Maximum Likelihood Estimation of Dietary Intake Distributions

Transcription

1 CARD Workig Papers CARD Reports ad Workig Papers Maximum Likelihood Estimatio of Dietary Itake Distributios Jeffrey D. Helterbrad Iowa State Uiversity Follow this ad additioal works at: Part of the Biometry Commos, ad the Huma ad Cliical Nutritio Commos Recommeded Citatio Helterbrad, Jeffrey D., "Maximum Likelihood Estimatio of Dietary Itake Distributios" (1992). CARD Workig Papers This Article is brought to you for free ad ope access by the CARD Reports ad Workig Papers at Iowa State Uiversity Digital Repository. It has bee accepted for iclusio i CARD Workig Papers by a authorized admiistrator of Iowa State Uiversity Digital Repository. For more iformatio, please cotact digirep@iastate.edu.

2 Maximum Likelihood Estimatio of Dietary Itake Distributios Abstract This paper applies maximum likelihood estimatio techiques to determie suitable models for dietary itake distributios. Hypothesis test results idicate that while the gamma ad Weibull models appear suitable for describig the itake distributios of some dietary compoets, a more flexible family of distributios is required i order to appropriately ecompass all dietary compoet distributios. Six utriets are cosidered i the aalysis icludig calcium, eergy, iro, protei, vitami A, ad vitami C. Based o chi-square goodess-of-fit tests, we coclude that the three parameter, geeralized gamma family of distributios accurately describes the distributios of all six dietary compoets. The additioal flexibility of this family results i large stadard errors for the parameter estimates. However, the stadard errors of the estimated percetage of the populatio below a specified level of utriet itake appear precise ad allow for substative coclusios regardig utritioal iadequacy to be made. Keywords Statistics Disciplies Biometry Huma ad Cliical Nutritio This article is available at Iowa State Uiversity Digital Repository:

3 Maximum Likelihood Estimatio of Dietary Itake Distributios Jeffrey D. Helterbrad Workig Paper 92-WP 98 August 1992 Ceter for Agricultural ad Rural Developmmt Iowa State UivErSity Ames, Iowa Jeffrey D. Heitert>rad is a graduate assista~ Departmet of Statistics, Iowa State Uiversity, Ames, Iowa. This paper was prepared for the Huma Nutritio Iformatio Service of USDA uder Research Support Agreemet

4 iii ABSTRACT This paper applies maximum likelihood estimatio techiques to determie suitable models for dietary itake distributios. Hypothesis test results idicate that while the gamma ad Weibull models appear suitable for describig the itake distributios of some dietary compoets, a more flexible family of distributios is required i order to appropriately ecompass all dietary compoet distributios. Six utriets are cosidered i the aalysis icludig calcium, eergy, iro, protei, vitami A, ad vitami C. Based o chi-square goodess-of-fit tests, we coclude that the three parameter, geeralized gamma family of distributios accurately describes the distributios of all six dietary compoets. The additioal flexibility of this family results i large stadard errors for the parameter estimates. However, the stadard errors of the estimated percetage of the populatio below a specified level of utriet itake appear precise ad allow for substative coclusios regardig utritioal iadequacy to be made.

5 l Itroductio A estimate of the distributio of itakes for a give dietary compoet ca be used to obtai estimates of the prevalece of iadequacy i the utritioal status of idividuals. Dietary iadequacies may result from either excessive or deficiet itakes of a dietary compoet over a log period of time. Dietary itake distributios are of iterest to utritioists ad policy makers for moitorig dietary ad utritioal status, for evaluatig food policies, ad i utritio ad dietary educatio programs. Estimatig dietary itake distributios typically ivolves estimatig distributios of usual itake. The usual daily itake of a dietary compoet for a idividual is a measure of the idividual's typical daily cosumptio rate durig a time period appropriate for the particular dietary compoet. Usual itake ca be viewed as a log-ru average of daily itakes for the idividual. There is clear evidece that may itake distributios are ot symmetric. Hece, distributios such as the gamma ad Weibull family are potetial models for usual itake distributio fittig. I prelimiary ivestigatios, Nusser et al. (1988) idicated that assumig that usual itakes follow a gamma distributio was appropriate for some, but ot all, dietary compoets. A similar result was obtaied usig the Weibull family. Thus, Nusser et al. suggested the geeralized gamma distributio as a potetially all-ecompassig family of distributios for dietary compoet distributios.

6 2 Properties of the Geeralized Gamma Distributio desity The geeralized gamma family is a three parameter distributio with -1-1.\tl-1-1,\ f(y)- [9r(tJ)j.\(8 y) exp(-(8 y) ), (1) y > 0 8 > 0. tl > 0.,\ > 0 where r(tj) - ~ttj-le-tdt ad 8, tj, ad.\ are parameters to be estimated. The geeralized gamma family has may familiar distributioal families as special cases. Examples iclude the expoetial (tj- 1,,\ - 1), gamma (,\ - 1), Weibull (tl- 1) ad chi-square (8-2, tl - /2,.\ - 1) distributios. I additio, the logormal family is a limitig special case (tl ~ ~) The cumulative distributio fuctio for the geeralized gamma distributio is F(y; e. tl.\)- rz(tl)/r(tj) (2) The r-th momet of Y (r-1, 2, 3,... ) ca be writte (3) If Y is a geeralized gamma variate, the ~ is distributed as a gamma with parameters 9,\ ad tj. This property of the geeralized gamma distributio is used i obtaiig startig estimates for the iterative procedures ecessary i estimatig geeralized gamma parameters.

7 3 The geeralized gamma distributio is a more geeral model tha the Weibull, expoetial, ad gamma. However, Hager ad Bai (1970) cocluded from their study that the Weibull model was about as flexible as the geeralized gamma distributio for sample sizes up to 200. Thus, give the complexity of the geeralized gamma distributio, ad some of the estimatio difficulties ecoutered, Hager ad Bai suggested that the Weibull assumptio was preferable to a geeralized gamma distributio for sample sizes up to 200. Maximum Likelihood Estimatio ad the Geeralized Gamma Distributio Stacy ad Mihram (1965), Hager ad Bai (1970), Parr ad Webster (1965), ad Pretice (1974) have examied maximum likelihood techiques to estimate the parameters of the geeralized gamma distributio. From the desity fuctio, the maximum likelihood equatios for idepedet observatios.:a be obtaied as follows. The likelihood fuctio is f(yl, Y2'..., y; e. fi, >.)- IT i-l -l -l,\fi-1 -l,\ [Bf(fi)J >.(B y.) exp(-(b y.) J. ~ ~ (4) Let L()- i f(y 1, y 2,..., y; B, fi, >.) be the log-likelihood fuctio. It follows that L() - i(>.) - >.fi l(b) - l[f(fi)] + (>.fi - l) 2: l(y.) i-l ~ (5)

8 4 The maximum likelihood equatios are obtaied by takig the first derivative of L() with respect to 8, ~ ad A, respectively, ad settig the resultig derivatives equal to zero. The equatios are -~ + A E (y./8) - 0, ~1 1 (6) -~(~) + A E i-l l(y./8) - 0, 1 (7) /A + ~ E i-l l(y./8) - 1 E i-1 A (y./8) l(y./8) ' (8) where ~(~)- d[lf(~)]/d~. Equatios (6), (7), ad (8) must be solved for 8 ad A simultaeously. Sice a closed form solutio for 8, ~ ad A is ot kow, a iterative techique is required to compute the estimators 8 ad A. Usig maximum likelihood techiques to estimate the geeralized gamma parameters requires good startig values for the parameters i order to obtai the appropriate estimates from the iterative procedure. Adequate computer resources are also eeded to calculate fuctios such asf(~), ~(~) ad ~ (~). Harter (1965) foud the umber of iteratios required for covergece teded to be large whe estimatig ~ ad A simultaeously. This is apparetly due to the high egative correlatio betwee the estimates of the two parameters. I additio, the approximate ormal distributio for ~ predicted by maximum likelihood theory was ot observed eve for samples of size 400 [Pretice (1974)]. To apply maximum likelihood, it is ecessary to develop a reliable iterative techique which will coverge to correct solutios, while keepig the estimates i the parameter space (8 > 0, ~ > 0, A > 0)

9 5 Hager ad Bai (1970) idicated that the Newto-Raphso method did ot appear to work well whe solvig for geeralized gamma parameters. Harter (1965) advised usig a hybrid of two differet iterative techiques to fid parameter estimates which coverge. A Algorithm for Maximum Likelihood Estimatio for the Geeralized Gamma Distributio A algorithm for estimatig geeralized gamma parameters for the usual itake data, requires two stages: 1) a algorithm providig good startig values ad 2) a iterative algorithm to compute accurate solutios from the give startig values. To obtai estimates of the geeralized gamma parameters, equatios for the first four cetral momets of the geeralized gamma distributio are eeded. They are ' 1 - E(yl - 6r ({l)f({l + >. ), (9) ' 2 - E( (y - p) l - 6 (r ({l)r({l + z>. ) - [r ({l)r <fl + >. ) J l (10) ' 3 - E( (y - p) J - 6 [r ({l)f({l + 3>. ) - 3r ({l)r({l + >. )r({l + 2>. ) (11)

10 6 ~ Et(y- ~) l- a [r (ft)f(ft + 4A ) (12) The startig value routie begis by usig a grid search scheme. Fiftee A values are chose from a rage of values believed to cotai the parameter. For a give power,. A, a startig value for the shape parameter, fta, is obtaied by solvig the secod momet equatio (10) evaluated at A. A iterative procedure for obtaiig a solutio for fta from (10) is the DBCPOL routie i IMSL, which also requires a startig value for fta Sice y- GG(8, ft, A) implies A A y - Gamma(B, ft), a iitial value for fta ca be derived as follows. variace of Usig the secod order Taylor series expasio, the mea ad A y ca be approximated by ( 13) ad (14) where a - 1

11 7 ad ad are the momets of y. Sice a gamma distributio,,1, y follows,1, -1,1, 2 f3 ~ [var(y ) J [E(y ) ] (15) Hece a iitial value for {3,1, is (16) where ' ' J.ll.l. ad J.l2.1. are evaluated at the sample momets 1'1 1'2 1'3, ad J.l4 Thus for each,1, {3.1.0 is used to start the DBCPOL subroutie ad {3,1, is computed. A startig value, e,~, is also calculated by evaluatig the first momet equatio (9) at.1. ad /3,1, The startig value algorithm computes fiftee vectors, (.1., {3,1,, e,~,), correspodig to the fiftee startig.1. values. The algorithm ext substitutes each of these vectors ito the log-likelihood equatio (5) for the geeralized gamma distributio, ad a value for the log-likelihood is calculated. The three largest log-likelihood values ad their correspodig,1, values are determied. A quadratic i,1, is fit to these three log-likelihood poits, ad the,1, correspodig to the maximum of the quadratic is take to be the startig value,.1.* for stage 2. Give.1.*, equatio (16) is used to compute {3.1.* ad the first momet equatio (9) is the solved to calculate e.l.*. The vector (.1.*, {3.1.*' e.l.*) serves as the startig value for the iterative maximum likelihood estimatio program.

12 8 To compute the maximum likelihood estimators, the oliear system A z (y./0) i-1 (17) z i-1 i.(y./0) - 0, 1 (18) F 3 (y; 0, ~. A) - /A + ~ Z i-1 i.(y./0) - 1 A Z (y./0) i.(y./0) i (19) is solved usig a modified Newto's method. This system correspods to (6), (7), ad (8). Newto's iterative method requires J(y; 0, ~. A) the Jacobia matrix of partial derivatives of F 1, F 2, ad r 3 with respect to 0, ~, ad A. The elemets of this matrix are (20) ( 21) A ( Z y.i. y. i A i.o Z y.). l 1 1- (22) (23) (24) Z i.(y./0) 1 i-1 (25) J 31 Cy; 0, ~. A)- -(/0) + -' 0 z y.) i-1 1

13 9 J 32 cy; e, ~. A) ~ ~ 1(y./8). l 1 1- (26) ( 2 7) 2 J 33 (y; e, ~..\) - (/A ) - -A A 2 e ( ~ y. (2 y. ) i (2 (28) The ( + 1)-st estimates of the Newto method satisfy the system, I I'.., ' e, e I IFl (y; e ' ~' A ) +l I ' ' -1 I ~+l! "j e lf2(y; ' ~' A ) r: -J l~+l F3(y; e ' ~' A ) ' where (8 ' ~' A ) are the.estimates at the th iteratio. At each iteratio, the log-likelihood is calculated usig (B ' ~ '.\ ) assure that the estimates are covergig to a solutio which maximizes the log-likelihood fuctio. The iteratio is assumed to have coverged A l'o. A "' 1\ A whe the maximum of ( le. B I 1~ ~ I 1.\.\ I} is less +l - ' "+l - " ' +l - tha Oce covergece is achieved, the maximum likelihood estimates are deoted by BMLE, ~MLE, ad.\mle Oe computatioal difficulty i this algorithm is the calculatio of ~(~) ad ~-(~). For real positive ~, ~(~) is a cocave icreasig fuctio which satisfies the followig relatios (Berardo, 1976) to ~(l) - - -y ' (29) ~(1 + ~) - ~(~) + 1/~ ' (30)

14 10 >/>(fj) - i{j - l l l l + + 0(_1_) (for large fj i.e. fj ~ oo) 2{J l2i 12oi 252i fj8 (31) >/>({J) - --y - ~ + O({J) (for small fj i.e.,b ~ 0) (32) where -y is Eulers costat. Equatio (32) is used to compute >/>(fj) for -5 {J < l. 0 X 10. The Stirlig expasio (31) is used i calculatig >/>(fj) for fj 2: 8.5 The recursive equatio (30) is used for X 10!0 fj < 8.5 ad gives values for >/>(fj) that are accurate to withi l.ox lo-s for fj i that rage. By differetiatig these cotiuous fuctios, the resultig equatios are used to calculate >J>'(fJ) for the three cases. A secod algorithm for fittig the parameters of the geeralized gamma distributio ca be costructed usig a reparameterizatio of the geeralized gamma desity based o the logarithms of the respose variable. This procedure was suggested by Pretice (1974). The log geeralized gamma variate follows a locatio-scale model. The desity fuctio uder the Pretice parameterizatio is f(x; a, a, q) - ( q "' 0) l 2 -(x 2a 2 - a) l (q - 0) ' where x - log y ad Note that, by this reparameterizatio, the geeralized gamma model has bee exteded such that q ca be egative. If -l/2 q - fj, x - log(y) follows a ormal model whe q - 0. The Pretice parameters (parameterizatio B) ca be

15 ll expressed i terms of the origial parameters (parameterizatio A) by the oliear mappig -l/2 q - fj ' a - 1 og [~(fj)/'] d a - (l/'fj 1 / 2 o +,., ", a " ). Also, the asymptotic variace of the maximum likelihood estimators q ad fj are related by var(q) 1 fj- 3 4 var(fj) Parameterizatio B leads to some useful results. First, previous authors foud that a large sample size was required whe tryig to discrimiate betwee fj- 1 ad {J.- 2 or 3 Pretice cocluded that o a log scale these distributios are very similar ad i fact, without a large sample size, fj - 1 is difficult to discrimiate from fj - "' Furthermore, whe q- 0 (fj close to ifiity), maximum likelihood estimates from parameterizatio A caot be obtaied by solvig the log likelihood equatios. Usig parameterizatio B, Pretice was able to obtai a log likelihood fuctio which exists for all q ad from which maximum likelihood estimates could be obtaied give a adequate sample size. By simulatio, Pretice showed that q coverged faster to its asymptotic ormal distributio tha fj A algorithm i SAS: PROC LIFEREG computes maximum likelihood estimates usig parameterizatio B. Applicatio of Maximum Likelihood to the Nutriet Itake Data The above algorithms (correspodig to parameterizatio A ad parameterizatio B) were applied to data collected i 1985 by the USDA i the Cotiuig Survey of Food Itakes by Idividuals (CFSII). The

16 12 survey collected daily dietary itakes from wome betwee 19 ad 50 years of age ad from their preschool childre. The data set used for our purposes was a subset of these data cotaiig four days of dietary itakes for 785 wome aged betwee 23 ad 50 years who were resposible for meal plaig withi the household ad who were ot pregat or lactatig durig the survey period. Six dietary compoets were aalyzed icludig calcium, eergy, iro, protei, vitami A, ad vitami C. The data to which the algorithm was applied were predicted 11 pseudo" usual itakes geerated from the origial data usig measuremet error techiques (Nusser et al. (1990)]. This methodology was desiged to adjust the observed itakes for the presece of measuremet error i the dietary itake data. The algorithm created by Nusser et al. (1990) ivolved several steps to produce pseudo usual itake values. First, the origial daily itakes were trasformed ito ormal space usig a oparametric trasformatio based o the iverse ormal cumulative distributio fuctio. I ormal space, the daily itakes were assumed to follow a measuremet error model, ad ormal theory was used to develop predictors of usual itakes i ormal space for each idividual. A iverse trasformatio was the applied to the predicted ormal usual itakes to produce the set of pseudo usual itakes i the origial space. Applyig the maximum likelihood estimatio algorithm to the pseudo usual itake data for five of the utriets produced the parameter estimates show i Table 1. Usig parameterizatio A, the algorithm failed to coverge to the maximum likelihood parameter estimates for

17 13 vitami A. Asymptotic stadard errors were computed as the iverse of the estimated iformatio matrix. The asymptotic stadard errors of the estimated coefficiets are large, idicatig the parameter estimates are relatively imprecise. This is due to the high correlatio betwee the parameters. For all utriets, the correlatio betwee ~ ad >. was betwee ad Hager ad Bai (1970) idicated that as ~ icreases away from oe, the asymptotic variace of ~ approaches ifiity. Table 1. Estimated Geeralized Gamma parameters for each dietary compoet. Calculated A A A Dietary Com:eoet 8 MLE ~MLE.l.MLE log-likelihood Calcium ( ) ( ) ( ) Eergy ( ) ( ) ( ) Iro ( ) ( ) ( ) Protei ( ) ( ) ( ) Vitami c ( ) ( ) ( )

18 14 Table 2. Estimated Geeralized Gamma parameters for each dietary compoet. Calculated Dietary ComEoet 0 MLE "MLE qmle log-likelihood Calcium ( ) ( ) ( ) Eergy ( ) ( ) ( ) Iro ( ) ( ) ( ) Protei ( ) ( ) ( ) Vitami A ( ) ( ) ( ) Vitami c ( ) ( ) ( ) Usig parameterizatio B, the LIFEREG procedure i SAS coverged to the parameter estimates i Table 2. The estimates of Table 1 trasformed ito the parameterizatio of Table 2 are very close to those of Table 2. The largest differece occurred for iro, where fi from parameterizatio A differs by 0.85 from the fi calculated by parameterizatio B. Also, the calculated log likelihoods obtaied by the two procedures are similar but ot idetical. Note that for vitami A, maximum likelihood estimates uder parameterizatio A would be fimle ~MLE ad ~ "MLE- e Thus, fimle is quite large,.ad as idicated by Pretice, this caused covergece problems for Vitami A whe the maximum likelihood estimatio algorithm based o parameterizatio A was used.

19 15 It is of iterest to test the fit of the hypothesized geeralized gamma distributios for each utriet. A chi-square goodess-of-fit statistic, usig twety~five mutually exclusive itervals over the rage of the pseudo usual itake data, was used as a test statistic. Observed frequecies for the pseudo usual itake data ad the expected frequecies from the hypothesized distributios were computed ad the test statistics, based o 21 degrees of freedom, are preseted i Table 3 for each of the six dietary compoets. Tests of size 0.05 idicate the geeralized gamma provides a satisfactory fit for all six of the dietary compoets. A plot of the hypothesized geeralized gamma ad empirical cumulative distributio fuctio for each utriet is icluded i Appedix A. Testig Hypotheses with the Maximum Likelihood Estimatio Algorithm Of iterest is whether or ot the Weibull or gamma distributios would provide a satisfactory fit for the pseudo usual itake data. Tests of these hypotheses ca be costructed usig the likelihood ratio test. Table 3. Goodess of Fit test for Geeralized Gamma Distributio. Compoet 2 X Calcium 27.3 Eergy 22.5 Iro 32.5 Protei 27.4 Vitami A 24.3 Vitami c 15.6 The a -.05 poit of the chi-square distributio with 21 df is 32.7.

20 16 Table 4. Estimated Weibull parameters for each dietary compoet. Dietary ComEoet 8 MLE,\MLE Calcium ( ) ( ) Eergy ( ) ( ) Iro ( ) ( ) Protei ( ) ( ) Vitami A l ( ) ( ) Vitami c ( ) ( ) Testig - l. The desity fuctio for the Weibull distributio is -1-1,\-1-1,\ fw(y) - 8,\(8 y) exp[(-8 y) ], which is the geeralized gamma desity with ~ - l. By costraiig the maximum likelihood estimator algorithm to estimate the parameters of the Weibull distributio, the parameter estimates for the six utriets listed i Table 4 were obtaied. Note that whe ~ is set equal to oe, the stadard errors of the parameter estimates decrease dramatically compared to those computed uder the geeralized gamma assumptio. To test whether the Weibull is a suitable family of distributios for each utriet, likelihood ratio tests were costructed. By the asymptotic properties of the likelihood ratio test, 2(log likelihoo~eib - log likelihoodggd) is asymptotically distributed

21 17 Table 5. Likelihood ratio test for Weibull ull hypothesis agaist Geeralized Gamma. Compoet 2 X Calcium 63.2 Eergy 58.0 Iro Protei 42.7 Vitami A Vitami c 24.1 The a -.OS poit of the chi-square distributio with l df is as chi-square with oe degree of freedom uder the ull hypothesis. The test statistics are preseted i Table 5. Tests of size 0.05 idicate that the Weibull family does ot adequately fit the distributio of ay of the dietary compoets aalyzed. That is, the geeralized gamma hypothesis domiates the Weibull hypothesis i the likelihood ratio test. Asymptotic chi-square goodess-of-fit tests for the hypothesized Weibull distributios were computed ad are listed i Table 6. These statistics also idicate that the hypothesized Weibull distributio does ot accurately describe the data for five of the dietary compoets, but is adequate for vitami C. Plots of the hypothesized Weibull ad empirical cumulative distributio fuctios are icluded i Appedix A. Testig A - l. The desity fuctio for the gamma distributio ca be writte -l -l p-l -1 fg(y) - [9r(p)] (9 y) exp[-9 y]

22 18 Table 6. Goodess of fit test for Weibull Distributio. Compoet 2 X Calcium Eergy Iro Protei Vitami A Vitami C The a poit of the chi-square distributio with 22 df is which is the geeralized gamma distributio with A - 1. Estimates of the gamma distributio parameters for the six dietary compoets are give i Table 7. Note that stadard errors are agai much lower tha those for the geeralized gamma distributio. Agai, 2(log likelihoodgam - log likelihoodggd) is asymptotically chi-square with oe degree of freedom uder the ull hypothesis. The Table 7. Estimated Gamma parameters for each dietary compoet. ' Dietary ComEoet 8 MLE.BMLE Calcium ( ) ( ) Eergy ( ) ( ) Iro ( ) ( ) Protei ( ) ( ) Vitami A ( ) ( ) Vitami c ( ) ( )

23 19 likelihood ratio test statistic for each utriet was computed ad is preseted i Table 8. The likelihood ratio test at size 0.05 idicates the gamma family caot be rejected as a adequate family to describe the distributio for the dietary compoets calcium, eergy, iro ad vitami C. A approximate chi-square goodess-of-fit test for the hypothesized gamma distributios were computed ad are listed i Table 9. Correspodig plots of the hypothesized gamma ad empirical cumulative distributio fuctio for each utriet are icluded i Appedix A. Chi-square tests of size 0.05 idicate that the hypothesized gamma distributios is satisfactory i describig the distributio of the data for five dietary compoets,.vitami A excluded. Table 8. Likelihood ratio test gamma ull hypothesis Geeralized Gamma. Compoet x 2 for agaist 2 X Calcium 0.6 Eergy 1.0 Iro 0.2 Protei 6.8 Vitami A 34.0 Vitami c 0.5 The a -.OS poit of the chi-square distributio with 1 df is 3.84.

24 20 Table 9. Approximate goodess-of-fit 2 test x for each dietary compoet gamma parameter estimates. Compoet 2 X Calcium Eergy Iro Protei Vitami A Vitami C The a -.05 poit of the chi-square distributio with 22 df is Usig the Estimated Usual Itake Distributios to Estimate the Prevalece of Nutritioal Iadequacy Estimated usual itake distributios ca be used to evaluate the utritioal status of a populatio of idividuals. The assessmet of dietary status withi a populatio typically ivolves compariso of observed dietary itakes with a measure of the requiremet for a particular utriet or food compoet (NRC, 1986). Oe commo method of compariso relies o a fixed poit cut-off requiremet level. The recommeded daily allowace for a dietary compoet, established by the Uited States Departmet of Agriculture Food ad Nutritio Board, is a example of such a cut-off poit. The cut-off method utilizes a stadard requiremet level as the criterio value, where idividuals with itakes below this stadard are said to be at utritioal risk. Sice the recommeded daily allowace levels are set sufficietly high to meet the kow utritioal eeds of early all healthy persos, a cut-off poit ofte is defied to be a proportio of the recommeded daily allowace. The proportio used by researchers ad policy makers differs

25 21 across utriets ad studies. The choice of a proportio, k, is a matter of judgmet, though the cut-off proportio is usually chose betwee 0.5 ad 0.8. Ay estimate of a at risk percetage is very sesitive to the value of the recommeded daily allowace proportio k. As this proportio decreases, the percetage of the populatio deemed at risk declies. Give a maximum likelihood estimate of the usual itake distributio ad a proportio k, estimates of the percetage of the populatio of wome aged who are at utritioal risk for a give dietary compoet ca be obtaied. The cumulative distributio fuctio for the geeralized gamma, defied-i equatio (2), is used to calculate the proportio of the populatio whose itakes fall below a specified level y. For the six dietary compoets ad for k- 0.5, 0.65, 0.8, the estimates of wome aged 23~50 at utritioal risk for a give dietary compoet usig the geeralized gamma models are preseted i Table 10.

26 22 Table 10. Compoet The estimated percetage at-risk (ad approximate stadard errors) for each dietary compoet usig geeralized gamma parameter estimates. Percetage at-risk for criteria k(rda) as the cut-off poit k RDA Calcium (mg) ( 1. 23) ( 1. 56) (1. 47) Eergy (kcal) 2, (0.88) ( 1. 44) ( 1. 46) Iro (mg) (1. 48) ( 1. 24) (0.74) Protei (g) (0.10) (0.26) (0.50) Vitami A ( IU) (0.85) (0.81) (1.17) Vitami C (mg) (0.78) ( 1. 08) ( 1. 26) Sice the distributio fuctio is oliear i the parameters, the stadard errors for the percetage estimates are approximated usig Taylor's theorem. Deote the estimated cumulative distributio fuctio for the geeralized gamma as F(y; 8, p, A). By Taylor's theorem we have F(y; 8, p, A)..;. F(y; 8, p, A) +df(y; ~O (3, A) (8-8) where (8, p, A) are the true parameters. Thus,

27 23 Table 11. Compoet The estimated percetage at risk (ad approximate stadard errors) for each dietary compoet usig gamma parameter estimates. Percetage at risk for criteria k(rda) as the cut-off poit k RDA. 5 Calcium (mg) (1. 21) Eergy (kcal) 2, (0.86) Iro (mg) ( 1. 40) Protei (g) (0.02) Vitami A (IU) (0.50) Vitami C (mg) (0. 72) (1. 41) (1.37) ( 1. 25) 0.67 (0.13) 8.33 (0.77) ( 1. 03) (1.37) ( 1. 38) (0.69) 3.59 (0.46) ( 1. 00) ( 1. 25) ' Var(F(y; 8, {J, A) - F(y; 8, A, fl)) where p- (8, {J, A). That is, ' Var(F(y; 8, {J, A) - F(y; 8, A, fl)) ~ df(y, e) dp V(p) df(y, p) dp where V(p) is the estimated covariace matrix of the parameter estimates. Sice the values of the partial derivatives at the true parameters are ot kow, the parameter estimates are used to

28 24 approximate the partial derivatives as well. Hece, the variace of the percetage estimates ca be approximated by Var[F(y; a, fj, >.)],;. [df df db' d{j' df ' ' ' ' df df df ' d>.] V(B, {J, >.) [do' d{j' d>.] where df df df] [do ' d{j' d>. are evaluated at the parameter estimates ad V(B, {J, >.) is the estimated covariace matrix of the parameter estimates. A relatively high percetage of the populatio is estimated to have utritioal deficiecies i calicum, eergy ad iro, ad a low percetage is estimated to be at risk with respect to protei ad Vitami A. The percetage of wome aged estimated to be at utritioal risk, based o the gamma models, are preseted i Table ll. The percetage at-risk estimates are similar uder the geeralized gamma ad gamma models, for all dietary compoets excludig Vitami A. Recall that the gamma model was foud ot to be a satisfactory model for the Vitami A data. Also, although the stadard errors for the geeralized gamma parameter estimates are large, the stadard errors of the percetages at-risk uder the geeralized gamma models are oly slightly larger tha the stadard errors computed uder the gamma model. Coclusio I this paper, estimatio of the geeralized gamma distributio has bee discussed. A algorithm has bee preseted which geerates startig values from the data, ad uses these startig values to obtai maximum likelihood estimators for the geeralized gamma parameters.

29 25 Maximum likelihood estimates of the parameters of the distributio of usual itakes were obtaied usig two parameterizatios. Likelihood ratio tests were used to test whether the distributio of usual itakes for selected dietary compoets could be reasoably described by the less complex Weibull ad gamma families. Fially, usig the maximum likelihood estimators of the geeralized gamma parameters, the percetage of wome aged at utritioal risk was estimated usig the cut-off method at differet proportios of the recommeded daily allowaces for the dietary compoets aalyzed i this study. This study idicates that for the CSFII data, the less complex gamma distributio provides a adequate fit for five of the utriets, vitami A excluded. Based o the chi-square goodess of fit tests, the geeralized gamma family appears to accurately represet all six utritioal compoet usual itake distributios. Although the stadard errors of the parameter estimates are large, the stadard errors of the statistic of iterest, percetage at-risk, are reasoable ad allow for substative coculsios regardig utritioal iadequacy to be made.

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