Reporting Conditional Effects and Predictions

Why Reporting a Conditional Result Takes More Than a Number

A coefficient from an interaction model does not usually summarize an effect for every person or predictor value. The estimated effect of one predictor can depend on a moderator, and a fitted outcome can depend on a whole predictor profile. A result such as “the effect was 1.45” is therefore incomplete unless readers know which effect, for whom or at what predictor values, in what units, and with what uncertainty.

A good report also indicates whether the relevant moderator value or predictor combination is represented in the observed data. A precisely calculated estimate at an unsupported profile may be an extrapolation rather than a useful description of the data. The aim here is to turn a conditional calculation into a concise, interpretable statement without implying more than the model supports.

Definition: A conditional effect is the model-implied change in the expected outcome associated with a change in a predictor, evaluated at a stated value or profile of other predictors. A fitted mean prediction is the model's estimated expected outcome at a stated predictor profile. Neither is automatically a causal effect.

What a Complete Results Statement Should Include

For an interaction result, identify the outcome and its units, the predictor effect or predicted mean being reported, and the moderator value or predictor profile at which it is evaluated. Give the estimate and an appropriate measure of uncertainty, usually a confidence interval, and describe the data range that supports the profile. If a result is a group contrast, specify which group is subtracted from which.

These details prevent several different quantities from being mistaken for one another. A conditional slope is a change in expected outcome per unit change in a predictor. A group contrast is a difference in expected outcomes between specified groups at a stated profile. A fitted mean prediction is an expected outcome at a profile. Their units and interpretations differ, even if each is calculated from the same fitted model.

For the worked examples, assume a linear conditional-mean model appropriate to the stated predictors. Assume observations are independent, the conditional mean is correctly specified over the reported data region, and the errors have mean zero and constant variance; for the displayed small-sample t intervals, also assume normally distributed errors conditional on the predictors. The examples use conventional model-based standard errors. In an applied analysis, the report should name the uncertainty method actually used if it differs.

Reporting rule: State the conditional quantity, the exact moderator value or predictor profile, its estimate and uncertainty in meaningful units, and the supported data range. Describe an association as an association unless the study design and assumptions justify a causal interpretation.

Uncertainty for a Conditional Estimate

Conditional effects and fitted means in a linear model are often linear combinations of estimated coefficients. If the coefficient vector is \(\hat{\boldsymbol{\beta}}\), a target quantity can be written as \(c^\mathsf{T}\hat{\boldsymbol{\beta}}\), where the vector \(c\) encodes the moderator value or predictor profile. Let \(\widehat{V}\) denote the estimated covariance matrix of the coefficient estimates. The estimated standard error of that target is obtained from its variance:

SE\!\left(c^\mathsf{T}\hat{\boldsymbol{\beta}}\right) = \sqrt{c^\mathsf{T}\widehat{V}c}.

This expression includes the variances of the relevant coefficients and their covariances. The covariance terms matter: adding coefficient estimates to form a conditional effect does not generally mean their standard errors can simply be added. With the stated model assumptions, a two-sided confidence interval takes the form:

c^\mathsf{T}\hat{\boldsymbol{\beta}} \ \pm\ t^* SE\!\left(c^\mathsf{T}\hat{\boldsymbol{\beta}}\right),

where \(t^*\) is the appropriate t critical value for the stated confidence level and degrees of freedom. The interval describes uncertainty in the conditional effect or fitted mean. It is not a range expected to contain a specified fraction of individual outcomes. Individual outcomes also vary around their conditional mean; that additional variability is included in a prediction interval for a new outcome.

A short justification

For a fixed profile, \(c\) is fixed, so the variance of a linear combination follows from the covariance matrix:

\operatorname{Var}\!\left(c^\mathsf{T}\hat{\boldsymbol{\beta}}\right) = c^\mathsf{T}\operatorname{Var}\!\left(\hat{\boldsymbol{\beta}}\right)c \ \approx\ c^\mathsf{T}\widehat{V}c.

Taking the square root gives the standard error. Standardizing the difference between the estimate and its target by this standard error yields a t statistic under the stated assumptions. Inverting the corresponding t interval gives the estimate plus or minus \(t^*\) standard errors. This is why the covariance between coefficients must be included when a conditional result combines them.

Worked Example: Reporting a Slope at a Stated Moderator Value

Suppose a hypothetical study models quiz score in points using weekly practice hours \(x\), sleep hours \(z\), and their interaction:

E(Y\mid x,z) = \beta_0+\beta_1x+\beta_2z+\beta_3xz.

Assume the fitted estimates are \(\hat{\beta}_1=3.20\) quiz points per practice hour and \(\hat{\beta}_3=-0.25\) quiz points per practice hour per sleep hour. At \(z=7\) sleep hours, calculate and report the conditional practice-hour slope. The estimated variances are \(\widehat{\operatorname{Var}}(\hat{\beta}_1)=0.16\) and \(\widehat{\operatorname{Var}}(\hat{\beta}_3)=0.0025\), and their covariance is \(-0.005\). Suppose the relevant critical value is \(t^*=2.02\).

Conditional estimate. Differentiating the fitted mean with respect to \(x\), or comparing its values as \(x\) changes, gives the slope \(\beta_1+\beta_3z\). At seven sleep hours, the estimate is:

\hat{\beta}_1+7\hat{\beta}_3 = 3.20+7(-0.25) = 1.45 \text{ quiz points per practice hour}.

Uncertainty. The variance of this sum includes the covariance term:

\begin{aligned} \widehat{\operatorname{Var}}(\hat{\beta}_1+7\hat{\beta}_3) &=0.16+7^2(0.0025)+2(7)(-0.005)\\ &=0.16+0.1225-0.0700\\ &=0.2125, \end{aligned} $$ $$ SE=\sqrt{0.2125}=0.461.

The 95% confidence interval is \(1.45\pm2.02(0.461)\), or approximately \(0.52\) to \(2.38\) quiz points per practice hour. Suppose the observed data include sleep values from 5 to 9 hours and, among students with 7 hours of sleep, practice values from 1 to 6 hours. The requested slope is evaluated at a supported moderator value.

Report: “Among students reporting 7 hours of sleep, each additional weekly practice hour was associated with an estimated 1.45-point higher quiz score (95% CI: 0.52 to 2.38 points per practice hour); students at this sleep level had observed practice hours from 1 to 6.” This states the conditional level, units, uncertainty, and relevant support without asserting that practice caused the score difference.

Reporting Fitted Mean Predictions at a Profile

A fitted mean prediction is tied to the complete profile used to calculate it. Naming only one predictor value can be insufficient if group membership or another interacting predictor also matters. State the profile explicitly, and make clear that the interval is for the expected outcome at that profile. Do not call it an interval for an individual outcome unless it is a prediction interval.

Worked Example: Reporting a Fitted Mean for One Group

Suppose a hypothetical interaction model predicts quiz score \(Y\), in points, from weekly practice hours \(x\), a group indicator \(G\), and their interaction:

E(Y\mid x,G) = \beta_0+\beta_1x+\beta_2G+\beta_3xG,

where \(G=0\) denotes Group A and \(G=1\) denotes Group B. The fitted estimates are \(\hat{\beta}_0=62\) points and \(\hat{\beta}_1=1.5\) points per practice hour. For Group A at \(x=4\) hours, suppose \(\widehat{\operatorname{Var}}(\hat{\beta}_0)=9\), \(\widehat{\operatorname{Var}}(\hat{\beta}_1)=0.04\), and \(\widehat{\operatorname{Cov}}(\hat{\beta}_0,\hat{\beta}_1)=-0.30\). Let \(t^*=2.01\).

Fitted mean. For Group A, \(G=0\), so the group and interaction terms are zero. The estimated mean score at four practice hours is:

\hat{E}(Y\mid x=4,G=0) = 62+1.5(4) = 68\text{ points}.

Uncertainty. This mean combines the intercept and slope estimates, so its estimated variance is:

\begin{aligned} \widehat{\operatorname{Var}}(\hat{\beta}_0+4\hat{\beta}_1) &=9+4^2(0.04)+2(4)(-0.30)\\ &=9+0.64-2.40\\ &=7.24, \end{aligned} $$ $$ SE=\sqrt{7.24}=2.69.

The 95% confidence interval for the fitted mean is \(68\pm2.01(2.69)\), or approximately 62.6 to 73.4 points. Suppose Group A students in the hypothetical data had practice values from 2 to 6 hours; four hours lies within that observed range.

Report: “For Group A students reporting 4 weekly practice hours, the estimated mean quiz score was 68.0 points (95% CI: 62.6 to 73.4); four hours was within the observed Group A practice range of 2 to 6 hours.” The interval describes uncertainty about the estimated mean for that profile, not the spread of scores among individual Group A students with four practice hours.

Reporting Conditional Group Contrasts

A group contrast at a moderator value is another conditional estimate, but it is not the same as a slope. Its direction depends on the subtraction order. A report should say which group is being compared with which, identify the moderator value, and give the contrast in outcome units. An interval that crosses zero indicates that the estimated contrast is uncertain in direction at the stated confidence level; it does not establish that the groups are identical.

Worked Example: Reporting a Group Difference at a Practice Level

Use the group-by-practice model from the preceding example. The conditional mean difference for Group B minus Group A at practice level \(x\) is \(\beta_2+\beta_3x\). Suppose the fitted estimates are \(\hat{\beta}_2=-4.0\) points and \(\hat{\beta}_3=1.2\) points per practice hour. At \(x=5\), suppose their estimated variances are 4.00 and 0.09, respectively, and their covariance is \(-0.30\). Let \(t^*=2.014\).

Conditional difference. At five weekly practice hours:

\hat{\beta}_2+5\hat{\beta}_3 = -4.0+5(1.2) = 2.0\text{ points}.

This is Group B's fitted mean minus Group A's fitted mean at the same practice level. The standard error is:

\begin{aligned} \widehat{\operatorname{Var}}(\hat{\beta}_2+5\hat{\beta}_3) &=4.00+5^2(0.09)+2(5)(-0.30)\\ &=4.00+2.25-3.00\\ &=3.25, \end{aligned} $$ $$ SE=\sqrt{3.25}=1.803.

The 95% confidence interval is \(2.0\pm2.014(1.803)\), or approximately \(-1.63\) to \(5.63\) quiz points. Suppose both groups have observed practice values in the range 3 to 7 hours, so the comparison at five hours lies in their common supported range.

Report: “At 5 weekly practice hours, the estimated mean quiz score in Group B was 2.0 points higher than in Group A (95% CI for Group B minus Group A: −1.63 to 5.63 points); five hours lay within both groups’ observed practice range of 3 to 7 hours.” Since the interval includes zero, these data do not determine the direction of the conditional group difference with 95% confidence. They do not show that the groups have exactly equal mean scores.

Common Mistakes and a Practical Reporting Sequence

A polished result need not reproduce the full model equation or every coefficient. It must, however, preserve the information needed to understand the conditional quantity. The following sequence is a useful final check.

Name the quantity.
Say whether the estimate is a slope, a group contrast, or a fitted mean.

Specify the conditioning profile.
Give the moderator value, group, and other predictor values that define the result.

Report estimate and uncertainty.
Include the estimate, a confidence interval or other stated uncertainty measure, and outcome-relevant units.

Check and state support.
Confirm that the profile lies in a region represented by the observed data and describe the relevant range or overlap.

Match the conclusion to the interval.
Describe the range of effects compatible with the interval; do not treat a wide interval crossing zero as proof of no effect.

A frequent mistake is to report “the practice effect was 1.45 points per hour” when that slope was evaluated only at 7 hours of sleep. The correction is to state the moderator value, because another sleep value can produce a different slope. Likewise, “the predicted score was 68” is incomplete without the group and practice-hour profile.

Another mistake is to interpret a confidence interval for a fitted mean as describing where most individual scores will fall. The correction is to identify what the interval quantifies: uncertainty about the estimated mean, rather than outcome-to-outcome variability. A third mistake is to report a conditional result at a profile that the data do not support as if it were an observed-data conclusion. A model can calculate such a value, but the report must disclose the unsupported profile and avoid presenting extrapolation as direct evidence.

Key takeaway: A conditional result is interpretable only when readers can see what is being estimated, at which profile, in what units, and with what uncertainty. State the supported range as part of the result, not as an afterthought.

Check Your Understanding

For each question, focus on the information a reader needs to interpret a conditional result.

A slope is estimated at a moderator value of 12. What details should a report include besides the numerical slope?
Why must the covariance between coefficient estimates be included when calculating the standard error of a conditional slope?
A fitted mean is 40 units with a 95% confidence interval from 36 to 44. Does this interval describe the range in which 95% of individual outcomes will fall? Explain.
A group contrast is defined as Group B minus Group A and estimated as −3 units at a stated predictor value. What does the negative sign mean?
A conditional estimate is calculated at a moderator value outside the observed range. What qualification should accompany its report?