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1 Author(s): Kerby Shedden, Ph.D., 2010 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution Share Alike 3.0 License: We have reviewed this material in accordance with U.S. Copyright Law and have tried to maximize your ability to use, share, and adapt it. The citation key on the following slide provides information about how you may share and adapt this material. Copyright holders of content included in this material should contact with any questions, corrections, or clarification regarding the use of content. For more information about how to cite these materials visit Any medical information in this material is intended to inform and educate and is not a tool for self-diagnosis or a replacement for medical evaluation, advice, diagnosis or treatment by a healthcare professional. Please speak to your physician if you have questions about your medical condition. Viewer discretion is advised: Some medical content is graphic and may not be suitable for all viewers. 1 / 26

2 Statistics in genetic association studies Kerby Shedden Department of Statistics, University of Michigan Sunday 3 rd February, / 26

3 The human genome The human genome consists of 23 chromosomes. Each chromosome is a single molecule that is a linear chain of four chemical units (bases), denoted A, T, G, C. The longest human chromosome is around 250 million bases long. The human genome in total is 3.2 billion bases long....aagtcaccata... Most cells contain two homologous copies of each chromosome, so there are 46 chromosomes in most cells. One member of each homologous pair of chromosomes is inherited from each parent....aagtcaccata... (chromosome 1 from mother)...aagtcagcata... (chromosome 1 from father) 3 / 26

4 The human genome 4 / 26

5 Functional role of the genome (Almost) all of a person s cells contain (almost) identical copies of the person s genome. Most of the genome does nothing. The most well-known functional part of the genome is the protein-coding genes. The protein coding genes contain the instructions for synthesizing proteins. Another important function of the genome is to control whether a given protein is synthesized in a certain situation. There are other functions of the genome, some poorly understood. 5 / 26

6 Genetic variation Around 99.9% of the genomes of two humans are identical (i.e. there is a difference every 1000 bases). This is called genetic variation, or genetic diversity. Variation in a protein coding gene may lead to a difference in the protein s structure and function. A characteristic of a person that is influenced by genetic makeup is called a trait, or phenotype. 6 / 26

7 Genetic measurement A locus is a position in the genome. A genotype is the genetic makeup of a person at a specific locus. Since there are two copies of each chromosome, a genotye consists of two bases, e.g. AT, GG, CT, etc. A genotype assay measures a genotype. See: 7 / 26

8 Alleles Genotypes at nearby loci are correlated. Often there are only two distinct states in a small region. For example, we observe a region on 4 distinct chromosomes, and see 2 alleles (denoted A and B ):...CCGGTGCG......ACCGTACC......ACCGTACC......CCGGTGCG... A allele B allele B allele A allele The genotype of a person at a given locus can be expressed as the alleles that are present at a given locus, on the two homologous chromosomes. Thus if there are two alleles at a locus, a person may have genotype AA, AB, or BB. The AA and BB genotypes are called homozygote genotypes, and the AB genotype is called a heterozygote genotype. 8 / 26

9 Complex diseases A trait that is entirely controlled by the genotype at a single locus is called Mendelian (after Gregor Mendel). Serious Mendelian diseases are rare, and the loci responsible for them are mostly known. A complex trait (complex disease) is influenced by genotypes at multiple loci, and possibly also by the environment. 9 / 26

10 Quantitative and qualitative traits A quantitative trait (e.g. height, blood pressure, BMI, intelligence), is a trait that can be measured on a quantitative scale. A qualitative trait (e.g. eye color, or having a disease such as diabetes) is a trait that is measured as a categorical variable. 10 / 26

11 Heritability We can learn whether a characteristic is genetically influenced without being able to measure the genes directly. Full siblings inherit their genomes from their (common) parents. They are 50% genetically identical. If a trait is genetically influenced, it is more likely that two siblings either both have the trait, or both lack the trait. But siblings also share a common environment. 11 / 26

12 Assessing heritability from familial data 80 Son's height Father's height Son s height Father s height 12 / 26

13 Assessing heritability from twin studies Twin studies may allow the genetic and environmental influences on a trait to be separated. Monozygotic (identical) twins have identical genomes. Dizygotic (fraternal) twins have genomes that are 50% identical (the same as full siblings). Monozygotic and dizygotic twins are presumed to experience identical environmental influences. Therefore, if monozygotic (MZ) twins are more similar in their trait status than dizygotic (DZ) twins, this suggests a genetic influence. 13 / 26

14 Assessing heritability of quantitative traits from twin studies If a quantitative trait is completely genetic, the MZ correlation will be 1, and the DZ correlation will be 0.5. If a trait is completely environmental, the MZ and DZ correlations will both be zero. 2 (MZ correlation DZ correlation) estimates the genetic contribution of a trait. It ranges between 0 and 1, and can be interpreted as the proportion of the variability of the trait that is explained by genetic factors. The correlation here is the intraclass correlation, and is almost the same as taking the usual Pearson correlation coefficient between the sibling trait values. 14 / 26

15 Testing genetic associations for quantitative traits Suppose we have a quantitative trait Y, such as height, and a locus that we believe might influence a person s value of Y. If there are two alleles at the locus (the locus is biallelic), we can code a person s genotype additively, Genotype X AA 0 AB 1 BB 2 Thus X is the number of B alleles that a person has. 15 / 26

16 Testing genetic associations for quantitative traits We can then fit the linear model E[Y X ] = α + βx, and look at the estimated value of β, ˆβ. If ˆβ is significantly different from zero, we might conclude that there is a genetic effect from this locus. We also might include non-genetic covariates (e.g. data on diet, gestational age): E[Y X ] = α + βx + γz. 16 / 26

17 Testing genetic associations for quantitative traits This is a good start, but there are several reasons this might produce a misleading conclusion: Confounding effects from other genetic loci Multiple testing Population stratification Genotyping errors We will return to some of these later. 17 / 26

18 Testing genetic associations for qualitative traits If the trait is qualitative, we can represent the data in a contingency table: AA AB BB D n 11 n 12 n 13 H n 21 n 22 n 23 D/H represents the values of a binary trait (e.g. D= disease and H= healthy ). We can assess the association between the genotypes and the trait using a test for associations in contingency tables, such as a χ 2 test, the Cochrane-Armitage trend test, or Fisher s exact test. Alternatively, we can use a regression technique such as logistic regression. 18 / 26

19 Additive and dominance effects So far we used this coding of the genetic data: Genotype X AA 0 AB 1 BB 2 This is called an additive coding, because it models each copy of the reference allele (B) as contributing the same amount to the trait. Under the linear model E[Y X ] = α + βx, the mean trait values for each genotype under the additive coding are Genotype X E[Y X ] AA 0 α AB 1 α + β BB 2 α + 2β 19 / 26

20 Additive and dominance effects Alternatively, a dominance coding for allele B would be Genotype X AA 0 AB 1 BB 1 and a dominance coding for allele A would be Genotype X AA 1 AB 1 BB 0 In this case, we treat either allele A or allele B as confering an effect, but there is no dose effect. That is, if B confers an effect, having one copy of B (genotype AB) is equivalent to having two copies of B (genotype BB). 20 / 26

21 Additive and dominance effects For a complex disease, there are assumed to be many different loci with genetic influences. Thus, when we use a model like E[Y X ] = α + βx, it is viewed as a working model. The working model is standing in for a much more complex data generating model. There is still some debate about whether it is important to consider dominance effects when studying complex traits. A common point of view is that even if there is a real dominance effect, it will be hard to see when we are unable to account for the effects of the other genetic effects. Furthermore, there are theoretical arguments showing that the additive model is a good approximation to the dominance model, even if the dominance effects are real. 21 / 26

22 Study designs for genetic association studies How is the study sample (i.e. the research subjects) obtained? Case/control study: consider a predetermined number of cases (people with the disease or trait of interest), and controls (people who do not have the trait or disease of interest). The numbers of cases and controls in the study sample do not reflect the proportion of cases in the population (i.e. the incidence). Population study: consider a representative sample of people from the population of interest. The distribution of the trait in the sample will be similar to the distribution of the trait in the population. 22 / 26

23 Study designs for genetic association studies Which genetic variants are to be considered? Confirmation/replication study: Consider a small number of variants that have been previously found to be associated with the trait, or for which there is a strong mechanistic hypothesis. Candidate gene study: Consider a moderate number of variants, for which there is some plausible basis to suspect an association may be present. Genome-wide association study: Consider a large, representative sample of relatively common genetic variants, regardless of whether there is any reason to suspect it is related to the trait. Complete sequencing study: Consider every locus in the genome, even if there is no evidence that it varies in the population. 23 / 26

24 Population stratification Population stratification (or population structure bias) is a particular type of confounding that occurs in genetic association studies. Allele frequencies for many genes differ by ancestry. For example, at a locus where the alleles are A and B, the allele frequencies may be as follows: Population A B Asian European Suppose our study population includes people of both European and Asian ancestry, and we are studying a trait whose prevalence differs by ancestry. For example, suppose that people with European ancestry tend to report having headaches more frequently than people with Asian ancestry. Then in this gene, the B allele will appear to be a genetic risk factor for headaches. But there are millions of loci where the allele frequencies differ between Eurpoeans and Asians, and it is very unlikely that most of them are involved with producing headaches. 24 / 26

25 Multiple testing If we carry out one test at the 0.05 level, there is a 5% chance of rejecting the null hypothesis if the null hypothesis is true. If we carry out k independent tests at the 0.05 level, and all the null hypotheses are true, the probability of rejecting at least one of the hypotheses (incorrectly) is: k. This is called the family-wise error rate (as opposed to the per-test error rate, which is 0.05). If k = 100, this value is greater than 99%. In genetic association studies, we often perform more than 100 tests. So we are almost guaranteed to find a false positive genetic association in this situation, even if there is no genetic influence on the trait at all. 25 / 26

26 Multiple testing The simplest way to address the issue of multiple testing is by using the Bonferroni correction. The idea of the Bonferroni correction is to carry out each individual test at a stricter significance level α < α, so that the family-wise error rate becomes the intended value α (e.g. α = 0.05). Recall that P(A or B) = P(A) + P(B) P(A and B), it follows that P(A or B) P(A) + P(B). Thus, the probability that at least one out of k tests is rejected is no bigger than kα. Thus we need kα α, so α α/k. If α = 0.05 and k = 100, we need α = This means that instead of rejecting when the test statistic T satisfies T > 2, when reject only when T > Equivalently, we can take all the single-test p-values, and multiply them by the number of tests. For example, if we observe p = it is adjusted to p = = / 26

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