Why Protein QTLs Need Assay-Aware Interpretation
A protein quantitative trait locus (protein QTL, or pQTL) is a genetic variant associated with variation in a measured protein phenotype. That phenotype is not the protein molecule in the abstract: it is a value produced by an assay. Affinity-based assays infer abundance from the binding of a reagent to a protein or epitope, while mass-spectrometry assays infer it from detected peptide signals. Both can be useful, but their reported values may use different scales, have different detection limits, and carry different measurement errors.
The key distinction is between a genetic association with the assay measurement and a genetic association with the amount of protein in a biological sample. These often align, but they need not. For example, a variant could alter an epitope recognized by an affinity reagent or a peptide used for quantification. In that case, the assay signal might change even if total protein abundance does not. A careful analysis therefore defines the measured phenotype, uses an effect scale that can be interpreted, and considers how assay properties could shape the result.
A Dosage Model for Measured Protein Abundance
For donor \(i\), let \(G_i\) be the number of copies of a designated effect allele, coded 0, 1, or 2. Let \(Y_i\) be the assay’s protein-abundance measurement, and let \(C_i\) denote measured covariates included in the model. A standard starting point is
Here, \(\alpha\) is an intercept, \(\beta\) is the per-allele association on the scale of \(Y_i\), and \(\gamma\) is a vector of covariate coefficients. The term \(u_i\) represents donor-level variation not explained by the included predictors, while \(e_i\) represents assay measurement error. Their sum is the regression error. Depending on the data, a model may use only one residual term; separating the terms is useful for thinking about biological variation and assay precision.
The dosage model assumes a linear mean relationship across the three genotype dosages: each additional effect-allele copy is associated with the same expected change \(\beta\), conditional on the covariates. It does not estimate separate heterozygote and homozygote effects. For the usual least-squares test and confidence interval, also assume independent donors, a correctly specified conditional mean, and residuals with mean zero given the full design. Normal, constant-variance residuals give the familiar exact small-sample \(t\)-based inference; with larger samples, the inference may instead rely on appropriate large-sample or robust methods.
For clarity, write \(D\) for the complete design matrix containing the genotypes, covariates, and intercept, and \(\varepsilon\) for the vector of total errors. A key conditional-mean assumption is \(E(\varepsilon\mid D)=0\). This conditions on the entire observed design, not merely on each donor’s genotype considered separately. Independence across donors, together with suitable donor-level conditional-mean assumptions, is one way to justify this condition; it should not be silently inferred from weaker assumptions.
Assay Scale Determines the Meaning of the Effect
The units and transformation of \(Y_i\) determine the units of \(\beta\). If \(Y_i\) is a log-base-2 abundance value, \(\beta\) is the expected change in log-base-2 units per allele. Under that scale, a one-unit increase corresponds to a twofold ratio, and a change of \(\beta\) corresponds to a ratio of \(2^\beta\). If \(Y_i\) is standardized, \(\beta\) is in standard-deviation units per allele. If it is a platform-specific normalized signal, the effect is in those assay units and may not have a direct conversion to concentration.
Transformations can help make a phenotype suitable for a linear model, but they do not make all assays comparable. Two platforms can use different calibration curves, normalization procedures, and protein-specific responses. Thus, a numerically larger coefficient on one platform need not mean a larger biological effect than a smaller coefficient on another. Report the phenotype scale and, where available, its calibration and normalization.
Worked Example: Interpreting a Log-Base-2 Effect
Suppose a hypothetical affinity assay reports a protein phenotype on a log-base-2 scale, and the fitted dosage coefficient is \(\hat{\beta}=0.30\) log-base-2 units per allele. Assume the model’s linear dosage interpretation is appropriate and the value is not dominated by censoring.
The estimated ratio in the measured abundance scale for one additional effect-allele copy is
Thus, each additional copy is associated with an estimated 1.231-fold measured abundance, or about a 23.1% higher value, conditional on the model’s covariates. This is an association on the assay’s log-base-2 scale. The calculation alone does not establish that the variant causes a 23.1% change in the amount of protein in the body; that interpretation also depends on what the assay measures.
Worked Example: Converting a Standardized Effect to a Log Scale
Suppose a hypothetical protein phenotype was standardized by dividing its log-base-2 abundance by a sample standard deviation of \(0.50\) log-base-2 units. A fitted genetic coefficient on the standardized phenotype is \(0.12\) standard deviations per allele. To express the corresponding effect on the original log-base-2 scale, multiply by the standard deviation used for standardization:
The associated ratio is \(2^{0.060}\approx 1.042\), or about 4.2% higher measured abundance per allele. This conversion is valid only if the stated standard deviation is the one used to create the standardized phenotype and the model uses the same linear dosage coding. Without those details, the standardized coefficient is interpretable in standard-deviation units, but not directly as a percentage or concentration change.
Detection Limits and Censored Measurements
An assay’s lower detection or quantification limit matters because a value below that limit is not necessarily zero. It may mean that the abundance is too low for reliable quantification, or that the assay returns only a bound. Replacing every below-limit result with zero, the limit itself, or an arbitrary small constant treats an unknown value as if it were measured exactly. The resulting phenotype can have a distorted mean and genotype relationship, particularly if the fraction below the limit differs by genotype.
A linear regression on observed quantitative values is most straightforward when measurements are available across the relevant range and the residual assumptions are reasonable. If a substantial fraction of values is censored, simply analyzing detected samples can also be misleading: it conditions on being detectable, and detectability may depend on abundance and genotype. The analysis should retain the detection-limit information and use a model appropriate to the measurement process, or otherwise make clear that the target is an association among the selected, detectable samples rather than the full population. A detection limit is a feature of the observation process, not evidence that the underlying protein amount equals zero.
Worked Example: How a Detection Floor Changes a Dosage Slope
Consider a hypothetical protein with two latent log-abundance measurements at each genotype dosage. The latent values are \(-1.2,-0.8\) for dosage 0; \(-0.2,0.2\) for dosage 1; and \(0.8,1.2\) for dosage 2. The true group means are \(-1,0,1\), which follow a linear increase of 1 unit per allele. Suppose the assay reports values below a floor of 0 as 0. The recorded values then have group means \(0,0,1\).
Using the six recorded values, the mean dosage is \(\bar{G}=1\), and the mean recorded phenotype is \(\bar{Y}=1/3\). The least-squares dosage slope is
For the numerator, the two dosage-0 observations contribute zero because their recorded values are zero and their centered dosages are \(-1\); the two dosage-1 observations also contribute zero; and the dosage-2 observations contribute \(1(0.8)+1(1.2)=2\). The denominator is \(2(1^2)+2(0^2)+2(1^2)=4\). The fitted slope of 0.50 is smaller than the latent group-mean slope of 1. This small illustration is not a method for estimating a censored-data effect; it shows why flooring values can change the association estimate.
Measurement Error: Precision, Bias, and Assay Response
If assay error is additive, has conditional mean zero given the full design, and is independent across donors, it contributes to residual variation. Provided the conditional mean model is otherwise correct, this outcome measurement error does not by itself attenuate the regression coefficient in the way that classical error in a predictor can. It generally makes the estimate less precise by increasing residual variance. The scale factor and the form of the assay response still matter: error that depends on genotype, abundance, or a binding interaction is not simply independent additive noise.
Protein-specific precision can differ substantially. A protein with high technical variability may yield a larger standard error and weaker evidence of association than a similarly sized effect measured precisely. Technical replicate measurements can help characterize repeatability, but averaging replicates does not correct a genotype-dependent assay response or a detection-floor problem. A QTL result should therefore be interpreted in light of the protein’s measurement properties rather than by treating all assay phenotypes as equally precise.
Worked Association Calculation
Worked Example: A Dosage Association on a Log-Base-2 Scale
In a hypothetical study, 12 unrelated donors have protein measurements above the assay’s quantification limit. There are four donors at each dosage, 0, 1, and 2. Their log-base-2 group means are 10.0, 10.4, and 10.8. Within each genotype group, the four deviations from that group mean are \(-0.2,-0.1,0.1,0.2\). Assume no additional covariates are needed for this illustration, and assume independent donors, normal errors with a common variance, and zero conditional mean given the complete genotype vector.
The group means increase by 0.4 for each dosage step, so the fitted dosage slope is \(\hat{\beta}=0.40\) log-base-2 units per allele. The residual sum of squares is three times the within-group sum of squared deviations:
There are \(n-2=10\) residual degrees of freedom for an intercept-and-slope model. The mean squared error is \(MSE=0.30/10=0.030\). The dosage sum of squares is
Therefore, the standard error and \(t\) statistic for testing \(H_0:\beta=0\) against a two-sided alternative are
With \(df=10\), the two-sided \(p\)-value is approximately \(0.00007\). A 95% confidence interval uses \(t_{0.975,10}\approx2.228\):
Under the stated model, the data provide evidence of a dosage association, with an estimated effect of 0.40 log-base-2 units per allele. The interval describes values compatible with the model and data under repeated sampling; it is not a 95% probability statement about this fixed parameter. Since \(2^{0.40}\approx1.320\), the point estimate corresponds to about 32.0% higher measured abundance per allele. The inference is conditional on the model assumptions and the assay-scale interpretation. The hypothetical result does not establish a causal effect or rule out an assay-specific explanation.
Why the Conditional-Mean Assumption Matters
A short least-squares justification shows why assumptions about the whole design matter. Let \(X\) denote the full design matrix and \(\boldsymbol{\beta}\) the vector of regression coefficients. If \(Y=X\boldsymbol{\beta}+\varepsilon\) and \(X\) has full column rank, the least-squares estimator is
Conditioning on \(X\), the entries of \(X\) are fixed, so
when \(E(\varepsilon\mid X)=0\). This establishes conditional unbiasedness under the specified linear model; it does not establish that the model is biologically correct or that the assay measures abundance without genotype-dependent distortion. In particular, knowing only that each error has mean zero given that donor’s genotype does not automatically establish a zero conditional mean given the entire genotype vector when donors may be dependent. State the stronger design-conditional assumption directly, or justify it with suitable assumptions about donor independence and the conditioning variables.
Common Mistakes and Practical Interpretation
- Calling the assay signal “protein concentration” without qualification. A normalized or standardized signal has its own units. State the measured phenotype and its scale before interpreting the coefficient.
- Assuming outcome measurement error always shrinks a genetic effect toward zero. Independent, mean-zero additive error in the outcome typically increases uncertainty rather than attenuating the slope. Genotype-dependent assay response can instead bias or create an association.
- Replacing below-limit values with zero and treating them as exact. A nondetection is not necessarily zero abundance. Retain the censoring information and use an analysis appropriate to the observation process when censoring is material.
- Interpreting a per-allele slope as a fully general genotype effect. The dosage model imposes a linear trend across 0, 1, and 2 copies. It does not allow an arbitrary heterozygote effect.
- Equating association with causation. A small \(p\)-value supports evidence against the specified null model; it does not by itself prove that the variant changes biological protein abundance or identify the mechanism.
Check Your Understanding
Answer each question using the assay scale and model assumptions in the explanation.
- A dosage coefficient is 0.25 on a log-base-2 abundance scale. What estimated fold ratio does this correspond to per allele, and what does that ratio describe?
- Why can independent, additive measurement error in the outcome increase the standard error without necessarily attenuating the dosage coefficient?
- An assay reports all values below a threshold as zero. Why is ordinary regression on these values not automatically an unbiased analysis of abundance?
- What assumption does a linear dosage model make about the effects for genotype dosages 0, 1, and 2?
- Why is a conditional-mean statement given the full design matrix stronger than separate statements conditional only on each donor’s genotype?