The SEQDESIGN Procedure

Fixed-Sample Clinical Trials

Maximum Likelihood Estimator
Score Statistic
One-Sample Test for Mean

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 $H_{0}: \theta = 0$ in a clinical trial, where $\theta$ is the parameter of interest. For example, to test whether a population mean $\mu$ is greater than a specified value $\mu _{0}$ , $\theta = \mu - \mu _{0}$ can be used with an alternative $\theta > 0$ .

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 $\alpha$ , 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 $y_{1}, y_{2}, \ldots , y_{n}$ of a random variable, the likelihood function for $\theta$ is

$L(\theta ) = \prod _{j=1}^{n} L_{i} (\theta )$

where $\theta$ is the population parameter and $L_{i}(\theta )$ is the probability or probability density of $y_{i}$ . 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 $\theta$ is the value $\hat{\theta }$ that maximizes the likelihood function for $\theta$ . Under mild regularity conditions, $\hat{\theta }$ is an asymptotically unbiased estimate of $\theta$ with variance $1/E_{\theta }( I(\theta ) )$ , where $I(\theta )$ is the Fisher information

$I(\theta ) = - \frac{ {\partial }^{2} \mr {log}( L(\theta ))}{\partial {\theta }^{2}}$

and $E_{\theta }(I(\theta ))$ is the expected Fisher information (Diggle et al., 2002, p. 340)

$E_{\theta }( I(\theta ) ) = - E_{\theta } \left( \frac{ {\partial }^{2} \mr {log}( L(\theta ))}{\partial {\theta }^{2}} \right)$

The score function for $\theta$ is defined as

$S( \theta ) = \frac{ \partial \, \mr {log}( L(\theta )) }{\partial \theta }$

and usually, the MLE can be derived by solving the likelihood equation $S( \theta ) = 0$ . Asymptotically, the MLE is normally distributed (Lindgren, 1976, p. 272):

$\hat{\theta } \sim N \left( \, \theta , \, \frac{1}{E_{\theta }( I(\theta ) )} \right)$

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

$I({\hat{\theta }}) = - \left( \frac{ {\partial }^{2} \mr {log}( L(\theta )) }{ \partial {\theta }^{2} } \, | \, \theta ={\hat\theta } \right)$

If the Fisher information $I(\theta )$ does depend on $\theta$ , 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,

$\hat{\theta } \sim N \left( \, \theta , \, \frac{1}{I} \right)$

where I is the information, either the expected Fisher information $E_{\theta =0} (I(\theta ))$ or the observed Fisher information $I{\hat{\theta }})$ .

So to test $H_{0}: \theta = 0$ versus $H_{1}: \theta \neq 0$ , you can use the standardized Z test statistic

$Z = \frac{\hat\theta }{\sqrt {\mr {Var}({\hat\theta })}} = {\hat\theta } \, \sqrt {I} \sim N \left( \, 0, \, 1 \right)$

and the two-sided p-value is given by

$\mr {Prob} (|Z| > |z_0|) = 1 - 2 \Phi (|z_0|)$

where $\Phi$ 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 $S= Z \sqrt {I}= \hat{\theta } I$ and

$S \sim N \left( \, 0, \, I \right)$

Score Statistic

The score statistic is based on the score function for $\theta$ ,

$S( \theta ) = \frac{ \partial \, \mr {log}( L(\theta )) }{\partial \theta }$

Under the null hypothesis $H_{0}: \theta = 0$ , the score statistic is the first derivative of the log likelihood evaluated at the null reference 0:

$S(0) = \frac{ \partial \, \mr {log}( L(\theta )) }{\partial \theta } \, | \, \theta =0$

Under regularity conditions, is asymptotically normally distributed with mean zero and variance $E_{\theta =0} (I(\theta ))$ , the expected Fisher information evaluated at the null hypothesis $\theta =0$ (Kalbfleisch and Prentice, 1980, p. 45), where $I(\theta )$ is the Fisher information

$I(\theta ) = - E \left( \frac{ {\partial }^{2} \, \mr {log}( L(\theta )) }{ \partial {\theta }^{2} } \right)$

That is, for large n,

$S(0) \sim N \left( \, 0, \, E_{\theta =0} (I(\theta )) \right)$

Asymptotically, the variance of the score statistic , $E_{\theta =0} (I(\theta ))$ , can also be replaced by the expected Fisher information evaluated at the MLE $\theta ={\hat{\theta }}$ ( $E_{\theta ={\hat{\theta }}} (I(\theta ))$ ), the observed Fisher information evaluated at the null hypothesis $\theta =0$ (, or the observed Fisher information evaluated at the MLE $\theta ={\hat{\theta }}$ ( $I({\hat{\theta }})$ ) (Kalbfleisch and Prentice, 1980, p. 46), where

$I(0) = - \left( \frac{ {\partial }^{2} \mr {log}( L(\theta )) }{ \partial {\theta }^{2} } \, | \, \theta =0 \right)$

$I({\hat{\theta }}) = - \left( \frac{ {\partial }^{2} \mr {log}( L(\theta )) }{ \partial {\theta }^{2} } \, | \, \theta ={\hat\theta } \right)$

Thus, asymptotically, for large n,

$S(0) \sim N \left( \, 0, \, I \right)$

where I is the information, either an expected Fisher information ( $E_{\theta =0} (I(\theta ))$ or $E_{\theta ={\hat{\theta }}} (I(\theta ))$ ) or a observed Fisher information ( or $I({\hat{\theta }})$ ).

So to test $H_{0}: \theta = 0$ versus $H_{1}: \theta \neq 0$ , you can use the standardized Z test statistic

$Z = \frac{S(0)}{\sqrt {I}}$

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 ${\hat\theta }= Z / \sqrt {I}= U(0) / I$ and

${\hat\theta } \sim N \left( \, 0, \, \frac{1}{I} \right)$

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 ${y}_{1}, {y}_{2}, \ldots , {y}_{n}$ are n observations of a response variable Y from a normal distribution

$y_{i} \sim N \left( \, \theta , \, {\sigma }^{2} \right)$

where $\theta$ is the unknown mean and ${\sigma }^{2}$ is the known variance.

Then the log likelihood function for $\theta$ is

$\mr {log} (L(\theta )) = \sum _{j=1}^{n} -\frac{1}{2} \frac{{(y_ j-\theta )}^{2}}{\sigma ^{2}} + c$

where c is a constant. The first derivative is

$\frac{ {\partial } \mr {log}( L(\theta )) }{ \partial {\theta } } = \frac{1}{\sigma ^{2}} \sum _{j=1}^{n} (y_ j - \theta ) = \frac{n}{\sigma ^{2}} ( {\overline{y}} - \theta )$

where ${\overline{y}}$ is the sample mean.

Setting the first derivative to zero, the MLE of $\theta$ is $\hat{\theta } = {\overline{y}}$ , the sample mean. The variance for $\hat{\theta }$ can be derived from the Fisher information

$I(\theta ) = - \frac{ {\partial }^{2} \mr {log}( L(\theta )) }{ \partial {\theta }^{2} } = \frac{n}{\sigma ^{2}}$

Since the Fisher information $I_0= I(\theta )$ does not depend on $\theta$ in this case, is used as the variance for $\hat{\theta }$ . Thus the sample mean ${\overline{y}}$ has a normal distribution with mean $\theta$ and variance ${\sigma }^{2}/n$ :

$\hat{\theta } = {\overline{y}} \sim N \left( \, \theta , \, \frac{1}{I_{0}} \right) = N \left( \, \theta , \, \frac{{\sigma }^{2}}{n} \right)$

Under the null hypothesis $H_{0}: \theta = 0$ , the score statistic

$S(0) = \frac{ \partial \, \mr {log}( L(\theta )) }{\partial \theta } | \theta =0 = \frac{n}{\sigma ^{2}} {\overline{y}}$

has a mean zero and variance

$I(\theta ) = - \frac{ {\partial }^{2} \mr {log}( L(\theta )) }{ \partial {\theta }^{2} } = \frac{n}{\sigma ^{2}}$

With the MLE $\hat{\theta }$ , the corresponding standardized statistic is computed as $Z= \hat{\theta } \sqrt {I_{0}} = {\overline{y}} / (\sigma / \sqrt {n})$ , which has a normal distribution with variance 1:

$Z \sim N \left( \, {\theta } \sqrt {I_{0}}, \, 1 \right) = N \left( \, \frac{\theta }{\sigma / \sqrt {n}}, \, 1 \right)$

Also, the corresponding score statistic is computed as $S= \hat{\theta } I_{0} = n {\overline{y}} / {\sigma }^{2}$ and

$S \sim N \left( \, {\theta } I_{0}, \, I_{0} \right) = N \left( \, \frac{n \theta }{{\sigma }^{2}}, \, \frac{n}{{\sigma }^{2}} \right)$

which is identical to computed under the null hypothesis $H_{0}: \theta = 0$ .

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.