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Wikipedia gives the following formula to calculate a "path of coefficient of relationship" between an ancestor $A$ and an offspring $O$:

$$ ho_{AO} = 2^{-n} left( frac{1+f_A}{1+f_O} ight)^{1/2} = left( frac{1}{2} ight)^n sqrt { frac{1+f_A}{1+f_O}}$$

, where $f_A$ and $f_O$ are the coefficient of inbreeding of the ancestor and the offspring respectively.

**Question**

Where does the term $sqrt { frac{1+f_A}{1+f_O}}$ comes from? Please explain why this multiplicative term is $sqrt { frac{1+f_A}{1+f_O}}$ and not something different such as $frac{f_A f_O}{2}$ for example.

## Inbreeding coefficients and coalescence times

This paper describes the relationship between probabilities of identity by descent and the distribution of coalescence times. By using the relationship between coalescence times and identity probabilities, it is possible to extend existing results for inbreeding coefficients in regular systems of mating to find the distribution of coalescence times and the mean coalescence times. It is also possible to express Sewall Wright's FST as the ratio of average coalescence times of different pairs of genes. That simplifies the analysis of models of subdivided populations because the average coalescence time can be found by computing separately the time it takes for two genes to enter a single subpopulation and time it takes for two genes in the same subpopulation to coalesce. The first time depends only on the migration matrix and the second time depends only on the total number of individuals in the population. This approach is used to find FST in the finite island model and in one- and two-dimensional stepping-stone models. It is also used to find the rate of approach of FST to its equilibrium value. These results are discussed in terms of different measures of genetic distance. It is proposed that, for the purposes of describing the amount of gene flow among local populations, the effective migration rate between pairs of local populations, M, which is the migration rate that would be estimated for those two populations if they were actually in an island model, provides a simple and useful measure of genetic similarity that can be defined for either allozyme or DNA sequence data.

## Chapter 6.10: Inbreeding coefficient and relationship

An animal is only inbred if its parents are related. The inbreeding level indicates the probability that an animal receives the same allele from both parents because they are related. In other words: it indicates the probability that an animal becomes homozygous for an allele that both parents share because they have a common ancestor. The inbreeding level of an individual animal is also called the inbreeding coefficient of that animal and can be calculated as:

F_{animal} = ½ * a_{between parents}

This simple formula indicates that it is easy to calculate the inbreeding coefficient of all animals in a population, as long as you know the additive genetic relationship between their parents. For example, the additive genetic relationship between a full brother and sister is 0.5. If they would be mated and have offspring, those offspring will be inbred. Their inbreeding coefficient would be ½ * 0.5 = 0.25. It means that for each locus the offspring will have a probability of 25% to be homozygous because its parents received the same alleles from their common ancestor. The more generations ago this common ancestor lived, the less the parents are related, so the lower the inbreeding coefficient.

*Important: An animal is inbred if, and only if, its parents are related!*

**INTERMEZZO: Why is F _{animal} = ½ * a_{between parents}?**

The inbreeding coefficient of an animal indicates the probability that the animal becomes homozygous because it inherits the same allele from both father and mother. For the animal to become homozygous, both parents need to have the same allele in the first place (= a_{between parents}). And then both parents need to pass it on to their offspring. That would result in

This would be correct in haploid organisms. However, animals are diploid: they each have two alleles per locus. So parents have two chances of sharing an allele. Therefore, the probability that their offspring becomes homozygous, expressed as the inbreeding coefficient, becomes

F_{animal} = 2 * a_{between parents} * ½ * ½ = ½ * a_{between parents}

## Methods

### The model

It is assumed that the inbreeding level of an individual is characterised by a single *f* value determined by its pedigree. All loci are assumed to be equally affected by inbreeding. Note that estimates of *f* based on known pedigrees may slightly differ from the true value of *f*, as founders are assumed to have an *f* of zero. However, provided the pedigrees are correct and several generations (three or more) deep, they should enable good estimates of true *f*. The distribution of *f* has probability density function *p*(*f*).

Let *h*_{i} be the heterozygosity (0 for a homozygote and 1 for a heterozygote at marker locus *i*). The expected heterozygosity *E*(*h*_{i}) at locus *i* is *h*_{0,i}(1−*f*) for a given *f*, where *h*_{0,i} is the genetic diversity at locus *i*, or the expected heterozygosity in the absence of inbreeding.

It is assumed that any correlation among *h*_{i} at different loci across individuals only arises due to within-population variation in *f*. In other words, no correlation will be expected in individuals that share the same value of *f*. This is usually true for unlinked loci. Note that 96% of locus pairs are unlinked in the Coopworth sheep data set presented in this study, with only 0.12% of locus pairs located within 10 cM of each other.

The term *H* refers to standardised MLH, calculated as the proportion of typed loci at which an individual was heterozygous divided by the population mean heterozygosity of those typed loci. This standardisation was initially used by Coltman et al (1999) in a study of Soay sheep, where individuals had been typed at different subsets of the same loci. The standardisation ensures that all individuals are measured on an identical scale and has subsequently been employed elsewhere (Slate et al, 2000 Amos et al, 2001). Standardised MLH is usually highly correlated with the more traditional nonstandardised version:

where *h̄*_{i} is the mean heterozygosity of all individuals typed at locus *i* and summation is across all typed loci.

#### The predicted distribution of heterozygosity

*f* has distribution with mean *E*(*f*) and variance *σ* 2 (*f*). We first derive the moments of the distribution of multiple-locus heterozygosity as functions *E*(*f*) and *σ* 2 (*f*). Then we will use these expressions to derive the expected correlations between *f* and heterozygosity, and between heterozygosity and fitness.

From the relationship above, the expected heterozygosity at locus *i* is

and, by definition, *E*(*H*)=1.

Given that *h*_{i} can only take the values 0 or 1, *h*_{i} 2 =*h*_{i}, the variance in *h*_{i} is

The covariance in heterozygosity among loci *i* and *j* is

The variance in *H* is

Then, after some algebra, we obtain

This expression involves several terms that depend on intrinsic characteristics of the set of loci analysed (number of loci, genetic diversity and how it varies across loci). In order to increase clarity, we can consider a first-order approximation that makes the simplifying assumption that all loci have the same genetic diversity *h*_{0}. The expression for the variance in *H* then becomes

where *L* is the number of typed loci. It is apparent that, as the number of typed loci increases, the third term alone will be a good approximation of the variance in *H*, which will be roughly proportional to the variance in inbreeding.

#### The predicted correlation in heterozygosity between two loci

The correlation in heterozygosity between two loci is obtained using the above expressions for the covariance and variance in heterozygosity at loci *i* and *j*:

The standard error of the estimated *r*(*h*_{i}*, h*_{j}) is (Zar, 1996)

#### The predicted correlation between individual heterozygosity and inbreeding coefficient

The covariance between heterozygosity and *f* is

The first term of this expression reduces to

so that the covariance between heterozygosity and *f* can be written as

The covariance between *H* and *f* is

and the correlation between *H* and *f* is

#### The predicted correlation between H and fitness traits

We assume the Morton et al (1956) model for the relationship between inbreeding and fitness traits such that the trait (or the logarithm of the trait) declines as a linear function of *f.* Thus, *W*=*a*−*bf*, where *W* is the trait, *a* is a constant and *b* is the inbreeding load. *b* can be estimated by linear regression of *W* on *f*.

If we assume that all correlations between heterozygosity and the trait arise as a result of inbreeding, then

*r*(*W*, *f*) is estimated by the regression of the trait on *f*, while *r*(*H, f*) is defined above (equation (4)).

### Application of the model to a real data set

Coopworth sheep were developed by crossing the Romney and Border Leicester breeds in New Zealand in the 1950s. The breed society was formally recognised in 1968 and Coopworths are now the second most numerous breed in New Zealand. We investigated a population of Coopworth sheep that was founded from six farms in the 1970s and has been the subject of divergent selection for backfat depth since 1981 (Morris et al, 1997). Subsequently, five F1 (fat × lean) sires have been backcrossed to both the fat and lean lines as part of an experiment to map QTL for morphological traits (Campbell et al, 2003). A total of 590 progeny were typed at up to 138 approximately evenly spaced microsatellite loci, spanning all 26 autosomes. Every individual was measured for a number of morphological traits and various potential explanatory variables were also recorded (see below). Inbreeding coefficients were calculated using the routine PROC INBREED, implemented in SAS (SAS Institute, Cary, NC, USA). For every individual, 7–10 generations of ancestors were known, dating back to the foundation of the selection lines, enabling accurate calculation of *f*. It was assumed that all founder individuals had an *f* of zero and were unrelated. MLH at all 138 loci was calculated, and converted to standardised MLH (see Coltman et al (1999) and above). Hereafter, MLH refers to the standardised version (*H* in the above model). Note that progeny were not genotyped at loci for which the sire was homozygous. Thus the genotype data file was only 73% complete – equivalent to an average of 101 genotypes per individual. The mean number of typed loci per half-sib family ranged from 98 (sire 603) to 106 (sires 610 and 616). In subsequent analyses, we report the expected relationship between *f* and MLH assuming that 101 loci were typed.

These data were then used to address three questions:

What is the relationship between *f* and MLH?

Is heterozygosity correlated between loci?

Do either *f* or MLH explain phenotypic variation?

The relationship between the two measures of inbreeding and 10 morphological traits (empty body weight, hot carcass weight, spleen weight, liver weight, heart weight, backfat depth at the 12th rib, tibia length, carcass length, *longissimus dorsi* weight and testes weight) was investigated. All traits appeared normally distributed (spleen weight was log-transformed), so univariate general linear modelling was employed.

The following terms were initially included in all models: sire, sex, rearing rank (litter size), maternal selection line (fat or lean), slaughter order (the first animal to be slaughtered on a given day is assigned rank 1, the next is assigned rank 2, etc.) and date of birth. All terms were factorial except slaughter order and date of birth, which were fitted as covariates. Initially, models were constructed with all terms fitted as both main effects and first-order interactions. Statistical significance of each term was assessed by F ratios. A minimal model was constructed by dropping all terms that were not significant at *P*<0.05. The minimal model was then used as a baseline model, to which genetic terms (*f* or MLH) were added. Both genetic terms were initially fitted as main effects and as interactions with sire. Note that terms containing *f* and MLH were not fitted in the same model. A significant interaction term would indicate between-sire variation in the number of segregating partially deleterious recessive alleles. The nine traits that were measurable in both sexes were positively correlated with each other (all correlations *P*<0.001), so multivariate analysis of variance (MANOVA) was also employed. Statistical analyses were implemented in S-plus 6.0 (Insightful, Seattle, WA, USA).

### Application of the model to other data sets

In addition to making a comparison between predictions from the model and observations in an extensive QTL mapping data set, we also examined the likely relationship between *f* and MLH in a number of other wild and domestic populations. This analysis was restricted to populations for which the mean and variance of *f* had been estimated, and for which descriptions of microsatellite marker variability were available. The analysis may not be exhaustive, but it does include a number of the best-known vertebrate populations for which inbreeding depression has been reported.

## Relationship between Coefficient of Inbreeding and Parasite Burden in Endangered Gazelles

Departamento de Ecología Evolutiva, Museo Nacional de Ciencias Naturales (CSIC), C/José Gutierrez Abascal 2, 28006–Madrid, Spain

Departamento de Ecología Evolutiva, Museo Nacional de Ciencias Naturales (CSIC), C/José Gutierrez Abascal 2, 28006–Madrid, Spain

Departamento de Ecología Evolutiva, Museo Nacional de Ciencias Naturales (CSIC), C/José Gutierrez Abascal 2, 28006–Madrid, Spain

### Abstract

**Abstract:** We studied the effects of inbreeding depression on parasite infection in three species of endangered gazelles: *Gazella cuvieri*, *G. dama,* and *G. dorcas*. Coefficients of inbreeding were calculated for all individuals because complete genealogies were available. The levels of inbreeding differ both intra- and interspecifically. We collected samples of feces and determined nematode infection by counting nematode eggs in the samples. At the interspecific level, the species with the highest mean levels of inbreeding ( *G. cuvieri*) had the highest levels of gastrointestinal parasites. Analyses done at the intraspecific level revealed a positive relationship between individual coefficient of inbreeding and parasite infection in *G. cuvieri*, but not in the species with the intermediate and lowest levels of inbreeding. Our findings suggest that high levels of inbreeding may make individuals more susceptible to parasitism, even under favorable environmental conditions, so this factor should be taken into account by those managing endangered species.

### Abstract

**Resumen:** Estudiamos los efectos ocasionados por la consanguinidad sobre el grado de infestación parasitaria en tres especies de gacelas en peligro de extinción: *Gazella cuvieri*, *G. dama* y *G dorcas*. Los coeficientes de consanguinidad fueron calculados para todos los individuos debido a que se encontraban disponibles genealogías completas. Los niveles de consanguinidad difieren tanto intra como interespecíficamente. Muestras de heces fueron colectadas y la infección por nemátodos se determinó por conteo de huevos en las muestras. A nivel interespecífico, la especie que posee el nivel promedio de consanguinidad más elevado ( *G. cuvieri*) tuvo los niveles más altos de parásitos intestinales. Los análisis efectuados a nivel intraespecífico revelaron la existencia de una relación positiva entre el nivel de consangunidad individual y la infestación por parásitos en *G. cuvieri*, pero no en las especies con niveles de consanguinidad medio y bajo. Nuestros resultados muestran que niveles elevados de consanguinidad podrían dar origen a una mayor susceptibilidad a los parásitos en los individuos, incluso bajo condiciones ambientales favorables, por lo que este factor deberá ser tomado en cuenta para el manejo de especies en peligro de extinción.

## Inbreeding Coefficient and Coefficient of Relationship - Biology

The second big change is that IC now has its own grid for displaying results as shown below: The grid displays:

- the minimum and maximum generations at which an ancestor appears
- the number of times it occurs
- the COI (Coefficient of Inbreeding)
- COR (Coefficient of Relationship).
- Percentage of Blood
- Line Breeding

Toolbar buttons provide the ability to sort on any of the fields making it easy to see which ancestors have the greatest influence (COR) or occur the most times. There's also a copy button for copying the grid contents to other applications like Excel.

We also have a Breed Planner which provides two side-by-side tables as above and also allows printing.

We get a lot of queries about inbreeding coefficient and to a lesser extent the relationship coefficient. BreedMate calculates the Wrights Inbreeding Coefficient.

**Wright's Inbreeding Coefficient (IC) **Inbreeding can be defined by either of the following two statements:

1. The probability that both genes of a pair in an individual are identical by descent, ie homozygous 2. The probable proportion of an individual's loci containing genes that are identical by descent **Wright's Coefficient of Relationship (RC) **A measure of pedigree relationship. The probable proportion of one individual's genes that are identical by descent to genes of a second individual The correlation between the breeding values of tw

individuals due to pedigree relationship alone.

**Number of generations used in calculation**

BreedMate allows you to select the number of generations used in the calculation. The way a generation limited calculation is done is, if an ancestor appeared within N generations then it was also included if it appeared past N generations. For example, if you calculate COI for

8 generations and ancestor "A" appears at gen 7, 10, 11 then all those "appearences will be included" whereas in 4.6 only its appearence at 7 is included.

Note if you set the generations large enough the COI will be the same for both 4.6 and 5.1+.

**Links **There are many articles relating to inbreeding coefficient on the internet. Note we do not necessarily approve or recommend any of the links listed:

## Inbreeding Coefficient and Coefficient of Relationship - Biology

**Inbreeding** is breeding between close relatives, whether plant or animal. If practiced repeatedly, it can lead to exposure of recessive, deleterious traits. This generally leads to a decreased fitness of a population, which is.

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NetPets - Your Complete Internet Pet Connection For Dogs . The **inbreeding** **coefficient** is a function of the number and location of the .

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Wright developed what is called the **inbreeding** **coefficient**. . The **inbreeding** **coefficient** thus specifically refers to those genes that are .

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Playing COI: Using **inbreeding** Coefficients. By C.A. Sharp. First printed in Double Helix Network . Via a formula called Wright's **Coefficient** of **Inbreeding**. .

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Online Medical Dictionary and glossary with medical definitions . The **coefficient** of **inbreeding**, symbolized by the letter F, is the probability .

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**inbreeding** **coefficient** results in a change in the performance. for . matings therefore, an **inbreeding** **coefficient** . **inbreeding** **coefficient** of the common .

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MU Extension, University of Missouri . Usefulness of **coefficient** of relationship information. Measurement of the degree of **inbreeding** .

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The **coefficient** of **inbreeding** is commonly symbolized as F x (called, "F of x" . CALCULATION OF **COEFFICIENT** OF **INBREEDING** IN MORE COMPLEX PEDIGREES .

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The **inbreeding** **coefficient** represents the probability that an offspring will . However, this maximum **inbreeding** **coefficient** of one cannot be achieved in human .

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By Sally L. Northcutt, Extension Beef Cattle Breeding Specialist and David S. Buchanan, Professor of Animal . FX = **inbreeding** **coefficient** of individual X, .

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Simply put, Cedric's **inbreeding** **coefficient** is found by calculating ½ to the . The **coefficient** of **inbreeding** is a fraction which when multiplied by 100 gives .

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**COEFFICIENT** OF **INBREEDING**. COI – an important factor of dog breeding . It is difficult to say when **inbreeding** **coefficient** is too high, but quite often .

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The **inbreeding** **coefficient**, as used in linebreeding, is a measurement of the . By considering the **inbreeding** **coefficient**, breeder can determine whether or not .

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The **coefficient** of kinship is defined as the probability that . Coefficients of **inbreeding** and relationship. American Naturalist 56:330-338 ^ "Kin Selection" .

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This number, the **inbreeding** **coefficient**, estimates the percentage . **inbreeding** **coefficient** does not identify whether the genes matching up are .

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**inbreeding** **coefficient** ( ′in′brēdiŋ ′kōə′fishənt ) ( genetics ) A measure of the rate of **inbreeding** or the degree to which an individual is inbred

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So Rover's **inbreeding** **coefficient** is 12.5%, based on the information we have. If we knew the **inbreeding** **coefficient** of Fido, then we would multiply .125 by (1 .

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Our pedigrees display the **Inbreeding** **Coefficient** for each dog in the first 4 . Each **Inbreeding** **Coefficient** is calculated from that dog's 10 generation pedigree. .

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**Inbreeding** **Coefficient**. Breeder's Guild. Embryo Transfer. Horse Health. Directories . To retrieve a lost User ID and/or Password, enter your email address. .

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Topics on dog breeding, with special reference to the Seppala Siberian . I wonder what the **inbreeding** **coefficient** looks like for contemporary Anadyrs! .

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## The distribution of individual inbreeding coefficients and pairwise relatedness in a population of mimulus guttatus

In order to infer population structure at the individual level, we estimated individual inbreeding coefficients and examined the relationship between geographical distance and genetic relatedness from polymorphic microsatellite data for a population of Mimulus guttatus that has an intermediate selfing rate. Expected heterozygosities for five microsatellites ranged from 0.79 to 0.93. The population inbreeding coefficient was calculated to be 0.19 (SE=0.023). A method-of-moments estimator developed by Ritland (1996b) was used to estimate the distribution of inbreeding among and relatedness between individuals of a natural population. The mean individual inbreeding coefficient (F=0.16) did not differ significantly from the population-level estimate. Most of the individuals appeared to be outbred, and there were very few plants that had estimated inbreeding coefficients greater than one-half. Individuals sampled from one transect showed significantly more inbreeding than individuals sampled along the other (P=0.005). There was no apparent relationship between interplant distance (range: 0-14 m) and mean genetic relatedness between individuals. These results represent the first application of polymorphic microsatellites to estimate fine-scale genetic population structure.

## What is the coefficient of relatedness between siblings?

Likewise, what is the coefficient of relatedness for the offspring of an individual? The **coefficient of relatedness** (r) is the fraction of alleles that two **individual** have in common. If two **individuals** are related, they **offspring** will be inbred. Ex.: **Relatedness** between parent & **offspring** is 1/2: the child inherits one-half its alleles from each parent.

Also know, what is the coefficient of relatedness between you and your first cousin?

It follows that **your relatedness coefficient with your** level-n **cousin** is equal to 1/2 2n + 1 . So, **your relatedness coefficient with your first cousin** is 1/8 **with your** second **cousin** is 1/32 **with your** third **cousin** is 1/128 and so on.

What does an inbreeding coefficient of 0.25 signify?

Chapter 6.10: **Inbreeding coefficient** and relationship. Their **inbreeding coefficient would** be ½ * 0.5 = **0.25**. It means that for each locus the offspring will have a probability of 25% to be homozygous because its parents received the same alleles from their common ancestor.

## Inbreeding Coefficient and Coefficient of Relationship - Biology

F measures the probability that two genes at any locus in an individual are identical by descent from the common ancestor(s) of the two parents. This means the degree to which two alleles are more likely to be homozygous (AA or aa) rather than heterozygous (Aa) in an individual, because the parents are related. Like R,F is a relative measure, in that there will be a certain level of homozygosity within the base population F simply estimates the increase from that initial level as a result of recent inbreeding.

The inbreeding coefficient of an individual is approximately half the relationship (R) between the two parents.This equivalence only applies to low levels of inbreeding in an otherwise outbred population.e.g.Two single first cousins normally have a relationship (R) of 1/8.If there has been no previous inbreeding, their children will have a coefficient of inbreeding of 1/16.With high levels of continuous inbreeding this relationship breaks down. e.g.Some strains of laboratory rats and mice have reached an F value of 1.0, resulting from a long history of close inbreeding but the coefficient of relationship (R) between any two members of the strain can never exceed 1.0. The mathematical reason for this is that although the basic formulae for R and F are Σ(1/2) n and Σ(1/2) n+1 respectively, as inbreeding within a line progresses, the correction terms applied to R for inbreeding (see here) gradually become more important and start to reduce the value of R below Σ(1/2) n . As F approaches 1.0, the correction terms for R also approach a maximum of (x 1/2) so that when F reaches 1.0 (complete homozygosity), R also becomes 1.0 and all members of the inbred line are identical.

The method of calculating the F coefficient of an individual is similar to that for the coefficient of relationship (R) between two collateral relatives, and involves the tracing of paths between the two parents via a common ancestor.The formula is as follows:-

Equation 1 method of calculating the F coefficient of an individual

Where F_{X} is the coefficient of inbreeding of individual X, n is the number of connecting links between the two parents of X through common ancestors and F_{A} is the coefficient of inbreeding of the common ancestor A.Thus, if the common ancestor is inbred, a minor calculation must be performed first to determine F_{A}, before the main calculation can take place.In the main calculation any coefficients of paths through inbred common ancestors can then be multiplied by (1 + F_{A}).

The only inbred common ancestor of the two parents (O and P) is I (his parents are single first cousins).

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