Check Interaction Plots Against Observed Predictor Coverage

Why Fitted Lines Need Observed Data

An interaction plot can make a change in slope look clear: two fitted lines may diverge as the focal predictor increases. But a fitted line is a model summary, not a picture of how densely the data cover every part of the predictor range. If one line is supported by many observations across that range while another is supported only near one end, the apparent contrast over the full range may depend heavily on extrapolation.

Checking observed predictor coverage means displaying the observed data alongside the fitted interaction lines and asking where observations actually occur. This check does not replace estimating or testing an interaction. It helps you judge what parts of the plotted pattern are supported by the observed predictor combinations and where the model is being asked to extend beyond them.

The Model and What an Interaction Plot Shows

For two quantitative predictors \(x\) and \(z\), consider the linear interaction model

Here \(Y_i\) is the outcome for observation \(i\), \(x_i\) and \(z_i\) are its predictor values, and \(\varepsilon_i\) is the error. The model specifies a linear conditional mean in the included terms: \(E(Y_i\mid x_i,z_i)=\beta_0+\beta_1x_i+\beta_2z_i+\beta_3x_iz_i\). For interpreting fitted lines as descriptions of the mean, assume this form is adequate over the predictor region being discussed and that observations provide relevant support there. Usual regression inference also relies on conditions such as independent errors and an appropriate error-variance model; normal errors are often used for exact small-sample inference. A coverage plot alone cannot verify those assumptions.

At a fixed value \(z=z_0\), the conditional fitted line as \(x\) changes is

Thus, the fitted slope for \(x\) at \(z_0\) is \(\hat{\beta}_1+\hat{\beta}_3z_0\). An interaction plot typically draws this line for each of several selected \(z\) values. When \(z\) is continuous, however, an observed point has its own \(z\) value; it does not automatically belong to every plotted line. A useful display therefore shows the points' actual \(x\) and \(y\) values and communicates their \(z\) values—for example, through color, symbols, or small panels. If a line is drawn for \(z_0\), highlight observations with \(z\) near \(z_0\), rather than implying that distant \(z\) values lie on that slice.

A Short Justification for Checking Coverage

The model's slope contrast between two fixed values \(z_1\) and \(z_2\) follows directly from the conditional slopes:

So the fitted lines differ in slope whenever \(\hat{\beta}_3\) and \(z_2-z_1\) are nonzero. This algebra describes the fitted model, not the distribution of observations. If few or no observations support the relevant combinations of \(x\) and \(z\), the lines can still be drawn there, but their plotted separation is not equally well anchored by observed data throughout the displayed range. The line equation alone cannot answer the coverage question; the observed predictor values must be inspected.

How to Read the Overlay

For continuous \(z\), there may not be many observations with exactly the selected value. A practical display can use narrow, clearly described bands around representative \(z\) values, or small panels showing ranges of \(z\). Bands should not be so broad that they mix substantially different fitted slopes, especially when the interaction is large. Alternatively, color points by their actual \(z\) values and draw selected conditional lines, while making clear that the line represents a fixed \(z\) slice. In every case, keep the observed pairs \((x,z)\) visible or otherwise summarize their distribution.

Joint Coverage and Predictor Association

Even when both predictors individually span wide ranges, the observed combinations of \(x\) and \(z\) may be limited. For example, if low \(x\) values occur almost exclusively with low \(z\) values and high \(x\) values almost exclusively with high \(z\) values, the data may contain little information about comparisons that hold one predictor fixed while changing the other. This is why inspecting the joint predictor coverage matters: a plot of the outcome against \(x\) alone can hide which \(z\) values accompany each part of the \(x\) range.

The fitted model still supplies conditional lines for any chosen \(z\), but a line at \(z=z_0\) is well supported only where observations near that \(z_0\) also cover the relevant \(x\) values. Empty regions in the observed \((x,z)\) combinations are not filled in by drawing more of the fitted line. If the predictors are associated, describe that pattern and distinguish observed combinations from model-based extensions.

Common Mistake: Treating Every Line Segment as Equally Evidenced

• Wrong reasoning: “The fitted lines are drawn from \(x=0\) to \(x=10\), so the slope difference is supported throughout that interval.”
• Correction: A line can be drawn anywhere the model equation can be evaluated. Check whether observations at the relevant \(z\) values actually cover each part of the \(x\) interval.
• Wrong reasoning: “Both predictors have a wide range, so all predictor combinations are represented.”
• Correction: Inspect their joint coverage. Strong association between predictors can leave large parts of the \((x,z)\) plane empty.
• Wrong reasoning: “A sparse region proves the interaction is absent.”
• Correction: Sparse coverage limits how confidently the plotted pattern can be interpreted in that region; it does not establish that the interaction is zero.