An Exact Maternal-Fetal Genotype Incompatibility (MFG) Test

An Exact Maternal-Fetal Genotype Incompatibility (MFG) Test Sonia L. Minassian,1 Christina G.S. Palmer,2 and Janet S. Sinsheimer1,3,4n 1Department of Biostatistics, University of California, Los Angeles, California 2Department of Psychiatry and Biobehavioral Sciences, University of California, Los Angeles, California 3Department of Human Genetics, University of California, Los Angeles, California 4Department of Biomathematics, University of California, Los Angeles, California The maternal-fetal genotype incompatibility (MFG) test can be used for a variety of genetic applications concerning disease risk in offspring including testing for the presence of alleles that act directly through offspring genotypes (child allelic effects), alleles that act through maternal genotypes (maternal allelic effects), or maternal-fetal genotype incompatibilities. The log-linear version of the MFG model divides the genotype data into many cells, where each cell represents one of the possible mother, father, and child genotype combinations. Currently, tests of hypotheses about different allelic effects are accomplished by an asymptotic MFG test, but it is unknown if this is appropriate under conditions that produce small cell counts. In this report, we develop an exact MFG test that is based on the permutation distribution of cell counts. We determine by simulation the type I error and power of both the exact MFG test and the asymptotic MFG test for four different biologically relevant scenarios: a test of child allelic effects in the presence of maternal allelic effects, a test of maternal allelic effects in the presence of child allelic effects, and tests of maternal-fetal genotype incompatibility with and without child allelic effects. These simulations show that, in general, the exact test is slightly conservative whereas the asymptotic test is slightly anti-conservative. However, the asymptotic MFG test produces significantly inflated type I error rates under conditions with extreme null allele frequencies and sample sizes of 75, 100, and 150. Under these conditions, the exact test is clearly preferred over the asymptotic test. Under all other conditions that we tested, the user can safely choose either the exact test or the asymptotic test. Genet. Epidemiol. 28:83–95, 2005. & 2004 Wiley-Liss, Inc. Key words: maternal-fetal interaction; gene-environment interaction; permutation test; small sample test; maternal effects; randomization test; gene-gene interaction; association test Grant sponsor: United States Public Health Service; Grant numbers: MH59490, MH66001, T32-HG02536. nCorrespondence to: Janet Sinsheimer, Ph.D., AV-617 Center for Health Sciences, Dept of Biomathematics, UCLA School of Medicine, Los Angeles, CA . E-mail: Received for publication 23 April 2004; Accepted 7 July 2004 Published online 14 September 2004 in Wiley InterScience () DOI: 10.1002/gepi.20027 INTRODUCTION The MFG test and other family-based associa tion tests are useful for a wide array of statistical genetic analyses including identification of dis ease risk alleles that act directly through offspring genotypes to increase disease risk (child allelic effects) [e.g., Schaid and Sommer, 1994; Spielman and Ewens, 1996; Horvath et al., 2001], identifica tion of alleles that act through maternal genotypes to increase offspring disease risk (maternal allelic effects) [Weinberg et al., 1998; Wilcox et al., 1998; Umbach and Weinberg, 2000], and identification of maternal-fetal genotype incompatibilities that increase offspring disease risk [Sinsheimer et al., 2003; Kraft et al., 2004; Cordell et al., 2004; Cordell, 2004]. Assessment of the significance of effects has relied on the MFG test statistic distribution’s asymptotic equivalence to a chi-square distribu tion. Because the chi-square distribution can be a poor approximation, for example in the case of the distribution of the TDT test statistic [Schaid, 1996; Lazzeroni and Lange, 1998; Whittaker and Thompson, 1999], it is important to critically examine the type I error rate of the asymptotic MFG test. The validity of the asymptotic approx imation to the MFG test statistic distribution has been examined through simulations [e.g., Schaid and Sommer, 1993; Weinberg, 1999; Sinsheimer et al., 2003; Cordell et al., 2004; Cordell, 2004]. The simulation results suggest that the significance values produced by the asymptotic MFG test are reasonable at least for the conditions studied. & 2004 Wiley-Liss, Inc. Theseexaminations, however, havebeenlimited toasignificancelevelof0.05,toasinglesetofrisk allele frequencies, and, when examining the maternal-fetal genotype incompatibility effect, largesamplesizes[Sinsheimeretal.,2003;Cordell etal.,2004]. Inprinciple,determiningsignificanceusingthe exact distributionof the test statistic shouldbe preferredtomakinganasymptoticapproximation to thedistribution. Exact testshavebeendevel oped to test for disease-associated child allelic effects[e.g.,Clevesetal.,1997;Morrisetal.,1997; LazzeroniandLange,1998;Whittakeretal.,1999; Dudbridge, 2003].Weareunawareof anyexact tests that canbeusedtotest formaternal allelic effects ormaternal-fetal genotype incompatibil ities that increase offspring disease risk. The purpose of this report is to develop an exact MFGtestandtoevaluateboththeexact testand asymptoticMFGtest under avarietyof condi tions. Webeginwiththelog-linearversionoftheMFG test that hasbeenusedto infer thepresenceof various genetic effects throughasymptotic like lihoodratio tests (LRT) [Sinsheimeret al., 2003]. TheMFGtest isageneralizationof thelog-linear model developed byWeinberg and colleagues [Weinbergetal.,1998].Bothmodelsusegenotype data fromparentsandasingleaffectedchildto testhypothesesabouttheroleofageneindisease. Foragivenlocus,allpossiblemother, father,and childgenotype triosaredelineated, producinga specificnumberofcellsintowhichobservedtrios canbeclassified.Thenumberofcellsisafunction ofthenumberofalleles;abi-alleliclocusproduces 15 genotype trio combinations and a tri-allelic locus produces 78 genotype trio combinations. Becausedataaredividedacrossarelativelylarge numberofcells, somecellswillhaveonlyafew or zero counts if the sample size is small or if one of the allele frequencies is small. Under these conditions, an exact test may be particularlywarrantedasanasymptotictestcan, in principle, produce inaccurate estimates of significance. Inthefollowingsection, themodelandasymp toticMFG test are brieflydescribed. Next, the exactMFGtest andcorrespondingpermutation schemaareintroducedforfourdifferenthypoth eses aboutmodel parameters: scenario (1) child alleliceffects in thepresenceofmaternal allelic effects; scenario(2)maternalalleliceffects inthe presence of child allelic effects; scenario (3) maternal-fetalgenotypeincompatibilityassuming no additional risk factors; and scenario (4) maternal-fetal genotype incompatibility in the presence of child allelic effects. We conduct simulation studies to evaluate the asymptotic andexactMFGtests.Weexamine theeffectsof varyingallelefrequencyandsamplesizeontypeI errorandpowerofbothtests. METHODS ASYMPTOTICMFGTEST Consider a locuswith twoalleles:D(the risk allele)andd(thenullallele). Ifsymmetricmating is assumed, then eachmother, father, and af fected-childtriocanbecategorizedintooneofsix matingtypesandintofifteendistinctcellsthatare summarized inTable I. Themost general case parenttriolog-linearmodelwewillconsideristhe MFGmodelwithelevenparameters:d1,d2,d3,d4, TABLEI.Case-parent triocombinationsandexpectedtriocountsforthegeneralMFGmodelwithabi-alleliclocus Cell Matingtype Mother, fathergenotype Childgenotype Expectedcounts 1 1 D/D,D/D D/D Z2r2d1 2 2 D/D,D/d D/D Z2r2d2 3 2 D/D,D/d D/d Z2r1d2 4 2 D/d,D/D D/D Z1r2d2 5 2 D/d,D/D D/d Z1r1d2 6 3 D/D,d/d D/d Z2r1d3 7 3 d/d,D/D D/d mr1d3 8 4 D/d,D/d D/D Z1r2d4 9 4 D/d,D/d D/d 2Z1r1d4 10 4 D/d,D/d d/d Z1d4 11 5 D/d,d/d D/d Z1r1d5 12 5 D/d,d/d d/d Z1d5 13 5 d/d,D/d D/d mr1d5 14 5 d/d,D/d d/d d5 15 6 d/d,d/d d/d d6 Minassianetal. 84 Exact MFG Test 85 d5, d6, r1, r2, Z1, Z2, and m [Scenario 1A, Sinsheimer et al., 2003]. The quantities d1,y,d6 are mating type parameters. The parameters r1 and r2 are the relative risks to the offspring associated with one and two copies of the D allele in the child’s genotype versus no copies of the D allele. The parameters Z1 and Z2 are the relative risks to the offspring associated with one and two copies of the D allele in the maternal genotype versus zero copies of the D allele in the maternal genotype. The parameter m is the relative risk to the offspring associated with an interaction between the maternal and fetal genotypes (i.e., a maternal fetal genotype incompatibility). A MFG incompat ibility can be defined in a variety of ways, but we focus on an incompatibility that occurs when the mother has genotype d/d and the child has genotype D/d. Palmer et al. [2002] recently detected a maternal-fetal incompatibility at the RHD locus in a schizophrenia sample using this specific definition of MFG incompatibility and form of the MFG test. Assuming Poisson distributed counts and in dependent trios, the natural logarithm of the expected number of counts in the ith trio combination, oi, (i 1,y, 15) is: lnoi lndj lnmIM d=d;C D=d lnZ1 IMD=d lnZ2 IMD=D lnr1 IC D=d lnr2 IC D=D ln2IMD=d;P D=d;C D=d for j 1;...;6 1 where I is an indicator function, and M, P, and C represent the maternal, paternal, and child’s genotypes respectively. The expected number of trio counts for the general model (equation (1)) for each of the 15 cells is given in Table I. An assumption of Hardy Weinberg Equilibrium (HWE) is not necessary. The additional factor of 2 is present in the expected value of cell 9 because, under the null hypothesis, one half of the children of heterozygous parents will be heterozygous (one quarter of the children will be d/d and one quarter of the children will be D/D). The LRT can be used to test hypotheses of interest. The general form of the likelihood, conditional that the child is affected is: Y 15 L i 1 P15 oi Nie i 1 oi; 2 where Ni is the observed number of counts in the ith cell and oi is expected number of counts in the ith cell. The LRT has an asymptotically chi-square distribution and is calculated in the usual manner, LRT 2lnLHa lnLH0 , where LH0 denotes the likelihood under the null hypothesis and LHa denotes the likelihood under the alternative hypothesis. EXACT MFG TEST The exact MFG test permutes trio counts based on their expected distribution under H0. Counts that contribute to the LRT are permuted between cells that have the same expected value under H0, but different expected values under a specified Ha, resulting in the permutation of counts only within a mating type. By permuting only within a mating type, we condition on the sum of the counts in each mating type. Our density is the product of binomial and multinomial probability mass functions and is equivalent to a ‘‘conditioning on exchangeable parental genotype’’ CEPG model [Cordell et al., 2004]. The significance of the observed LRT, LRTobs, can be determined exactly on the basis of the permutation distribution of the data. With small sample sizes, the distribution of the LRT is discrete and, thus, an exact test will tend to be conservative [Lancaster, 1961; Berger, 2000]. To partially adjust for this phenomenon, the mid p value [Lancaster, 1961], Pmid, can be used where Pmid is X Pmid 1 2 i2A X ILRTi4LRTobs Nperm i2A ! ILRTi LRTobs Nperm : 3 A is the set of all possible data sets; i indexes the possible data sets; LRTi is the corres ponding LRT for i; I is the indicator function; and Nperm is the total number of possible data sets. Calculation of significance for the exact MFG test requires enumeration of all possible trio combinations within mating types, which can be computationally cumbersome as the sample size increases. However, the permutation distribution can be approximated through Monte Carlo simulations. In this case, we estimate the mid p value for the exact test with equation (3) where A is now the set of sampled data sets and Nperm is the total number of sampled data sets.