# What's the exact meaning of and how to derive the coefficient of the path from sire to offspring?

According to Wright's Coefficients of Inbreeding and Relationship,

the coefficient of the path from sire to offspring is given as

$p_{o \cdot s}=\frac{1}{2}\frac{\sqrt{1+F_s}}{\sqrt{1+F_o}}$

But how to get this formula?

And what is the exact　meaning of this path? It's the path from sire's genetype to offspring's corresponding genetype, or just from one allele in sire's locus to offspring's one allele in the corresponding?

Furthermore, Wright gives $F_o=\frac{1}{2}r_{sd}\sqrt{1+F_s}\sqrt{1+F_d}$

From Wright's another paper Correlation and Causation we know $d_{o \cdot s}+d_{o \cdot d}+d_{o \cdot \bar{sd}}=1$

in which

$d_{o \cdot s}=p_{o \cdot s}^2$,

$d_{o \cdot d}=p_{o \cdot d}^2$,

$d_{o \cdot \bar{sd}}=2p_{o \cdot s}p_{o \cdot d}r_{sd}$

Thus

$\frac{1}{4}(1+F_s)+\frac{1}{4}(1+F_d)+\frac{1}{2}r_{sd}\sqrt{1+F_s}\sqrt{1+F_d}=1+F_o$

$\implies F_s+F_d=2$

Since $F_s,F_d\le 1$, we have $F_s=F_d=1$......

Incredible. But what's wrong with it?

• I think it must be forwarded to Math.StackExchange rather than stagnant here May 8 '16 at 8:04

In The Method of Path Coefficients(Wright1934) Wright showed that the path linked two objects is the path linked their genetic constructions. Probabily "genetic construction" means sequences of loci rather than a locus. However, if we consider a genotype of a locus as a random varible, the total genetic construction of a germ cell will be a random vector. Hence something like correlation or covariance between genetic constructions of germ cells is usually in form of matrix or tensor rather than a real number $$F$$(which is Wright actually showed). Moreover, if germ cells are made of several chromosomes, the correlation between corresponded pairs of loci in two germ cells may be not equal to each other. (e.g. chromosomeI of $$G_1$$ and $$G_2$$ are from the same ancestor, but chromosomeII are not)

So I guess things were worked out in this way.

Suppose $$G_1=(g_0,g_1,\dots,g_n)$$,$$G_2=(g'_0,g'_1,\dots,g'_n)$$. If $$g_i$$ is mth chromosome's kth locus and original from an ancestor germ cell $$G_A$$. And let us only modify $$G_A$$'s mth chromosome's kth locus, then $$g_i$$ would get modified; $$g'_i$$ would also get modified if only $$g'_i$$ is also original from $$G_A$$; loci on $$G_1$$ or $$G_2$$ in other chromosomes or in mth chromosome but other loci would not get modified anyway. Hence $$\rho(g_i,g_j)=\rho(g_i,g'_j)$$ for any $$j \neq i$$. Therefore $$cor(G_1,G_1)=cor(G_2,G_2)=\mathcal{I}\dots(1)$$ $$cor(G_1,G_2)=diag(\rho_{g_0,g'_0},\rho_{g_1,g'_1},\dots,\rho_{g_n,g'_n})\dots(2)$$

Since $$1=\sum\limits_{i}{p^2_{G_1\cdot g_i}}+\sum\limits_{i\neq j}p_{G_1\cdot g_i}\rho_{g_i,g_j}p_{G_1\cdot g_j}$$ and by $$(1)$$, $$1=\sum\limits_{i}{p^2_{G_1\cdot g_i}}$$. hence $$p_{G_1\cdot g_i}=\frac{1}{\sqrt{n}}$$ if we weighted every $$g_i$$ the same. And $$p_{g_i\cdot G_1}=\rho_{g_i, G_1}=\rho_{G_1, g_i}=p_{G_1\cdot g_i}=\frac{1}{\sqrt{n}}$$, Analogously to $$G_2$$. Therefore $$\rho_{G_2,G_1}=\sum\limits_{i,j}p_{g_i\cdot G_1}\rho_{g'_j, g_i}p_{G_2\cdot g'_j}=\sum\limits_{i}\frac{1}{n}\rho_{g'_i, g_i}=E(\rho_{g'_i, g_i})$$. i.e. in this case the correlation between germ cells are just the mean of correlations between their corresponding loci.

So similarly the coefficient of the path from a sire to its offspring is also the coefficient of the path linked their genetic constructions. and in detail is from sire's genetic construction to its components, then from components to offspring's construction's components, then to offspring's construction.

The derivation was given in the article. In fig6 the coefficient of the path from a sire to its offspring can be gotten from the product $$p_{o \cdot G_s}p_{G_s \cdot s}$$.

$$p_{G_s \cdot s}=\sqrt{\frac{1+F_s}{2}}$$; $$p_{o \cdot G_s}=\sqrt{\frac{1}{2(1+F_o)}}$$. So $$p_{o \cdot s}=\frac{1}{2}\frac{\sqrt{1+F_s}}{\sqrt{1+F_o}}$$

If $$F_s=F_d=F_o=r_{sd}=0$$, then $$d_{o \cdot s}=p^2_{o\cdot s}=d_{o\cdot d}=\frac{1}{4}$$, $$d_{o \cdot \bar{sd}}=0$$ . Thus $$d_{o \cdot s}+d_{o \cdot d}+d_{o \cdot \bar{sd}}=\frac{1}{2}<1$$. This result means parents' genetic constructions are not sufficient to determine their offsprings'. But it is very normal the same parents have both sons and daughters. Which causes their difference in gender are random factors.