Within-Family Benchmarks for Score Prediction

Why Compare Population and Within-Family Associations?

A polygenic score can predict a trait across a population because people with different scores also differ in many other ways. Some differences may reflect genetic effects indexed by the score; others may reflect population structure, shared family circumstances, or pathways linking parental genotypes to offspring environments. Comparing population-based associations with associations among siblings helps assess how sensitive the prediction is to sources of between-family variation.

The comparison is a benchmark, not a causal verdict. Siblings share parents and much of their environment, but they differ in inherited alleles and in experiences. A within-family association therefore removes some between-family differences while retaining other sources of association and adding practical limitations, such as measurement error in sibling score differences.

Definition: A population-based score association is the regression association between a polygenic score and an outcome across individuals in a population or evaluation sample, possibly adjusted for measured covariates. A within-family score association is the association between differences in scores and differences in outcomes among relatives in the same family, commonly estimated with a family fixed effect. Both describe association in a specified sample and model; neither, by itself, establishes a causal effect.

Models and the Within-Family Contrast

Let \(Y_{if}\) be the outcome for individual \(i\) in family \(f\), and let \(P_{if}\) be that individual's polygenic score. A population-based linear model can be written as

Y_{if} = \alpha + \beta_{\mathrm{pop}} P_{if} + \boldsymbol{\gamma}^{\mathsf T}\mathbf{C}_{if} + \varepsilon_{if}.

Here, \(\mathbf{C}_{if}\) denotes included covariates, such as age, sex, or ancestry principal components, and \(\beta_{\mathrm{pop}}\) is the fitted outcome difference associated with a one-unit score difference, conditional on those covariates and the linear model. The regression describes the adjusted association in the analyzed population. Covariate adjustment can address measured differences, but it does not guarantee that all relevant population structure or familial pathways have been removed.

A family fixed-effects model gives each family its own intercept:

Y_{if} = \alpha_f + \beta_{\mathrm{within}} P_{if} + \boldsymbol{\gamma}^{\mathsf T}\mathbf{C}_{if} + \varepsilon_{if}.

Subtracting each family's mean from the individual values eliminates \(\alpha_f\). The fitted slope is consequently identified by score and outcome variation within families, rather than by differences between family means. For a sibling pair, this is equivalent to modeling the outcome difference as a function of the score difference, with covariate differences included when needed.

Y_{if} - \bar{Y}_{f} = \beta_{\mathrm{within}}(P_{if} - \bar{P}_{f}) + \boldsymbol{\gamma}^{\mathsf T}(\mathbf{C}_{if} - \bar{\mathbf{C}}_{f}) + (\varepsilon_{if} - \bar{\varepsilon}_{f}).

In these expressions, \(\bar{Y}_{f}\), \(\bar{P}_{f}\), and \(\bar{\mathbf{C}}_{f}\) are family-specific means. The fixed effect absorbs factors shared by members of a family, whether measured or not, to the extent that those factors are constant within that family. It cannot remove factors that differ between siblings.

Model conditions: A linear slope is a useful summary when the conditional association is reasonably represented as linear over the observed score range. Within-family estimation also requires score variation within families; families whose members have identical scores provide no information about the score slope. Inference should account for the family-level dependence of observations, commonly by using standard errors clustered by family. These conditions support estimation of an association, not a causal interpretation.

Estimating a Sibling-Difference Association

For a set of sibling pairs, orient each pair consistently and define \(\Delta P_f=P_{1f}-P_{2f}\) and \(\Delta Y_f=Y_{1f}-Y_{2f}\). If the model has a shared family intercept and no pair-specific intercept after differencing, the within-family slope can be estimated by regressing \(\Delta Y_f\) on \(\Delta P_f\). With no additional covariate differences, the least-squares slope through the origin is

\hat{\beta}_{\mathrm{within}} = \frac{\sum_f \Delta P_f \Delta Y_f}{\sum_f (\Delta P_f)^2}.

The zero intercept follows from differencing out the shared family intercept; it is not an instruction to force the intercept to zero in an unrelated population regression. If covariates vary between siblings, their differences must also be included in the model or otherwise addressed.

Worked Example: Calculating a Within-Family Slope

Suppose a hypothetical study records a score and a quantitative outcome for four sibling pairs. The score is on the same fixed scale for all siblings, and differences are calculated as sibling 1 minus sibling 2.

Pair	\(\Delta P_f\) (score units)	\(\Delta Y_f\) (outcome units)	\(\Delta P_f\Delta Y_f\)	\((\Delta P_f)^2\)
1	-1	-2	2	1
2	0	-1	0	0
3	1	1	1	1
4	2	3	6	4

The score differences are not all zero, so the pairs provide within-family score variation. The numerator is \(2+0+1+6=9\), and the denominator is \(1+0+1+4=6\). Therefore,

\hat{\beta}_{\mathrm{within}} = \frac{9}{6} = 1.50 \quad \text{outcome units per score unit}.

In this hypothetical sample, a one-unit higher score difference is associated with a 1.50-unit higher outcome difference under the fitted through-origin model. Four pairs are far too few to support a stable general conclusion; the calculation illustrates the estimator, not a reliable prediction result. The shared family intercept cancels in each pair difference, but the calculation does not eliminate nonshared sibling differences that may also be related to both score and outcome.

What the Comparison Can Reveal

Population and within-family estimates can differ because they use different sources of variation. The population slope draws on both between-family and within-family differences. The within-family slope draws on within-family differences alone. For example, a population association may be larger if score differences between families align with ancestry-related environmental differences or with parental characteristics that also relate to the outcome. A smaller within-family association can therefore flag sensitivity to between-family pathways or structure.

The comparison is most informative when the score, outcome, sample definition, covariate strategy, and score units are aligned. If the population score is standardized to one standard deviation in the evaluation sample, but the within-family score is separately standardized using sibling differences, the resulting slopes are expressed per different score units. Their numerical sizes cannot be compared directly without conversion. Reporting the score scale and the within-family score variation makes the contrast interpretable.

A useful conceptual separation writes the score as a family mean plus an individual deviation:

P_{if} = \bar{P}_{f} + (P_{if} - \bar{P}_{f}).

The family fixed-effects slope uses the second component. A model that includes both components can estimate separate within-family and between-family associations, although interpreting the between-family component remains subject to confounding by factors that differ across families. A single pooled population slope is not generally a simple average of the two slopes: the relationship depends on the amount of variation in each component, covariates, and the model specification.

Worked Example: Interpreting Different Population and Within-Family Estimates

Suppose a hypothetical evaluation reports slopes for the same quantitative outcome and the same score, measured in overall evaluation-sample standard deviations. The population model gives \(0.30\) outcome standard deviations per score standard deviation, with \(SE=0.02\). A sibling fixed-effects model gives \(0.20\), with \(SE=0.05\). Both models use the same outcome scale and adjustment for age and sex; the population model also adjusts for ancestry principal components. Assume the within-family standard error is clustered by family.

Using a normal approximation, the population 95% confidence interval is

0.30 \mathbin{\pm} 1.96(0.02) = 0.30 \mathbin{\pm} 0.0392 = (0.2608,\ 0.3392).

The within-family 95% confidence interval is

0.20 \mathbin{\pm} 1.96(0.05) = 0.20 \mathbin{\pm} 0.098 = (0.102,\ 0.298).

The point estimates differ by \(0.10\) outcome standard deviations per score standard deviation. The within-family interval is wider, consistent with less precise estimation from the smaller amount of score variation among siblings. The intervals overlap, but overlap or nonoverlap is not a substitute for a formal test of the difference: because the estimates may be statistically dependent, a test or interval for their difference should use the covariance between estimates. The observed discrepancy is a reason to examine sensitivity to between-family variation, not proof that the population association is biased by a particular amount.

Limits of the Benchmark

Within-family estimation controls shared family factors, but several limitations remain. First, siblings do not share every relevant exposure. A sibling-specific environment, measurement process, or other factor may be associated with both score and outcome. Second, parental genotypes can relate to offspring environments through family pathways that are not fully represented by a sibling's own score; the fixed effect does not make every such pathway irrelevant. Third, assortative mating and other features of how families form can affect the relationship between genetic scores and family characteristics. The size and direction of a population–within-family difference are therefore not a unique diagnosis of population stratification.

Within-family estimates can also be less precise because siblings have more similar scores than unrelated individuals. A score measured with error can create an especially important issue: differencing may leave substantial error relative to the true score difference, attenuating the within-family slope. This is one reason a smaller within-family estimate does not automatically imply that the population estimate is wholly spurious.

Worked Example: How Score Error Can Attenuate a Within-Family Slope

Consider a hypothetical setting in which \(\Delta G\) is the true sibling score difference and the true difference model is \(\Delta Y=0.50\Delta G+u\). Assume \(\Delta G\) has variance \(1\), \(u\) is uncorrelated with \(\Delta G\), and the measured score difference is \(\Delta P=\Delta G+e\), where measurement error \(e\) is independent of both \(\Delta G\) and \(u\), with variance \(1\). The regression slope of \(\Delta Y\) on the measured \(\Delta P\) is

\frac{\operatorname{Cov}(\Delta P,\Delta Y)} {\operatorname{Var}(\Delta P)} = \frac{\operatorname{Cov}(\Delta G+e,\ 0.50\Delta G+u)} {\operatorname{Var}(\Delta G+e)} = \frac{0.50\operatorname{Var}(\Delta G)} {\operatorname{Var}(\Delta G)+\operatorname{Var}(e)} = \frac{0.50(1)}{1+1} =0.25.

The expected measured-score slope is \(0.25\) outcome units per measured score-difference unit, half the true slope of \(0.50\) in this example. The result follows from the stated independent-error assumptions; real score error need not satisfy them exactly. It demonstrates why attenuation is a plausible contributor to a smaller within-family estimate, not a universal correction factor.

A Practical Comparison and Common Pitfall

A careful analysis should define the target outcome and score scale, fit population and within-family models that are as comparable as possible, report estimates with uncertainty, and describe what variation identifies each coefficient. It should also report the number of families and the within-family score variation, since a large individual count does not guarantee substantial information about sibling contrasts.

Specify comparable quantities.
Use the same score construction and outcome definition. State the score unit, outcome unit, sample, covariates, and any differences between the population and sibling analyses.

Estimate both associations.
Fit the population regression and the family fixed-effects regression. Account for family clustering in within-family inference and report uncertainty for each estimate.

Interpret the contrast cautiously.
Describe whether the estimates differ in magnitude or precision and identify plausible explanations, including between-family structure, familial pathways, nonshared confounding, and measurement error.

Common mistake: “The within-family estimate is smaller, so the entire population association was caused by population stratification.” This conclusion does not follow. The difference can also reflect limited precision, score measurement error, nonshared sibling confounding, differences in model specification, or multiple familial pathways. The correction is to call the within-family estimate a sensitivity benchmark and state which sources of variation each estimate uses. Neither coefficient alone establishes a causal effect.

Key takeaway: A population-based score association uses differences across individuals, including differences between families. A within-family association uses differences among relatives. Their discrepancy can reveal sensitivity to between-family variation, but it does not identify a single bias mechanism or provide a definitive causal estimate.

Check Your Understanding

Use the models and examples above to answer the following questions.

What variation identifies the slope in a family fixed-effects model, and which shared factor does the family intercept absorb?
Why should population and within-family slopes be reported on the same score scale before comparing their magnitudes?
A within-family estimate is smaller and less precise than a population estimate. Give two explanations other than population stratification that could contribute to this pattern.
In the measurement-error example, what assumptions made the measured-score slope equal to one-half the true slope?
Why is it incorrect to describe a within-family association as a definitive causal effect?