Confidence Intervals for Predicting Lumber Strength Properties Based on Ratios of Percentiles From Two Weibull Populations

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1 United Sttes Deprtment of Agriculture Forest Service Forest Products Lbortory Reserch Pper FPL RP 606 Confidence Intervls for Predicting Lumber Strength Properties Bsed on Rtios of Percentiles From Two Weibull Popultions Richrd A. Johnson Jmes W. Evns Dvid W. Green

2 Abstrct Rtios of strength properties of lumber re commonly used to clculte property vlues for stndrds. Although originlly proposed in terms of mens, rtios re being pplied without regrd to position in the distribution. It is now known tht lumber strength properties re generlly not normlly distributed. Therefore, nonprmetric methods re often used to derive property vlues. In some situtions, estimting properties bsed on prmetric estimte is required. For these situtions, the three-prmeter Weibull distribution looks promising. To use this pproch, procedures for estimting confidence intervls for rtios of per- centiles from two Weibull popultions re needed. In this study, we employed the lrge smple properties of mximum likelihood estimtors to obtin confidence intervl for the rtio of 00α-th percentiles from two different threeprmeter Weibull distributions. The coverge probbilities were investigted by computer simultion study. We concluded tht the procedure hs considerble promise, but mny questions remin to be nswered. Keywords: three-prmeter Weibull, confidence intervls, rtio of percentiles August 003 Johnson, Richrd A.; Evns, Jmes W.; Green, Dvid W Confidence intervls for predicting lumber strength properties bsed on rtios of percentiles from two Weibull popultions. Res. Pp. FPL-RP-606. Mdison, WI: U.S. Deprtment of Agriculture, Forest Service, Forest Products Lbortory. 8 p. A limited number of free copies of this publiction re vilble to the public from the Forest Products Lbortory, One Gifford Pinchot Drive, Mdison, WI This publiction is lso vilble online t Lbortory publictions re sent to hundreds of librries in the United Sttes nd elsewhere. The Forest Products Lbortory is mintined in coopertion with the University of Wisconsin. The United Sttes Deprtment of Agriculture (USDA) prohibits discrimintion in ll its progrms nd ctivities on the bsis of rce, color, ntionl origin, sex, religion, ge, disbility, politicl beliefs, sexul orienttion, or mritl or fmilil sttus. (Not ll prohibited bses pply to ll progrms.) Persons with disbilities who require lterntive mens for communiction of progrm informtion (Brille, lrge print, udiotpe, etc.) should contct the USDA s TARGET Center t (0) (voice nd TDD). To file complint of discrimintion, write USDA, Director, Office of Civil Rights, Room 36-W, Whitten Building, 400 Independence Avenue, SW, Wshington, DC , or cll (0) (voice nd TDD). USDA is n equl opportunity provider nd employer.

3 Confidence Intervls for Predicting Lumber Strength Properties Bsed on Rtios of Percentiles From Two Weibull Popultions Richrd A. Johnson, Sttisticin Deprtment of Sttistics, University of Wisconsin, Mdison, Wisconsin Jmes W. Evns, Supervisory Mthemticl Sttisticin Dvid W. Green, Supervisory Reserch Generl Engineer Forest Products Lbortory, Mdison, Wisconsin Introduction The rtio of two property estimtes is commonly used in engineering design codes to estblish llowble properties. For exmple, dry green rtios (the rtio of smll cler specimens dried to % moisture content to mtched green specimens) re given in ASTM D555 for number of timber species (ASTM 997b) nd my be used to clculte llowble properties. Also, n E/G rtio of 6 is ssumed in ASTM D95 (ASTM 997c) for djusting the flexurl modulus of elsticity (MOE) bsed on ny spn to depth rtio nd severl loding modes. As third exmple, ASTM D45 specifies strength rtios in tension prllel to grin re 55% of the corresponding bending strength rtios (ASTM 997). Strength rtios re the strength tht wood with defect (like knots) is expected to hve compred with the strength of cler piece of wood. So ASTM D45 is sying tht defect lowering the bending strength to 80% of cler piece ( strength rtio of 80%) would lower the tensile strength to 44% (0.80 times 0.55) of the cler wood specimen bending strength. Rtios often re used when it is not prcticl, or perhps not possible, to conduct tests for ll combintions of fctors (such s grdes, sizes, test modes, environmentl conditions) tht my ffect llowble properties. In wood engineering, the usul prctice hs been to conduct extensive tests for one combintion of fctors nd to develop rtios for djusting llowble properties from the mesured combintion of fctors to different set of fctors. A primry exmple of this problem is estimting tensile strength prllel to the grin. Due to experimentl difficulties in determining the strength of wood stressed in tension prllel to the grin, reltively little dt exist on which to bse llowble tensile properties. Thus, tensile strength hs historiclly been estimted s percentge of bending strength (Gllign nd others 979). A similr pproch is tken in estimting the strength of lumber t vrious moisture content levels (ASTM 997, b). Most of these rtios were originlly estblished by nlysis of men trends. However, they re being pplied without regrd to position in the distribution. Recent studies hve begun to focus on the rtio of properties t other percentile levels, especilly the fifth percentile. These include studies of the effect of rte of loding on tensile strength (Gerhrds nd others 984), the effect of moisture content on flexurl strength (Aplin nd others 986; McLin nd others 984), nd the effect of redrying on the strength of CCA-treted lumber (Brnes nd Mitchell 984). It ws once commonly ssumed tht lumber strength properties were normlly distributed. It is now generlly recognized tht lumber strength properties re not normlly distributed. Usully, nonprmetric methods re now used for deriving llowble lumber properties (ASTM 997c). In some situtions, however, prmetric estimte of lumber properties for relibility-bsed design is required. Bsed on empiricl evidence (Aplin nd others 986, Bodig 977, Hoyle nd others 979, McLin nd others 984, Pierce 976, Wrren 973), it is the three-prmeter Weibull tht emerges s serious cndidte. To use this prmetric pproch to estimte rtios of lumber properties, it is necessry to develop procedures for estimting confidence intervls for rtios of percentiles from two Weibull popultions. Typiclly the vribility of rtio estimtors cn be very lrge. An underestimte of the property in the denomintor nd corresponding overestimte of the property in the numertor of the rtio cn result in lrge estimte of the rtio. Correspondingly, n overestimte in the denomintor nd n underestimte in the numertor cn produce smll estimte of rtio. Since mny decisions regrding ASTM stndrds re bsed on confidence intervls ssocited with n estimte of the rtio of properties, it is importnt to develop the best confidence limits on this rtio. The objective of this pper ws to develop nd evlute procedures to crete confidence intervls tht hve pproximtely 90% or 95% nominl coverge for the rtio of percentiles from two different

4 three-prmeter Weibull popultions. In n erlier study, Johnson nd Hskell (984) investigted lrge smple tolernce bounds nd confidence intervls for percentiles from single three-prmeter Weibull distribution. In the Procedures section of this pper, we develop three lrge smple confidence intervl procedures bsed on the erlier work of Johnson nd Hskell nd one procedure bsed on order sttistics. Using simultion techniques, we then evlute the performnce of lrge smple pproximtion to the distribution of n estimte of percentile rtio. Procedures Method Let us ssume we hve two popultions, both of which follow Weibull distribution. Let X,..., X n be smple of size n from the first Weibull popultion, which hs the cumultive distribution function F( x) = exp[ {( x c ) / b} ], x c with probbility density function f(x) nd 00α-th percentile given by / [ ln( )] α = c + b α Let Y,...,Y n be smple of size n from the second Weibull popultion, which hs the cumultive distribution function Gy ( ) exp[ {( y c ) / b} ], y c be mximum likelihood estimtes obtined from the two smples. In the single smple setting, Johnson nd Hskell (984) derived lrge smple pproximtion for confidence intervls of percentiles from Weibull distribution. Their 00( γ)% confidence intervl for the 00( α) popultion percentile is if (, b, c z σ / n + z σ / n α γ α α α γ α ) occurs t the interior of Ω, where Ω is the prmeter spce (see Johnson nd Hskell (984) for more thorough discussion of the prmeter spce) nd where α, α, α b c α α α α α σα =,, b c I b α c b / / = [ ln( α)] ln[ ln( α)],[ ln( α)], = nd I is the observed Fisher informtion mtrix. The informtion mtrix cn be estimted by with probbility density function g(y) nd 00α-th percentile given by n I ln ( ;, n = f xi b, c) / n i wj w k [ ln( )] α = c + b α jk,,,3 Then the rtio of the popultion 00α-th percentiles is given by Let α α = c + b [ ln( α)] c + b [ ln( α)] (, b, c ) / / where (w, w, w 3 ) = (, b, c). The second-order prtil derivtives of the log-likelihood going into this estimted informtion mtrix re n xi c x I ln i c = n i= b b n ( + ) xi c I = b b n i= b nd (, b, c )

5 n n ( ) xi c I33 = ( ) n i= ( xi c) b n i= b n x i c I = + b b b n i= n xi c xi c + ln bni= b b n n xi c I3 = + i ( i ) n = x c b n i= b n xi c xi c + ln bni= b b n xi c I3 = b n i= b So we cn estimte the vrince by α α, α, α α α = b c I ( bc,, b ) α c σ ( bc,, ) If the estimted shpe prmeter hs vlue, plcing it on the boundry of the prmeter spce, Johnson nd Hskell (984) recommend using modified mximum likelihood solution for the two-prmeter exponentil distribution. In this cse, we tke α zγ σ α / n α + zγ σ α / n P α γ α + zγ σ α / n α α zγ σ α / n when neither shpe prmeter estimte equls. The inequlities of the rtios will hold unless α zγ σα / n ws negtive. Since ll vlues of rndom vrible with Weibull distribution re ssumed to be greter thn or equl to the loction prmeter (c 0), ny negtive estimte of the lower confidence limit of α cn be replced by 0, which would mintin the inequlity. If either or both shpe prmeter estimtes equl, then the modified mximum likelihood estimte nd confidence intervl bsed on the two-prmeter exponentil distribution cn be used. Since ny combintion of γ nd γ tht meet our restriction cn be chosen, the procedure bove cn led to vriety of confidence intervls for the sme two sets of dt. We could produce shortest confidence intervl by serching for the vlues of γ nd γ tht produce the shortest confidence intervl. However, it is not cler tht we would chieve the desired coverge with shortest confidence intervl nd with the coverge dependent on the estimtes of α, α, σ α, nd σ α s well s γ nd γ. It is lso not cler tht this procedure would led to simple confidence intervl for our rtio. Method Insted, it might be better to employ mximum likelihood estimtors to obtin lrge smple pproximtion to the distribution of α α α X() b[ ln( ) ( n ) ( ln( ))] = + α + α From Tylor expnsion, we see tht nd its estimted vrince (Johnson nd Hskell 984) to obtin the confidence intervl α z b ln ( α) ( n ) + ln ( α) γ n + n n α α γ ( ) + z b ln ( α) ( n ) + ln ( α) n + n n ( ) If we choose our confidence levels for ech of two percentile estimtes (which we denote by ( γ ) nd ( γ )) such tht ( γ )( γ ) = ( γ), then by independence c + b ln ( α ) α = α c + b ln ( α ) α + α ( ) ( ) higher order terms α α α α + α α α This suggests the lrge smple pproximtion Vr Vr( ) Vr( ) α = α α + 4 α α α α Then lrge smple pproximte ( γ) 00% confidence intervl for α / α is given by α ± z ( / n) α ( 4 / γ σ α + σ α n ) α α α 3

6 Method 3 Since X ( r ) nd Y ( s ) re independent, Note tht similr procedure could be obtined by X considering ( r ) P < z Y( s) = y = P[ X( r) < zy ] Y( s) ln( α / α) = ln( α / α) + ( α α) α So ( α α) + higher order terms X α ( r ) P < z = P[ X 0 ( r) zy] gs ( y Y < )dy ( s) If we exponentite the resulting intervl for ln( α / α ), we cn get confidence intervl tht would be expected to work where well if the norml distribution is good pproximtion to Method 4 ln( iα ), i =, Finlly, we cn get conservtive confidence intervl bsed on order sttistics. If the smple size is sufficiently lrge, it is well known tht r nd s cn be selected so tht the order sttistics X ( r ) nd X ( s ) stisfy If Y ( r ) nd Y ( s ) stisfy PX [ < α < X ] γ ( r) ( s) PY [ < α < Y ] γ ( r) ( s) nd n n j ( r ) < = j j= r P[ X zy] F ( zy)[ F( zy)] = ( s )!( n s)! n j n! g ( y) G ( y) g( y)[ G( y)] s s n s where X hs cumultive distribution F nd Y hs cumultive distribution G. The result cn be simplified further by writing j j j i F = ( F + ) = ( ) ( F) i j i= 0 i where ( γ )( γ ) γ, then γ PX [ < < X, Y < < Y ( r ) α ( s ) ( r ) α ( s ) ] X X P < < Y Y ( r) α ( s) ( s ) α ( r ) Since rndom vribles with Weibull distribution ssume vlues greter thn or equl to the loction prmeter, which is greter thn or equl to zero, ll of the order sttistics will be greter thn or equl to zero, which mintins the ine- qulity. So the intervl X Y X, Y ( r ) ( s ) ( s ) ( r ) hs probbility t lest γ of covering α / α Although four order sttistics re involved in the confidence limits, it is possible to express the exct coverge probbility in terms of single numericl clcultion. X X X X P P P Y Y Y Y ( r) α ( s) ( r) α ( s) α < < = < < ( s ) α ( r ) ( s ) α ( r ) α Thus, given the true prmeters for the two Weibull distributions, we could clculte n exct coverge probbility for our confidence intervl. Computer Simultion Of the four methods of developing confidence intervls for rtios of Weibull percentiles, method ppers to offer the most promise of giving unique solution tht is not encumbered by hving to pick two significnce levels whose product is the significnce level we wnt for our intervl. To evlute the sttisticl properties of this lrge smple confidence intervl procedure using simultion study, we generted n ordered smple of uniform rndom numbers, using IMSL (987) subroutines nd then converted these to Weibull vrites from specified Weibull popultion. Ech vlue of these observtions could, for instnce, represent the modulus of rupture for hypotheticl piece of lumber. A second smple ws generted in similr mnner from nother Weibull distribution. Mximum likelihood estimtes of the Weibull prmeters nd corresponding percentile estimtes for the nd, 5th, 0th, 30th, 50th, 70th, 90th, 95th, nd 98th percentiles were clculted for ech smple. The 5th percentile ws of primry interest since design vlues for strength properties re often 5th percentile estimtes. The two sets of mximum likelihood estimtors were then 4

7 combined ccording to the confidence intervl procedure detiled in the Procedures section. Nine runs were used to evlute the coverge of the lrge smple confidence intervls for the rtio of the percentiles from two different three-prmeter Weibull distributions. Erlier studies (Johnson nd Hskell 983, 984) indicted tht smple sizes of t lest 00 re required to obtin reltively smll vrinces nd somewht ccurte norml pproximtions to the percentile estimtes in the single popultion cse. For ech run of our simultion, the smple sizes n = 00 nd n = 00 were used. There were 50 replictions t ech set of conditions. The nine runs were one-ninth run of fctoril design s shown in Tble. We selected the prticulr vlues s shown in Tble. These prticulr vlues were selected fter exmintion of the prmeter estimtes in McLin nd others (984) nd Aplin nd others (986), which re typicl of lumber industry pplictions. The prticulr loction/scle rtios were chieved by keeping the loction prmeters fixed. Specificlly, we selected loction = loction = 0.5 which ment the scle prmeters ssocited with the coded vlues were s shown in Tble 3. Results nd Conclusions Tbles 4 nd 5 present the empiricl coverge proportions nd the verge length of the intervls for both the pproximte 95% nd 90% confidence intervls. From Tble 4, we see tht, t the lower percentiles, the coverge of the 95% intervls is poorest for run 8, run, nd run 5, in tht order. The shpe prmeter for the first popultion is.75 for ll these cses, but this cnnot be pinpointed s the single cuse becuse so mny other prmeters re chnging. Becuse the runs were designed s discussed in section 3 to be frctionl fctoril with min effects shpe, shpe, loction /scle, nd loction /scle, we cn nlyze them s such to get n ide of the sensitivity of our coverge probbilities to the prmeters. Tble 6 gives the men squre errors of the four min effects on the response vrible coverge probbilities. The lrger the men squre error, the more sensitive the result ws to tht effect. Tble 7 gives the rnkings of the effects from Tble 6. Tble 7 shows tht for percentiles in the til of our distribution, our 95% confidence intervls re most sensitive to the shpe prmeter of the Weibull distribution in the numertor. Tble Prmeters for the nine runs Run Shpe Shpe Loction / scle Loction / scle Tble Vlues selected for the fctoril design Coded vlues Weibull prmeter 0 Shpe Shpe Loction /Scle Loction /Scle Tble 3 Scle prmeters obtined with set loction prmeters Coded vlues for loction/scle Weibull prmeter 0 Scle Scle In view of the coverges for single popultion percentile presented in Johnson nd Hskell (984), this ws lmost better thn could be expected. We gin nlyzed the coverge probbilities s frctionl fctoril design. We see in Tble 8 tht our 90% confidence intervls re most sensitive to the shpe prmeter of the Weibull distribution in the numertor. Wht should we conclude from ll this? This is very smll study, but it does indicte tht the procedure hs considerble promise. A gret del more work should be completed before we cn be sure tht the promise is relity. The pttern is not so distinctive for the 90% confidence intervls (Tble 5). Most importntly, the proportion of intervls tht cover the true rtio of popultion percentiles is quite close to the nominl vlue in most cses considered. 5

8 Tble 4 Proportion of 95% intervls tht cover the true rtio of percentiles (verge length is given s the second entry) Popultion percentiles (lph) Run n = 00, n = 00, 50 replictions. Tble 5 Proportion of 90% intervls tht cover the true rtio of percentiles (verge length is given s the second entry) Popultion percentiles (lph) Run n = 00, n = 00, 50 replictions. 6

9 Tble 6 Men squre errors for the effects of the Weibull prmeters from nlysis of the proportion of 95% intervls tht cover the true rtio of percentiles Popultion percentiles (lph) Shpe Shpe Loction / scle Loction / scle n = 00, n =00, 50 replictions. Tble 7 Reltive importnce of the effects of the Weibull prmeters from nlysis of the proportion of 95% intervls tht cover the true rtio of percentiles b Popultion percentiles (lph) Shpe 4 3 Shpe Loction /scle Loction /scle Importnce is rted on scle of to 4, being most importnt nd 4 lest. b n = 00, n = 00, 50 replictions. Tble 8 Reltive importnce of the effects of the Weibull prmeters from nlysis of the proportion of 90% intervls tht cover the true rtio of percentiles b Popultion percentiles (lph) Shpe 4 Shpe Loction /scle Loction /scle Importnce is rted on scle of to 4, being most importnt nd 4 lest. b n = 00, n = 00, 50 replictions. 7

10 Some of the things tht need to be done or questions we should try to nswer re s follows:. We need to look t wider rnge of shpe, scle, nd loction prmeters in wy tht llows us to look for interctions in the vribles. For wood-relted pplictions, shpe prmeters from to 0 would be good rnge. Since the shpe prmeter is relted to the coefficient of vrition of dt from two-prmeter Weibull, cn we model the reltionship of vribility of the dt nd the needed smple size to get intervls of certin width on rtios of percentiles?. It would simplify the simultion work if we could show tht rtios of prmeters (like loction to scle) were the importnt fctors. 3. The number of replictions in this study ws much too smll. With the vribility tht comes with estimting Weibull prmeters, we might need,000 to 0,000 replictions to get four deciml plce ccurcy on our coverge probbilities. We certinly need to find out how mny replictions it tkes to get highly repetble results. 4. We need to look t different smple sizes. How good is this procedure when we hve 400 in ech distribution of 40? 5. We need to look t the design spects of the problem. If we hve 300 green wood specimens, how mny should we dry to estimte dry/green rtio for the mteril? 6. Cn we get closed form solution for the problem? If we go to two-prmeter Weibull distribution, how does this chnge things? 7. We lso need to look t the other methods for confidence limits. How well does the nonprmetric procedure (method 4) work? If we use method 3, how do confidence intervls it produces compre with the method intervls? Is there n esy wy in method to pick our confidence levels to get shortest confidence intervl? 8. We could lso expnd the problem to other distributionl forms. Both the norml nd lognorml distributions re very useful in wood-relted reserch. Clerly severl questions could be looked t, nd the results in this pper imply tht the procedure does hve promise nd should be investigted further. Literture Cited Aplin, E.N.; Green, D.W.; Evns, J.W.; Brrett, J.D The influence of moisture content on the flexurl properties of Dougls Fir dimension lumber. Res. Pp. FPL RP 475. Mdison, WI: U.S. Deprtment of Agriculture, Forest Service, Forest Products Lbortory. 3 p. ASTM Stndrd prctice for estblishing structurl grdes nd relted llowble properties for visully-grded lumber. ASTM D Phildelphi, PA: Americn Society for Testing nd Mterils. ASTM. 997b. Stndrd test methods for estblishing cler wood strength vlues. ASTM D Phildelphi, PA: Americn Society for Testing nd Mterils. ASTM. 997c. Stndrd prctice for evluting llowble properties for grdes of structurl lumber. ASTM D Phildelphi, PA: Americn Society for Testing nd Mterils. Brnes, H.M.; Mitchell, P.H Effect of post-tretment drying schedule on the strength of CCA-treted southern pine dimension lumber. Forest Products Journl. 34(6): Bodig, J Bending properties of Dougls Fir-Lrch nd Hem-Fir dimension lumber. Spec. Publ. No Fort Collins, CO: Colordo Stte University, Deprtment of Forestry nd Wood Science. Gllign, W.L.; Gerhrds, C.C.; Ethington, R.L Evolution of tensile design stresses for lumber. Gen. Tech. Rep. FPL GTR 8. Mdison, WI: U.S. Deprtment of Agriculture, Forest Service, Forest Products Lbortory. Gerhrds, C.C.; Mrx, C.M.; Green, D.W.; Evns, J.W Effect of loding rte on tensile strength of Dougls-fir by 6 s. Forest Products Journl. 34(4): 3 6. Hoyle, R.J., Jr.; Gllign, W.L.; Hskell, J.H Chrcterizing lumber properties for truss reserch. In: Proceedings, Metl plte wood truss conference, 979 Nov. 3 6, St. Louis, MO. Mdison, WI: Forest Prod. Res. Soc.: IMSL, Inc User s mnul: Fortrn subroutine librry. Houston, TX: IMSL, Inc. Johnson, R.A.; Hskell, J.H Smpling properties of estimtors of Weibull distribution of use in the lumber industry. The Cndin Journl of Sttistics. (): Johnson, R.A.; Hskell, J.H An pproximte lower tolernce bound for the three-prmeter Weibull pplied to lumber property chrcteriztion. Sttistics nd Probbility Letters. : McLin, T.E.; DeBonis, A.L.; Green, D.W.; Wilson, F.J.; Link, C.L The influence of moisture content on the flexurl properties of southern pine dimension lumber. Res. Pp. FPL RP 447. Mdison, WI: U.S. Deprtment of Agriculture, Forest Service, Forest Products Lbortory. 40 p. Pierce, C.B The Weibull distribution nd the determintion of its prmeters for ppliction to timber strength dt. Current Pp. CP/76. Princes Risborough, Aylesbury, Buckinghmshire: Princes Risborough Lbortory. Build. Res. Estb. Wrren, W.G Estimtion of the exclusion limit for dimension lumber. In: Proceedings, IUFRO Division V meeting, Sept. nd Oct. 973, South Afric, Vol. :