Best examples of power analysis for ANOVA examples in real research

Starting with real examples of power analysis for ANOVA examples

Let’s skip the theory lecture and start where most analysts start: with a messy real question.

Imagine a researcher planning a three‑group clinical trial for blood pressure control. She knows she’ll use a one‑way ANOVA to compare mean systolic blood pressure across the three drugs. The very first thing she needs is an example of power analysis for ANOVA that looks like her situation: multiple groups, continuous outcome, and uncertainty about how big the difference will be.

That’s the point of this article. We’re going to line up several examples of power analysis for ANOVA examples from different fields—medicine, public health, education, psychology, and business analytics—and unpack how the decisions are made.

Example 1: One‑way ANOVA in a three‑arm clinical trial

Scenario. A cardiology team wants to compare three antihypertensive drugs: A, B, and C. Outcome: change in systolic blood pressure (mmHg) after 12 weeks. They’ll use a one‑way ANOVA with 3 groups.

Planning questions:

• What is the smallest difference in mean blood pressure that matters clinically?
• How variable is blood pressure change in similar patients?

They review prior trials from sources like the National Institutes of Health and find:

• Standard deviation (SD) of change in systolic blood pressure ≈ 12 mmHg
• They care about detecting a 6 mmHg difference between at least two drugs

For ANOVA, this translates into an effect size f (Cohen’s f). For a balanced design with 3 groups, a 6 mmHg difference with SD 12 mmHg corresponds to roughly f ≈ 0.25, which is often labeled a “medium” effect.

They set:

• Significance level α = 0.05 (two‑sided)
• Desired power = 0.80
• Number of groups = 3

Using G*Power or the pwr.anova.test function in R, they plug in:

• f = 0.25
• α = 0.05
• power = 0.80
• k = 3 groups

The software returns a required total sample size of about 159 patients (53 per group). The team adds a 10–15% buffer for dropout and decides to recruit 180 patients.

This is one of the cleanest and most common examples of power analysis for ANOVA examples: a simple one‑way design, equal groups, and a continuous clinical outcome.

Example 2: Marketing A/B/C/D test with unequal group sizes

Scenario. A marketing analyst at a retail company wants to compare four email subject lines on click‑through rate (CTR). Outcome: proportion of users who click; but the analyst plans to treat CTR as approximately continuous (percentage) and use a one‑way ANOVA on user‑level data.

The twist: different lists have different sizes, so group sizes will be unequal.

From historical data:

• Mean CTR ≈ 4%
• Standard deviation at user level ≈ 19 percentage points (binary outcome, so SD ≈ √[p(1−p)])

The marketing team will be happy if they can detect an increase from 4% to 6.5% in at least one subject line. At the user level, that’s a modest effect, but across thousands of users it’s detectable.

They approximate a medium effect size, f ≈ 0.20, and plan:

• α = 0.05
• power = 0.90 (they want higher power because the test is cheap to run)
• k = 4 groups

Using R or G*Power with f = 0.20, α = 0.05, power = 0.90, k = 4, they get a total required sample size of around 520 users. Because traffic is high, they simply allocate about 130 users per subject line.

Here, the example of power analysis for ANOVA shows how even modest differences in CTR become detectable with large samples. In digital marketing, these ANOVA‑based power calculations are common because experiments run continuously.

Example 3: Two‑way ANOVA in an education study (teaching method × school type)

Scenario. An education researcher wants to compare three teaching methods (traditional lecture, flipped classroom, and project‑based learning) across two types of schools (urban and suburban). Outcome: standardized math test scores.

This is a two‑way ANOVA with:

• Factor A: teaching method (3 levels)
• Factor B: school type (2 levels)
• Interest in both main effects and the interaction

From pilot data at a regional university (Harvard Graduate School of Education and similar sources often publish effect size benchmarks), they estimate:

• SD of test scores ≈ 15 points
• Meaningful main effect of teaching method ≈ 5 points difference in means
• Meaningful interaction effect ≈ 6–7 points difference in how methods perform across school types

They translate this into:

• Main effect of teaching method: f ≈ 0.20
• Interaction effect: f ≈ 0.25

Because interaction effects are usually harder to detect, they power the study for f = 0.25 on the interaction:

• α = 0.05
• power = 0.80
• 3 × 2 design → 6 cells

Using G*Power’s “ANOVA: Fixed effects, special, main effects and interactions” option, they find they need about 30–35 students per cell, or roughly 200 students in total.

This is one of the best examples of power analysis for ANOVA examples when you have multiple factors and you care a lot about interactions—very common in education and social science research.

Example 4: Repeated‑measures ANOVA in a weight‑loss program

Scenario. A public health team wants to compare two weight‑loss programs over time: Program X (digital coaching) and Program Y (in‑person group sessions). Outcome: weight in pounds, measured at baseline, 3 months, 6 months, and 12 months.

Design:

• Between‑subjects factor: program (2 levels)
• Within‑subjects factor: time (4 repeated measures)
• Planned analysis: repeated‑measures ANOVA with a program × time interaction

They consult data from prior obesity interventions summarized by the CDC and Mayo Clinic. They estimate:

• SD of weight at each time point ≈ 25 lb
• Correlation between repeated measures ≈ 0.70
• A meaningful program × time effect: a 10 lb difference in weight loss at 12 months between programs

Using a repeated‑measures power approach (e.g., G*Power’s “ANOVA: Repeated measures, between factors”), they set:

• f = 0.20 for the interaction (moderate effect)
• α = 0.05
• power = 0.80
• groups = 2
• measurements = 4
• correlation among repeated measures = 0.70

The software suggests about 40–50 participants per program, so a total of 80–100 participants. Accounting for attrition over 12 months (often 20–30% in weight‑loss trials), they plan to recruit 130–140 participants.

This repeated‑measures case is a more advanced example of power analysis for ANOVA because you have to think about correlations across time, not just variance at a single time point.

Example 5: One‑way ANOVA in a psychology lab experiment

Scenario. A psychology lab wants to study the effect of sleep deprivation on reaction time. Participants are randomly assigned to:

• 0 hours sleep deprivation (control)
• 24 hours awake
• 36 hours awake

Outcome: mean reaction time on a cognitive task (milliseconds). Analysis: one‑way ANOVA with 3 groups.

From past literature and meta‑analyses hosted on platforms like PubMed at NIH, they find:

• SD of reaction time ≈ 80 ms
• They care about detecting a 40 ms difference between at least two groups

This yields an effect size f ≈ 0.50 (large). They plan:

• α = 0.05
• power = 0.90 (lab time is expensive; they want strong evidence)
• k = 3

Using G*Power with f = 0.50, α = 0.05, power = 0.90, k = 3, the required total sample size is around 66 participants (22 per group). They add a small buffer for data quality issues and decide to recruit 75 participants.

This is a classic lab‑based example of power analysis for ANOVA examples where effect sizes are often larger than in field studies, so required samples are smaller.

Example 6: Three‑way ANOVA in a manufacturing quality study

Scenario. An industrial engineer is studying defect rates in a manufacturing process. Factors:

• Machine type (2 levels)
• Operator experience (3 levels: novice, intermediate, expert)
• Shift (2 levels: day, night)

Outcome: number of defects per 1,000 units, treated as approximately continuous. Planned analysis: three‑way ANOVA.

The engineer is especially interested in the machine × experience interaction: whether experienced operators can offset issues with older machines.

Using historical quality data, they estimate:

• SD of defects ≈ 4 defects/1,000 units
• A meaningful interaction effect: 2 defects/1,000 units difference between combinations

This translates into f ≈ 0.25 for the interaction. They plan:

• α = 0.05
• power = 0.80
• 2 × 3 × 2 design → 12 cells

Using software that supports factorial ANOVA power (e.g., SAS PROC GLMPOWER, R’s pwr2 or simulation‑based approaches), they find they need about 12–15 observations per cell, or roughly 150–180 observations.

This manufacturing case shows another example of power analysis for ANOVA where interactions across multiple factors drive the sample size decision.

Example 7: Public health nutrition trial with cluster‑level ANOVA

Scenario. A public health team wants to test three versions of a school cafeteria menu to improve fruit and vegetable intake. Randomization is at the school level, not the student level.

Design:

• Factor: menu type (3 levels)
• Outcome: mean servings of fruits and vegetables per student per day, aggregated at the school level
• Analysis: one‑way ANOVA on school‑level means

Clustered designs bring in the intraclass correlation coefficient (ICC). From prior school‑based nutrition research (e.g., trials referenced by the CDC Nutrition program), they estimate:

• ICC ≈ 0.05
• SD of servings at student level ≈ 1.2
• On average 200 students per school

They first compute the design effect:

Design effect = 1 + (m − 1) × ICC

where m is the average cluster size (students per school).

Here:

Design effect ≈ 1 + (200 − 1) × 0.05 ≈ 1 + 9.95 ≈ 10.95

If this were a simple one‑way ANOVA without clustering, they might need 30 students per group to detect a moderate effect. But after accounting for clustering, they instead think in terms of schools.

Using cluster‑level power methods (for example, via R packages like CRTsize or simulation), they determine they need about 10 schools per menu type (30 schools total), with at least 150–200 students per school, to detect a difference of 0.3 servings with 80% power.

Even though the analysis is still an ANOVA on three groups, the power analysis has to handle clustering. This is one of the more advanced examples of power analysis for ANOVA examples you’ll see in modern public health research.

How software supports these examples of power analysis for ANOVA examples

Across these real examples, the workflow is similar even when the designs differ:

• Choose the ANOVA model type. One‑way, two‑way, three‑way, repeated‑measures, or mixed.
• Specify effect sizes. Often in terms of Cohen’s f, derived from expected mean differences and standard deviations.
• Set α and desired power. Commonly α = 0.05 and power between 0.80 and 0.90.
• Enter group numbers and structure. Number of groups, number of factors, repeated measures, and any correlation assumptions.

Popular tools in 2024–2025 include:

• G*Power: Free, widely used for ANOVA power analysis in psychology, medicine, and social sciences.
• R: Packages like pwr, pwr2, and simulation‑based approaches for complex designs.
• SAS, Stata, SPSS: Built‑in procedures for power and sample size for ANOVA and mixed models.

These tools make it relatively quick to turn an informal research idea into a precise example of power analysis for ANOVA with a defensible sample size.

Why these examples matter in 2024–2025

If you scan recent clinical trials on ClinicalTrials.gov or education and psychology journals, you’ll notice a pattern: reviewers and regulators are far less forgiving of underpowered studies than they were a decade ago. Underpowered ANOVA designs lead to:

• Non‑significant results that are impossible to interpret
• Overestimation of effects in the few significant results you do get
• Wasted time, money, and participant effort

The best examples of power analysis for ANOVA examples—like the ones above—share a few traits:

• They use realistic effect sizes, often grounded in prior data or meta‑analysis.
• They acknowledge attrition, clustering, or repeated measures.
• They document their assumptions clearly in protocols and pre‑registrations.

In other words, they treat power analysis as part of the scientific argument, not a checkbox.

FAQ: Common questions about examples of power analysis for ANOVA

Q1. Can you give a simple example of power analysis for ANOVA with two groups?
Technically, a two‑group comparison is usually handled with a t‑test. But you can still frame it as a one‑way ANOVA with 2 levels. Suppose you compare a control and a treatment group on blood glucose levels, want to detect a 10 mg/dL difference, expect an SD of 20 mg/dL, and aim for 80% power at α = 0.05. That corresponds to Cohen’s d = 0.5 and f ≈ 0.25. Using ANOVA power tools, you’d get a total sample size of roughly 128 participants (64 per group).

Q2. How do I choose the effect size for my power analysis?
Most real examples of power analysis for ANOVA examples use one of three strategies: (1) pull SDs and mean differences from prior studies or pilot data; (2) use standardized benchmarks (small, medium, large) from sources like Cohen’s guidelines; or (3) work backward from a difference that would actually change practice or policy and see what effect size that implies.

Q3. Do I need equal group sizes for ANOVA power analysis?
No. Many real examples include unequal group sizes due to practical constraints. However, unequal sizes generally reduce power for a given total sample size. When group sizes are very imbalanced, it’s better to use software that can handle specific allocation ratios.

Q4. Are simulation methods better than formula‑based power analysis for complex ANOVA designs?
For simple one‑way or two‑way ANOVA, standard formulas and tools like G*Power are fine. For more complex designs—cluster‑randomized trials, three‑way interactions, or non‑normal outcomes—simulation in R or similar tools often gives more realistic power estimates.

Q5. Where can I find more worked examples of power analysis for ANOVA?
Many graduate‑level statistics courses hosted by universities (.edu domains) publish lecture notes and worked examples. Searching for “ANOVA power analysis examples site:.edu” often surfaces detailed walkthroughs from statistics departments, which can be adapted to your own field.

In practice, the best way to learn is to take one of the examples of power analysis for ANOVA examples above that looks similar to your study, copy its structure, and then swap in your own expected means, standard deviations, and design details. From there, refine the assumptions with domain experts and pilot data until the sample size feels scientifically and logistically realistic.