Fixed-Effects Mean Trajectories

Why Specify a Mean Trajectory?

A longitudinal study records a biomarker repeatedly, so its mean can change over follow-up. A fixed-effects mean trajectory describes the average pattern the model assigns to that biomarker as time changes. It gives a concise answer to questions such as “What is the expected biomarker level at two years?” and “How much does the expected level change per year?”

The trajectory is a model for the mean, not a claim that every participant follows the same line. A mixed model can allow individual measurements to deviate from that shared mean. Here the focus is the fixed-effects part: how to construct and interpret a simple linear mean function.

The Linear Fixed-Effects Mean

Definition: For subject $i$ at measurement $j$, let $Y_{ij}$ be the biomarker value and $t_{ij}$ the follow-up time, measured in stated time units. A simple linear fixed-effects mean function is $$ \mu(t_{ij}) = \beta_0 + \beta_1 t_{ij}. $$ Here $\mu(t)$ is the population mean at time $t$, $\beta_0$ is the mean at $t=0$, and $\beta_1$ is the mean change in the biomarker per one unit of time. The coefficients are fixed effects because they describe shared features of the modeled population mean.

A basic mixed-model representation adds subject-level random effects and measurement error:

Y_{ij} = \beta_0 + \beta_1 t_{ij} + \mathbf{z}_{ij}^{\mathsf T}\mathbf{b}_i + \varepsilon_{ij}.

The vector $\mathbf{b}_i$ represents the subject-level random effects, and $\mathbf{z}_{ij}$ specifies how they enter the observation. This tutorial does not require a particular random-effects structure. For the population mean interpretation, assume that the random effects have conditional mean zero given the measurement times, and that the residual has conditional mean zero given the random effects and times. Then

E(Y_{ij}\mid t_{ij}) = \beta_0+\beta_1t_{ij} + \mathbf{z}_{ij}^{\mathsf T}E(\mathbf{b}_i\mid t_{ij}) + E(\varepsilon_{ij}\mid \mathbf{b}_i,t_{ij}) = \beta_0+\beta_1t_{ij}.

Thus the fixed-effects part is the population mean trajectory under these assumptions. It is not the trajectory of a particular subject, whose random effects can shift measurements away from the mean.

What the Coefficients Say

The intercept $\beta_0$ is the modeled mean at time zero, in the biomarker’s measurement units. Its interpretation depends on what time zero represents. If zero is the start of follow-up, it is the modeled mean at the start of follow-up. If zero is outside the observed time range, the intercept is an extrapolated reference value and may have little practical meaning.

The slope $\beta_1$ is the mean change associated with a one-unit increase in time. Its units are biomarker units per time unit. A negative slope describes a decreasing mean; a positive slope describes an increasing mean. In this simple model the slope is constant over the modeled period: the expected change over an interval of length $\Delta t$ is $\beta_1\Delta t$.

After fitting the model, the fitted population trajectory uses estimated coefficients:

\widehat{\mu}(t) = \widehat{\beta}_0+\widehat{\beta}_1t.

This equation provides a fitted mean at any time within the range where the linear assumption is reasonable. A fitted value is not an individual prediction: it does not include a particular subject’s random effects or a new measurement’s residual variation.

Worked Example: Interpreting a Biomarker Slope

Suppose a hypothetical study models a biomarker in milligrams per liter (mg/L) against time in years since follow-up began. The fitted fixed effects are $\widehat{\beta}_0=12.4$ mg/L and $\widehat{\beta}_1=-0.80$ mg/L per year. Calculate and interpret the fitted means at 0, 2.5, and 5 years.

The fitted trajectory is $\widehat{\mu}(t)=12.4-0.80t$. At the start of follow-up, $\widehat{\mu}(0)=12.4-0.80(0)=12.4$ mg/L. At 2.5 years, $\widehat{\mu}(2.5)=12.4-0.80(2.5)=12.4-2.0=10.4$ mg/L. At 5 years, $\widehat{\mu}(5)=12.4-0.80(5)=12.4-4.0=8.4$ mg/L.

The fitted mean decreases by 0.80 mg/L per year. Under this model, the mean at five years is 4.0 mg/L lower than at the start. These are fitted population means; they do not imply that each subject’s biomarker decreases by exactly 0.80 mg/L per year.

Justifying the Population Mean Interpretation

The mean interpretation follows by averaging the mixed-model equation over the subject-level and measurement-level deviations. In particular, if $E(\mathbf{b}_i\mid t_{ij})=\mathbf{0}$ and $E(\varepsilon_{ij}\mid \mathbf{b}_i,t_{ij})=0$, then

\begin{aligned} E(Y_{ij}\mid t_{ij}) &= E\!\left(\beta_0+\beta_1t_{ij} +\mathbf{z}_{ij}^{\mathsf T}\mathbf{b}_i+\varepsilon_{ij} \mid t_{ij}\right)\\ &= \beta_0+\beta_1t_{ij} +\mathbf{z}_{ij}^{\mathsf T}E(\mathbf{b}_i\mid t_{ij}) +E(\varepsilon_{ij}\mid t_{ij})\\ &= \beta_0+\beta_1t_{ij}. \end{aligned}

The fixed-effects mean therefore does not require normally distributed random effects or residuals as a matter of algebra. Distributional assumptions may be needed for particular likelihood-based estimation and inference procedures, but they are distinct from the conditions that make the displayed conditional expectation equal the fixed-effects line.

The assumptions about the observation process also matter. If visits occur more often when biomarker values are unusually high or low, the mean among recorded visits may not represent the intended population trajectory without appropriate modeling or other justification. The interpretation here concerns the population mean under a model and observation-process assumptions that support that target.

Worked Example: Computing a Fitted Mean at an Intermediate Time

In a hypothetical study, time is measured in months and the biomarker in units per liter. Suppose $\widehat{\beta}_0=5.20$ units/L and $\widehat{\beta}_1=0.35$ units/L per month. What is the fitted mean at 6 months, and how should the slope be interpreted?

Substitute $t=6$ into the fitted mean: $\widehat{\mu}(6)=5.20+0.35(6)=5.20+2.10=7.30$ units/L.

The fitted population mean at six months is 7.30 units/L. The slope says that the fitted mean increases by 0.35 units/L for each additional month over the period represented by the model. For example, the fitted difference between month 2 and month 6 is $0.35(6-2)=1.40$ units/L. This is a difference between modeled means, not a causal effect of time.

Assumptions, Diagnostics, and Alternatives

A straight-line mean is a useful starting point, but it is a substantive shape assumption. For its interpretation to be credible, the following conditions should be considered:

Linearity over the relevant follow-up period. The expected biomarker change per time unit is assumed constant. The assumption need only be plausible over the period being modeled; a straight line should not be extended automatically beyond it.
Correct time and outcome scales. Time must be expressed in known units, and the outcome scale must be stated. Changing years to months changes the numerical slope, not the underlying fitted pattern.
Mean-zero deviations for the population interpretation. The random effects and residuals must average to zero under the conditioning used to define the target mean. If deviations are systematically associated with measurement times, the simple expectation argument may fail.
A suitable observation process. The measurements used must support inference about the target population trajectory. Informative observation schedules can alter which outcomes are represented among observed records.

A practical diagnostic is to inspect the observed data and residual patterns against time. A curved residual pattern, such as residuals tending positive at both ends of follow-up and negative in the middle, suggests that a straight-line mean may miss curvature. Systematic changes in residual spread can indicate that other aspects of the model also need attention. Subject-level plots can reveal that the common line is a poor summary of the range of observed patterns, though variation between subjects alone does not establish that the population mean is nonlinear.

A linear mean can be compared with alternatives. Visit-specific means treat each scheduled visit as its own mean and can represent irregular shapes, but do not provide a single constant rate of change and may be less useful between visits. Separate regressions at each visit do not automatically account for the repeated measurements on the same subjects. A linear mixed model combines a specified mean shape with a model for within-subject dependence; the mean function and the dependence structure answer different questions.

The most important limitation is that a simple line can conceal changes in rate. It can also give misleading predictions when used outside the observed follow-up range. If diagnostics show persistent curvature, the remedy is to consider a more flexible mean function and assess it against the scientific question and available data, rather than to interpret the linear slope as a universal rate.

Worked Example: Comparing Two Follow-Up Times

A hypothetical biomarker is measured in micrograms per milliliter, and time is measured in years. A fitted linear mean has $\widehat{\beta}_0=18.0$ micrograms/mL and $\widehat{\beta}_1=-1.25$ micrograms/mL per year. Calculate the fitted means at 1.5 and 4 years, then compare them.

At 1.5 years, $\widehat{\mu}(1.5)=18.0-1.25(1.5)=18.0-1.875=16.125$ micrograms/mL. At 4 years, $\widehat{\mu}(4)=18.0-1.25(4)=18.0-5.0=13.0$ micrograms/mL.

The fitted difference is $13.0-16.125=-3.125$ micrograms/mL. Equivalently, the time interval is $4-1.5=2.5$ years and the slope-based change is $(-1.25)(2.5)=-3.125$ micrograms/mL. The agreement follows because the model has a constant slope. The comparison describes fitted population means at the two times, not the change for every participant.

Common Mistakes

Calling the intercept “the baseline mean” without checking time zero. It is the mean at $t=0$; that is baseline only when zero was defined as the baseline time.
Dropping the slope’s units. A coefficient of $-0.80$ is incomplete without saying, for example, “mg/L per year.”
Interpreting the fixed slope as each subject’s slope. The fixed effect describes the modeled population mean. Individual trajectories may differ.
Treating a fitted mean as an individual prediction. The fitted mean excludes subject-specific deviations and measurement error.
Assuming repeated measurements guarantee a correct trajectory. Repeated observations do not make a linear mean appropriate, and an observation schedule related to outcomes may affect the target represented by the observed data.

Key takeaway: In the linear mean $\mu(t)=\beta_0+\beta_1t$, the intercept is the modeled population mean at time zero and the slope is the constant mean change per time unit. Interpret both in context, and check that the time range, observation process, and observed pattern support the model.

Check Your Understanding

Answer each question, then compare with the solutions below.

A fitted mean is $\widehat{\mu}(t)=9.0+0.6t$, with time in months and biomarker in units/L. What is the fitted mean at month 5, and what does the slope mean?
In a model with $\beta_0=14$ and time measured from the start of follow-up, what does $\beta_0$ represent? What changes if time zero is outside the observed range?
A colleague says, “The fitted slope is $-2$ mg/L per year, so every subject’s biomarker falls by 2 mg/L each year.” Identify and correct the mistake.
What residual pattern over time would make a linear population mean questionable?

Solutions. 1. $\widehat{\mu}(5)=9.0+0.6(5)=12.0$ units/L. The fitted population mean increases by 0.6 units/L per month. 2. $\beta_0$ is the modeled population mean at the start of follow-up because that time is zero. If zero is outside the observed range, it is an extrapolated mean and may not have a useful direct interpretation. 3. The fixed slope describes the modeled population mean change, not each subject’s change; subject-level deviations can produce different individual patterns. 4. A systematic curved pattern, such as residuals positive at both ends and negative in the middle, suggests the straight-line mean misses curvature.