Calculating Predictions From Factor Model Coefficients

From Coefficients to Predictions

A fitted model’s coefficients become useful when they are translated into predictions for particular profiles. With treatment coding, each coefficient is defined relative to a reference category. To predict for a profile, start with the intercept, add the coefficients for its nonreference categories, and include any interaction coefficients that apply to that profile. The calculation is simple once the coding is made explicit; omitting an applicable term or adding one for the reference category changes the prediction.

This tutorial focuses on calculating fitted values from supplied treatment-coded coefficients. It does not require fitting a model. The outcome and prediction scale matter: in a linear model the coefficient calculation is on the outcome scale, while in a generalized linear model it first gives a linear predictor that may need to be transformed to obtain a response-scale prediction.

Building a Prediction Row

Consider a linear model with one factor, whose reference level is Direct and whose other levels are Broker and Online. Its treatment-coded form is

Here, \(I(\text{Broker})\) is 1 if the profile is Broker and 0 otherwise; \(I(\text{Online})\) is defined similarly. For Direct, both indicators are 0, so its fitted value is the intercept. For Broker, the Broker coefficient is added, and for Online, the Online coefficient is added. A factor profile cannot simultaneously be both Broker and Online, so only one of these indicators is 1.

Predictions When Factors Interact

With two factors and an interaction, the model includes an additional coefficient for combinations of nonreference levels. Suppose Plan has levels Basic and Plus, with Basic as reference, and Zone has levels North and South, with North as reference. A linear model with their interaction can be written as

The interaction indicator is 1 only for the Plus-South profile. It is 0 for each of the other three profiles. Thus, the Plus coefficient describes the Plus-versus-Basic difference in the reference zone, North; the South coefficient describes the South-versus-North difference for the reference plan, Basic. The interaction coefficient adjusts the combination of those two nonreference levels.

Why the Indicator Calculation Works

A treatment-coded model is a linear combination of columns in its design matrix. For a specified profile, each factor indicator has a known value, either 0 or 1. Substituting those values into the model gives the profile’s row of the design matrix, denoted by \(x\). Multiplying that row by the estimated coefficient vector \(\widehat{\beta}\) gives the fitted value.

For the two-factor example, the row for Plus-South is \(x^\mathsf{T}=(1,1,1,1)\), corresponding to the intercept, Plus indicator, South indicator, and interaction indicator. The coefficient vector is \(\widehat{\beta}^\mathsf{T}=(410,60,-25,18)\), so

This derivation explains why profile prediction is addition of the applicable terms: the fitted value is the dot product of the profile’s design row and the model’s coefficient vector. If the model uses a different contrast scheme, the design row and coefficient meanings change; the same dot-product principle still applies, but the treatment-coded shortcut must not be used without checking the contrasts.

The difference-in-differences identity follows from the same model. Let the four cell predictions be \(m_{BN},m_{PN},m_{BS},m_{PS}\), where the first subscript denotes Basic or Plus and the second denotes North or South. Subtracting the Basic prediction from the Plus prediction in each zone gives

Subtract the first difference from the second. The main-effect coefficient \(\widehat{\beta}_{P}\) cancels, leaving \(\widehat{\beta}_{PS}\). Rearranging gives

This is a useful arithmetic check: a calculated interaction coefficient should equal the difference between the two group differences.

Linear Predictors and Response-Scale Predictions

For a generalized linear model, the coefficient calculation first produces the linear predictor, not necessarily the predicted mean on the outcome scale. In a binomial logistic model, the link is the logit: \(\eta=\log\{p/(1-p)\}\). The response-scale prediction is \(p=1/(1+\exp(-\eta))\). Therefore, check whether an R prediction is requested on the link scale or the response scale.

Practical Checks and Exam Traps

A reliable prediction calculation starts by writing down the reference levels and the model terms. Then record which indicators equal 1 for the requested profile. This prevents a common error: treating the reference category as though it had its own coefficient to add. The reference contribution is already represented by the intercept.

Before accepting an R check, confirm that the requested factor values are valid levels and that their reference levels match those used when fitting the model. A prediction for an unseen level is not obtained by assigning it a coefficient of zero; it is generally unavailable from the fitted model without additional assumptions or a changed model. Also distinguish a prediction on the link scale from one on the response scale for a GLM.

These steps make coefficient-based predictions auditable: every term has a clear reason for being included or excluded. They also provide a quick way to diagnose disagreement with R. A discrepancy often indicates a different reference level, an omitted interaction, or a mismatch between link and response scales.