Evidence-based practice statistics: practice problems with step-by-step solutions

1Numbers behind clinical decisions

Evidence-based practice means using research results to make decisions about care. To read a study you need a small set of numbers: how good a test is, how common a disease is, and how much a treatment changes the risk.

Almost all of them come from a 2 × 2 table: two groups in the rows and two outcomes in the columns. Once you know where a, b, c and d sit, every formula is a simple fraction.

2Diagnostic tests

Put the test result in the rows and the true disease status (from the reference standard) in the columns: a = true positives, b = false positives, c = false negatives, d = true negatives.

• Sensitivity = a/(a + c): of the people with the disease, the share that tests positive.
• Specificity = d/(b + d): of the people without the disease, the share that tests negative.
• PPV = a/(a + b): of the positive tests, the share that is truly ill.
• NPV = d/(c + d): of the negative tests, the share that is truly healthy.

Sensitivity and specificity look down the columns; predictive values look along the rows.

3Prevalence and predictive values

Prevalence is the share of a population that has a disease at one moment. Incidence counts new cases among the people at risk over a period, or per person-time (for example per 1000 person-years).

Sensitivity and specificity belong to the test. Predictive values also depend on the prevalence: a good test used where a disease is rare gives many false positives.

A quick way to see this is to imagine 10 000 people. With a prevalence of 1%, 100 are ill; a test with 90% sensitivity finds 90 of them. With 95% specificity, 5% of the 9900 healthy people also test positive: 495 false positives. So the PPV is 90/585, only about 15%.

Likelihood ratios update a probability through odds: odds = p/(1 − p), post-test odds = pre-test odds × LR, and probability = odds/(1 + odds).

4Treatment effects: RR, ARR and NNT

In a trial, the event rate in the treated group is EER and in the control group CER.

• Relative risk RR = EER/CER.
• Absolute risk reduction ARR = CER − EER.
• Relative risk reduction RRR = ARR/CER = 1 − RR.
• Number needed to treat NNT = 1/ARR, always rounded up to a whole patient. For harm, NNH = 1/(EER − CER).

In a case-control study the researcher chooses how many cases and controls to include, so risks cannot be calculated. Use the odds ratio OR = ad/(bc) instead.

The same RRR gives a much larger NNT when the baseline risk is low: a 25% reduction of a 2% risk is an ARR of 0.5%, an NNT of 200.

5A worked example

Follow every step and predict the next one before you open it. All values are made up for practice.

6Common mistakes and exam questions

• Mixing up sensitivity (down the column) and PPV (along the row).
• Treating the PPV as a fixed property of the test.
• Multiplying a probability by a likelihood ratio without converting to odds.
• Rounding the NNT down, or using 1/RRR instead of 1/ARR.
• Calculating a relative risk from a case-control study.

Easy: sensitivity, specificity, PPV, NPV, accuracy, prevalence and incidence. Medium: likelihood ratios, RR, ARR, RRR, NNT and NNH, odds ratios and incidence rates. Hard: PPV at another prevalence, post-test probability, expected counts in a screened population and NNT from a baseline risk.