Comparing Lambda, QQ Plots, and LDSC

Why Compare More Than One Inflation Diagnostic?

A GWAS can produce many test statistics, and no single summary reveals every feature of their distribution. A genomic inflation factor reports one central tendency; a quantile-quantile (QQ) plot displays the distribution across ranks; and LD score regression asks whether chi-square statistics vary systematically with linkage disequilibrium (LD) score. Their conclusions can differ without any calculation being wrong. The important question is what each diagnostic is sensitive to, and what assumptions support its interpretation.

These diagnostics are most informative when applied to association statistics that have already been computed using an appropriate model and calibration procedure. A diagnostic is not a substitute for checking phenotype and genotype quality, model specification, or the study design. Nor does a pattern alone establish whether its source is confounding, polygenic signal, or another feature of the analysis.

Three Summaries, Three Different Questions

Definition: For $m$ approximately independent or suitably calibrated one-degree-of-freedom association tests with chi-square statistics $\chi_j^2$, the genomic inflation factor is $$ \lambda = \frac{\operatorname{median}_j(\chi_j^2)}{0.4549364}. $$ The denominator is the median of a $\chi_1^2$ random variable under the null. Thus, $\lambda$ compares the observed median statistic with its null expectation.

Lambda is a one-number summary of the middle of the test-statistic distribution. It is not a measure of the largest association, and it does not reveal how the upper tail behaves. Its usual null reference also presumes that the test statistics have the intended null distribution. Dependence among variants and widespread genuine effects complicate a simple interpretation of its value.

A QQ plot sorts observed association p-values and compares them with their expected order under the null. If $p_{(i)}$ is the $i$th smallest of $m$ p-values, a common expected value is $i/(m+1)$. Plotting $-\log_{10}(p_{(i)})$ against $-\log_{10}(i/(m+1))$ places small p-values at the upper right. Under a well-calibrated global null, points should broadly track the reference diagonal, allowing for sampling variation. Departure across many ranks and departure limited to the extreme tail are different visual patterns, but neither pattern identifies a cause by itself.

Definition: In the standard LD score regression mean model for one-degree-of-freedom chi-square statistics, $$ E(\chi_j^2) \approx I + b\,\ell_j, $$ where $\ell_j$ is the LD score for variant $j$, $I$ is the fitted intercept, and $b$ is the fitted slope. Under the model, the slope is related to polygenic signal, while the intercept represents the expected baseline at LD score zero. Interpreting either quantity requires the assumptions and reference data to be adequate.

LD score regression uses how association statistics vary with LD score, rather than only their overall median or their rank. In the standard interpretation, a positive slope is consistent with polygenic signal distributed across variants, while an elevated intercept can indicate average inflation not explained by the LD-score-dependent component. As covered in the earlier material on the LDSC intercept, the intercept is not a definitive or universally unbiased estimate of confounding.

Worked Examples: Different Patterns, Different Messages

Worked Example: Elevated Lambda With a Polygenic-Looking Trend

Hypothetical data. Suppose a GWAS has a median one-degree-of-freedom chi-square statistic of $0.546$. A QQ plot follows the diagonal through much of its range and bends upward progressively in the upper ranks. An LD score regression fit has intercept $1.02$ and slope $0.020$ chi-square units per LD-score unit.

The genomic inflation factor is $$ \lambda = \frac{0.546}{0.4549364} \approx 1.20. $$ Thus the median statistic is about 20% above the null median on this scale. That observation alone does not show why it is elevated.

The regression mean model gives a fitted mean of $1.02 + 0.020(10)=1.22$ at LD score 10 and $1.02 + 0.020(50)=2.02$ at LD score 50. The positive slope means the fitted mean statistic rises with LD score. Together with a QQ plot whose departure builds toward the tail, this pattern is compatible with distributed polygenic signal. It does not prove that confounding is absent: model fit, ancestry matching, and other assumptions still need scrutiny.

Worked Example: A Broad Shift With Little LD-Score Trend

Hypothetical data. Suppose the median chi-square statistic is $0.50043$, the QQ plot sits above the diagonal across a broad range of ranks, and the fitted LD score regression intercept is $1.08$ with slope $0.001$.

The median-based calculation is $$ \lambda = \frac{0.50043}{0.4549364} \approx 1.10. $$ This indicates a median above the null reference, but it does not determine whether the reason is systematic miscalibration, widespread small effects, or a mixture.

At LD score 20, the fitted regression mean is $1.08 + 0.001(20)=1.10$; at LD score 60, it is $1.08 + 0.001(60)=1.14$. The fitted slope is small relative to the intercept in this hypothetical example, while the QQ plot suggests a broad shift rather than a departure confined to the strongest ranks. This combination warrants checks of calibration and potential confounding, but it is not a proof that the intercept has isolated confounding. LD score regression assumptions and fit must be assessed before assigning that interpretation.

Worked Example: Near-Null Lambda With an Extreme Tail

Hypothetical data. Suppose the median chi-square statistic is $0.456$, while the QQ plot closely tracks the diagonal until the most extreme ranks, where several observed p-values are much smaller than expected. For example, at a rank with expected p-value $0.001$, the observed p-value is $0.00001$. An LD score regression fit has intercept $1.01$ and slope $0.015$.

The median-based calculation is $$ \lambda = \frac{0.456}{0.4549364} \approx 1.0023, $$ or $1.00$ to three significant figures. The central part of the distribution is therefore close to its null median. But lambda does not report the tail: at the stated rank, the observed p-value is 100 times smaller than its expected value.

The fitted LD score mean is $1.01 + 0.015(10)=1.16$ at LD score 10 and $1.01 + 0.015(50)=1.76$ at LD score 50. A positive fitted slope and an extreme QQ tail can coexist with a near-one lambda when relatively few statistics are large. The tail could reflect real associations, technical problems, or both; these summaries alone do not decide which. The slope also requires careful interpretation if a small number of strong signals disproportionately influence the fit.

Why the Diagnostics Can Disagree

The contrast follows from what each calculation retains. Lambda uses only the median, so changing a small number of extreme statistics may barely affect it. A QQ plot retains the ordering of all p-values and can show whether departure is broad or concentrated near the tail, but it does not use LD scores and cannot assign a cause to the shape. LD score regression uses the relationship between chi-square statistics and LD scores. It can separate a fitted LD-dependent trend from an intercept under its model, but it is not a general-purpose test that identifies every source of error.

A short justification shows why a median can be elevated by polygenic signal. If a sizeable fraction of variants have nonzero effects, their test-statistic distribution is no longer the central $\chi_1^2$ null distribution. When those effects shift enough statistics upward that the middle order statistic also rises, the numerator of lambda increases, even if the null tests are otherwise calibrated. Conversely, if only a small fraction of variants have strong effects, the middle order statistic may remain near its null value while the upper QQ tail departs sharply. Therefore neither $\lambda>1$ nor $\lambda\approx1$ by itself resolves the source or presence of association signal.

\text{median affected by many shifted tests} \ \Longrightarrow\ \lambda \text{ can rise}; \qquad \text{few extreme tests} \ \not\Longrightarrow\ \text{large change in the median}.

This is a qualitative argument, not a universal threshold rule: the outcome depends on the number and strength of effects, the number of tests, dependence among variants, and calibration. It explains why the three diagnostics should be read together rather than treated as interchangeable.

Assumptions, Diagnostics, and Limitations

The usual interpretations require attention to several conditions. The chi-square statistics must use the intended degrees of freedom and be adequately calibrated under the null. The variants and their summary statistics should be appropriately filtered and aligned with the LD scores used in regression. The LD reference should represent the study population sufficiently well, and the regression mean model should be a reasonable description of the relationship being fitted. Sample overlap, ancestry structure, model misspecification, and outlying associations can complicate interpretation. These considerations are especially important because an apparently precise fitted intercept or slope does not repair a violated model assumption.

Useful diagnostics include examining whether the QQ departure is broad or tail-specific, inspecting whether the LD score regression relationship is approximately linear, checking for influential variants, and comparing results under justified quality-control choices. A marked change in the slope or intercept after removing a few extreme statistics may signal sensitivity rather than a stable genome-wide pattern. Likewise, a poor linear trend or an intercept that changes substantially with reference-panel choice should weaken confidence in a simple LDSC interpretation.

The main alternatives answer different questions. Lambda is fast and compact, but discards almost all distributional detail. A QQ plot is more revealing about the ranks and tails, but has no LD adjustment and does not diagnose causes. LD score regression adds an LD-dependent model, but its interpretation depends on assumptions and reference quality. None is an automatic correction: dividing statistics by lambda, interpreting an intercept as a complete confounding estimate, or treating a diagonal-looking QQ plot as proof of validity can all mislead.

Common Mistake

A common mistake is to conclude, “Lambda is near one, so the GWAS is well calibrated and has no important signals.” The reasoning confuses a central summary with the whole distribution. A near-one median can coexist with a pronounced tail, as in the third worked example. The correction is to inspect the QQ plot and, where suitable, the LD score regression fit, then investigate data quality and model assumptions. The converse mistake is to treat an elevated lambda as proof of confounding: widespread polygenic signal can also elevate the median.

Key takeaway: Lambda describes the median, a QQ plot describes ordered departures across the distribution, and LD score regression describes how mean chi-square statistics vary with LD score. Interpret their agreement or disagreement in light of assumptions; none alone establishes the cause of inflation.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A hypothetical GWAS has median chi-square $0.4549364$. What is lambda, and what does it summarize?
Why might a QQ plot show a strong upper-tail departure when lambda is close to one?
An LD score regression fit has a high intercept and nearly flat slope. What does that pattern suggest, and what does it not prove?
Why should a positive LD score regression slope not automatically be interpreted as proof of polygenic signal?
Which diagnostic most directly shows whether departure is concentrated in the upper ranks: lambda, a QQ plot, or the LD score regression slope?

Solutions. (1) $\lambda=0.4549364/0.4549364=1.00$; it summarizes the median relative to its one-degree-of-freedom null expectation. (2) A small subset of extreme statistics can move the tail while leaving the median nearly unchanged. (3) It is consistent with a broad baseline shift not explained by an LD-dependent trend, but it does not prove that confounding is the sole source or that the intercept cleanly measures it. (4) The interpretation relies on the regression assumptions, appropriate LD scores, and a fit not dominated by misspecification or outliers. (5) The QQ plot directly displays departures by rank.