Contrast Confidence Intervals and Their Interpretation

Why Put an Interval Around a Contrast?

A regression question often concerns more than one coefficient at a time. A researcher might want to compare two estimated effects, or combine several effects with specified weights. A single coefficient interval does not directly describe such a comparison. A confidence interval for a linear contrast does: it gives a range of plausible values for the particular combination of coefficients that answers the question, in the units of the response.

For instance, if two regression coefficients both describe changes in the same response per unit of a predictor, their difference is also measured in response units per predictor unit. If the contrast instead combines coefficients that describe response levels, its units are the response units. The weights determine the comparison, but do not change the underlying units when they are dimensionless.

Definition and Model Conditions

Let a regression model have coefficient vector $\boldsymbol{\beta}=(\beta_1,\ldots,\beta_k)^\mathsf{T}$, estimated by $\hat{\boldsymbol{\beta}}$. Choose fixed weights $c_1,\ldots,c_k$. The corresponding linear contrast and its estimate are

L=\sum_{j=1}^{k}c_j\beta_j \qquad\text{and}\qquad \hat L=\sum_{j=1}^{k}c_j\hat\beta_j.

The contrast is a single target quantity formed from the specified coefficients. For example, setting weights to $1$ and $-1$ gives a difference; other weights produce a weighted sum. A contrast must be estimable from the regression model, so the data and model must contain information to estimate that particular combination.

Definition: A confidence interval for a linear contrast is an interval procedure for the unknown value $L$, constructed from the estimate $\hat L$, its standard error $SE(\hat L)$, and a critical value. In the usual regression $t$ procedure, a $100(1-\alpha)\%$ interval is $$ \hat L\ \pm\ t_{1-\alpha/2,\,df}\,SE(\hat L). $$

This $t$ interval is exact under the usual normal linear model conditions: the mean response is correctly represented by the specified linear model; errors are independent, have mean zero and a common variance; and errors are normally distributed. The contrast must be estimable, and $df$ is the residual degrees of freedom for the fitted model. These assumptions support the regression $t$ inference used here. The standard error $SE(\hat L)$ is the estimated standard deviation of the contrast estimator, calculated using the regression fit and the chosen weights.

The confidence level is chosen in advance, commonly 95%, so $\alpha=0.05$. The critical value $t_{1-\alpha/2,\,df}$ is the $1-\alpha/2$ quantile of the $t$ distribution with $df$ degrees of freedom. It is positive. The estimate, standard error, and critical value all need to refer to the same contrast and fitted model.

Why the Interval Has This Form

Under the stated model conditions, the standardized estimation error for an estimable contrast follows a $t$ distribution with the regression residual degrees of freedom. Thus, with probability $1-\alpha$, the standardized error falls between the two symmetric critical values:

P\left( -t_{1-\alpha/2,\,df} \leq \frac{\hat L-L}{SE(\hat L)} \leq t_{1-\alpha/2,\,df} \right)=1-\alpha.

By symmetry, the same event can be written with $L-\hat L$ in the numerator and the bounds in the same order. Since $SE(\hat L)>0$, multiplying by it preserves both inequality directions. Adding $\hat L$ to all three parts gives the interval form:

-t_{1-\alpha/2,\,df} \leq \frac{L-\hat L}{SE(\hat L)} \leq t_{1-\alpha/2,\,df} $$ $$ -t_{1-\alpha/2,\,df}SE(\hat L) \leq L-\hat L \leq t_{1-\alpha/2,\,df}SE(\hat L) $$ $$ \hat L-t_{1-\alpha/2,\,df}SE(\hat L) \leq L \leq \hat L+t_{1-\alpha/2,\,df}SE(\hat L).

The result is a procedure that would capture the fixed contrast $L$ in $100(1-\alpha)\%$ of repeated samples, provided the model conditions hold. For the particular data set, the calculated interval either contains $L$ or it does not; the confidence level is not a probability assigned to $L$ after seeing the interval.

Worked Contrast Intervals

Worked Example: A Difference Between Two Coefficients

Suppose a hypothetical regression compares two effects on a response measured in milligrams per litre, with both coefficients expressing response change per unit of their respective predictors. The estimated coefficients are $\hat\beta_1=4.8$ and $\hat\beta_2=2.1$. The research question concerns their difference $L=\beta_1-\beta_2$. The fitted model has $df=24$, and the standard error reported for this contrast is $SE(\hat L)=1.10$ milligrams per litre per predictor unit. Assume the usual regression $t$ conditions stated above are appropriate.

The estimated contrast is

\hat L=\hat\beta_1-\hat\beta_2=4.8-2.1=2.70.

For a 95% interval, $t_{0.975,24}\approx2.064$. The margin of error is

$$ 2.064(1.10)=2.2704. $$

Therefore, the interval is

2.70\pm2.2704 = (2.70-2.2704,\ 2.70+2.2704) = (0.430,\ 4.97).

We are 95% confident that the difference between these two predictor effects is between 0.430 and 4.97 milligrams per litre per predictor unit. The interval is entirely above zero, so it indicates a positive difference at the corresponding two-sided 5% test level. It does not establish that changing either predictor causes a response change; the interpretation is about the modeled regression coefficients.

Worked Example: A Weighted Comparison

Consider a hypothetical model with three coefficients describing response levels in milligrams per litre under three conditions. Suppose the target is the average of the first two coefficient values minus the third: $L=0.5\beta_1+0.5\beta_2-\beta_3$. The estimated coefficients are $\hat\beta_1=12$, $\hat\beta_2=15$, and $\hat\beta_3=9$ milligrams per litre. The contrast standard error is $1.80$ milligrams per litre, and the residual degrees of freedom are 18. Assume the coefficients and contrast are estimable and the usual regression conditions hold.

Apply each weight to its corresponding estimate:

\hat L =0.5(12)+0.5(15)-1(9) =6+7.5-9 =4.50\ \text{mg/L}.

For a 95% interval, $t_{0.975,18}\approx2.101$. The margin of error is

2.101(1.80)=3.7818\ \text{mg/L}.

Thus,

4.50\pm3.7818 = (4.50-3.7818,\ 4.50+3.7818) = (0.718,\ 8.28)\ \text{mg/L}.

The interval estimates the difference between the average of the first two modeled coefficient values and the third. Under the model and its assumptions, that contrast is plausibly between 0.718 and 8.28 mg/L. The weights describe the scientific comparison; they are not probability weights. Because the weights sum to zero, adding the same constant to all three coefficient values would leave this comparison unchanged.

Worked Example: An Interval That Includes Zero

In a hypothetical regression, a pre-specified contrast in mean response is $\beta_1-2\beta_2$, measured in points. Its estimate is $\hat L=-1.2$ points, its standard error is $1.50$ points, and the model has 15 residual degrees of freedom. Assume the contrast is estimable and the usual regression $t$ conditions are reasonable.

For a 95% interval, $t_{0.975,15}\approx2.131$, so the margin of error is

2.131(1.50)=3.1965\ \text{points}.

The interval is

-1.2\pm3.1965 = (-1.2-3.1965,\ -1.2+3.1965) = (-4.40,\ 2.00)\ \text{points}.

The data are compatible, at the 95% confidence level, with contrast values from about $-4.40$ to $2.00$ points. Because zero lies inside the interval, the corresponding two-sided test of $H_0:L=0$ at $\alpha=0.05$ would not reject $H_0$. This does not show that the contrast equals zero: values on both sides of zero remain compatible with the data, including effects that may matter scientifically.

Interpreting and Reporting the Interval

A useful interpretation names the contrast, gives its direction, and includes the units. For example: “We are 95% confident that the difference in the two modeled effects is between 0.430 and 4.97 milligrams per litre per predictor unit.” This is more informative than saying merely that “the contrast is significant” or reporting two endpoints without explaining what they measure.

For a conventional two-sided interval, zero provides a useful reference when the null hypothesis is $H_0:L=0$. An interval wholly above zero supports positive values for the contrast at the matching test level; an interval wholly below zero supports negative values. An interval containing zero indicates that zero is among the values not ruled out by this procedure. It does not establish equality, and it does not tell us whether any nonzero values in the interval are practically important.

The confidence level describes the long-run performance of the method, not the probability that the fixed parameter lies in the interval already calculated. If the same sampling process and interval procedure were repeated many times under the model conditions, about 95% of the resulting 95% intervals would contain the fixed contrast.

Common Mistakes

Using the standard error of one coefficient for a contrast. A contrast combines coefficient estimates, so its standard error must correspond to that complete weighted combination. Use $SE(\hat L)$, not automatically the standard error of one coefficient.
Forgetting the weights or their signs. For $L=\beta_1-2\beta_2$, the estimate is $\hat\beta_1-2\hat\beta_2$, not $\hat\beta_1+2\hat\beta_2$. Apply every weight exactly as specified.
Reporting endpoints without units or context. State what is being compared and what a one-unit change means for the contrast’s units.
Calling a zero-containing interval proof of no difference. The interval says zero is compatible with the data at the chosen confidence level; it does not show that the true contrast is exactly zero.
Interpreting confidence as a probability about the fixed contrast. The 95% refers to the coverage rate of the procedure over repeated samples, not a 95% chance that this particular interval contains the fixed value.

A contrast interval matters because it puts uncertainty around the exact scientific comparison of interest, rather than around separate coefficients in isolation. Its usefulness depends on defining the comparison correctly, using the matching standard error and degrees of freedom, and explaining the resulting range in meaningful units.

Check Your Understanding

For each question, identify the contrast or interpret the interval carefully.

If $\hat\beta_1=6.0$, $\hat\beta_2=2.5$, and the target is $\beta_1-\beta_2$, what is the estimated contrast?
A 95% interval for a contrast in response points is $(-1.4,\ 3.2)$. What does inclusion of zero imply about the matching two-sided test of $H_0:L=0$, and what does it not prove?
In a weighted contrast, why must the weights be applied to both the estimated coefficients and the definition of the target parameter?
Explain in repeated-sampling terms what it means for an interval procedure to have 95% confidence coverage.
A contrast interval is reported as $(0.8,\ 2.6)$ milligrams per litre. What information should be added to make its interpretation scientifically clear?