Implications of binomial distribution in the daily orthopaedic practice

Deep infection is a devastating complication in the orthopaedic surgery. It usually requires a revision surgery and a risk for further complications increases greatly. Functional outcome is usually much worse after deep infection compared to uneventful primary surgery especially if removal of implants is required to eridicate the infection. Deep infection are often systematically recorded to give some insight to overall quality of surgical activity.

Prevalance of deep infection is usually monitored and reported for certain periods. Yearly quarter is a common time frame for which the prevalance is reported. Quite often administration or other personnel responsible for quality and safety gets worried if the prevalance is higher than usually. It is of course human to think about possible reasons why infection prevalence might have been higher than usual during some period. I think better knowledge on random variation and simulation data is helpful here.

Let´s assume that the 5-year average prevalence for a deep infection is 1.5% for certain orthopaedic procedure in my hospital. It is reasonable to assume that the true population value for the risk of infection is 1.5% and it varies around that estimate. We can model this as a binomial probability density.

Below is a histogram for 10,000 “blocks” of 500 hypothetical surgeries with expected risk of infection being 1.5%. Histogram peaks at 0.014 which is closest to the expected value of 1.5% or 0.015. This means that majority of block including 500 surgeries have infection rate of around 1.5% and ~4% of the blocks have >2.5% infection rate.

If we restrict our analysis to blocks of 200 surgeries things gets more interesting.

Again, histograms peaks at a prevalence of 0.015. But when infection rate is analyzed in blocks of 200 surgeries, ~8% of the blocks have infection rate of 2.5% or more. And almost ~3% of the blocks have infection rate of 3% or more which more than double to the true underlying prevalence!

This is not particle physics but more like a remainder that we should not underestimate how random variation affects our daily practice.

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