<i>An Introduction to Probability Theory and Its Applications</i>

Feller, William; Morse, Philip Μ.

doi:10.1063/1.3062516

Cited by 3,329 publications

(3,666 citation statements)

References 0 publications

Supporting

Mentioning

3,600

Contrasting

Unclassified

Order By: Relevance

“…Let us further remark that, in principle, even finer error estimates (of Berry-Esséen type, cf. [7], Ch. XVI) can be obtained beyond the asymptotics in Theorem 1, but this becomes technically more involved.…”

Section: Large Deviationsmentioning

confidence: 99%

How T-cells use large deviations to recognize foreign antigens

2008

View full text Add to dashboard Cite

A stochastic model for the activation of T-cells is analysed. T-cells are part of the immune system and recognize foreign antigens against a background of the body's own molecules. The model under consideration is a slight generalization of a model introduced by Van den Berg, Rand and Burroughs in 2001 [18], and is capable of explaining how this recognition works on the basis of rare stochastic events. With the help of a refined large deviation theorem and numerical evaluation it is shown that, for a wide range of parameters, T-cells can distinguish reliably between foreign antigens and self-antigens.

show abstract

Section: Large Deviationsmentioning

confidence: 99%

How T-cells use large deviations to recognize foreign antigens

2008

View full text Add to dashboard Cite

show abstract

“…; p r Þ 0 , where t is the vector of counts of susceptibility mutations associated with each of the r marker alleles, such that P r i¼1 t i ¼ t and p is the vector of marker allele frequencies. For a fixed number of susceptibility mutations, the number of historical scenarios considered for LRT power computations is the number of outcomes for the multinomial distribution [Feller, 1957].…”

Section: Multiple Historical Scenariosmentioning

confidence: 99%

On the advantage of haplotype analysis in the presence of multiple disease susceptibility alleles

Morris

Kaplan

2002

Genetic Epidemiology

329

214

View full text Add to dashboard Cite

We investigated the effect of multiple susceptibility alleles at a single disease locus on the statistical power of a likelihood ratio test to detect association between alleles at a marker locus and a disease phenotype in a case-control design. Using simplifying assumptions to obtain the joint frequency distribution of marker and disease locus alleles, we present numerical results that illustrate the impact of historical variation of initial associations between marker alleles and susceptibility alleles on the power of a likelihood ratio test for association. Our results show that an increase in the number of susceptibility alleles produces a decrease in power of the likelihood ratio test. The decrease in power in the presence of multiple susceptibility alleles, however, is less for markers with multiple alleles than for markers with two alleles. We investigate the implications of this observation for tests of association based on haplotypes made up of tightly linked single-nucleotide polymorphisms (SNPs). Our results suggest that an analysis based on haplotypes can be advantageous over an analysis based on individual SNPs in the presence of multiple susceptibility alleles, particularly when linkage disequilibria between SNPs is weak. The results provide motivation for further development of statistical methods based on haplotypes for assessing the potential for association methods to identify and locate complex disease genes.

show abstract

“…For simplicity, let us assume that the allelic effects are all additive. Without loss of generality we can assume that the allelic affects of A is one and the effect of B, is zero, for all i 1, 2,..., n. We shall further assume that the allele frequencies at the ith locus are p (for the allele Under the assumption that the loci are independently segregating in the population, the joint distribution of (X, Y) can be evaluated by the bivariate probability generating function method (Feller, 1950); i.e., the probability that X = r and Y= k is given by Prob(X=r, Y=k) where = Coefficient of ss in G(s1, s2), G(s1, s2) = H (q+2pq1s1s2+ps), (2) for r = 0, 1,2, . .…”

Section: Distribution Of Heterozygosity By Phenotype Classesmentioning

confidence: 99%

Biochemical heterozygosity and phenotypic variability of polygenic traits

Chakraborty¹

1987

Heredity

View full text Add to dashboard Cite

An Introduction to Probability Theory and Its Applications

Cited by 3,329 publications

References 0 publications

How T-cells use large deviations to recognize foreign antigens

How T-cells use large deviations to recognize foreign antigens

On the advantage of haplotype analysis in the presence of multiple disease susceptibility alleles

Biochemical heterozygosity and phenotypic variability of polygenic traits

Contact Info

Product

Resources

About