Scenario

• Duration of remission after chemotherapy (exponential distribution)

• Mean Duration: 5 months

• Our aim is to estimate mean remission duration of the underlying population

Two random samples (size = 100)

• Both the random samples are different

• Blue Sample (Mean): 4.8525941

• Red Sample (Mean): 4.9722859

Distribution (1000) of sample means (Sample size: 10)

Distribution of Sample Means with Sample Size

Larger the sample size

• Distribution nearer to normal (Central Limit Theorem)

• Narrower is the spread of distribution. More accurate is the estimate.

• No change in the mean value of the distribution

Requirements for smple size calculation for 1 sample

• Minimum desired precision (Margin of error) (\(\Delta\))

• Underlying standard deviation (\(sd\)) (measure of spread, may require pilot study)

• Width of confidence interval (95% CI, 90% CI, 99% CI)

• For 95% CI

\[ n = (sd * Z_{0.975} / \Delta)^{2} \\Z_{0.975} = 1.96 \]

Example

• A survey is carried out to estimate the mean height of a population in a city

• A pilot survey was carried out with 50 people and standard deviation of 50 cm was estimated

• We want to estimate the mean height with a margin of error of 10 cm

• n = ((50 * 1.96) / 10)^2 = 96

Scenario

• A new drug (Drug 2) has been invented as chemotherapy which is being tested against standard of care (Drug 1)

• Outcome of interest is the duration of remission

• Say, duration of remission for standard of care (Drug 1) is exponentially distributed with mean of 5 months

• Duration of remission for new drug (Drug 2) is exponentially distributed with mean of 10 months

Assessing difference between Drug 1 and Drug 2

• Difference between means between Drug 2 and Drug 1 (Drug 2 - Drug 1)

• Drug 2 is better by 5 months than Drug 1 (A VALUE WHICH IS NOT KNOWN IN REAL LIFE)

• We define a clinically significant difference between Drug 1 and Drug 2 as 3 months (POPULATION CHARACTERISTICS)

Simulation

• Population under Null hypothesis: difference = 0 (5m, 5m)

• Population under Clinically significant difference hypothesis (population with minimum difference of clinically significant difference): difference = 3 (5m, 8m)

Sample size = 10

• Green dashed line: sample mean

• Red line: 97.5% of the null population (determines the region of rejection)

• Sample in the region of rejection: assumed that sample doesnot belong to the null population

• Assuming that null population is the truth, probability of committing error that the sample doesnot belong to the null population: Type I error (Alpha) (5%)

• Upper panel: Population belonging to minimum clinically significant difference

• Assuming the the above population is the truth, the probability of correctly inferring that the sample belongs to above population is the POWER (80%)

• Assuming that the null population is the truth, the probability that the sample same or more extreme to the present sample belongs to the null population is p value

Sample size = 100

• p value decreases to so called statistically significant level, just by increasing the sample size

  • FALLACY of P VALUE: Some Other Day!!

• Power of study increases by increasing the sample size

• Estimates estimate the population parameter more precisely by increasing the sample size

Sample size = 1000

Download …

Downloadable from http://biostat.mc.vanderbilt.edu/wiki/Main/PowerSampleSize as pssetup3.exe file

Cite the package …

Dupont WD, Plummer WD: ‘Power and Sample Size Calculations: A Review and Computer Program’, Controlled Clinical Trials 1990; 11: 116-28

or

Dupont WD, Plummer WD: ‘Power and Sample Size Calculations for studies involving Linear Regression:’, Controlled Clinical Trials 1998; 19: 589-601

Slides can be obtained from …

https://sumprain.netlify.com/files/html/sample_ahrr/presentation_ahrr.html