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180214P - CONFIDENCE INTERVALS and THEIR INTERPRETATIONS

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Prepared by Professor Omar Hasan Kasule Sr. MB ChB (MUK), MPH (Harvard), DrPH (Harvard) Chairman of the KFMC IRB


1.0 INTERVAL ESTIMATES  1

  • In Interval estimation the confidence interval is stated as the lower confidence level and the upper confidence level using usual or customary confidence of 95%. There are three definitions of 95%CI
  • Definition 1: In a common-sense way the 95% confidence interval (CI) means that we are 95% sure that the true value of the parameter is within the interval. 


2.0 INTERVAL ESTIMATES 2

  • Definition 2: The probability that the true parameter lies in the confidence interval is 95%. Expressed mathematically, the 95% confidence interval for parameter q is defined as Pr (a < q < b) = 0.95. 
  • Definition 3: A third way of describing the 95% is to imagine taking repeated samples from a population and computing a mean for each sample. Ninety-five percent (95%) of the sample means will be within the 95% CI. 


3.0 CONFIDENCE INTERVAL FOR A MEAN 

  • The standard deviation (square root of the variance), the commonest measure of variation, assesses the degree of dispersion above and below the mean
  • The 95% CI for the mean is computed as mean +/- {1.96 ns2 /n-1} / n1/2 where n = sample size, s =sample standard deviation (standard error).
  • The percentage of observations covered by mean +/- 1 s is 66.6%, 
  • The percentage of observations covered by mean +/- 2 s is 95%, 
  • The percentage of observations covered by mean +/- 4 s is virtually 100%. 


4.0 CONFIDENCE INTERVAL FOR A PROPORTION

• The 95% CI for a proportion is computed as ps +/- 1.96 {(ps qs / n)}1/2 where ps = sample proportion, qs = 1 - ps, , and n = sample size. The variance of a proportion can be computed as {p(1-p)/n} 

• Approximately The 95% confidence intervals for a proportion: proportion +/- 2 SD


5.0 CONFIDENCE INTERVAL FOR AN ODDS RATIO

• Complicated mathematical formulas can be used to compute the 95%CI of a proportion

• For example the odds ratio can be written as 2.5(1.2,3.0). The confidence interval is from 1.2 to 3.0.


6.0 INTERPRETATION OF THE 95% CONFIDENCE INTERVAL

• Narrow CI indicates precision

• Wide CI indicates imprecision


7.0 VALIDITY AND PRECISION

• Validity tells us how well an instrument measures what it is supposed to measure. The mean is a measure of validity (parameter of location). 

• The standard deviation is a measure of precision (spread). 

• Validity and precision are both desirable but may not always be achieved simultaneously. 

• A valid measurement may not be precise. A precise measurement may not be valid.