Estimating Age-specific Prevalence of Testosterone Deficiency in Men Using Normal Mixture Models

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1 Journal of Data Scence 7(2009), Estmatng Age-specfc Prevalence of Testosterone Defcency n Men Usng Normal Mxture Models Yungta Lo Mount Sna School of Medcne Abstract: Testosterone levels declne as men age. There s lttle consensus on what testosterone levels are normal for agng men. In ths paper, we estmate age-specfc prevalence of testosterone defcency n men usng normal mxture models when no generally agreed on cut-off value for defnng testosterone defcency s avalable. The Box-Cox power transformaton s used to determne whch transformaton s most approprate for correctng skewness n data and best suts normal mxture dstrbutons. Parametrc bootstrap tests are used to determne the number of components n a normal mxture. Key words: Box-Cox power transformaton, EM algorthm, normal mxture, parametrc bootstrap, testosterone level. 1. Introducton Testosterone s a male sex hormone that helps mantan bone mass, fat dstrbuton, male har patterns, muscle mass and strength, sex drve and sperm producton throughout male adult lfe. Testosterone levels declne as men age. Harman et al.(2001) reported that most of change n male testosterone levels occurs before age 50. Some reported effects of testosterone defcency n men nclude decreased energy, reduced muscle mass and strength, decreased cogntve functon, less sexual nterest or potency, ncreased breast sze, and a depressed mood. What are normal testosterone levels for healthy older men? There s no general agreement on what testosterone levels are normal for healthy agng men. A recent endocrne socety annual andropause consensus meetng suggested that 300 nanograms per declter (ng/dl) s the lower lmt for normal testosterone levels, and total testosterone levels less than 200 ng/dl clearly ndcate hypogonadsm n healthy young men. Harman et al.(2001) reported age-specfc prevalences of hypogonadsm of 12, 19, 28, and 49 percent for men n ther 50s, 60s 70s, and 80s, respectvely. They defned hypogonadsm as total testosterone levels < 325

2 204 Yungta Lo ng/dl. Mohr et al.(2005) used data from men aged 40 to 79 years to ad n the dagnoss of testosterone defcency. They defned total testosterone levels less than the age-specfc 2.5th percentle as abnormally low testosterone levels; thus about 2.5 % of men would have abnormally low testosterone n ther male agng study. The correspondng age-specfc cut-off values are 251, 216, 196, and 156 ng/dl for men n ther 40s, 50s, 60s, and 70s, respectvely. The prevalence of testosterone defcency s conventonally estmated usng a pre-specfed cut-off value to classfy each serum total testosterone. When there s a gold standard for defnng or determnng testosterone defcency, the conventonal approach wll n general gve a good estmate of the testosterone defcency prevalence. When there s no gold standard, prevalence estmates obtaned usng the conventonal approach to defne defcency status wll often vary wth the defnton used. Establshng cut-offs that are generally agreed s dffcult or mpossble when there s no clear separaton between normal testosterone levels and abnormal testosterone levels. As testosterone levels declne gradually wth age n men, a subgroup of men wll have testosterone defcency and have sgns and symptoms related to hypogonadsm whereas a subgroup wll stll have testosterone levels wthn the normal range throughout ther lfetmes causng no sgnfcant problems at all. If we assume that serum total testosterone samples are taken from a mxture of men who have testosterone defcency and those who do not have testosterone defcency, the prevalence of testosterone defcency can be estmated through fttng a two-component mxture model to the total testosterone levels. Unlke the conventonal approach for estmatng prevalence, mxture models do not need a pre-specfed cut-off value to classfy each serum total testosterone. Mxture models have been used to estmate the dsease prevalence n other studes n the absence of a gold standard measure for a defntve dagnoss of dsease status; see, for example, Gay (1996), Pfeffer, Gal and Brown (2000), and Pfeffer et al.(2008). In ths paper, we use mxture models to account for ndvdual testosterone levels that can arse ether from a subpopulaton wth testosterone defcency or from a subpopulaton wthout testosterone defcency, and to estmate the agespecfc rate of testosterone defcency n men n a partcular age group when no generally agreed on cut-off value for that age group s avalable. Mxture models are often used to model data that arse from heterogeneous populatons where the subpopulaton to whch an ndvdual observaton belongs s unknown. The component dstrbutons can arse ether from the same or dfferent parametrc famles. For examples of applcatons of mxture models, see Evertt and Hand (1981), Ttterngton, Smth and Makov (1985), McLachlan and Basford (1988), and McLachlan and Peel (2000). The Box-Cox famly of power transformaton s used to transform data to a mxture of normal dstrbutons (Box

3 Age-specfc Prevalence of Testosterone Defcency 205 and Cox, 1964). The maxmum lkelhood estmators of the unknown parameters are obtaned through applcaton of the EM algorthm and a grd search over a range of possble power transformatons (Dempster, Lard and Rubn, 1977). We llustrate the use of the Box-Cox power transformaton n mxture models to estmate age-specfc prevalence of testosterone defcency wth a sample of serum total testosterone levels obtaned from an HIV at-rsk agng men s prospectve study. Data and model descrpton are gven n Secton 2. Parametrc bootstrap tests for determnng the number of components n a normal mxture dstrbuton are gven n Secton 3. In Secton 4, we adjust effects of covarates on total testosterone levels. Dscussons and concludng remarks are presented n Secton 5. Fgure 1: Hstogram of total testosterone levels from 404 men n ther 50s 2. Data and Model Descrpton Serum total testosterone levels were measured n duplcate usng tme-resolved mmunofluorometrc assays (DELFIA; Pharmaca) n 404 men who were 50 to 59 years of age and currently not takng androgens from a cohort of HIV at-rsk agng men s prospectve study; detals on the study desgn are gven n Klen et al.(2005). We wsh to estmate the age-specfc rate of testosterone defcency n men aged 50 to 59 wth or at rsk for HIV nfecton usng a two-component normal mxture model when no generally agreed on cut-off for testosterone defcency exsts. The mean (± standard error) total testosterone level was ± ng/dl wth a medan (range) value of (2-2070) ng/dl. Of these 404 men, 180

4 206 Yungta Lo (44.6%) had total testosterone levels greater than 300 ng/dl, 91(22.5%) between 200 and 300 ng/dl, and 133 (32.9%) less than 200 ng/dl. The hstogram of total testosterone levels s gven n Fgure 1. From nspecton of Fgure 1, t appears that the dstrbuton of total testosterone levels s skewed to the rght and does not seem to suggest a bmodal dstrbuton that may support usng a mxture of two normal dstrbutons to ft the data and estmate proporton of men wth testosterone defcency. It s noted that skewness s an nherent part of mxture dstrbutons, n partcular, when the component dstrbutons are not well separated. If evdence for a mxture of two normal dstrbutons s confounded by skewness, an approprate transformaton on total testosterone levels s often suffcent to make the normal mxture dstrbuton approprate for the transformed data. As conventonal transformatons such as logarthms or square roots are not suffcent to provde an approprate adjustment for skewness n the data, the Box-Cox power transformaton s used to determne whch transformaton s most approprate for correctng skewness and best suts normal mxture dstrbutons. Mxture models of transformed normal components by the Box-Cox power transformaton have been used n other studes (MacLean, Morton and Elston, 1976; Schork and Schork, 1988; Gray, 1994; Guterrez et al., 1995). Let Y denote the total testosterone level obtaned from subject and n = 404 be the number of men n ths study. Assume that there exsts a real number λ such that the Box-Cox power transformaton on Y denoted as Y (λ) s dstrbuted as a two-component normal mxture dstrbuton, the probablty densty functon of Y s gven as follows where f(y ; θ) = {πf 1 (y (λ) ; µ 1, σ1) 2 + (1 π)f 2 (y (λ) ; µ 2, σ2)}y 2 λ 1, (2.1) { y (λ) (y λ = 1)/λ f λ 0, log y f λ = 0, λ s the unknown Box-Cox power transformaton parameter, θ = (π, µ 1, µ 2, σ1 2, σ2 2 ) s a vector of unknown parameters, f ( ) s the normal densty functon wth mean µ and varance σ 2, π > 0 s the mxng proporton, and the last term on the rghthand sde of (2.1) corresponds to the Jacoban of the transformaton from y to y (λ). To ensure dentfablty of θ for a gven value of λ, we assume wthout loss of generalty that µ 1 < µ 2. Kefer and Wolfowtz (1956) noted that the lkelhood functon of a normal mxture wth unequal varances s unbounded above. Day (1969) argued that spurous maxmzers may occur f some component dstrbuton has a very small varance relatve to others when the correspondng cluster contans few data ponts suffcent close together. Hathaway (1985) suggested

5 Age-specfc Prevalence of Testosterone Defcency 207 usng the constrant that mn,j (σ /σ j ) c > 0, c (0, 1] to rule out spurous maxmzers and showed that there exsts a constraned global maxmzer of the lkelhood functon on the constraned parameter space. To cope wth unboundedness of the lkelhood functon and avod spurous maxmzers, we further assume that mn(σ 1 /σ 2, σ 2 /σ 1 ) c > 0, for some c n the unt nterval. The mxng proporton π represents that a subgroup π of these men have testosterone defcency whereas a subgroup 1 π of these men do not have testosterone defcency. The log-lkelhood functon of y 1,... y n s gven by L(λ, θ; y) = n =1 [log{πf 1 (y (λ) ; µ 1, σ1) 2 + (1 π)f 2 (y (λ) ; µ 2, σ2)} 2 + (λ 1) log y ]. (2.2) For a fxed λ, the last term on the rght-hand sde of (2.2), (λ 1) log y, s just a constant and maxmzaton of L can be acheved by maxmzng the loglkelhood functon of the transformed data, y (λ) 1,... y(λ) n gven n the frst two terms on the rght-hand sde of (2.2) through applcaton of the EM algorthm wth some modfcaton subject to the constrant mn(σ 1 /σ 2, σ 2 /σ 1 ) c. Thus, the maxmzer of L can be found through use of the EM algorthm and a grd search over a range of possble λ values. It can be shown that the maxmum lkelhood estmator ˆλ s that value of λ that maxmzes (2.2). Alternatvely, the maxmum lkelhood estmator for λ can be found usng Newton s method n the M step of the EM algorthm to maxmze L. A grd of λ [ 2, 2] were used n our attempt to determne whch transformaton s most approprate for the normal mxture dstrbuton. Snce the lkelhood of mxture dstrbutons often has multple modes, there s no guarantee that one or few startng values wll fnd the correct root. Thode, Fnch and Mendell (1988) suggested usng several startng values to search for all possble modes. It s noted that problems assocated wth the lkelhood functon on the unconstraned parameter space can occur when c s small. How small can c be to ensure that the constraned parameter space contans the true values of the parameters. Lo (2005) reported based on a smulaton study that c = 0.25 appears to work well for normal mxture dstrbutons wth unequal varances. In ths study, we set the lower bound for the constrant on the component varances c = For each λ value, one hundred sets of startng values for θ were used n our search for the global rather than local maxmzer through applcaton of the EM algorthm subject to the constrant that mn(σ 1 /σ 2, σ 2 /σ 1 ) For the selecton of startng values, See, for example, Thode, Fnch and Mendell (1988). Fttng model (2.1) to our n = 404 measurements on total testosterone levels yelded the maxmum lkelhood estmates (± standard error) ˆλ = 0.3, ˆπ = ± 0.09, ˆµ 1 = ± 0.97, ˆµ 2 = ± 0.23, ˆσ 1 = 4.06 ± 0.69 and ˆσ 2 =

6 208 Yungta Lo 3.00 ± 0.24,. The approxmate 95% confdence nterval for the Box-Cox power transformaton parameter s gven by 0.05 λ The estmated mxng proporton ˆπ = gave an estmated 18.5% of men aged 50 to 59 wth or at rsk of acqurng HIV havng testosterone defcency, and an estmated 81.5% of these men wth normal testosterone levels. Mean total testosterone levels were ng/dl n men from the frst component dstrbuton and ng/dl n men from the second component dstrbuton after transformng Y back to ts orgnal scale, Y = (1 + λy (λ) ) 1/λ. The hstogram and densty curve of Y (ˆλ=0.3) are gven n Fgure 2. The fgure shows that the two-component normal mxture dstrbuton fts the data well. Fgure 2: Hstogram of Y ˆλ=0.3 wth a ftted densty for the two-component normal mxture dstrbuton 3. Testng for the Number of Components The Box-Cox power transformaton may remove skewness from the data such that a sngle normal dstrbuton may be approprate for the data. On the other hand, there may exst a Box-Cox power transformaton such that a mxture of three normal dstrbutons (for example, among men n the low testosterone group, some have very low testosterone levels) may gve a better ft to the data. We wsh to test whether a mxture of two transformed normal dstrbutons fts the data sgnfcantly better than a sngle transformed normal dstrbuton or to test whether a mxture of three transformed normal dstrbutons fts the data better than a mxture of two transformed normal dstrbutons. Note that the value of the Box-Cox power transformaton λ for the null model s, n general,

7 Age-specfc Prevalence of Testosterone Defcency 209 dfferent from that for the alternatve model. As the null model s, n general, not nested wthn the alternatve model, the lkelhood rato statstc for testng the number of components does not have the classc χ 2 reference dstrbuton. Even f both null and alternatve models have the same value of λ, the asymptotc χ 2 theory does not hold as under the null hypothess, the mxng proporton s on the boundary of the parameter space and the parameters are not dentfable under the null model. For a thorough account of breakdown n regularty condtons under whch classc asymptotc theory holds for the lkelhood rato test, refer to Ttterngton, Smth and Makov (1985) and McLachlan and Basford (1988). Several studes have been conducted to nvestgate the asymptotc dstrbuton of the lkelhood rato statstc for testng the number of components n a mxture model, see, for example, Dacunha-Castelle and Gassat (1999), Lemdan and Pons (1999), Chen, Chen and Kalbfesch (2001), and Lo, Mendell and Rubn (2001). These approaches can be used to determne the number of components when both null and alternatve models have the same power transformaton. When power transformaton under the null model s dfferent from that under the alternatve model, these approaches may not be sutable for determnng the number of mxture components wthout approprate modfcaton to account for Box-Cox power transformaton parameters. Parametrc bootstrap tests are hence used to test whether total testosterone levels after a Box-Cox power transformaton have arsen from a sngle normal dstrbuton, a mxture of two normal dstrbutons, or a mxture of three normal dstrbutons. The lkelhood rato statstc s defned as LR = 2(L 0 L 1 ) where L 0 and L 1 are the log lkelhood functons maxmzed under the null and alternatve hypotheses, respectvely. Inferences are drawn based on the observed bootstrap p-value that s defned as the proporton of replcates of LR that are as extreme as or more extreme than the value of LR obtaned from the observed data. The null hypothess wll be rejected f the observed bootstrap p-value s less than or equal to a pre-specfed sgnfcance level. For each bootstrap sample, the maxmum lkelhood estmates of the parameters under the null and alternatve models are obtaned n turn through a grd search of λ [ 1, 1] and the EM algorthm, usng 100 sets of startng values, subject to the constrant on the component varances. We frst tested the null hypothess that data have arsen from a sngle normal dstrbuton after a Box-Cox power transformaton aganst the alternatve hypothess that the data have arsen from a mxture of two normal dstrbutons after another Box-Cox power transformaton. One thousand bootstrap samples of sze n = 404 were generated from a random varable Y such that Y (λ) N(µ, σ 2 )

8 210 Yungta Lo where λ, µ, and σ were estmated by the maxmum lkelhood estmates ˆλ = 0.4, ˆµ = , and ˆσ = obtaned from the observed data. Among 1000 replcates of LR, there was none n excess of 19.24, the observed value of LR, so the observed bootstrap p-value was 0.0. Thus, there s strong evdence that there are at least two subpopulatons. The hstogram of 1000 bootstrap replcates of the lkelhood rato statstc s gven n Fgure 3(a). Fgure 3: Hstograms of 1000 bootstrap replcates of LR for testng (a) a sngle normal versus a two-component normal mxture and (b) a two-component normal mxture versus a three-component normal mxture. The vertcal lne s the observed value of LR. We then tested the null hypothess that data have been drawn from a mxture of two normal dstrbutons after a Box-Cox power transformaton aganst the alternatve hypothess that the data have been drawn from a mxture of three normal dstrbutons after another Box-Cox power transformaton. One thousand bootstrap samples of sze n = 404 were generated from a random varable Y such that Y (λ) πn(µ 1, σ 2 1) + (1 π)n(µ 2, σ 2 2) where λ, π, µ 1, µ 2, σ 1, and σ 2 were estmated by the maxmum lkelhood estmates ˆλ = 0.3, ˆπ = 0.185, ˆµ 1 = 13.19, ˆµ 2 = 14.57, ˆσ 1 = 4.06, and ˆσ 2 = 3.00 obtaned from the observed data. Among 1000 replcates of the test statstc LR, there were 80 n excess of 8.72, the observed value of LR, so the observed bootstrap p-value was Thus, there s no sgnfcant evdence that more than two subpopulatons are requred. The hstogram of 1000 bootstrap replcates of the lkelhood rato statstc s gven n Fgure 3(b).

9 Age-specfc Prevalence of Testosterone Defcency Adjustment for Covarates Of 404 men aged 50 to 59, 231 were black, 224 had human mmunodefcency vrus (HIV) nfecton, 295 had hepatts C vrus (HCV) nfecton, 168 had both HIV nfecton and HCV nfecton, 48 were njecton drug users, 291 smoked cgarettes, and 128 used psychotropc medcatons. The medan (range) body mass ndex (BMI) was 26.3 ( ) kg/m 2 ; 157 men were overweght wth a BMI beteen 25 and 30 kg/m 2, and 87 men were obese wth a BMI greater than 30 kg/m 2. To adjust effects of these factors on total testosterone levels, the mxture model gven n (2.1) can be easly extended to a mxture of two normal lnear regresson models by lettng Y (λ) = x T β 1 + ɛ 1 wth probablty π, Y (λ) = x T β 2 + ɛ 2 wth probablty 1 π where x s a vector of predctors, β 1 and β 2 are vectors of unknown parameters, and the error terms ɛ k N(0, σk 2 ) for k =1 and 2. The log-lkelhood functon of y 1,... y n s then gven as follows: L(λ, θ; y) = n =1 [log{πf 1 (y (λ) ; µ 1, σ1) 2 + (1 π)f 2 (y (λ) ; µ 2, σ2)} 2 + (λ 1) log y ], (4.1) where µ 1 = x T β 1 and µ 2 = x T β 2. Smlarly, for a fxed value of λ, the maxmum lkelhood estmators for π, β 1, β 2, σ1 2, and σ2 2 can be obtaned by maxmzng the log-lkelhood functon n (4.1) through applcaton of the EM algorthm subject to the constrant on the component varances. The covarance matrx of the model parameters can be estmated by the method proposed by Lous (1982). Detals on dervaton of the covarance matrx can be found n Thompson, Smth and Boyle (1998) and Turner (2000) when usng the EM algorthm to fnd the maxmum lkelhood estmators. We began our modelng wth a model ncludng all the covarates and an nteracton between HIV nfecton and HCV nfecton, and an nteracton between HIV nfecton and njecton drug use n the frst component dstrbuton and n the second component dstrbuton. A stepwse approach along wth the Bayesan nformaton crteron (BIC) and lkelhood rato tests was used for model selecton. The parameter estmates and ther correspondng standard errors for the fnal model are gven n Table 1. The estmated mxng proporton ˆπ = 0.29 was dfferent from that n the model wthout adjustment for covarates gven n (2.1). The regresson model for µ 1 ncluded terms for black, BMI, njecton drug use, and use of psychotropc medcatons. Ths suggests that among men n the frst

10 212 Yungta Lo component populaton, Afrcan Amercan men were lkely to have hgher total testosterone levels; ncreased BMI, njecton drug use, and use of psychotropc medcatons were assocated wth lower total testosterone levels. The regresson model for µ 2 ncluded terms for BMI, cgarette smokng, HCV nfecton, HIV nfecton, and an nteracton between HCV nfecton and HIV nfecton. Among men n the second component populaton, ncreased BMI and smokng cgarettes were assocated wth decreased total testosterone levels. Both HCV nfecton and HIV nfecton were assocated wth lower total testosterone levels. However, there was a sgnfcant nteracton between HCV and HIV; the effect of HCV nfecton on testosterone was modfed by HIV nfecton. Age was not sgnfcantly assocated wth total testosterone levels. Compared to the mxture regresson model, the estmated means of two components n (1) appear to be very close when the covarates were not ncluded. Table 1: Parameter estmates and standard errors for the mxture of two normal lnear regresson models ftted to the testosterone data wth ˆλ = 0.3 Varable Estmate Estmated Z-value p-value standard error π Intercept Black BMI Injecton drug use < Use of psychotropc medcatons σ Intercept BMI < Smokng HCV < HIV HCV x HIV σ Dscusson We have demonstrated the use of a mxture dstrbuton wth two normal components for estmatng age-specfc prevalence of testosterone defcency n men and the use of a mxture of two normal lnear regresson models for adjustng effects of covarates on testosterone levels. The age-specfc prevalence of testosterone defcency n men aged 50 to 59 wth or at rsk for HIV nfecton was estmated to be about 29% after adjustng for covarates. Ths estmated prevalence rate s lower than 61.9% estmated by usng the cut-off pont 325

11 Age-specfc Prevalence of Testosterone Defcency 213 ng/dl suggested by Harmen et al. (2001), 55.4% usng the cut-off pont 300 ng/dl suggested by the recent endocrne socety annual andropause consensus meetng, or 37.6% usng the cut-off pont 216 ng/dl suggested by Mohr et al. (2005) for men n ther 50s as the lower lmt of the normal range for testosterone levels n these HIV at rsk men. Due to study desgn, only the age-specfc prevalence n men aged 50 to 59 can be estmated n ths study. However, the mxture model can be used to estmate age-specfc prevalence n men, for example, n ther 60s and 70s, respectvely when there are a suffcent number of men n these two age groups. In ths study, we found that njecton drug use and use of psychotropc medcatons were assocated wth decreased total testosterone levels. Low testosterone levels have been reported n opate addcts (Ccero, 1980). Opates may act by suppressng the hypothalamc-ptutary-gonadal axs, a reproductve hormonal axs n men that conssts of hypothalamus, ptutary gland, and tests (Ccero, 1980). Use of psychotropc medcatons may lead to sexual dysfuncton, ncludng testosterone defcency resultng from hyperprolactnema (Dobs et al., 1988). Early n the HIV/AIDS epdemc, hypogonadsm was common n men wth AIDS (Ferr, Bertozz and Zgnego, 2002). Low testosterone levels n men wth HIV nfecton have been correlated wth advanced mmunodefcency. We observed an assocaton between lower testosterone levels and HIV nfecton n HCV unnfected men. Hepatts C vrus nfecton had the strongest assocaton wth lower total testosterone levels n the second component populaton. Ths may suggest that severe lver dsease s assocated wth low testosterone levels. Ferr et al. (2002) reported that plasma levels of total and free testosterone were generally lower n 207 patents wth hepatts C than n 2010 Italan men prevously evaluated for erectle dysfuncton. Increased body mass was assocated wth lower total testosterone levels n ths study. Obesty assocated wth low total testosterone levels has been reported by Gapstur et al. (2002) n ther study of assocatons of age, obesty, and race wth serum androgen concentratons n young men. Glass et al. (1977) reported that serum total testosterone levels decrease wth ncreasng body mass because of decreasng sex hormone-bndng globuln. Hgher testosterone levels n Afrcan Amercan men than n whte men have been reported by Ells and Nyborg (1992) and Wnters et al. (2001), but partcpants were generally younger than the sample of men studes here. We found smokng cgarettes was sgnfcantly assocated wth decreased testosterone levels. A lterature revew on cgarette smokng and male reproducton found reports varously suggestng that testosterone may be unchanged, or sgnfcantly elevated or decreased n smokers (Vne, 1996). We dd not fnd age sgnfcantly assocated wth testosterone levels. Perhaps, ths s because most of change n male testosterone levels occurs before age 50 (Harman et al., 2001). At only ten

12 214 Yungta Lo years, the range s not great enough to provde power to detect assocatons wth age. A further nvestgaton of these seems warranted. After estmatng the unknown parameters and testng for the number of components, mxture models can be used to cluster ndvduals nto one of two subpopulatons. The classfcaton can be acheved by makng use of posteror probabltes of populaton membershp of each observaton as calculated n the E step of the EM algorthm. Indvdual observatons wll be assgned to the component dstrbuton to whch t has the hghest estmated posteror probablty of belongng. As these estmated posteror probabltes may have lmted relablty n small samples, they may not gve a satsfactory assgnment of the data. When sample szes are large, mxture models can also be used to ad n determnaton of cut-off ponts for dentfyng men wth testosterone defcency. A cut-off pont can be chosen to attan, for example, a specfcty rate n the range of 90% to 99% to lmt the number of false postve results as treatment of older HIV and HCV nfected men wth testosterone may be assocated wth greater rsk than treatment of younger patents. Both senstvty and specfcty are calculated based on the selected mxture model under the assumpton that the model correctly dentfes two dstnct populatons and the component populaton wth lower testosterone levels represents men wth testosterone defcency; see Thompson, Smth, and Boyle (1998) for detals and related references. For example, among njecton drug and psychotropc medcaton users, respectve cut-off ponts to have a specfcty of 90% for detectng testosterone defcency, are 104 ng/dl for black non-smokng men aged 50 to 59 wth HIV nfecton and HCV nfecton, and 223 ng/dl for black non-smokng men aged 50 to 59 wthout HIV nfecton and HCV nfecton. The correspondng senstvty rates are 87% and 99%, respectvely. An average value was used for BMI. We have used mxture models assumng that the mxng proporton s ndependent of covarates as we are nterested n estmatng the age-specfc prevalence rate n a partcular age group. If subject-specfc probabltes of havng testosterone defcency are of nterest nstead, a logstc lnk functon that allows mxng proportons dependng on ndvdual characterstcs and some predctors can be ncorporated nto mxture models. Acknowledgments The author would lke to thank the referee and the edtor for ther comments and suggestons that mproved the presentaton of ths paper. References

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15 Age-specfc Prevalence of Testosterone Defcency 217 Thompson, T. J., Smth, P. J. and Boyle, J. P. (1998). Fnte mxture models wth concomtant nformaton: assessng dagnostc crtera for dabetes. Appled Statstcs 47, Turner, T. R. (2000). Estmatng the propagaton rate of a vral nfecton of potato plants va mxtures of regressons. Appled Statstcs 49, Vne, M. F. (1996). Smokng and male reproducton: a revew. Internatonal Journal of Andrology 19, Wnters, S. J. et al. (2001). Testosterone, sex hormone-bndng globuln, and body composton n young adult Afrcan Amercan and Caucasan men. Metabolsm 50, Receved August 23, 2007; accepted October 29, Yungta Lo Department of Communty and Preventve Medcne Mount Sna School of Medcnene One Gustave L. Levy Place, Box 1057 New York, NY ungta.lo@mssm.edu