A BOOTSTRAP ANALYSIS OF TEMPERATURE EFFECTS ON BEAN LEAF BEETLE EGG HATCH TIMES

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1 Libraries Cnference n Applied Statistics in Agriculture th Annual Cnference Prceedings A BOOTSTRAP ANALYSIS OF TEMPERATURE EFFECTS ON BEAN LEAF BEETLE EGG HATCH TIMES Kenneth J. Kehler Fllw this and additinal wrks at: Part f the Agriculture Cmmns, and the Applied Statistics Cmmns This wrk is licensed under a Creative Cmmns Attributin-Nncmmercial-N Derivative Wrks 4.0 License. Recmmended Citatin Kehler, Kenneth J. (1994). "A BOOTSTRAP ANALYSIS OF TEMPERATURE EFFECTS ON BEAN LEAF BEETLE EGG HATCH TIMES," Cnference n Applied Statistics in Agriculture. This is brught t yu fr free and pen access by the Cnferences at. It has been accepted fr inclusin in Cnference n Applied Statistics in Agriculture by an authrized administratr f. Fr mre infrmatin, please cntact cads@k-state.edu.

2 Applied Statistics in Agriculture 215 A BOOTSTRAP ANALYSIS OF TEMPERATURE EFFECTS ON BEAN LEAF BEETLE EGG HATCH TIMES Kenneth J. Kehler Department f Statistics Iwa State University, Ames, Iwa Abstract The bean leaf beetle (Certma trifurcata) is a significant sybean pest in the Midwest. The pssibility f reducing crp damage by disrupting the synchrny between emergence f F2 adults and the availability f yung green pds requires an increased understanding f bean leaf beetle (BLB) phenlgy. An imprtant cnsideratin in predicting emergence f F2 adults is the influence f temperature n the rates f egg and larval develpment. In this article we cncentrate n the analysis f data frm a cntrlled study f the effects f temperature n bth the hatch time distributin and the viability f BLB eggs. Several temperature levels are cnsidered with a different number f egg chrts expsed t each f several temperature levels. Hatch time bservatins fr individual eggs are subject t bth interval and right censring, and inspectin schedules vary acrss chrts. Limited failure ppulatin (LFP) mdels with Weibull hatch time distributins are used t estimate parameters f the hatch time distributin and the prprtin f viable eggs at each temperature. These estimates are used in the subsequent weighted least squares estimatin f curves fr predicting the prprtin f viable eggs and the inverse median f the hatch time distributin as functins f temperature. Btstrap prcedures are used t estimate variances that prperly accunt fr within chrt crrelatins. The imprtance f replicating experiments fr different temperature levels is illustrated. Key Wrds: Btstrap estimatin, censred data, limited failure survival mdels. 1. Intrductin In Iwa, tw generatins f BLB adults develp each year (Smelser and Pedig, 1991). Overwintered BLB adults clnize sybean fields sn after seedling emergence and lay eggs which develp int Fl adults. Because Fl adults are usually present while sybeans are still in vegetative grwth stages, feeding by Fl adults is usually cnfined t fliage, resulting in little if any yield reductin (Smelser and Pedig, 1992). Eggs laid by the Fl adults develp int F2 adults which typically emerge in early August when sybeans are in grwth stage R5. F2 adults supplement their leaf diet with uter green tissues f stems, peduncles, and pds. Since BLB adults prefer green immature pds t dry mature pds, the degree t which feeding by F2 adults reduces yield depends n the

3 216 relative phenlgies f the beetles and the crp. Disrupting the synchrny between emergence f F2 adults and availability f green pds is a prmising BLB management strategy that requires an increased understanding f BLB phenlgy. An imprtant cnsideratin in predicting the emergence f F2 adults is the influence f temperature n egg and larval develpment rates (e.g. Herzg et al and Turner 1979). This article is restricted t the analysis f temperature effects n egg hatch times and viability, but the basic methd culd be equally well applied t the analysis f varius stages f larval develpment t analyze the effects f temperature n ttal develpment time frm egg depsit t the emergence f F2 adults. T quantify the effects f temperature n the viability and develpmental velcity f BLB eggs, multiple chrts f eggs were incubated at specific temperatures. Each chrt cnsisted abut 20 eggs laid in the same day and incubated in a specific rearing cup. In this case, existence f within chrt crrelatins deviates frm the usual assumptin made in survival and life event analysis that each respnse time is independent f any ther respnse time (e.g. Cx and Oakes, 1984, r Kalbfleisch and Prentice, 1980). Variatin in cnditins within a single temperature chamber results in a tendency fr respnses frm eggs within a single rearing cup t be mre similar than respnses frm eggs in ther cups incubated at the same temperature. Additinal crrelatins arise frm genetic similarities f eggs laid by a single beetle, but eggs laid by individual adult females culd nt be identified because eggs were cllected frm cages cntaining mre than ne adult female. Cnsequently, unequal crrelatins amng eggs in a single chrt arising frm eggs laid by the same r different beetles cannt be directly mdelled. Btstrap res amp ling prcedures are used t btain standard errrs fr estimates f parameters in the temperature curves that prperly accunt fr effects f within chrt crrelatins. An additinal cmplicatin is that sme eggs are nt viable, i.e., they never hatch, and the prprtin f viable eggs culd be affected by temperature. Due t varying inspectin schedules, final inspectin times are nt the same fr all egg chrts, and sme viable eggs may nt be bserved t hatch prir t the final inspectin times fr their respective chrts. A limited failure ppulatin (LFP) apprach is used t allw fr this pssibility. The results indicate, hwever, that few eggs hatched after the final inspectin times. 2. Experimental Prcedures Eggs were cllected frm BLB adults reared at 30 C under cnstant daily phtperids cnsisting f 16 hurs f light and 8 hurs f darkness. Chrts, cnsisting f abut 20 eggs, cllected n the same day and placed in a single rearing cup, were randmly assigned t incubatin at ne f the prescribed temperatures. Multiple chrts were incubated at the same temperature in a single temperature chamber. Each egg was cllected n the day it was laid, and the same phtperid f 16 hurs f light per day was used fr all egg chrts. In mst cases, all eggs in a single rearing cup (chrt)

4 Applied Statistics in Agriculture 217 were btained frm a single cage f adults, but achieving the gal f abut 20 eggs per cup smetimes made it necessary t put eggs frm different cages int a single cup. Each cage cntributed eggs t nly ne chrt. Numbers f eggs varied amng chrts, but eighty percent f the chrts cntained between 18 and 22 eggs. The extremes were 4 eggs in ne cup and 38 eggs in anther. Hatch times were recrded fr individual eggs. Results fr eggs in different chrts are assumed t be independent, but there are substantial crrelatins amng results fr eggs within individual chrts. Within chrt crrelatins need nt be hmgeneus acrss chrts nr even fr different pairs f eggs in any single chrt. Hatch times were recrded fr individual eggs as the number f days after the eggs were laid with the day n which the egg was laid taken t be day O. Althugh the time f day was kept reasnably cnstant, the set f days n which inspectins were dne varied acrss chrts, and n chrt was inspected mre than nce n any single day. Cnsequently, hatch times are interval censred with the shrtest inspectin intervals crrespnding t a single day and ther inspectin intervals cnsisting f tw r mre days. The data fr 20 f the 111 egg chrts incubated at 2SOC in the first year f the study are displayed in Table 1. Data are nt displayed fr the first 6 days after eggs were laid because n eggs were bserved t hatch during that perid. Dts represent days n which inspectins were nt made. The value 4 at day 8 fr chrt 1, fr example, indicates that 4 eggs hatched between 6 and 8 days after the eggs were laid, and the value 11 at day 10 indicates that 11 eggs hatched between 9 and 10 days after the eggs were laid. Since this chrt was nt inspected after day 10, hatch times fr any eggs that hatched after day 10 are right censred. These data prvide a clear indicatin f the variatin in inspectin times amng egg chrts. The bserved numbers f viable eggs, reprted in the last clumn f Table 2, prvide lwer bunds n the actual numbers f viable eggs in the individual chrts. 3. Hatch Time Mdel Fr a particular temperature, the data cnsist f bservatins n individual eggs in m independent chrts where the hatch times and censring times fr the 11j eggs in the i-th chrt fall int K(i)+l disjint time intervals, ([~,k-l,ti,k): k = 1,2,..., K(i)+l}, determined by the inspectin schedule fr that particular chrt. By cnventin, ~,=O, ti,k(i) + 1 = 00 and ti,k(i) is the last inspectin time fr the i-th chrt. The survivr functin S(t) is the cnditinal prbability that an egg des nt hatch befre time t, given that it is a viable egg. Denting the prbability f a viable egg by 7r, the uncnditinal prbability that a randmly selected egg hatches in time interval [~,k,ti,k+l) is and the uncnditinal prbability that a randmly selected egg des nt hatch befre the final inspectin time is (1)

5 218 the sum f the prbability f randmly selecting a nnviable egg and the prbability f selecting a viable egg that des nt hatch befre the final inspectin time. Meeker (1987) and Kehler and McGvern (1990) refer t this as a limited failure ppulatin (LFP) mdel in applicatins where failure times are bserved, but it can als be used t mdel life event times such as the egg hatch times cnsidered in this study. Estimates f 71' and parameters in the survivr functin are btained by maximizing (2) (3) where n ik is the number f eggs in the i-th chrt that hatch in the k-th inspectin interval fr that chrt and CK(i) is the number f eggs in the i-th chrt that are right censred at the final inspectin time fr that chrt. Althugh (3) is prprtinal t the likelihd functin fr a mdel that incrrectly assumes that each egg respnds independently f any ther egg, the resulting estimatrs are cnsistent when Set) is crrectly specified. Huster, et al (1989) refer t this as an independence wrking mdel (IWM) apprach. Unlike the situatin where the crrect likelihd is maximized, the inverse f the infrmatin matrix btained frm the secnd partial derivatives f the lgarithm f (3) is generally nt a cnsistent estimatr f the cvariance matrix f the parameter estimates. Cnsequently, we use btstrap resampling t btain cnsistent estimates f cvariance matrices. Preliminary analyses f the bserved hatch times indicated that tw-parameter Weibull distributins with different starting times prvide adequate mdels fr Set) fr the three incubatin temperatures. A different set f parameters is estimated fr each temperature level. The survivr functin fr a tw-parameter Weibull distributin with knwn starting time A is and the median hatch time is Set) = exp{-[(t-a)/]'y}, t>a, 0>0, 1'>0, (4) M = A + 0 [In(2)]1/'Y (5) The inverse f the median hatch time is ften used as an indicatr f egg develpment rate. Different starting times are used t indicate the earliest pssible hatch dates at different incubatin temperatures. Since the LFP mdel allws fr the pssibility that viable eggs may hatch after the final inspectin time, sme care must used in its applicatin. Early terminatin f

6 Applied Statistics in Agriculture 219 inspectin results in a large prprtin f unhatched eggs at the final inspectin time and unreliable estimatin f mdel parameters. When inspectin is terminated t early, the IWM likelihd (3) is quite flat in a large regin cntaining the maximum, because it is nt clear whether (2) shuld be made large by using a small value f 7r r by using a large value f 7r and adjusting the parameters in the survivr functin t make Set) large at the final inspectin time. Based n a Mnte Carl study f the case f cmpletely independent respnses, Meeker (1987) recmmends that inspectin shuld cntinue until at least 80 percent f the eventual failures (hatches f viable eggs) have ccurred. Unpublished simulatins we perfrmed indicate that this is als a reasnable guideline when respnses are crrelated. Premature terminatin f inspectin tends t result in estimates f 7r that are t clse t 1. In the present study, the final inspectin times were selected s that nearly all f the viable eggs wuld be bserved t hatch and the 80% guideline is clearly satisfied. 4. Btstrap Estimatin It fllws frm Huber (1967) that maximizatin f L in (3) prduces cnsistent parameter estimates with large sample nrmal distributins. The inverse f the matrix f secnd partial derivatives f 10g(L), hwever, yields variance and cvariance estimates that tend t be t small in the presence f psitive intra-chrt crrelatins. Mre accurate estimates f cvariance matrices and cnfidence intervals with apprpriate cverage prbabilities are btained frm a btstrap prcedure that apprximates the selectin f new samples frm the ppulatin by sampling with replacement frm the riginal sample. Efrn (1979, 1982) reviews and develps basic thery underlying the btstrap and ther resampling prcedures fr data btained frm simple randm samples. The BLB egg hatch study is smewhat different because it invlves chrts f eggs with pssible intra-chrt crrelatins. Fllwing the apprach used by Kehler and McGvern (1990), the btstrap prcedure is applied by using chrts as the basic sampling units, rather than individual eggs. Cnsider hatch time data fr m chrts f BLB eggs incubated at a particular temperature. Each iteratin f the btstrap prcedure mimics a new replicatin f a study f hatch times fr m chrts f BLB eggs maintained at the same temperature. This is dne as fllws: (a) (b) (c) Use simple randm sampling with replacement t select a sample f m chrts frm the riginal set f m chrts. Maximize (3) t btain parameter estimates fr the Weibull LFP mdel, say D,,y, and -n-, frm the btstrapped sample frm step (a). Repeat steps (a) and (b) B times, where B represents the ttal number f btstrapped samples. In the present analysis, B = 1000.

7 220 This prduces B independent sets f parameter estimates, /:Jj = (D j, ~j, iy, i = 1,2,...,B, ne set fr each f the B btstrapped samples. Distributinal prperties f these B estimates are used t apprximate the crrespnding prperties f the parameter estimates cmputed frm the riginal sample. Fr example, the sample cvariance matrix fr the parameter estimates evaluated frm the riginal data is estimated as V = B (B_l)-l I (/:Jj - /:J+)(/:Jj - /:J+)T i=l Since resampling f egg chrts is dne with replacement with prbability 11m assigned t each f the m chrts in the riginal sample, any particular btstrapped sample f m chrts may cntain multiple cpies f sme egg chrts and n cpies f ther egg chrts frm the riginal sample. Btstrapped samples need nt cntain the same number f individual eggs as the riginal sample, and this resampling prcedure autmatically accunts fr variatin in egg chrt sizes. In this case n subsampling is dne within egg chrts selected fr a btstrapped sample, because egg chrts were usually cnstructed frm all f the eggs laid by the adult females in a particular cage n a specific day. 5. Analysis f the Initial Experiments In the first year f the study, egg chrts were incubated at l()'c, 20 C, 25 C, and 30 C. Fur temperature chambers were used, ne fr each temperature. Since n BLB eggs were bserved t hatch at loc, daily develpment rates were estimated nly fr the ther three temperatures. The number f chrts and number f individual BLB eggs maintained at each temperature are presented in Table 2. Based n the experience f the entmlgists, values f A in frmulas (4) and (5) were specified as 3, 5, and 9 days at 20 C, 25 C, and 30 C, respectively. N eggs were bserved t hatch prir t the specified A fr any f the respective temperatures, but increasing any f the A values wuld vilate this criterin. It is pssible t use smaller values f A. This wuld affect the estimates f 0 and "I, but it wuld have relatively little affect n estimates f 7f r M because large numbers f eggs were examined and final inspectin times were used that allwed ver 90 % f the viable eggs t hatch. Estimates f the prbability f viable eggs (7f), the hatch time distributin parameters (0 and "I), and the median time t hatch fr viable eggs (M), btained by maximizing (3), are presented in Table 3 fr each f the three temperatures. The estimates f 7f increase as the temperature increases and are nly slightly higher than the lwer bunds prvided by the bserved prprtins f hatched eggs reprted in the last clumn f Table 2. Estimates f the Weibull shape parameters are relatively unaffected by temperature.

8 Applied Statistics in Agriculture 221 Estimates f the scale parameter and median hatch time decrease as temperature increases, indicating that eggs tend t hatch sner and the variability in hatch times decreases as incubatin temperature increases frm 20 0 e t 30 e. Parameter estimates btained by maximizing (3) fr the Weibull mdel defined by (4) are listed in the third clumn f Table 3, and the fifth clumn gives estimates adjusted fr btstrap estimates f bias. The last clumn f Table 3 shws the btstrap estimates f the standard deviatins. Fr cmparative purpses, the furth clumn f Table 3 displays standard errrs btained frm inverting the matrix f secnd partial derivatives f the IWM lg-likelihd. Accunting fr within chrt crrelatin with the btstrap prcedure increases the estimated standard errrs by at least a factr f 1.9 in all cases. The btstrapped standard errr fr the median hatch time fr viable BLB eggs at each temperature is cmputed by applying frmula (5) t the estimates f 0 and'y frm each f the 1000 btstrap samples t btain 1000 btstrapped estimates f the median. The btstrapped standard errr is simply cmputed as the standard deviatin f thse 1000 btstrapped estimates f the median. The standard errr fr the median reprted in the furth clumn f Table 3 is cmputed by applying the delta methd t the cvariance matrix fr the estimates f 0 and 'Y btained by inverting the negative f the matrix f estimated secnd partial derivatives f the IWM lg-likelihd. Results fr the inverse median can be cmputed in a similar fashin. T further examine the effects f temperature n the prprtins f viable BLB eggs, 1000 sets f estimates f the prprtins f viable eggs at 200e, 2ye, and 30 0 e were cnstructed by randmly matching estimates frm the three sets f btstrapped samples. The methd f least squares was used t fit a quadratic plynmial t each f the 1000 sets f estimates. The frm f the mdel is ir = {3 + {31 (T - 25) + {32 (T - 25f + E, (6) where T = 20, 25, 30 represents temperature, ir is the btstrapped estimate f the prprtin f viable eggs at the crrespnding temperature, and E is a randm errr. The mean and standard deviatin f the resulting 1000 btstrapped estimates fr (32 are and , respectively, which suggests that the quadratic term is nt needed. The frmula fr the linear regressin f the prprtin f viable BLB eggs n temperature is ir = (T - 25), fr 20 :::;; T :::;; 30. (7) The btstrapped standard errrs fr the estimated intercept and slpe shwn in (7) are and , respectively. The same prcedure was used t examine the effects f temperature n the inverse f the median hatch time fr viable BLB eggs. Previus experience with beetle egg

9 222 hatch times suggested that the percentage daily develpment rate, defined as the inverse f the median hatch time multiplied by 100 %, shuld be linearly related t temperature. The inverses f the btstrapped estimates f the median hatch times were regressed n temperature fr 1000 sets f btstrapped estimates. The estimated regressin frmula IS 100% Median hatch time (T - 25), 20 ~ T ~ 30. (8) The btstrapped standard errrs fr the intercept and slpe are and 0.012, respectively, indicating that bth parameters are psitive. Fitting a quadratic mdel t the inverses f the median hatch times resulted in an estimated cefficient f fr the quadratic temperature term with a standard errr f Cnsequently, a straight line seems t adequately reflect the effect f temperature n the inverse median hatch time (develpment rate) fr viable BLB eggs. 6. The Need fr Replicatin In the initial set f three experiments all egg chrts incubated at a particular temperature were maintained in a single temperature chamber. Only three temperature chambers were used, ne fr each f the three temperatures. Cnsequently, btstrap standard errrs cmputed in the previus sectin essentially reflect nly variatin within and amng chrts within temperature chambers. Variatin amng replicated experiments fr a single temperature dne in different temperature chambers cannt be reliably assessed frm these data, and such variatin is nt taken int accunt in the previus analyses f temperature effects n inverse medians and prprtins f viable eggs. This was reslved by perfrming tw additinal replicatins f the experiment at 20 e, 25 e, and 30 e. In additin, sets f three replicatins were perfrmed at 18 C, 22 C, 28 C, and 32 e. Each f these experiments was perfrmed in a different temperature chamber. These experiments were smaller than the riginal experiments, using fewer egg chrts with abut 20 eggs per chrt. The number f chrts, ttal number f eggs bserved t hatch, and ttal number f eggs used in each experiment are listed in Table 4. The fifth clumn f Table 4 gives the estimate f the prprtin f viable eggs btained frm maximizing (3) with the Weibull survivr functin defined by (4). Crrespnding estimates f inverse median hatch times multiplied by 100%, referred t as daily develpment rates, are listed in clumn 7 f Table 4. Btstrap estimates f standard errrs f these estimates are presented in clumns 6 and 8, respectively. The squares f the inverses f these standard errrs were used as weights in weighted least squares estimatin f regressin mdels cmputed with the GLM prcedure in the SAS cmputer package.

10 Applied Statistics in Agriculture 223 The estimated prprtins f viable eggs, frm clumn 5 f Table 4, are pltted against temperature in Figure 1. The estimates frm the riginal three experiments appear as slid dts. It appears that the linear trend suggested by these three pints is a spurius result that culd be attributed t temperature chamber variatin. Weighted least squares regressin applied t the pints n this plt did nt prduce a significant slpe (p-value = 0.84). The R-squared value was Between 18 C and 32 C, there is n apparent affect f temperature n the viability f BLB eggs. The percentage daily develpment rate f BLB eggs, given by the inverse f the median hatch time multiplied by 100%, is affected by temperature. Estimates frm clumn 7 n Table 4 are pltted against temperature in Figure 2. The pints crrespnding t the riginal three experiments are displayed with slid dts. Figure 2 als shws the weighted least squares estimate f a simple linear regressin mdel. Fr this mdel, R-squared is 0.95 and the slpe is significant (p-value <.0001). The estimated frmula is 100% Median hatch time (T - 25), 18 ::; T ::; 32. (9) Adding quadratic and cubic terms t. the mdel yields nly a slight imprvement in the R-squared value frm t 0.967, but bth the quadratic term (p-value =.0026) and the cubic term (p-value =.0031) are significant. This suggests that the accuracy f the straight line mdel begins t deterirate utside f the temperature range frm 20 C t 30 C. A referee nted that the weights used in the weighted least squares regressin analysis are estimated which leads t underestimatin f variability f parameter estimates. It is pssible that a secnd applicatin f a btstrap r sme ther resampling prcedure culd be helpful, but this wuld greatly increase the cmputatinal burden and this suggestin has nt yet been investigated. 7. Summary This article presents a methd fr analyzing life event r survival data bserved frm chrts f respndents where results fr members f any single chrt can be crrelated but members f different chrts respnd independently. This methd des nt require the specificatin f a jint survivr functin fr members f a single chrt, nr des it require the specificatin f a mdel fr within chrt crrelatins, which are especially difficult t mdel when sme members f a chrt may nt be viable in the sense that the mnitred life event never ccurs. Only the frm fr a marginal distributin fr survival r life event times fr viable members f the verall ppulatin is required. In this applicatin egg hatch times were either interval r right censred, but the btstrap

11 224 prcedure als accmmdates left censred and exact time data. Inspectin schedules d nt have t be the same fr all chrts and the prcedure autmatically accunts fr variatin in chrt sizes. An LFP apprach was used t accunt fr the pssibility that sme viable eggs may nt be bserved t hatch. Btstrap estimatin was used t btain apprpriate estimates f cvariance matrices fr parameters estimated frm a single experiment. This infrmatin was used in a weighted least regressin analysis f the effects f temperature n the viability and develpment rates f BLB eggs. Replicatin f experiments in different temperature chambers revealed a spurius crrelatin between temperature and prprtin f viable eggs suggested by the three riginal experiments. Similar prcedures can be used t mdel temperature effects n BLB larval develpment times. Ultimately, the results f these studies will be used t cnstruct mdels fr predicting peak emergence times and ppulatin sizes f F2 BLB adults frm sample infrmatin n the time eggs are laid by the Fl generatin. Cmbining this with predictins f the crp stage at the time f peak emergence f F2 adults will enable mre cst effective and envirnmentally sund management strategies t be develped. Acknwledgements This research was supprted by the Natinal Cancer Institute, grant number 1 ROI CA Al, and the Iwa Agriculture and Hme Ecnmics Experiment Statin, Ames, Iwa; Prject Ns and The BLB egg hatch experiments were perfrmed by Michael R. Zeiss as part f his dissertatin research at Iwa State University, under the directin f Prfessr Larry P. Pedig, Department f Entmlgy. References Cx, D. R. and Oakes, D. (1984). Analysis f Survival Data. New Yrk: Chapman Hall. Efrn, B. (1979). The Reitz Lecture. Btstrap methds: anther lk at the jackknife. Annals f Statistics, 7, Efrn, B. (1982). The Jackknife, the Btstrap and Other Resampling Plans, Sciety fr Industrial and Applied Mathematics, Philadelphia. Herzg, D. C., Eastman, C. E. and Newsm, L. D. (1974). Labratry rearing f the bean leaf beetle. Jurnal f Ecnmic Entmlgy, 67, Huber, P. J. (1967). The behavir f maximum likelihd estimatrs under nnstandard cnditins. Prceedings f the Fifth Berkeley Sympsium,

12 Applied Statistics in Agriculture 225 Huster, W. J., Brkmeyer, R. and Self, S. G. (1989). Mdelling paired survival data with cvariates. Bimetrics, 45, Kalbfleisch, J. D. and Prentice, R. L. (1980). The Statistical Analysis f Failure Time Data, New Yrk: Wiley. Kehler, J. K. and McGvern, P. G. (1990). Applicatin f the LFP survival mdel t smking cessatin data. Statistics in Medicine, 9, Meeker, W. Q. (1987). Limited failure ppulatin life tests: applicatins t integrated circuit reliability. Technmetrics, 29, Smelser, R. B. and Pedig, L. P. (1991). Phenlgy f the bean leaf beetle, Certma trifurcata (Fster), n sybean and alfalfa in central Iwa. Envirnmental Entmlgy, 20, Smelser, R. B. and Pedig, L. P. (1992). Bean leaf beetle (Cleptera: Chrysmelidae) herbivry n leaf, stem, and pd cmpnents f sybean. Jurnal f Ecnmic Entmlgy, 85, Turner, L. A. (1979). Binmics f the bean leaf beetle, Certma trifurcata (Fster) in Illinis. Unpublished Ph.D. dissertatin, University f Illinis, Urbana-Champaign. Table 1. Hatch results fr 20 f the 111 bean leaf beetle egg chrts incubated at 25 C. Observed number Days after eggs were laid Number f Egg f viable chrt eggs eggs

13 226 Table 2. Numbers f bean leaf beetle egg chrts maintained at each temperature. Number Number Percent f f bserved Temperature chrts eggs t hatch 20 C C C Table 3. Estimates f parameters and standard errrs fr Weibull LFP mdels fr BLB eggs incubated at three different temperatures. MLE's assuming independent respnses Btstrap estimates Std. Std. Temperature Parameter Estimate errr Estimate errr 20 C Scale (0) Shape (y) Prprtin viable (7f ) Median hatch time (M) C Scale (0) Shape (y) Prprtin viable ( 7f) Median hatch time (M) C Scale (0 ) Shape (y) Prprtin viable (7f ) Median hatch time (M)

14 Applied Statistics in Agriculture 227 Table 4. Estimates f the prprtin f viable eggs (1f') and the daily develpment rate (M- 1 x 100%) fr 21 BLB egg hatch experiments. Estimated Ttal Number Estimated % daily Standard Temper- Number number f eggs Prprtin Standard develpment errr f ature f f bserved f viable errr f (UN{) (11M), ("C) chrts eggs t hatch eggs (i-) 7f x 100% x 100%

15 228 Fig 1. Estimated Prprtin f Viable Eggs 1.0 p. r p. r ~ t i 0 n r-r.,...,...,..._T""T"""~"""T""T""T""'T".,...,...,..._T""T"""~"""T""T""T""'T".,...,...,...T"""T_,...,...""'T""'T""'T"""I" Temperature (degrees C)

16 Applied Statistics in Agriculture 229 Fig 2. BLB Egg Daily Develpment Rates P 16 e r 14 c e 12 n t Temperature (degrees C)