Data sources

We consider the situation in which ascertainment is based on all incident cases in the target population during the enrollment period (the recruitment window). This then leads to observed data on sibships with a proband, and his/her siblings who have been at risk for the disease under investigation. In our application, the observed data consist of four parts:

1. Age at (possible) onset of T1D among all members of the ascertained families, who have been at risk during the follow-up period. Here, only diagnoses reported before the age of 15 years are considered. A total of 768 families are included in the recruited group.
2. HLA genotypes of the members of the ascertained nuclear families.
3. The numbers of the individuals in the background population at risk, comprising of the individuals who were alive during the calendar period included in the recruitment window (y = 1987, 1988, 1989) and belonging to the age groups a = 0,…,14.
4. The HLA genotypes from approximately 20,000 unrelated, healthy Finns. This population reference group thus corresponds to the control individuals utilized in case-control association studies.

The Lexis diagram displayed in Figure 1 illustrates the ascertainment procedure for sibships and the data needed to construct the likelihood for an absolute risk model. Let i = 1, …, I index all the ascertained families, j = 1, …, J_i indexing the individuals (siblings including the proband) in the ith family, and δ_ij being the indicator for right censoring of the follow-up (δ_ij = 1 if individual j in family i was diagnosed with T1D in the recruitment window, and δ_ij = 0 if right censored. Let X_ij represent the age at onset or the age at a censoring event. Further, for the siblings in the family data, let be the phase unknown marker alleles over the set of loci of interest, where l = 1,…, N_L, and let be phased alleles where p = m, f index the parents of the ith family. The alleles received from the mother (m) and the father (f) are indexed accordingly, and an analogous indexing is later used for haplotypes as well. Parental genotypes are denoted by . In order to actually model risks associated with haplotypes, it is assumed that the haplotype phases have been resolved computationally prior to the risk analysis. The HLA-A, HLA-B and HLA-DRB1 loci are included in the current data of the HLA region. The observed data of the ascertained sibships (Parts 1 and 2) are thus collectively represented by the set

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Figure 1. Lexis diagram of the population-based ascertainment of Finnish sibships with Type 1 diabetes.

An example of an ascertained sibship included in the population-based disease registry data, because an individual (proband) younger than 15 years who was diagnosed with T1D during the recruitment period 1.1.1987–30.4.1989.

https://doi.org/10.1371/journal.pone.0006836.g001

The T1D families were collected from the nationwide T1D registry in Finland, ascertained within the DiMe study [10] through a child under the age of 15 years and diagnosed between January 1, 1987 and April 30, 1989. The Childhood Diabetes in Finland (DiMe) study was a large population-based genetic-epidemiologic family study of T1D. Nationwide, all T1D cases under the age of 15 in Finland were diagnosed. The cut-off age was chosen purely for practical reasons. Newly diagnosed children under the age of 15 years with T1D were hospitalized in the pediatric wards in Finland and therefore were easier to recruit than older subjects with T1D. T1D status was checked against the data of the National Drug Registry. Of the 801 cases in the study 800 had also been registered in the Drug Registry and one person died soon after the diagnosis. The participation rate in the study was approximately 95%. Parents and siblings of the 801 probands were also asked to participate. Extensive questionnaires were filled in and blood samples were taken from participants. Probands, their parents and siblings were HLA genotyped at A, B, C and DR loci using conventional serology (768 families). Details of study procedures, especially data collection, are described elsewhere [10]. For the ascertained families we constructed a follow-up of T1D until 31.12.2001, by updating the T1D status and the date of diagnosis from the National Hospital Discharge Registry (Part 1). An individual was thus considered censored if he/she reached the age of 15 without having been diagnosed to have T1D, or was less than 15 years old and was not yet diagnosed on 31.12.2001.

In summary, we have for the comprehensive statistical analysis of HLA-A, HLA-B, and DRB1 loci, a total of 768 ascertained families comprising 1,944 probands or siblings. HLA genotypes were available from 1,342 probands or siblings (684 families, see Table S1). The DiMe study protocol has been described in detail elsewhere [14] and it has been approved by the ethics committees of the participating hospitals. Informed consent was obtained from the families taking part in the study. The HLA genotyping (Part 2) of these families was done in the National Public Health Institute Laboratory using classical serology. Haplotypes within the sibships in the ascertained family data were established using the SimWalk2 software [15], based on the available information about the parental HLA genotypes.

In order to include demographic information in the likelihood (Part 3), we assume that the sizes of the birth cohorts are known for the relevant time period. Here they are denoted by (see Figure 1). Let the starting and end points of the recruitment window be denoted by c₀ and c₁, respectively, and let w be the maximum age at which a subject may be ascertained (here less than 15 years). As the ascertainment is population-based, with a negligible magnitude of missing cases during the recruitment period, it is possible to incorporate in the risk model information about all individuals who were born during a certain calendar time interval and who had passed the recruitment window without being ascertained. This corresponds to all subjects born in the population between the calendar time points c₀-w and c₁. We divide subjects born in the general population such that they could have become probands but did not, into three sets according to the recruitment window as shown in Fig. 1. Subjects born during the time interval (c₀-w, c₁-w) are in the sequel indexed by k₁, respectively subjects born during (c₁-w, c₀) are in the sequel indexed by k₂, and finally, subjects born during the recruitment period (c₀, c₁) are indexed by k₃. The demographic information allows us to treat these individuals systematically in the risk model via the known sizes of the birth cohorts.

The genetic information concerning HLA for the demographic data (N_b) is in our model formulation inferred from the genotypic data in the population reference sample (Part 4). From the BMDR we obtained comparable genotypes for HLA-A, HLA-B and DRB1 loci for 19,386 unrelated individuals. The BMDR database includes healthy individuals who had agreed to volunteer for possible bone marrow donation. The Finnish BMDR registry is owned by the Finnish Red Cross Blood Service and it is not a public database. The primary purpose of this registry is to search for potential bone marrow donors for transplantations. All individuals who have joined the BMDR registry have given their written consent for anonymous registry based research. Anybody full filling the health criteria (roughly equivalent to those required for blood donation) and willing to donate stem cells (or “bone marrow”) for stem cell transplantation can join the registry. The registry does not accept joining that is based on, or motivated, by ‘targeted’ donation to e.g. relative or friend only, but the registree must be willing to donate to any patient. The bone marrow transplantations between related individuals are handled separately from the BMDR. Hence, we had no reason to believe that members of the BMDR are more related to each other than general population. From subjects who had given written consent, about 10 mL of peripheral blood was drawn for standard HLA typing. HLA typing included the standard serological HLA-A and -B typings and either serological (early samples) or DNA-based DRB1 typing. Genotype consistent HLA haplotypes for the BMDR data were constructed with the PHASE software [16]. In order to be able to compare HLA genotypes/haplotypes between different data sources or typing methods, all HLA types were converted to pooled alleles (see supplementary material), if necessary, to the serological main specificities according to the official HLA nomenclature [17]. Let be the set of genotypes for the unrelated individuals in the BMDR database, using a notation analogous to the familial data. These reference individuals are known not to have acquired T1D before age w. The genotype and haplotype frequencies for this reference population are collectively denoted by and , respectively.

Modeling of genetic risks and of the family-based ascertainment procedure

Likelihood expression

An assumption of the conditional independence of individual disease onset times, conditionally given the model parameters, leads to the following likelihood expression for the combined set of data: (2)

The first factor in this likelihood expression is the direct contribution of the DiMe family data, with i = 1, …, I indexing all the ascertained families, j = 1, …, J_i indexing the individuals (siblings including the proband) in the ith family, and δ_ij being the indicator for right censoring of the follow-up (δ_ij = 1 if individual j in family i was diagnosed with T1D in the recruitment window, and δ_ij = 0 if right censored, cf. Figure 1).

The next three factors in (2) are the contributions of the individuals, indexed here with k₁, k₂ and k₃, in the background population. These indexes represent the non-ascertained individuals, who could have been diagnosed with T1D in the “ascertainment window”, but who were in fact not ascertained to the DiMe sample. These non-ascertained individuals are considered individually in the “full likelihood” function (2). Let be the age of individual k₁, who was born at , at the beginning of the recruitment period. Similarly and are the ages of an individual k₂ born at at the beginning and at the end of the recruitment interval, and is the age of an individual born at at the end of the recruitment interval. Then , and are the corresponding probabilities of not being diagnosed with T1D in that interval, expressed here as a function of the parameters β of the hazard model (1).

Note that since the hazard model (1) also contains in its arguments the genotype/haplotype of the considered individual, which is unknown for individuals of types k₁, k₂ and k₃, they need to be integrated away from the corresponding expressions of the survival function. For the distribution of the genotypes/haplotypes we made, for computational reasons, a shortcut and used their empirical frequencies in the BMDR data base as the prior. (A theoretically more satisfying solution could have been to postulate the Dirichlet(1,…,1) prior for the frequencies and then update their estimates by sampling, concurrently with the estimation of other model parameters and using the BMDR genotypes/haplotypes as data. However, in view of the size of this data base we thought that doing so would not be worth the extra effort). Note also that, with the genotype/haplotype information being integrated away, all individuals of type k₁ born at the same time are treated as exchangeable, and therefore the corresponding product in (2) becomes a power, where the exponent is the number of individuals (excluding those belonging to the DiMe families) born at . These numbers are obtained directly from the demographic data.

Bayesian inference

Next, we describe how the likelihood (2) can be calculated and how the resulting parameter estimates are obtained. Applying the Bayesian approach to statistical inference, the likelihood is complemented with the joint prior distributions of all model parameters and latent variables of interest. Here this is done by following the principles of hierarchical Bayesian modeling and by specifying the distributions appearing on the right hand side of the following expression(3)in the OpenBugs MCMC software [18], where G^miss are the missing HLA genotypes/haplotypes in the ascertained sibships. The prior distributions used for the model parameters β and missing HLA genotypes were specified as follows. The age specific log baseline hazards were assigned independent Gaussian priors according to log(λ_a) with mean -8 and precision (reciprocal of variance) equal to 0.000001, where a = 0,…,14. The genotype/haplotype effects were assigned independent truncated normal (−10,10) prior distributions with mean 0.0 and precision 0.000001. This is, effectively, the uniform distribution over the interval (−10, 10). Particularly when noting that β is in the hazard model (1) in the exponent, this interval can be said to cover more than adequately all plausible values of genotype or haplotype effect.

The prior of the missing HLA genotypes in the family-based data was treated according to available parental HLA genotype information. Two distinct situations of missing HLA genotypes can be identified among the ascertained sibships:

1. Both parents were genotyped, but not all children. Here Mendel's law of segregation was utilized to assign the two parental alleles with equal probabilities, and then applying the method of data augmentation, as a part of the resulting Bayesian computation.
2. Genotype information was missing from at least one parent and no children were genotyped. Here, the logically consistent way to treat the missing data problem would involve using Bayes' formula, with the haplotype frequencies of the parents of the ascertained families as a “prior” probability distribution, while Mendel's law defines the likelihood. However, in the 40 ascertained families, where parental HLA information was missing and some sibs were genotyped, the inference was implemented as in 1), despite the fact that some inconsistent genotypes may then be obtained. This was done in order to avoid a further increase in the computational burden of implementing the already complex model when using OpenBugs software [18].

In the final implementation we first obtained the posterior mean of the baseline hazard for T1D using a model without the HLA effects. In the reported analyses of the HLA genotype and haplotype effects, the baseline hazard was held fixed at this posterior mean. This strategy was chosen in view of the fact that the Finnish birth cohorts, approximately 60,000 each and effectively forming the risk sets of our population based analysis, are so large that the joint estimation of the baseline hazards with the other model parameters would have hardly led to numerically different estimates. The reported analyses are based on 5,000 iterations (5,000 burn-in), and the estimates of genotype effects passed the convergence diagnostics criteria of Geweke [19] (R-program package CODA). In order to describe the age dependency in the HLA genotypes, we calculated the genotype and haplotype specific predictive disease free survival functions and its' 95% credible intervals. It is the expectation of disease free survival function, with respect to the joint posterior distribution of all model parameters [20].