Bayesian latent trait modeling of migraine symptom data

Abstract

Definition of disease phenotype is a necessary preliminary to research into genetic causes of a complex disease. Clinical diagnosis of migraine is currently based on diagnostic criteria developed by the International Headache Society. Previously, we examined the natural clustering of these diagnostic symptoms using latent class analysis (LCA) and found that a four-class model was preferred. However, the classes can be ordered such that all symptoms progressively intensify, suggesting that a single continuous variable representing disease severity may provide a better model. Here, we compare two models: item response theory and LCA, each constructed within a Bayesian context. A deviance information criterion is used to assess model fit. We phenotyped our population sample using these models, estimated heritability and conducted genome-wide linkage analysis using Merlin-qtl. LCA with four classes was again preferred. After transformation, phenotypic trait values derived from both models are highly correlated (correlation = 0.99) and consequently results from subsequent genetic analyses were similar. Heritability was estimated at 0.37, while multipoint linkage analysis produced genome-wide significant linkage to chromosome 7q31-q33 and suggestive linkage to chromosomes 1 and 2. We argue that such continuous measures are a powerful tool for identifying genes contributing to migraine susceptibility.

Appendix 1

DIC3 is the difference between twice the posterior mean deviance and the deviance of estimated η

$$ {\rm{DIC}} =2\overline{D(\eta)}-D(\tilde{\eta}) $$

(4)

In the DIC3 proposed by Celeux et al. (2006), when the likelihood has a closed form, the first term can be approximated using M simulated values, $\eta^{(1)},{\ldots},\eta^{(M)},$ where $\eta^{(m)}=(p^{m}, \lambda^{m})$ from an MCMC chain.

$$ \begin{aligned} \overline{D(\eta)} &={\mathbb{E}}_{\eta}[-2 \log f(y|\eta)|y] \\ &\approx -{\frac{2} {M}} \sum_{m=1}^{M} \log f(y|\eta^{(m)})\\ \end{aligned} $$

(5)

The second term of Eq. 3 we used here is the posterior expectation, ${\mathbb{E}}[f(y|\eta)|y]$ which is also approximated using the parameters of an MCMC chain.

$$ \begin{aligned} D(\tilde{\eta})&=-2\log \hat{f}(y)=-2\log {\mathbb{E}}_{\theta}[f(y|\eta)|y] \\ &\approx -2 \log {\frac{1} {M}} \sum_{m=1}^{M} f(y|\eta^{(m)})\\ \end{aligned} $$

(6)

From Eqs. 5 and 6, Eq. 3 is the expanded form of Eq. 4. In the Bayesian LCA model, $f(y|\eta^{(m)})$ is

$$ f(y|\lambda^{(m)}, p^{(m)})=\sum_{k=1}^{K}p^{(m)}_{k} \prod_{i}^{n}\prod_{j}^{J}\left (\lambda^{(m)}_{kj}\right)^{y_{ij}}\left(1-\lambda^{(m)}_{kj}\right)^{1-y_{ij}} $$

and the posterior mean deviance is

$$ \overline{D(p, \lambda)}=-{\frac{2} {M}}\sum_{m=1}^{M} \log \sum_{k=1}^{K}p^{(m)}_{k} \prod_{i}^{n}\prod_{j}^{J}\left(\lambda^{(m)}_{kj}\right)^{y_{ij}}\left(1-\lambda^{(m)}_{kj}\right)^{1-y_{ij}} $$

and $D(\hat{\eta})$ is

$$ D(\hat{p}, \hat{\lambda})= -2\log \left \{{\frac{1} {M}}\sum_{m=1}^{M} \sum_{k=1}^{K}p^{(m)}_{k} \prod_{i}^{n}\prod_{j}^{J}\left(\lambda^{(m)}_{kj}\right)^{y_{ij}}\left(1-\lambda^{(m)}_{kj}\right)^{1-y_{ij}}\right \}. $$

For the Bayesian IRT model, the likelihood is

$$ f(y|\theta, a, b)=\prod_{i}^{n} \prod_{j=1}^{J} \left[{\frac{e^{a_{j}(\theta_{i}-b_{j})}} {1+e^{a_{j}(\theta_{i}-b_{j})}}}\right]^{yij} \left[1-{\frac{e^{a_{j}(\theta_{i}-b_{j})}} {1+e^{a_{j}(\theta_{i}-b_{j})}}}\right]^{1-yij} $$

therefore $\overline{D(\eta)}$ is

$$ \overline{D(\theta, a, b)}=-{\frac{2}{M}} \sum_{m=1}^{M} \log \prod_{i}^{n} \prod_{j=1}^{J} \left[{\frac{e^{a^{(m)}_{j}(\theta^{(m)}_{i}-b^{(m)}_{j})}} {1+e^{a^{(m)}_{j}(\theta^{(m)}_{i}-b^{(m)}_{j})}}}\right]^{yij} \left[1-{\frac{e^{a^{(m)}_{j}(\theta^{(m)}_{i}-b^{(m)}_{j})}} {1+e^{a^{(m)}_{j}(\theta^{(m)}_{i}-b^{(m)}_{j})}}}\right]^{1-yij} $$

and $D(\hat{\eta})$ is

$$ D(\hat{\theta}, \hat{a}, \hat{b})= -2\log \left\{{\frac{1} {M}} \sum_{m=1}^{M} \prod_{i}^{n} \prod_{j=1}^{J} \left[{\frac{e^{a^{(m)}_{j}(\theta^{(m)}_{i}-b^{(m)}_{j})}} {1+e^{a^{(m)}_{j}(\theta^{(m)}_{i}-b^{(m)}_{j})}}}\right]^{yij} \left[1-{\frac{e^{a^{(m)}_{j}(\theta^{(m)}_{i}-b^{(m)}_{j})}} {1+e^{a^{(m)}_{j}(\theta^{(m)}_{i}-b^{(m)}_{j})}}}\right]^{1-yij} \right\}. $$