Statis-Gravity | Clinical Biostatistics & Visualization Suite

Power & Sample Size Determination

A-priori sample size planning & post-hoc power calculation

Study Design & Statistical Test:

Calculation Objective:

Expected Mean 1 (Control):

Expected Mean 2 (Intervention):

Standard Deviation (Pooled SD):

Significance Level (α):

Desired Statistical Power (1 - β):

Available Sample Size per Group (n):

Effect Size (Cohen's d)

Standardized contrast

Sample Size per Group (n)

Equal 1:1 allocation

Total Sample Size (N)

Total study enrollment

Statistical Power (1 - β)

1 - Type II error rate

Fisher's Exact / CC Sample Size

Fleiss / Casagrande-Pike-Smith

Clinical Risk Profile

ARR, RR & NNT

Study Protocol Justification (Grant / Ethics / IRB Ready)

⚙️ Simple Randomization Parameters

Uniform Odd/Even Mapping

Group A Label (Odd Numbers)

Group B Label (Even Numbers)

Next Participant ID (Auto-Incrementing)

🛡️ Bias & Modulo Prevention: Uses window.crypto.getRandomValues() with rejection sampling (rejection threshold 4,294,967,200) to ensure true cryptographic uniformity across 1–100, eliminating modulo bias.

🎯 Prominent Assignment & Live Balance

No allocation yet

🎲

Ready to Randomize

Click the generate button to assign the next subject via cryptographically secure odd/even number draw.

Total (N)

Group A

0 (0%)

Group B

0 (0%)

Ratio (A:B)

1:1

Group A Balance (50%) Group B Balance (50%)

Methodological Warning: In smaller samples (N < 100), simple randomization naturally experiences random drift leading to noticeable sample size imbalances. If strict balance is essential throughout enrolment, switch to Calculator 2: Block Randomization.

⚙️ Block Randomization Parameters

Permuted Balanced Sequences

Block Size (Even Number)

Planned Sample Size (Target N)

Group A Label

Group B Label

🔒 Allocation Concealment Safeguard: Assignments are strictly concealed behind a digital veil until the investigator registers the participant. Shuffling inside each block is executed via Fisher-Yates with crypto.getRandomValues to eliminate selection and ordering bias.

🛡️ Allocation Concealment & Block Tracker

Block #1 Active

🔒

Allocation Concealed

Participant #1 is ready for assignment. Allocation is masked to prevent investigator selection bias.

Active Block Allocation Progress: Slot 0 of 4

Enrolled / Target

0 / 40

Group A

0 (0%)

Group B

0 (0%)

Completed Blocks

Group A Balance (50%) Group B Balance (50%)

📋 Session Audit Trail & Allocation Log

Permanent verifiable chronological sequence of participant assignments for current session

0 Records

# ID	Timestamp	Method	Random Draw / Slot	Group Code	Assigned Label
🎲 No participants randomized yet Click "Generate / Assign Next Participant" above to initiate sequence allocation.

Dataset Input

Paste numeric values or load clinical sample

Numeric Series (comma or newline separated):

Sample Size (n)

Range: --

Mean (M)

SD: ±--

Median (50%)

50th Percentile

Mode

Interquartile Range (IQR)

Q1: -- | Q3: --

Std Dev (SD)

Var: --

95% CI for Mean

[--, --]

SEM: --

Skewness (G₁)

Kurtosis (Excess G₂)

Outliers (Tukey's)

Normality Test (Jarque-Bera)

JB Stat: -- | p = --

Frequency Distribution & Normal Curve Fit Binned Histogram & Parametric Gaussian Fit

Box & Whiskers Plot Median, IQR Box, Tukey Fences & Outliers

Violin Density Plot Kernel Density Estimation (KDE) & Quartiles

Descriptive Statistical Summary (ICMJE & APA Ready)

Two-Group Comparison

Automated assumption-driven test selection & execution

Paired Samples (Pre vs Post / Repeated Measures / Matched)

Active Design: Independent Cohorts (Two Separate Groups) (Uncheck for separate groups; check for repeated measures on same subjects)

Statistical Test Selection:

Group 1 Label:

Group 2 Label:

Group 1 Observations:

Group 2 Observations:

Test Statistic

p-Value

Effect Size (Cohen's d)

Mean Difference

Comparative Cohort Distribution Error Bars: 95% Confidence Interval (Mean ± 95% CI)

Error Bars / Dispersion:

🔬 Statistical Assumptions & Automated Decision Engine

Comprehensive diagnostic validation: Pairing, Normality (Jarque-Bera), & Homoscedasticity (F-Test)

Automated Test Recommendation & Decision Rationale: Run hypothesis test to evaluate assumptions and obtain data-driven recommendation.

1. Normality Distribution --

2. Equality of Variances --

Decision Pipeline: -- ➔ -- ➔ -- ➔ --

Clinical / Publication Summary (APA Style)

Multi-Cohort & Repeated Measures Analysis

Automated assumption-driven test selection & execution

Paired / Repeated Measures Design (Within-Subjects / Repeated Timepoints / Matched Subjects)

Active Design: Independent Cohorts (Between-Subjects) (Uncheck for separate groups; check for repeated measures on same subjects)

Statistical Test Selection:

Cohorts & Observation Data:

Cohort 1:

n = 10

Cohort 2:

n = 10

Cohort 3:

n = 10

Test Statistic (F / H)

p-Value

Effect Size (η² / ω² / ε²)

Design & Degrees of Freedom

Multi-Cohort Variance & Dispersion Error Bars: 95% Confidence Interval (Mean ± 95% CI)

Error Bars / Dispersion:

🔬 Statistical Assumptions & Automated Decision Engine

Comprehensive diagnostic validation: Pairing, Normality (Jarque-Bera across all cohorts), & Homoscedasticity (Levene's Test)

Automated Test Recommendation & Decision Rationale: Run multi-group analysis to evaluate assumptions and obtain data-driven recommendation.

1. Normality Distribution (All Cohorts) --

2. Equality of Variances (Levene's Test) --

Decision Pipeline: Design: Independent ➔ Normality: Testing... ➔ Variances: Testing... ➔ Recommended: --

🔬 Post-Hoc Pairwise Contrasts

Pinpoints which specific cohorts contribute to the statistically significant difference

Pairwise Contrast	Difference (ΔM / Med)	Std. Error	Contrast Stat	Adjusted p-Value	95% CI of Difference	Effect Size	Significance
Run multi-group analysis to view post-hoc pairwise contrasts

Multi-Cohort & Post-Hoc Clinical Summary

🎯 Observational Cohort & PSM Settings

Control for baseline confounding in non-randomized clinical studies

Observational Dataset (CSV / TSV): N = 120 patients

Treatment (Binary 0/1):

Primary Outcome:

Confounding Baseline Covariates: Selected: 4

⚙️ Matching Algorithm & Caliper Specifications

Caliper Width (× SD logit PS): Austin 2011 standard: 0.20

Enforce Common Support Discard off-support tails

Matched Cohorts

Causal Effect (ATT)

95% Confidence Interval

Max Post-Match |SMD|

🔬 Balance & Overlap Diagnostics

Love Plot (SMD reduction) & Propensity Score Overlap Distribution

📊 Baseline Covariate Balance Diagnostics

Comparison of Unadjusted vs. Matched Standardized Mean Differences (|SMD| < 0.10 threshold)

Baseline Covariate	Unadjusted Means (T vs C)	Pre-Match SMD	Matched Means (T vs C)	Post-Match SMD	Variance Ratio	% Bias Reduction	Balance Status
Run Propensity Score Matching to view covariate balance diagnostics

⚖️ Causal Treatment Effect (ATT) vs. Unadjusted Estimate

Quantifies confounding bias removed by Propensity Score Matching

Analytical Estimate	Point Estimate	Std. Error	Test Statistic	p-Value	95% Confidence Interval	Clinical Interpretation
Run analysis to view causal treatment effect comparison

💻 Reproducible Statistical Code Scripts

Expert production-grade scripts with full methodology and inline biostatistical comments

# Select a language tab above to generate full reproducible code...

Causal Inference & STROBE Observational Summary

🌐 Multivariate EDA Controls

Dimensionality reduction for Continuous, Categorical, & Mixed Data

Dimensionality Reduction Technique: Balances continuous and categorical blocks via standardized indicator scaling (Pagès 2004).

Dataset Matrix (CSV / TSV):

N = 120 observations 📂 Upload

Active Variables & Types:

📐 Projection Axes & Grouping

X Axis:

Y Axis:

Z Axis (3D):

Stratification Color Group:

Show Biplot Vectors

Technique & Cohort

FAMD

N = 120

Dim 1 Variance (λ₁)

Dim 2 Variance (λ₂)

Displayed Plane Inertia

🔬 Interactive Factor Space

Projection of observations, variable correlation vectors, and modality centroids

📊 Eigenvalues & Inertia Decomposition

Kaiser-Guttman criterion (λ > 1.0) and cumulative explained variance

Dimension	Eigenvalue (λ)	Variance Explained (%)	Cumulative Variance (%)	Kaiser Criterion
Run analysis to view eigenvalue decomposition

💡 Automated Natural-Language Findings & Drivers

Real-time interpretation of subspace variance, influential drivers, and outlier observations

Execute factor decomposition to generate automated clinical and statistical interpretation.

💻 Reproducible Statistical Code Scripts

Production-grade scripts utilizing scikit-learn/prince (Python) and FactoMineR/factoextra (R)

# Select a language tab above to generate full reproducible code...

Multivariate Dimensionality Reduction Academic Summary

2x2 Contingency & Diagnostic Matrix

Choose between Diagnostic Accuracy vs Clinical Study

Analysis Objective / Framework:

Test \ Ref

Gold Standard (+)

Gold Standard (-)

New Test (+)

True Positive (TP)

False Positive (FP)

New Test (-)

False Negative (FN)

True Negative (TN)

Evaluates index test accuracy against the reference standard: Sensitivity, Specificity, PPV, NPV, Overall Accuracy (Wilson Score 95% CIs), and Likelihood Ratios.

Sensitivity (TPR)

95% CI: [--]

Specificity (TNR)

95% CI: [--]

Positive Predictive Value (PPV)

95% CI: [--]

Negative Predictive Value (NPV)

95% CI: [--]

Overall Diagnostic Accuracy

95% CI: [--]

Likelihood Ratios (LR+ / LR-)

Youden's J: --

Clinical Diagnostic / Epidemiological Summary

Bivariate Relationship

Pearson, Spearman & OLS Regression

Independent Variable X (e.g. Cranial Circumference):

Dependent Variable Y (e.g. Brain Volume):

Pearson r

R-Squared (R²)

Spearman Rank (ρ)

p-Value

Fitted Regression Equation

y = mx + c

Scatter Plot & Regression Fit

🛡️ Assumption Diagnostics & Test Decision Engine

Automated statistical verification of normality, outliers, homoscedasticity & linear vs U-shaped non-linear curvature

Recommended Method --

1. Normality of X

2. Normality of Y

3. Outlier / Leverage

4. Homoscedasticity

5. Curve Morphology

Correlation & Regression Clinical Statement

Biomarker & Diagnostic Test

Continuous score and true disease status (0 or 1)

Biomarker Value, True Status (1 line per subject):

ROC Area Under Curve (AUC)

Optimal Cutoff (Youden J)

Sensitivity

Specificity

Positive LR (+LR)

Negative LR (-LR)

Empirical ROC Curve & Optimal Youden J

Diagnostic Accuracy Statement

Computer-Generated Distributions & Probability Densities

Explore theoretical probability density functions (PDF) vs empirical simulated samples

Statistical Distribution:

Sample Size (N): 500

Generated Data Series (First 50 Preview):

Sample Mean

Theo: --

Sample SD

Theo: --

Skewness (G₁)

Kurtosis (G₂)

Normality Test

JB p = --

Empirical Histogram with Theoretical PDF Overlay

Clinical & Mathematical Context

Central Limit Theorem (CLT) Interactive Simulation

Observe how the distribution of sample averages (x̄) converges to a Gaussian bell curve regardless of parent population shape

The Core Principle of the Central Limit Theorem: Even if the raw patient population is strongly non-normal (e.g. uniform, exponential survival curves, or bimodal responder groups), the distribution of sample means x̄ drawn from that population will naturally conform to a normal distribution N(μ, σ²/n) as the sample size n increases (typically n ≥ 30).

Non-Normal Parent Population:

Sample Size per Draw (n): 30

n=2 (Skewed) n=30 (Parametric Zone) n=100 (Tight)

Simulated Means (k)

Each of size n = 30

True Pop Mean (μ)

5.00

Pop SD: 2.89

Observed Mean (x̄̄)

|Diff|: --

Theoretical SE

0.527

Formula: σ / √n

Observed SE (s_x̄)

Convergence: --

Normality (x̄)

JB Test: --

1. Parent Population Distribution

2. Sampling Distribution of Mean (x̄)

Student's t-Distribution Convergence to Standard Normal N(0, 1)

Demonstrating how heavier Student's t tails merge into the Standard Gaussian curve as sample size n increases

William Sealy Gosset's 1908 Discovery ("Student"): When estimating the population mean μ using the sample standard deviation s instead of true σ, small samples introduce extra stochastic uncertainty. This creates heavy/fat tails in the t-distribution. As sample size n grows (and degrees of freedom ν = n - 1 → ∞), Slutsky's theorem guarantees that s → σ and the t-distribution becomes mathematically indistinguishable from N(0, 1).

Sample Size (n) / Degrees of Freedom (ν = n - 1): n = 4 (ν = 3)

n=2 (Extreme Tails) n=10 (Phase I) n=31 (Classic Threshold ν=30) n=61 n=150 (Identical)

Presets:

Degrees of Freedom (ν)

Sample size n = 4

Two-Tailed t_crit (α=0.05)

3.182

vs z = 1.960 (+62.3%)

Peak Density f(0)

0.3676

Normal: 0.3989 (-7.8%)

Tail Risk P(|T|>1.96)

14.5%

Normal: 5.0% (Type I Risk)

Max Discrepancy

0.0313

sup |f_t(x) - φ(x)|

Excess Kurtosis

∞

Fat-tailed (ν ≤ 4)

Student's t(ν=3) Density Curve vs Standard Normal N(0, 1)

Highlight Tail Excess Area

Why This Matters in Clinical Research:

When conducting clinical trials with small cohorts (e.g. pilot neurosurgical studies with n < 30), using the Gaussian critical value z = 1.960 instead of Gosset's t-critical cutoff leads to serious Type I error inflation (false positives). For n = 4, testing at z = 1.96 yields an actual false-positive rate of 14.5% rather than the intended 5.0%. Once sample size reaches n ≥ 31 (ν ≥ 30), t_crit drops to 2.042 (only 4.2% wider than 1.960), making the normal approximation clinically acceptable.

Two-Sample Overlap, Dispersion (SD vs. SEM) & Alpha Significance Boundary

Explore how sample curves overlap and separate as SD, SEM (via n), and difference (Δ) change, why α = 0.05 defines significance, and what happens when α is adjusted

The Fundamental Distinction Between SD, SEM, and Alpha (α): Standard Deviation (SD) measures real biological variability between individual patients—it does not shrink as sample size grows. Standard Error (SEM = SD/√n) measures our uncertainty in the mean—it shrinks steadily as n increases. Two patient populations can overlap by 70% (large SD), yet their treatment means can separate completely (small SEM). Significance level α sets our tolerance for false-positive claims (Type I error). At α = 0.05, the required separation boundary is 1.96 standard errors. Tightening α to 0.01 pushes this boundary out to 2.58 SEs; relaxing α to 0.10 pulls it inward to 1.65 SEs.

Mean Difference (Δ = μ₂ - μ₁): 2.00

Δ=0 (Null) Δ=1.5 (Moderate) Δ=3.5 (Large) Δ=6.0

Significance Level (α): α = 0.050 (z = 1.960)

α=0.001 (0.1%) α=0.01 α=0.05 (Standard) α=0.10 α=0.20

🔗 Link Group 2 to Group 1 (Homoscedastic)

🔵 Group 1 (Control / Baseline) SD₁ = 2.50 | SEM₁ = 0.625

Patient SD (σ₁): 2.50

Standard Error (SEM₁): 0.625

Sample Size (n₁): n₁ = 16

Formula: SEM₁ = SD₁ / √n₁

🟣 Group 2 (Treatment / Test) SD₂ = 2.50 | SEM₂ = 0.625

Patient SD (σ₂): 2.50

Standard Error (SEM₂): 0.625

Sample Size (n₂): n₂ = 16

Formula: SEM₂ = SD₂ / √n₂

Alpha Presets:

View Mode:

Significance Decision

SIGNIFICANT

p = 0.031 < α = 0.050

Observed vs Δcrit

Δ = 2.00

Δcrit = 1.80 (Boundary)

Group 1 Spread (SD₁ | SEM₁)

2.50 | 0.625

n₁ = 16 subjects

Group 2 Spread (SD₂ | SEM₂)

2.50 | 0.625

n₂ = 16 subjects

Distribution Overlap

11.0%

Patient Overlap: 68.9%

Precision & Cutoff

t = 2.042

SE_diff: 0.884 | df: 30.0

Sampling Distributions of Means (SEM): Overlap & Alpha Boundary

Mathematical & Clinical Explanation:

Loading overlap simulation explanation...

⚡ Simulation 5: Statistical Power (1 − β), Type II Error (β), SD & SEM Interactive Visualizer

Interactively explore the mechanical relationship between biological noise (SD), sample precision (SEM = SD/√n), Type II error (β), and Statistical Power (1 − β). See why clinical trials become underpowered and how sample size governs statistical discovery.

Power Presets: Effect Presets:

View Mode:

🧬 Biological Noise & Sample Precision SD = 4.00 | SEM = 0.500

Patient Standard Deviation (SD, σ): 4.00

Biological heterogeneity / measurement noise. Higher SD requires more subjects.

Standard Error of the Mean (SEM): 0.500

Estimation precision (SEM = SD / √n). Shrinking SEM separates curves and skyrockets Power.

Sample Size per Group (n): n = 64 (N = 128)

Formula: n = (SD / SEM)² | Total trial size N = 2n subjects

🎯 Power, Type II Error & Target Effect Power = 80.2% | β = 19.8%

Statistical Power (1 − β): 80.2%

Probability of detecting a true effect (True Positive Rate). Regulatory standard: ≥ 80%.

Type II Error Rate (Beta, β): 19.8%

Risk of missing a true difference (False Negative Rate: β = 1 − Power).

Mean Diff (Δ): 2.00

Cohen's d = 0.50

Alpha (α): 0.050

z_crit = 1.960

Statistical Power (1 − β)

80.2%

Target: ≥ 80% (Adequate)

Type II Error (β)

19.8%

False Negative Risk

Required Sample Size

n = 64

Total N = 128 subjects

Effect Size (Δ & d)

Δ = 2.00

Cohen's d = 0.50 (Medium)

Sample Precision

SEM = 0.500

SE_diff: 0.707 (σ·√(2/n))

Significance Cutoff

xcrit = 1.386

α = 0.050 (Type I: 5.0%)

Dual Distribution: Null Hypothesis H₀ vs True Effect H₁ (Power & Beta Shading)

Mathematical & Clinical Explanation:

Loading power calculation explanation...

Teaching Report: Distributions, CLT, Student's t & Alpha Significance Simulation

Select a distribution above or run the Central Limit Theorem and Student's t simulations to generate educational outcomes.

🧠 Bayesian Statistics & Logic — The Geometry of Changing Beliefs

Inspired by 3Blue1Brown (Grant Sanderson) — Visualizing Bayes' Theorem via 1×1 unit square area, natural counts, and odds updating

Classic Puzzles & Presets:

🌱 Prior Belief & Population Base Rates Prior: 4.8% (1:20.0)

Prior Probability P(H): 4.8%

Initial degree of belief before observing evidence (Steve: 1 librarian per 20 farmers = 4.76%).

Representative Population Sample (N): N = 210 people

Representative count (e.g. 10 librarians + 200 farmers = 210). Framing as counts drops error rate to 0%.

🔍 Evidence Likelihoods & False Alarms P(E|H): 40% | P(E|¬H): 10%

Likelihood P(E|H) (True Positive Rate): 40.0%

Probability of evidence if H is true (e.g. 40% of librarians fit meek/tidy description).

False Positive Rate P(E|¬H) (False Alarm): 10.0%

Probability of evidence if H is false (e.g. 10% of farmers fit meek/tidy description).

Prior Probability P(H)

4.8%

Prior Odds: 1 : 20.0

Likelihood P(E|H)

40.0%

True Positive Probability

False Alarm P(E|¬H)

10.0%

False Positive Probability

Total Evidence P(E)

11.43%

Marginal / Shaded Area

Bayes Factor (LR)

4.00×

Substantial Evidence

POSTERIOR P(H|E)

16.7%

Belief Shift: +11.9%

3Blue1Brown 1×1 Unit Square: Visualizing Bayes' Theorem as Proportions of Area

3Blue1Brown Geometric Insight & Clinical Logic:

Loading Bayesian statistics explanation...

📺 Video Reference: Grant Sanderson (3Blue1Brown) — "Bayes theorem, the geometry of changing beliefs" (YouTube)

Teaching Report: Bayesian Statistics, Logic & 3Blue1Brown Geometric Belief Updating

Loading Bayesian statistics clinical and educational narrative...

Power & Sample Size Determination

🎲 Clinical & Research Randomisation Suite

⚙️ Simple Randomization Parameters

🎯 Prominent Assignment & Live Balance

⚙️ Block Randomization Parameters

🛡️ Allocation Concealment & Block Tracker

📋 Session Audit Trail & Allocation Log

Dataset Input

Two-Group Comparison

🔬 Statistical Assumptions & Automated Decision Engine

Multi-Cohort & Repeated Measures Analysis

🔬 Statistical Assumptions & Automated Decision Engine

🔬 Post-Hoc Pairwise Contrasts

🎯 Observational Cohort & PSM Settings

🔬 Balance & Overlap Diagnostics

📊 Baseline Covariate Balance Diagnostics

⚖️ Causal Treatment Effect (ATT) vs. Unadjusted Estimate

💻 Reproducible Statistical Code Scripts

🌐 Multivariate EDA Controls

🔬 Interactive Factor Space

📊 Eigenvalues & Inertia Decomposition

💡 Automated Natural-Language Findings & Drivers

💻 Reproducible Statistical Code Scripts

2x2 Contingency & Diagnostic Matrix

Bivariate Relationship

🛡️ Assumption Diagnostics & Test Decision Engine

Biomarker & Diagnostic Test

Computer-Generated Distributions & Probability Densities

Central Limit Theorem (CLT) Interactive Simulation

Student's t-Distribution Convergence to Standard Normal N(0, 1)

Two-Sample Overlap, Dispersion (SD vs. SEM) & Alpha Significance Boundary

⚡ Simulation 5: Statistical Power (1 − β), Type II Error (β), SD & SEM Interactive Visualizer

🧠 Bayesian Statistics & Logic — The Geometry of Changing Beliefs