TABLES TO FACILITATE FITTING SB CURVES. N.L. Johnson* J.0. Kitchen. Department of Statistics University of North Carolina at Chapel Hill

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1 * This rsarch was supportd in part by th ~1athmatics Division of th Unitd Stats Air Forc Offic of Scintific Rsarch undr Contract No. AFOSR and th National Instituts of Halth, Institut of Gnral Hdica1 Scincs undr Grant No. 5R01GM TABLES TO FACILITATE FITTING SB CURVES by N.L. Johnson* With an Appndix on Calculation of th Tabls by J.0. Kitchn Dpartmnt of Statistics Univrsity of North Carolina at Chapl Hill Institut of Statistics MimD Sris No. 683 l4ay 1970

2 TABLES TO FACILITATE FITTING by SB CURVES N. L~ Johnson.* llnivpsity of NOl'th CaroUna at Chapt Bitt With an Appndix on Calculation of th Tabls by J. O. Kitchn, llnivrity of North Caf'OUna at Chapl, Bitt 1. DESCRIPTIOO OF TABLES. Th SB family of frquncy curvs is dfind (Johnson, 1949) by th distribution of x, whr z = y+ 6 log[(x-~)/(~+a-x)] (l) (~ < x < ;+A) is a unit normal variabl. Th paramtrs ; and A afft only th location and sal, rspctivly, of th distribution. Th shap of th distribution of x is th sam as that of a variabl y for which z y + o og[y/(l-y)] (2) This rsarch was supportd in part by th Mathmatics Division of th Unitd Stats Air For Offic of Scintific Rsarch undr Contract No. AFOSR-68-l4l5 and th National Instituts of Halth, Institut of Gnral Mdical Scincs undr Grant No. 5ROlGM-12868

3 2 (0 < y < 1) is a unit normal variabl. This shap is thrfor dtrmind by th valus of y and 6. By convntion, w mak 6 positiv. Th r-th momnt about zro of y is (3) An xplicit xprssion «56) of Johnson (1949» can b givn for lji and by rpatd us of th rlationship (4) formula for lj t can b obtaind. r xcpt for occasional us as chcks on computation. Ths ar, howvr, too complicatd, Formula for lj t can also b obtaind using th rlationship r (5) This is vn lss convnint than (4). In ordr to fit an SB curv by making th first four momnts agr with thos of a givn distribution, th first, and most difficult, stp is to dtrmin y and 6 to giv th rquird valus of th momnt-ratio. 1(31 and 6 2 Tabl 3 givs valus of y and 6 for 16 1 = 0.00(0.05)2.00 with 6 at intrvals of 0.1. For a givn valu of 2 113, th smallst tabulatd 6 valu is th smallst quantity 2 1 1~ x (intgr) such that 6 > <ra;:-)2+1. A fw valus of 6, clos to th lognormal lin, ar 2 2 omittd at th uppr nd of th rang of possibl valus. For such valus, a lognormal fit would usually b vry narly-th sam as an SB fit, and would also b simplr.

4 3 Onc th appropriat valus of y and 6 hav bn dtrmind, ~ and A can b obtaind from th formula standard dviation of x =- AO; man of x... ; + AlJi whr a standard dviation of y. Tabl 3 includs valus of lji and a to aid ths calculations. If th sign of.ra;:- b changd (to -), y should b rplacd by -y, and lji by (l-lji>. Th valus of 6 and a rmain unaltrd. Th prsnt tabls of y and 0 ar similar in form to arlir tabls rlating to th Su class of frquncy curvs (Johnson, 1965). If x' has an Su distribution, thr ar paramtr valus y', 6' (>O), ~', A'(>O) such that (6) has a unit normal distribution. Comparison of (1) and (6) shows that if x has an SB distribution thn has an Su distribution with y' =- y, cs' == a, ~' == 0, A' Th prsnt Tabl 3 also includs valus of lj' and 6. Th corr 1.ponding valus for Su distributions ar much asir to calculat, and so it was not ncssary to giv ths valus in th arlir Tabls. Th mthods usd in calculating Tabl 3 ar dscribd in th Appndix.

5 4 2. INTERPOLATIOO. It will usually b ncssary to intrpolat with rspct to both ~ and 6 2 Excpt nar th lognormal lin (i.. for th highst valus of 13 2 for givn ), rpatd univariat intrpolation using Evrtt's scond cntral diffrnc formula should giv rsults corrct to -within about 2 units in th last dcimal plac shown.. For valus of ~ gratr than 1, linar intrpolation is adquat ovr about th lowst valus of 13 2 " Whn y and/or 0 ar larg (gratr than 1, say) it is -1-2 advantagous to 1Iltrpolat for yo and 15, rathr than for y and 6 dirctly_ Intrpolation with rspct to <Sa (rathr than a) can also b usful" Th vry slow initial variation of lli, as 13 2 incrass from ( ) with ~ fixd, is notabl. Thr is a rmarkably flat maximum which gradually movs away from th lowst 13 2 valu as ~ incrass., W will now us th prsnt tabls to fit an SB distribution to th data of Exampl 2 (p.169) of Johnson (1969). For ths data w hav th sampl valus: man = standard dviation = ~ = =

6 5 obtain Using Evrtt's cntral scond diffrnc intrpolation formula, w y = ; o == corrsponding to th sampl valus of ~ and 13 2 (Linar intrpolation givs y == ; 0 = Dirct calculation givs y == ; o == ) Dirct calculation and scond diffrnc intrpolation both giv ~i == , a == ai == ). Thn ~ and A (linar intrpolation givs ~i == , ar stimatd from th formula t A = ; A == whnc A == " (By comparison, Johnson (1949), using a mthod basd on quating crtain obsrvd and fittd prcntil points, obtaind th valus y = ; IS == ; ~ == 15.0; A = 36.5.) Tabl 1 shows th original data togthr with fittd frquncis from a Typ I curv (Prtorius, 1930), th SB curv from Johnson (1949), and th SB curv fittd by momnts. Th fit of th lattr is rlativly poor in th tails, but quit good in th major part of th distribution.

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8 7 4. Blt'ODALITY. SB curvs ar bimodal if othrwis thy ar unimodal. Som valus of 13 and on th boundary btwn unomodal and bimodal curvs ar shown in Tabl 2 (which is an xtnsion of DrapT (1952, F1g. 2». For a givn 61' curvs ar bimodal if 8 2 is lss than th valu shown in this tabl. TABLE 2 Boundar'y of Bimodal SB CU1'tXl (12)

9 8 5. FITTING AN Sa DISTRIBtI11~, USING 00 KM:NTS. It oftn happns that th rang of variation of x is known -- that is, ~ and A ar known -- and it is only ncssary to stimat y and 15. A simpl way to do this is to us th first and scond momnts of u = log[(x-~)/(~+a-x)], taking 15 = (standard dviation of u)-l y = - 15 (man u) Howvr, th momnts of u may b difficult to valuat (for a thortical distribution) or tdious to comput (for an mpirical distribution). In th lattr cas, thr may b additional problms associatd with unqual group-widths for u, consqunt upon qual group-widths for x. In such cass th first two momnts of y = (x-f~)/a may b usd. Th valus of 1.1' and a in Tabl 3 giv som indication of appropriat valus 1 of y and 15. Furthr tabls, giving y and 15 with argumnts 1.Ii, a at rgular intrvals ar now bing computd. REFERENCES DRAPER, J. (1952) Proprtis of distributions rsulting from crtain simpl transformations of th normal distribution, Biomtpi.ka~ 39, JOHNSON, N. L. (1949) Systms of frquncy curvs gnratd by mthods of translation, Biommka, 36, JOHNSON, N. L. (1965) Tabls to facilitat fitting Su frquncy curvs, Biomtrika~ 52, P"RETORIUS, s. J. (1930) Skw bivariat frquncy surfacs, xamind in th light of numrical illustrations, Biomtrika, 22,

10 9 TABLE 3 ~ = 0.00 a y rs;: a y ~ ljt 1 a (y =OJ lji 0.5 in all cass)

11 10 Ii\ = 0.15 ~ y 0 ll' 1 a 8 2 y 0 ll' 1 a ; I

12 11 r;:- ca 0.25 li' = y 6 \.I' a 6 2 y 6 \.1' a

13 12 ~ ~ y 0 ]..I' a 8 2 y 6 ]..I' a

14 JB;" 0.45 Il CIl y 0 11' 0'. 6 2 y 0 11' 0'

15 14 ~ ;a; y t5 11' a 6 2 y 0 11' a

16 15 ~ a 0.65 ~ y IS lit CJ 13 2 y ts ljt CJ ~

17 r;: ~ y 6 2 ll' a 8 y 15 2 ll' a '; L

18 rat 0.85 I6i" y <5 lj' a 6 2 y <5 lj' a ,

19 18 ~ D 0.95 ~ = y c5 jj' C y 15 l1' C

20 19 ~ = 1.05 ra;: y <5 u' a 13 2 y <5 u' a ~

21 5.2 ~ ~ Y IS lji a 8 y c5 lji a

22 ~ III 1.25 ~ /3 y 2 0 ~' a /3 y 15 2 lj' a

23 ~ III 1.35 ~ y 15 lit s 13 y 6 lit s

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25 ~ = 1.55 ~ = y 2 ~ lj' (1 13 y 2 <5 lj' ( S '

26 ~ Bi" (32 y 6 }.I' <1 6 y 6 }.I' < i lS

27 ~ lsi" y 6 lj' a B 2 y 6 lj' a O.U

28 27 ~ III 1.85 ~ y 15 ).1' a 8 2 y 15 lj' a

29 28 ~ = 1.85 (cont.) IBi" 1.90 (cont.) I

30 29 ~:II 1.95 ~ y 6 lj' y 6 ll' lll ' ~

31 y: 30 ~ (cont.) ra; (cont.) S

32 31 APPENDIX CAlCULATION OF THE TABLES In ordr to gnrat ths tabls, it was first ncssary to find a satisfactory mthod of computing ~ and 6 for th various combinations of valus of y and <5 that wr likly to occur. To do this, 2 th paramtr spac was dividd into four rgions dfind as follows (1) <5.. 0 (2) 0 < 0 < 0.5 (3) 0.5 s <5 < 2 (4) <5 ~ 2. For <5... 0, ~ and 6 wr obtaind using th formulas 2 whr For th rmaining thr rgions, th following mthod was usd. It was ncssary to calculat whr n.. 16<5 + Y and int{x).. intgral part of x.

33 32 W hav and so th first trm in (A.l) 18 ngligibl. For th third trm in brackts in (A.l), w not that if t ~ n thn and so (/2'iT)-l f' {l+(y-t)/c5}-r -%t 2 dt n with an rror lss than - 16 This lattr intgral was valuatd using a standard subroutin. Th scond trm in (A. 1) was valuatd using a 32 point Lgndr-Gauss quadratur, for 0 < 6 < 0.5. A 64 point Hrmit-Gauss quadratur was usd for th rgion 0.5 ~ 6 < 2 and a 24 point Hrmit-Gauss quadratur for th rgion 15 ~ 2. Th rsulting valus of ~ and 6 wr compard with th tabls 2 in Johnson (1949) and closly xamind for consistncy. All wr foood to b satisfactory to fiv dcimal placs. As a first stp in finding y and 6 as functions of ~ and (32' a tabl of valus of (31 and 6 2 was computd for a pr-dtrmind st of (y, 15) pairs which covrd th dsird rang of r;:- and 6 2 Th curvs corrsponding to fixd 0 and fixd y whn plottd in th (31' 6 2 spac ar approx1matly linar ovr

34 33 small rgions and this approximat linarity was satisfactory for most of th itrations to b dscribd in th nxt paragraph. To find th valus of y and 0 that corrspond to a dsird point p D (~, 13 ), th tabl gnratd abov was sarchd until th 'quadrilatral' containing.! was found. This 'quadrilatral' was thn sub 2 dividd, first with rspct to 0 and thn with rspct to y to find a smallr 'quadrilatral' containing X. Th valus of ~ and 6 for 2 th midpoint with rspct to 0 and y of this nw 'quadrilatral' wr thn compard to th valus for X. If both coordinats wr corrct to within a spcifid tolranc ( ), ths valus of 0 and y wr accptd as th dsird valus. Othrwis, th procss was continud until this condition was achivd. Occasionally, bcaus of th non-linarity of th boundaris of th 'quadrilatrals' usd, X was found to b in a crtain 'quadrilatral' but not in any of th 'quadrilatrals' which rsultd from th subdivision procss. In ths situations, th original 'quadrilatral' was nlargd slightly and th subdividing procss continud. Whn rquird, on such corrction was usually sufficint. Th final print-out gav valus of rs;:- and 6 2 at th vt'tics of th last 'quadrilatral' as wll as thos at th cntr. This nabld final adjustmnts to b mad, spcially whn dciding whthr to 'roundoff' upwards or downwards. All of th calculations wr prformd on an IBM 360/75 computr using a look partition of mmory. With th xcption of th valuation of th cumulativ normal function which was don using an IBM supplid routin, all of th calculations wr don using doubl prcision arithmatic. About 220 minuts of computr tim wr rquird to do all of th computing including th ncssary prliminary invstigations and dbugging.