For comparative treatment efficacy (CTE) trials, the primary endpoints often are measured on a continuous scale. The sample mean (sample standard deviation), the sample median (sample interquartile range), or the sample geometric mean (sample coefficient of variation) serve as reasonable descriptive statistics in such circumstances.

The sample mean (sample standard deviation) is suitable if the data are normally distributed or symmetric without heavy tails. The sample median (sample interquartile range) is suitable for symmetric or asymmetric data. The sample geometric mean (sample coefficient of variation) is suitable when the data are log-normally distributed.

Usually, two-sample *t-tests* or Wilcoxon rank tests are applied to compare the two randomized groups. In some instances, baseline measurements (prior to randomized treatment assignment) of the primary endpoints are taken.

Suppose \(Y_{i1}\) and \(Y_{i2}\) denote the baseline and final measurements of the endpoint, respectively, for the \(i^{th}\) subject, \(i = 1, 2, \dots, n\). Instead of statistically analyzing the \(Y_{i2}s\), there could be an increase in precision by analyzing the change (or gain) in the response, namely, the \(Y_{i}s\) where \(Y_i = Y_{i2} - Y_{i1}\).

Suppose that the variance for each \(Y_{i1}\) and \(Y_{i2}\) is \(\sigma^2\) and that the correlation between \(Y_{i1}\) and \(Y_{i2}\) is \(ρ\) (we assume that subjects are independent of each other but that the pair of measurements within each subject are correlated).

This leads to

\(Var(Y_{i2} - Y_{i1}) = Var(Y_{i2}) + Var(Y_{i1}) - 2Cov(Y_{i2},Y_{i1}) = 2\sigma^2(1 - \rho)\)

Therefore,

\(Var(Y_{i2}) = \sigma^2 \text{ and } Var(Y_{i2} - Y_{i1}) = 2\sigma^2(1 - \rho)\)

If \(ρ> \dfrac{1}{2}\), which often is the case for repeated measurements within patients, then \(Var(Y_{i2} - Y_{i1}) < Var(Y_{i2})\). Thus, there may be more precision if the \(Y_{i2} - Y_{i1}\) are analyzed instead of the \(Y_{i2}\). This happens all the time. Using the patient as their own control is a good thing. We are interested in the differences that are occurring, therefore we will subtract the treatment period measurements from the baseline data for the patient. A two-sample t test or the Wilcoxon rank sum test can be applied to the change-from-baseline measurements if the CTE trial consists of two randomized groups, such as placebo and an experimental therapy.

An alternative approach with baseline measurements is analysis of covariance (ANCOVA). In this situation, the baseline measurement, \(Y_{i1}\), serves as a covariate, so that the final measurement for a subject is adjusted by the baseline measurement. A linear model that describes this for a two-armed trial with placebo and experimental treatment groups is as follows. The expected value for the \(i^{th}\) patient, \(i = 1, 2, \dots, n\), is:

\( E(Y_{i2}) = \mu_p + T_i( \mu_E - \mu_P) + Y_{i1}\beta \)

where \(\mu_P\) is the population mean for the placebo group, \(\mu_E\) is the population mean for the experimental treatment group, \(T_i = 0\) if the \(i^{th}\) patient is in the placebo group and 1 if in the experimental treatment group, and \(\beta\) is the slope for the baseline measurement.

The expectations for subjects in the placebo group and experimental treatment group, respectively, can be rewritten as:

\( E(Y_{i2} - Y_{i1}\beta) = \mu_P \text{ and } E(Y_{i2} - Y_{i1}\beta) = \mu_E \)

These expectations are analogous to the expectations for the change-from baseline measurements:

\( E(Y_{i2} - Y_{i1}) = \mu_P \text{ and } E(Y_{i2} - Y_{i1}) = \mu_E\)

The only difference between the two approaches is that in the change-from-baseline measurements, \(\beta\) is set equal to 1.0. In the ANCOVA approach, \(\beta\) is estimated in the analysis and may differ from 1.0. Thus, ANCOVA approach is more flexible and can yield slightly more statistical power and efficiency.