54  Survey: Single proportion

54.1 Introduction

Surveys are a standard and straightforward study methodology when one wishes to investigate the prevalence of a condition or the average within a population. In this example, we will determine the sample for a population proportion.

54.2 Scenario

An investigator wishes to determine the prevalence of anaemia among children under 5 years old in a community. To do this, he needs to obtain a representative sample. Thus, he is confronted with the challenge of knowing how many children must be samples.

54.3 Requirements

To determine the sample size required, the investigator needs to pre-define and specify the following:

  1. Estimated prevalence: This can be obtained from a previous study evaluating the proportion in the same population, a similar population, or one reasonably close to the population to be investigated. If none of these is obtainable, the proportion that divides the maximum sample size should be chosen.
  2. Margin of Error: This expresses how precise one wants his estimate to be. An error of 0.05, for instance, will represent an allowable error of 5%.
  3. Confidence level: A confidence level represents the degree of certainty or probability that the population parameter (such as the actual proportion) falls within the computed confidence interval. For instance, if we were to calculate confidence intervals for 100 distinct samples, we would anticipate that approximately 95 would contain the actual population value if we used a 95% confidence level. The traditional confidence level for most studies is 95%.
  4. The Z-value, also known as the Z-score, corresponds to the number of standard deviations a data point is from the mean in a standard normal distribution (bell curve). Common Z-values include 1.96 for the 95% confidence level, meaning 95% of the data falls within 1.96 standard deviations from the mean in a normal distribution. Others include 2.576 for the 99% Confidence Level and 1.645 for the 90% confidence level.
  5. Here, we use the Cochrane formula(Richter 1963).

\[n = \frac{Z^2 P(1-P)}{e^2}\]

54.4 Determination

For the study above, a background search indicated that 45% of children less than 5 years old had anaemia in a survey done 2 years ago in the same population. The investigator decided to estimate this to a precision of 5% and use the 95% confidence limit.

Code 
p <- 0.45
a <- 0.05
e <- 0.05

n = (qnorm(1-a/2)^2)*p*(1-p)/(e^2)
n = ceiling(n)

Slotting these into the formula, we have a minimum of 381 children required for the study.