Sample Size & Power - Interactive Study Guide

The Big Idea

Sample size calculation isn't just a requirement for IRBs, it's your insurance policy. It ensures you have enough statistical "muscle" to find a difference if it truly exists, preventing you from wasting resources on an underpowered study.

Alpha

The "False Positive" rate. Usually 0.05. The risk you're willing to take of finding a result by chance.

1-β

Power

The probability of detecting a real effect. Usually 0.80 or 0.90. Higher power needs more subjects.

Delta

The Effect Size. The smallest clinically meaningful difference you want to detect between groups.

Sigma

Standard Deviation. The "noise" in your data. More variability makes it harder to see the effect.

Sample Size Drivers

Direct Correlation (n ↑)

Increase Confidence Level (↓ Alpha) n Increase
Increase Statistical Power n Increase
Increase Variability (σ) n Increase

Inverse Correlation (n ↓)

Increase Effect Size (Δ) n Decrease
Use Paired/Matching Design n Decrease

Parameters

Alpha ($\alpha$) / False Alarm 0.05

Sample Size ($n$) 30

Effect Size ($\Delta$) 0.8

Outcomes

Type I Error ($\alpha$)

False Positive

5.0%

Type II Error ($\beta$)

Missed Opportunity

--%

Power ($1 - \beta$)

Success Rate

--%

Adjust sliders to see how reality changes...

Null ($H_0$)

Alt ($H_1$)

Standard Error Units

Reject $H_1$

Reject $H_0$

Formula Decision Tree

Start at the top to identify the correct statistical test and sample size formula.

Step 1 What is your Primary Endpoint?

Means (Continuous)

e.g., Blood Pressure, IQ, BMI

Step 2: How many groups?

1 Group

$$n = \left( \frac{(z_{\alpha} + z_{\beta})\sigma}{\Delta} \right)^2$$

One-Sample T-Test

Paired?

YES (Paired T-Test)

$$n = \left( \frac{(z_{\alpha} + z_{\beta})\sigma_d}{\Delta} \right)^2$$

NO (Indep. T-Test)

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

Proportions (Categorical)

e.g., Cured/Not Cured, Yes/No

Step 2: How many groups?

1 Group

$$n = \frac{(z_{\alpha}\sqrt{P_0(1-P_0)} + z_{\beta}\sqrt{P_1(1-P_1)})^2}{(P_1-P_0)^2}$$

One-Sample Z-Test

2 Independent Groups

$$n = \frac{(z_{\alpha} + z_{\beta})^2 [P_1(1-P_1) + P_2(1-P_2)]}{(P_1-P_2)^2}$$

Chi-Square / Z-Test

Sigma ($\sigma$) for Means

For Independent designs, $\sigma$ is the spread of the raw data. For Paired designs, $\sigma_d$ is the spread of the differences. $\sigma_d$ is usually smaller, increasing power.

Variance in Proportions

Unlike means, proportions don't need a separate $\sigma$. The variance is intrinsic: $\text{Var} = P(1-P)$. Variance is highest when $P = 0.5$ (50%).

Universal Sample Size Estimator

Supports both Means (Continuous) and Proportions (Categorical).

Required N

per group

1. Select Study Design

Effect Size ($\Delta$) 5.0

Minimum difference you want to detect (Mean 1 - Mean 2).

Std. Deviation ($\sigma$) 15.0

Expected spread of data in the groups.

Proportion 1 ($P_1$) 0.50

Expected proportion in Group 1 (or Alternative Hypothesis).

Proportion 2 ($P_2$ / $P_0$) 0.40

Expected proportion in Group 2 (or Null Hypothesis).

Alpha ($\alpha$)

Power ($1-\beta$)

Formula Preview

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

Z-Score Sum $(z_\alpha + z_\beta)^2$: 7.84

Variance / Spread: 225

Effect Squared: 25

Note: Result is always rounded up to the nearest integer.

Example 1: Paired Mean (Continuous)

Reducing tumor size with a new drug

Paired T-Test

The Scenario

You are testing if a drug reduces tumor size by 5mm. You match mice by genetics and age (Paired Design). Historical data suggests the standard deviation of differences ($\sigma_d$) is 4mm.

$\alpha = 0.05$ Power = 80%

The Math

n = \left( \frac{(1.96 + 0.84) \times 4}{5} \right)^2 = 5.01 \approx 6

Example 2: Independent Mean (Continuous)

Comparing BMI in a pilot study

Indep. T-Test

The Scenario

You want to detect a Medium Effect Size (Cohen's $d = 0.5$) for an independent group study.

The Math

n = \frac{2(1.96 + 0.84)^2}{0.5^2} = 62.7 \approx 63

Example 3: Comparing Two Proportions

Vaccine Efficacy Trial

Chi-Square

The Scenario

You expect the standard treatment to cure 40% ($P_2 = 0.40$) of patients. You hope the new drug cures 60% ($P_1 = 0.60$). How many subjects per group do you need for 80% Power at $\alpha=0.05$?

The Math

n = \frac{(1.96+0.84)^2 [0.6(0.4) + 0.4(0.6)]}{(0.6-0.4)^2} = \frac{7.84(0.48)}{0.04} \approx 95

Conclusion:

You need 95 patients per group (190 total).

1. Non-Adherence

Adjusts for "Drop-Outs" (subjects leaving treatment) and "Drop-Ins" (controls starting treatment).

n_{adj} = \frac{n}{(1 - R_o - R_i)^2}

$R_o$ = Rate of Drop-outs
$R_i$ = Rate of Drop-ins

2. Clustering (ICC)

Necessary when randomizing groups (clinics, schools) rather than individuals.

n_{adj} = n \times [1 + (m-1)\rho]

$m$ = Average cluster size
$\rho$ = Intraclass Correlation Coefficient (ICC)