Confidence Intervals For SNP Effects

What a SNP-Effect Interval Reports

A SNP association estimate is not a perfectly known effect. It is calculated from a sample and varies across samples, so a point estimate alone does not show how precisely the association has been estimated. A confidence interval pairs that estimate with a range of values compatible with the data under a specified model and standard-error procedure.

The scale depends on the outcome model. For a quantitative trait, the interval is usually expressed in the units of the trait per coded allele. For a case-control trait fit by logistic regression, the coefficient is on the log-odds scale; exponentiating the coefficient gives an odds ratio, and exponentiating the interval endpoints gives an interval on the odds-ratio scale.

Definition: A two-sided \(100(1-\alpha)\%\) confidence interval for a parameter is an interval-valued procedure that, under its stated assumptions, contains the fixed parameter in a proportion \(1-\alpha\) of repeated samples. For a Wald interval, the estimate is shifted in both directions by a critical value times its estimated standard error.

The confidence level describes the long-run coverage of the procedure, not the probability that a particular calculated interval contains the fixed parameter. Once the data have been observed, the interval either contains that parameter or it does not; the parameter is not treated as random in this interpretation.

Quantitative-Trait Coefficient Intervals

Suppose a quantitative trait \(Y_i\) is modeled using a coded genotype \(G_i\) and covariates \(Z_i\). In an additive model, \(G_i\) is commonly coded as the number of copies of a specified allele, 0, 1, or 2. The coefficient \(\beta_G\) is the expected trait difference associated with one additional coded allele, conditional on the included covariates and the model.

Y_i = \beta_0 + \beta_G G_i + Z_i^\mathsf{T}\gamma + \varepsilon_i

The coefficient is an association conditional on this model; it is not by itself a causal effect. For a large-sample Wald interval, let \(\hat{\beta}_G\) be the estimated coefficient and \(SE(\hat{\beta}_G)\) its standard error. The interval is

\hat{\beta}_G \pm z_{1-\alpha/2}SE(\hat{\beta}_G)

Here \(z_{1-\alpha/2}\) is the corresponding standard-normal quantile; for a 95% interval it is approximately 1.96. With normally distributed errors and the usual homoskedastic linear-model assumptions, a finite-sample interval can instead use a \(t\) critical value with the appropriate residual degrees of freedom. In large GWAS samples, normal critical values are commonly used. If residual variance is not constant, the standard error should use a method appropriate to that setting, such as a heteroskedasticity-robust estimate. The interval's coverage depends on the validity of the model and standard-error procedure.

Worked Example: A Quantitative-Trait Effect in Trait Units

Suppose a hypothetical additive GWAS model estimates a SNP coefficient of \(\hat{\beta}_G=0.42\) mg/dL per coded allele, with a heteroskedasticity-robust standard error of \(0.16\) mg/dL per allele. Assume the large-sample normal approximation is appropriate and construct a 95% interval.

The interval is

0.42 \pm 1.96(0.16) = 0.42 \pm 0.3136

Thus the endpoints are \(0.42-0.3136=0.1064\) and \(0.42+0.3136=0.7336\). Rounded to three significant figures, the 95% confidence interval is 0.106 to 0.734 mg/dL per coded allele.

Under the fitted model and standard-error assumptions, the interval procedure has 95% coverage over repeated samples. In this sample, the estimated conditional association is positive, and the interval excludes zero. This is evidence against a zero coefficient at the corresponding two-sided 5% level; it does not establish that the SNP causes a change in the trait.

Intervals From Logistic Regression

For a binary case-control outcome, a logistic model specifies the probability of being a case conditional on genotype and covariates. In the model below, \(\beta_G\) is the change in log odds of being a case for one additional coded allele, holding the included covariates fixed. Its exponentiation is the per-allele odds ratio.

\log\left(\frac{P(Y_i=1\mid G_i,Z_i)} {1-P(Y_i=1\mid G_i,Z_i)}\right) = \beta_0+\beta_GG_i+Z_i^\mathsf{T}\gamma

When the logistic maximum-likelihood estimate is sufficiently well behaved and its large-sample approximation is reasonable, form the Wald interval on the log-odds scale first. Then exponentiate each endpoint. This preserves the ordering of the endpoints because the exponential function is increasing.

\left[ \hat{\beta}_G-z_{1-\alpha/2}SE(\hat{\beta}_G),\ \hat{\beta}_G+z_{1-\alpha/2}SE(\hat{\beta}_G) \right] \quad\longrightarrow\quad \left[ \exp\{\hat{\beta}_G-z_{1-\alpha/2}SE(\hat{\beta}_G)\},\ \exp\{\hat{\beta}_G+z_{1-\alpha/2}SE(\hat{\beta}_G)\} \right]

On the odds-ratio scale, the null value is 1, corresponding to a log-odds coefficient of 0. An odds-ratio interval entirely above 1 indicates a positive estimated association in the coded-allele direction; an interval entirely below 1 indicates a negative one. These interpretations are conditional on the model and do not imply causation.

Worked Example: Converting a Logistic Interval to an Odds-Ratio Interval

In a hypothetical case-control analysis, suppose the per-allele log-odds estimate is \(\hat{\beta}_G=0.262\), with standard error \(0.105\). Assume the estimate is finite and stable, and the large-sample Wald approximation is suitable.

First calculate the 95% interval for the log-odds coefficient:

0.262 \pm 1.96(0.105) =0.262\pm0.2058

The log-odds endpoints are \(0.0562\) and \(0.4678\). The estimated per-allele odds ratio is \(\exp(0.262)\approx1.300\). Exponentiating the endpoints gives

[\exp(0.0562),\exp(0.4678)] \approx [1.058,1.596]

The 95% odds-ratio interval is approximately 1.058 to 1.596 per coded allele. It excludes 1, so the corresponding two-sided Wald test rejects a zero log-odds coefficient at the 5% level. The interpretation is that the estimated odds of case status are higher per additional coded allele, conditional on the model covariates; the interval quantifies uncertainty in that odds ratio, not in a risk ratio.

Why Interval Width Depends on the Standard Error

For a symmetric Wald interval on the coefficient scale, its total width is twice the critical value times the standard error. At a fixed confidence level, a larger standard error therefore produces a wider interval. A wider interval reflects less precision in the estimate; it does not by itself show that the estimated effect is larger or smaller.

\text{Width}=2z_{1-\alpha/2}SE(\hat{\beta}_G)

For an odds-ratio interval, the endpoints are exponentiated, so the interval is generally asymmetric around the estimated odds ratio. Its width on that scale depends on both the standard error and the location of the log-odds estimate. The log-scale interval remains symmetric around \(\hat{\beta}_G\).

Worked Example: A Wider Interval and an Unchanged Estimate

Consider two hypothetical estimates of the same quantitative-trait coefficient, each equal to 0.20 trait units per allele. Suppose their standard errors are 0.05 and 0.10, and use a 95% normal critical value of 1.96.

For the first estimate, the interval is \(0.20\pm1.96(0.05)=0.20\pm0.098\), or [0.102, 0.298] trait units per allele. Its width is \(0.298-0.102=0.196\).

For the second estimate, the interval is \(0.20\pm1.96(0.10)=0.20\pm0.196\), or [0.004, 0.396] trait units per allele. Its width is \(0.396-0.004=0.392\), twice as large because the standard error doubled. The point estimate did not change; only its estimated precision changed. The second interval barely excludes zero under the stated rounded values, so rounding should not replace checking the unrounded endpoints when making a decision.

Intervals, Tests, and Alternatives

A two-sided Wald test of \(H_0:\beta_G=0\) at significance level \(\alpha\) uses the statistic \(\hat{\beta}_G/SE(\hat{\beta}_G)\) and compares its absolute value with \(z_{1-\alpha/2}\). The corresponding \(100(1-\alpha)\%\) Wald interval excludes zero exactly when that test rejects, apart from differences due to rounding or to using different procedures. For a logistic coefficient, the equivalent interval check is whether the odds-ratio interval excludes 1.

This equivalence applies when the interval and test use the same model, standard-error method, approximation, and two-sided significance level. It does not make an interval a substitute for stating the effect and its uncertainty. Nor does an interval that includes the null prove that the null is true: it says the data do not rule out that value at the stated level using that procedure.

Approach	How it forms an interval	Important consideration
Wald	Estimate plus or minus a critical value times its standard error	Simple and common, but relies on a suitable normal approximation
Profile likelihood	Uses likelihood values across candidate parameter values	Can behave better than Wald intervals when the likelihood is asymmetric, but still depends on a well-defined model fit
Bootstrap	Repeats estimation across resampled data sets	Can reflect sampling variability with fewer distributional assumptions, but depends on a suitable resampling scheme and stable refitting

These alternatives do not remove the need to check whether the model and data support reliable inference. In particular, sparse genotype counts or separation in logistic regression can make estimates and standard errors unstable or prevent a finite maximum-likelihood estimate. In those settings, a routine Wald interval may be misleading. A quantitative-trait interval can also be unreliable if the standard error fails to account for the residual structure relevant to the analysis. The earlier material on heteroskedasticity and standard errors is relevant when choosing a linear-model standard error.

Common Pitfalls and Diagnostics

Reading a confidence interval as a probability about the fixed parameter. The 95% refers to the long-run coverage of the procedure, not a 95% probability that this particular interval contains the parameter.
Exponentiating only the logistic point estimate. An odds-ratio interval requires exponentiating both log-odds endpoints; the estimate alone is not an interval.
Using the wrong null value. A quantitative coefficient is compared with zero; an odds ratio is compared with one.
Treating a wide interval as evidence of a small effect. Width describes precision, while the location of the interval describes the range of effect values compatible with the data and procedure.
Assuming an interval addresses every source of uncertainty. A standard confidence interval is conditional on the specified model, coding, sample, and standard-error method. A nominal single-SNP interval does not automatically provide simultaneous coverage across a genome-wide scan.

Useful checks include confirming the effect units and allele coding, verifying which standard-error method was used, and checking whether the model fit is stable enough for the chosen approximation. For logistic models, inspect for very large estimates or standard errors and sparse genotype-outcome combinations. An interval is a concise uncertainty summary, not a diagnostic that guarantees the underlying analysis is sound.

Worked Example: An Interval That Includes the Null

Suppose a hypothetical quantitative-trait analysis estimates \(\hat{\beta}_G=-0.08\) units per coded allele, with standard error \(0.06\). Under a suitable large-sample normal approximation, the 95% interval is

-0.08\pm1.96(0.06) =-0.08\pm0.1176 =[-0.1976,0.0376]

Because the interval includes zero, the corresponding two-sided 5% Wald test does not reject \(H_0:\beta_G=0\). The estimate is negative, but the data and procedure are also compatible with a small positive coefficient. This is not evidence that the true coefficient is zero, and it is not a 95% probability statement about the parameter.

Key takeaway: Report a SNP estimate with its interval, scale, units, and model context. The standard error governs Wald-interval width; the interval's null-value comparison aligns with a two-sided test only when both use the same assumptions and procedure.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A quantitative-trait coefficient is 0.30 units per allele with standard error 0.10. Calculate a 95% normal Wald interval and state whether it excludes zero.
A logistic coefficient interval is [−0.20, 0.10]. Give its odds-ratio interval and identify whether it excludes the null.
What happens to the width of a 95% symmetric Wald interval if the standard error is multiplied by 1.5 while the estimate and critical value stay fixed?
In frequentist terms, what does a 95% confidence level mean?

Solutions. (1) \(0.30\pm1.96(0.10)=0.30\pm0.196\), giving [0.104, 0.496] units per allele; it excludes zero. (2) Exponentiate both endpoints: \([\exp(-0.20),\exp(0.10)]\approx[0.819,1.105]\). It includes 1, so it does not exclude the logistic null. (3) Width is \(2(1.96)SE\), so multiplying \(SE\) by 1.5 multiplies the interval width by 1.5. (4) Across repeated samples, 95% of intervals constructed by that procedure contain the fixed parameter when the assumptions hold.