| Course | HCR 557 Clinical Research Design and Methods |
|---|---|
| Module | Module 4 |
| Paper type | Sample size calculation paper |
| Length | About 753 words, 5 pages |
| Format | APA 7 student paper |
| School | Arizona State University |
| Program | MS in Clinical Research Management |
| Updated | October 2026 |
Free sample paper for HCR 557 Module 4
How Many Patients? Sample Size and Dropout Planning for a Placebo-Controlled Trial in Lateral Epicondylitis
Student Name
MS in Clinical Research Management, Arizona State University
HCR 557: Clinical Research Design and Methods
Instructor Name
Month Day, Year
How Many Patients? Sample Size and Dropout Planning for a Placebo-Controlled Trial in Lateral Epicondylitis
The Study
Participants with chronic lateral epicondylitis are randomized 1:1 to an investigational oral drug or a matching placebo, taken for eight weeks, with participants and assessors blinded. The primary endpoint is treatment success at 12 weeks, defined as complete recovery or much improvement on a six-point global rating of change scale. A secondary endpoint is the change in a 100-point patient-rated pain and function score.
Hypotheses
Null hypothesis: the proportion of participants with treatment success at 12 weeks is the same in the drug and placebo groups. Alternative hypothesis: the proportions differ. The test is two-sided, which allows the drug to be shown worse as well as better.
Assumptions
| Parameter | Value | Reason |
|---|---|---|
| Significance level (alpha) | 0.05, two-sided | Conventional for a confirmatory trial |
| Power (one minus beta) | 80%, with 90% shown | 80% is the usual minimum; 90% for a sponsor wanting more certainty |
| Placebo success rate | 40% | Recovery on placebo rises steeply over time in this condition; in a published factorial trial, placebo injection groups ranged from 10% to 39% recovered at 4 weeks and 85% at 26 weeks (Coombes et al., 2013), so 40% at 12 weeks is a reasonable midpoint |
| Drug success rate | 60% | A 20-point absolute difference, the smallest change clinicians would consider worth a new drug |
| Allocation | 1:1 | Simplest and most efficient |
| Expected dropout | 20% | Conservative for a 12-week trial with an oral drug and frequent visits |
Calculation for the Primary Endpoint
For two independent proportions, the sample size per group is n = [z for alpha/2 x square root of (2 x p-bar x q-bar) + z for beta x square root of (p1q1 + p2q2)]^2 / (p1 minus p2)^2, where p-bar is the average of the two proportions and q is one minus p (Browner et al., 2023).
With p1 = 0.40 and p2 = 0.60, p-bar is 0.50, so 2 x p-bar x q-bar = 0.50 and its square root is 0.7071. The sum p1q1 + p2q2 = 0.24 + 0.24 = 0.48, with a square root of 0.6928. For alpha 0.05 two-sided, z = 1.96, and for 80% power, z = 0.8416.
Numerator: (1.96 x 0.7071 + 0.8416 x 0.6928)^2 = (1.3859 + 0.5831)^2 = 1.9690^2 = 3.877. Denominator: (0.20)^2 = 0.04. Result: 3.877 / 0.04 = 96.9, rounded up to 97 per group, or 194 in total.
At 90% power, z = 1.2816, so the numerator becomes (1.3859 + 0.8879)^2 = 5.170, and n = 129.3, rounded up to 130 per group, or 260 in total.
Adjusting for Dropout
If 20% of enrolled participants are expected to drop out, the number to enroll is the evaluable number divided by 0.80. At 80% power, 97 / 0.80 = 121.25, rounded up to 122 per group, 244 in total. At 90% power, 130 / 0.80 = 162.5, or 163 per group, 326 in total. Dividing by the retention rate, rather than adding 20% to the evaluable number, is the correct method (Noordzij et al., 2010), because 20% of the enrolled group, not of the evaluable group, will be lost.
Secondary Endpoint Check
For the continuous pain and function score, assume a clinically important difference of 10 points and a standard deviation of 20 points. With n = 2 x (z for alpha/2 + z for beta)^2 x (SD / difference)^2, the result at 80% power is 2 x (2.8016)^2 x 4 = 62.8, or 63 per group, 79 after dropout. Because this is smaller than the primary requirement, the trial sized for the primary endpoint will also have adequate power for the secondary one, if the assumed standard deviation holds.
Recommendation and Sensitivity
The recommended enrollment is 244 participants, 122 per group. If the placebo success rate turns out to be 50% rather than 40%, a 20-point difference would need a similar number, but a smaller true difference would quickly require many more participants, so the protocol should plan a blinded check of the pooled success rate at mid-enrollment.
| Scenario | Per group, evaluable | Per group, enrolled (20% dropout) | Total enrolled |
|---|---|---|---|
| Primary, 80% power | 97 | 122 | 244 |
| Primary, 90% power | 130 | 163 | 326 |
| Secondary, 80% power | 63 | 79 | 158 |
Conclusion
A trial of 244 participants provides 80% power to detect a 20-point difference in 12-week treatment success, after allowing for one participant in five to leave the study. Because every number above rests on stated assumptions, each one is written down with its reason, so a reviewer can challenge any of them.
References
Browner, W. S., Newman, T. B., Cummings, S. R., Grady, D. G., Huang, A. J., Kanaya, A. M., & Pletcher, M. J. (2023). Designing clinical research (5th ed.). Wolters Kluwer.
Coombes, B. K., Bisset, L., Brooks, P., Khan, A., & Vicenzino, B. (2013). Effect of corticosteroid injection, physiotherapy, or both on clinical outcomes in patients with unilateral lateral epicondylalgia: A randomized controlled trial. JAMA, 309(5), 461-469. https://doi.org/10.1001/jama.2013.129
Noordzij, M., Tripepi, G., Dekker, F. W., Zoccali, C., Tanck, M. W., & Jager, K. J. (2010). Sample size calculations: Basic principles and common pitfalls. Nephrology Dialysis Transplantation, 25(5), 1388-1393. https://doi.org/10.1093/ndt/gfp732
HCR 557 Module 4 instructions, in plain terms
Paper 2 is due in Module 4 of HCR 557, the module on sample size, statistics and the midterm. The syllabus asks for sample size calculations for a given study, with consideration of dropout rates. Your instructor may give you the study or point you to the course's practice protocol; either way, the paper needs the same parts: the study and its primary endpoint, null and alternative hypotheses, every assumption with a reason, the formula and the arithmetic and an adjustment for participants expected to leave the study. Chapter material in Browner's Designing Clinical Research supplies the formulas for proportions and means and the reasoning about alpha, power and effect size. Show the calculation step by step rather than reporting only a number from an online calculator, because what earns the marks is visible reasoning about what drives the number. Like the other two papers, it is worth 100 points.
How this HCR 557 Module 4 example is built
The sample begins by describing the study and its primary and secondary endpoints. Hypotheses come next, stated in words. A table lists six assumptions with a reason for each, and the placebo response rate is grounded in a published trial. The primary calculation works the two-proportion formula by hand at 80% and 90% power, showing each intermediate value. The dropout section divides by the retention rate and explains why that is correct. A secondary-endpoint check confirms the trial is also powered for the continuous outcome. A summary table shows all scenarios, and the recommendation adds a sensitivity point about the placebo rate before a two-sentence conclusion.
HCR 557 Module 4 rubric: what earns full marks
Paper 2 is graded out of 100 points. Credit goes to clearly stated hypotheses, justified assumptions for alpha, power, effect size and baseline rate, the correct formula for the endpoint type, accurate arithmetic shown step by step, a correct dropout adjustment and a clear final recommendation. Points are lost when assumptions are given without reasons, when the formula does not match the endpoint (a means formula for a proportion, for example), when the dropout adjustment adds a percentage instead of dividing by retention, when numbers are not rounded up and when the result is reported from software with no explanation. Readers value sensitivity analysis because it shows the writer understands how fragile a sample size can be when the assumed rates are wrong.
HCR 557 Module 4 help: mistakes that cost marks
Write down the endpoint type first; it decides the formula. Justify each assumption with a source or a clinical argument. Show every step of the arithmetic and round up at the end. Divide by the retention rate to adjust for dropout. Check at least one other power level or effect size to show sensitivity. Present the results in a table. If you used software, still show the formula. If you are unsure which formula fits your endpoint, the desk can check your setup before you calculate. State the software and version if you checked your arithmetic with one, but keep the hand calculation as the main evidence, because the grader is assessing your reasoning, not the tool's output.
Write yours, or have the desk draft it
This paper is an original model document written by our desk, not a submitted student paper and not an official Arizona State University document. Read it for the moves, then write your own to the instructions in your classroom. If you want one built to your exact prompt and rubric, the first custom sample is free and arrives in 24 to 48 hours.
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HCR 557 Module 4 questions, answered
Where can I find a free HCR 557 Module 4 sample paper?
The complete Paper 2 sample on this page sizes a placebo-controlled trial in lateral epicondylitis and adjusts for dropout.
How do you adjust sample size for dropout?
Divide the evaluable sample by the expected retention rate; with 20% dropout, divide by 0.80 and round up.
What do you need to calculate a sample size?
The endpoint type, alpha, power, the expected control rate or variability and the smallest difference worth detecting.
Why round sample size up?
Rounding down would leave the study slightly underpowered, so any fraction becomes another participant.
What happens to sample size if the expected difference is smaller?
It rises sharply, because the required number grows with the inverse square of the difference.