2.5 Completeness of the data
Measurement error concerns the people who are measured; the other threat at the collection stage is the people who are not. Non-response in a survey, refusal in a case-control study and loss to follow-up in a cohort remove people from the data, and if the reasons are related to exposure or disease, the people left are no longer a random sample.
Proposition 2.5.1. In a population in which a fraction \(r\) responds, let \(\mu _R\) and \(\mu _N\) be the means (or prevalences) of a variable among responders and non-responders. The population mean is \(\mu = r\mu _R + (1 - r)\mu _N\), and the responders’ mean has bias \[ \mu _R - \mu = (1 - r)(\mu _R - \mu _N) . \]
Proof. \(\mu _R - \mu = \mu _R - r\mu _R - (1 - r)\mu _N = (1 - r)(\mu _R - \mu _N)\). □
The bias is the product of the non-response rate and the difference between responders and non-responders. A high response rate limits it, but cannot remove it if the two groups are very different, and the second factor cannot be estimated from the responders alone.
Example 2.5.2. In a survey with \(70\%\) response, the prevalence of a condition among responders is \(0.12\). If a follow-up of a sample of non-responders finds a prevalence of \(0.25\) among them, the population prevalence is \(0.7 \times 0.12 + 0.3 \times 0.25 = 0.159\), and the responders understated it by \(0.3 \times (0.12 - 0.25) = -0.039\), about a quarter of the true value. People who are ill are often less likely to take part in a health survey, which makes this direction of bias common.
Missing values within the records of those who do take part raise the same issue. Data are missing completely at random (MCAR) if whether a value is missing is unrelated to any variable, observed or not; missing at random (MAR) if it depends only on observed variables; and missing not at random otherwise. An analysis of complete records only is unbiased under MCAR. Under MAR the missingness can be corrected for by using the observed variables it depends on (weighting, or multiple imputation); when values are missing not at random no analysis of the data alone can correct for it, which is why the protocol should aim to prevent missing data rather than plan to analyse it.
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