Heterogeneity LOD Scores for Genetically Distinct Families

Why Allow Linkage to Differ Across Families?

A conventional parametric linkage analysis evaluates whether the observed pedigree data are more likely under linkage at a specified recombination fraction than under no linkage. That comparison can be poorly suited to a collection of families if the marker is linked to the disease locus in some families but not in others. For example, a disorder may have more than one genetic cause, so families with the same clinical diagnosis need not share the same disease locus.

A heterogeneity LOD score, or HLOD, extends the likelihood comparison by allowing a proportion of families to be linked and the remaining families to be unlinked. It does not require the analyst to label each family as linked or unlinked in advance. Instead, the likelihood averages over those possibilities for each family and estimates the linked-family proportion from the combined data.

Definition: Let $L_i(\theta)$ be the likelihood of the observed data in family $i$ under linkage at recombination fraction $\theta$, and let $L_i(1/2)$ be that family's likelihood under no linkage. If $\alpha$ is the proportion of families assumed to be linked, the family-level mixture likelihood is $$ L_i(\theta,\alpha)=\alpha L_i(\theta)+(1-\alpha)L_i(1/2), $$ where $0\leq\alpha\leq1$ and $0\leq\theta\leq1/2$. For independent families, the HLOD is $$ \operatorname{HLOD}=\max_{\substack{0\leq\theta\leq1/2\\0\leq\alpha\leq1}} \log_{10}\left\{\prod_{i=1}^{m} \frac{\alpha L_i(\theta)+(1-\alpha)L_i(1/2)} {L_i(1/2)}\right\}. $$

The parameter $\alpha$ is a mixing proportion: in the model, each family is treated as arising from a linked component with weight $\alpha$ or an unlinked component with weight $1-\alpha$. It is not a statement that a particular family has been definitively classified. The linked component uses a common $\theta$ in this formulation, while the unlinked component uses $\theta=1/2$.

Rewriting the Likelihood in Convenient Form

For each family, define the likelihood ratio at a fixed recombination fraction as $r_i(\theta)=L_i(\theta)/L_i(1/2)$. Dividing the mixture likelihood by the unlinked likelihood gives a contribution of $1-\alpha+\alpha r_i(\theta)$. Thus, at fixed $\theta$, the heterogeneity log-likelihood ratio is

H(\theta,\alpha)= \log_{10}\left\{\prod_{i=1}^{m} \left[1-\alpha+\alpha r_i(\theta)\right]\right\} =\sum_{i=1}^{m}\log_{10}\left[1+\alpha\{r_i(\theta)-1\}\right].

This form makes the role of each family visible. If $r_i(\theta)>1$, that family favors linkage at the specified $\theta$; if $r_i(\theta)<1$, it favors the unlinked likelihood; and if $r_i(\theta)=1$, it contributes no preference between the two. The mixture keeps every factor nonnegative and lets the data determine how much weight to assign to the linked component.

The maximization over $\alpha$ is usually a one-dimensional profile-likelihood problem at each fixed $\theta$. For interior values, differentiating the natural logarithm of the product (which has the same maximizer as its base-10 logarithm) gives the score equation

\sum_{i=1}^{m} \frac{r_i(\theta)-1}{1+\alpha\{r_i(\theta)-1\}}=0.

The maximizer may instead occur at the boundary $\alpha=0$ or $\alpha=1$. In particular, $\alpha=0$ gives a likelihood ratio of exactly 1 and an HLOD of 0 for every $\theta$. Also, setting $\theta=1/2$ makes every $r_i=1$, again giving a score of 0 for every $\alpha$. Therefore, when both parameters are maximized over their stated ranges, the maximized HLOD cannot be below 0. That fact does not by itself establish evidence for linkage; it follows partly because the no-linkage model is included among the possibilities.

Worked Examples

Worked Example: Opposing Family Evidence

Suppose two hypothetical, independent pedigrees have likelihood ratios $r_1(\theta)=3$ and $r_2(\theta)=1/3$ at a particular fixed $\theta$. The first family favors linkage threefold relative to no linkage; the second favors no linkage threefold relative to linkage. Find the maximizing linked-family proportion at this $\theta$, and calculate the corresponding score.

The likelihood ratio as a function of $\alpha$ is

R(\alpha)=[1+2\alpha]\left[1-\frac{2}{3}\alpha\right] =1+\frac{4}{3}\alpha-\frac{4}{3}\alpha^2.

Differentiating the product gives $R'(\alpha)=\frac{4}{3}-\frac{8}{3}\alpha$. Setting this equal to zero gives $\hat{\alpha}=1/2$. Since $R''(\alpha)=-8/3<0$, this is a maximum. Substitution yields

R(1/2)=[1+2(1/2)]\left[1-\frac{2}{3}(1/2)\right] =2\cdot\frac{2}{3}=\frac{4}{3}.

The fixed-$\theta$ heterogeneity score is therefore $\log_{10}(4/3)\approx0.1249$. The estimate $\hat{\alpha}=0.5$ reflects a balance in the likelihood between these two families under the mixture model. It does not mean that the analysis has identified one specific family as linked and the other as unlinked.

Worked Example: An Interior Proportion With Three Families

Consider three hypothetical independent pedigrees at a fixed $\theta$, with family likelihood ratios $r_1=1/2$, $r_2=1/2$, and $r_3=4$. Find the maximizing $\alpha$ and the score at this $\theta$.

The mixture likelihood ratio is

R(\alpha)=\left(1-\frac{\alpha}{2}\right)^2(1+3\alpha).

The derivative of its natural logarithm is $-1/(1-\alpha/2)+3/(1+3\alpha)$. Setting it to zero gives \[ -\frac{1}{1-\alpha/2}+\frac{3}{1+3\alpha}=0. \] The denominators are positive for $0\leq\alpha\leq1$, so cross-multiplication gives $3(1-\alpha/2)=1+3\alpha$, and hence $\hat{\alpha}=4/9$. The second derivative of the log likelihood is \[ -\frac{1}{2(1-\alpha/2)^2}-\frac{9}{(1+3\alpha)^2}<0, \] so the stationary point is a maximum. At that value,

R(4/9)=\left(\frac{7}{9}\right)^2\left(\frac{7}{3}\right) =\frac{343}{243}\approx1.412.

Thus $H(\theta,4/9)=\log_{10}(343/243)\approx0.1497$. This is a fixed-$\theta$ profile calculation: the full HLOD still requires maximizing over $\theta$ as well. The arithmetic uses the exact likelihood ratio $343/243$ before rounding the displayed result.

Worked Example: Profiling the Proportion for a Candidate Recombination Fraction

Suppose three hypothetical families have $r_1(\theta)=9$, $r_2(\theta)=1/5$, and $r_3(\theta)=1/5$ at a candidate recombination fraction. Find the maximizing linked-family proportion and the resulting fixed-$\theta$ score.

The likelihood ratio is

R(\alpha)=(1+8\alpha)\left(1-\frac{4}{5}\alpha\right)^2.

The derivative of its natural logarithm is \[ \frac{8}{1+8\alpha}-\frac{8/5}{1-4\alpha/5}. \] At $\alpha=1/3$, the first term is $8/(11/3)=24/11$, and the second is $(8/5)/(11/15)=24/11$. The derivative is zero. The second derivative is \[ -\frac{64}{(1+8\alpha)^2} -\frac{32/25}{(1-4\alpha/5)^2}<0, \] so this interior value maximizes the likelihood. Substituting,

R(1/3)=\frac{11}{3}\left(\frac{11}{15}\right)^2 =\frac{1331}{675}\approx1.972.

The profile score at this candidate $\theta$ is $\log_{10}(1331/675)\approx0.2949$. To report the overall HLOD, an analysis would repeat the maximization over $\alpha$ across the allowed $\theta$ values and retain the largest score. This example calculates the proportion maximization at one candidate $\theta$, not a claim that this candidate necessarily gives the global maximum.

How the Full HLOD Is Obtained

The definition requires a joint maximization over $\alpha$ and $\theta$. A practical approach is to fix a candidate $\theta$, maximize the mixture likelihood over $0\leq\alpha\leq1$, and then compare the resulting profile scores across the allowed recombination fractions. A numerical optimizer or a sufficiently fine search can be used, with boundary values checked explicitly. At each candidate $\theta$, the family likelihoods must be calculated under the same specified penetrance, disease-allele-frequency, marker, and pedigree model.

Calculate family likelihoods.
For each family, obtain $L_i(\theta)$ at candidate recombination fractions and $L_i(1/2)$ under no linkage, using the same model assumptions.

Form family mixture contributions.
At each fixed $\theta$, calculate $1-\alpha+\alpha r_i(\theta)$ for every family and multiply these contributions.

Profile over the linked-family proportion.
Maximize the product over $0\leq\alpha\leq1$, checking any interior solution and both boundaries.

Maximize over recombination fraction.
Compare the profiled likelihood ratios across $0\leq\theta\leq1/2$, including the unlinked value, and take the base-10 logarithm of the largest ratio.

The family contributions are multiplied because the model treats the pedigrees as independent observations. If families are dependent, for example because relatives are represented across pedigrees, this product does not describe their joint likelihood without an appropriate dependence model.

Interpretation and Common Pitfalls

A positive HLOD means that the best-fitting mixture of linked and unlinked families has a likelihood greater than the all-unlinked likelihood under the specified model. The estimated $\hat{\alpha}$ is the fitted mixing proportion for the linked component. It is not a posterior probability that any particular family is linked, nor does it prove that the families have distinct disease genes. Heterogeneity is one explanation represented by the model; conclusions remain conditional on the pedigree and disease assumptions used to calculate the likelihoods.

The HLOD is a likelihood-ratio statistic on a base-10 logarithmic scale. For example, an HLOD of 0.3 corresponds to a maximized likelihood ratio of $10^{0.3}$, approximately 2.00, relative to the all-unlinked likelihood. That is a ratio of likelihoods, not a p-value or a probability that linkage is true. Interpretation of statistical significance requires appropriate calibration for the model and search; the numerical HLOD alone is not a probability statement.

Incorrect: “The estimated $\alpha$ is 0.6, so 60% of these observed families are certainly linked.” Correction: $\hat{\alpha}$ is a fitted mixture proportion. It does not assign certain status to each family.
Incorrect: “Add each family's ordinary LOD score and multiply the sum by $\alpha$.” Correction: The mixture is formed on each family's likelihood scale, $1-\alpha+\alpha r_i(\theta)$, and those factors are multiplied before taking the logarithm.
Incorrect: “A positive HLOD means the marker is linked in every family.” Correction: The alternative explicitly permits an unlinked component; the fitted score concerns the combined likelihood under that mixture.
Incorrect: “A maximized HLOD is automatically significant because it is at least zero.” Correction: A score of zero is available under the null boundary. Evidence and significance require an appropriate reference distribution or testing procedure, not just the sign of the maximized score.

Heterogeneity modeling matters because pooling families under an assumption that all share linkage can dilute evidence when only a subset is linked. Conversely, a fitted mixture can be sensitive to the family likelihoods and model assumptions: a few families with strong linkage evidence can influence $\hat{\alpha}$ substantially. Reporting the HLOD together with the maximizing $\hat{\theta}$, $\hat{\alpha}$, the model assumptions, and the family-level likelihood evidence gives a more informative account than reporting a single score alone.

Key takeaway: The HLOD compares the all-unlinked likelihood with a family-level mixture of linked and unlinked likelihoods, maximizing over both the recombination fraction and the linked-family proportion. It summarizes evidence for linkage under heterogeneity, but does not by itself classify families or establish a causal genetic mechanism.

Check Your Understanding

Use the mixture likelihood and its interpretation to answer these questions.

For a family with $r_i(\theta)=1$, what is its mixture contribution for any value of $\alpha$, and why?
Two families have likelihood ratios 2 and $1/2$ at a fixed $\theta$. Write the combined likelihood ratio as a function of $\alpha$.
Why is $\hat{\alpha}$ not, by itself, a probability that a named family is linked?
What happens to the mixture likelihood ratio when $\theta=1/2$, regardless of $\alpha$?
Explain why multiplying family mixture contributions differs from adding family LOD scores and then multiplying by $\alpha$.