Clinical trial

Independent trial safety monitoring reviews accumulating serious outcomes according to a prespecified plan and may recommend stopping or modifying a clinical trial when harm or benefit changes the ethical balance. Complete ascertainment of deaths and other severe outcomes is needed even if the trial is sized for a different endpoint. Pooled evidence and prespecified sequential thresholds can make rare-event monitoring useful without presenting a nonsignificant harm comparison as safety.

Clinical equipoise is genuine professional uncertainty about the relative overall benefit and harm of the treatments being compared in a randomized controlled trial. A favorable intermediate endpoint does not by itself resolve uncertainty about mortality. Ethical design requires informed consent, meaningful benefit-risk assessment and monitoring of serious harms; an underpowered safety comparison is not evidence that harm is absent.

A clinical trial comparison must define a common time origin and observation window for both assigned treatments. A probability measured from allocation cannot be replaced by one measured from a later treatment-dependent release or discharge. The latter may describe a different time interval and a selected population. A numerical sample size cannot be obtained by silently treating those probabilities as interchangeable.

Eligibility criteria specify which individuals may enter a clinical trial, defining its enrolled population and excluding individuals for whom the intervention or measurements are unsuitable. Screening applies those criteria before randomization. The treatment comparison's external validity depends on how this population relates to the population of interest.

The sample size is the number of observational units included in a study or assigned to a specified study arm.

For equal independent binomial samples, the displayed normal-approximation size per group targets a two-sided significance level $\alpha$ and power $1-\beta$ at specified alternative rates, with $\bar{p}=(p_1+p_0)/2$. Its first variance term sets the null critical value and its second the alternative distribution. A numerical sample size is not determined by an effect ratio alone; power, sidedness, allocation and baseline rates must also be specified. Rare-event discreteness and clustered designs can materially change the achieved power.

Suppose each collaborating centre contributes $n_0$ independently randomized participants per arm with specified event probabilities. At 50% statistical power, the usual normal-approximation alternative quantile is zero. The required size per arm is $2\bar{p}(1-\bar{p})z_{\ast }^2/(p_A-p_B{)}^2$, with $z_{\ast }=z_{1-\alpha /2}$ for a two-sided test or $z_{1-\alpha }$ for a one-sided test. Divide by the per-centre arm size and round up. Endpoints must use the same time origin and follow-up window, and cluster dependence requires an additional adjustment.

A sample size calculation for a binary endpoint requires its own baseline probability, not merely the rate of a related outcome. A fatal-event probability equals the probability of the underlying event times its conditional fatality probability; knowing only that product does not identify the incidence of all events. Thus a mortality rate can size a mortality trial but cannot determine the size for all fatal and nonfatal events without further assumptions. The significance level, allocation, target statistical power and effect definition must also be specified.

For a one-sided normal-mean test of size $\alpha$ with known standard deviation $\sigma$, a positive alternative ${\delta }^{\ast }$ and target statistical power $1-\beta >\alpha$, the required sample size is the ceiling of ${\sigma }^2(z_{1-\alpha }+z_{1-\beta }{)}^2/({\delta }^{\ast }{)}^2$. This follows by shifting the standard normal quantile in the alternative rejection probability.

The statistical power of a test at an alternative parameter value is the probability that the test rejects its null hypothesis at that value.

For two independent samples of size $n$ with event probabilities $p_0,p_1$, put $\delta =p_0-p_1$, $\bar{p}=(p_0+p_1)/2$, $s_0^2=2\bar{p}(1-\bar{p})/n$ and $s_1^2=[p_0(1-p_0)+p_1(1-p_1)]/n$. The normal approximation to a two-sided level-$\alpha$ test has powerFor rare events, independent Poisson distributions approximate the counts. Conditional on their total, the null count allocation is binomial distribution with probability one half; under the alternative the allocation probability is $p_0/(p_0+p_1)$. Averaging its exact rejection probability over the alternative total supplies a discrete power calculation.

// Target: probability-theory.bigb

A group sequential design analyzes accumulating data at prespecified interim analyses and may stop early for efficacy or futility.

A futility boundary specifies an interim result below which a clinical trial stops without rejecting the null hypothesis. Removing only potential rejection paths cannot increase the Type I error. With a finite boundary and a nonsingular joint normal test-statistic distribution, it strictly reduces that error relative to the corresponding fixed-final-boundary test.

In a two-stage normal trial, conditioning on a first-stage sample mean exceeding $c$ raises that mean. With stage standard error $s$, mean parameter $\delta$, and combined mean $S$, the estimator bias after continuation is $E(S\mid X\ge c)-\delta =(s/2)\lambda ((c-\delta )/s)$, where $\lambda$ is the upper-tail Inverse Mills ratio. It is positive and tends to zero as continuation becomes virtually certain.

In a two-stage normal trial, invert the pooled estimator's conditional mean after continuation to correct its selection shift. The root is unique when the pooled stage-1 weight satisfies $0<w<1$, because its conditional-mean map has derivative between $1-w$ and one. This root also maximizes the corresponding conditional likelihood. It is distinct from exact conditional unbiasedness, which is supplied by a suitable Rao-Blackwell theorem construction.

For two independent stage means $X,Y\sim N(\delta ,s^2)$ and continuation $X\ge c$, let $S=(X+Y)/2$ and $v=s/\sqrt{2}$. The second-stage mean is conditionally unbiased. The Rao-Blackwell theorem gives $E(Y\mid S,X\ge c)=S-v\lambda ((c-S)/v)$, using $X\mid S\sim N(S,v^2)$ and truncation. This is a conditionally unbiased estimator using both stages through their combined sample mean.

The conditional error principle permits adaptation of a testing design if, conditional on the observed interim information, the modified future rule's Type I error does not exceed the original rule's conditional future rejection probability. Averaging then preserves the original unconditional error bound. For a completed single-look test that has not rejected, there is no future conditional error budget under that original design.

An alpha-spending function assigns the cumulative Type I error budget to successive information fractions in a group sequential design. Its total at the final analysis is the overall significance level. The rejection boundaries must be calibrated using the joint distribution of the sequential test statistics; independent fixed-level tests do not automatically spend the assigned budget correctly.

An interim analysis examines data before a study reaches its planned maximum sample size.

A sequential stopping boundary specifies values of an accumulating test statistic that trigger early stopping for efficacy, futility, or harm.

At information levels $I_1<\cdots <I_J$, standardized efficient test statistics have unit variances, means proportional to $\sqrt{I_j}$, and correlations $\mathrm{Corr}(Z_j,Z_k)=\sqrt{I_j/I_k}$ for $j\le k$.

A two-stage clinical trial design permits early stopping after an interim analysis for futility or efficacy and otherwise continues to a prespecified final decision boundary.

Neyman allocation assigns sample sizes in proportion to outcome standard deviations to minimize variance.

Response-adaptive randomization updates treatment probabilities using outcomes observed during a trial.

In a two-treatment urn design, begin with $a$ balls of each type, draw and replace a ball to allocate treatment, then add $b$ balls of the same type after success or of the opposite type after failure. This response-adaptive randomization favours treatments with more observed successes. For RPW$(1,1)$, type-0 probability after counts $(s_0,f_0,s_1,f_1)$ is $(1+s_0+f_1)/(2+s_0+f_0+s_1+f_1)$.