Why Diagnose a Fitted Partition?
A partitioned LD score regression model can fit the observed association statistics reasonably well while giving uncertain or unstable estimates for individual annotations. One reason is collinearity: annotation-specific LD score columns may contain nearly the same information. Another is model misspecification: the included predictors, their assumed relationships to the expected statistics, or the error structure may not adequately represent the data. These problems matter because a numerical coefficient is not automatically a well-identified or reliable annotation-specific estimate.
Diagnostics do not prove that a model is correct or incorrect. They help locate potential weaknesses, indicate which conclusions are sensitive to the design, and guide checks of the annotations and reference data. In particular, a small residual summary or a plausible intercept is not sufficient evidence that the partition is adequate.
Model and Diagnostic Quantities
For SNP \(j\), let \(\chi_j^2\) denote its association chi-square statistic, \(N\) the relevant sample size, and \(\ell(j,c)\) the LD score for SNP \(j\) constructed from annotation \(c\). A common stratified LD score regression mean model is
Here \(a\) is the intercept, \(C\) is the number of annotation columns, and \(\tau_c\) is the coefficient associated with annotation \(c\), conditional on the other included columns. Weighted regression fits this relationship across SNPs, using positive regression weights. Write \(y_j=\chi_j^2\), \(\widehat y_j\) for the fitted value, and \(r_j=y_j-\widehat y_j\) for the residual. A diagnostic concerns the design matrix of LD score columns as well as how the residuals behave across SNPs.
The relevant correlations are those among the predictors actually used in the regression, typically the annotation-specific LD scores, not merely correlations among the original annotation labels. LD-score construction can alter the relationships between columns. A pairwise correlation is a useful first check, but it cannot reveal every dependency involving three or more columns. Scale differences also affect some numerical diagnostics, so compare standardized columns or otherwise account for scale when assessing near-dependence.
Collinearity: Strong Prediction, Weak Attribution
When columns are highly correlated, many combinations of their coefficients can yield nearly the same fitted values. The model may therefore predict the association statistics adequately but estimate the separate annotation coefficients imprecisely. Estimates can change substantially after a small change in the sample, SNP set, or model specification, even if predictions change very little. This is not evidence that the underlying annotations have suddenly acquired different biology; it is evidence that the data and design may not distinguish their conditional contributions well.
In a two-predictor regression, after centering and standardizing the predictors, the variance inflation factor for either coefficient is \(1/(1-r^2)\), where \(r\) is their correlation. This describes the increase in coefficient variance relative to an otherwise comparable design with uncorrelated predictors. It is a diagnostic quantity, not a universal pass-fail threshold. In larger models, inspect the full design as well: singular values or a condition number can reveal a near-dependency that pairwise correlations miss.
Worked Example: Quantifying a Correlated Pair of LD Score Columns
Suppose two standardized annotation-specific LD score columns have correlation \(r=0.98\) across the SNPs used in a hypothetical regression. The two-predictor variance inflation factor is
The corresponding standard-error inflation factor, relative to the uncorrelated comparison, is approximately \(\sqrt{25.25}=5.025\). Thus the design gives much less precise separation of these two coefficients than an uncorrelated design would, all else equal. This calculation does not show that either annotation is irrelevant, nor does it establish that the model is misspecified. It identifies a reason to be cautious about interpreting the two conditional coefficients separately.
Worked Example: Coefficients Shift While Predictions Stay Similar
Consider two hypothetical annotation columns \(x_1\) and \(x_2\) that are close for every SNP, with \(|x_1-x_2|\leq 0.05\). One fit estimates their coefficients as \(0.12\) and \(0.01\); a small perturbation of the data gives estimates \(0.08\) and \(0.05\). Both coefficient pairs sum to \(0.13\).
For the first fit, the two-column contribution is \(0.12x_1+0.01x_2\). For the second it is \(0.08x_1+0.05x_2\). Their difference is
Because \(|x_1-x_2|\leq0.05\), the absolute difference in these contributions is at most \(0.04(0.05)=0.002\) for each SNP. The fitted contributions are therefore very close even though the individual coefficients shifted. This pattern is consistent with collinearity: the sum-like direction is better determined than the contrast between the annotations. It does not, on its own, say which coefficient is biologically meaningful.
Residual Patterns and Misspecification
Residuals help assess whether the fitted mean relationship leaves systematic structure unexplained. Examine them against fitted values, relevant LD score columns, SNP or genomic position, and useful groupings such as bins of an annotation or LD score. Patterns such as a trend, curvature, changes in spread, or persistent differences across genomic blocks can suggest that the specified mean structure or data handling needs review. For example, curvature against a predictor can be consistent with a non-linear relationship that the fitted linear form does not capture.
A residual pattern is a clue, not a unique diagnosis. It might reflect an omitted predictor, an unsuitable functional form, problematic SNPs, or other features of the data and estimation. Conversely, a plot without an obvious pattern cannot prove adequacy. SNP-level residuals also should not be treated as independent evidence just because there are many SNPs; dependence induced by genomic structure matters when judging how much weight to give an apparent pattern.
Worked Example: A Residual Trend Across an LD Score
Suppose a hypothetical model is fitted to four SNP groups ordered from lower to higher values of an LD score. Their group-average residuals, in chi-square-statistic units, are \(0.20,\ 0.10,\ -0.10,\) and \(-0.20\). The residuals decrease across the ordered groups, even though their unweighted average is
The zero average does not remove the pattern: the low-LD-score groups have positive residuals and the high-LD-score groups have negative residuals. This suggests checking whether the fitted relationship captures the association between the expected statistic and LD score adequately, and whether an omitted feature correlated with that LD score could be involved. Four group averages are only a diagnostic illustration, not enough to establish a particular cause or quantify its importance.
What the Intercept Can and Cannot Tell You
In this regression, the intercept \(a\) is the expected chi-square statistic when the modeled LD score contribution is zero, under the fitted mean model. It is commonly examined as a diagnostic for inflation not represented by the modeled polygenic contribution. Its interpretation depends on the model and data: it is not a direct, exhaustive measure of every possible source of bias, and an apparently plausible estimate does not certify that the annotation model is adequate. Omitted or poorly represented structure can affect the fitted relationship, including its intercept.
An intercept that changes noticeably when annotation columns are added or removed is a reason to inspect the specification and data, not a stand-alone proof of a particular failure. Likewise, an intercept near one does not rule out collinearity: correlated columns can make individual coefficients unstable while leaving fitted values and the intercept fairly similar.
Worked Example: Intercept and Residual Checks Answer Different Questions
Suppose a hypothetical weighted regression of \(\chi_j^2\) on annotation-specific LD scores estimates an intercept of \(1.03\), with a standard error of \(0.04\). The estimate is \(0.03\) above one, or \(0.75\) standard errors above one, since
This calculation describes the estimate relative to one in standard-error units; it is not a test or a conclusion that the model is adequate. If residual plots still show a systematic trend across an annotation LD score, that pattern warrants investigation despite the intercept estimate. Conversely, a residual plot without a clear trend would not make the intercept a complete check of all model assumptions. The intercept and residual diagnostics address related but distinct aspects of the fit.
Why the Weighted Residual Mean Is Not a Fit Test
A technical detail matters when interpreting residual summaries from weighted least squares. With an intercept, the weighted residual sum is zero up to numerical precision. This does not generally imply that the ordinary, unweighted residual mean is zero. Neither a zero weighted mean nor a zero ordinary mean establishes model adequacy.
Worked Example: Weighted Residuals Can Have a Nonzero Ordinary Mean
Consider four hypothetical residuals \(0.20,\ -0.10,\ 0.20,\ -0.10\), with weights \(1,\ 2,\ 1,\ 2\). Their weighted sum is
The sum of the weights is \(1+2+1+2=6\), so the weighted residual mean is \(0/6=0\). But the ordinary residual mean is
Thus an intercept in this weighted fit guarantees the weighted, not necessarily the ordinary, residual mean is zero. Both summaries can conceal structure among residuals, so inspect residuals in relation to predictors and relevant groupings as well.
A Practical Diagnostic Sequence
A useful diagnosis proceeds from the design to the fit and then to sensitivity. The objective is not to produce a single numerical pass-fail label, but to understand which features of the model are well supported by the data.
Review correlations among the annotation-specific LD score columns and check for near-dependencies involving multiple columns. Account for scale when using condition numbers or related diagnostics.
Compare estimates and uncertainty under reasonable small changes to the data or specification. Large coefficient changes paired with small changes in fitted values are consistent with weak separation of correlated annotations.
Plot or summarize residuals against fitted values, predictors, genomic position, and meaningful groups. Look for systematic patterns, while remembering that their cause is not identified by the plot alone.
Record its estimate and uncertainty, and note whether it changes materially across reasonable specifications. Neither a plausible intercept nor a zero weighted residual mean proves adequacy.
Common Mistakes
- Reading unstable coefficients as precise annotation differences. If correlated columns can substitute for one another, individual conditional coefficients may be poorly identified even when fitted values are stable. Describe that limitation rather than treating a coefficient shift as a reliable biological contrast.
- Using a high pairwise correlation as a complete diagnosis. Pairwise correlations can flag a problem, but dependencies may involve several columns together. Inspect the full design and its stability.
- Calling any residual pattern proof of a specific omitted factor. A pattern indicates that the fit merits investigation; it does not uniquely identify its source.
- Claiming that an intercept makes the ordinary residual mean zero. In weighted least squares with an intercept, it is the weighted residual sum, and hence the weighted residual mean, that is zero up to numerical precision. The ordinary mean need not be zero.
- Treating a zero residual mean or a plausible intercept as proof of fit. These are limited summaries. Systematic residual structure and coefficient sensitivity can remain.
Check Your Understanding
Use the model, calculations, and diagnostic distinctions from this tutorial.
- Why can fitted values remain similar while two annotation coefficients change substantially?
- For two standardized predictors with correlation \(r=0.90\), calculate the two-predictor variance inflation factor using \(1/(1-r^2)\).
- What can a systematic residual trend across an annotation LD score suggest, and why does it not uniquely identify the cause?
- In a weighted least-squares fit with an intercept, which residual sum is constrained to zero up to numerical precision?
- Why does an intercept estimate near one not establish that the annotation model is adequate?