Analysis of Serial Measurements

A common study design in clinical research gives patients an intervention and then observes what happens over time — measuring blood glucose repeatedly after a glucose load, tracking blood pressure during dialysis, or following tumour volume across treatment cycles. The result is serial measurements: a sequence of observations on each subject at multiple time points. Analysing such data correctly is less straightforward than it may appear, and a widely used approach — testing for group differences separately at each time point — is flawed on both statistical and clinical grounds. The remedy is a two-stage method of summary measures, first described in this context by Matthews et al. (1990), which reduces each subject's response curve to a single clinically meaningful number and then compares those numbers between groups with standard statistical tests.

The Problem with Time-Point Analysis

Displaying group means at each time point, with error bars and a significance marker at each one, is a familiar format in clinical journals. It is, however, inappropriate for three reasons.

First, the mean curve may not represent a typical individual. When subjects vary in the shape, timing, or magnitude of their response — some peaking early, some late, some with multiple peaks — the group mean may correspond to no individual's actual profile. Clinically important variation among subjects is hidden.

Second, repeated measurements on the same subject are correlated. A value at one time point predicts the next; successive tests are not independent. Significant results at adjacent time points carry far less independent information than they appear to. If measurements were taken every minute instead of every thirty, the number of significant $P$ values would multiply with no real increase in information about the group difference.

Third, the approach imposes an artificial dichotomy on data that change continuously. Biological responses do not switch from non-significant to significant and back at discrete instants; dividing the time axis into "significant" and "not significant" segments is statistically and clinically misleading.

Types of Serial Data

Before choosing a summary measure it is useful to characterise the shape of the response over time. Two broad patterns are common.

In a peaked response the variable starts from a baseline, rises (or falls) to an extreme value, and then returns toward baseline. Drug concentration after an oral dose, blood glucose after a glucose challenge, and heart rate during exercise all follow this pattern. The clinically relevant questions are typically about the size of the peak, the total response, and possibly how quickly the peak is reached.

In a growth response the variable increases or decreases steadily over the observation period without returning toward its starting value. Tumour growth under treatment, lung function decline in occupational exposure, and weight change during a diet intervention are examples. Here the key question is usually the rate of change.

Choosing a Summary Measure

The table below lists the summary measures available in MedCalc, with a brief description of each:

More than one summary measure can be used to capture different aspects of the same response — for example, both AUC and C_max for a pharmacokinetic curve — but rarely more than two or three. Each measure should target a distinct feature of the response, and the measures should ideally be chosen before the data are examined, to avoid selecting whichever measure produces the most striking difference between groups.

Area Under the Curve

For peaked data, the AUC quantifies the total exposure to whatever the variable measures. It is calculated using the trapezium rule: for each consecutive pair of observations $(y_i, y_{i+1})$ at times $(t_i, t_{i+1})$, the area of the trapezoid beneath them is $(t_{i+1} - t_i)(y_i + y_{i+1})/2$. Summing over all consecutive pairs gives the total AUC. When the time intervals between measurements are equal, the AUC is proportional to the simple mean of the observations; when intervals are unequal, the trapezium rule correctly weights each observation by the width of the interval it spans.

MedCalc offers three baseline options for the AUC: zero, the first observation, or the minimum value. The time-weighted average — the AUC divided by the total duration of the observation period — is a useful summary when measurements cover different lengths of time across subjects.

Statistical Analysis of Summary Measures

Once a summary measure has been computed for each subject, the values are treated as ordinary independent observations and analysed with standard methods. For two groups, a two-sample $t$-test or Welch's $t$-test is used if the summary measures are approximately normally distributed; the Mann–Whitney test otherwise. For three or more groups, one-way ANOVA or the Kruskal–Wallis test is appropriate. If the distribution is right-skewed, a logarithmic transformation is often sufficient to achieve normality, and the group comparison is then expressed as a ratio of geometric means with a confidence interval — a natural scale for many pharmacokinetic quantities.

MedCalc applies normality testing automatically and selects the appropriate parametric or non-parametric test, or offers log-transformation as an option.

An important consequence of the summary-measures approach is that the effective sample size is the number of subjects, not the total number of observations. Measuring each subject ten times does not provide ten times as much information as measuring once; the strong correlation between repeated measurements within a subject makes the raw count of observations largely illusory. The summary measures give a more honest account of how much independent information the study contains.

MedCalc computes all summary measures described here, with automatic or user-specified statistical testing, in its Serial measurements procedure.