STAT 509 Lesson 8: Sample Size and Power

Vocabulary terms and definitions related to sample size, statistical power, and design considerations for clinical trials, including means, proportions, survival analysis, and adjustments for multiplicity and adherence.

Precision

The central theme of sample size calculation; refers to the closeness of an estimator to the true population value, often measured by confidence interval width.

Statistical Power

The probability that a hypothesis test will correctly reject the null hypothesis at a specified significance level ($\alpha$) for a clinically meaningful effect size.

$$z_{0.975}$$

The critical value approximately equal to $1.96$; it is used for a two-sided test with $\alpha=0.05$.

$$z_{1-\beta}$$

The standard normal percentile corresponding to power ($1-\beta$); common values include $0.842$ for $80\%$ power and $1.282$ for $90\%$ power.

SAS PROBIT function

A function that generates standard normal percentiles; for example, $\text{PROBIT}(0.99)$ yields approximately $2.33$.

Treatment mechanism (TM) study

A study where sample size is typically not a major issue because many measurements are taken on a small number of subjects.

0 of 3 toxicity rule

A dose-escalation design rule where the next cohort is escalated to the next dose if none of the three subjects in the current cohort experience toxicity.

2 of 6 stopping rule

A design rule that terminates the study whenever at least two of six subjects at the same dose level experience the toxicity/effect.

Efficacy

The demonstration that a tested dose is effective at ameliorating the treated condition, a requirement for drug approval by the U.S. FDA.

Conservative planning value ($p^*$)

Using $p^*=0.5$ for sample size calculations because it maximizes the variance $p(1-p)$, providing the largest/most conservative sample size ($n$).

Quadratic precision relationship

The mathematical principle where the required sample size ($n$) is proportional to $1/\Delta^2$, meaning halving the margin of error ($\Delta$) requires four times as many subjects.

Exact binomial confidence interval

An interval that uses binomial probabilities rather than normal approximations; it is preferred when normal approximation adequacy is uncertain due to small sample size or extreme proportions.

Sample size for mean confidence interval

Calculated by the formula $n = [\frac{z_{1-\alpha/2} \times \sigma}{\Delta}]^2$, where $\Delta$ is the desired half-width and $\sigma$ is the population standard deviation.

Comparative treatment efficacy (CTE) trial

A trial designed to compare independent treatment groups with respect to a primary clinical endpoint.

Allocation ratio (AR)

The ratio of treatment group sizes ($AR = n_1/n_2$); equal allocation is $AR=1$.

Effect size in SD units

Used in power calculations for means, defined as $\Delta/\sigma$, where $\Delta$ is the treatment difference and $\sigma$ is the common standard deviation.

PROC POWER

The SAS procedure used for power and sample-size calculations for continuous, binary, and survival outcomes.

Hazard rate ($h(t)$)

The instantaneous risk of an event at time $t$, conditional on the subject surviving event-free up to that time.

Hazard ratio ($\Lambda$)

The ratio of two treatment-group hazard functions, assumed to be constant over time under proportional hazards.

Logrank total number of events ($E$)

The primary quantity driving sample size for survival studies; for equal allocation, $E = 4 \times [\frac{z_{1-\alpha/2} + z_{1-\beta}}{\ln(\Lambda)}]^2$.

Expanded Safety (ES) trial

A Phase IV study specifically designed to estimate the frequency of uncommon adverse events that may have gone undetected in earlier phases.

Poisson rare-event model

A model where the probability of observing exactly $d$ events among $m$ subjects with rate $\lambda$ is $P(D=d) = \frac{\exp(-m \times \lambda) \times (m \times \lambda)^d}{d!}$.

Detection probability ($\gamma$)

The target probability of detecting at least one rare event, used to determine cohort size $m \ge \frac{-\ln(1-\gamma)}{\lambda}$.

Bonferroni correction

A multiplicity adjustment for $K$ primary comparisons where each is tested at $\alpha/K$ to control the overall false-positive rate.

Composite endpoint

A single statistical endpoint constructed from multiple clinical events (e.g., time to stroke, MI, or death) that does not require multiplicity adjustment.

Intention-to-treat (ITT) analysis

An analysis principle where subjects are compared according to their originally randomized group, regardless of adherence, discontinuation, or adding other therapies.

Therapeutic dropout

A participant assigned to the active therapy arm who stops taking that therapy but may continue to provide data.

Drop-in

A participant assigned to the placebo arm who adds or switches to an active therapy.

Nonadherence adjustment formula

A method to increase sample size based on a dilution factor: $N^* = N / (1-d_1-d_0)^2$, where $d_1$ and $d_0$ are discontinuation and drop-in rates.

Withdrawal adjustment formula

Calculated as $N_{\text{adjusted}} = N / (1-W)$, where $W$ is the anticipated proportion of subjects leaving the study before providing key outcome data.