## Randomization Models

As was indicated in the preceding chapter, randomization ensures that patients selected from the target patient population constitute a representative sample of the target patient population. Therefore, statistical inference can be drawn based on some probability distribution assumption of the target patient population. The probability distribution assumption depends on the method of randomization under a randomization (population) model. As a result a study without randomization will result in the violation of the probability distribution assumption, and consequently no accurate and reliable statistical inference on the study medicine can be drawn.

Lachin (1988a) provides a comprehensive summary of the randomization basis for statistical tests under various models. His observations are discussed below.

### Population Model

Cochran (1977) points out that the validity of statistical inference by which clinicians can draw conclusions for the patient population is based on the selection of a representative sample drawn from the patient population by some random procedure. This concept is called the population model (Lehmann, 1975; Lachin, 1988a). Suppose that for a certain disease, a clinical trial is planned to investigate the efficacy and safety of a newly developed therapeutic agent compared to an inert placebo. Under the population model we can draw two samples independently with equal chance at random from the (infinitely large) patient population. One sample consists of nT patients, and the other sample consists of nP patients. We denote these two samples by sample T and sample P, respectively. The nT patients in sample T will receive the newly developed agent, while the inert placebo is given to the nP patients in sample P. If the patient population is homogeneous with respect to the inclusion and exclusion criteria specified in the protocol, we do not expect that the responses of clinical endpoints for a particular patient will have anything to do with those of other patients. In other words, they are statistically independent of one another. For a homogeneous population, a common (population) distribution can be used to describe the characteristics of the clinical responses. That is, they are assumed to have identical distribution. Hence the clinical responses of the nI patients (I = T, P) are said to have an independent and identical distribution (i.i.d). Therefore, optimal statistical inference can be precisely obtained. For example, with respect to hypotheses (2.6.1) regarding the detection of the difference between the new agent and the placebo, the common two-sample t-test is the optimal testing procedure (Armitage and Berry, 1987).

As mentioned above, randomization in clinical trials involves random selection of the patients from the population and random assignment of patients to the treatments. Under the assumption of a homogeneous population, the clinical responses of all patients in the trial, regardless of sample T or sample P, are independent and have the same distribution. Lachin (1988a) points out that the significance level (i.e., the probability of type I error) and the power (i.e., the probability of correctly rejecting a false null hypothesis) will not be affected by random assignment of patients to the treatments as long as the patients in the trial represent a random sample from the homogeneous population. Furthermore suppose that we split a random sample into two subsamples; the statistical inferential procedures are still valid even if the one-half of the patients are assigned to the test drug and the other half to the placebo.

### Invoked Population Model

In clinical trials, we usually select investigators first and then select patients at each selected investigator's site. At each selected study site, the investigator will usually enroll qualified patients sequentially. A qualified patient is referred to as a patient who meets the inclusion and exclusion criteria and has signed the informed consent form. As a result, neither the selection of investigators (or study centers) nor the recruitment of patient is random. However, patients who enter a trial are assigned to treatment groups at random. In practice, the collected clinical data are usually analyzed as if they were obtained under the assumption that the sample is randomly selected from a homogeneous patient population. Lachin (1988a) refers to this process as the invoked population model because the population model is invoked as the basis for statistical analysis as if a formal sampling procedure were actually performed. In current practice, the invoked population model is commonly employed for data analysis for most clinical trials. It, however, should be noted that the invoked population model is based on the assumption that it is inherently untestable.

Note that one of the underlying assumptions for both the population model and the invoked population model is that the patient population is homogeneous. This assumption, however, is not valid in most clinical trials. In practice, we can employ the technique of stratified sampling to select samples according to some prespecified covariates to describe the differences in patient characteristics. The idea of stratification is to have homogeneous subpopulations with respect to the prespecified covariates (or patient characteristics). In many clinical trials, it is almost impossible to use a few covariates to describe the differences among heterogeneous subpopulations due to the complexity of patient characteristics and disease conditions. In addition, patients who are enrolled at different times may not have similar relevant demographic and baseline characteristics. In other words, the patient population is time-heterogeneous population in which the patient's characteristics are a function of the time when they enter the trial. The impact of the heterogeneity due to the recruitment time on the results of a clinical trial is well documented in the literature. For example, Byar et al. (1976) indicate that in a study conducted by the Veterans Administration Cooperative Urological Research Group in 1967, the survival rate of the patients who entered earlier in the study was worse than of those who enrolled later in the study. Therefore it is not uncommon for patient characteristics to change over time even if the population is homogeneous at one time point. The above discussion indicates that the assumption of the population model or the invoked population model may not be valid.

### Randomization Model

As discussed above, for current practice, although the study site selection and patient selection are not random, the assignment of treatments to patients is usually performed based on some random mechanism. Thus, treatment comparisons can be made based on the so-called randomization or permutation tests introduced in the mid-1930s (Fisher, 1935). To illustrate the concept of permutation tests, we consider the following hypothetical data set concerning endpoint changes from baselines in peak urinary flow rate (mL/s) after three months of treatment for patients with benign prostate hyperplasia:

Table 4.2.1 All Possible Ranks for the Two Patients in the Placebo Group Based on Conditional Permutation

Possible Ranks Sum of Ranks

Table 4.2.1 All Possible Ranks for the Two Patients in the Placebo Group Based on Conditional Permutation

Possible Ranks Sum of Ranks

 1,