Linkage Analysis: An Application of the Likelihood Ratio Test

by Debbie Goldwasser STAT600

November 8,2004

Topics for Discussion

Mendel’s Contribution to the Understanding the Distribution of Genetic Material in Genetic CrossesWhat is the Goal of Linkage Analysis? Why is a Likelihood Approach Appropriate? Optimal?How is Linkage Analysis Performed?Abraham Wald’s Contribution to the Optimization of Linkage Analysis Methods

Mendel: Brief Biography

Gregor Johann Mendel was born on July 22, 1822, in Heinzendorf, AustriaThe only son of a peasant farmer, his talents were recognized and he attended the Olmutz Philosophical Institute as a young manAt age 21, Mendel entered the Augustinian monastery of St. Thomas in Brunn, Austria, a site of impressive learning in many areas of studyIt was as a monk that Mendel developed an interest in the natural sciences and gained recognition as a particularly well-received teacher among his studentsPublished his landmark paper “Experiments in Plant Hybridization” in 1865, in which he laid the experimental foundation for the laws of independent assortment and the law of segregationAfter Mendel’s death in 1882, his work was rediscovered in 1902, after which his ideas gained widespread recognition for their relevance in explaining basic mechanisms of heredity.

Mendel and His Peas: The Law of Segregation

Isolated pure breeds of plants with complementary traits then crossed them to generate hybrids.Defines “Dominant” and “Recessive” properties: dominant properties constitute the entire character of the hybrid whereas recessive properties are lost in the hybrid generation (i.e. Axa --A) Crossed hybrids and found a ratio of 3:1 between dominant and recessive traits (I.e. (3:1 A:a)Key insight lie in Mendel’s ability to distinguish between dominant forms of the hybrid cross. The ratio of variant dominant forms to invariant dominant forms is 2:1 (Aa is Aa OR A in a ratio of 2:1)Therefore he concluded that the ratio of peas resulting form the hybrid cross has a true ratio of 1:2:1 (A:Aa:a)These findings eventually led to the law of segregation which in the year 2004 states that “diploid organisms possess genes in pairs, and only one member of this pair is transmitted to each offspring”.

Mendel and His Peas: The Law of Independent Assortment

Next, Mendel looked at hybrid crosses from doubly constant multiple traits (i.e. AbXaB)Again, he defines the recessive properties as those that are lost in the hybrid generation (lowercase letters)The phenotypic ratio resulting from the hybrid cross is 9:3:3:1 for combinations of traits (AB:Ab:aB:ab)Again, Mendel distinguished these offspring by their ability to generate variant forms in order to find a true ratio of (AB,Ab,ABb,AaB,AaBb,Aab,aBb,aB,ab) among the hybrids to be 1:1:2:2:4:2:2:1:1The law of independent assortment in the year 2004 states that alleles at different loci segregate independently of each other.

Relevance of Mendel’s Findings

Mendel’s findings form the basis for the study of genetics. It has been proved that genes, in fact, do lie on chromosomes, of which we receive a full set from both of our parents, resulting in a total of two copies each. In the parent generation, unlinked genes segregate independently of each other during meiosis and gamete formation.We can model the distribution of genes transmitted to offspring as a series of Bernoulli trials. Each copy of a gene is transmitted with probability ½The law of independent assortment is the null hypothesis we will test in linkage analysis. We will thus test the assumption that neutral genetic material and a disease gene segregate independently. If we suspect that a particular gene and disease trait do not segregate independently, then we refer to them as “linked” A likelihood approach is used because the null hypothesis is a fixed parameter.

The Law of Likelihood:

As quoted by Royall in “On the Probability of Observing Misleading Statistical Evidence”: Hacking (1965) defines the law of likelihood:If one hypothesis, H1, implies that a random variable X takes the value x with probability f1(x), while another hypothesis, H2, implies that the probability is f2(x), then the observation X=x is evidence supporting H1 over H2 if f1(x) > f2(x), measures the strength of that evidence.

A Neyman-Pearson approach asks, for a given hypotheses, how likely is it that this data may have occurred? An evidential likelihood approach asks, given the data, which of my hypotheses best explains the data?

)2|(/)1|(: xfxfRatioLikelihoodDefine

The likelihood ratio test can favor results that the Neyman-Pearson test would

rejectCase Example: Let the critical value for the likelihood ratio = R.

RNXNX

NX

XobservedNSuppose

NXH

NXH

/1)]1,(|21Pr[/)]1,(|21Pr[

1)]1,(|21Pr[

65.11,30,05.:

)1,(:2

)1,(:1

The Universal Bound Links Neyman-Pearson Results with Likelihood Ratio

Tests

RxxfRxxf

xfRxfx

xRfxfxRxfxfxDomain

xxf

xfRxfxf

truexfrejectH

xfxfRatioLikelihoodDefine

/1)2|()/1()1|(

)2|()/1()1|(:{

)}1|()2|(:{)})1|(/)2|(:{:

)1|(

)]1|(|))1|(/)2|(Pr[

])1|(1|1Pr[

)2|(/)1|(:

Pr(Misleading Evidence) + Pr(Weak Evidence) +Pr(Strong Evidence)=1

Conclusion: The Probability of Misleading Evidence is Bounded by 1/R where R is the selected critical value for the likelihood test. As the sample size becomes large the Pr(Misleading Evidence) )2/1)ln2(( k

Linkage Analysis: Intuitive Case Example of Linkage:

Dd dd

Dd dd DdDd dd Dddd Dd dd dd

1 2 9876543 10

Suppose Children 2,5,6,9,10 have the disease and children 1,3,4,7,8 do not. Intuitively conclude that Gene D is strongly related to the disease.

Linkage Analysis: Intuitive Case Example of Non-Linkage:

Suppose Children 1,3,5,7,9 have the disease and children 2,4,6,8,10 do not.

Intuitively conclude that gene F is not related to the disease.

Ff ff

Ff ff FfFf ff Ffff Ff ff ff

1 2 9876543 10

Mathematical Model: Key FeaturesCase: We have identified a disease that we are certain has a genetic component. Therefore, we assume that a gene or genes relating to the disease exist. Therefore, it or they must be located on one of the 23 chromosomes. Our job is to find them!

Linkage analysis entails testing the hypothesis that an unknown disease gene is at a near position to a known piece of genetic material by looking at the segregation ratio from the children.

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F

d?

f

D?

F

d?

f

D?

f

d?

F

D? d?fF F

D?F

d?f

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d?

Null Hypothesis (Independence): Gamete Probabilities:

2/2/2/)1( 2/)1(

4/1 4/1 4/1 4/1

Alternative Hypothesis (Linkage): Gamete Probabilities:

LOD Scores for a Single Family:

)]2/1(/)([log)( 10 LLZ LOD SCORE:

NRR

R

NL

)1()(

)5(.log)1(log)()(log)( 101010 NRNRZ

givenZ ,)( 21

NR

Zlod

/ˆ

)ˆ()(max

Typically, a curve is plotted for the range of theta values ranging from 0 to 0.5.

Assumption: We can determine with certainty which children result from a recombination event under H1 (i.e. the informative parent is phase-known)

Why the Computation of LOD Scores is More Complicated in Practice

• The sample size for a particular family is too small to achieve significance from a single family. (i.e. probability of 3 Heads in a row is 1/8). N is constrained by the number of children in a given family

• A simple way of combining multiple data sets is given below. However, this statistic is not readily interpretable for several reasons:

•Data is not distributed i.i.d. because family/pedigree data comes in a variety of sizes.

•The phase of the data for a set of parents is not always known. Oftentimes, this depends on gaining extended pedigree information (i.e. grandparents), which is not always available

K

iiiii NRNRZ

1101010 )5(.log)1(log)()(log)(

Abraham Wald: Brief Biography

•Born into a Jewish intellectual family in Hungary in 1902, Wald was home-schooled through primary and secondary schools by his parents

•Attended the University of Cluj (Romania) and demonstrated outstanding ability in mathematics

•Continued his studies at the University of Vienna with Karl Menger and was awarded his doctorate in 1931.

•Continued his research while serving as a mathematics tutor to the wealthy Karl Schlesinger, a leading banker and economist. Wald developed an interest in economics and econometrics

•After the Nazi occupation of Austria in 1938, Wald’s position became tenuous, so he emigrated to the United States to become a Fellow of the Carnegie corporation studying statistics at Columbia University under Hotelling

•Made key contributions in the area of decision theory, time series, sequential analysis.

Abraham Wald’s Theory on Sequential Probability Testing

Wald: “By a sequential test of a statistical hypothesis is meant any statistical test procedure which gives a specific rule at any stage of the experiment for making one of the following three decisions: (1) to accept the hypothesis being tested (null hypothesis) (2) to reject the null hypothesis, (3) to continue the experiment by making an additional observations”.

Wald invented the topic of sequential analysis in response to the demand for more efficient methods (i.e. reduced cost) of industrial quality control during World War II.

Decision Rule in Wald’s Sequential Probability Testing Scheme

AppBpeat

BppAcceptH

AppAcceptH

ddggpp

ddggpp

ionContradictddddggSuppose

dpgpgpgg

dpgpgpgg

dChoose

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xxpgxxpgxxpgg

mm

mm

mm

mm

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mmmm

mmmm

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2/11:

2/11:

,...,(,...,(/(,...,(

,...,(,...,(/(,...,(

Sequential Probability Testing and Linkage Analysis

Wald claims that his test is most efficient when used for testing a simple hypothesis against a single alternative. S and S* are two different methods of testing with the same strength (Type I and Type II Error rates).

Efficiency:)]|(),|([

*)]|(*),|([

10

10

SnESnEMax

SnESnEMax

It is desirable when searching for potential biomarkers to minimize the time to a decision, given the large amount of potential gene targets.

Mendel’s youthful reflections…

Yes, his laurels shall never fade,though time shall suck down by its vortexWhole generations into the abyss,Though naught but moss grow fragmentsShall remain of the epochIn which the genius appeared…May the might of destiny grant meThe supreme ecstasy of earthly joyThe highest goal of earthly destinyThat of seeing, when I arise from the tomb,My art thriving peacefullyAmong those who are to come after me.

Gregor Mendel, circa 1830-1840

References

Hodge, Susan E. (Spring 2004) Course Notes, “Theoretical Genetic Modeling” Columbia University

Morton, Newton E.,(1955) Sequential Tests for the Detection of Linkage, American Journal of Human Genetics 7:277-318

Wald, Abraham, Sequential Tests of Statistical Hypotheses Annals of Mathematical Statistics 1945; 6: 117–186

Mendel, Gregor, Experiments in Plant Hybridization, Proceedings of the Natural History Society,1865;

Sham, Pak (1998): Statistics in Human Genetics, Arnold Publishers (London)

Royall, Richard, On the Probability of Observing Misleading Statistical Evidence, Journal of the American Statistical Association; 2000; 95:760-780.