Deriving The Linear Regression SNP Estimate

From Dosage to a Regression Estimate

In an additive quantitative-trait analysis, the SNP dosage \(G_i\) records the number of effect alleles for individual \(i\), usually as 0, 1, or 2; an imputed dosage may take intermediate values. The regression coefficient is estimated from the way the trait values vary with dosage. This tutorial derives that estimate and its standard error for a model with an intercept and one dosage predictor, without additional covariates.

The dosage slope is not calculated by comparing only the average trait values in three genotype groups. Least squares uses all observed dosage-trait pairs and chooses the line that minimizes the sum of squared vertical residuals. The resulting standard error quantifies sampling uncertainty under specified error assumptions; those assumptions are also what make the conventional t-test appropriate.

The slope \(\beta_1\) is the change in the model’s mean trait value for a one-unit increase in dosage, with units of trait per dosage copy. The intercept \(\beta_0\) is the model’s mean at dosage zero. The error \(\varepsilon_i\) is the difference between the observed trait and its conditional mean. Least squares estimates the intercept and slope by minimizing the residual sum of squares.

Deriving the Least-Squares Estimates

Write \(\bar{G}\) and \(\bar{Y}\) for the sample means of dosage and trait. Define the centered sums of squares and cross-products \(S_{GG}=\sum_{i=1}^n(G_i-\bar{G})^2\) and \(S_{GY}=\sum_{i=1}^n(G_i-\bar{G})(Y_i-\bar{Y})\). The least-squares objective for proposed coefficients \(b_0,b_1\) is \(Q(b_0,b_1)=\sum_{i=1}^n(Y_i-b_0-b_1G_i)^2\).

These formulas follow by minimizing \(Q\). Differentiating with respect to \(b_0\) and setting the derivative to zero gives \(\sum_i(Y_i-b_0-b_1G_i)=0\), hence \(b_0=\bar{Y}-b_1\bar{G}\). Substituting that expression into the objective leaves a function of the slope alone:

Differentiating this expression and setting the derivative to zero gives \(-2S_{GY}+2b_1S_{GG}=0\). Because \(S_{GG}>0\), the unique minimizing slope is \(S_{GY}/S_{GG}\), and substituting it into the intercept expression gives the stated estimate. This also shows why a monomorphic SNP, with the same dosage for everyone, has no estimable slope in this model: its \(S_{GG}\) is zero.

Residual Variance and the Standard Error

For each individual, the fitted value is \(\widehat{Y}_i=\widehat{\beta}_0+\widehat{\beta}_1G_i\), and the residual is \(e_i=Y_i-\widehat{Y}_i\). The residual sum of squares is \(SSE=\sum_i e_i^2\). With an intercept and one estimated slope, the residual variance estimate divides \(SSE\) by \(n-2\), the residual degrees of freedom:

Here \(s_e^2\) estimates the common error variance \(\sigma^2\). The standard error is an estimate of the standard deviation of the slope estimate across repeated samples under the model. A larger residual variance increases the standard error; greater spread in the observed dosages, measured by \(S_{GG}\), decreases it.

The reason for this formula can be seen by treating the observed dosages as fixed. Substituting the model into the slope formula and using \(\sum_i(G_i-\bar{G})=0\) gives \(\widehat{\beta}_1-\beta_1=\sum_i(G_i-\bar{G})\varepsilon_i/S_{GG}\). Under independent errors with variance \(\sigma^2\), the variance of this weighted sum is \(\sigma^2\sum_i(G_i-\bar{G})^2/S_{GG}^2=\sigma^2/S_{GG}\). Replacing the unknown \(\sigma^2\) with \(s_e^2\) gives the estimated standard error above. Under the normal-error assumption, the standardized estimation error divided by the estimated residual standard deviation has a \(t\) distribution with \(n-2\) degrees of freedom. The two degrees of freedom are used to estimate the intercept and slope.

The Usual t-Test and Confidence Interval

To test whether the dosage coefficient is zero, use \(H_0:\beta_1=0\) against a specified alternative, commonly \(H_a:\beta_1\ne0\). Under the model assumptions, the test statistic has a \(t\) distribution with \(n-2\) degrees of freedom when \(H_0\) is true. The two-sided p-value is the probability, under that null model, of a t statistic at least as far from zero as the observed statistic.

A \(100(1-\alpha)\%\) confidence interval for the slope is \(\widehat{\beta}_1\pm t_{1-\alpha/2,n-2}SE(\widehat{\beta}_1)\), where \(t_{1-\alpha/2,n-2}\) is the corresponding quantile of the \(t\) distribution. Its coverage interpretation concerns repeated samples under the model: the procedure produces intervals that contain the fixed coefficient at the stated rate when the assumptions hold.

Assumptions, Diagnostics, and Alternatives

The exact t reference distribution requires the stated normal, independent, equal-variance errors and a correctly specified linear conditional mean. Normality is not required for the least-squares formulas themselves, but it is required for the exact finite-sample t result. With sufficiently large samples, t-based inference may be approximately valid under weaker distributional conditions, but that is not a guarantee when errors are strongly dependent, highly influential, or severely heteroskedastic.

Useful checks include plotting residuals against fitted values and dosage, looking for changing residual spread or a systematic curve, and identifying observations with unusually large residuals or leverage. A residual-versus-fitted plot with a funnel shape suggests unequal variance; curvature suggests that a straight-line mean may be inadequate. These checks can reveal warning signs but cannot establish independence or prove that errors are normal. In genomic data, related individuals or other sources of sample dependence can make the independent-error standard error inappropriate.

When the assumptions fail, the slope formula can still describe the least-squares line, but the conventional standard error and t calibration may be unreliable. Heteroskedasticity-robust standard errors address unequal variances under suitable independence conditions; resampling or permutation approaches require their own validity conditions. These are alternative inferential procedures, not changes to the least-squares slope itself. A dosage with very little variation also yields a small \(S_{GG}\), and therefore potentially a large standard error; no statistical method can recover information absent from the observed genotypes.

In the simple model with an intercept, testing the slope is equivalent to testing zero linear correlation between dosage and trait: both tests are based on the same centered cross-product and have the same t statistic. This equivalence is specific to the one-predictor setting. It does not mean that correlation is a substitute for a regression model in settings where additional predictors or a different conditional mean are needed.

Common Mistake

A common error is to divide \(SSE\) by \(n\), or by \(n-1\), when estimating residual variance for this regression. The correction is to use \(n-2\): two parameters, the intercept and slope, have been estimated from the same observations. A second error is to use the resulting t-test automatically whenever a slope can be calculated. The slope is algebraically available whenever \(S_{GG}>0\), but its usual standard error and t-test rely on the error assumptions; calculation alone does not validate inference.