The SEQDESIGN Procedure

Fixed-Sample Clinical Trials

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 $\text{[math]}$ in a clinical trial, where $\text{[math]}$ is the parameter of interest. For example, to test whether a population mean $\text{[math]}$ is greater than a specified value $\text{[math]}$ , $\text{[math]}$ can be used with an alternative $\text{[math]}$ .

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 $\text{[math]}$ , the sample sizes required by one-sided and two-sided tests are different. Refer to Senn (1997, pp. 161–167) for a detailed description of issues involving one-sided and two-sided tests.

For independent and identically distributed observations $\text{[math]}$ of a random variable, the likelihood function for $\text{[math]}$ is

$\text{[math]}$

where $\text{[math]}$ is the population parameter and $\text{[math]}$ is the probability or probability density of $\text{[math]}$ . 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 $\text{[math]}$ is the value $\text{[math]}$ that maximizes the likelihood function for $\text{[math]}$ . Under mild regularity conditions, $\text{[math]}$ is an asymptotically unbiased estimate of $\text{[math]}$ with variance $\text{[math]}$ , where $\text{[math]}$ is the Fisher information

$\text{[math]}$

and $\text{[math]}$ is the expected Fisher information (Diggle et al. 2002, p. 340)

$\text{[math]}$

The score function for $\text{[math]}$ is defined as

$\text{[math]}$

and usually, the MLE can be derived by solving the likelihood equation $\text{[math]}$ . Asymptotically, the MLE is normally distributed (Lindgren 1976, p. 272):

$\text{[math]}$

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

$\text{[math]}$

If the Fisher information $\text{[math]}$ does depend on $\text{[math]}$ , the observed Fisher information is recommended for the variance of the maximum likelihood estimator (Efron and Hinkley 1978, p. 457).

Thus, asymptotically, for large $\text{[math]}$ ,

$\text{[math]}$

where $\text{[math]}$ is the information, either the expected Fisher information $\text{[math]}$ or the observed Fisher information $\text{[math]}$ .

So to test $\text{[math]}$ versus $\text{[math]}$ , you can use the standardized $\text{[math]}$ test statistic

$\text{[math]}$

and the two-sided $\text{[math]}$ -value is given by

$\text{[math]}$

where $\text{[math]}$ is the cumulative standard normal distribution function and $\text{[math]}$ is the observed $\text{[math]}$ 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 $\text{[math]}$ statistic, the score statistic $\text{[math]}$ and

$\text{[math]}$

Score Statistic

The score statistic is based on the score function for $\text{[math]}$ ,

$\text{[math]}$

Under the null hypothesis $\text{[math]}$ , the score statistic $\text{[math]}$ is the first derivative of the log likelihood evaluated at the null reference $\text{[math]}$ :

$\text{[math]}$

Under regularity conditions, $\text{[math]}$ is asymptotically normally distributed with mean zero and variance $\text{[math]}$ , the expected Fisher information evaluated at the null hypothesis $\text{[math]}$ (Kalbfleisch and Prentice 1980, p. 45), where $\text{[math]}$ is the Fisher information

$\text{[math]}$

That is, for large $\text{[math]}$ ,

$\text{[math]}$

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

$\text{[math]}$

Thus, asymptotically, for large $\text{[math]}$ ,

$\text{[math]}$

where $\text{[math]}$ is the information, either an expected Fisher information ( $\text{[math]}$ or $\text{[math]}$ ) or a observed Fisher information ( $\text{[math]}$ or $\text{[math]}$ ).

So to test $\text{[math]}$ versus $\text{[math]}$ , you can use the standardized $\text{[math]}$ test statistic

$\text{[math]}$

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 $\text{[math]}$ statistic, the MLE statistic $\text{[math]}$ and

$\text{[math]}$

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 $\text{[math]}$ are $\text{[math]}$ observations of a response variable Y from a normal distribution

$\text{[math]}$

where $\text{[math]}$ is the unknown mean and $\text{[math]}$ is the known variance.

Then the log likelihood function for $\text{[math]}$ is

$\text{[math]}$

where $\text{[math]}$ is a constant. The first derivative is

$\text{[math]}$

where $\text{[math]}$ is the sample mean.

Setting the first derivative to zero, the MLE of $\text{[math]}$ is $\text{[math]}$ , the sample mean. The variance for $\text{[math]}$ can be derived from the Fisher information

$\text{[math]}$

Since the Fisher information $\text{[math]}$ does not depend on $\text{[math]}$ in this case, $\text{[math]}$ is used as the variance for $\text{[math]}$ . Thus the sample mean $\text{[math]}$ has a normal distribution with mean $\text{[math]}$ and variance $\text{[math]}$ :

$\text{[math]}$

Under the null hypothesis $\text{[math]}$ , the score statistic

$\text{[math]}$

has a mean zero and variance

$\text{[math]}$

With the MLE $\text{[math]}$ , the corresponding standardized statistic is computed as $\text{[math]}$ , which has a normal distribution with variance $\text{[math]}$ :

$\text{[math]}$

Also, the corresponding score statistic is computed as $\text{[math]}$ and

$\text{[math]}$

which is identical to $\text{[math]}$ computed under the null hypothesis $\text{[math]}$ .

Note that if the variable Y does not have a normal distribution, then it is assumed that the sample size $\text{[math]}$ is large such that the sample mean has an approximately normal distribution.