Calculating Sample Size and Power (Dr Shreedhar).pptx

Sample size is a crucial aspect of research design, determining the minimum number of participants required to detect a meaningful effect while ensuring statistical validity and ethical feasibility. An inadequate sample size may lead to unreliable results, while an excessively large sample increases costs and complexity. Key factors influencing sample size include P-value, power, confidence interval, margin of error, effect size, and variability. The P-value and alpha (α) set the threshold for statistical significance, with lower alpha requiring a larger sample. Power (1 - β) represents the probability of detecting a true effect, typically set at ≥80%, meaning a 20% chance of missing a real effect (Type II error). A narrow confidence interval (CI) ensures a more precise estimate, whereas higher variability in data necessitates a larger sample to maintain accuracy.

Different statistical formulas are used to estimate sample size depending on study design. For continuous variables, sample size is calculated based on the mean difference and standard deviation between groups. For proportions, calculations consider the expected event rate in study groups. Case-control and cohort studies use different approaches to compare qualitative and quantitative variables, while animal studies apply the resource equation method to determine an optimal number of subjects. Additionally, researchers adjust for potential dropouts and confounding factors, following the 10% rule, which recommends increasing the sample size by 10% per confounder. Software tools such as G*Power, OpenEpi, and nQuery simplify these calculations, ensuring that the study remains statistically sound while being cost-effective and ethically appropriate.

1. Calculating Sample Size And Power Dr. Shreedhar Angadi Junior Resident 3 Department of Pharmacology & Therapeutics King George’s Medical University, Lucknow, U.P., India E-mail: [email protected] 16-01-2025 Dr Shreedhar Angadi 1

2. Content 16-01-2025 Dr Shreedhar Angadi 2

3. Learning Objectives Understand sample size and power importance Identify parameters influencing calculations Apply formulas for sample size estimation Explore software tools for calculations 16-01-2025 Dr Shreedhar Angadi 3

4. Introduction Sample Size is the number of participants included in a study to represent a population It determines the minimum number of participants required to detect a clinically relevant treatment effect It Ensures validity, reliability, and ethical balance Too Small: Invalid results, poor population coverage Too Large: Unnecessary cost, ethical concerns, false significance 16-01-2025 Dr Shreedhar Angadi 4

5. PARAMETERS REQUIRED FOR SAMPLE SIZE CALCULATION 16-01-2025 Dr Shreedhar Angadi 5

6. 1.P value P value: The P-value is the calculated probability of obtaining results as extreme as those observed in your study, assuming the null hypothesis is true Alpha (α): Pre-set threshold for statistical significance Common α Value: 0.05 (5% risk of Type I error) P value and α : P value < α: Reject null hypothesis α and Sample Size : Lower alpha requires larger sample size 16-01-2025 Dr Shreedhar Angadi 6

7. 2.Power It is the probability that a statistical test will correctly reject the null hypothesis when it is false It is the probability of detecting a true effect Power = 1 - Type II error (β) Standard : Power ≥ 80% ( β ≤ 20%) Influencing Factors : Sample size, Effect size, Variability 16-01-2025 Dr Shreedhar Angadi 7

8. 3.Confidence Interval It is a range within which the true value of the population parameter lies Example : BP reduction= 10 mmHg (95% CI: 8–12mmHg) Confidence Level : The percentage (or probability) that the confidence interval contains the true population parameter across many samples Confidence level = 1 - α Factors Affecting CI: Sample Size, Confidence Level , Variability CI and Precision: Narrow CI = More precise estimate 16-01-2025 Dr Shreedhar Angadi 8

9. 4.Margin of Error (MOE) It represents the range within which we expect the true population parameter to lie, based on our sample data It indicates how much the sample estimate is expected to differ from the true population value due to random sampling Expression : ± Deviation from population mean Example : BP reduction: 10 mmHg ± 2 mmHg CI and MOE : MOE is half the width of the confidence interval Factors Affecting MOE : Sample Size, Confidence Level ,Variability 16-01-2025 Dr Shreedhar Angadi 9

10. 5.PRECISION It refers to how consistently an estimate or measurement can be reproduced It indicates the degree of variability or consistency in repeated measurements or estimates. In other words, it is how close repeated results are to each other Example : Blood pressure readings like 120 mmHg,121 mmHg,119 mmHg Precision vs. Accuracy Factors Affecting Precision : Sample size, Measurement tools , Consistency of data 16-01-2025 Dr Shreedhar Angadi 10

11. 6.Effect Size (ES) It describes the magnitude of the difference or relationship between two groups, treatments, or variables It helps in understanding how big the effect is rather than just whether the effect exists (which is what p-value indicates) Types: Cohen’s d : Compares 2 groups(e.g., treatment vs placebo) Pearson’s r : Measures the strength of a linear relationship between 2 variables Odds Ratio : Quantifies the odds of an event in one group compared to other Example : Drug A reduces BP by 10 mmHg (mean difference) with a SD of 15 mmHg 16-01-2025 Dr Shreedhar Angadi 11

12. 7.Variability It refers to how much the data points in a dataset differ from the mean or central value It is a measure of the spread or dispersion in the data Example: FBS - Group 1: 90, 91, 92, 89, 88 mg/dL and Group 2: 70, 110, 85, 120, 95 mg/dL Variability and Power 16-01-2025 Dr Shreedhar Angadi 12 Type Description Range Difference between maximum and minimum values Variance Average of squared deviations, measures overall spread Standard Deviation (SD) Square root of variance, same units as data, measures spread

13. Feature One-Tailed Test Two-Tailed Test Test Direction Tests for effect in one direction (greater or smaller) Tests for effect in both directions (greater or smaller) Critical Region Only on one side of the distribution (either left or right) Critical regions on both sides of the distribution Hypothesis Null hypothesis is tested against a specific direction (e.g., > or <) Null hypothesis is tested for deviations in both directions (e.g., ≠) Type of Research Used when you have a specific direction in mind for the effect Used when the effect could go in either direction Significance Level Split across one tail (α = 0.05 in one tail) Split across two tails (α = 0.025 in each tail ) 16-01-2025 Dr Shreedhar Angadi 13 8.Two-Tailed vs One-Tailed Test

14. 9.Event Rate It is the proportion of individuals who experience a specific event in the total population or in a particular group. Formula: Example: 50 out of 100 patients experience side effects : ER=50% Role of ER in sample size calculation : L ower ER requires a larger sample size to detect a difference ER in different types of studies: 16-01-2025 Dr Shreedhar Angadi 14 Cross-Sectional Studies Represents the prevalence of a condition in the population. Clinical Trials To refer to Proportion of participants experiencing an adverse event, disease progression, or treatment response.

15. 10.Dropout Rate It is the percentage of participants who fail to complete the study due to various reasons, such as side effects, lack of compliance, or personal reasons Formula : Importance : High dropout rate = larger sample size needed. Adjusted Sample Size : Impact on Study Design: Reduces statistical power Bias in results Increased Cost and Time 16-01-2025 Dr Shreedhar Angadi 15

16. Rule of Thumb : Critical Z-Scores for Confidence Levels Confidence Level (%) Critical Zα​ Score - (Two-Tailed) Critical Zα/2 -Score (One-Tailed) Application 90% 1.645 1.28 Exploratory studies or less stringent precision requirements. 95% 1.96 1.645 Most common in research for balance of precision and certainty. 99% 2.576 2.33 Critical studies where high certainty is required. 99.9% 3.291 3.09 Rarely used; for extremely high precision requirements. 16-01-2025 Dr Shreedhar Angadi 16

17. FORMULAS FOR SAMPLE SIZE CALCULATION 16-01-2025 Dr Shreedhar Angadi 17

18. Formula for Sample Size : Example: A junior resident is conducting a thesis study to evaluate the effect of a new antidiabetic drug on the HbA1c levels of patients with Type 2 Diabetes Mellitus (T2DM). In a pilot study, the drug resulted in a mean HbA1c reduction of 2%, with a standard deviation (SD) of 4%. The resident sets the alpha level at 5% for a two-tailed test. Sample Size :16 If the possible dropout is 20%, then the adjusted sample ( N adj ): ? 20 16-01-2025 Dr Shreedhar Angadi 18 1.For Single Group Mean Z alpha-1.96,SD:4,d=2

19. 2.For Comparing Two Means Formula for Sample Size : Example: A researcher is conducting a randomized placebo-controlled trial to assess a new drug's effect on hemoglobin levels in iron-deficiency anemia patients. A pilot study showed a 2 g/dL increase in hemoglobin with a standard deviation of 4 g/dL. The study uses a 5% alpha level (Zα/2 = 1.96), 80% power (Zβ = 0.84), and a 1:1 allocation ratio. Sample Size :62 participants(Each group=31) If 1:2 allocation rate , then (N)=? 93 Participants (Placebo 31+Treatment 6 2 ) 16-01-2025 Dr Shreedhar Angadi 19 r=1,Z beta=0.84,SD=4,d=2

20. 3.For Single Group Proportions Formula for Sample Size : Example : A researcher is evaluating the efficacy of a new antibiotic in preventing postoperative staphylococcal infections at the incision site. Data shows a prevalence of such infection is 70% . The researcher aims to detect a 10% reduction in the infection rate, considering this a significant outcome. Researcher fixed the alpha level at 5% (for two-tailed) and the study is powered at 80% . Sample Size: 9 Participants 16-01-2025 Dr Shreedhar Angadi 20 p(proportion of events in a population)=70%=0.7 q(Proportion of non events) =1-p=(1-0.7)=0.3 d=70%-10%=60%=0.6

21. 4.Comparing Two Proportions Formula for Sample Size : Example : A researcher is conducting a study to compare the effectiveness of a new antibiotic against standard treatment for preventing postoperative staphylococcal infections at the incision site . Literature review shows that 15% of patients receiving standard treatment develop infections, while previous pilot study shows 5% of patients receiving the new antibiotic develop infections. Sample Size: 80 16-01-2025 Dr Shreedhar Angadi 21 P1=0.05, p2=0.15, p=(p1+p2/2)=0.1, d=(p1-p2)=10%=0.1

22. A . For Qualitative Variables (Proportions) Formula: A researcher is conducting a case-control study to investigate the association between smoking (risk factor) and lung cancer . Previous studies show that the proportion of smoking exposure in the lung cancer (case) group is 0.4 and in the control (non-cancer) group is 0.2 . The study is set with a 5% alpha level, 80% power, and equal numbers of participants in both groups. Sample Size (N) = 82 (41 cases, 41 controls) 16-01-2025 Dr Shreedhar Angadi 22 5.For Case-Control Studies P1=0.4, p2=0.2, p=0.3 , d=0.2,

23. B . For Quantitative Variables Formula: Example : A researcher is studying the association between cognitive decline (MMSE scores) and Alzheimer’s disease. The expected difference in MMSE scores between the Alzheimer’s group and healthy controls is 4 points, with a standard deviation of 6 points. Using a 1:1 case-control ratio, 95% confidence (Z = 1.96), and 80% power (Z = 0.84), the required sample size for each group is? Sample size: 35 participants (35 cases and 35 controls) 16-01-2025 Dr Shreedhar Angadi 23 r=1, s=6, d=4

24. Formula : Example : A researcher is conducting a cohort study to evaluate the impact of regular 30-minute walking on cardiovascular mortality . According to previous literature, the proportion of cardiovascular mortality is 20% among those who do regular walking and 40% for those who do not walk regularly. The study is set with a 5% alpha level, 80% power, and equal numbers in both groups. Sample Size : 92 participants (46 per group) 16-01-2025 Dr Shreedhar Angadi 24 6.Sample Size for Cohort Studies p1=proportion of events in non exposed group=0.4 P2=proportion of events in exposed group=0.2 P=p1+p2/2 = 0.3 m=ratio of exposed to unexposed participants=1 d=p1-p2=0.4-0.2=0.2

25. Formula : Guidelines for E : Optimum Size : 10≤E≤20 If E<10 : Add more animals If E>20 : Reduce sample size 16-01-2025 Dr Shreedhar Angadi 25 7. For Animal studies :Resource equation method

26. Example Calculation: Initial Case: Groups: 4 (positive control, negative control, low-dose, high-dose) Animals per group: 6 Total animals: 4×6=24 E=24−4= 20 (Appropriate sample size) Adjusted Case: Animals per group: 7 Total animals: 4×7=28 E=28−4= 24 (Too large, reduce sample size) 16-01-2025 Dr Shreedhar Angadi 26

27. The 10% Rule Initial Sample size calculations assume a simple relationship (exposure → outcome) i.e., no confounders are considered Confounders can distort results if not adjusted The 10% Rule: “ Increase the sample size 10% for each confounder added” Ensures study accuracy and power Example: Initial Sample Size: 100 participants Confounders: Age, gender, smoking status (3) Adjusted Sample Size: 133 participants 16-01-2025 Dr Shreedhar Angadi 27

28. Software Type Software Name Link/Description Free Software 1.G*Power http://www.gpower.hhu.de 2.OpenEpi OpenEpi Menu 3.R Packages https://cran.r-project.org/web/packages/pwr Paid Software a. PASS (Power Analysis and Sample Size Software) https://www.ncss.com/software/pass b. nQuery How to use nQuery - Calculate sample size and optimize your trials c.SPSS (Sample Power) Power Analysis - IBM Documentation d.STATA (power) https://www.stata.com/features/power-and-sample-size/ 16-01-2025 Dr Shreedhar Angadi 28 Software-Based Sample Size Calculation

29. Summary Sample Size : Balances validity, reliability, and ethics. Power : Ensures detection of true effects. Key Parameters : P value, Power , CI, MOE, ES, Variability. Adjustments : Account for dropouts and variability. Tools : G*Power, OpenEpi , and nQuery streamline calculations. Outcome : Accurate, ethical, cost-effective research design 16-01-2025 Dr Shreedhar Angadi 29

30. REFERENCES Mehta T. Basic Course in Biomedical Research Handbook. 1st ed. Chennai: Notion Press; 2021. Gupta KK, Attri JP, Singh A, Kaur H, Kaur G. Basic concepts for sample size calculation: critical step for any clinical trials! Saudi J Anaesth . 2016;10:328-31. Hazra A, Gogtay N. Biostatistics series module 5: Determining sample size. Indian J Dermatol. 2016;61:496-504. Charan J, Biswas T. How to calculate sample size for different study designs in medical research? Indian J Psychol Med. 2013;35:121-6. Bujang MA. A step-by-step process on sample size determination for medical research. Malays J Med Sci. 2021;28:15-27. Das S, Mitra K, Mandal M. Sample size calculation: basic principles. Indian J Anaesth . 2016;60:652-6. 16-01-2025 Dr Shreedhar Angadi 30

31. THANK YOU 16-01-2025 Dr Shreedhar Angadi 31

32. What is precision in research? How does the effect size influence sample size? Differentiate between a one-tailed and a two-tailed test. What parameters are needed to calculate sample size? List some software tools used for sample size estimation. 16-01-2025 Dr Shreedhar Angadi 32 Questions