Determination of Sample Size

Mr. Arvind Kumar

Determination of Sample Size

  • Determining the correct sample size is a critical aspect of any research study.
  • The sample size refers to the number of participants or observations included in the study.
  • A well-chosen sample size ensures that the results of the study are statistically significant and provide reliable conclusions.
  • If the sample size is too small, the study may lack statistical power and fail to detect a true effect.
  • If it’s too large, it could waste resources and introduce unnecessary complexity.
  • The adequacy of the sample size in a research study is determined by several important factors that help ensure reliable, accurate, and meaningful results.
  • These factors help researchers make sure that the study is powerful enough to detect real effects while minimizing errors.

 

Here’s a more detailed look at the factors influencing sample size:

1. Degree of Difference (Effect Size)

  • The effect size refers to the magnitude of the difference or relationship that the study aims to detect. A larger effect size (e.g., a big difference between two groups) can be detected with a smaller sample size because the difference is more obvious. However, when the effect size is small (e.g., minor differences), a larger sample size is required to detect those small differences reliably.

  • Example: In clinical trials, if a treatment produces a significant improvement in health, a smaller sample size will likely detect the effect, but a slight improvement would need a larger sample to be identified.

2. Type I Error (Alpha)

  • Type I error (denoted as α) occurs when the null hypothesis is incorrectly rejected—this is also known as a false positive. The significance level (α) is the threshold at which this error is considered acceptable (e.g., 0.05 means a 5% chance of making a Type I error).

  • A stricter alpha level (such as 0.01) means that the researcher is reducing the risk of Type I errors but will need a larger sample size to maintain power. A more stringent alpha level increases the precision required in the study, hence the sample size must be larger.

  • Example: If you are setting a significance level of 0.01 (1% chance of Type I error), you will need a larger sample size compared to a 0.05 significance level.

3. Confidence Level

  • The confidence level indicates how confident the researcher wants to be about the results of the study. Common confidence levels are 95% or 99%, which correspond to a 5% or 1% risk of error, respectively. A higher confidence level (e.g., 99%) means that the study must be more precise, requiring a larger sample size.

  • Example: If you want to be 99% confident that your estimate is correct, the required sample size will be larger than if you were only aiming for 95% confidence.

4. Type II Error (Beta)

  • Type II error (denoted as β) happens when the null hypothesis is incorrectly accepted—this is also called a false negative. The power of the study is the probability of detecting a true effect when it exists (1 – β). A higher power means a lower risk of Type II error.

  • A common target for power is 80% (i.e., an 80% chance of detecting a true effect), but a higher power (e.g., 90%) will require a larger sample size. Increasing the power decreases the likelihood of a Type II error and improves the ability to detect small effects.

  • Example: If you’re aiming for 90% power, your sample size will need to be larger compared to a study aiming for 80% power.

5. Power of the Test

  • Power is the ability of the study to detect a true effect when it exists. Higher power means a higher probability of correctly rejecting the null hypothesis when it’s false. Most studies aim for a power of at least 80%, which means that there’s an 80% chance of detecting a true effect if one exists.

  • A larger sample size increases the power of the test by reducing the variability in the sample, which improves the precision of the estimates.

  • Example: For detecting subtle effects, researchers may aim for 90% power, which requires a larger sample size than 80% power to reduce the risk of missing a true effect.

6. Variation of Results (Population Variability)

  • The variation or spread in the population, usually measured as the standard deviation, affects the sample size needed. If the population is highly variable (wide spread in data), a larger sample size is required to estimate the population parameter accurately.

  • If the population variability is small (narrow spread of data), a smaller sample size can still provide accurate estimates.

  • Example: If you are measuring something with high variability, such as income or weight, a larger sample size is needed to get reliable results.

7. Z Value (Standard Normal Distribution)

  • The Z value corresponds to the desired confidence level and is derived from the standard normal distribution. It represents the number of standard deviations from the mean that corresponds to the chosen confidence level. For instance, a 95% confidence level corresponds to a Z value of 1.96, and a 99% confidence level corresponds to a Z value of 2.58.

  • The Z value affects the sample size calculation: the higher the Z value (for higher confidence), the larger the sample size required.

  • Example: A 99% confidence level requires a Z value of 2.58, which increases the required sample size compared to a 95% confidence level, which only requires a Z value of 1.96.

 

Sample Size Determination

  • Sample size determination is the process of calculating the minimum number of participants needed in a study to achieve reliable and valid results.
  • It depends on several factors, such as the type of study, the expected effect size, the level of precision desired, and the power of the test.
  • The goal is to ensure that the sample size is large enough to detect the effects of interest while minimizing the risk of errors.

The formula for determining sample size varies depending on the research design. Here is a general approach:

Factors Affecting Sample Size:

  1. Confidence Level: The probability that the population parameter will lie within the specified confidence interval. Typically, a 95% confidence level is used.

  2. Margin of Error (Precision): The acceptable range of error around the sample estimate (often referred to as the “confidence interval”).

  3. Population Variability: The degree of variation in the population from which the sample is drawn. Higher variability requires a larger sample size to achieve the same precision.

  4. Statistical Power (1-β): The likelihood of detecting a true effect when it exists, typically set at 80% or 90%.

Sample Size Calculation Formula (for Proportions):

n = Z2×p×(1−p) / E2

Where:

  • n = required sample size

  • Z = Z-score (for a 95% confidence level, Z = 1.96)

  • p = estimated proportion of the population

  • E = margin of error (e.g., 0.05 for a 5% margin)

Example:

If you want to estimate the proportion of people who support a particular political candidate with 95% confidence and a margin of error of 5%, and you estimate the proportion of supporters to be 0.50, the sample size would be:

n=(1.96)2×0.50×0.50(0.05)2=384

So, you would need a sample size of 384 individuals to estimate the proportion with the desired precision.

Estimating Sample Size with Absolute Precision

  • In certain studies, researchers need to estimate the sample size to achieve a specific level of precision or accuracy.
  • Absolute precision means that the estimate is expected to be within a certain amount of error from the true population value.
  • For example, in a health survey estimating average blood pressure, absolute precision could mean that the mean blood pressure estimate will be within 2 mmHg of the true population mean.

To achieve absolute precision, researchers need to consider:

  • The variability of the measure in the population (standard deviation).

  • The margin of error desired.

  • The confidence level that the result will be within the specified margin of error.

 

Absolute Size of Sample is Important, Not the Sampling Fraction

The absolute size of the sample is generally more important than the sampling fraction, especially in large populations. While the sampling fraction (the proportion of the population sampled) is a useful metric in smaller populations, the total sample size plays a more critical role in determining the reliability of the results in larger populations.

For example:

  • In a small population (less than 500), a higher sampling fraction (such as 50% or more) is often necessary for accurate results.

  • In a large population (several thousand), even a small fraction of the population (e.g., 5-10%) can still result in a large sample size that provides reliable estimates.

 

Two-sample Situations

In some research designs, such as when comparing two different groups, two-sample situations arise. This involves determining the sample size for comparing two means or proportions (e.g., comparing the effectiveness of two treatments).

For two-sample situations, the sample size is calculated by considering:

  1. The effect size (the difference you expect to detect between the two groups).

  2. The variance within each group.

  3. The desired power (usually 80% or 90%).

  4. The significance level (commonly 0.05).

Formula for Two-sample t-test (for comparing means):

n = 2×(Zα/2+Zβ)2×σ2 / d2

Where:

  • Zα/2 = Z-value for the significance level (e.g., for 95% confidence, Z = 1.96)

  • Zβ = Z-value for power (e.g., for 80% power, Z = 0.84)

  • σ2 = variance of the population

  • d = the minimum difference you want to detect (effect size)

Example:

If you are comparing the effectiveness of two drugs in reducing blood pressure and you want to detect a difference of 5 mmHg with 95% confidence and 80% power, you would calculate the sample size based on the variability in blood pressure measurements and the desired effect size.

Sample Size Calculation for Various Epidemiological Studies

In epidemiological studies, determining the appropriate sample size is essential for ensuring that findings are reliable and statistically significant. The sample size depends on the type of study, the outcome being measured (e.g., disease occurrence, risk factors), and the research question.

For Cohort Studies:

For studies that examine the relationship between exposure and outcome (such as a cohort study), the sample size is often based on:

  • The anticipated risk of the outcome (e.g., disease prevalence).

  • The relative risk (the ratio of the risk in the exposed group compared to the unexposed group).

  • The desired precision in estimating the risk or relative risk.

For Case-Control Studies:

Case-control studies often require determining the sample size based on:

  • The odds ratio (the likelihood of exposure in cases vs. controls).

  • The exposure rate in the population.

  • The desired statistical power and confidence level.

 

Sample Size Calculations for Measuring One Variable

When studying one variable, such as estimating a mean (e.g., average blood pressure, average income), the sample size depends on:

  • The expected mean and standard deviation of the variable.

  • The desired margin of error for the estimate.

  • The confidence level (usually 95%).

Formula for Estimating Mean:

n = Z2×σ2 / E2

Where:

  • Z = Z-score for the desired confidence level.

  • σ = Standard deviation of the population.

  • E = Desired margin of error.

Example:

If you want to estimate the average income in a city with a margin of error of $1000, a standard deviation of $5000, and a 95% confidence level, you would calculate the sample size to ensure your estimate is within the desired range.