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Nonlinear Fractional Polynomials for Estimating Long-Term Persistence of Induced Anti-HPV Antibodies: AHierarchical Bayesian ApproachMehreteab Aregay, Ziv Shkedy, Geert Molenberghs, Marie-Pierre David & Fabian TibaldiAccepted author version posted online: 22 Apr 2014.Published online: 27 Aug 2014.

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Nonlinear Fractional Polynomialsfor Estimating Long-Term Persistenceof Induced Anti-HPV Antibodies:A Hierarchical Bayesian Approach

Mehreteab AREGAY, Ziv SHKEDY, Geert MOLENBERGHS, Marie-Pierre DAVID,and Fabian TIBALDI

When the true relationship between a covariate and anoutcome is nonlinear, one should use a nonlinear meanstructure that can take this pattern into account. In thisarticle, the fractional polynomial modeling framework,which assumes a prespecified set of powers, is extendedto a nonlinear fractional polynomial framework (NLFP).Inferences are drawn in a Bayesian fashion. The proposedmodeling paradigm is applied to predict the long-termpersistence of vaccine-induced anti-HPV antibodies. Inaddition, the subject-specific posterior probability to beabove a threshold value at a given time is calculated. Themodel is compared with a power-law model using the de-viance information criterion (DIC). The newly proposedmodel is found to fit better than the power-law model. Asensitivity analysis was conducted, from which a relativeindependence of the results from the prior distributionof the power was observed. Supplementary materials forthis article are available online.

Key Words: Deviance information criterion; Fractional polyno-mial model; Power-law model.

1. Introduction

Human papillomavirus (HPV) infection is a necessarycause of cervical cancer. Even though 90% of the HPVinfections are cleared within two years (Goldstein et al.2009), persistent infection will lead to the developmentof cervical cancer and other anogenital cancers (Ho et al.1998). There are 120 HPV types which are identified andindexed by a number (Chaturvedi and Maura 2010). Ofmore than 40 HPV types, HPV-16 and HPV-18 causeabout 70% of the cervical cancers (Munoz et al. 2003).

To protect against persistent HPV infection, manyscientists synthesized a virus-like particle (VLP) vaccine(Zhou et al. 1991; Kirnbauer et al. 1992). There are twotypes of vaccines available on the market, Cervarix andGardasil,1 that prevent infection with HPV-16/18 andmay lead to further decrease in cervical cancer (Kahn2009). Several authors employed different models to pre-dict long-term immunity following this vaccine and/ornatural infection. Fraser et al. (2007) proposed a modifiedpower-law model, which imposes an asymptote onto theantibody level, whereas David et al. (2009) implemented

1Cervarix is a registered trademark of the GlaxoSmithKline group ofcompanies and Gardasil is a registered trademark of Merck and Co Inc.

C© American Statistical AssociationStatistics in Biopharmaceutical Research

August 2014, Vol. 6, No. 3DOI: 10.1080/19466315.2014.911201

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a piece-wise model, which assumes a constant antibodydecline within a specified period. Recently, Aregay et al.(2013) showed that the power-law model of Fraser et al.(2007) and David et al. (2009) can be formulated usingfractional polynomial (FP), which is a data-driven methodto predict long-term persistence and to estimate the timepoint above a given threshold.

Royston and Altman (1994) extended conventionalpolynomial regression to the aforementioned FPs, a moreflexible method. It has been shown that FPs are frequentlyamong the least biased smoothing methods for fittingnonlinear exposure effects (Govindarajulu et al. 2009).Several researchers applied FPs to nonlinear longitudinaldata (Long and Ryoo 2010). Unsurprisingly, there arealso some limitations to FP functions. Some of them aresufficiently flexible to capture a nonlinear function andpossible sensitivity to extreme values at either end of thedistribution of a covariate (Royston and Sauerbrei 2008).Although Royston and Sauerbrei (2008) argued that theset {−2,−1,−0.5, 0, 0.5, 1, 2, 3} is oftentimes sufficientto approximate all powers in the interval [−2, 3], theremay be reasons to extend it (Shkedy et al. 2006; Aregayet al. 2013).

To incorporate prior information, Bove and Held(2011) implemented an FP model that combines vari-able selection and “parsimonious parametric modeling”(Royston and Altman 1994) of the covariate effects, withBayesian methods for univariate data. Auranenn et al.(1999) fitted a hierarchical Bayesian regression model topredict the duration of immunity to Hemophilus influenzatype b. In the context of long-term antibody persistenceafter vaccination of Hepatitis A, Van Damme et al. (1994)used the geometric mean antibody titer (GMT) to showthat the predicted duration of antibody persistence for in-activated hepatitis A vaccine is estimated to be at least20 years. Wiens et al. (1996) proposed to estimate dura-tion of protection for the Hepatitis A vaccine based onan individual level instead of the GMT level. In the cur-rent article, we combined the approaches of Van Dammeet al. (1994) and Wiens et al. (1996) and used a hierar-chical Bayesian model from which we estimated both themean antibody titer and subject-specific profiles. Long-term persistence is estimated using information obtainedfrom subject-specific profiles.

In this article, we extend the FP framework to non-linear longitudinal data using a hierarchical Bayesian ap-proach. The method is applied to predict the long-termpersistence of vaccine-induced anti-HPV-16 and anti-HPV-18 antibodies, as well as to predict the proportionof subjects above a threshold value. Many researchers(Fraser et al. 2007; David et al. 2009; Aregay et al. 2013)have been focusing on the prediction of long-term im-munity but no attention was given to the subject-specificprobability of being above a threshold for a given time

point. In contrast with the model-based long-term predic-tion, which treats subjects as above threshold or not at anygiven time point, a subject-specific probability quantifiesthe uncertainty about the subject protection status at anytime point.

The article is organized as follows. Section 2 intro-duces the dataset and a brief exploratory data analysis.The hierarchical Bayesian model used to predict the prob-ability of being above a threshold is discussed in Section3. We apply the proposed model to the data in Section 4and discuss the results in Section 5.

2. Motivating Case Study

The dataset used encompasses 390 healthy womenwho received the HPV-16/18 AS04-adjuvanted vaccine(Cervarix). They were enrolled in the initial multicen-ter study (HPV-001, NCT00689741) and took part ina follow-up study for three additional years (HPV-007,NCT00120848). Their blood samples were evaluated forthe presence of anti-HPV-16/-18 antibodies at months7, 12, 18, and then annually up to 6.4 years. The ageof the women ranged over 15–25 years and they camefrom North America (USA and Canada) and Brazil.An enzyme-linked immunosorbent assay (ELISA), whichwas developed in-house by GlaxoSmithKline (GSK), wasused to check the presence of anti-HPV-16/-18 antibodies.The assay cut-off value was 8 EU/mL and 7 EU/mL foranti-HPV-16 and anti-HPV-18 antibodies, respectively.

The number of blood samples for each visit (cate-gorized month; the continuous time in month was cat-egorized into 12 time points) was obtained. For anti-HPV-16 antibodies, there was a drop in the number ofblood samples in the month range 25–32. Month 7 re-veals the highest number of blood samples and there-after a small increase from 364 to around 366 was shownin month 12 and 18, but there was a rapid decrease to89 in the months 25–32. The number of blood samplesranges between 130 and 234 in the month range 69–74,followed by a decrease to 66 blood samples in months be-yond 75. A similar pattern was observed for anti-HPV-18antibodies.

Figure 1 shows randomly selected individual profilecurves for the antibody mean and threshold value forboth HPV-16 and HPV-18 and reveals a high amount ofvariability between subjects, as well as that there are afew subjects who cross a threshold value. The evolutionsof mean antibody over time for both HPV-16 and HPV-18are shown in Figure 2. In both studies, it can be seen thatthe decline in the antibody level was higher in the firstfew months followed by a moderate decrease until theend of the follow-up period.

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Nonlinear Fractional Polynomials for Estimating Long-Term Persistence of Induced Anti-HPV Antibodies

Figure 1. Random selection of 30 individual profiles for HPV-16 (left panel) and for HPV-18 (right panel).

3. Modeling Mean Antibody UsingHierarchical Subject-Specific Models

3.1 Model Formulation

In previous studies of the decline in antibody levelafter induced vaccination against HPV, Fraser et al. (2007)and David et al. (2009) employed a power-law (PL) modelto estimate the persistence of anti-HPV level. We start thissection with a brief discussion the modeling approachesfor longitudinal anti-HPV antibodies data and then we

formulate the hierarchical Bayesian model used for theanalysis presented in this article. Let Yi j denote the logantibody level of subject i measured at time j, and considerthe model

Yij = f (tij) + εij, (1)

(i = 1, . . . , n; j = 1, . . . , ti ). Here, f (tij) is the mean logantibody titer, assumed to be time-dependent, and εij isthe random error term for subject i at time j, assumed to benormally distributed, εij ∼ N (0, σ 2

ε ). The mean structureof such a subject-specific power-law model, discussed by

Figure 2. Mean structure of HPV-16 and HPV-18.

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Fraser et al. (2007), David et al. (2009), and Aregay et al.(2013) could be formulated as

f (tij) = (β0 + b0i ) + (β1 + b1i ) ln tij. (2)

Note that the parameters β0 and β1 are the population(fixed) parameters, while b0i and b1i are subject-specificrandom intercept and random slope, respectively, b0i ∼N (0, σ 2

b0) and b1i ∼ N (0, σ 2

b1).

Aregay et al. (2013) pointed out that the power-lawmodel can be formulated as first-order fractional polyno-mials with p = 0 using the Box-Tidwell transformation(Box and Tidwell 1962). For the general case with p �= 0Aregay et al. (2013) used a first-order fractional polyno-mial mixed model to model the long-term persistence ofinduced anti-HPV antibodies in which the mean structureis given by

f (tij) = (β0 + b0i ) + (β1 + b1i )tpij . (3)

Within the fractional polynomials framework, the un-known power in (3) are estimated by a grid search over theprespecified sequence p1 ≤ p2 ≤ · · · ≤ pm . Note that,for a given value of p, the mean structure in (3) is linearwith respect to t p

ij . Aregay et al. (2013) compared the first-and second-order fractional polynomials for the datasetdiscussed in Section 2 and they found that the first-orderfractional polynomial in (3) was the best model. In whatfollows, we formulate a hierarchical Bayesian model that,in contrast with the FP framework, is estimating the FPmodel (3) as a nonlinear model (NLFP).

At the first stage of the hierarchical model we specifya normal likelihood model, that is,

Yij ∼ N(

f (tij), σ2ε

),

where f (tij) has the mean structure defined in (3). In thesecond stage of the model we specify a prior model forthe unknown parameters in the model. We consideredindependent normal prior distributions for the populationparameters,

βk ∼ N(μk, σ

2βk

), for k = 0, 1.

The subject-specific intercept and slope are assumed tofollow a bivariate normal distribution with a mean vectorof zeros and variance-covariance structure Db, that is,[

b0i

b1i

]∼ MVN

([0

0

],

Db =[

σ 2b0

ρb0b1σb0σb1

ρb0b1σb0σb1 σ 2b1

]).

A uniform prior distribution was defined for the powerp ∼ U(a, b). A sensitivity analysis for the choice of a andb is performed and is presented in Section 4.1.

To complete the specification of the hierarchicalmodel, the following hyperprior distributions were spec-ified. A noninformative independent normal prior distri-

bution (Gelman 2006) for μk and a gamma prior distri-bution (Gelman 2006; Spiegelhalter et al. 2002) for theprecision parameters, that is, σ−2

βk∼ G(0.01, 0.01), and

σ−2ε ∼ G(0.01, 0.01). However, Gelman (2006) argued

that this prior is informative because of its shape andsuggested to use instead a uniform prior on the hierar-chical standard deviation. We have considered a uniformprior distribution for the standard deviation but the resultdoes not change much compared with the result obtainedfrom the assumption of an inverse gamma prior for thevariance. An inverse Wishart prior distribution for thecovariance matrix Db was employed, that is,

D−1b ∼ Wishart(RD, 2), (4)

where RD is 2 × 2 identity matrix. We notice that the casein which Db is diagonal, that is, the random intercept andslope are independent, implies that

b0i ∼ N(0, σ 2

b0

)and b1i ∼ N

(0, σ 2

b1

),

with hyperprior distribution given by σ−2bk

∼ G(0.01,

0.01) for k = 0, 1.Model formulation for the modified power-law model

is given in Section 3 in the online supplementary webappendix of this article.

3.2 The Probability Above a Threshold

All studies discussed above were conducted to as-sess the long-term protection after vaccination. Such anassessment can be done by comparing the model-basedempirical Bayes’ subject-specific predictions at a giventime point to a prespecified threshold τ . Based on this ap-proach, Fraser et al. (2007), David et al. (2009), and Are-gay et al. (2013) concluded that 76%, 100%, and 99.7%of the vaccinated subjects will be above the thresholdlevel near life long persistence of anti-HPV-16 antibodiesafter vaccination.

One of the motivations for using the hierarchicalBayes’ model presented in the previous section is to calcu-late the probability of being above a threshold for a giventime point. Note that, in contrast with the approaches dis-cussed by Fraser et al. (2007), David et al. (2009), andAregay et al. (2013), here we wish to quantify the un-certainty about being above threshold or not, rather thanjust the status of each subject. Hence, for each subject,we estimate the probability of being above a prespecifiedthreshold. Let Zij be an indicator latent variable, repre-senting above threshold or not status of the ith subject attime tij:

Zij

={

1 Yij > τ, an antibody level above a threshold,

0 Yij ≤ τ, an antibody level below a threshold.

(5)

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Nonlinear Fractional Polynomials for Estimating Long-Term Persistence of Induced Anti-HPV Antibodies

Hence, the subject-specific probability of being above agiven threshold value at time tij is given by πij = P(Yij >

τ ). We notice that to estimate the proportion of subjectsabove a threshold in the sample, Fraser et al. (2007),David et al. (2009), and Aregay et al. (2013) derived thevalue of Zij as

Zij ={

1 f (tij) > τ,

0 f (tij) ≤ τ.(6)

Hence, the proportion of subjects above a threshold attime tij was estimated by

p j =n j∑

i=1

Zij/n j ,

where n j is the number of subjects at the jth time point.The proposed hierarchical Bayes’ model allows us to es-timate the probability πij using Monte Carlo integration.In practice, πij can be obtained from the WinBUGS pro-gram by using thestep( ) function. The function createsa binary variable that counts the number of MCMC sim-ulations for which, Yij > τ is true. Hence, the 0/1 valuesfrom the step( ) function can be used to compute πij.For example, consider an MCMC simulation with M sam-ples of step(Yij) from which the first T are the burn-insamples, πij is estimated using Monte Carlo integrationby

1

M

T +M∑t=T +1

Z (t)ij .

This implies that the hierarchical Bayesian model enablesestimation of both the quantities Zij and πij.

4. Data Application

4.1 Long-Term Prediction Using Subject-SpecificNonlinear Fractional Polynomials

To estimate the subject evolution of the log antibod-ies within the follow-up period, the NLFP model was fit-ted using the R2WinBUGS package (Sturtz, Ligges, andGelman 2005). A Markov chain Monte Carlo (MCMC)simulation of 10,000 iterations, from which the first 1000were considered burn in and discarded from the analy-sis, was used to estimate the model parameters. Modelselection was done using the deviance information cri-terion (DIC; Spiegelhalter et al. 2002) and convergencewas checked using trace plots, estimated potential scalereduction factor (R) and Brooks, Gelman, and Rubin’s(BGR) statistics (Gelman and Rubin 1992). The traceplot and BGR statistic indicate convergence for all modelparameters. Furthermore, the estimated potential scalereduction factor (R) for all the parameters were close to

one which indicates convergence for all model parameters(see online supplementary web appendix).

Initially, we fitted an NLFP that assumes the randomintercept and the random slope to be independent. Be-cause we have prior knowledge of the power p, which isp = −1.25 from Aregay et al. (2013), a uniform priordistribution for the power p ∼ U(−1.4,−1.2) was used.A sensitivity analysis for the prior of p will be discussedin the section below. The posterior mean for the powerwas estimated to be −1.356 for anti-HPV-16 antibodies,while −1.259 for anti-HPV-18 antibodies. In the secondstage, an NLFP that assumes the random intercept and therandom slope to be correlated was applied. For this model,the posterior means are estimated to be equal to −1.332for anti-HPV-16 antibodies, whereas it is −1.243 for anti-HPV-18 antibodies. For both anti-HPV-16 and anti-HPV-18 antibodies, the DIC for the correlated random-effectsmodel was smaller than the DIC for the independent ran-dom effects model. Hence, the former is to be preferred.Additionally, the posterior mean for the correlation be-tween the random effects is equal to −0.413 (95% cred-ible interval: [−0.511,−0.307]) for anti-HPV-16 anti-bodies and −0.596 (credible interval [−0.677,−0.508])for anti-HPV-18 antibodies. Parameter estimates forthe posterior mean of the fixed effects are shown inTable 1.

To assess whether the results depend on the priordistribution of the power p, a sensitivity analysis wasperformed using different values for a and b. This showsthat the results do not depend on the prior distributionchosen (see Table 2).

Different threshold values (τ ) were used: 29.8 EU/mL(1.474 log (EU/mL)) and 417.8 EU/mL (2.621 log(EU/mL)) for anti-HPV-16 antibodies and 22.6 EU/mL(1.355 log (EU/mL)) and 279.3 EU/mL (2.446 log(EU/mL)) for anti-HPV-18 antibodies (Fraser et al. 2007;David et al. 2009; Aregay et al. 2013). Unless other-wise specified in the text, authors will focus their analy-sis on the low threshold values, that is, 29.8 EU/mL foranti-HPV-16 antibodies and 22.6 EU/mL for anti-HPV-18antibodies.

Figure 3 shows the long-term posterior predicted pop-ulation means with 95% predictive intervals, indicatingthat on average all of the subjects have antibody levelsabove a threshold level for near life time.

Figure 4 shows the observed and posterior predictionfor the antibody levels for selected months within theestimation period. The posterior prediction densities andobserved densities are similar, indicating that the NLFPis fitting the data very well over the follow-up period.The posterior predictive densities for 30 and 50 years atdifferent threshold values are shown in Figure 5. Notethat the posterior predictive densities for 30 years and50 years are almost the same.

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Statistics in Biopharmaceutical Research: August 2014, Vol. 6, No. 3

Tabl

e1.

Com

pari

son

ofpo

wer

-law

mod

el,n

onli

near

frac

tion

alpo

lyno

mia

lmod

elw

ith

ρ12

=0

and

nonl

inea

rfr

acti

onal

poly

nom

ialm

odel

wit

hρ

12�=

0fo

ran

ti-H

PV

-16

and

anti

-HP

V-1

8an

tibo

dies

HP

V-1

6

PL

NL

FP

wit

hp∼

duni

f(−1

.4,−

1.2)

;ρ12

=0

NL

FP

wit

hp∼

duni

f(−1

.4,−

1.2)

;ρ12

�=0

Para

met

ers

mea

nsd

MC

erro

r95

%C

Im

ean

sdM

Cer

ror

95%

CI

mea

nsd

MC

erro

r95

%C

I

β0

4.10

40.

033

0.00

1(4

.042

,4.1

71)

2.60

40.

024

0.00

1(2

.559

,2.6

51)

2.60

10.

025

0.00

1(2

.555

,2.6

49)

β1

−0.3

680.

008

3.83

E-4

(−0.

386,

−0.3

51)

13.9

70.

824

0.03

9(1

2.21

,15.

29)

13.3

20.

941

0.04

8(1

1.45

,15.

03)

σb 0

0.53

20.

029

3.56

E-4

(0.4

75,0

.592

)0.

4431

0.01

70.

0001

(0.4

13,0

.478

)0.

4605

0.01

730.

0001

(0.4

28,0

.496

)σ

b 10.

140

0.00

89.

05E

-5(0

.126

,0.1

56)

4.58

50.

359

0.01

13(3

.866

,5.2

64)

4.53

90.

3821

0.01

6(3

.805

,5.2

94)

σε

0.23

50.

004

3.50

3E-5

(0.2

27,0

.242

)0.

1847

0.00

33.

19E

-5(0

.178

,0.1

91)

0.18

380.

003

2.5E

-5(0

.178

,0.1

89)

ρb 0

b 1−0

.581

0.04

85.

62E

-4(−

0.66

8,−0

.488

1)−0

.413

0.05

25.

1E-4

(−0.

511,

−0.3

07)

p−1

.356

0.03

10.

001

(−1.

398,

−1.2

87)

−1.3

320.

037

0.00

2(−

1.39

,−1.

254)

DIC

429.

51−7

56.5

4−7

77.4

54

HP

V-1

8

PL

NL

FP

wit

hp∼

duni

f(−1

.4,−

1.2)

;ρ12

=0

NL

FP

wit

hp∼

duni

f(−1

.4,−

1.2)

;ρ12

�=0

Para

met

ers

mea

nsd

MC

erro

r95

%C

Im

ean

sdM

Cer

ror

95%

CI

mea

nsd

MC

erro

r95

%C

I

β0

4.07

90.

024

0.00

1(4

.032

,4.1

25)

2.46

80.

022

0.00

1(2

.425

,2.5

12)

2.46

50.

021

0.00

1(2

.422

,2.5

07)

β1

−0.3

890.

006

0.00

01(−

0.40

2,−0

.377

)12

.07

0.18

60.

0034

(11.

71,1

2.44

)11

.79

0.54

50.

026

(10.

97,1

3.08

)σ

b 00.

347

0.01

70.

0002

(0.3

15,0

.379

)0.

446

0.01

50.

0001

(0.4

17,0

.477

)0.

446

0.01

50.

0001

(0.4

17,0

.476

)σ

b 10.

079

0.00

50.

0001

(0.0

68,0

.089

)3.

149

0.16

90.

003

(2.8

22,3

.485

)3.

069

0.21

30.

007

(2.6

82,3

.527

)σ

ε0.

219

0.00

43.

21E

-5(0

.213

,0.2

26)

0.17

10.

003

2.71

E-5

(0.1

65,0

.178

)0.

169

0.00

32.

54E

-5(0

.164

,0.1

75)

ρb 0

b 1−0

.396

0.06

58.

85E

-4(−

0.51

5,−0

.259

)−0

.596

0.04

34,

4E-4

(−0.

677,

−0.5

08)

p−1

.259

0.03

20.

002

(−1.

326,

−1.2

05)

−1.2

370.

024

0.00

1(−

1.29

3,−1

.202

)D

IC34

.5−1

182.

65−1

232.

97

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Nonlinear Fractional Polynomials for Estimating Long-Term Persistence of Induced Anti-HPV Antibodies

Tabl

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Figure 3. Long-term prediction with posterior predictive interval over 50 years for anti-HPV-16 (left panel) and anti-HPV-18 antibodies (rightpanel).

Figure 4. The densities of the posterior predictions (solid line) and observed values (dashed line) of the antibody level at the categorized months,that is, M63–M68 and M75–. . ., for anti-HPV-16 (top figure) and anti-HPV-18 antibodies (bottom figure).

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Nonlinear Fractional Polynomials for Estimating Long-Term Persistence of Induced Anti-HPV Antibodies

Figure 5. The densities of posterior predictions of the antibody level over 30 and 50 years for anti-HPV-16 and anti-HPV-18 antibodies. Verticalsolid line and bold dashed line indicate the thresholds, 29.8 EU/mL and 417.8 EU/mL (left panel), and 22.6 EU/mL and 279.3 EU/mL (right panel),respectively.

Figure 6. Observed proportion and model-based proportion above different threshold values (τ = 1.474 and τ = 1.355 (solid line) and τ = 2.621and τ = 2.446 (dashed line) for anti-HPV-16 (left panel) and anti-HPV-18 antibodies (right panel) within (top figure) and after (bottom figure) thefollow up period, using the power-law and NLFP models.

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Figure 7. Long-term (50 years) prediction with posterior predictive interval of some selected subjects for anti-HPV-16 antibodies.

A comparison of the observed and model-based pro-portion within the follow-up period using the power-lawand NLFP model is shown in Figure 6 at the top. Usingthe NLFP, we observe from the lower panels that the pro-portion of subjects who are above the threshold value was99.7% (389 out of 390 subjects) for anti-HPV-16 antibod-ies, while it was 99.5% (388 out of 390 subjects) for anti-HPV-18 antibodies over 50 years. If we use τ = 2.621and τ = 2.446, the proportion above the threshold willdecrease to 48.9% and 52.6% for anti-HPV-16 and anti-HPV-18 antibodies, respectively. These results agree withthose reported in David et al. (2009) and Aregay et al.(2013).

4.2 Estimation of Subject-Specific Probability to beAbove a Threshold

As we mentioned in Section 3.2, the hierarchicalBayesian model allows us to estimate the posterior prob-ability of being above a threshold for a given time point.

First, we discuss the results obtained for the anti-HPV-16antibodies.

The posterior mean for some selected subjects isshown in Figure 7. Note that the first subject (8650) has aposterior predictive value below the threshold level whilethe other subjects have values above this one. The esti-mated posterior mean for πij above the threshold level forthese subjects is equal to 0.25, 1, 1, 1, respectively.

Using the methodology described in Fraser et al.(2007), David et al. (2009), and Aregay et al. (2013),subject 8650 is classified as having a predicted mean be-low the threshold level for 50 years. However, using thecurrent model πi50 is equal to 0.25. In other words, thecurrent model provides a measure for uncertainty for eachsubject.

Figure 8 shows the histogram of the posterior proba-bility of being above threshold for anti-HPV-16 antibod-ies. It shows that 93 (23.8%) subjects have a posteriorprobability of being above the threshold τ = 2.621 equalto 0, for 193 (49.5%), 0 ≤ πij ≤ 1 and the rest (104, 26.7%) have a probability of being above the threshold equal

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Figure 8. The posterior probability of being above threshold value 2.621, at 10, 20, 30, and 50 years for anti-HPV-16 antibodies.

Figure 9. HPV-16: Subject-specific sorted posterior probability of being above threshold 2.621 (left panel) and posterior probability of beingabove threshold 2.621 over 10 years for subjects who were classified as above threshold and below threshold (right panel). The index represents thenumber of subjects.

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Figure 10. HPV-18: Subject-specific posterior probability of beingabove threshold 2.446 over 10 years for subjects who were classified asabove threshold and below threshold. The index represents the numberof subjects.

to 1; πij = 1 for 50 years. The sorted posterior proba-bilities of being above the threshold for all subjects inthe trial are shown in left panel of Figure 9 and theposterior probabilities of being above the threshold for10 years for subjects who have posterior predicted meanabove/below the threshold are shown in the right panelof Figure 9 and illustrate the main difference betweenthe analysis presented in this article to the analysis dis-cussed in Fraser et al. (2007), David et al. (2009), andAregay et al. (2013). As mentioned before, all authorsused a model-based classification procedure. In contrast,the right panel of Figure 9 shows πij for subjects whowere classified as above/below the threshold by Davidet al. (2009) and Aregay et al. (2013). We clearly seethat among the 199 (51.02%) subjects who were classi-fied as above the threshold, 102 subjects have πij = 1,while 97 subjects have 0.5 < πij < 1. On the other hand,among the 191 (48.98%) subjects who were classified asbelow the threshold, 89 subjects have πij = 0 while 102subjects have 0 < πij < 0.5 over 10 years. This indicatesthat some of the subjects who were classified as above thethreshold in David et al. (2009) and Aregay et al. (2013),are surrounded by some uncertainty.

The hierarchical model allows us to calculate theproportion of individuals for which the probability ofbeing above a threshold is more than α. For instance,over 50 years, if we use τ = 2.621, 190 subjects for anti-HPV-16 antibodies have a posterior probability of beingabove a threshold more than 0.5. On the other hand, ifwe use the lower thresholds, τ = 1.474, all of the sub-

Figure 11. An illustrative probability of being above a threshold plot,which shows the comparison of two vaccines.

jects but one have a posterior probability above 0.5 over50 years.

Figure 10 shows the subject-specific posterior prob-ability of being above a threshold for anti-HPV-18 anti-bodies. Similar to the results obtained for anti-HPV-16antibodies, among the 212 (54.4%) subjects who wereclassified as above the threshold, we can see that 133 sub-jects have πij = 1 while 79 subjects have 0.48 < πij < 1.For an elaborate presentation of the results obtained foranti-HPV-18 antibodies, we refer to Section 1 in the on-line supplementary web appendix.

There are two vaccines against HPV available in themarket. It is difficult for many medical experts to chooseamong them. Our method can be used to compare twoor more vaccines using the posterior probability of beingabove threshold for a given time point. To underscorethis, we created an illustrative figure with vaccine 1 andvaccine 2 (Figure 11). From the plot, we can see thatvaccine 1 is better than vaccine 2.

5. Concluding Remarks

In this article, we proposed an extension of the frac-tional polynomial model discussed by Aregay et al. (2013)to nonlinear fractional polynomial using a hierarchicalBayesian model. We have shown that the model can beused to calculate a subject-specific probability of beingabove a threshold and to predict the long-term persis-tence of vaccine induced anti-HPV-16/18 antibodies. TheBayesian perspective of the fractional polynomial was im-plemented by assuming a uniform prior distribution forthe power. The NLFP is more flexible than the fractional

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Nonlinear Fractional Polynomials for Estimating Long-Term Persistence of Induced Anti-HPV Antibodies

polynomial, which assumes prespecified fractional pow-ers. It can easily be extended to include multiple covari-ates. We have conducted sensitivity analysis, establishingthat the results do not depend on the prior distribution ofthe power.

Furthermore, using the current method, the uncer-tainty of lying above a threshold can be calculated forsubjects who were classified as above/below a thresholdin David et al. (2009) and Aregay et al. (2013). Subjectswho were classified previously as above a threshold havesome uncertainty of being above the threshold also in thisstudy (Figure 9; Figure 10).

For both HPV-16 and HPV-18, the main findings showthat the posterior probability of being above the thresholdvalue is equal to one for 97.5% subjects (380 out of390 subjects) over 50 years. We note that one can alsoestimate the probability of being above a given threshold,proposed by David et al. (2009) and Aregay et al. (2013),by simulating from a normal distribution with mean f (tij)

and residual variance, σ 2ε .

One of our objectives was to obtain the long-termindividual prediction of being above a threshold. Hence,the posterior individual predictive mean was calculated.It was found that 389 out of 390 subjects had posteriorpredicted mean above a threshold level for 50 years foranti-HPV-16 antibodies while 388 out of 390 subjects foranti-HPV-18 antibodies. If we use τ = 2.621, the propor-tion of subjects above this threshold for 50 years was ap-proximately 48.9% for anti-HPV-16 antibodies, whereas52.6% for anti-HPV-18 antibodies with τ = 2.446. Theseresults were similar to previous findings that were ob-tained from the same dataset by Aregay et al. (2013). Wewere able to show that the posterior predicted mean wasabove the threshold level for 50 years.

Model comparison between the nonlinear fractionalpolynomial and power-law model was done using the de-viance information criterion. For both anti-HPV-16 andanti-HPV-18 antibodies, the NLFP was to be preferred.To evaluate the performance of the prediction over theestimation period, the model-based proportions and ob-served proportions for both models were obtained. TheNLFP returned proportions more similar to the observedproportion than the power-law model. Hence, in this work,the NLFP model fits to the data better than the power-lawmodel within the follow-up period. Moreover, to assessthe robustness of the inferences, the modified power-law(MPL) model was implemented. The NLFP fits the databetter than the MPL model. On the other hand, the resultsobtained from the MPL and NLFP models are compara-ble (see online supplementary web appendix). However,this does not automatically mean it does the same out-side the range of the observed data. Rather, this can beascertained only with long-term follow up.

In summary, we have shown that the fractional poly-nomial framework can be extended to a nonlinear frac-tional polynomial approach by assuming a prior distribu-tion on the power using Bayesian approach. In this study,more than 99% of the subjects who were vaccinated withHPV-16/18 AS04-adjuvanted vaccine, had higher chanceof having antibody level above the threshold level for50 years. Moreover, we have discussed that subjects whowere classified as above a threshold in David et al. (2009)and Aregay et al. (2013) approach, will not automaticallybe classified above a threshold in this study. We have alsoshown that the performance of different vaccines can becompared using the posterior probability of being abovea threshold for a given time point. We would like to rec-ommend that further work be done to define a thresholdantibody level associated with protection.

Supplementary Materials

The supplementary materials discuss the probability of protection foranti-HPV-18 antibodies, model diagnostics, and how to model the meanantibody using modified power-law models.

Acknowledgments

Mehreteab Aregay, Ziv Shkedy and Geert Molenberghs gratefully ac-knowledge support from IAP research Network P7/06 of the BelgianGovernment (Belgian Science Policy). The authors thank the study par-ticipants and clinical investigators from the Phase IIb primary efficacystudy (NCT00689741). Finally, they thank the laboratory personnel fortheir contribution in performing the assays.

[Received March 2013. Revised January 2014.]

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About the Authors

Mehreteab Aregay, I-BioStat, Katholieke Universiteit Leu-ven, B-3000 Leuven, Belgium (E-mail: mehreteab.aregay@med.kuleuven.be). Ziv Shkedy (E-mail: ziv.shkedy@uhasselt.be)and Geert Molenberghs (E-mail: geert.molenberghs@uhasselt.be), I-BioStat, Universiteit Hasselt, B-3590 Diepen-beek, Belgium. Marie-Pierre David (E-mail: marie-pierre.david@gsk.com) and Fabian Santiago Tibaldi (E-mail: fabian.s.tibaldi@gsk.com), GlaxoSmithKline Biologicals, 1330 Rixensart,Belgium.

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