The case-parents design, comprising affected offspring and their two parents, permits testing and estimation of offspring- and maternally-mediated genetic effects [Weinberg et al., 1998; Wilcox et al., 1998]. Offspring effects are evaluated by the apparent over-transmission (compared to the Mendelian expectation of 0.5) of deleterious alleles from heterozygous parents to affected offspring. Maternal genotype effects are assessed through deviations from genotype mating symmetry in the case-parents data under the assumption of mating symmetry in the source population. That is, a deleterious allele that acts via the mother will be more prevalent in mothers than in fathers of affected offspring. The key mating-symmetry assumption cannot, however, be evaluated using case-parent triads alone.

Another design that can be used to disentangle offspring and maternal genetic effects ascertains affected children and their mothers as well as a random sample of unaffected children and their mothers. In such a case-mother/control-mother design, the offspring-mother pairs are the units for analysis. Both offspring and maternal effects are assessed through case-control comparisons; no mating symmetry assumption is required. Unlike the case-parents design, however, this design is vulnerable to bias due to the existence of genetically distinct subpopulations. Weinberg and Umbach [2005] describe the strengths and weaknesses of both the population-based case-mother/control-mother and the family-based case-parents design.

To bring together the advantages of both family-based and population-based approaches, Nagelkerke et al. [2004], Epstein et al. [2005] and Weinberg and Umbach [2005] have proposed various hybrid designs that combine both kinds of data. One hybrid approach [Weinberg and Umbach, 2005] enrolls case-parent and control-parent triads and genotypes case-parent triads but only the parents of control offspring (case-parent triad/control parents design). This hybrid design greatly improves power for the evaluation of offspring and maternally-mediated genetic effects. It allows testing for bias from population stratification; and, if bias is detected, confounding is avoided by using only the case-parent triads. In addition, the assumption of parental mating symmetry in the population at large can be tested, and maternal genetic effects can be validly estimated even if mating symmetry is rejected [Weinberg and Umbach, 2005].

Let p denote the frequency of the minor or ‘variant’ allele. Which allele is designated as the ‘variant’ has no effect on estimation or testing beyond the mathematical inversions. Let M, F and C represent the number of variant alleles (0,1,2) carried by the mother, father and child, respectively. D is an indicator variable for disease status, which is 1 for case families and 0 for control families.

Following Schaid and Sommer [1993], we define nine different mating types based on the number of variant copies carried by the mother and the father. These mating types along with their possible offspring genotypes lead to 15 possible (M,F,C) categories, and, consequently, one can imagine two 15-cell multinomial distributions of offspring and parental genotypes, one for control triads and one for case triads. In hybrid designs, typically the full (M,F,C) data are recorded for case families whereas only partial data are collected from control families; here, only (M,C) data. Initially we assume that any genotyping called for by the design is complete, but missing-data methods can be employed if some genotypes are missing [Weinberg, 1999a].

For control families, expected counts in the 15-cell multinomial can be modeled using Mendelian transmission probabilities and mating type parameters (μ_mf), which are proportional to the frequencies of mother-father pairs with M=m and F=f in the source population. The expected counts for control-mother dyads (the last 7 lines of Table I) arise from the 15-cell multinomial by summing counts across the genotypes of possible fathers. The distribution of control-mother dyads has two noteworthy features: First, the (M,C) cells (0,2) and (2,0) are not possible, leaving only seven cells with non-zero expected counts. Second, and less obviously, the following relationship is a consequence of Mendelian transmission alone: when M=1, the expected count for C=1 is the sum of the expected counts for C=0 and C=2. This constraint reduces the available degrees-of-freedom contributed by the seven control-dyad cells from six to five.

For case families, expected counts in the 15-cell multinomial involve not only mating-type parameters and Mendelian probabilities but also four genetic relative risk parameters. We denote these relative risk parameters as follows: R₁ (R₂) is the relative risk for offspring carrying 1 (respectively, 2) copies of the variant compared to offspring carrying none; S₁ (S₂) is the relative risk for offspring whose mother carries 1 (respectively, 2) copies of the variant allele compared to offspring whose mother carries none. Combining the 15 cells for case-parent triads with the seven cells for control-mother dyads yields a 22-cell multinomial for the proposed design (Table I).

For the case-parents design and for the hybrid design that uses parents of controls, the multinomial expected cell counts are all products of parameters and can be fitted using log-linear Poisson regression. Because the expected counts for control-mother dyads involve sums of parameters, however, the expected counts in Table I for the 22-cell multinomial are not themselves log-linear. A straightforward way to proceed with fitting either model is to regard the 22-cell multinomial as a version of the full 30-cell multinomial (15 cells each for case families and control families) that is missing genotype data for fathers of controls by design. Thus, one can use missing-data methods like the Expectation-Maximization (EM) algorithm [Dempster et al., 1977] in conjunction with a log-linear model for the 30-cell multinomial. Use of the EM algorithm also allows inclusion of any case-parent triads or control-mother dyads with missing genotypes that may arise through genotyping failure or incomplete ascertainment. For valid analysis, one must be able to assume that these genotype data are missing at random conditional on disease status and the observed genotypes. When control fathers are missing only by design, this assumption is satisfied without doubt because all of them are missing.

Assuming a multiplicative model for risk and no bias from population structure, a log-linear model for the full 30-cell multinomial (corresponding to Table I, column 5) would be: I_{( )} is an indicator function which takes a value of 1 if the parenthetical condition is met and 0 otherwise. β₁ and β₂ denote the natural logarithms of the offspring genetic relative risks, R₁ and R₂, respectively; and α₁ and α₂ denote the natural logarithms of maternal genetic relative risks, S₁ and S₂, respectively. γ corresponds to the natural logarithm of the normalizing factor B. The offset, denoted Off_mfc is the constant multiplier (1, ½ or ¼) given in Table I, column 5. In this model, the nine mating-type parameters, μ_mf, are common to both cases and controls and can be interpreted as proportional to the mating type frequencies in the source population.

This model is fundamental to the analysis of data from our proposed design. Modifications of the model by omitting or including certain additional terms allow one to construct likelihood ratio tests (LRTs) of hypotheses about the genetic relative risk parameters or to test the assumptions about bias from population stratification or about mating symmetry. In addition, models addressing maternal-fetal incompatibility can be constructed by including two additional relative risk parameters [Sinsheimer et al., 2003]. All tests of assumptions that we subsequently develop under the four risk parameters in model (1) can be developed and applied without difficulty when these two additional risk parameters are present.

Our power calculations are based on the noncentrality parameter for a four-df chi-squared LRT. We calculated the noncentrality parameter as the LRT statistic constructed by treating expected counts under the specified alternative hypothesis as data [Agresti, 1990]. Values of the noncentrality parameter can be translated to power values using the noncentral χ² distribution with four degrees of freedom. To calculate expected counts, we considered populations with allele frequencies ranging over the interval from zero to one where parental mating-type frequencies obeyed Hardy-Weinberg equilibrium and, consequently, exhibited mating symmetry. (Note Hardy-Weinberg equilibrium is simply a convenience here; it is not needed for the validity of our analyses.) We based all our calculations on 150 case families and 150 control families (the case-parents design used only the 150 case families). We plotted the noncentrality parameters as a function of allele prevalence and included horizontal reference lines corresponding to specific levels of power for a four-df LRT at alpha level 0.05. When the noncentrality parameter exceeds a given reference line, the LRT’s power exceeds the specified power. Transformation of the noncentrality parameter to a different planned sample size with the same case:control ratio (say, for K cases) is accomplished by multiplying the noncentrality parameter values from our figure by K/150. For any alpha level, the ratio of two noncentrality parameters corresponds to the relative efficiencies of the corresponding two designs, i.e., the ratio of the sample sizes needed to achieve the same power.

We considered four distinct risk scenarios. The first scenario included a gene-dose effect of the fetal genotype but no effect of the maternal genotype, specifically, R₁, R₂, S₁, S₂, were 2, 3, 1, 1, respectively. The second scenario included a gene-dose effect of the maternal genotype and no effect of the fetal genotype, specifically, R₁, R₂, S₁, S₂, were 1, 1, 2, 3, respectively. The third involved recessive effects of both the fetal and maternal genotypes (R₁, R₂, S₁, S₂ were 1, 2, 1, 3, respectively). The final scenario included a recessive effect of the fetal genotype and a dominant effect of the maternal genotype (R₁, R₂, S₁, S₂, were 1, 3, 2, 2, respectively). In all four scenarios, either hybrid design exhibited better power across the range of allele frequencies than both the case-parent triads and the case-mother/control-mother design (Figure 1). In addition, the hybrid using parents of controls always performed somewhat better than the hybrid using control-mother dyads, reflecting that the former design provides more mating-type information from control families. Analyses of the hybrid designs that enforced mating symmetry generally provided more power than those that did not, but the difference in power depended on the scenario. In the scenario with maternal but without offspring effects (Figure 1, panel b), the power of hybrid designs when mating symmetry was not enforced was the same or close to that of the case-mother/control-mother design. The relative efficiency of the case-parents design compared to the case-mother/control-mother design depended on the particular scenario.

With 150 case families and 150 control families, this population structure posed a serious challenge to the proposed hybrid design under model (1) by producing biased relative risk estimates and an inflated type I error rate for testing genetic effects (actual α-level = 0.43 at nominal α = 0.05). Testing for the presence of bias due to population structure under mating symmetry at α = 0.10, the power was 0.42 and 0.54 for the four- and the one-df test, respectively, corresponding to noncentrality parameters of 3.71 and 3.03. Doubling the sample size doubles the noncentrality parameters, yielding power 0.69 and 0.79, respectively. All these values were near those achieved with the hybrid design using control parents where the analogous tests yielded power of 0.41, 0.53, 0.68 and 0.79, respectively [Weinberg and Umbach, 2005].

The primary motivation for considering a hybrid design that uses control-mother dyads instead of a hybrid design that uses parents of controls is a practical one: recruiting mothers of controls is easier than recruiting both mothers and fathers. Thus, larger sample sizes should be easier to achieve and possible bias from selective refusals of fathers (or of mothers with concerns about paternity) can be reduced. On the other hand, control-mother pairs provide less direct information about mating-type parameters in the underlying population than do control parent pairs. This feature is reflected in our results on power. The hybrid with control parents provided slightly more power for testing offspring and maternal genetic effects than did the hybrid with control-mother dyads.

In practice, the best design might be a hybrid of the two hybrid designs considered here. One could plan to recruit the parents of controls but, when the father is unavailable, try to recruit the control-mother dyad instead. Again, missing-data methods like the EM algorithm can be used for inference. Presumably the power for such a design would correspond to noncentrality parameters that fall between the curves of the two hybrid designs (Figure 1). This hybrid-of-hybrids design would be tantamount to a design where case-parent triads are augmented by control-parent triads but missing data is dealt with in the analysis. The only advantage of genotyping control offspring when both parents are available, however, is the capacity to test the basic assumption of Mendelian transmission, an assumption that is not often questioned.

For studying both offspring and maternal genetic effects on disease risk, hybrid designs that use control parents or control-mother dyads offer additional benefits over the case-parents design. The case-parents design is capable of studying the maternal genetic effects under the assumption of mating symmetry in the source population but offers no way to check that assumption. Hybrid designs provide information not only for checking that assumption but also for studying maternal genetic effects even when mating symmetry is not satisfied. These features were delineated previously for the hybrid that uses control parents [Weinberg and Umbach, 2005], and we have shown here that they are maintained when control-mother pairs are substituted for control parents. Two different phenomena can masquerade as maternal genetic effects if not properly accounted for: mating asymmetry in the underlying population; and imprinting, where the effect of a transmitted allele depends on the parent of origin. After detecting a maternal effect, the investigator should ideally exclude the possibility that it was wholly or partly attributable to one of these other sources. With a case-parents design, mating symmetry in the underlying population must remain an assumption. The case-mother/control-mother design implicitly tolerates mating asymmetry. With either hybrid design, mating symmetry can be checked; and, if rejected, maternal effects can be studied under a model that accommodates mating asymmetry. Imprinting can be accommodated by the case-parents design using a log-linear model that incorporates parent-of-origin effects [Weinberg, 1999b]. Although we have not considered it here, this same modeling tactic could be adapted for the hybrid designs or for the case-mother/control-mother design as well.

An appealing feature of hybrid designs that involve either parents of controls or control-mother pairs is the ability to check key assumptions. The corresponding weakness is that these tests, particularly the key test for bias from population structure, appear to have limited power. For example, with 150 case and control families, we had little power to detect bias from population structure under a scenario which, if it went undetected, would induce substantial bias in inference. Power seemed somewhat better for detecting mating asymmetry in our examples; but we could certainly produce other examples where power was more limited. Larger sample sizes would help, of course. With a hybrid design, a simple practical precaution is to compare risk estimates under models that do and do not make a particular assumption. With concern about population structure, comparing relative risk estimates from the hybrid design with those from its case-parent triad component alone might provide reassurance. Similarly, with concern about mating symmetry, one could compare estimates from the mating-symmetry and the mating-asymmetry models.

The hybrid designs that we have considered will be useful for studies of complex phenotypes such as a birth defect where environmental factors will likely also play a role. The log-linear model for the 30-cell multinomial for case triads and control triads together can in principle be extended to incorporate environmental factors assessed categorically. Consequently, environmental risks and gene-by-environment interactions can be evaluated under either hybrid design. Postulating appropriate models for checking bias from population structure might be challenging, however, because exposure prevalence could change across genetically distinct subpopulations.

We have also demonstrated the potential increase in power that the hybrid designs can achieve when bias from population stratification is absent. The relative gain in efficiency (as measured by the ratio of the curves) depends on the underlying scenario. We based these power comparisons on equal numbers of case families in each design reflecting that the number of case families available is often a limitation when designing a study. The hybrid designs considered here require genotyping five individuals per case compared to only three or four, respectively, for the case-parents or the case-mother/control-mother designs. To compare designs when each of them genotyped 750 individuals (the number measured in the hybrid designs in Figure 1), one would modify Figure 1 by multiplying the noncentrality curves for the case-parents design by 5/3 and those for case-mother/control-mother design by 5/4 (curves for hybrid designs remain the same). On that per-genotype basis, hybrid designs were not the most efficient in every scenario that we considered; however, they were more efficient than case-mother/control-mother designs in general and more efficient than the case-parents design in realistic scenarios where genotypes were missing at random. In our view, any disadvantage in power per genotype of hybrid designs compared to the case-parents design is more than offset by the greater flexibility that hybrid designs offer for checking assumptions and evaluating effects of exposures. The cost of genotyping should not be the investigator’s sole concern.