Handling Non-Linear Trajectories in Longitudinal Data

Both trajectories are non-linear, but:

is constant.

Therefore, a non-linear time trend does not automatically imply a time-varying treatment effect.

Visit as a Categorical Variable in Clinical Trials

In many confirmatory clinical trials, scheduled visit is treated as categorical rather than continuous.

This is often particularly attractive when:

• Visits are irregularly spaced.
• The primary comparison is defined at a specific endpoint visit.
• The shape of the trajectory is not itself scientifically important.
• There is no compelling reason to assume a particular mathematical form.
• The number of scheduled visits is moderate.

For example, Week 12 and Week 24 can have independent effects without assuming that the difference between Weeks 12 and 24 follows the same rate of change as earlier intervals.

Continuous Time vs. Categorical Time

Handling Irregular Measurement Times

One major advantage of continuous-time modeling is that time can be represented using the actual measurement time rather than merely the scheduled visit.

Suppose participants have observations at:

• Day 0
• Day 13
• Day 29
• Day 57
• Day 91

A continuous-time model can use the actual day value.

Categorical visit coding would treat observations according to visit labels, which may be preferable when the estimand is explicitly visit based.

Example: Rapid Early Response Followed by Plateau

Consider a clinical outcome measured over 24 weeks. The observed group means suggest:

A linear model might estimate an average slope of approximately:

But that does not mean the treatment changes the outcome by 0.44 units every week.

The actual trajectory suggests much faster change during the first several weeks.

A quadratic or spline model can represent this changing rate.

Comparing Candidate Models

Model Selection Should Not Be Based Only on P-Values

Suppose a quadratic term has a p-value of 0.03. That does not automatically mean the quadratic model is the correct clinical model.

Likewise, a non-significant quadratic term does not prove that the true trajectory is linear.

Model selection should consider:

• Scientific plausibility
• Graphical fit
• Residual behavior
• Clinical interpretability
• Prespecified analysis strategy
• Stability of estimates
• Model complexity
• Ability to estimate the desired treatment effect

Residual Diagnostics

Residual plots are particularly useful for identifying an inappropriate time specification.

For a linear model, plot residuals against time.

If residuals display a systematic pattern such as:

• Positive residuals early
• Negative residuals in the middle
• Positive residuals late

then the fitted line may be missing curvature.

A successful non-linear model should reduce systematic structure in the residuals.

Observed vs. Fitted Trajectories

A particularly useful diagnostic is to overlay:

• Observed mean values
• Linear fitted trajectory
• Quadratic fitted trajectory
• Spline fitted trajectory

This allows investigators to determine whether increased model complexity actually provides a meaningful improvement in the trajectory.

Mixed Model Covariance and Non-Linear Time Are Different Decisions

Two separate modeling decisions are sometimes confused.

The first is the mean trajectory:

The second concerns the within-subject covariance:

A non-linear mean trajectory does not determine the appropriate covariance structure.

For example, a study might use:

• A quadratic mean trajectory
• Random intercepts and slopes
• An unstructured residual covariance

or:

• A spline mean trajectory
• Random intercepts
• AR(1) residual covariance

These are separate components of the longitudinal model.

R: Linear Mixed Model

A basic linear mixed model can be fitted using lmer() from the lme4 package.

library(lme4)

fit_linear <- lmer(
  change ~ week + treatment + week:treatment +
    (1 + week | subject),
  data = dat
)

summary(fit_linear)

This model allows:

• A linear population-average time trend
• A treatment effect
• A treatment-by-time interaction
• Subject-specific random intercepts
• Subject-specific random slopes

R: Quadratic Mixed Model

To allow curvature:

fit_quad <- lmer(
  change ~ week + I(week^2) +
    treatment +
    week:treatment +
    I(week^2):treatment +
    (1 + week | subject),
  data = dat
)

summary(fit_quad)

The fixed-effect component now allows each treatment group to have a quadratic trajectory.

Centering Time in R

dat$week_c <- dat$week - 12

fit_quad <- lmer(
  change ~ week_c + I(week_c^2) +
    treatment +
    week_c:treatment +
    I(week_c^2):treatment +
    (1 + week_c | subject),
  data = dat
)

Here, zero corresponds to Week 12. The intercept therefore has a more clinically meaningful interpretation.

R: Categorical Visit

dat$visit <- factor(dat$visit)

fit_visit <- lmer(
  change ~ visit * treatment +
    (1 | subject),
  data = dat
)

summary(fit_visit)

This model estimates treatment differences separately across visits without requiring a linear or polynomial trajectory.

R: Natural Cubic Splines

The splines package provides a convenient way to fit natural cubic splines.

library(splines)

fit_spline <- lmer(
  change ~ ns(week, knots = c(4, 8, 16)) *
    treatment +
    (1 | subject),
  data = dat
)

summary(fit_spline)

The treatment interaction allows the trajectory to differ between groups.

R: Estimated Treatment Differences

For a flexible model, one useful approach is to estimate marginal means at specific times.

library(emmeans)

emm <- emmeans(
  fit_spline,
  ~ treatment | week,
  at = list(
    week = c(2, 4, 8, 12, 24)
  )
)

pairs(emm)

This produces treatment comparisons at clinically meaningful time points.

SAS: Linear Mixed Model

In SAS, the standard procedure for many linear mixed-effects longitudinal models is PROC MIXED.

proc mixed data=dat;
    class subject treatment;
    
    model change =
        week
        treatment
        week*treatment
        / solution ddfm=kr;
        
    random intercept week
        / subject=subject
          type=un;
run;

This specifies random intercepts and random slopes with an unstructured covariance matrix for the random effects.

SAS: Quadratic Time

data dat2;
    set dat;
    
    week_c = week - 12;
    week_c2 = week_c * week_c;
run;

proc mixed data=dat2;
    class subject treatment;
    
    model change =
        week_c
        week_c2
        treatment
        week_c*treatment
        week_c2*treatment
        / solution ddfm=kr;
        
    random intercept week_c
        / subject=subject
          type=un;
run;

The centered quadratic model allows the average trajectory to curve over time while retaining participant-specific random intercepts and slopes.

SAS: Categorical Visit

proc mixed data=dat;
    class subject treatment visit;
    
    model change =
        visit
        treatment
        visit*treatment
        / solution ddfm=kr;
        
    random intercept
        / subject=subject;
run;

This approach is often very useful when the primary estimand is defined at specific scheduled visits.

SAS: Spline Basis Variables

SAS can generate spline basis variables using procedures such as PROC TRANSREG, or analysts can construct basis variables using appropriate spline functions and then fit those variables in PROC MIXED.

A conceptual workflow is:

/* Generate or define spline basis variables first */

proc mixed data=dat_spline;
    class subject treatment;
    
    model change =
        spline1
        spline2
        spline3
        treatment
        spline1*treatment
        spline2*treatment
        spline3*treatment
        / solution ddfm=kr;
        
    random intercept
        / subject=subject;
run;

The exact basis construction should be aligned with the prespecified spline method and knot locations.

Piecewise Models in R

A piecewise model can be constructed explicitly.

dat$week_early <- pmin(dat$week, 8)
dat$week_late  <- pmax(dat$week - 8, 0)

fit_piecewise <- lmer(
  change ~ week_early +
    week_late +
    treatment +
    week_early:treatment +
    week_late:treatment +
    (1 + week_early | subject),
  data = dat
)

summary(fit_piecewise)

The coefficient for week_early describes the early slope. The coefficient for week_late represents the change in slope after the Week 8 knot.

Piecewise Models in Clinical Interpretation

Suppose the early slope is \(-1.5\) units per week and the post-knot slope is \(-0.2\).

The clinical interpretation is not simply that the outcome changes at \(-0.2\) after Week 8. Rather:

• Before Week 8, the estimated rate of change is \(-1.5\) units per week.
• After Week 8, the trajectory becomes much flatter.

This can be much easier to communicate than a high-dimensional spline model.

Non-Linear Trajectories and Baseline Adjustment

Clinical longitudinal models often adjust for baseline outcome. Suppose \(Y_{i0}\) is baseline and \(Y_{ij}\) is a post-baseline measurement. A model can include baseline as a covariate:

The exact parameterization depends on the estimand and analysis strategy.

Baseline adjustment should not be confused with the decision about whether time is linear or non-linear.

Non-Linearity Can Be Different by Treatment Group

One of the most important questions is whether the treatment groups have the same trajectory shape.

A model without a treatment-by-time interaction assumes that treatment shifts the trajectory without changing its shape.

A model with treatment-by-time interaction allows the treatment effect to change over time.

For a general function \(f(t)\):

where \(g(t)\) represents the time-varying component of the treatment effect.

What If the Trajectory Is Exponential?

Some biological processes naturally suggest exponential behavior. A simple exponential decay model is:

where:

• \(A\) is the asymptotic level.
• \(B\) describes the initial displacement from that level.
• \(k\) controls the rate of decay.

This can be appropriate when the underlying scientific mechanism strongly supports exponential behavior.

However, such models are more restrictive than splines and can require nonlinear estimation.

Mechanistic vs. Statistical Non-Linearity

There are two broad reasons to use a non-linear trajectory model.

Statistical flexibility

The objective is to describe the observed trajectory without imposing an unnecessary shape. Splines are often useful here.

Mechanistic modeling

The objective is to represent a biological or pharmacological mechanism. Examples include:

• Exponential decay
• Logistic growth
• Emax models
• Compartmental models
• Indirect-response models

Mechanistic models are especially important in pharmacokinetics, pharmacodynamics, and quantitative systems pharmacology.

Modeling a Plateau

A plateau is one of the most common clinical trajectory patterns. A simple quadratic model can approximate a plateau over a limited follow-up period.

But if the asymptotic behavior is scientifically important, a model with an explicit asymptote may be preferable.

For example:

The trajectory approaches \(A+B\) as \(t\) increases.

The advantage is mechanistic interpretability. The disadvantage is that the model imposes a specific functional form.

Don't Overinterpret the Quadratic Coefficient

Suppose a quadratic mixed model estimates:

It is tempting to say: "The treatment has a negative quadratic effect of 0.04."

That is usually not the most useful clinical interpretation.

Instead, calculate predicted outcomes or treatment contrasts across time. For example:

This directly communicates how the treatment effect evolves over time.

Non-Linearity and Missing Data

Longitudinal studies frequently contain missing measurements. A flexible trajectory model does not automatically solve missing-data problems.

Investigators should consider:

• Why observations are missing
• The assumed missing-data mechanism
• Whether the likelihood-based model is appropriate
• Sensitivity analyses
• Pattern-mixture or selection models when appropriate

Under a missing-at-random assumption, mixed models estimated by maximum likelihood can use incomplete longitudinal records without requiring last-observation-carried-forward imputation.

Non-Linearity and Unequal Follow-Up

Participants may have different numbers of observations. One participant might have measurements through Week 24, while another might withdraw at Week 12.

Mixed-effects models can naturally accommodate different numbers of observations per participant under their assumed missing-data framework.

This is one reason mixed models are widely used for longitudinal clinical data.

Non-Linearity and Unequally Spaced Visits

Unequally spaced visits create another reason to think carefully about the time variable.

Suppose visits occur at Days 1, 7, 28, 84, and 168. Treating visit number as:

implicitly assumes equal spacing.

Using actual time:

preserves the actual temporal distances.

Which approach is appropriate depends on whether the scientific estimand is visit-based or time-based.

A Practical Model Selection Workflow

How to Choose Among the Main Approaches

Common Mistakes

1. Assuming every longitudinal trajectory is linear. Many clinical outcomes have rapid early changes followed by stabilization.
2. Adding polynomial terms without examining the data. A quadratic term should have a scientific or empirical rationale.
3. Using a high-degree polynomial to fit every fluctuation. This can produce unstable and poorly interpretable trajectories.
4. Confusing random slopes with non-linear trajectories. Random slopes allow subjects to have different linear rates; they do not create curvature.
5. Interpreting spline coefficients directly. Spline basis coefficients are usually not clinically intuitive.
6. Choosing knots solely because they improve a p-value. Knot selection should be scientifically and statistically defensible.
7. Ignoring treatment-by-time interaction. If treatment effects evolve differently over time, a common trajectory may be inadequate.
8. Focusing only on the time-term p-value. The clinical question is often better answered using estimated treatment differences at relevant times.
9. Confusing mean trajectory with covariance structure. These are separate components of a longitudinal model.
10. Using actual time automatically. Actual elapsed time is not always the appropriate estimand when the analysis is explicitly visit based.
11. Assuming a flexible model solves missing-data problems. The missing-data mechanism still requires consideration.
12. Overfitting the trajectory. More flexibility does not necessarily produce a better clinical model.

A Detailed Example: Linear vs. Quadratic vs. Spline

Suppose a randomized clinical trial measures change from baseline through Week 24. The investigator fits three models.

The spline has the smallest AIC, but that does not automatically mean it should be selected.

The quadratic model may be preferable if:

• Its fitted trajectory is nearly identical to the spline.
• The quadratic shape is clinically plausible.
• The parameters are easier to communicate.
• The prespecified analysis plan favors a parsimonious model.

Conversely, the spline may be preferable if the quadratic model systematically misses clinically important changes in slope.

Comparing Trajectories at Specific Times

For a flexible trajectory model, investigators often want estimates at a set of clinically important times. For example:

At each time, estimate:

and its confidence interval.

This transforms a complicated trajectory model into clinically interpretable results.

Plotting the Estimated Trajectory

A publication-quality longitudinal analysis should usually include a plot of the estimated trajectories.

A useful figure can show:

• Observed group means
• Model-based predicted means
• 95% confidence intervals
• Time on the horizontal axis
• Outcome or change from baseline on the vertical axis

The figure often communicates the non-linear pattern more effectively than a table of regression coefficients.

Trajectory Modeling and the Estimand

Before selecting a model, define what quantity the study is trying to estimate. Possible estimands include:

• Treatment difference at Week 12
• Treatment difference at Week 24
• Average treatment difference across follow-up
• Difference in slopes
• Difference in trajectory shape
• Time to maximum treatment response
• Area under the outcome trajectory

Different estimands can lead to different preferred models.

When Categorical Time Is the Better Choice

Categorical time is often underestimated because it is statistically simple.

Suppose the trial's primary objective is the treatment difference at Week 12. If measurements occur at six scheduled visits, categorical visit can provide direct estimates at each visit without imposing an artificial curve.

This is particularly useful in confirmatory clinical trials where scheduled visits and prespecified contrasts are central to the analysis.

When a Continuous Non-Linear Time Model Is Better

Continuous non-linear time is especially useful when:

• The actual timing of observations varies.
• The trajectory itself is scientifically important.
• The study contains many observation times.
• A smooth response curve is desired.
• There is interest in estimating changes between scheduled visits.
• A biologically meaningful functional form exists.

Non-Linear Time and Random Slopes

Suppose the population trajectory is quadratic. It may still be reasonable to allow participants to have different linear rates of change:

This model says:

• The population trajectory is curved.
• Participants vary in baseline level.
• Participants vary in their linear rate of change.
• The curvature is shared through the fixed-effect structure.

That can be a sensible compromise when there are insufficient data to support subject-specific curvature.

Should You Add a Random Quadratic Effect?

In principle, a model could include:

so that each participant has an individual quadratic component.

But this substantially increases model complexity.

The data must support estimation of:

• Random intercept variance
• Random slope variance
• Random quadratic variance
• Covariance between these random effects

With limited numbers of observations per participant, this can lead to singular fits or unstable covariance estimates.

Model Convergence Matters

Flexible longitudinal models can encounter computational problems. Common warnings include:

• Non-convergence
• Singular covariance matrices
• Boundary estimates
• Very large standard errors
• Highly correlated parameters

These are not merely technical annoyances. They can indicate that the proposed model is too complex for the available data.

A simpler, stable model is often preferable to a theoretically richer model that cannot be estimated reliably.

Centering and Scaling for Polynomial Models

Suppose time ranges from 0 to 365 days. Using:

produces values as large as:

Centering and scaling can improve numerical behavior. For example:

now expresses time approximately in months relative to Day 180.

The exact transformation should be chosen so that coefficients and predictions remain interpretable.

Reporting a Non-Linear Longitudinal Analysis

A statistical report should explain:

• Why a non-linear trajectory was considered
• How time was parameterized
• Whether time was centered or scaled
• How knots were selected if splines were used
• Whether treatment-by-time interactions were included
• The random-effects specification
• The residual covariance structure
• The method used for estimation
• How missing observations were handled
• Which time points were used for treatment contrasts

Example Statistical Methods Language

A concise methods description might state:

For a spline model, the corresponding description would specify the spline type and knot locations.

What to Include in a Statistical Analysis Plan

For an important longitudinal analysis, the SAP should specify:

• Outcome definition
• Baseline definition
• Scheduled visits
• Time scale
• Primary estimand
• Fixed effects
• Time parameterization
• Treatment-by-time interaction
• Random effects
• Covariance structure
• Estimation method
• Degrees-of-freedom method where applicable
• Missing-data assumptions
• Planned contrasts
• Confidence interval method
• Graphical displays
• Sensitivity analyses

Sensitivity Analyses

When the trajectory is uncertain, sensitivity analyses can compare reasonable alternative specifications. For example:

• Categorical visit
• Linear time
• Quadratic time
• Piecewise linear time
• Restricted cubic spline

The goal is not to select whichever model gives the most favorable treatment result.

Instead, the purpose is to assess whether the substantive conclusion is robust to plausible trajectory assumptions.

Avoiding Data-Driven Model Shopping

Suppose an analyst tries:

• Linear time
• Quadratic time
• Cubic time
• Four-knot spline
• Five-knot spline
• Piecewise model with three knots

and then selects the model with the smallest p-value for treatment.

This can introduce substantial bias.

Model flexibility should be justified independently of the observed treatment result whenever possible.

Non-Linear Trajectories in Regulatory Clinical Trials

In a regulatory clinical-trial setting, the model should be chosen to support a clearly defined estimand.

A highly flexible trajectory can be useful for understanding the longitudinal profile, but the primary efficacy conclusion may still be based on a prespecified visit-specific contrast.

This is an important distinction:

The trajectory model is a statistical tool used to estimate the desired clinical quantity.

A Complete Decision Framework

The Most Important Distinction: Shape vs. Heterogeneity

Longitudinal modeling involves at least two distinct sources of complexity.

The first is trajectory shape:

which determines whether the average response is linear, curved, piecewise, smooth, or otherwise non-linear.

The second is between-subject heterogeneity:

which describes how participants differ from one another.

A strong longitudinal model addresses both without confusing them.

A Practical Example of the Distinction

Imagine two patients:

Both patients show a curved trajectory. They also differ substantially in baseline level.

A useful model may therefore need:

• A non-linear fixed time effect to describe curvature
• A random intercept to describe baseline heterogeneity
• Possibly a random slope to describe differences in rate of change

These are complementary rather than competing ideas.

Recommended Default Strategy

There is no universal best model, but a practical strategy for many longitudinal clinical analyses is:

Summary Comparison

Key Takeaways

Non-linear longitudinal trajectories are common in clinical research. A linear time effect is convenient, but it should not be used automatically when the underlying trajectory clearly changes slope.

Several approaches are available:

• Categorical time avoids imposing a functional form.
• Quadratic models provide a simple representation of curvature.
• Piecewise models are useful when slope changes occur at meaningful time points.
• Splines provide flexible smooth trajectories.
• Mechanistic models are appropriate when biological knowledge supports a specific functional form.

These approaches can all be embedded within a mixed-effects framework.

The most important conceptual distinction is that non-linear time effects describe trajectory shape, while random effects describe between-subject heterogeneity.

References

Fitzmaurice, G.M., Laird, N.M. & Ware, J.H. (2011). Applied Longitudinal Analysis, 2nd ed. Wiley.
Verbeke, G. & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer.
Pinheiro, J.C. & Bates, D.M. (2000). Mixed-Effects Models in S and S-PLUS. Springer.
Harrell, F.E. (2015). Regression Modeling Strategies, 2nd ed. Springer.
Ruppert, D., Wand, M.P. & Carroll, R.J. (2003). Semiparametric Regression. Cambridge University Press.
Diggle, P., Heagerty, P., Liang, K.-Y. & Zeger, S. (2002). Analysis of Longitudinal Data, 2nd ed. Oxford University Press.
Little, R.J.A., et al. (2012). The prevention and treatment of missing data in clinical trials. New England Journal of Medicine, 367, 1355–1360.