Detailed STAT 509 Lesson 8: Sample Size and Power

A complete set of VOCABULARY style flashcards covering Lesson 8 on Sample Size and Power, including formulas, statistical concepts, and SAS procedures.

Precision

The central theme of sample size calculation; it describes how close an estimator is to the true population value.

Validity

A characteristic of a study that ensures correct results; the lesson notes that this does not necessarily depend on sample size.

Unbiasedness

The quality of an estimator where its expected value equals the true population parameter; it does not necessarily depend on sample size.

Desired Precision

One of the two major reasons to calculate sample size, specifically intended for an estimator or confidence interval.

Statistical Power

One of the two major reasons to calculate sample size, specifically intended to achieve a desired probability for a hypothesis test.

$$z_{1-\alpha/2}$$

The normal percentile commonly used for a two-sided $\alpha$-level test in sample-size formulas.

$$z_{1-\alpha}$$

The normal percentile that replaces $z_{1-\alpha/2}$ for a one-sided $\alpha$-level test.

$$z_{1-\beta}$$

The normal percentile used in sample-size formulas to represent a test with power $1-\beta$.

Common $\beta$ values

Values of $0.20$ and $0.10$, which correspond to statistical power levels of $0.80$ and $0.90$ respectively.

$\alpha = 0.05$ two-sided z-value

$$z_{0.975} \approx 1.96$$

$\alpha = 0.05$ one-sided z-value

$$z_{0.95} \approx 1.645$$

$\alpha = 0.01$ two-sided z-value

$$z_{0.995} \approx 2.576$$

$\alpha = 0.01$ one-sided z-value

$$z_{0.99} \approx 2.33$$

80% power z-value

$$z_{0.80} \approx 0.842$$

90% power z-value

$$z_{0.90} \approx 1.282$$

SAS PROBIT function

A function used to generate standard normal percentiles, such as $PROBIT(0.99)$ giving approximately $2.33$.

Distributional approximation issue

A potential problem where a calculation is based on a t statistic assuming exact normality even though the outcome is only approximately normal.

Loss-to-follow-up

A consideration in recruitment planning because not all enrolled subjects will provide complete data.

Protocol deviation

A reality where some subjects do not take the assigned treatment or add another treatment, which should be reflected in sample-size planning.

Treatment mechanism (TM) study

Studies where sample size is often not a major issue because only a few subjects are enrolled but many measurements are taken on each.

Dose-finding (DF) sample size drivers

Design algorithms such as modified Fibonacci or continual reassessment, rather than conventional fixed sample-size calculations.

Modified Fibonacci

A design algorithm used to drive sample size for dose-finding or dose-ranging studies.

Continual reassessment

A design algorithm used for dose-finding studies instead of fixed sample-size calculations.

0 of 3 toxicity rule

An escalation rule where the next cohort is moved to a higher dose if none of the three subjects experience toxicity.

1 of 3 toxicity rule

An escalation rule where the next cohort of three is treated at the same dose if one of the initial three experiences the effect.

2 of 6 stopping rule

A rule to terminate a dose-finding study whenever at least two of six subjects at the same dose level experience the effect.

Final DF sample size

A value that is not fixed because it depends on observed patient outcomes and escalation/stopping rules.

Efficacy

The requirement that a tested dose is effective at ameliorating the treated condition, which the FDA requires before drug approval.

Phase II efficacy role

The stage of clinical testing that evaluates the potential for efficacy.

Phase III efficacy role

The stage of clinical testing that confirms efficacy.

Safety and activity studies

An alternative term for safety-and-efficacy studies used in the lesson.

Confidence interval width

A reflection of precision that is a function of sample size.

$$p$$

Notation for the true unknown population success proportion in binomial trials.

$$n$$

Notation for the sample size.

$$r$$

Notation for the number of observed successes.

$$\hat{p}$$

The point estimate of a binomial proportion, calculated as $\frac{r}{n}$.

Large-sample Confidence Interval for $p$

$\hat{p} \pm z_{1-\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

$$p^*$$

The planning value or educated guess for a population proportion used before a study begins.

$\delta$ (Margin of error)

The desired half-width of a confidence interval.

Sample size for Confidence Interval for $p$

$$n = \frac{z_{1-\alpha/2}^2 \times p^* \times (1-p^*)}{\delta^2}$$

Conservative planning value for $p$

$p^* = 0.5$, as it maximizes the product $p(1-p)$ and results in the largest required sample size.

Quadratic precision relationship

The principle that the required sample size $n$ is proportional to $\frac{1}{\delta^2}$.

Halving margin of error

The act of reducing the desired margin of error by half, which requires approximately four times as many subjects.

Exact binomial CI

A method that uses binomial probabilities rather than normal approximation, recommended when expected successes and failures are small.

Exact lower confidence limit (p_L)

The value chosen so the upper-tail binomial probability at the observed number of successes equals $\alpha/2$.

Exact upper confidence limit (p_U)

The value chosen so the lower-tail binomial probability at the observed number of successes equals $\alpha/2$.

SAS PROC FREQ

A procedure used to provide exact and asymptotic confidence intervals for a binomial proportion.

Example 8.1 normal 95% CI truncation

The practice of setting a lower limit of $0.00$ even if the formula results in $-0.01$, because probabilities must lie between $0$ and $1$.

Discarding ineffective treatment rule

The decision to terminate a study when the exact confidence interval for $p$ no longer contains the minimally acceptable success rate.

$$\bar{x}$$

Notation for the sample mean.

$$s^2$$

Notation for the sample variance, which estimates the population variance $\sigma^2$.

Large-n approximation for t percentiles

The assumption that for $n \ge 60$, the t percentile is close enough to the corresponding z percentile to suggest large-scale precision.

Sample size for mean Confidence Interval

$n = [\frac{z_{1-\alpha/2} \times \sigma}{\delta}]^2$, rounded up to the nearest whole number.

Planning $\sigma$

A value for the population standard deviation that must be estimated from prior information or pilot data for sample size calculations.

Comparative Treatment Efficacy (CTE) trial

A trial that compares independent treatment groups with respect to a primary clinical endpoint.

$\Delta$ (Two-mean context)

The treatment difference in means, defined as $\mu_1 - \mu_2$.

Two-mean z statistic

$$Z = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sigma \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

AR (Allocation Ratio)

The ratio $n_1/n_2$ used to determine group balance in a study design.

Equal-allocation per-group sample size

$$n = \frac{2 \times \sigma^2 \times (z_{1-\alpha/2} + z_{1-\beta})^2}{\Delta^2}$$

Effect size in SD units

The ratio $\frac{\Delta}{\sigma}$ used to characterize the magnitude of the treatment difference.

Halving detectable difference (Delta)

An action that quadruples the required sample size because $n$ is proportional to $\frac{1}{\Delta^2}$.

Small-sample t refinement

The iterative process of using t percentiles instead of z percentiles when solving for $n$, as the percentile depends on the degrees of freedom.

SAS PROC POWER

The specific SAS procedure used for power and sample-size calculations in Lesson 8.

FEV1

Forced Expiratory Volume in one second; a continuous response used as an endpoint in the lesson's examples.

Unequal allocation consequence

The statistical fact that a $2:1$ allocation requires a larger total sample size than a $1:1$ allocation for the same power and effect size.

Normalization Adequacy (Two proportions)

The rule of thumb that normal or chi-square approximations are useful when expected cell counts are greater than about $5$.

$\bar{p}$ (Weighted average proportion)

$\frac{AR \times p_1 + p_2}{AR + 1}$, where $AR$ is the allocation ratio.

Fisher's exact test

A test used for comparing proportions when expected cell counts are small.

TWOSAMPLEFREQ

A SAS syntax statement in PROC POWER used to specify parameters for comparing two frequencies/proportions.

Time-to-event endpoint

A clinical response measuring the duration until an event occurs, analyzed using survival-analysis methods.

Hazard rate $h(t)$

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

Hazard ratio (Lambda)

The ratio of two treatment-group hazard functions.

Proportional hazards assumption

The assumption that the ratio of hazards between two groups remains constant over time.

Increasing hazard

An interpretation where the instantaneous risk of the event or death increases over the course of follow-up.

Decreasing hazard example

Adult Respiratory Distress Syndrome (ARDS), where the risk of death may be highest early in the disease.

$E$ (Required events)

The total number of events needed for a logrank test to achieve specified power.

Equal-allocation event formula

$$E = 4 \times [\frac{z_{1-\alpha/2} + z_{1-\beta}}{\ln(\Lambda)}]^2$$

Censoring

The phenomenon in survival studies where some participants do not experience the event before the study ends.

Constant hazard model

A model where survival is exponential and the hazard rate $h(t) = \lambda$ is constant.

Exponential survival function

$$S(t) = e^{-\lambda \times t}$$

TWOSAMPLESURVIVAL

A statement in SAS PROC POWER used with $TEST=LOGRANK$ to solve for sample size in time-to-event studies.

Accrual time ($A$)

The duration over which subjects are recruited into a study.

Follow-up time ($F$)

The duration subjects are observed after recruitment to check for endpoints.

Expanded Safety (ES) trial

A Phase IV study designed to estimate the frequency of uncommon adverse events.

Poisson model for rare adverse events

A model where events $D$ are following a distribution with mean $m \times \lambda$, where $m$ is the number of subjects and $\lambda$ is the rate.

$P(D=0)$ (Poisson)

$$e^{-(m \times \lambda)}$$

$P(D \ge 1)$ (Poisson)

$$1 - e^{-(m \times \lambda)}$$

Cohort size for detection probability $\gamma$

$m \ge \frac{-\ln(1-\gamma)}{\lambda}$, rounded up to the nearest person.

Bonferroni correction

A multiplicity adjustment where each of $K$ primary comparisons is tested at $\frac{\alpha}{K}$.

Multiple endpoints: AND rule

The logic that if success requires all primary endpoints to be significant, a multiplicity adjustment is not necessary.

Composite endpoint

A single analysis combining multiple events (e.g., stroke, MI, death) that does not require a multiplicity adjustment for the components.

Intention-to-treat (ITT) analysis

A method where subjects are compared according to their originally randomized group regardless of adherence or discontinuation.

Therapeutic dropout

A participant who stops their assigned active therapy but remains in the study to provide data.

Drop-in

A participant assigned to a control group who begins taking the active or a more effective therapy during the study.

Nonadherence adjustment formula

$N^* = \frac{N}{(1 - d_1 - d_0)^2}$, where $d_1$ and $d_0$ are discontinuation and drop-in rates.

Withdrawal adjustment formula

$N_{adjusted} = \frac{N}{1 - W}$, where $W$ is the anticipated proportion of subjects withdrawing.

Iterative planning step 1

Determine the null and alternative hypotheses tied to the primary outcome.

Iterative planning step 3

Define the study population and assess primary-outcome variability and the clinically important difference.

Iterative planning step 7

Select a sample size and plot power curves over plausible parameter values.

Survival-study key quantity

The principle that power is driven strongly by the number of observed events ($E$), not just the number of subjects enrolled ($n$).