family fund in the regin) Figures: Figure 1: Plt f residuals against predicted values fr mdel (1.6) using the OLSE Figure 2: Plt f square rt transfrmed medicinal species against ttal species per family Figure 3: Plt f residuals against predicted values fr mdel (1.6) using the squared rt transfrmed data 8

9 Table 1. ANOVA fr (1.6) fr the cmbined data Surce DF SS MS F P Regressin Residual Errr Ttal Table 2. P values f testing H 0 in (1.5) fr the cmbined data Predictr Cef StDev T P X X X X X X Table 3. P values f testing H 0 in (1.5) fr the transfrmed cmbined data Predictr Cef StDev T P X 1= =2 X 1 1=2 X 2 1=2 X 3 1=2 X 4 1=2 X Table 4. Slpe parameter estimates with R-squared values Merman's mdel Original data Transfrmed data Regin Cnstant Slpe R 2 Slpe R 2 Slpe R 2 NA KO CH KA EC SH

10 Table 5. Five mst ppular families f medicinal species (empty cell means n family fund in the regin) Regin NA KO KA CH EC SH Family ( a, b, c ) ( a, b, c ) ( a, b, c ) ( a, b, c ) ( a, b, c ) ( a, b, c ) Asteraceae ( 1, 1, 2 ) ( 2, 3, 27 ) ( 1, 1, 1 ) ( 1, 1, 1 ) ( 45, 46, 46 ) ( 8, 8, 13 ) Apiaceae ( 2, 2, 4 ) ( 5, 5, 6 ) ( 11, 13, 58 ) ( 5, 5, 6 ) ( 56, 54, 62 ) ( 6, 5, 4 ) Ericaceae ( 3, 3, 1 ) ( 17, 17, 30 ) ( 51, 50, 43 ) ( 13, 13, 16 ) ( 78, 77, 84 ) ( 41, 41, 41 ) Rsaceae ( 4, 5, 9 ) ( 13, 19, 64 ) ( 37, 73, 81 ) ( 4, 4, 5 ) ( 40, 39, 37 ) Ranunculaceae ( 5, 4, 3 ) ( 4, 4, 8 ) ( 3, 3, 4 ) ( 17, 17, 14 ) ( 75, 72, 87 ) Salicaceae ( 6, 6, 5 ) ( 134, 134, 133 ) ( 39, 37, 25 ) ( 38, 42, 43 ) Lamiaceae ( 8, 8, 14 ) ( 3, 2, 4 ) ( 4, 4, 28 ) ( 2, 2, 2 ) ( 93, 91, 98 ) ( 1, 1, 1 ) Slanaceae ( 14, 14, 19 ) ( 24, 24, 24 ) ( 6, 6, 6 ) ( 3, 3, 7 ) ( 16, 20, 39 ) ( 9, 12, 16 ) Liliaceae ( 20, 20, 43 ) ( 1, 1, 2 ) ( 5, 5, 3 ) ( 12, 12, 22 ) ( 35, 33, 28 ) ( 103, 102, 112 ) Araliaceae ( 27, 27, 25 ) ( 6, 6, 1 ) ( 38, 37, 30 ) ( 140, 139, 139 ) ( 95, 94, 100 ) ( 59, 56, 71 ) Iridaceae ( 30, 30, 44 ) ( 35, 36, 42 ) ( 54, 53, 46 ) ( 6, 6, 3 ) Mraceae ( 34, 34, 30 ) ( 21, 21, 21 ) ( 9, 8, 5 ) ( 114, 116, 101 ) ( 117, 117, 109 ) ( 93, 99, 57 ) Araceae ( 39, 39, 58 ) ( 8, 7, 3 ) ( 90, 89, 87 ) ( 7, 7, 4 ) ( 1, 1, 1 ) ( 109, 111, 60 ) Balsaminaceae ( 82, 82, 84 ) ( 67, 64, 56 ) ( 27, 27, 18 ) ( 86, 84, 87 ) ( 3, 3, 5 ) ( 61, 58, 73 ) Zingiberaceae ( 95, 95, 96 ) ( 21, 20, 14 ) ( 130, 130, 137 ) ( 7, 7, 8 ) ( 4, 4, 5 ) Lganiaceae ( 213, 213, 124 ) ( 111, 111, 117 ) ( 31, 31, 22 ) ( 52, 52, 70 ) ( 4, 4, 2 ) ( 95, 93, 105 ) Rutaceae ( 231, 232, 188 ) ( 11, 9, 5 ) ( 36, 36, 60 ) ( 72, 77, 83 ) ( 42, 41, 38 ) ( 92, 91, 63 ) Euphrbiaceae ( 236, 236, 108 ) ( 12, 11, 11 ) ( 2, 2, 2 ) ( 21, 24, 53 ) ( 90, 103, 54 ) ( 17, 20, 33 ) Piperaceae ( 239, 239, 241 ) ( 96, 96, 100 ) ( 105, 108, 91 ) ( 27, 39, 44 ) ( 2, 3, 3 ) Malvaceae ( 240, 240, 187 ) ( 119, 118, 127 ) ( 13, 12, 27 ) ( 19, 21, 39 ) ( 5, 5, 3 ) ( 3, 2, 2 ) Fabaceae ( 253, 253, 186 ) ( 7, 8, 26 ) ( 85, 94, 91 ) ( 138, 140, 102 ) ( 2, 2, 7 ) ( 54, 98, 47 ) Cyperaceae ( 254, 254, 255 ) ( 136, 136, 136 ) ( 95, 95, 95 ) ( 142, 142, 141 ) ( 102, 104, 58 ) ( 5, 7, 9 ) Lecythidaceae ( 6, 6, 4 ) ( 35, 35, 39 ) Table 6: Five least ppular families f medicinal species (empty cell means n family fund in the regin) Regin NA KO KA CH EC SH Family ( a, b, c ) ( a, b, c ) ( a, b, c ) ( a, b, c ) ( a, b, c ) ( a, b, c ) Paceae ( 1, 1, 2 ) ( 2, 2, 2 ) ( 1, 1, 1 ) ( 1, 1, 2 ) ( 67, 68, 68 ) ( 17, 14, 59 ) Cyperaceae ( 2, 2, 1 ) ( 1, 1, 1 ) ( 6, 6, 6 ) ( 3, 3, 4 ) ( 16, 14, 60 ) ( 112, 110, 108 ) Fabaceae ( 3, 3, 72 ) ( 132, 131, 113 ) ( 16, 7, 10 ) ( 7, 5, 43 ) ( 117, 117, 112 ) ( 63, 19, 70 ) Scrphulariaceae ( 4, 4, 37 ) ( 14, 10, 35 ) ( 86, 80, 40 ) ( 134, 134, 124 ) ( 45, 46, 39 ) ( 102, 102, 105 ) Rubiaceae ( 5, 5, 9 ) ( 73, 64, 57 ) ( 91, 90, 46 ) ( 74, 57, 71 ) ( 9, 4, 63 ) ( 106, 104, 102 ) Brassicaceae ( 9, 9, 133 ) ( 85, 65, 56 ) ( 4, 4, 5 ) ( 121, 123, 119 ) Orchidaceae ( 10, 10, 51 ) ( 3, 3, 3 ) ( 83, 84, 90 ) ( 2, 2, 1 ) ( 1, 1, 2 ) ( 5, 5, 50 ) Acanthaceae ( 12, 12, 4 ) ( 67, 71, 76 ) ( 87, 87, 42 ) ( 46, 41, 55 ) ( 71, 70, 70 ) ( 96, 94, 79 ) Gesneriaceae ( 16, 16, 3 ) ( 9, 9, 15 ) ( 105, 104, 88 ) ( 105, 107, 109 ) Melastmataceae ( 17, 17, 8 ) ( 10, 10, 18 ) ( 68, 31, 67 ) ( 4, 4, 24 ) Brmeliaceae ( 23, 23, 19 ) ( 4, 4, 3 ) ( 10, 10, 7 ) ( 2, 2, 2 ) Myrsinaceae ( 24, 24, 5 ) ( 90, 92, 98 ) ( 129, 129, 113 ) ( 80, 82, 85 ) (,, ) Arecaceae ( 37, 37, 88 ) ( 40, 41, 48 ) ( 5, 6, 6 ) ( 110, 109, 102 ) ( 1, 1, 1 ) Sapindaceae ( 40, 40, 69 ) ( 71, 74, 82 ) ( 9, 10, 8 ) ( 101, 100, 86 ) ( 81, 77, 76 ) ( 7, 8, 4 ) Urticaceae ( 41, 41, 136 ) ( 16, 16, 53 ) ( 2, 2, 2 ) ( 118, 117, 109 ) ( 101, 102, 98 ) ( 103, 103, 106 ) Annnaceae ( 44, 44, 68 ) ( 45, 46, 45 ) ( 5, 6, 4 ) ( 94, 95, 85 ) Saptaceae ( 46, 46, 94 ) ( 11, 11, 5 ) ( 8, 8, 30 ) ( 3, 3, 3 ) Tiliaceae ( 60, 60, 96 ) ( 10, 11, 4 ) ( 17, 17, 17 ) ( 19, 19, 30 ) ( 86, 88, 93 ) ( 35, 36, 21 ) Cecrpiaceae ( 106, 106, 75 ) ( 6, 7, 5 ) ( 25, 26, 54 ) Lauraceae ( 194, 192, 180 ) ( 107, 107, 102 ) ( 34, 35, 35 ) ( 6, 7, 13 ) ( 3, 3, 1 ) ( 64, 64, 63 ) Chrysbalanceae ( 197, 198, 216 ) ( 12, 12, 9 ) ( 14, 15, 5 ) Mraceae ( 224, 224, 228 ) ( 118, 118, 118 ) ( 92, 93, 96 ) ( 31, 29, 44 ) ( 2, 2, 10 ) ( 24, 18, 60 ) Cucurbitaceae ( 230, 230, 222 ) ( 58, 56, 58 ) ( 5, 5, 4 ) ( 97, 97, 77 ) ( 66, 50, 66 ) ( 11, 11, 53 ) Clusiaceae ( 233, 232, 229 ) ( 74, 72, 72 ) ( 80, 81, 87 ) ( 23, 23, 28 ) ( 4, 5, 3 ) ( 9, 9, 25 ) Anacardiaceae ( 239, 240, 242 ) ( 117, 117, 125 ) ( 3, 3, 3 ) ( 113, 112, 112 ) ( 31, 33, 25 ) ( 39, 42, 27 ) Saxifragaceae ( 240, 239, 230 ) ( 4, 4, 5 ) ( 84, 85, 91 ) ( 43, 44, 33 ) Salicaceae ( 252, 252, 253 ) ( 5, 5, 6 ) ( 26, 26, 28 ) ( 106, 108, 120 ) 10

11 (Asteraceae, CH) Residuals (Asteraceae, NA) (Orchidaceae, CH) (Paceae, CH) (Cyperaceae, NA) (Paceae, NA) Predicted Y Figure 1: Plt f residuals against predicted values fr mdel (1.6) using the OLSE. 11

12 Nrth America Krea Kashmir Chiapas Highlands Ecuadr Shuar Figure 2: Plt f the square rt transfrmed medicinal species against ttal species per family 12

13 Predicted squared rt f Y Residuals Figure 3: Plt f residuals against predicted values fr mdel (1.4) using the square rt transfrmed data 13

14 Appendix: Number f medicinal species and ttal species per family, fr angisperms in Shuar (Med= the number f medicinal species per family, Ttal= the ttal number f species per family) Family Med Ttal Family Med Ttal Family Med Ttal Acanthaceae 4 14 Cucurbitaceae 2 14 Nyctaginaceae 1 3 Agavaceae 0 2 Cyclanthaceae 1 4 Ochnaceae 0 2 Alaceae 0 1 Cyperaceae 7 18 Olacaceae 0 3 Amaranthaceae 5 9 Discreaceae 0 1 Onagraceae 0 2 Anacardiaceae 0 1 Elaecarpaceae 0 2 Orchidaceae 3 23 Annnaceae 2 5 Ericaceae 2 7 Oxalidaceae 1 2 Apiaceae 4 4 Euphrbiaceae 6 22 Passiraceae 0 3 Apcynaceae 1 4 Fabaceae Phytlaccaceae 0 1 Aquifliaceae 1 1 Flacurtiaceae 2 10 Piperaceae 9 18 Araceae 5 30 Fumariaceae 0 1 Plantaginaceae 1 1 Araliaceae 0 1 Gentianaceae 0 1 Paceae 4 22 Arecaceae 0 26 Gesneriaceae 4 7 Plygalaceae 1 1 Aristlchiaceae 1 1 Haemdraceae 1 1 Plygnaceae 1 4 Asclepiadaceae 1 3 Helicniaceae 0 3 Prtulacaceae 1 1 Asteraceae 9 28 Hernandiaceae 0 1 Ranunculaceae 0 1 Balanphraceae 0 1 Hippcrateaceae 0 1 Rhamnacae 1 1 Balsaminaceae 0 1 Lacistemaceae 0 1 Rsaceae 1 2 Begniaceae 3 6 Lamiaceae Rubiaceae 9 30 Bignniaceae 3 11 Lauraceae 3 14 Rutaceae 1 7 Bixaceae 1 1 Lecythidaceae 2 6 Sapindaceae 0 7 Bmbacaceae 1 6 Lemnaceae 0 1 Saptaceae 0 16 Braginaceae 1 10 Liliaceae 0 4 Scrphulariaceae 3 5 Brmeliaceae 0 20 Lasaceae 0 1 Smilacaceae 1 3 Brunelliaceae 0 1 Lganiaceae 0 3 Slanaceae Burseraceae 1 9 Lranthaceae 2 4 Staphyleaceae 0 1 Cactaceae 1 4 Lythraceae 0 3 Sterculiaceae 2 9 Campanulaceae 1 2 Malpighiaceae 2 4 Symplcaceae 0 1 Cannaceae 0 2 Malvaceae 7 9 Thephrastaceae 0 3 Capparaceae 1 2 Marantaceae 1 7 Tiliaceae 0 2 Caricaceae 2 5 Marcgraviaceae 0 1 Ulmaceae 1 4 Carycaraceae 0 2 Melastmataceae 5 38 Urticaceae 3 5 Caryphyllaceae 1 1 Meliaceae 2 10 Verbenaceae 5 11 Cecrpiaceae 1 7 Menispermaceae 0 2 Vilaceae 1 3 Chrysbalanceae 0 4 Mnimiaceae 0 1 Viscaceae 0 2 Clusiaceae 1 11 Mraceae 5 26 Vitaceae 1 3 Cmbretaceae 1 1 Musaceae 1 1 Vchysiaceae 0 1 Cmmelinaceae 3 8 Myricaceae 0 1 Xyridaceae 0 1 Cnvlvulaceae 0 4 Myristicaceae 1 9 Zingiberaceae 5 7 Cstaceae 3 3 Myrtaceae

15 Lianfen Qian* Department f Mathematical Sciences Flrida Atlantic University Bca Ratn, FL U. S. A. Sujin Wang Department f Statistics Texas A& M University Cllege Statin, TX U.S.A. * The crrespnding authr. 15