Introduction

As the number of markers used by the forensic community increases, considerations of linked markers become increasingly important. Weight of evidence calculations for these markers are not affected by this if a direct match between a trace and a suspect is considered and the markers are in linkage equilibrium, but in kinship analysis, the recombination probabilities between adjacent markers generally need to be taken into account. This means that likelihood ratios cannot be obtained as the product of likelihood ratios per locus anymore, and as a result, the computations are far more complex than they are for independent markers. Software for such calculations will be more complex as well, and it is naturally of importance that such software is thoroughly validated.

In a previous paper [1], the first of a series of which the second paper [2] dealt with mixtures rather than with kinship, we derived general properties of likelihood ratios for pairwise kinship calculations on a single marker. These results of course immediately extend to independent markers. In that case any pairwise non-inbred relationship can be summarized by IBD coefficients (κ 0, κ 1, κ 2) and we showed that the expected value of any likelihood ratio testing for a specified pairwise kinship, is independent of the allele frequencies when the actual relationship is the one investigated by the LR. We also gave explicit expressions for the variance, which does depend on the allele frequencies and is minimal when these are all equal.

In this paper, we extend the results of [1] in several directions. First, we rederive the results of [1] in a simpler and general way, allowing also to obtain expected values and variances in the case where the true relationship is different from the one investigated by the LR. We then generalize these results to an arbitrary number of linked markers, supposed to be in linkage equilibrium. We show that the expected values of the LR is still independent of the allele frequencies, and depends only on the recombination probabilities and the number of alleles carried by each locus. A pairwise relationship is now characterized by joint IBD probabilities on all considered loci, instead of by a single vector of IBD coefficients, and this allows to discriminate between pedigrees that are indistinguishable on a single locus.

We also generalize the expression of the variance in [1] to this general case. Finally we show that for the recombination rate ρ ∈ [0,0.5], the expected values of the likelihood ratios are non-increasing monotonously with increasing ρ.

These results are not always immediately relevant to casework in a given particular case, as the LR obtained in a case summarizes the evidence obtained and no further information can be extracted regarding the distinction between the investigated hypotheses. They rather apply for predicting the average evidence to be expected over all cases. Also, the exact expressions that we provide can be used for validation of software. By considering all possible genotypic combinations, it may be verified that the software produces the correct answers since these can be computed precisely using the expressions obtained in this paper. This was done for the Familias 3 software, cf. [3, 4]. The results in the present paper allow an extension of that validation towards linked markers, and towards the situation where the actual relatedness between the individuals is not the one investigated by the likelihood ratio. Since the resulting LR distributions are typically very skewed and have a very large variance, taking a random sample of genotypes and computing the LR’s on these is generally insufficient.

Theory

Review of previous results

On a single locus, non-inbred pairwise relationships can be described by the parameters κ = (κ 0, κ 1, κ 2) denoting the probability that two individuals share 0, 1 or 2 alleles identical by descent (IBD), respectively. For illustration, note that a paternity case corresponds to κ = (0,1,0) while for siblings κ = (0.25,0.5,0.25). The general inbred case would require nine parameters as described in [4, 5] and is not considered in this paper.

Let LR be the likelihood ratio comparing H P with H D where H P states relatedness according to (κ 0, κ 1, κ 2) and H D states unrelatedness (or equivalently, relatedness according to (κ 0, κ 1, κ 2)=(1,0,0)). In order to view the LR as a random variable it is useful to think about the hypotheses themselves as random variables that take values in the set of possible evidence E. In this case, the evidence consists of a pair of DNA profiles (g 1, g 2). We denote these random variables by \(\mathcal {H}_{P}\) and \(\mathcal {H}_{D}\). Thus, \(P(\mathcal {H}_{P}=(g_{1}, g_{2}))=P((g_{1}, g_{2})\mid H_{P})\) and \(P(\mathcal {H}_{D}=(g_{1}, g_{2}))=P((g_{1}, g_{2})\mid H_{D})\). The likelihood ratio of (g 1, g 2) ∈ E is then by definition

$$ LR_{\mathcal{H}_{P},\mathcal{H}_{D}}(g_{1}, g_{2})=\frac{P(\mathcal{H}_{P}=(g_{1}, g_{2}))}{P(\mathcal{H}_{D}=(g_{1}, g_{2}))}. $$
(2.1)

Note that we can now view an obtained likelihood ratio as an outcome of the LR as a random variable since

$$LR_{\mathcal{H}_{P},H_{\mathcal{D}}}((g_{1}, g_{2}))=P((g_{1}, g_{2}) \mid H_{P})/P((g_{1}, g_{2}) \mid H_{D}). $$

By composition, it now makes sense to write \(LR_{\mathcal {H}_{P},\mathcal {H}_{D}}(\mathcal {H}_{P})\) and we abbreviate this by \(LR(\mathcal {H}_{P})\). The expected value of \(LR(\mathcal {H}_{P})\), derived in [1], can be written

$$\begin{array}{@{}rcl@{}} E[LR(\mathcal{H}_{P})]&=&\alpha L^{2}+\beta L+(1-\alpha-\beta) \text{ where}\\ &&\alpha=\frac{{\kappa_{2}^{2}}}{2} ,\beta=\frac{{\kappa_{1}^{2}}+4\kappa_{1}\kappa_{2}+2{\kappa_{2}^{2}}}{4}\\ &&\text{ and } L=\text{length of allelic ladder.} \end{array} $$
(2.2)

The example on page 417 of [1] explains some of the fundamental mathematical concepts in a tutorial example. A SNP marker is considered, and so L = 2. For a parent–child relationship, corresponding to κ 0 = 0, κ 1 = 1 and κ 2 = 0, the above formula then gives

$$E[LR(\mathcal{H}_{P})]=\frac{1}{4}L+\left( 1-\frac{1}{4}\right)=1.25 $$

as is verified in a specific numerical example in the mentioned tutorial example. Not all specifications of κ correspond to a possible pedigree, Thompson [6] explains why the following restriction

$$\begin{array}{@{}rcl@{}} {\kappa_{1}^{2}}\geq 4\kappa_{0}\kappa_{2} \end{array} $$

is needed in addition to the obvious κ 0 + κ 1 + κ 2 = 1.

In [1] we calculated the variance of \(LR(\mathcal {H}_{P})\) explicitly for a few specific cases, but no general formula was presented.

Generalization

Before we proceed to considering pairs of possibly linked markers, we recast the results of [1] in a setting that will be useful when considering several markers.

Expectation

Extending the framework of [1], suppose that the true relation between individuals is not necessarily the same as the one tested for in the likelihood ratio: the LR compares κ = (κ 0, κ 1, κ 2) to unrelated whereas the true relationship is given by \(\kappa ^{\prime }=(\kappa ^{\prime }_{0},\kappa ^{\prime }_{1},\kappa ^{\prime }_{2})\). Let \(\mathcal {H}_{\kappa }\) draw random genotypes for individuals related via κ, i.e., \(P(\mathcal {H}_{\kappa }=(g_{1},g_{2}))=P(g_{1},g_{2} \mid H_{\kappa })\) where H κ is the hypothesis that specifies that the persons with genotypes g 1 and g 2 are related with IBD coefficients κ. We note that

$$ P(\mathcal{H}_{\kappa}=(g_{1},g_{2}))=\sum\limits_{i=0}^{2} \kappa_{i} P(\mathcal{H}_{i}=(g_{1},g_{2})), $$
(2.3)

where \(\mathcal {H}_{i}\) corresponds to κ such that κ i = 1. Let L R κ be the LR that compares the hypotheses of being related according to κ, versus being unrelated, i.e.,

$$LR_{\kappa}(g_{1},g_{2})=\frac{P({\mathcal H}_{\kappa}=(g_{1},g_{2}))}{P({\mathcal H}_{0}=(g_{1},g_{2}))}.$$

Letting

$$LR_{0}=LR_{(1,0,0)}; LR_{1}=LR_{(0,1,0)}, LR_{2}=LR_{(0,0,1)},$$

we have

$$ LR_{\kappa}= \kappa_{0} LR_{0} + \kappa_{1}LR_{1}+\kappa_{2} LR_{2}. $$
(2.4)

Note that L R 0≡1, L R 1 is the PI (Paternity Index, testing for being parent–child versus unrelated), and L R 2 is the LR testing for identity (versus unrelated).

From (2.3) and (2.4) follows the decomposition

$$\begin{array}{@{}rcl@{}} E[LR_{\kappa}(\mathcal{H}_{\kappa^{\prime}})]=\sum\limits_{i,j=0}^{2} \kappa_{i}\kappa^{\prime}_{j} E[LR_{i}(\mathcal{H}_{j})], \end{array} $$
(2.5)

which we can also write as a matrix product

$$ E[LR_{\kappa}(\mathcal{H}_{\kappa^{\prime}})]=\kappa\cdot M \cdot (\kappa^{\prime})^{t}. $$

A direct argument (cf. [1]) gives

$$\begin{array}{@{}rcl@{}} M= \left( \begin{array}{lll} 1 & 1 & 1 \\ 1 & (L+3)/4 & (L+1)/2 \\ 1 & (L+1)/2& L(L+1)/2 \end{array}\right). \end{array} $$
(2.6)

Plugging these expressions into (2.5) we get

$$\begin{array}{@{}rcl@{}} E[LR_{\kappa}(\mathcal{H}_{\kappa^{\prime}})]&=&\kappa_{0}+(1-\kappa_{0})\kappa_{0}^{\prime}+\kappa_{1}\kappa^{\prime}_{1}\frac{L+3}{4}\\ &&+(\kappa^{\prime}_{1}\kappa_{2}+\kappa_{1}\kappa^{\prime}_{2})\frac{L+1}{2}\\ &&+\kappa_{2}\kappa^{\prime}_{2}\frac{L(L+1)}{2}. \end{array} $$
(2.7)

Note that

$$ E[LR_{\kappa}(\mathcal{H}_{\kappa^{\prime}})]=E[LR_{\kappa^{\prime}}(\mathcal{H}_{\kappa})] $$
(2.8)

is symmetric in κ and κ ′.

If κ = κ ′ this reduces to

$$\begin{array}{@{}rcl@{}} E[LR_{\kappa}(\mathcal{H}_{\kappa})]&=&\kappa_{0}(2-\kappa_{0})+{\kappa_{1}^{2}}\frac{L+3}{4}+2\kappa_{1}\kappa_{2}\frac{L+1}{2}\\ &&+{\kappa_{2}^{2}}\frac{L(L+1)}{2}, \end{array} $$
(2.9)

which is readily seen to be the same expression as (2.2).

Variance

We can use the same decomposition to facilitate the computation of the variance. We have

$$\begin{array}{@{}rcl@{}} E[LR_{\kappa}(\mathcal{H}_{\kappa})^{2}]&=&\sum\limits_{i,j}\kappa_{i}\kappa_{j}E[(LR_{i} \cdot LR_{j})(\mathcal{H}_{\kappa})]\\ &=&\sum\limits_{i,j,k}\kappa_{i}\kappa_{j}\kappa_{k} E[(LR_{i} \cdot LR_{j})(\mathcal{H}_{k})]. \end{array} $$
(2.10)

Note that

$$\begin{array}{@{}rcl@{}} E[(LR_{i} \cdot LR_{j})(\mathcal{H}_{k})]&=&E[(LR_{i} \cdot LR_{k})(\mathcal{H}_{j})]\\ &=&E[(LR_{j} \cdot LR_{i})(\mathcal{H}_{k})] \end{array} $$
(2.11)

is symmetric in i, j, k. Suppose that i≤j≤k; if i = 0 then \(E[(LR_{0} \cdot LR_{j})(\mathcal {H}_{k})]=E[LR_{j}(\mathcal {H}_{k})]\) which we have already determined with (2.6). The only combinations still to consider are

$$(i,j,k) \in \{(1,1,1), (1,1,2), (1,2,2), (2,2,2)\}. $$

In the previous paper [1] it was shown that

$$ E[LR_{1}(\mathcal{H}_{1})^{2}]=\frac{5L+3}{8}+\frac{1}{16}\sum\limits_{a<b}\frac{p_{a}}{p_{b}}+\frac{p_{b}}{p_{a}}, $$
(2.12)

and similar, direct arguments, give

$$ E[LR_{1}(\mathcal{H}_{2})^{2}]=\frac{L(L+7)}{8}+\frac{1}{8}\sum\limits_{a<b}\frac{p_{a}}{p_{b}}+\frac{p_{b}}{p_{a}}, $$
(2.13)
$$ E[(LR_{1}LR_{2})(\mathcal{H}_{2})]=\sum\limits_{a}\frac{1}{p_{a}}+\frac{1}{4}\sum\limits_{a<b}\frac{1}{p_{b}}+\frac{1}{p_{a}}, $$
(2.14)
$$ E[LR_{2}(\mathcal{H}_{2})^{2}]=\sum\limits_{a}\frac{1}{{p_{a}^{2}}}+\sum\limits_{a<b}\frac{1}{2p_{a}p_{b}}. $$
(2.15)

Using these identities, \(E[LR(\mathcal {H}_{P})^{2}]\) can be obtained for any choice of κ, and then the variance is obtained by subtracting \({\left (E[LR(\mathcal {H}_{P})]\right )}^{2}\). The representation (2.5) extends to

$$\begin{array}{@{}rcl@{}} E[LR_{\kappa}(\mathcal{H}_{\kappa^{\prime}})^{2}] =\sum\limits_{i,j,k}\kappa_{i}\kappa_{j}\kappa_{k}^{\prime} E[(LR_{i} \cdot LR_{j})(\mathcal{H}_{k})]. \end{array} $$
(2.16)

With this formula the variance of L R κ , when in reality the tested individuals are related according to κ ′, can be calculated.

Several linked markers

We next generalize (2.2): consider a pair of linked markers M 1, M 2 with L 1 and L 2 alleles separated by a recombination probability of ρ. Some notation is needed now, since κ does not suffice to describe a binary relationship on more than one locus. Consider a pair of individuals connected by a pedigree \(\mathcal {P}ed\). Let I 1 and I 2 be the random variables describing the number of IBD alleles between the individuals on the two loci, and let κ i = P(I 1 = i) = P(I 2 = i) be the IBD probabilities capturing the described relatedness by \(\mathcal {P}ed\) on a single locus. Let \(\kappa _{i_{1},i_{2}}(\rho )=P(I_{1}=i_{1}, I_{2}=i_{2} \mid \mathcal {P}ed)\) denote the probability that a pair of individuals related according to \(\mathcal {P}ed\) share i 1 and i 2 alleles IBD for the first and second marker respectively. Obviously \(\kappa _{i_{1},i_{2}}(\rho )\) is a function of the recombination rate ρ ∈ [0,0.5]. Note the extremes: \(\kappa _{i_{1},i_{2}}(0.5)=\kappa _{i_{1}}\kappa _{i_{2}}\) and \(\kappa _{i_{1},i_{2}}(0)=\delta _{i_{1},i_{2}}\kappa _{i_{1}}\) where \(\delta _{i_{1},i_{2}}=1\) if i 1 = i 2 and 0 otherwise.

Let \(\mathcal {H}_{j_{1},j_{2}}\) be the random variable that draws the genotypes of two individuals on both loci, such that there are j 1 pairs of IBD alleles on the first locus and j 2 on the second. Similarly let \(\mathcal {H}^{(i)}_{j}\) select two genotypes on locus i with j pairs of IBD alleles. That is

$$\begin{array}{@{}rcl@{}} P\Big(\mathcal{H}_{j_{1},j_{2}}\!&=&\!\left.\left.\left( \left( g_{1}^{(1)},g_{1}^{(2)}\right), \left( g_{2}^{(1)},g_{2}^{(2)}\right)\right)\right.\right)\\ \!&=&\! P\left( \mathcal{H}_{j_{1}}^{(1)}\,=\,\left( g_{1}^{(1)},g_{2}^{(1)}\right)\right) P\left( \mathcal{H}_{j_{2}}^{(2)}\,=\,\left( g_{1}^{(2)},g_{2}^{(2)}\right)\right), \end{array} $$

where \(g_{i}=\left (g_{i}^{(1)},g_{i}^{(2)}\right )\) is the genotype of person i on the two loci. Note that in the above formula, the assumption of linkage equilibrium is used. If the loci would not be in linkage equilibrium, we would need conditional probabilities for the genotypes on one of the loci, given the genotypes on the other locus.

The assumption of linkage equilibrium means that the likelihood ratio factorizes, since

$$\begin{array}{@{}rcl@{}} LR_{i_{1},i_{2}}(g_{1},g_{2}) &:=& \frac{P(\mathcal{H}_{i_{1},i_{2}}=(g_{1},g_{2}))}{P\,(\mathcal{H}_{0,0}\;=(g_{1},g_{2}))} \\ &=& \frac{P\!\left( \mathcal{H}^{(1)}_{i_{1}}\,=\,\left( g_{1}^{(1)},g_{2}^{(1)}\right)\!\right)\!P\!\left( \mathcal{H}^{(2)}_{i_{2}}\,=\,\left( g_{1}^{(2)},g_{2}^{(2)}\right)\!\right)}{P\!\left( \mathcal{H}^{(1)}_{0}\,=\,\left( g_{1}^{(1)},g_{2}^{(1)}\right)\!\right)\!P\!\left( \mathcal{H}^{(2)}_{0}\,=\,\left( g_{1}^{(2)},g_{2}^{(2)}\right)\!\right)} \\&=& LR^{(1)}_{i_{1}}\left( g_{1}^{(1)},g_{2}^{(1)}\right)LR^{(2)}_{i_{2}}\left( g_{1}^{(2)},g_{2}^{(2)}\right). \end{array} $$

Therefore

$$ E[LR_{i_{1},i_{2}}(\mathcal{H}_{j_{1},j_{2}})]\,=\,E\left[LR^{(1)}_{i_{1}}\left( \mathcal{H}^{(1)}_{j_{1}}\right)\right]E\left[LR^{(2)}_{i_{2}}\left( \mathcal{H}^{(2)}_{j_{2}}\right)\right] $$
(2.17)

where the expectations on the right hand side are defined by (2.7). The general formula for linked markers can be written

$$\begin{array}{@{}rcl@{}} E[LR_{\mathcal{P}ed}(\mathcal{H}_{\mathcal{P}ed^{\prime}})]&&=\sum\limits_{j_{1},j_{2},i_{1},i_{2}=0}^{2} \kappa_{i_{1},i_{2}}(\rho)\kappa^{\prime}_{j_{1},j_{2}}(\rho)\\ &&\times E\left[LR^{(1)}_{i_{1}}(\mathcal{H}_{j_{1}})\right]E\left[LR^{(2)}_{i_{2}}(\mathcal{H}_{j_{2}})\right],\\ \end{array} $$
(2.18)

where the \(\kappa _{i_{1},i_{2}}(\rho )\), resp. \(\kappa ^{\prime }_{j_{1},j_{2}}(\rho )\) are derived from the pedigrees \(\mathcal {P}ed\), resp. \(\mathcal {P}ed^{\prime }\).

The above formula readily extends to more than two markers. Indeed, considering n markers let \(\vec {\rho }=(\rho _{1},\dots ,\rho _{n-1})\) where ρ i is the recombination probability between loci i and i+1, and let \(\kappa _{i_{1},\dots ,i_{n}}(\vec {\rho })=P(I_{1}=i_{1},\dots ,I_{n}=i_{n} \mid \mathcal {P}ed)\). Then

$$\begin{array}{@{}rcl@{}} E[LR_{\mathcal{P}ed}(\mathcal{H}_{\mathcal{P}ed^{\prime}})]&=&\sum \kappa_{i_{1},\dots,i_{n}}(\rho)\kappa^{\prime}_{j_{1},\dots,j_{n}}(\rho) \\ &&\times E [LR_{i_{1},\dots,i_{n}}(\mathcal{H}_{j_{1},\dots,j_{n}})]\\ &=& \sum \kappa_{i_{1},\dots,i_{n}}(\rho)\kappa^{\prime}_{j_{1},\dots,j_{n}}(\rho) \\ &&\times {\prod}_{k=1}^{n}E\left[LR^{(k)}_{i_{k}}(\mathcal{H}_{j_{k}})\right]. \end{array} $$
(2.19)

This formula generalizes (2.5). Thus, for any number of loci, the expected value of the likelihood ratio \(LR_{\mathcal {P}ed}\), realized on members of a possibly different pedigree \(\mathcal {P}ed^{\prime }\) does not depend on the allele frequencies but only on the lengths of the allelic ladders of the loci and the recombination probabilities.

It also follows that if pedigrees are indistinguishable for single loci because the IBD coefficients are the same on single loci (as is the case for half-siblings, grandparent–grandchild and uncle–nephew) then the expected LRs involving these relationships are the same for fully linked loci, as well as for independent loci. For recombination rates between 0 and 0.5 this will not be the case any more.

The variance of \(LR_{\mathcal {P}ed}(\mathcal {H}_{\mathcal {P}ed^{\prime }})\) can be computed similarly to (2.10). Suppose pedigree \(\mathcal {P}ed\) corresponds to IBD probabilities \(\kappa _{I}(\rho )=\kappa _{i_{1},\dots ,i_{n}}(\rho )\) and pedigree \(\mathcal {P}ed^{\prime }\) to IBD probabilities \(\kappa ^{\prime }_{J}(\rho )\). Then we have

$$\begin{array}{@{}rcl@{}} E[LR_{\mathcal{P}ed} (\mathcal{H}_{\mathcal{P}ed^{\prime}})^{2}]\!&=&\!\sum\limits_{I,J,L}\!\kappa_{I}(\rho)\kappa_{J}(\rho)\kappa^{\prime}_{L}(\rho)\\ &&\!\times E[(LR_{I}\cdot LR_{J})(\mathcal{H}_{L})]\\\!&=&\! \sum\limits_{I,J,L}\kappa_{I}(\rho)\kappa_{J}(\rho)\kappa^{\prime}_{L}(\rho)\\ && \!\!\!\times\!\! \prod\limits_{k=1}^{n}\!E\! \left[\!\left( \!LR^{(k)}_{i_{k}}\!\!\cdot\! LR^{(k)}_{j_{k}}\!\right)\!\left( \!\mathcal{H}^{(k)}_{l_{k}}\!\right)\!\right]. \end{array} $$
(2.20)

The above expression allows to calculate the variance explicitly as a function of the allele frequencies and the recombination probabilities, but the expressions of course quickly become cumbersome. Table 1 contains expected values and standard deviations for a particular choice of markers and some possible relationships.

Table 1 Expectation and standard deviation for LR investigating various relationships for a pair of markers with 2 and 3 and alleles separated by ρ = 0,0.25,0.5
Full size table

In Appendix A we prove that

$$\begin{array}{@{}rcl@{}} f(\rho)=E[LR(\mathcal{H}_{P})] \text{ decreases with increasing } \rho, \end{array} $$
(2.21)

for a pair of linked markers.

The last three equations, i.e., (2.19)–(2.21) summarize our main findings in the most general framework. The first shows that the expectation of the LR only depends on allele frequencies via the lengths of the allelic ladders. It can also be used to derive exact expressions. Generally, expected values should be accompanied by variances and for this (2.20) can be used. Allele frequencies now matter as products of LR’s are involved. The variance is minimal for equal allele frequencies. The last result, (2.21) is explored further towards the end of “Completely unlinked and linked markers”. It shows that more strongly linked markers provide more information on average. However, this finding should be interpreted cautiously as will be pointed out in the discussion. In particular (2.21) does not tell us how ρ influences the exceedance probabilities \(P(LR_{\mathcal {P}ed}(\mathcal {H}_{\mathcal {P}ed}) \geq t)\) for fixed t, a quantity which in casework may be of greater practical importance than the expected value of the likelihood ratio.

Completely unlinked and linked markers

If the markers are unlinked corresponding to ρ = 0.5, or completely linked (ρ = 0) the expected value of LR simplifies for general pairwise relationships. In the former case, the IBD matrix is

$$\begin{array}{@{}rcl@{}} \left( \begin{array}{lll} \kappa_{0} \kappa_{0}&\kappa_{0} \kappa_{1} & \kappa_{0} \kappa_{2}\\ \kappa_{1} \kappa_{0}&\kappa_{1} \kappa_{1} & \kappa_{1} \kappa_{2}\\ \kappa_{2} \kappa_{0}&\kappa_{2} \kappa_{1} & \kappa_{2} \kappa_{2}\\ \end{array}\right) \end{array} $$

and from (2.9)

$$\begin{array}{@{}rcl@{}} E[LR_{0.5}(\mathcal{H}_{P})]&= &\left( \kappa_{0}(2-\kappa_{0})+ {\kappa_{1}^{2}}\tfrac{L_{1}+3}{4}+2\kappa_{1}\kappa_{2}\tfrac{L_{1}+1}{2}\right.\\ &&\quad\left.+{\kappa_{2}^{2}}\tfrac{L_{1}(L_{1}+1)}{2} \right) \\ && \times \left( \kappa_{0}(2-\kappa_{0})+{\kappa_{1}^{2}}\tfrac{L_{2}+3}{4}+2\kappa_{1}\kappa_{2}\tfrac{L_{2}+1}{2}\right.\\ &&\qquad\left.+{\kappa_{2}^{2}}\tfrac{L_{2}(L_{2}+1)}{2} \right). \end{array} $$
(2.22)

For completely linked markers, the IBD matrix is

$$\begin{array}{@{}rcl@{}} \left( \begin{array}{lll} \kappa_{0} &0 & 0\\ 0 &\kappa_{1} & 0\\ 0 &0 & \kappa_{2}\\ \end{array}\right) \end{array} $$

and

$$\begin{array}{@{}rcl@{}} E[LR_{0}(\mathcal{H}_{P})]&=&\kappa_{0}(2-\kappa_{0})+ {\kappa_{1}^{2}} \tfrac{L_{1}+3}{4} \tfrac{L_{2}+3}{4}\\ &&+2\kappa_{1}\kappa_{2} \tfrac{L_{1}+1}{2}\tfrac{L_{2}+1}{2}\\ &&+{\kappa_{2}^{2}} \tfrac{L_{1}(L_{1}+1)}{2}\tfrac{L_{2}(L_{2}+1)}{2}. \end{array} $$
(2.23)

In view of the result summarized in (2.21), the expected likelihood ratio for a pair of markers is bounded from below by (2.22) and from above by (2.23).

Likelihood ratio and weight of evidence

We have seen that the expected value of a likelihood ratio \(LR_{\kappa }(\mathcal {H}_{\kappa ^{\prime }})\) does not depend on the allele frequencies. This finding has some wide ranging implications and these are explored further in this section as well as in the discussion.

A very rare allele causes a small number of LR’s to be very large, giving rise to a large variance. As the frequency of that alleles becomes smaller, the variance increases while the expectation remains the same. This means that, if we reduce the allele frequency f of an allele until it vanishes completely, the expected value has a discontinuity at f = 0. However, the variance of the LR will go to infinity as f goes to zero. From a practical point of view this means that we cannot determine the expected value of the LR for a certain population, since unless we have genotyped the whole population, we cannot know with certainty all the alleles on the loci under consideration. The purpose of this section is to argue that the logarithm of the LR is better behaved in this respect, and to connect this quantity to information theory.

First we note that since \(P(LR_{\kappa }(\mathcal {H}_{\kappa })>0)=1\), \(E[\log (LR_{\kappa }(\mathcal {H}_{\kappa }))]\) is well defined. For most of this discussion the choice of base for the logarithm is immaterial. To fix notation, let log denote the natural logarithm and log10 the base-10 logarithm. Contrary to \(E[LR_{\kappa }(\mathcal {H}_{\kappa })]\), the expected value of \(\log (LR_{\kappa }(\mathcal {H}_{\kappa }))\) will be a function of all allele frequencies. The log10(L R) is sometimes called the weight of evidence [8], measuring the amount of information in the evidence, with the ban as unit. We will argue now that indeed log(L R) is related to the information theoretic concept of entropy, the entropy of a probability distribution with outcome probabilities p a being defined as \(-\sum p_{a} \log (p_{a})\).

For instance, if we set κ = (0,0,1) then we reduce kinship to genetic identity, and \(E[LR(\mathcal {H}_{P})]\) is equal to L(L+1)/2 (cf. (2.2)), which is the number of genotypes on a locus with L alleles. On the other hand \(E[\log (LR(\mathcal {H}_{P})]=-\log (2)(1-{\sum }_{a}{p_{a}^{2}})-2{\sum }_{a}p_{a}\log (p_{a})\), so the expected weight of evidence is given by a sum involving the expected heterozygosity on the locus and its entropy. Similarly, for the parent–child likelihood ratio PI we have, after some rewriting,

$$\begin{array}{@{}rcl@{}} E[\log(PI(\mathcal{H}_{P}))] &=& -2\log(2)\left( 1-\sum\limits_{a}{p_{a}^{2}}\right)\\ &&+\sum\limits_{a} p_{a}({p_{a}^{2}}-p_{a}-1)\log(p_{a}) \\ &&+ \sum\limits_{a < b} p_{a}p_{b}(p_{a}+p_{b})\log(p_{a}+p_{b}). \end{array} $$

We see that this expression again is a sum containing a term corresponding to the expected heterozygosity \(1-{\sum }_{a}{p_{a}^{2}}\), and terms similar to the entropy \(-{\sum }_{a} p_{a}\log (p_{a})\). If, in this expression, one of the allele frequencies goes to zero, we simply get the expression corresponding to one allele less on the ladder. In other words, \(E[\log (PI(\mathcal {H}_{P}))]\) is continuous in the allele frequencies, whereas \(E[PI(\mathcal {H}_{P})]\) is not. Essentially this is because \(\lim _{x\to 0}x\log (x)=0\) but \(\lim _{x\to 0}\frac {x}{x}=1\). Thus, an allele that becomes rarer still contributes the same to the expected LR, but less to the expected log(L R).

Similarly, the variance is well defined and continuous on the log scale. This means that contrary to \(E[LR(\mathcal {H}_{P})]\), we can estimate \(E[\log (LR(\mathcal {H}_{P}))]\) from a sample of population alleles: the presence of rare alleles does not influence \(E[\log (LR(\mathcal {H}_{P}))]\) very much. Also, \(E[\log (LR(\mathcal {H}_{D}))]<0\), but note that it is −∞ whenever there are situations possible under H D where H P can be proven to be impossible. For non-inbred pairwise kinship comparisons this will only affect parent–child comparisons and direct matching (monozygous twins).

In our opinion this phenomenon connects to the interpretation of log(L R) as a measure of information. For example, intuitively adding an independent new set of loci with allele frequencies identical to ones already investigated would correspond to doubling the amount of genetic information that we expect to find, and of course the expected log(L R) will be twice as large as for the original set, whereas the expected LR itself is squared. Information content is measured in bans (also called hartleys), where by definition one ban corresponds to the information content carried by a random variable with ten possible and equally likely outcomes. A ban is therefore equal to log2(10)≈3.322 bits of information.

The entropy, i.e., information carried by a set of N equally likely outcomes is log10(N) bans. Just as N is the number of potential outcomes, we could speculatively look at \(E[LR(\mathcal {H}_{P})]\) as the potential evidence in favor of the true relationship, whereas \(E[\log (LR(\mathcal {H}_{P}))]\) is the average weight of evidence that is actually obtained. For example, suppose we consider \(E[\log (PI(\mathcal {H}_{P}))]\) on a locus with L alleles. Regardless of their frequencies, we have \(E[PI(\mathcal {H}_{P})]=(L+3)/4\), i.e., the expected PI for parents and children increases linearly with L, the number of alleles. For the expected log(P I), we assume that all alleles are equally frequent: all have population frequency 1/L. Then one can show, using the above formula, that

$$\lim\limits_{L\to\infty} E[\log(PI(\mathcal{H}_{P}))]/\log(L)=1. $$

In particular \(\lim _{L\to \infty } E[\log (PI(\mathcal {H}_{P}))]/\log (E[PI(\mathcal {H}_{P})])=1\) as well. Thus, the expected weight of evidence grows with the logarithm of the number of alleles, and not linearly with it. In Fig. 1 we plot \(E[\log _{10}(PI(\mathcal {H}_{P}))]\) as function of L.

Fig. 1
figure 1

Expected \(\log _{10}(PI(\mathcal {H}_{P}))\) on a locus with L equally frequent alleles

Full size image

To summarize, we can distinguish between LR, the (multiplicative) Bayes factor between the prior and posterior odds on the hypothesis, and the weight of evidence log(L R), the (additive) term between the prior and posterior log-odds. The LR and log(L R) are of course computationally equivalent in any particular case. However the corresponding random variable behaves differently in terms of expectation and variance. The expected LR seems to summarize the information potential, which is independent of the allele frequencies. The expected log(L R) measures the average amount of information obtained.

Approximation by a normal distribution

The reasoning below is relevant for power calculations. In many cases, \(\log (LR(\mathcal {H}_{P}))\) being the sum of independent random variables can be expected to be reasonably well approximated by a normal distribution (cf. [7]), especially around its expected value. Suppose that this normal distribution has expectation μ and variance σ 2. Then

$$\begin{array}{@{}rcl@{}} m&=&E[LR(\mathcal{H}_{P})]=\exp(\mu+\sigma^{2}/2), \\ v&=&Var [LR(\mathcal{H}_{P})]= (\exp(\sigma^{2})-1)\exp(2\mu+\sigma^{2}) \end{array} $$

can be calculated exactly and we can estimate

$$\begin{array}{@{}rcl@{}} \mu&=&\log \left( \frac{m}{\sqrt{1+\frac{v}{m^{2}}}}\right), \\ \sigma^{2}&=&\log\left( 1+\frac{v}{m^{2}}\right). \end{array} $$

If \(\log (LR(\mathcal {H}_{P})) \sim N(\mu , \sigma ^{2})\) then \(\log (LR(\mathcal {H}_{D})) \sim N(-\mu , \sigma ^{2})\), and moreover σ 2=2μ (cf. [8]). This means that, when \(\log (LR(\mathcal {H}_{P}))\) is normally distributed, then so is \(\log (LR(\mathcal {H}_{D}))\) and the average weight of evidence in favor of H P when H P is true is equal to the average weight of evidence against H P when H D is true; the variances are also the same and are determined by the mean. In the above notation note that v = m 2(m−1), and that m = exp(2μ) and v = (exp(2μ)−1)exp(4μ).

Examples

The first example demonstrates how the exact variance for one marker is obtained.

Example 3.1 (Variance for symmetric relationships)

For grandparent – grandchild, half-siblings and uncle–nephew, \(\kappa _{0}=\kappa _{1}=\frac {1}{2}\) and therefore these relationships are statistically equivalent for one marker and we refer to them as symmetric. From (2.2), \(E[LR(\mathcal {H}_{P})]=(L+15)/16\). In order to find the variance, we use (2.10)–(2.15)

$$\begin{array}{@{}rcl@{}} E[LR(\mathcal{H}_{P})^{2}]&=&\frac{1}{8}E[(LR_{0}\cdot LR_{0})(\mathcal{H}_{0})]\\ &&+ \frac{1}{8}E[(LR_{0}\cdot LR_{0})(\mathcal{H}_{1})] \\ &&+\frac{1}{8}E[(LR_{0}\cdot LR_{1})(\mathcal{H}_{0})] \\ &&+\frac{1}{8}E[(LR_{0}\cdot LR_{1})(\mathcal{H}_{1})]\\ &&+\frac{1}{8}E[(LR_{1}\cdot LR_{0})(\mathcal{H}_{0})]\\&&+\frac{1}{8}E[(LR_{1}\cdot LR_{0})(\mathcal{H}_{1})] \\&&+\frac{1}{8}E[(LR_{1}\cdot LR_{1})(\mathcal{H}_{0})]\\ && +\frac{1}{8}E[(LR_{1}\cdot LR_{1})(\mathcal{H}_{1})]\\ & =& \frac{4}{8}+\frac{3}{8}\frac{L+3}{4}+\frac{1}{8}E[LR_{1}(\mathcal{H}_{1})^{2}], \end{array} $$

from which we obtain

$$\begin{array}{@{}rcl@{}} E[LR(\mathcal{H}_{P})^{2}]&=&\frac{11L+53}{64}+\frac{1}{128}\sum\limits_{a<b}\frac{p_{a}}{p_{b}}+\frac{p_{b}}{p_{a}}\\ &=&\frac{11L+53}{64}+\frac{1}{128}s. \end{array} $$

A similar calculation for first cousins appears in Appendix B.

The above sum, denoted s, will appear repeatedly below. Sometimes, we write s i where i refers to the marker with allele frequency vector p (i). The true relationship may differ from the one assumed in the numerator of the likelihood ratio as exemplified next.

Example 3.2 (Expectation and variance for symmetric relationships when parent–offspring is true)

Using (2.7) and (2.16) we find

$$\begin{array}{@{}rcl@{}} E[LR_{(0.5,0.5,0)}(\mathcal{H}_{1})]&=&\frac{L+7}{8}=\mu,\\ Var[LR_{(0.5,0.5,0)}(\mathcal{H}_{1})]&=&\frac{1}{4}\left( \frac{9L+23}{8}+\frac{1}{16}s\right)-\mu^{2}. \end{array} $$

Consider next pairs of linked markers.

Example 3.3 (The base relationships)

We derive explicit formulae for expectation and variance of the likelihood ratio for a pair of linked markers for the base relationships unrelated, parent–offspring and monozygous twins. Trivially, the expected LR for the unrelated case is 1 while the variance is 0. For parent–offspring, linkage is known not to matter and therefore

$$E[LR(\mathcal{H}_{P})]=\frac{L_{1}+3}{4}\frac{L_{2}+3}{4} $$

from (2.2) as is confirmed by (2.18). Moreover,

$$E[LR(\mathcal{H}_{P})^{2}] =\left( \frac{5L_{1}+3}{8}+\frac{s_{1}}{16}\right)\left( \frac{5L_{2}+3}{8}+\frac{s_{2}}{16}\right). $$

Finally, for MZ twins, linkage is again irrelevant and

$$\begin{array}{@{}rcl@{}} E[LR(\mathcal{H}_{P})]&=&\frac{L_{1}(L_{1}+1)}{2}\frac{L_{2}(L_{2}+1)}{2},\\ E[LR(\mathcal{H}_{P})^{2}]&=&\left( \sum\limits_{a}{\left( \frac{1}{p_{a}^{(1)}}\right)}^{2}+\sum\limits_{a<b}\frac{1}{2p_{a}^{(1)}p_{b}^{(1)}}\right) \\&&\times \left( \sum\limits_{a}{\left( \frac{1}{p_{a}^{(2)}}\right)}^{2}+\sum\limits_{a<b}\frac{1}{2p_{a}^{(2)}p_{b}^{(2)}}\right). \end{array} $$

The next example gives the expected likelihood ratio for symmetric relationships for a pair of linked markers.

Example 3.4 (Grandparent–grandchild, half-siblings, uncle–nephew)

The joint IBD probabilities are given in Table 2. Equation (2.18) gives the expected likelihood ratios as function of ρ for grandparent–grandchild, half-siblings and uncle–nephew respectively

$$\begin{array}{@{}rcl@{}} \mu_{\text{Grand}}&=&\frac{3}{4}+\frac{1}{64}(L_{1}+3)(L_{2}+3)+\frac{1}{64}(L_{1}-1)\\ &&\times(L_{2}-1)\rho(\rho-2),\\ \mu_{\text{Half}}&=& \frac{3}{4}+\frac{1}{64}(L_{1}+3)(L_{2}+3) \\&&+ \frac{1}{16}(L_{1}-1)(L_{2}-1)\rho(\rho-1)(\rho^{2}-\rho+1),\\ \mu_{\text{Uncle}}&=& \frac{3}{4}+\frac{1}{64}(L_{1}+3)(L_{2}+3)\\&& + \frac{1}{256}(L_{1}-1)(L_{2}-1)\rho \left( 4 \rho^{2}-8 \rho +5\right)\\ &&\times \left( 4 \rho^{3}-8 \rho^{2}+5 \rho -4\right). \end{array} $$

Figure 2 shows the expected LR for the preceding three relationships, for several choices of number of alleles per locus.

Table 2 IBD probabilities for grandparent–grandchild, half-siblings and uncle–nephew relationships
Full size table
Fig. 2
figure 2

Expected LR for (L 1, L 2)=(10,15) (lower curves), (15,15) (middle), (10,30) (top)

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In general, it is cumbersome to derive exact expressions for the variance in the linked cases. However, below we present an example.

Example 3.5

Consider next the grandparent–grandchild relationship, when the true relation is parent–child. In that case \(\kappa ^{\prime }_{L}(\rho )=1\) if L = (1,1) and zero otherwise, so we have

$$\begin{array}{@{}rcl@{}} E[LR_{\mathcal{P}ed}(\mathcal{H}_{\mathcal{P}ed^{\prime}})^{2}]&=&\sum\limits_{i_{1},i_{2},j_{1},j_{2}}\kappa_{i_{1}i_{2}}(\rho)\kappa_{j_{1}j_{2}}(\rho)\\ &&\times {\prod}_{k=1}^{2}E\left[(LR^{(k)}_{i_{k}}\cdot LR^{(k)}_{j_{k}})(\mathcal{H}^{(k)}_{1})\right]. \end{array} $$

Let A i = (L i +3)/4 and \(z_{i}=(L_{i}+5)/8+{\sum }_{a<b}p_{a}^{(i)}/p_{b}^{(i)}+p_{b}^{(i)}/p_{a}^{(i)}.\) Then the above expectation may be written

$$\begin{array}{@{}rcl@{}} \frac{(1-\rho)^{2}}{4}&+& \frac{1}{4} \left( 2\rho(1-\rho)(A_{1}+A_{2})+\left( 2{(1-\rho)}^{2}\right.\right.\\ &&\qquad+\left.\left.2\rho^{2}\right)A_{1}A_{2}\right)\\ &+&\frac{1}{4} \left( \rho^{2}(z_{1}+z_{2})+2\rho(1-\rho)(z_{1}A_{2}+z_{2}A_{1})\right.\\ &&\qquad\left.+ (1-\rho)^{2}z_{1}z_{2} \right). \end{array} $$

If ρ = 0 this simplifies to (1+2A 1 A 2 + z 1 z 2)/4.

The last two examples go beyond a pair of linked markers.

Example 3.6 (Several loci)

If we take 5 loci with 15 alleles each, then we can use (2.19) to calculate expected values \(E[LR_{\mathcal {P}ed}(\mathcal {H}_{\mathcal {P}ed})]\) for the pedigrees half-siblings, uncle–nephew and grandparent–grandchild. To compute \(\kappa _{i_{1},\dots ,i_{n}}\) we use that \(P(I_{k}=i_{k} \mid I_{k-1}=i_{k-1},\dots ,I_{1}=i_{1},\mathcal {P}ed)=P(I_{k}=i_{k} \mid I_{k-1}=i_{k-1}, \mathcal {P}ed)\) and \(P(I_{1}=i_{1} \mid \mathcal {P}ed)=\kappa _{i_{1}}\). Furthermore, for these three relationships where κ = (1/2, 1/2, 0), clearly I k = 2 is impossible. For grandparent–grandchild we have P(I k = x∣I k−1 = x)=(1−ρ), for half-siblings \(P(I_{k}=x \mid I_{k-1}=x, \mathcal {P}ed)=\rho ^{2}+(1-\rho )^{2}\) and for uncle–nephew \(P(I_{k}=x \mid I_{k-1}=x, \mathcal {P}ed)=1 - \rho (4 \rho ^{2} - 8 \rho + 5)/2\), where in all cases x ∈ {0, 1}. This produces the graphs of Fig. 3. We observe that the expected likelihood ratio is much larger for ρ = 0 (where it is equal to 59145/128≈462), than for ρ = 0.5 (where it is equal to 759375/32768 ≈ 23). As ρ increases the expected LR seems to decrease exponentially fast.

Fig. 3
figure 3

Expected LR for sibs with the number of alleles for the two markers indicated

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Example 3.7

(Sibs) Let R = ρ 2+(1−ρ)2. Then we may write the kappa matrix (a similar table for the conditional distribution is in [9]),

$$\begin{array}{@{}rcl@{}} \left( \begin{array}{lll} \tfrac{1}{4} R^{2} &\tfrac{1}{2}R(1-R)&\tfrac{1}{4}{(1-R)}^{2} \\ \tfrac{1}{2}R(1-R) &\tfrac{1}{2}(1-2R(1-R))&\tfrac{1}{2}R(1-R) \\ \tfrac{1}{4} {(1-R)}^{2} & \tfrac{1}{2}R(1-R)&\tfrac{1}{4} R^{2} \end{array}\right).\end{array} $$
(3.1)

Explicitly, we have (Fig. 4)

$$\begin{array}{@{}rcl@{}} E[LR(\mathcal{H}_{P})]&=& \frac{1}{64}\left( 2 {L_{1}^{2}}+L_{1} L_{2}+13 L_{1}+2 {L_{2}^{2}}+13 L_{2}+33\right)\\ &&+ \frac{1}{64}(L_{1}-1)(L_{2}-1)\left( 4R+2(L_{1}+L_{2}+2)\right.\\ &&\qquad\left. R^{2}+L_{1}L_{2}R^{4} \right). \end{array} $$
(3.2)
Fig. 4
figure 4

Expected LR for half-siblings (dotted), uncle–nephew (dashed) and grandparent–grandchild (black), for 5 loci with 15 alleles each

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For more than two markers, the formula generalizes following (2.19). For example, if we consider again 5 linked loci with the same number of alleles and the same recombination frequencies between them, we obtain \(E[LR_{\mathcal {P}ed}(\mathcal {H}_{\mathcal {P}ed})]\) as plotted in Fig. 5.

Fig. 5
figure 5

Expected LR for siblings based on 5 linked loci with each 10 (dotted), 15 (dashed), or 20 (black) alleles

Full size image

Real markers

The markers SE33 and D6S1043 are closely linked. The impact of linkage on forensic calculations has been discussed [10]. Table 3 shows the expected values for this pair of markers and includes ρ = 0.044, a reported value, and gives completely linked and unlinked values for comparison.

Table 3 Expected LR for SE33 (55 alleles) and D6S1043 (22 alleles) for ρ = 0,0.044,0.5
Full size table

Discussion

This is the last of three papers devoted to mathematical properties of the likelihood ratio. The two first [1, 2] addressed kinship and mixture applications respectively. The current paper extends the part of [1] dealing with expectation and variance in various directions. The framework differs allowing for simpler arguments. We have shown that the expected likelihood ratio \(E_{\rho }[LR_{\mathcal {P}ed}(\mathcal {H}_{\mathcal {P}ed^{\prime }})]\) is a function which we may denote \(f_{\vec {\rho }}(L_{1},L_{2},\dots ,L_{n})\). This means that f does not depend on the allele frequencies other than through L i which is the number of alleles with non-zero frequency. It is discontinuous as a function of the allele frequencies, at the point corresponding to an allele having frequency zero.

At the same time, if an allele frequency tends to zero then the variance \(var[LR_{\mathcal {P}ed}(\mathcal {H}_{\mathcal {P}ed^{\prime }})]\) will go to infinity by (2.20) and (2.12)–(2.15). Heuristically, a very rare allele in the population causes a very small number of LR’s to be very large, and this does not affect the expected LR but it does increase the variance. At the point when the allele has disappeared altogether, we suddenly lose its contribution and get the corresponding smaller expected LR and variance.

As we have seen as well in “Likelihood ratio and weight of evidence,” the expected logarithm of the likelihood ratio is better behaved in this respect. It depends on all the allele frequencies, and is continuous in them. This, to us, indicates that it is more natural to see log(L R) as the weight of evidence than the LR itself. However, the calculation of log(L R) of course must go through the same likelihood calculations as for the LR, and in order to validate such calculations it can be of practical use to have the explicit required expressions as provided in this paper.

We have also seen that, for a pair of linked markers, the expected likelihood ratio increases as the recombination rate decreases. We note however that, this does not mean that \(P(LR_{\mathcal {P}ed}(\mathcal {H}_{\mathcal {P}ed}) \geq t)\) increases for all fixed t. Indeed, consider for example a pedigree with individuals related according to IBD coefficients (1/2,1/2,0). If ρ = 0 then for the pedigrees considered in this example LR=1/2 as soon as on at least one locus, no alleles are shared, since this means that no alleles are shared IBD. Suppose we now add more loci, still connected to each other with recombination rate equal to zero. In half of the cases, related individuals will share no IBD alleles, but they may share alleles identical by state by chance. As the number of loci increases, the probability that there is a locus without identical alleles will increase towards 1/2. Thus, about half of the truly related pairs will obtain L R = 1/2 whereas the other half obtains a high LR. Thus, the LR distribution will reduce to the distribution it has for a single, very polymorphic locus. The other extreme, corresponding to independent loci with ρ = 1/2 between them, behaves very differently. It is much more likely that L R>1 but the average LR is smaller than for ρ<1/2. Also, for unrelated individuals the LR may be as small as (1/2)n with n the number of considered loci, whereas it cannot be smaller than 1/2 with completely linked loci.

In this article exact expressions for the variance have also been obtained. As for the LR itself, we need not assume that the true relationship coincides with the one being tested for by the likelihood ratio. We conjecture that the variance of the likelihood ratio for a pair of linked markers, similar to the expectation, is a non-decreasing function of the recombination rate ρ. The expected likelihood ratio measured in standard deviation units, i.e., \(g(\rho )=E[LR(\mathcal {H}_{P})]/SD[LR(\mathcal {H}_{P})]\) is, however, not generally a monotone function as can be seen from Table 3.

The relevance of the current paper goes beyond the theoretical findings discussed above. The results may be used to check calculations and implementations, which may sometimes be difficult to do by simulations. The function textit{moment} of theverb@R@ package BookEKM freely available from http://familias.name/book.html calculates expected values and variances exactly and most calculations of this paper are included as examples in the documentation. These values can in turn be used to estimate the distribution of the log likelihood ratio relevant for power calculations.