The generalized linear mixed model for linkage

The actual test is one-sided, so negative values of T are set to 0 and the resulting statistic T⁺ follows the usual 0.5χ₀ + 0.5χ₁ mixture under the null hypothesis. The weight w _i given to a specific sibling pair depends not only on the segregation parameters β, σ², and ρ but also on its covariate values. The weight w _i in the linkage statistic given by Eq. (1) is as follows: in a sib pair with individuals indexed by j = 1, 2, let ${\psi }_j^{\text{'}}=E\left(y_j|X_j,b_j=0\right)=\frac{e^{X_j\beta }}{1+e^{X_j\beta }},{\psi }_j^{\text{'}\text{'}}=\mathrm{var}\left(y_j|X_j,b_j=0\right)=\frac{e^{X_j\beta }}{{\left(1+e^{X_j\beta }\right)}^2}$, and ${\nu }_j={\left({\sigma }^2{\psi }_j^{\text{'}\text{'}}\right)}^{-1}$, the weight given to this sibling pair in the test statistic is given by ${\nu }_1{\nu }_2{\left\{\left(1+{\nu }_1\right)\left(1+{\nu }_2\right)-{\rho }^2\right\}}^{-2}\times \left[\begin{array}{l}\left\{\left(1+{\nu }_1\right)\left(1+{\nu }_2\right)+{\rho }^2\right\}\left(y_1-{\psi }_1^{\text{'}}\right)\left(y_2-{\psi }_2^{\text{'}}\right)\\ -\rho \left(1+{\nu }_2\right){\left(y_1-{\psi }_1^{\text{'}}\right)}^2-\rho \left(1+{\nu }_1\right){\left(y_2-{\psi }_2^{\text{'}}\right)}^2\\ +\rho {\left({\sigma }^2{\nu }_1{\nu }_2\right)}^{-1}\left\{\left(1+{\nu }_1\right)\left(1+{\nu }_2\right)-{\rho }^2\right\}\end{array}\right]$.

Estimation of segregation parameters

Population data are required in order to obtain estimates of the segregation parameters β, σ², and ρ. In fact, full maximum-likelihood estimation is often a difficult exercise for this type of data so we advocate the use of an approximate method of estimation which has been applied here [2]. Based on an arbitrary value ρ = 0.8, we estimate β and σ² so as to best reflect prevalence of RA in men and women (0.5% and 1.5%) as well as the overall recurrence risk ratio λ _s = 6. The choice of those estimates does not affect the asymptotic validity (type I error) of the method, indeed the test statistic has mean 0 and variance 1 under the null hypothesis, and only slightly influences the efficiency of the test.

For anti-CCP levels, we first used the available measurements in NARAC to estimate the distribution of anti-CCP levels in RA patients (we chose one affected sib at random per family) and in healthy individuals (only 11 unrelated individuals) separately. The estimation was carried out separately in the two groups using a nonparametric method (Gaussian kernel). Based on an overall prevalence of 1% for RA, we used Bayes formula to derive the probability of an individual to be a RA patient conditional upon his/her anti-CCP level. Given the very limited data available in healthy patients, the resulting curve exhibited high instability, especially for high anti-CCP levels, so we eventually smoothed the results by fitting a logistic curve.

Adjustment for sex

For a correlation between random effects ρ = 0.8, the segregation estimation lead to a variance σ² = 2.5 and sex effect β =(-6.43, 0.73). Taking male-male pairs of affected individuals as the reference, the relative weights were 0.97 in female-female pairs and 0.99 in male-female affected pairs. Thus, there was little difference in weighing of pairs of different kinds. Discordant pairs provided very little information for traits with a prevalence like RA (relative weight of -0.01 for female-female pairs). The results of the analysis hardly differed from an unadjusted analysis that only incorporated affected pairs and are therefore not presented.

Adjustment for anti-CCP levels

Based on the same parameters ρ and σ² as for sex, the logistic curve describing the effect of anti-CCP level X on the probability of developing RA was estimated as $\frac{e^{-14+0.2X}}{1+e^{-14+0.2X}}$. Anti-CCP level is a special type of covariate for RA because it is actually biologically related to the disease. The purpose of our analysis was therefore to assess whether linkage peaks for RA could be entirely accounted for by the anti-CCP measurements. The remaining peaks or potential new peaks could be attributed to a different component of the disease.

The test yielded very extreme relative weights (see Fig. 1) in which affected pairs with both individuals having high anti-CCP values became negligible compared to affected pairs with both individuals having low anti-CCP values, while pairs with very different anti-CCP values became discordant (i.e., were given negative weights). The genome-wide scan is presented in Figure 2. Initial suggestive peaks present on chromosomes 1, 9, 10, and 18 have disappeared, so they can be entirely attributed to anti-CCP levels. Peaks on chromosomes 6, 7, 8, 12, and 16 remain or are enhanced so they cannot be completely attributed to anti-CCP levels.