Combining Multiple Marker Measurements

Why Several Marker Measurements Need a Joint Contribution

A subject in a joint model may contribute several longitudinal marker values, not just one. Each value is informative about the subject’s latent trajectory, and that trajectory is linked to the subject’s event process through shared random effects. The likelihood must therefore account for all the marker records together, while retaining the dependence structure implied by those shared effects.

The key construction is simple once its conditioning is clear: if measurements are conditionally independent given the subject’s random effects and trajectory, their conditional densities multiply. But after the random effects are integrated out, those same measurements are generally dependent. Multiplying their separate marginal densities would therefore give the wrong longitudinal contribution.

Definition: Let $Y_{ij}$ be the marker value for subject $i$ at time $t_{ij}$, for $j=1,\ldots,n_i$. Let $b_i$ denote that subject’s random effects and $m_i(t_{ij};b_i)$ the modelled marker trajectory at the measurement time. Under the conditional Gaussian measurement model, $$ Y_{ij}\mid b_i \sim N\!\left(m_i(t_{ij};b_i),\sigma^2\right), $$ and the $Y_{ij}$ are conditionally independent across measurement times given $b_i$. The subject’s longitudinal likelihood contribution conditional on $b_i$ is $$ L_{Yi}(b_i)=\prod_{j=1}^{n_i} \frac{1}{\sqrt{2\pi\sigma^2}} \exp\!\left[-\frac{\{Y_{ij}-m_i(t_{ij};b_i)\}^2}{2\sigma^2}\right]. $$

Here the trajectory may contain fixed covariates as well as random effects; those components are included in $m_i$. The model assumes a common residual variance $\sigma^2$, Gaussian measurement errors, and no remaining dependence between distinct measurements once the random effects and trajectory have been conditioned on. The measurement times are treated as fixed or ignorable for this likelihood contribution. The formula describes the marker records under analysis; it does not specify a model for a potentially informative observation schedule.

From Conditional Densities to a Likelihood

The product follows from the conditional-independence assumption. Given $b_i$, the joint density of the observed marker vector $\mathbf Y_i=(Y_{i1},\ldots,Y_{in_i})$ factors into its one-measurement densities:

f(\mathbf Y_i\mid b_i) =\prod_{j=1}^{n_i} f(Y_{ij}\mid b_i).

Substituting the Gaussian density gives the stated $L_{Yi}(b_i)$. On the log scale, multiplication becomes addition:

\ell_{Yi}(b_i) =-\frac{n_i}{2}\log(2\pi\sigma^2) -\frac{1}{2\sigma^2}\sum_{j=1}^{n_i} \{Y_{ij}-m_i(t_{ij};b_i)\}^2.

This expression shows precisely how each record contributes: one Gaussian normalization term and one squared residual. If $\sigma^2$ is estimated, its normalization term must be retained; it is not a constant with respect to that parameter. For fixed $\sigma^2$, the residual sum of squares determines how the conditional likelihood changes with the trajectory and random effects.

Worked Example: Three Gaussian Measurements at Fixed Random Effects

Suppose a hypothetical subject has three marker measurements. At a particular value of the random effects, the fitted trajectory values are 10, 12, and 14 marker units, while the observed values are 11, 13, and 16 marker units. Let the conditional residual variance be $\sigma^2=4$ marker-units squared. The residuals are $1,1,2$, so their squared sum is $1+1+4=6$.

Conditional independence gives $$ L_{Yi}(b_i) =\left(\frac{1}{\sqrt{2\pi(4)}}\right)^3 \exp\!\left[-\frac{6}{2(4)}\right] =\left(\frac{1}{\sqrt{8\pi}}\right)^3\exp(-6/8). $$

Since $1/\sqrt{8\pi}\approx0.1994711$ and $\exp(-0.75)\approx0.4723666$, $$ L_{Yi}(b_i)\approx0.1994711^3(0.4723666)\approx0.0037490. $$

This is a conditional density for the three measurements, with units of inverse marker-units cubed. It is not a probability that the subject’s measurements equal these exact values; continuous observations have densities, not point probabilities.

Conditional Independence Is Not Marginal Independence

The product in $L_{Yi}(b_i)$ is conditional on $b_i$. If $b_i$ is random with density $p(b_i)$, the marginal longitudinal density is obtained by integrating the entire product:

f(\mathbf Y_i)=\int L_{Yi}(b_i)\,p(b_i)\,db_i.

A shared random intercept, for example, shifts all measurements for a subject together. Even if the measurement errors are independent conditional on that intercept, measurements tend to be associated marginally because they share the same uncertain intercept. Thus, in general, $$ \int \prod_j f(Y_{ij}\mid b_i)p(b_i)\,db_i \ne \prod_j \int f(Y_{ij}\mid b_i)p(b_i)\,db_i. $$ The integration must be performed after multiplying the conditional densities.

Worked Example: A Shared Random Intercept Creates Dependence

Consider the hypothetical model $Y_{ij}=b_i+\varepsilon_{ij}$, where $b_i\sim N(0,1)$, and the errors are conditionally independent with $\varepsilon_{ij}\sim N(0,0.25)$. For two measurements, take observed values $Y_{i1}=1$ and $Y_{i2}=0$.

At $b_i=0$, the conditional means are both zero. The two conditional densities are $$ f(1\mid b_i=0)=\frac{1}{\sqrt{2\pi(0.25)}}\exp\!\left[-\frac{1^2}{2(0.25)}\right]\approx0.10798, $$ and $$ f(0\mid b_i=0)=\frac{1}{\sqrt{2\pi(0.25)}}\approx0.7979. $$ Their conditional product is $0.10798(0.7979)\approx0.08616$, using the unrounded Gaussian densities.

After integrating over $b_i$, the two measurements have marginal variances $1.25$ and covariance $1$. Their joint covariance matrix is $$ \Sigma= \begin{pmatrix} 1.25&1\\ 1&1.25 \end{pmatrix}, \qquad |\Sigma|=1.25^2-1^2=0.5625. $$

For the observed vector $(1,0)^\mathsf{T}$, the quadratic form is $1.25/0.5625=2.2222$. The correct joint marginal density is therefore $$ \frac{1}{2\pi\sqrt{0.5625}}\exp(-2.2222/2) \approx0.06986. $$

By contrast, each marginal variance is $1.25$. Multiplying the two marginal densities gives approximately $0.2392(0.3568)=0.08535$, which is not the joint density. That product incorrectly treats the measurements as marginally independent.

Using a Random-Intercept-and-Slope Trajectory

The same likelihood construction applies when the trajectory changes with time through multiple random effects. For example, with a random intercept $b_{0i}$ and random slope $b_{1i}$, a linear trajectory can be written as $m_i(t;b_i)=\beta_0+\beta_1t+b_{0i}+b_{1i}t$. At a fixed $b_i$, calculate this fitted value at every measurement time, find each residual, and use the same product of Gaussian densities. No new likelihood rule is needed; the trajectory determines the residuals.

Worked Example: Measurements Along a Random-Slope Trajectory

Suppose, hypothetically, that $\beta_0=5$, $\beta_1=1$, $b_{0i}=0$, and $b_{1i}=1$, with residual variance $\sigma^2=1$. The three measurement times are 0, 1, and 2; observed marker values are 5, 7, and 8. The fitted trajectory values are $5,7,9$, so the residuals are $0,0,-1$ and their squared sum is 1.

The conditional likelihood is $$ L_{Yi}(b_i) =(2\pi)^{-3/2}\exp(-1/2) =\phi(0)\phi(0)\phi(-1) \approx0.03851, $$ where $\phi$ is the standard normal density. The number is a density in inverse marker-units cubed. It is relatively high at these random effects because the observations are close to the fitted trajectory; it does not by itself describe the likelihood after accounting for uncertainty in $b_i$.

Combining the Marker and Event Contributions

In the shared-random-effects model, the longitudinal contribution is combined with the event or censoring contribution conditional on the same $b_i$. Let $L_{Ti}(b_i)$ denote the subject’s event-process contribution under the specified event and censoring model. The subject likelihood contribution after integrating out the random effects is

L_i=\int L_{Yi}(b_i)L_{Ti}(b_i)p(b_i)\,db_i.

The order matters: multiply the two conditional contributions at a common random-effects value, then integrate over that value. The event contribution is not integrated separately and then multiplied by a separately integrated marker likelihood, because that would discard the shared random effects that connect the two processes.

Worked Example: Forming the Joint Integrand at a Fixed Random-Effects Value

Return to the three-measurement example, for which $L_{Yi}(b_i)\approx0.0037490$ at the specified random-effects value. Suppose the event submodel gives a conditional event or censoring contribution $L_{Ti}(b_i)=0.12$ per unit time at that same value. The product in the integrand is $$ L_{Yi}(b_i)L_{Ti}(b_i) =0.0037490(0.12)\approx0.0004499. $$

This is the conditional joint contribution before multiplying by the random-effects density and integrating over $b_i$. Its units include inverse marker-units cubed and inverse time. It is not the final subject likelihood: the latter averages this product over the random-effects distribution.

Assumptions, Diagnostics, and Common Mistakes

The product likelihood is appropriate when the measurement errors are Gaussian with the stated variance and are independent conditional on the random effects and trajectory. A shared random effect can account for some within-subject association, but it does not automatically account for every pattern in repeated measurements. For example, serial residual correlation remaining after conditioning on the trajectory violates the conditional-independence assumption.

Useful checks include plotting conditional residuals against time and fitted marker values, looking for systematic trends, changing spread, or runs of residuals with the same sign. Strong non-normal tails or outliers may also make the Gaussian density a poor description. These checks do not prove the assumptions, but patterns can show where the model is inadequate. Numerical integration should also be checked for stability when evaluating the marginal likelihood; a stable conditional product does not guarantee accurate integration.

A main alternative is to specify a multivariate Gaussian residual distribution with a non-diagonal covariance matrix. That approach allows residual dependence beyond the shared random effects, but changes the density from a product of univariate terms to a joint multivariate density. The independent-error product should not be used merely for convenience if the conditional residuals show clear dependence.

Common mistake: “The measurements are conditionally independent, so their marginal densities can be multiplied.” This omits the dependence induced by integrating over shared random effects. Multiply the conditional densities first, then integrate their product.
Common mistake: dropping Gaussian normalization factors while estimating $\sigma^2$. Those factors depend on $\sigma^2$ and belong in the likelihood.
Common mistake: using a different random-effects value for each marker record. A subject has one shared vector $b_i$ for the subject-level contribution; each fitted value uses that same vector evaluated at its own measurement time.

Key takeaway: Under conditional Gaussian measurement errors, form one density per marker record and multiply them at the same subject-specific random-effects value. In the joint model, multiply that conditional longitudinal contribution by the conditional event contribution and integrate the product over the shared random effects.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

A subject has four marker records, and conditional independence is assumed. How many Gaussian density factors appear in the conditional longitudinal likelihood?
Why is it generally incorrect to multiply the marginal densities of two measurements that share a random intercept?
With three residuals $1,-1,2$, what squared-residual sum enters the log likelihood?
What likelihood structure is appropriate if residuals remain serially correlated after conditioning on the random effects and trajectory?

Solutions. (1) Four, one for each measurement time. (2) Integrating over their shared random intercept induces marginal dependence; integrate the product of conditional densities instead. (3) $1^2+(-1)^2+2^2=6$. (4) A model with an appropriate joint residual covariance structure, such as a multivariate Gaussian residual model, rather than the conditionally independent product.