Stat 531 Statistical Genetics I Homework 4

Transcription

1 Stat 531 Statistical Genetics I Homework 4 Erik Erhardt November 17, Duerr et al. report an association between a particular locus on chromosome 12, D12S1724, and in ammatory bowel disease (Am. J. Hum. Genet. 63: , 1998). Their study included 115 sibling pairs who were each affected with inflammatory bowel disease. Suppose that the number of siblings sharing 0, 1, and 2 alleles IBD at the D12S1724 locus was 20, 61, and 34, respectively. Use the four nonparametric tests we discussed in class for affected sib pairs to determine whether in ammatory bowel disease is linked to locus D12S1724. There are n = 115 Sib Pairs affected, with r 0 = 20, r 1 = 61, r 2 = 34 giving ˆp 0 = 20 = , ˆp = 61 = , ˆp = 34 = Under H 115 0, p 0 = 1, p 4 1 = 1, p 2 2 = 1. We test these proportions under H 4 0 against related but different alternatives with four tests. 1. Mean Test H 0 : p 1 + 2p 2 = 1 (no linkage), H a : p 1 + 2p 2 > 1 (linkage). t m = (ˆp 1 + 2ˆp 2 ) 1 1 2n = ( (0.2957)) 1 1 2(115) = = 1.85 Pr(Z > 1.85) = , which is moderate evidence against H 0 in favor of linkage of inflammatory bowel disease to locus D12S Proportions Test t p = ˆp n H 0 : p 2 = 1 4 (no linkage), H a : p 2 > 1 4 (linkage). = (115) = = 1.13 Pr(Z > 1.13) = , which is insufficient evidence against H 0.

2 Erik Barry Erhardt 2/15 3. MinMax Test H 0 : 0.275p 1 + p 2 = (no linkage), H a : 0.275p 1 + p 2 > (linkage). Z = (0.275ˆp 1 + ˆp 2 ) = n = = 1.56 (0.275(0.5304) ) Pr(Z > 1.56) = , which is weak evidence against H 0 in favor of linkage of inflammatory bowel disease to locus D12S χ 2 Test H 0 : p 0 = 1 4, p 1 = 1 2, p 2 = 1 4 (no linkage), H a : not H 0 (linkage). X = 2 (O i E i ) (20 4 = ) E i=0 i 4 = = (61 2 ) ( ) Pr(X > ) > 0.10, which is insufficient evidence against H 0. As a summary, the two more powerful tests display evidence for linkage, while the other two tests provide insufficient evidence for linkage.

3 Erik Barry Erhardt 3/15 2 Wiggs et al. (Am. J. Hum. Genet. 74: , 2004) report the results of a genomewide scan in the search for genes responsible for earlyonset glaucoma. They identify two chromosomal regions in which a gene (or genes) that confer susceptibility to glaucoma may lie, one on chromosome 9 and one on chromosome 20. We will examine the chromosome 9 data for two of the pedigrees in their study (pedigrees 25 and 26, given in Figure 2 of the text). Our goal is to determine whether there is linkage to any region of chromosome 9 in these two pedigrees, using any appropriate methods for parametric or nonparametric linkage that we have discussed in class so far. (a) Description of the rationale for the study and for the inclusion of pedigrees in the sample. A risk factor for Glaucoma is having a family history of the disease, so inclusion of pedigrees should give us more statistical power since we have inheritance information available. While JOAG is inherited as an autosomal dominant trait, and has an associated gene, myocilin (MYOC), POAG is understood to be inherited as a complex trait. Indeed, the study finds by doing a genomewide screen that linkage is found on both chromosomes 9 and 20. Also, MYOC mutations may contribute to both JOAG and POAG, with different MYOC mutations causing different disease severities. MYOC mutations are highly penetrant for the early-onset disease. Using pedigrees will also allow us to look at mutations once identifying possible disease locus locations.

4 Erik Barry Erhardt 4/15 (b) Test parametrically for linkage to chromosome 9 using both singlepoint and multi-point methods. State and justify any assumptions you make in determining the model underlying your analysis. First the assumptions are presented then the results using these assumption are presented. The assumptions are rendered in the sghw4-2.dat file, presented in part here and in whole in the appendix. We assume there is no heterogeneity among linked families. We specify one liability class which provides the expected penetrances of the disease under the Mendelian Autosomal dominant mode of inheritance for DD, Dd and dd. In this case it is not 1, 1, 0 since pedigree I member 4 had strand recombination for his last two alleles, which causes computational difficulties. Therefore, the values specified were close to 1 when at least one disease allele is present and near 0 when no disease allele is present. Inference seemed somewhat sensitive to the choice of these values. Both 0.99 and 0.95 are close to 1, but differ slightly since I would expect higher penetrance if DD vs. Dd. 1 << NO. OF LIABILITY CLASSES << PENETRANCES For the specification of gene frequencies, the sample gene frequencies were calculated from the two pedigrees (MatLab code in appendix) and used as surrogates for the population frequencies. No attempt was made to allow for frequences of genes not observed in the sample. 3 5 # D9S << GENE FREQUENCIES (relative frequencies in pedigree) 3 7 # D9S << GENE FREQUENCIES 3 8 # D9S << GENE FREQUENCIES 3 5 # D9S << GENE FREQUENCIES 3 5 # D9S << GENE FREQUENCIES 3 3 # D9S << GENE FREQUENCIES 3 5 # D9S << GENE FREQUENCIES 3 4 # D9S << GENE FREQUENCIES 3 3 # D9S << GENE FREQUENCIES 3 3 # D9S << GENE FREQUENCIES

5 Erik Barry Erhardt 5/15 The genetic distances were specified on the Kosombi scale using the supplemental table provided, column 5: gen.loc estimated sex-averaged genetic location from the p-telomere in centimorgans. For our ten markers the chromosomal locations are 90.99, 93.16, 95.32, 95.32, 96.58, 96.58, 99.68, , , Marker distances are calculated using the distances between each of these positions. For adjacent markers having the same position, a distance of 0.01 is used. The second marker is not in the table so the average position of the first and third markers is used. The first value (1) is for the disease marker << RECOMB VALUES (From Kosombi file) Additional general specifications can be found in the sghw4-2.dat file in the appendix. Table 1 on page 7 provides LOD scores for both multi-point and singlepoint parametric analyses of these two pedigrees. Both methods (multipoint) [single-point] assign the highest LOD score of ( ) [ ] to the last marker, D9S160, and the second highest LOD score of ( ) [ ] to the penultimate marker, D9S271. The plots in Figure 1 on page 6 summarize the multi-point parametric analysis from Genehunter. The first plot displays the LOD scores and suggests that the disease may be associated with the last two markers already mentioned, or the seventh marker, D9S1851.

6 Erik Barry Erhardt 6/15 LOD LOD plot Pedigree file: sghw4-2.ped cm D9S278 D9S1815 D9S1781 D9S1841 D9S1803 D9S196 D9S1851 D9S176 D9S271 D9S160 Thu Nov 4 13:23: Z-all NPL plot Pedigree file: sghw4-2.ped cm D9S278 D9S1815 D9S1781 D9S1841 D9S1803 D9S196 D9S1851 D9S176 D9S271 D9S160 Thu Nov 4 13:23: Information Content Information content Pedigree file: sghw4-2.ped cm D9S278 D9S1815 D9S1781 D9S1841 D9S1803 D9S196 D9S1851 D9S176 D9S271 D9S160 Thu Nov 4 13:23: Figure 1: GENEHUNTER plots from multi-point analysis providing (1) parametric LOD scores, (2) nonparametric NPL scores, and (3) information content.

7 Erik Barry Erhardt 7/15 rel. position Chromosome 9 LOD score information (from 90.99) marker locus Multi-point Single-point content 0.00 D9S D9S D9S D9S D9S D9S D9S D9S D9S D9S Table 1: Genehunter Multi-point and Single-point parametric analysis for linkage.

8 Erik Barry Erhardt 8/15 (c) Test nonparametrically for linkage to chromosome 9 using both Genehunter and SimIBD. State and justify any assumptions you make. We assume there is no heterogeneity among linked families. Table 2 on page 9 provides Genehunter s NPL scores and p-values for both multi-point and single-point nonparametric analyses and SimIBD s APM (Affected Pedigree Member) and AC (Affected Comparison) Weighted ZObs and p-values for these two pedigrees. The p-value is used to locate the marker loci demonstrating the most evidence for linkage. Both methods of Genehunter (multi-point) [single-point] and SimIBD {AC} suggests marker D9S176 with p-values ( ) [ ] {0.001}, while SimIBD APM suggests two markers are more significant than D9S176, first the first marker D9S278 (p-value 0.262), then D9S1803 (p-value 0.343), finally D9S176 (pvalue 0.379). The second plot in Figure 1 on page 6 summarizes the multi-point nonparametric analysis from Genehunter, displaying the NPL scores and suggests that the disease may be associated with the fifth through eighth markers, with just slightly more evidence on the eighth marker D9S176.

9 Erik Barry Erhardt 9/15 rel. pos. Chrom. 9 Genehunter NPL score and p-value SimIBD APM SimIBD AC (cm) marker Multi-point Single-point Single-point Single-point (from 90.99) locus NPL p-value NPL p-value W. ZObs p-value W. ZObs p-value 0.00 D9S D9S D9S D9S D9S D9S D9S D9S D9S D9S Table 2: Genehunter Multi-point and Single-point nonparametric and Sim- IBD nonparametric APM and AC analysis for linkage.

10 Erik Barry Erhardt 10/15 (d) Compare the results of all three methods (parametric Genehunter, nonparametric Genehunter, SimIBD). Are they the same? Different? Discuss any discrepancies. To help us see the differences Table 3 on page 10 provides LOD scores for parametric analyses and p-values of nonparametric analyses for all methods considered. Large LOD scores and small p-values indicate evidence for linkage. For each method the two loci showing the strongest evidence have been highlighted. We observe that the Genehunter parametric methods agree with each other, the Genehunter nonparametric methods and SimIBD AC agree only at the most likely locus, and the Genehunter and SimIBD APM method do not agree. The parametric and nonparametric methods do no agree. This could indicate model misspecification in the parametric analysis. Misspecification could occur if heterogeneity exists, in which case inclusion of the heterogeneity parameter may be required, or if penetrance was misspecified in the liability class. However, an attempt was not made to make the parametric and nonparametric methods agree since such strategies do not rely solely on a priori information and so inevitably bias the results without theory and knowledge supporting those decisions. Small sample size could be another explanation for observed differences. A large difference at marker D9S1803 is between Genehunter s parametric single-point method smallest LOD score ( ) and SimIBD APM second smallest p-value (0.343), giving almost opposite inference for this locus. rel. pos. Chrom. 9 Genehunter LOD scores and NPL p-values SimIBD APM SimIBD AC (cm) marker LOD scores NPL p-values p-value p-value (from 90.99) locus Multi Single Multi Single Single Single 0.00 D9S D9S D9S D9S D9S D9S D9S D9S D9S D9S Table 3: Summary of Genehunter Multi-point and Single-point nonparametric and SimIBD nonparametric APM and AC analysis for linkage.

11 Erik Barry Erhardt 11/15 (e) Compare your results to the results in the paper. What factors contribute to the differences in your results from those reported by the authors? Wiggs s Table 2 presents two-point (Genehunter single-point) LOD scores. Their two most highly significant markers under the assumption of homogeneity were D9S196 (LOD=2.40) and D9S1803 (LOD=2.10), our sixth and fifth markers. My parametric method had very low LOD scores for these markers. In fact, our lowest LOD score for parametric single-point at marker D9S1803 is the marker Wiggs had for his second highest. My Genehunter multi-point nonparametric analysis matches most closely with Wiggs s results. Some explanation of differences observed. The Wiggs, et. al., paper presents, inferred from their Figure 1, results for map locations 15.0 through cm. We observed data only for map locations through Also, Wiggs had data from 25 pedigrees while we had available pedigrees from only two families. Therefore, they had more data from which to make inferences, both in chromosome length and number of samples.

12 Erik Barry Erhardt 12/15 Extra Credit Under the assumptions discussed in class for the derivation of the IBD distribution in pairs of siblings, show that t = w 1ˆp 1 + ˆp 2 E H0 (w 1ˆp 1 + ˆp 2 ) VarH0 (w 1ˆp 1 + ˆp 2 ) = w 1ˆp 1 + ˆp 2 1w w w n Under H 0, p 0 = 1, p 4 1 = 1, p 2 2 = 1 with ˆp 4 0, ˆp a 2 N ( 1, n) and a ˆp1 N ( 1, ) 1 2 4n. The last piece we need is Cov(ˆp1, ˆp 2 ) = ˆp 1 ˆp 2. Using these facts, n the derivation is simple. t = (w 1ˆp 1 + ˆp 2 ) E H0 (w 1ˆp 1 + ˆp 2 ) VarH0 (w 1ˆp 1 + ˆp 2 ) = (w 1ˆp 1 + ˆp 2 ) ( 1 2 w ) w 2 1 Var H0 (ˆp 1 ) + Var H0 (ˆp 2 ) + 2w 1 Cov H0 (ˆp 1, ˆp 2 ) = w 2 1 w 1ˆp 1 + ˆp 2 1w ( ) ( 1 ) 4n w 1 2( 1 4) 16n 1 n = w 1ˆp 1 + ˆp 2 1w n w w 16n 16n 1 = w 1ˆp 1 + ˆp 2 1w w w n

13 Erik Barry Erhardt 13/15 Appendix Matlab code used for the above analysis %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% 2. Genehunter and SimIBD analysis %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% sghw4-2.ped (Pedigree file for Genehunter) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% Matlab counting frequencies of alleles in sample to use in sghw4-2.dat file a = [ ]; f=zeros(1,8); for i=1:10 f(1)=length(find(a(:,2*i-1:2*i)==1)); f(2)=length(find(a(:,2*i-1:2*i)==2)); f(3)=length(find(a(:,2*i-1:2*i)==3)); f(4)=length(find(a(:,2*i-1:2*i)==4)); f(5)=length(find(a(:,2*i-1:2*i)==5)); f(6)=length(find(a(:,2*i-1:2*i)==6)); f(7)=length(find(a(:,2*i-1:2*i)==7)); f(8)=length(find(a(:,2*i-1:2*i)==8)); f1=f/48; disp(f1) %sum(f1) end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% Genehunter and SimIBD sghw4-2.dat file << NO. OF LOCI, RISK LOCUS, SEXLINKED (IF 1) PROGRAM << MUT LOCUS, MUT RATE, HAPLOTYPE FREQUENCIES (IF 1) << AFFECTATION, NO. OF ALLELES << GENE FREQUENCIES 1 << NO. OF LIABILITY CLASSES << PENETRANCES << less than one since last alleles ped member 4 affected but suspect strand had recomb, but 3 5 # D9S << GENE FREQUENCIES (relative frequencies in pedigree) 3 7 # D9S << GENE FREQUENCIES 3 8 # D9S << GENE FREQUENCIES

14 Erik Barry Erhardt 14/ # D9S << GENE FREQUENCIES 3 5 # D9S << GENE FREQUENCIES 3 3 # D9S << GENE FREQUENCIES 3 5 # D9S << GENE FREQUENCIES 3 4 # D9S << GENE FREQUENCIES 3 3 # D9S << GENE FREQUENCIES 3 3 # D9S << GENE FREQUENCIES 0 0 << SEX DIFFERENCE, INTERFERENCE (IF 1 OR 2) << RECOMB VALUES (From Kosombi file) (second marker not in table, so ave << REC VARIED, INCREMENT, FINISHING VALUE %%%% Running Genehunter %%%% Note: alias gh ~salter/genprog/genehunter/gh.sun gh run in.in quit %%%% in.in file photo sghw4-2.out map function kosambi units cm single point off postscript output on load markers sghw4-2.dat use scan pedigrees sghw4-2.ped total stat sghw4-2_npl_plot_all.ps y sghw4-2_lod_plot_all.ps y sghw4-2_info_content_all.ps y single point on scan pedigrees sghw4-2.ped total stat %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% sghw4-2_simibd.ped (Pedigree file for SimIBD) Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: 1 Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: Per: Ped: 2 Per: 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% Running SimIBD (run once for each locus) %%%% Note: alias ibd ~salter/genprog/simibd/simibd -a -b ibd sghw4-2_simibd.ped sghw4-2.dat b cat simibd.out > sghw4-2_simibd.out cat simibd.aff.out > sghw4-2_simibd_aff.out cat simapm.out > sghw4-2_simapm.out ls -l *.out rm sim*.out ibd sghw4-2_simibd.ped sghw4-2.dat b cat simibd.out >> sghw4-2_simibd.out cat simibd.aff.out >> sghw4-2_simibd_aff.out

15 Erik Barry Erhardt 15/15 cat simapm.out ls -l *.out rm sim*.out >> sghw4-2_simapm.out ibd sghw4-2_simibd.ped sghw4-2.dat b cat simibd.out >> sghw4-2_simibd.out cat simibd.aff.out >> sghw4-2_simibd_aff.out cat simapm.out >> sghw4-2_simapm.out ls -l *.out rm sim*.out