Calc Sample Size

Calc-Sample-Size Skill

You are assisting a medical researcher with sample size and power calculations. Guide the user through test selection using the decision tree, generate reproducible code in R (primary) and Python (alternative), interpret effect sizes clinically, and produce IRB-ready justification text.

Reference Files

• Formulas: ${CLAUDE_SKILL_DIR}/references/formulas.md -- mathematical formulas, R/Python functions, effect size conventions
• Observational cohort precision branch: ${CLAUDE_SKILL_DIR}/references/observational_cohort.md
• Justification prose exemplars: ${CLAUDE_SKILL_DIR}/references/justification_examples.md -- reviewer-safe IRB/Methods justification paragraphs per design (proportions, means, DTA precision, survival/log-rank, ICC agreement, non-inferiority), each stating the five required elements; load when producing the justification text
• Existing R template: See analyze-stats skill at references/templates/sample_size.R for the 7 original tests

Read formulas.md before generating calculation code. For retrospective observational cohorts with a fixed extract, also read references/observational_cohort.md and report event budget / confidence-interval precision instead of forcing a prospective recruitment-style power calculation.

Cross-Skill References

• design-study calls calc-sample-size when a sample size justification is needed during study design.
• calc-sample-size output feeds into write-protocol and write-paper (Methods section).
• Detailed formulas and references are in ${CLAUDE_SKILL_DIR}/references/formulas.md.

---

Decision Tree

When the user requests a sample size calculation, walk them through this tree interactively. Ask one question at a time. Do not assume answers.

What is your primary outcome?
|
+-- Binary (yes/no, positive/negative)
|   |
|   +-- Paired data (same subjects, two methods)?
|   |   +-- YES --> [5] McNemar test
|   |   +-- NO  --> How many groups?
|   |       +-- 2 groups, superiority     --> [4] Two-proportion comparison (chi-square)
|   |       +-- 2 groups, non-inferiority --> [10] Non-inferiority / equivalence
|   |       +-- Multivariable model       --> [9] Logistic regression
|   |
+-- Continuous (measurement, score)
|   |
|   +-- How many groups?
|       +-- 2 groups  --> [6] Independent t-test
|       +-- 3+ groups --> [8] One-way ANOVA
|
+-- Time-to-event (survival, recurrence)
|   |
|   +-- Two groups, unadjusted      --> [7] Log-rank test
|   +-- Multivariable / adjusted HR  --> [7] Log-rank (Schoenfeld) + [11] Cox EPV
|
+-- Agreement (inter-rater, reproducibility)
|   |
|   +-- Continuous measurements --> [2] ICC
|   +-- Categorical ratings     --> [3] Kappa
|
+-- Diagnostic accuracy (Se, Sp, AUC precision)
    |
    +--> [1] Diagnostic accuracy (precision-based)

---

Supported Tests

Test 1: Diagnostic Accuracy (Sensitivity/Specificity Precision)

When to use: Estimating required sample size for desired precision of sensitivity or specificity in a diagnostic accuracy study.

Required parameters (ask the user):

Effect size interpretation: The CI half-width determines precision. A half-width of 0.05 means the 95% CI for sensitivity will be within +/-5 percentage points. Narrower CIs require larger samples.

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Test 2: ICC Agreement (Bonett 2002)

When to use: Inter-rater or intra-rater agreement for continuous measurements (e.g., tumor size, angle measurement).

Required parameters:

Effect size interpretation: ICC < 0.50 = poor, 0.50-0.75 = moderate, 0.75-0.90 = good, > 0.90 = excellent (Koo & Li, 2016).

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Test 3: Kappa Agreement (Donner & Eliasziw 1992)

When to use: Inter-rater agreement for categorical ratings (e.g., BI-RADS category, lesion present/absent).

Required parameters:

Effect size interpretation: Kappa < 0.20 = slight, 0.21-0.40 = fair, 0.41-0.60 = moderate, 0.61-0.80 = substantial, 0.81-1.00 = almost perfect (Landis & Koch, 1977).

---

Test 4: Two-Proportion Comparison (Chi-Square)

When to use: Comparing proportions between two independent groups (e.g., AI detection rate vs. conventional detection rate).

Required parameters:

Effect size interpretation: Cohen's h = 2 arcsin(sqrt(p1)) - 2 arcsin(sqrt(p2)). Small = 0.20, medium = 0.50, large = 0.80.

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Test 5: McNemar Test (Paired Proportions)

When to use: Paired binary outcomes (e.g., two readers reading same cases, before/after on same patients).

Required parameters:

Effect size interpretation: The ratio p10/p01 (discordant ratio) drives the required sample size. Larger asymmetry in discordant pairs means fewer subjects needed. Only discordant pairs contribute information.

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Test 6: Independent t-Test

When to use: Comparing means between two independent groups (e.g., lesion size in malignant vs. benign).

Required parameters:

Effect size interpretation: Cohen's d = mean_diff / pooled_sd. Small = 0.20, medium = 0.50, large = 0.80. In clinical terms, d = 0.50 means the groups differ by half a standard deviation.

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Test 7: Survival / Log-Rank Test (Schoenfeld 1981)

When to use: Comparing survival or time-to-event between two groups (e.g., treatment vs. control, RFA vs. surgery).

Required parameters:

Effect size interpretation: HR < 1 favors treatment. HR = 0.50 means treatment halves the hazard (strong effect). HR = 0.80 is a modest 20% reduction. The Schoenfeld formula calculates required number of events, then inflates for expected event probability and dropout.

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Test 8: One-Way ANOVA (NEW)

When to use: Comparing means across 3 or more independent groups (e.g., comparing AI model performance across 3 architectures, comparing measurement accuracy across multiple readers).

Required parameters:

Help user estimate Cohen's f:

• If the user knows group means and pooled SD: f = sigma_means / pooled_SD
• If the user knows eta-squared: f = sqrt(eta_sq / (1 - eta_sq))
• Benchmarks: small = 0.10, medium = 0.25, large = 0.40

R function: pwr::pwr.anova.test(k, f, sig.level, power) Python equivalent: statsmodels.stats.power.FTestAnovaPower().solve_power(effect_size, nobs, alpha, power, k_groups)

Effect size interpretation: Cohen's f = 0.25 (medium) means the group means span about half a pooled SD. In clinical terms, this is typically a meaningful difference across treatment arms or measurement methods.

---

Test 9: Logistic Regression (NEW)

When to use: Multivariable binary outcome models (e.g., predicting malignancy from multiple imaging features). Two approaches are provided.

Required parameters:

Approach A: Peduzzi Rule of Thumb (EPV >= 10)

• N_events = 10 * n_predictors
• N_total = N_events / event_rate
• Simple, widely cited, conservative. Use as a minimum baseline.

Approach B: Hsieh (1989) Formula

• Uses the OR of interest for the primary predictor to calculate a more precise sample size.
• Accounts for correlation with other predictors via R-squared adjustment.

Always report both approaches and recommend the larger N.

Effect size interpretation: OR = 1.5 is a small-to-moderate effect; OR = 2.0 is moderate; OR = 3.0+ is large. The Peduzzi rule ensures model stability; the Hsieh formula targets power for the primary predictor.

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Test 10: Non-Inferiority / Equivalence (NEW)

When to use: Demonstrating that a new method is not worse than the standard by more than a pre-specified margin (non-inferiority) or that two methods are equivalent within a margin (equivalence / TOST).

Required parameters:

Key guidance for margin selection:

• The margin must be clinically justified and smaller than the effect of the reference treatment vs. placebo.
• Common approach: margin = 50% of the established treatment effect (preservation of effect).
• For proportions: absolute difference margin (e.g., delta = 0.10 means new method can be at most 10 percentage points worse).
• For continuous: margin in the same unit as the outcome.

Non-inferiority (one-sided test):

• H0: new - reference <= -margin (new is inferior)
• H1: new - reference > -margin (new is non-inferior)
• Alpha is one-sided (typically 0.025).

Equivalence (TOST):

• H0: |new - reference| >= margin
• H1: |new - reference| < margin
• Two one-sided tests, each at alpha (typically 0.05 overall).

Effect size interpretation: The margin defines the largest clinically acceptable difference. A smaller margin requires a larger sample. Always justify the margin based on clinical reasoning and prior literature.

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Test 11: Cox Regression EPV (Events Per Variable)

When to use: Multivariable Cox proportional hazards models — ensuring enough events for stable model estimates. Same EPV logic as logistic regression (Test 9), applied to time-to-event outcomes.

Required parameters:

Formula:

N_events = EPV × n_predictors
N_total = N_events / event_rate
N_adj = N_total / (1 - attrition_rate)

EPV guidelines:

• EPV >= 10: minimum for stable estimates (Peduzzi et al., 1995)
• EPV >= 20: recommended for reliable CI coverage and type I error control
• EPV < 5: model likely unstable — reduce predictors or use penalized methods

Effect size interpretation: The EPV rule ensures model stability, not power for a specific HR. If the user also needs power for detecting a specific HR, combine with Test 7 (log-rank/Schoenfeld) and report the larger N.

Always report both approaches (EPV minimum + Schoenfeld power, if HR is available) and recommend the larger N.

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Scope Limitations

Supported

The 11 tests listed above cover the vast majority of sample size calculations needed in medical imaging research, diagnostic accuracy studies, and clinical trials.

NOT Supported

The following designs require specialized software or biostatistician consultation:

• Adaptive trials (group-sequential, sample size re-estimation)
• Cluster-randomized trials (design effect, ICC-based inflation)
• Bayesian sample size determination
• Crossover designs
• Multi-endpoint correction (mention Bonferroni adjustment if asked, but do not compute corrected sample sizes)

If the user requests any of these, respond:

"This design requires specialized tools beyond this skill's scope. Consider using G*Power software (free, https://www.psychologie.hhu.de/gpower), PASS software, or consulting a biostatistician for [specific design]."

---

Workflow

Phase 1: Understand the Study

1. Ask the user to describe their study briefly (design, primary outcome, groups). 2. Walk through the decision tree to identify the appropriate test. 3. Confirm the selected test with the user before proceeding.

Phase 2: Collect Parameters

1. Present the parameter table for the selected test. 2. For each parameter without a user-provided value, explain what it means and offer the default. 3. Help the user estimate effect sizes from:

• Prior literature (ask for references)
• Pilot data
• Cohen's conventions (as a last resort, with a note that convention-based estimates are less precise)

Phase 2b: Retrospective Study — Experience-Based Sample Size Justification

For retrospective studies, formal power analysis is often impractical because the dataset already exists. In these cases, an experience-based justification is acceptable for IRB and many journals. Offer this path when the user describes a retrospective design.

Two approaches:

Approach A: Institution Volume-Based

Estimate N from the number of examinations performed at the institution during the study period.

Total exams in period × prevalence of target condition × (1 - exclusion rate) = Expected N

• Ask the user for: annual exam volume for the modality, study period length, estimated

prevalence, and expected exclusion rate

• This gives a realistic upper bound for N

IRB justification template:

Based on approximately [X] [modality] examinations performed annually at [institution],

and an estimated prevalence of [condition] of [Y]%, we anticipate identifying approximately

[N] eligible patients over the [Z]-year study period. After accounting for an estimated

[W]% exclusion rate (due to [reasons]), we expect a final sample of approximately [N_adj]

patients for analysis.

Approach B: Prior Study-Based

Use sample sizes from published studies with similar designs as justification.

• Search for 3-5 comparable studies and report their sample sizes
• The user's N should be in the same range or larger
• Cite the specific studies in the IRB justification

IRB justification template:

Previous studies evaluating [similar topic] with [similar design] enrolled [N1] (Author1

et al., Year), [N2] (Author2 et al., Year), and [N3] (Author3 et al., Year) patients.

Our anticipated sample of [N] patients is [comparable to / larger than] these prior studies.

When to Use Formal Calculation Instead

Even for retrospective studies, a formal sample size calculation is preferred when:

• The study is prospective or will prospectively enroll a subset
• The primary analysis involves hypothesis testing (not just estimation)
• The journal explicitly requires power analysis (check Instructions for Authors)
• The IRB requires it for approval

In these cases, proceed to Phase 3 with the appropriate test from the decision tree.

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Phase 3: Calculate and Report

1. Read ${CLAUDE_SKILL_DIR}/references/formulas.md for the exact formula. 2. Generate the R code (primary) and Python code (alternative). 3. Run the R code via Bash to produce the actual result. 4. Present results in the output format below.

Phase 4: Sensitivity Analysis (Optional)

If the user is uncertain about parameters, offer a sensitivity table showing N across a range of plausible values (e.g., varying effect size or power from 0.80 to 0.90).

---

Output Format

Always structure the final output as follows:

## Sample Size Calculation Report

### Study Design
[1-2 sentence summary of the design and test selected]

### Parameters
| Parameter | Value | Source |
|-----------|-------|--------|
| ... | ... | user / literature / convention |

### Result
- **Required sample size**: N = [value]
- **With [X]% attrition adjustment**: N_adj = [value]

### R Code (Reproducible)

[complete, self-contained R script]

Dependencies: [list packages]

Run: Rscript sample_size_calc.R

### Python Code (Alternative)

[complete, self-contained Python script]

Dependencies: [list packages]

Run: python sample_size_calc.py

### IRB Justification Text
> A sample of [N] participants is required to detect [effect description] with [power]% power
> at a [one/two]-sided significance level of [alpha], assuming [key assumptions].
> Accounting for an estimated [X]% attrition rate, we plan to enroll [N_adj] participants.
> This calculation is based on [formula/method reference].

### Effect Size Interpretation
[Cohen's benchmark classification + clinical meaning in the context of this study]

---

IRB Justification Text Guidelines

The IRB text must: 1. State the required N clearly. 2. Name the statistical test and its formula source. 3. Specify all assumed parameters (effect size, alpha, power). 4. State the attrition adjustment and final enrollment target. 5. Cite the methodological reference (e.g., "Schoenfeld, 1981" for survival). 6. Use formal, third-person language suitable for an ethics board.

---

Communication Rules

• Communicate with the user in their preferred language.
• Use English for all statistical terminology, effect size names, and test names.
• Be explicit about assumptions and their impact on the result.
• When the user provides vague effect size estimates, flag the uncertainty and suggest a sensitivity analysis.
• Never fabricate references. Cite only verified methodological sources from formulas.md.

Anti-Hallucination

• Never fabricate file paths, URLs, DOIs, or package names. Verify existence before recommending.
• Never invent journal metadata, impact factors, or submission policies without verification at the journal's website.
• If a tool, package, or resource does not exist or you are unsure, say so explicitly rather than guessing.

Sample Size Formulas Reference

Mathematical formulas, R/Python implementations, and effect size conventions for all 10 supported tests.

---

Test 1: Diagnostic Accuracy (Sensitivity/Specificity Precision)

Formula

For estimating a single proportion p with desired 95% CI half-width w:

n_positive = ceil( (z_{alpha/2} / w)^2 * p * (1 - p) )
n_total = ceil( n_positive / prevalence )

Where:

• z_{alpha/2} = 1.96 for alpha = 0.05
• p = expected sensitivity (or specificity)
• w = desired CI half-width
• prevalence = disease prevalence in the study population

R Implementation

z <- qnorm(1 - alpha / 2)
n_pos <- ceiling((z / ci_half_width)^2 * se_expected * (1 - se_expected))
n_total <- ceiling(n_pos / prevalence)
n_adj <- ceiling(n_total / (1 - attrition_rate))

Python Implementation

from scipy.stats import norm
import math

z = norm.ppf(1 - alpha / 2)
n_pos = math.ceil((z / ci_half_width)**2 * se_expected * (1 - se_expected))
n_total = math.ceil(n_pos / prevalence)
n_adj = math.ceil(n_total / (1 - attrition_rate))

R Package

Base R (no additional packages needed).

Key Reference

Buderer NMF. Statistical methodology: I. Incorporating the prevalence of disease into the sample size calculation for sensitivity and specificity. Acad Emerg Med. 1996;3(9):895-900.

---

Test 2: ICC Agreement (Bonett 2002)

Formula

Fisher z-transformation approach:

z_1 = 0.5 * ln((1 + rho_1) / (1 - rho_1))    # expected ICC
z_0 = 0.5 * ln((1 + rho_0) / (1 - rho_0))    # null ICC
n = ceil( ((z_{alpha} + z_{beta}) / (z_1 - z_0))^2 + 3 )

Where:

• rho_1 = expected ICC
• rho_0 = null hypothesis ICC
• z_{alpha} = qnorm(1 - alpha) for one-sided, qnorm(1 - alpha/2) for two-sided
• z_{beta} = qnorm(power)

R Implementation

# Option A: MKpower package (preferred)
library(MKpower)
result <- sampleSize.ICC(rho0 = icc_null, rho1 = icc_expected,
                          k = n_raters, alpha = alpha, power = power)
n <- result$n

# Option B: Manual Fisher z (if MKpower unavailable)
z_exp  <- 0.5 * log((1 + icc_expected) / (1 - icc_expected))
z_null <- 0.5 * log((1 + icc_null) / (1 - icc_null))
n <- ceiling(((qnorm(1 - alpha) + qnorm(power)) / (z_exp - z_null))^2 + 3)

Python Implementation

from scipy.stats import norm
import math
import numpy as np

z_exp  = 0.5 * np.log((1 + icc_expected) / (1 - icc_expected))
z_null = 0.5 * np.log((1 + icc_null) / (1 - icc_null))
z_alpha = norm.ppf(1 - alpha)
z_beta  = norm.ppf(power)
n = math.ceil(((z_alpha + z_beta) / (z_exp - z_null))**2 + 3)

R Package

MKpower (preferred), or base R for manual calculation.

Key Reference

Bonett DG. Sample size requirements for estimating intraclass correlations with desired precision. Stat Med. 2002;21(9):1331-1335.

---

Test 3: Kappa Agreement (Donner & Eliasziw 1992)

Formula

Z-test approximation for testing kappa_1 vs kappa_0:

n = ceil( ((z_{alpha/2} + z_{beta})^2 * kappa_1 * (1 - kappa_1)) / (kappa_1 - kappa_0)^2 + 1 )

Where:

• kappa_1 = expected kappa
• kappa_0 = null hypothesis kappa
• pe = expected chance agreement = (po - kappa) / (1 - kappa)

R Implementation

n_kappa <- ceiling(
  ((qnorm(1 - alpha / 2) + qnorm(power))^2 *
     kappa_expected * (1 - kappa_expected)) /
    (kappa_expected - kappa_null)^2 + 1
)

Python Implementation

from scipy.stats import norm
import math

z_a = norm.ppf(1 - alpha / 2)
z_b = norm.ppf(power)
n = math.ceil(((z_a + z_b)**2 * kappa_exp * (1 - kappa_exp)) /
              (kappa_exp - kappa_null)**2 + 1)

R Package

Base R. Also available via kappaSize package.

Key Reference

Donner A, Eliasziw M. A goodness-of-fit approach to inference procedures for the kappa statistic: confidence interval construction, significance-testing and sample size estimation. Stat Med. 1992;11(11):1511-1519.

---

Test 4: Two-Proportion Comparison (Chi-Square)

Formula

Based on Cohen's h effect size (arcsine transformation):

h = 2 * arcsin(sqrt(p1)) - 2 * arcsin(sqrt(p2))
n_per_group = ceil( ((z_{alpha/2} + z_{beta}) / h)^2 )

R Implementation

library(pwr)
h <- ES.h(p1, p2)
result <- pwr.2p.test(h = h, sig.level = alpha, power = power)
n_per_group <- ceiling(result$n)
n_total <- n_per_group * 2

Python Implementation

from statsmodels.stats.power import NormalIndPower
import numpy as np
import math

h = 2 * np.arcsin(np.sqrt(p1)) - 2 * np.arcsin(np.sqrt(p2))
analysis = NormalIndPower()
n_per_group = math.ceil(analysis.solve_power(effect_size=h, alpha=alpha, power=power,
                                              ratio=1, alternative='two-sided'))
n_total = n_per_group * 2

R Package

pwr

Key Reference

Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Lawrence Erlbaum Associates; 1988.

---

Test 5: McNemar Test (Paired Proportions)

Formula

n = ceil(
  (z_{alpha/2} * sqrt(p01 + p10) + z_{beta} * sqrt(p01 + p10 - (p10 - p01)^2))^2
  / (p10 - p01)^2
)

Where:

• p01 = P(Method A negative, Method B positive)
• p10 = P(Method A positive, Method B negative)

R Implementation

n_mc <- ceiling(
  (qnorm(1 - alpha / 2) * sqrt(p01 + p10) +
     qnorm(power) * sqrt(p01 + p10 - (p10 - p01)^2))^2 /
    (p10 - p01)^2
)

Python Implementation

from scipy.stats import norm
import math

z_a = norm.ppf(1 - alpha / 2)
z_b = norm.ppf(power)
disc_sum = p01 + p10
disc_diff = p10 - p01
n = math.ceil((z_a * math.sqrt(disc_sum) +
               z_b * math.sqrt(disc_sum - disc_diff**2))**2 / disc_diff**2)

R Package

Base R. Also available via exact2x2 package.

Key Reference

Connor RJ. Sample size for testing differences in proportions for the paired-sample design. Biometrics. 1987;43(1):207-211.

---

Test 6: Independent t-Test

Formula

d = mean_diff / pooled_sd    (Cohen's d)
n_per_group = ceil( 2 * ((z_{alpha/2} + z_{beta}) / d)^2 )

R Implementation

library(pwr)
d <- mean_diff / pooled_sd
result <- pwr.t.test(d = d, sig.level = alpha, power = power, type = "two.sample")
n_per_group <- ceiling(result$n)
n_total <- n_per_group * 2

Python Implementation

from statsmodels.stats.power import TTestIndPower
import math

d = mean_diff / pooled_sd
analysis = TTestIndPower()
n_per_group = math.ceil(analysis.solve_power(effect_size=d, alpha=alpha, power=power,
                                              ratio=1, alternative='two-sided'))
n_total = n_per_group * 2

R Package

pwr

Key Reference

Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Lawrence Erlbaum Associates; 1988.

---

Test 7: Survival / Log-Rank Test (Schoenfeld 1981)

Formula

Step 1 -- Required number of events:

d = ceil( ((z_{alpha/2} + z_{beta}) / ln(HR))^2 )

Step 2 -- Total sample size:

lambda_ctrl = ln(2) / median_ctrl
lambda_trt  = lambda_ctrl * HR
p_event_ctrl = 1 - exp(-lambda_ctrl * follow_up)
p_event_trt  = 1 - exp(-lambda_trt * follow_up)
avg_p_event  = (p_event_ctrl + p_event_trt) / 2
n_total = ceil( d / avg_p_event )

R Implementation

n_events <- ceiling((qnorm(1 - alpha / 2) + qnorm(power))^2 / (log(hr))^2)
lambda_ctrl <- log(2) / median_ctrl
lambda_trt  <- lambda_ctrl * hr
p_event_ctrl <- 1 - exp(-lambda_ctrl * follow_up)
p_event_trt  <- 1 - exp(-lambda_trt * follow_up)
avg_p_event  <- (p_event_ctrl + p_event_trt) / 2
n_total <- ceiling(n_events / avg_p_event)
n_adj <- ceiling(n_total / (1 - drop_rate))

Python Implementation

from scipy.stats import norm
import math
import numpy as np

z_a = norm.ppf(1 - alpha / 2)
z_b = norm.ppf(power)
n_events = math.ceil((z_a + z_b)**2 / np.log(hr)**2)
lambda_ctrl = np.log(2) / median_ctrl
lambda_trt = lambda_ctrl * hr
p_event_ctrl = 1 - np.exp(-lambda_ctrl * follow_up)
p_event_trt = 1 - np.exp(-lambda_trt * follow_up)
avg_p_event = (p_event_ctrl + p_event_trt) / 2
n_total = math.ceil(n_events / avg_p_event)
n_adj = math.ceil(n_total / (1 - drop_rate))

R Package

Base R. Also available via gsDesign, survival packages.

Key Reference

Schoenfeld DA. The asymptotic properties of nonparametric tests for comparing survival distributions. Biometrika. 1981;68(1):316-319.

---

Test 8: One-Way ANOVA

Formula

Using Cohen's f effect size:

f = sigma_between / sigma_within

Where:

• sigma_between = SD of group means
• sigma_within = pooled within-group SD

From eta-squared: f = sqrt(eta_sq / (1 - eta_sq))

The sample size per group is obtained via the F-test power formula (implemented in pwr).

R Implementation

library(pwr)
result <- pwr.anova.test(k = k, f = f, sig.level = alpha, power = power)
n_per_group <- ceiling(result$n)
n_total <- n_per_group * k
n_adj <- ceiling(n_total / (1 - attrition_rate))

Python Implementation

from statsmodels.stats.power import FTestAnovaPower
import math

analysis = FTestAnovaPower()
n_per_group = math.ceil(analysis.solve_power(effect_size=f, nobs=None, alpha=alpha,
                                              power=power, k_groups=k))
n_total = n_per_group * k
n_adj = math.ceil(n_total / (1 - attrition_rate))

R Package

pwr

Key Reference

Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Lawrence Erlbaum Associates; 1988. (Chapter 8: F tests for ANOVA)

---

Test 9: Logistic Regression

Approach A: Peduzzi Rule of Thumb (EPV >= 10)

N_events = 10 * p   (where p = number of predictor variables)
N_total = N_events / event_rate

This ensures at least 10 events per predictor variable (EPV), the widely accepted minimum for stable logistic regression estimates.

Approach B: Hsieh (1989) Formula

For detecting a specific odds ratio (OR) for a continuous or binary predictor:

# For a binary predictor (proportion B exposed):
n_unadj = ((z_{alpha/2} + z_{beta})^2) / (B * (1-B) * ln(OR)^2)

# Adjusted for correlation with other predictors:
n_adj = n_unadj / (1 - R^2)

Where:

• OR = odds ratio of interest
• B = proportion of subjects exposed (for binary predictor) or use p*(1-p) for the outcome
• R^2 = multiple correlation of the predictor with other covariates

For a continuous predictor, the formula uses the standard normal density.

R Implementation

# Approach A: Peduzzi
n_events_peduzzi <- 10 * n_predictors
n_total_peduzzi <- ceiling(n_events_peduzzi / event_rate)

# Approach B: Hsieh (binary predictor)
z_a <- qnorm(1 - alpha / 2)
z_b <- qnorm(power)
n_hsieh <- ceiling((z_a + z_b)^2 /
                     (event_rate * (1 - event_rate) * log(or_interest)^2))
n_hsieh_adj <- ceiling(n_hsieh / (1 - r2_other))

# Report the larger of the two
n_final <- max(n_total_peduzzi, n_hsieh_adj)

Python Implementation

from scipy.stats import norm
import math
import numpy as np

# Approach A: Peduzzi
n_events_peduzzi = 10 * n_predictors
n_total_peduzzi = math.ceil(n_events_peduzzi / event_rate)

# Approach B: Hsieh (binary predictor)
z_a = norm.ppf(1 - alpha / 2)
z_b = norm.ppf(power)
n_hsieh = math.ceil((z_a + z_b)**2 /
                     (event_rate * (1 - event_rate) * np.log(or_interest)**2))
n_hsieh_adj = math.ceil(n_hsieh / (1 - r2_other))

n_final = max(n_total_peduzzi, n_hsieh_adj)

R Package

Base R. Also see powerMediation package for Hsieh's formula.

Key References

• Peduzzi P, Concato J, Kemper E, Holford TR, Feinstein AR. A simulation study of the number of events per variable in logistic regression analysis. J Clin Epidemiol. 1996;49(12):1373-1379.
• Hsieh FY. Sample size tables for logistic regression. Stat Med. 1989;8(7):795-802.

---

Test 10: Non-Inferiority / Equivalence

Non-Inferiority (Proportions)

n_per_group = ceil(
  ((z_{1-alpha} + z_{beta})^2 * (p_ref*(1-p_ref) + p_new*(1-p_new))) /
  (margin - |p_new - p_ref|)^2
)

When assuming p_new = p_ref (no true difference):

n_per_group = ceil( ((z_{1-alpha} + z_{beta})^2 * 2 * p*(1-p)) / margin^2 )

Note: alpha is one-sided (typically 0.025) for non-inferiority.

Non-Inferiority (Continuous)

n_per_group = ceil( ((z_{1-alpha} + z_{beta})^2 * 2 * sd^2) / margin^2 )

Equivalence / TOST (Proportions)

n_per_group = ceil(
  ((z_{1-alpha/2} + z_{beta})^2 * 2 * p*(1-p)) / margin^2
)

Note: For TOST, each one-sided test uses alpha (e.g., alpha = 0.05 for overall 5% type I error).

Equivalence / TOST (Continuous)

n_per_group = ceil( ((z_{1-alpha} + z_{beta})^2 * 2 * sd^2) / margin^2 )

R Implementation

# Non-inferiority for proportions (assuming no true difference)
z_a <- qnorm(1 - alpha)  # one-sided, alpha = 0.025
z_b <- qnorm(power)
p <- p_reference
n_per_group <- ceiling(((z_a + z_b)^2 * 2 * p * (1 - p)) / margin^2)
n_total <- n_per_group * 2

# Non-inferiority for continuous
n_per_group <- ceiling(((z_a + z_b)^2 * 2 * sd^2) / margin^2)
n_total <- n_per_group * 2

# Equivalence / TOST for continuous
n_per_group <- ceiling(((z_a + z_b)^2 * 2 * sd^2) / margin^2)
n_total <- n_per_group * 2

n_adj <- ceiling(n_total / (1 - attrition_rate))

Python Implementation

from scipy.stats import norm
import math

# Non-inferiority for proportions (assuming no true difference)
z_a = norm.ppf(1 - alpha)  # one-sided, alpha = 0.025
z_b = norm.ppf(power)
p = p_reference
n_per_group = math.ceil(((z_a + z_b)**2 * 2 * p * (1 - p)) / margin**2)
n_total = n_per_group * 2

# Non-inferiority for continuous
n_per_group = math.ceil(((z_a + z_b)**2 * 2 * sd**2) / margin**2)
n_total = n_per_group * 2

# Equivalence / TOST for continuous
# Same formula; the difference is in the hypothesis and alpha interpretation
n_per_group = math.ceil(((z_a + z_b)**2 * 2 * sd**2) / margin**2)
n_total = n_per_group * 2

n_adj = math.ceil(n_total / (1 - attrition_rate))

R Package

Base R. Also see TrialSize, PowerTOST packages.

Key References

• Julious SA. Sample sizes for clinical trials with normal data. Stat Med. 2004;23(12):1921-1986.
• Schuirmann DJ. A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. J Pharmacokinet Biopharm. 1987;15(6):657-680.

---

Test 11: Cox Regression EPV (Events Per Variable)

Formula

N_events = EPV × k    (where k = number of predictor variables)
N_total = ceil( N_events / event_rate )
N_adj = ceil( N_total / (1 - attrition_rate) )

Where:

• EPV = events per variable (minimum 10, recommended 20)
• k = number of predictors in the Cox model
• event_rate = proportion of subjects experiencing the event

R Implementation

n_events <- epv * n_predictors
n_total <- ceiling(n_events / event_rate)
n_adj <- ceiling(n_total / (1 - attrition_rate))

cat(sprintf("EPV = %d, Predictors = %d\n", epv, n_predictors))
cat(sprintf("Required events = %d\n", n_events))
cat(sprintf("Total N = %d (event rate = %.0f%%)\n", n_total, event_rate * 100))
cat(sprintf("Adjusted N = %d (attrition = %.0f%%)\n", n_adj, attrition_rate * 100))

Python Implementation

import math

n_events = epv * n_predictors
n_total = math.ceil(n_events / event_rate)
n_adj = math.ceil(n_total / (1 - attrition_rate))

R Package

Base R (no additional packages needed).

Key References

• Peduzzi P, Concato J, Feinstein AR, Holford TR. Importance of events per independent variable in proportional hazards regression analysis. II. Accuracy and precision of regression estimates. J Clin Epidemiol. 1995;48(12):1503-1510.
• Vittinghoff E, McCulloch CE. Relaxing the rule of ten events per variable in logistic and Cox regression. Am J Epidemiol. 2007;165(6):710-718.

---

Cohen's Effect Size Conventions

Important: Cohen's conventions are rules of thumb. Always prefer effect sizes estimated from prior literature or pilot data. When conventions are used, explicitly state this limitation in the IRB justification.

---

Common Attrition Rates by Study Type

Note: Attrition rates vary widely by disease, population, and follow-up duration. Use study-specific estimates when available.

---

Key References

1. Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.

• Foundation for effect size conventions (d, f, h, w) and power analysis methodology.

2. Schoenfeld DA. The asymptotic properties of nonparametric tests for comparing survival distributions. Biometrika. 1981;68(1):316-319.

• Formula for required number of events in log-rank test.

3. Bonett DG. Sample size requirements for estimating intraclass correlations with desired precision. Stat Med. 2002;21(9):1331-1335.

• ICC-based sample size using Fisher z-transformation.

4. Hsieh FY. Sample size tables for logistic regression. Stat Med. 1989;8(7):795-802.

• Power-based sample size for detecting a specific odds ratio.

5. Peduzzi P, Concato J, Kemper E, Holford TR, Feinstein AR. A simulation study of the number of events per variable in logistic regression analysis. J Clin Epidemiol. 1996;49(12):1373-1379.

• EPV >= 10 rule for logistic regression model stability.

6. Donner A, Eliasziw M. A goodness-of-fit approach to inference procedures for the kappa statistic: confidence interval construction, significance-testing and sample size estimation. Stat Med. 1992;11(11):1511-1519.

• Sample size for testing kappa against a null value.

7. Buderer NMF. Statistical methodology: I. Incorporating the prevalence of disease into the sample size calculation for sensitivity and specificity. Acad Emerg Med. 1996;3(9):895-900.

• Prevalence-adjusted sample size for diagnostic accuracy.

8. Julious SA. Sample sizes for clinical trials with normal data. Stat Med. 2004;23(12):1921-1986.

• Comprehensive non-inferiority and equivalence sample size formulas.

9. Schuirmann DJ. A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. J Pharmacokinet Biopharm. 1987;15(6):657-680.

• TOST (two one-sided tests) procedure for equivalence testing.

10. Koo TK, Li MY. A guideline of selecting and reporting intraclass correlation coefficients for reliability research. J Chiropr Med. 2016;15(2):155-163.

• ICC interpretation benchmarks.

11. Landis JR, Koch GG. The measurement of observer agreement for categorical data. Biometrics. 1977;33(1):159-174.

• Kappa interpretation benchmarks.

12. Connor RJ. Sample size for testing differences in proportions for the paired-sample design. Biometrics. 1987;43(1):207-211.

• McNemar test sample size formula.

13. Peduzzi P, Concato J, Feinstein AR, Holford TR. Importance of events per independent variable in proportional hazards regression analysis. II. Accuracy and precision of regression estimates. J Clin Epidemiol. 1995;48(12):1503-1510.

• EPV >= 10 rule for Cox regression model stability.

14. Vittinghoff E, McCulloch CE. Relaxing the rule of ten events per variable in logistic and Cox regression. Am J Epidemiol. 2007;165(6):710-718.

• Evidence that EPV 5-10 may be acceptable with careful validation.

Sample-size justification — worked prose exemplars (IRB / Methods)

Reviewer-safe justification paragraphs for the prospective designs this skill computes, complementing formulas.md (the math) and the retrospective/experience-based templates in SKILL.md Phase 2b. Each shows the same five elements a methods reviewer or IRB looks for; fill the [brackets] from the actual calculation — never invent the inputs. These are synthetic teaching models of standard justification structure, not copied text.

The five elements (every justification states all five)

1. The primary outcome and its effect/precision target (with the source of the assumed value — prior study, pilot, or a minimal clinically important difference, cited). 2. α (and one- vs two-sided) and power (or the target CI half-width for a precision aim). 3. The test/method the calculation matches (it must match the planned primary analysis). 4. The resulting n, including any inflation for attrition/clustering/multiplicity. 5. The software/approach used (named, so it is reproducible).

Exemplars by design

Two proportions (superiority). "Assuming an event rate of [p1] in the control arm (from [cited source]) and a clinically meaningful absolute reduction to [p2], a two-sided α = 0.05 and 80% power require [n/arm] per arm (two-proportion test). Allowing for [d]% attrition, we will enrol [N]."

Two means (superiority). "To detect a between-group difference of [Δ] in [outcome] (SD [s], from [source]; standardised effect [d]) with two-sided α = 0.05 and 90% power requires [n/arm] per arm (two-sample t-test)."

Diagnostic accuracy (precision aim). "To estimate sensitivity with a 95% CI half-width of [w], assuming sensitivity [Se] and disease prevalence [π], [N_total] participants ([N_diseased] with the target condition) are needed; the specificity target is checked separately from the non-diseased count [N_nondiseased] and is satisfied at this n. The aim is precision, not power — no comparison hypothesis is tested." (Cite Hajian-Tilaki / Buderer.)

Survival (log-rank). "To detect a hazard ratio of [HR] (median [m1] vs [m2]) with two-sided α = 0.05 and 80% power, [E] events are required (Schoenfeld); with an accrual of [a] over [t] and follow-up [f], this needs [N] participants. The event count, not N, drives power."

Agreement / reliability (ICC). "Assuming a true ICC of [ρ] with [k] raters per subject, [n] subjects give a 95% CI half-width of [w] (a precision aim, Bonett/Walter) — or, framed as assurance, [n] subjects so the 95% CI lower bound exceeds the minimally acceptable [ρ0]. State which of the two aims you used; report the CI target, not power."

Non-inferiority. "With a non-inferiority margin of [m] (justified clinically and by [regulatory/prior] precedent), assuming true equivalence and a control rate [p], one-sided α = 0.025 and 90% power require [n/arm] per arm. The margin and its rationale are pre-specified."

Discipline

• The calculation must match the planned primary analysis (do not power for a t-test and

analyse with a mixed model); state any clustering (design effect) or multiplicity adjustment.

• For a precision aim (DTA, agreement, single-arm), report the CI half-width, not power —

do not invent a comparison hypothesis to manufacture a power statement.

• Never reverse-engineer the effect size from an achievable n (post-hoc justification); the

assumed effect comes from a cited source or an MCID. Post-hoc/observed power is uninformative.

Observational Cohort Sample Size: Precision Branch

Use this reference when a cohort study is extraction-constrained or when the goal is precision rather than formal power.

When To Use

• Retrospective hospital-based cohort with all eligible records included.
• National database cohort where the sample is fixed by release permissions.
• Sparse-event outcome where the real question is whether confidence intervals are narrow enough to support a claim.

Core Checks

1. Primary estimand: risk difference, odds ratio, hazard ratio, incidence rate, or mean difference. 2. Event count: total events and events per modeled parameter. 3. Precision target: acceptable confidence interval half-width or maximum acceptable upper bound. 4. Model complexity: covariate count relative to events. 5. Fragility: subgroup and sensitivity analyses with smaller denominators.

Rule-of-Thumb Gates

Manuscript Phrase

"The study size was determined by all eligible participants in the released cohort. We therefore evaluated statistical informativeness by the number of outcome events, events per modeled parameter, and the width of the 95% confidence intervals around the primary estimate rather than by a conventional a priori recruitment target."

Minimal R Sketch

events <- 130
parameters <- 8
epv <- events / parameters

rate <- 130 / 250000
poisson.test(130, T = 250000)$conf.int

Reviewer-Safe Interpretation

• If events are sparse, do not write "no association" without discussing the upper confidence bound.
• If EPV is below 10, state that estimates are exploratory or use penalized/sparse models.
• If a subgroup has few events, report it descriptively even when the p-value looks attractive.