### Fixed-Sample Clinical Trials

Subsections:

A clinical trial is a research study in consenting human beings to answer specific health questions. One type of trial is a treatment trial, which tests the effectiveness of an experimental treatment. An example is a planned experiment designed to assess the efficacy of a treatment in humans by comparing the outcomes in a group of patients who receive the test treatment with the outcomes in a comparable group of patients who receive a placebo control treatment, where patients in both groups are enrolled, treated, and followed over the same time period.

A clinical trial is conducted according to a plan called a protocol. The protocol provides detailed description of the study. For a fixed-sample trial, the study protocol contains detailed information such as the null hypothesis, the one-sided or two-sided test, and the Type I and II error probability levels. It also includes the test statistic and its associated critical values in the hypothesis testing.

Generally, the efficacy of a new treatment is demonstrated by testing a hypothesis in a clinical trial, where is the parameter of interest. For example, to test whether a population mean is greater than a specified value , can be used with an alternative .

A one-sided test is a test of the hypothesis with either an upper (greater) or a lower (lesser) alternative, and a two-sided test is a test of the hypothesis with a two-sided alternative. The drug industry often prefers to use a one-sided test to demonstrate clinical superiority based on the argument that a study should not be run if the test drug would be worse (Chow, Shao, and Wang, 2003, p. 28). But in practice, two-sided tests are commonly performed in drug development (Senn, 1997, p. 161). For a fixed Type I error probability , the sample sizes required by one-sided and two-sided tests are different. See Senn (1997, pp. 161–167) for a detailed description of issues involving one-sided and two-sided tests.

For independent and identically distributed observations of a random variable, the likelihood function for is

where is the population parameter and is the probability or probability density of . Using the likelihood function, two statistics can be derived that are useful for inference: the maximum likelihood estimator and the score statistic.

#### Maximum Likelihood Estimator

The maximum likelihood estimate (MLE) of is the value that maximizes the likelihood function for . Under mild regularity conditions, is an asymptotically unbiased estimate of with variance , where is the Fisher information

and is the expected Fisher information (Diggle et al., 2002, p. 340)

The score function for is defined as

and usually, the MLE can be derived by solving the likelihood equation . Asymptotically, the MLE is normally distributed (Lindgren, 1976, p. 272):

If the Fisher information does not depend on , then is known. Otherwise, either the expected information evaluated at the MLE () or the observed information can be used for the Fisher information (Cox and Hinkley 1974, p. 302; Efron and Hinkley 1978, p. 458), where the observed Fisher information

If the Fisher information does depend on , the observed Fisher information is recommended for the variance of the maximum likelihood estimator (Efron and Hinkley, 1978, p. 457).

Thus, asymptotically, for large n,

where I is the information, either the expected Fisher information or the observed Fisher information .

So to test versus , you can use the standardized Z test statistic

and the two-sided p-value is given by

where is the cumulative standard normal distribution function and is the observed Z statistic.

If the BOUNDARYSCALE=SCORE is specified in the SEQDESIGN procedure, the boundary values for the test statistic are displayed in the score statistic scale. With the standardized Z statistic, the score statistic and

#### Score Statistic

The score statistic is based on the score function for ,

Under the null hypothesis , the score statistic is the first derivative of the log likelihood evaluated at the null reference 0:

Under regularity conditions, is asymptotically normally distributed with mean zero and variance , the expected Fisher information evaluated at the null hypothesis (Kalbfleisch and Prentice, 1980, p. 45), where is the Fisher information

That is, for large n,

Asymptotically, the variance of the score statistic , , can also be replaced by the expected Fisher information evaluated at the MLE (), the observed Fisher information evaluated at the null hypothesis (, or the observed Fisher information evaluated at the MLE () (Kalbfleisch and Prentice, 1980, p. 46), where

Thus, asymptotically, for large n,

where I is the information, either an expected Fisher information ( or ) or a observed Fisher information ( or ).

So to test versus , you can use the standardized Z test statistic

If the BOUNDARYSCALE=MLE is specified in the SEQDESIGN procedure, the boundary values for the test statistic are displayed in the MLE scale. With the standardized Z statistic, the MLE statistic and

#### One-Sample Test for Mean

The following one-sample test for mean is used to demonstrate fixed-sample clinical trials in the section One-Sided Fixed-Sample Tests in Clinical Trials and the section Two-Sided Fixed-Sample Tests in Clinical Trials.

Suppose are n observations of a response variable Y from a normal distribution

where is the unknown mean and is the known variance.

Then the log likelihood function for is

where c is a constant. The first derivative is

where is the sample mean.

Setting the first derivative to zero, the MLE of is , the sample mean. The variance for can be derived from the Fisher information

Since the Fisher information does not depend on in this case, is used as the variance for . Thus the sample mean has a normal distribution with mean and variance :

Under the null hypothesis , the score statistic

has a mean zero and variance

With the MLE , the corresponding standardized statistic is computed as , which has a normal distribution with variance 1:

Also, the corresponding score statistic is computed as and

which is identical to computed under the null hypothesis .

Note that if the variable Y does not have a normal distribution, then it is assumed that the sample size n is large such that the sample mean has an approximately normal distribution.