The TWAS Estimand and Its Interpretation

What Does a TWAS Association Describe?

A transcriptome-wide association study (TWAS) asks whether a trait is associated with the component of gene expression that can be predicted from genetic variation. The target is therefore not, in general, the effect of changing a person's measured expression. It is an association between the trait and genetically predicted expression, defined by a prediction model and evaluated in a specified population.

That distinction matters because measured expression also reflects non-genetic influences, such as cell state, environment, and measurement variation. A genetic prediction represents a particular component of expression, on a scale determined by its model. The TWAS estimand describes how the trait varies with that predicted component; it does not by itself say what would happen if expression were experimentally increased.

Definition: Let $G$ denote the genotype information used for a gene, and let $w$ be a fixed vector of prediction weights. Genetically predicted expression is $X^G=w^\mathsf{T}G$. For a continuous trait $Y$, the marginal TWAS estimand is the population linear-projection slope $$ \beta_{\mathrm{TWAS}}=\frac{\operatorname{Cov}(X^G,Y)}{\operatorname{Var}(X^G)}, $$ provided $\operatorname{Var}(X^G)>0$. It is the change in the best linear prediction of $Y$ associated with a one-unit increase in $X^G$, in the specified population and on the specified scales.

Here, $w$ is treated as fixed when defining the target. It may have been obtained from expression and genotype data in a training sample, but the estimand concerns the prediction rule as applied to the target population. The details of how prediction weights are obtained are separate from what their association with a trait means.

The formula is a population quantity, not a result calculated from one particular dataset. In an analysis, a sample slope is used to estimate it. Both the population and the prediction rule matter: changing the target population can change the distribution of genotypes and traits, while changing $w$ changes the predicted-expression variable itself. Thus, two analyses bearing the same gene name need not target precisely the same parameter if their weights, samples, or measurement scales differ.

From the Estimand to a Regression Interpretation

For a continuous trait, the marginal estimand can be stated as the slope in a population linear projection:

Y=a+\beta_{\mathrm{TWAS}}X^G+\varepsilon, \qquad E(\varepsilon)=0,\quad \operatorname{Cov}(X^G,\varepsilon)=0.

This is a description of the best linear association, not a claim that the conditional mean of $Y$ must be exactly linear in $X^G$. The intercept $a$ is the projection's intercept, and the residual $\varepsilon$ is uncorrelated with the predictor by the definition of a linear projection. No causal interpretation follows from these properties.

In some analyses, covariates are included. The corresponding estimand is then the coefficient of predicted expression in the population linear projection of $Y$ on $X^G$ and those covariates. It describes an association conditional on the specified covariates, and can differ from the marginal slope. The covariates, target population, and trait scale should therefore be stated when interpreting a reported coefficient.

Interpretation: If $Y$ is measured in millimeters of mercury and $X^G$ in the prediction model's expression units, a coefficient of 2.0 means that a one-unit higher genetically predicted expression is associated with a 2.0 mmHg higher trait value in the stated analysis. It does not mean that experimentally raising measured expression by one unit would raise the trait by 2.0 mmHg.

Why the Regression Slope Has the Covariance Form

The covariance expression follows directly from the definition of a least-squares linear projection. For a population with finite second moments, choose $a$ and $b$ to minimize $E[(Y-a-bX^G)^2]$. Differentiating with respect to $a$ and setting the result to zero gives $a=E[Y]-bE[X^G]$. Differentiating with respect to $b$ gives $E[X^G(Y-a-bX^G)]=0$. Substituting the expression for $a$ yields:

\begin{aligned} 0 &=E[(X^G-E[X^G])(Y-E[Y]-b(X^G-E[X^G]))]\\ &=\operatorname{Cov}(X^G,Y)-b\operatorname{Var}(X^G). \end{aligned}

When $\operatorname{Var}(X^G)>0$, solving this equation gives $b=\operatorname{Cov}(X^G,Y)/\operatorname{Var}(X^G)$. This derivation establishes the slope's association interpretation. It does not establish that predicted expression causes the trait.

Worked Example: Calculating a Sample Association Slope

Suppose a hypothetical analysis applies a fixed prediction model to four individuals. Their predicted expression values are 0, 1, 2, and 3 arbitrary expression units, and their trait values are 2, 3, 3, and 6 mmHg. Treat these observations only as a small illustration of the slope calculation, not as evidence from a real study.

The sample means are $\bar{x}^G=(0+1+2+3)/4=1.5$ expression units and $\bar{y}=(2+3+3+6)/4=3.5$ mmHg. The sample regression slope is:

\hat{\beta} = \frac{\sum_i (x_i^G-\bar{x}^G)(y_i-\bar{y})} {\sum_i (x_i^G-\bar{x}^G)^2} = \frac{2.25+0.25-0.25+3.75}{2.25+0.25+0.25+2.25} = \frac{6}{5} =1.2.

The sample slope is 1.2 mmHg per predicted-expression unit. In this hypothetical sample, individuals differing by one unit in predicted expression differ by 1.2 mmHg in the fitted linear trait value. The slope is a sample estimate of a population association parameter; these four observations do not establish the size or certainty of that population parameter.

Predicted Expression Is Not Observed Expression

Let $X$ denote measured expression and $X^G$ denote expression predicted from genotype. They are different variables. Measured expression can include genetically predictable variation as well as variation not captured by the prediction model. A regression of $Y$ on $X$ therefore targets an association with observed expression, whereas a regression of $Y$ on $X^G$ targets an association with the genetically predicted component.

The two associations can differ in size, direction, or uncertainty. This is not a contradiction: their predictors differ, and so do the parameters being described. Even when measured expression is used to train a genetic prediction model, the prediction and the measured value are not interchangeable in the target analysis.

Worked Example: Two Predictors, Two Associations

In a second hypothetical four-person dataset, a fixed model gives predicted-expression values $0,1,2,3$ units. The observed expression values are $1,3,2,6$ units, and the trait values are $2,4,3,10$ units. Calculate the two sample slopes to see how changing the predictor changes the association being summarized.

For predicted expression, $\bar{x}^G=1.5$ and $\bar{y}=4.75$. The centered cross-products sum to $4.125+0.375-0.875+7.875=11.5$, and the centered predicted-expression squares sum to $5$. Thus:

\hat{\beta}_{\text{predicted}} =\frac{11.5}{5}=2.3 \quad\text{trait units per predicted-expression unit}.

For observed expression, $\bar{x}=3$. Its centered cross-products with the trait sum to $5.5+0+1.75+15.75=23$, and its centered squares sum to $4+0+1+9=14$. Therefore:

\hat{\beta}_{\text{observed}} =\frac{23}{14}\approx1.643 \quad\text{trait units per observed-expression unit}.

The numerical slopes differ because the predictors differ. The first summarizes association with the model-predicted genetic component; the second summarizes association with measured expression. Neither slope, by itself, is an estimate of the causal effect of intervening on expression.

Scale and Analysis Choices Shape Interpretation

A coefficient is meaningful only with its predictor and outcome scales. Predicted expression may be reported in unstandardized model units, standardized units, or another chosen scale. If the predictor is rescaled, the numerical coefficient changes even though the underlying fitted association can be represented equivalently.

Worked Example: Interpreting a Rescaled Prediction

Suppose a hypothetical analysis reports a slope of 0.8 trait units per unit of predicted expression, where the predicted-expression standard deviation in the target population is 0.25 units. Under the same linear model, a one-standard-deviation increase corresponds to 0.25 predicted-expression units, so the associated fitted trait difference is:

0.8\ \frac{\text{trait units}}{\text{expression unit}} \times 0.25\ \text{expression units} =0.20\ \text{trait units}.

If instead the prediction is recorded on a scale twice as large, the numerical slope is halved: $0.8/2=0.4$ trait units per rescaled unit. The product of the slope and the corresponding predictor change remains 0.20 trait units. The unit change must be made explicit; a coefficient without the prediction scale is difficult to interpret.

Outcome processing matters in the same way. A coefficient for a trait measured in its original units is not numerically the same as a coefficient for a standardized trait. For non-continuous outcomes, the association parameter and its scale depend on the chosen model; the continuous-trait slope above should not be silently interpreted as an odds ratio or as another type of effect measure.

Adjustment choices also define the target. For example, a coefficient from a model that includes ancestry-related covariates is a conditional association given those covariates in that model, rather than the unadjusted marginal association. Adjustment can be important, but it does not turn an association coefficient into a causal effect. The coefficient should be interpreted in the context of the precise model and target population used.

Association Is Not a Causal Effect

The word “effect” is sometimes used informally when describing a regression coefficient, but it can suggest an intervention that the estimand does not encode. A TWAS coefficient, as defined here, quantifies how the trait is associated with genetically predicted expression. Genotype-based prediction does not by itself guarantee that the association operates through the expression of the named gene. Genetic variants may relate to the trait through other pathways, and the prediction can capture only part of expression variation.

Accordingly, a TWAS association is evidence of a relationship between a specified genetic prediction and a trait under a specified analysis. Additional assumptions and evidence are needed for claims about a causal expression effect. A small p-value, if one is calculated, concerns evidence against a statistical null for the association parameter; it does not establish the probability that a causal interpretation is true.

Common mistake: “The TWAS coefficient is positive, so increasing the gene's measured expression will increase the trait.” This treats predicted expression as observed expression and treats an association as an intervention effect. The correction is to state that the coefficient describes the trait association with genetically predicted expression, with its units and analysis conditions. A causal claim requires additional justification.

What to Report

A clear interpretation identifies the predictor, the trait, the scale, and the population or analysis context. Where relevant, it also states whether the estimate is marginal or adjusted. These details prevent a coefficient from being mistaken for a universal gene effect.

Name the gene's genetically predicted expression as the predictor, rather than simply saying “expression” if measured expression was not analyzed.
Give the outcome scale and the predicted-expression scale, including whether either variable was standardized.
State whether the reported association is marginal or conditional on specified covariates.
Describe the result as an association unless a causal interpretation has been supported by additional assumptions and evidence.

Check Your Understanding

Use the definitions and distinctions in this tutorial to answer the following questions.

In one sentence, define the marginal continuous-trait TWAS estimand and state what a one-unit increase in its predictor means.
Why can the association of a trait with measured expression differ from its association with genetically predicted expression?
A coefficient is reported in trait units per standardized predicted-expression unit. What information is needed to interpret it in the original expression scale?
Explain why adding covariates can change the estimand being reported, even when the gene and trait are unchanged.
Give the correction to the claim, “A positive TWAS coefficient proves that raising observed expression causes the trait to rise.”