Error factors and confidence intervals
An error factor turns a study estimate (the single ratio the study reports) into a confidence interval, the range of values around it where the true value could lie.
Remember
For the lower bound, divide the study estimate by the error factor, and for the upper bound, multiply it by the error factor. The interval shows how uncertain the estimate is and does not remove that uncertainty.
There is some doubt with any study, because studies cannot guarantee to be correct. So a confidence interval gives the range of values around the study estimate where the true value could lie.
The interval is worked out with an error factor, a number greater than 1 that expresses the proportional uncertainty around the study estimate. The method keeps the uncertainty in proportion to the size of the estimate, so the bounds come from dividing and multiplying rather than adding and subtracting. To find the lower bound, divide the study estimate by the error factor. To find the upper bound, multiply it by the error factor. Because the error factor is greater than 1, dividing gives a smaller number and multiplying gives a larger one.
The study estimate is a ratio measure, and which one depends on the study design. A case-control study (one that begins with cases and controls and then looks back at what people were exposed to) uses the odds ratio, a ratio derived from the odds of that exposure in cases and controls. With the error factor method, a cohort study (one that follows people over time and counts new cases of disease as they arise) uses the incidence rate ratio, which compares the incidence rate in one group with the incidence rate in the other.
Take a case-control study with an odds ratio of 2.4 and an error factor of 1.6. The lower bound is 2.4 divided by 1.6, which is 1.5. The upper bound is 2.4 multiplied by 1.6, which is 3.84.
A wider interval means a less precise estimate. Interpret the interval against the null value of 1, which for a ratio is the value for no effect.
Key facts
- Error factor
- a number greater than 1 that expresses the proportional uncertainty around a study estimate.
- Lower bound
- divide the study estimate by the error factor.
- Upper bound
- multiply the study estimate by the error factor.
- Case-control study
- the confidence interval uses the odds ratio.
- Cohort study
- with the error factor method, the confidence interval uses the incidence rate ratio.
- Wider interval
- a less precise estimate.
- Null value
- for a ratio measure the null value is 1, so an interval containing 1 means the data are compatible with no effect and the result is not statistically significant.
Exam traps
- An error factor does not correct systematic bias, which comes from flaws in how a study was designed or carried out. It only describes the uncertainty from random error (chance).
- A confidence interval describes random error rather than removing it, because it shows how wide the uncertainty is.
- Multiplying by the error factor does not give the lower bound. Because the error factor is greater than 1, multiplying gives the upper limit and dividing gives the lower limit.
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