Estimability Under Sparse and Irregular Sampling

Why Sparse Sampling Makes Timing Matter

A QSP experiment may have few measurements because sampling is burdensome, costly, or constrained by the biology. With sparse observations, the times that are measured can determine whether the data capture distinct parts of a response trajectory or repeatedly sample nearly the same feature. Uneven spacing is not automatically a defect: a late observation may add substantial information about a slowly changing process, while several closely spaced observations can add little information about a parameter that affects the long-term response.

The relevant question is not simply how many observations are available. It is how the predicted measurements change when a parameter changes, relative to the observation error. Fisher information summarizes this sensitivity locally near a parameter value; a profile likelihood shows how the best fit changes as a parameter is fixed across a wider range. Their agreement is informative, but depends on the model and likelihood being appropriate.

Assumptions and a Simple Trajectory Model

To make the comparison concrete, consider a hypothetical response measured at time \(t_i\), modeled as a linear trajectory over the observation window:

Here \(\beta_0\) is the response at time zero, \(\beta_1\) is the change in response per unit time, and \(\sigma\) is the known observation-error standard deviation. For the calculations below, observation times are fixed, errors are independent with a common known variance, and the linear mean model is assumed correct over the sampled interval. These assumptions make the likelihood calculations exact for this example. A biological QSP model may be nonlinear, and its information calculations may then be local approximations around fitted parameter values.

For a model with mean predictions \(\mu_i(\boldsymbol{\theta})\) and independent Gaussian errors of known standard deviations \(\sigma_i\), the Fisher information matrix for parameters \(\boldsymbol{\theta}\) is

Each observation contributes according to its parameter sensitivities and its error variance. For the linear trajectory, the sensitivity row at time \(t_i\) is \((1,t_i)\), so the information matrix is \(I=X^\mathsf{T}X/\sigma^2\), where \(X\) has rows \((1,t_i)\). With a full-rank design, the covariance matrix of the maximum-likelihood estimates is exactly \(I^{-1}\) under this model.

Profile Likelihood for a Parameter of Interest

Fisher information is a local summary. A profile likelihood directly evaluates the fit available at different fixed values of a parameter while allowing other parameters to adjust. For the slope \(\beta_1\), the profile likelihood is obtained by fixing \(\beta_1\) and maximizing the likelihood over the intercept \(\beta_0\). Under the stated Gaussian model, the maximizing intercept at each fixed slope is \(\bar{Y}-\beta_1\bar{t}\).

The calculation has a particularly transparent form. Let \(S_{ty}=\sum_i(t_i-\bar{t})(Y_i-\bar{Y})\) and \(S_{xx}=\sum_i(t_i-\bar{t})^2\). After profiling out the intercept, the residual sum of squares as a function of the slope is a quadratic. Completing the square gives the following likelihood-ratio result:

This is not merely a local approximation for this linear model with independent Gaussian errors and known variance. The maximum-likelihood slope is exactly normally distributed with variance \(\sigma^2/S_{xx}\), and the likelihood-ratio statistic at a fixed slope has an exact \(\chi^2_1\) distribution under the model. The 95% cutoff is approximately 3.841. Different error assumptions or model misspecification can invalidate that calibration.

What Irregular Timing Can and Cannot Tell You

The previous comparison shows why observation count alone is inadequate. For the linear trajectory, the slope uncertainty depends on the spread of the sampled times around their mean. Yet a broad time span is not a universal design objective: a nonlinear response can change rapidly over one interval and slowly over another, and parameters may affect different parts of the trajectory in different ways. For a nonlinear model, the information contribution at a sampled time depends on the local sensitivity vector there, not simply on the distance between observation times.

A schedule can also leave parameters highly correlated. In the linear example, the off-diagonal entries of the information matrix reflect the relation between intercept and slope sensitivities. A design that estimates one parameter well may still constrain another combination poorly. For nonlinear QSP models, inspect information in the parameter combinations of interest and use profiles to see whether nuisance parameters can compensate as a parameter is varied.

Diagnostics, Alternatives, and Limitations

A useful comparison begins with assumptions and ends with a check against the observed data. For each candidate schedule, inspect the predicted sensitivity at its observation times, the rank and eigenstructure of the information matrix, and profile shapes for parameters that matter scientifically. A very flat profile indicates weak constraint; a sharply asymmetric or bounded profile warns that a symmetric local standard error may summarize the uncertainty poorly. The profile and Fisher information need not give similar impressions in a nonlinear model, particularly when the profile is curved or the optimum lies near a parameter bound.

For a fixed schedule and the simple known-variance linear Gaussian model, the profile likelihood and Fisher information are two equivalent views of slope precision: the profile curvature is \(S_{xx}/\sigma^2\), which is the inverse of the slope variance. In more complex models, Fisher information is often convenient for screening or comparing local precision, while profile likelihood can reveal nonlinear changes in fit and compensation by nuisance parameters. Neither approach repairs an incorrect mean model, ignored correlation, or an incorrect observation-error model.

Common warning signs include nearly redundant sensitivity vectors at the sampled times, an information matrix with a very small eigenvalue, and a profile that remains nearly level over a wide parameter range. Sparse data also limit what can be checked about trajectory shape: a model may fit the few observed points while behaving implausibly between them. If errors are heteroscedastic, the simple common-\(\sigma\) formula is not appropriate; if measurements are correlated, treating each as independent can overstate information. Those features must be represented in the likelihood used for the comparison.

A schedule comparison is therefore conditional: it describes precision if the assumed model, error structure, and parameter region are credible. It does not by itself prove that the design will resolve every biological mechanism, nor that a narrow interval is scientifically useful. Profile likelihoods and Fisher information are complementary tools for making the consequences of sparse, uneven sampling explicit.