PSM and biostatistics: the calculations worth being able to do in your head

Ask a group of final-year students which part of Preventive and Social Medicine they like least and the answer is usually the same: the numbers. Sensitivity, specificity, odds ratio, relative risk, standardised rates. They look like the kind of thing you'd need a calculator for and a statistics degree to understand.

They aren't. Almost every calculation PSM asks for is arithmetic you could do in a minute with a pencil, built on a small number of ideas. The difficulty is not the maths. It's remembering which fraction goes where, and that is a problem of understanding, not of memory.

This article walks through the calculations worth being able to do quickly, using small invented numbers so you can follow every step. If you can do these, you can do most of what the subject asks.

Start with the two-by-two table

Nearly every diagnostic-test calculation comes from a single table. Draw it every time. It costs ten seconds and prevents most mistakes.

Imagine a test for a disease, applied to 1000 people. We know who truly has the disease and who doesn't, from a gold-standard test. The test result is either positive or negative.

Hold on to this layout. Everything below is a ratio of the cells.

Let's fill it with numbers. Suppose 100 people have the disease and 900 don't. The test is positive in 90 of the 100 diseased, and positive in 45 of the 900 healthy. So a = 90, c = 10, b = 45, d = 855.

Sensitivity and specificity

Sensitivity is the proportion of diseased people the test correctly identifies: a divided by (a + c). Here, 90 out of 100, which is 90%. It answers the question: if you have the disease, how likely is the test to catch it?

Specificity is the proportion of healthy people the test correctly clears: d divided by (b + d). Here, 855 out of 900, which is 95%. It answers: if you don't have the disease, how likely is the test to say so?

A way to remember which is which: sensitivity is about the diseased column, specificity is about the healthy column. Both are properties of the test, and they don't change with how common the disease is.

A common mnemonic is SnNout and SpPin. A highly Sensitive test, when Negative, rules out disease, because it rarely misses. A highly Specific test, when Positive, rules in disease, because it rarely produces false alarms.

Predictive values: the ones that depend on prevalence

Patients don't ask "how sensitive is the test?" They ask "I tested positive, so how likely is it that I have the disease?" That's the positive predictive value: a divided by (a + b). Here, 90 out of 135, which is about 67%.

The negative predictive value is d divided by (c + d): 855 out of 865, about 99%.

The important point is that predictive values change with prevalence. In our table, 10% of the people had the disease. If the same test were used in a population where only 1% had it, the number of true positives would fall and false positives would dominate, and the PPV would drop sharply. That's why a screening test can have excellent sensitivity and specificity and still produce mostly false positives in a low-prevalence population. It is a favourite exam trap.

Try it: same test, 10,000 people, 1% prevalence. That's 100 diseased and 9,900 healthy. True positives: 90% of 100 is 90. False positives: 5% of 9,900 is 495. PPV = 90 out of 585, about 15%. The test hasn't changed. The population has.

Incidence and prevalence

These two cause more confusion than anything else in the chapter, so let's keep it simple.

Prevalence is how many people have the disease right now, as a proportion of the population. It's a snapshot. Number of existing cases divided by the population.

Incidence is how many new cases appear over a period of time. It's a movie. Number of new cases divided by the population at risk, over a stated period.

An analogy: think of a bath. Prevalence is how much water is in it. Incidence is the rate at which the tap is running. A disease can have a high prevalence and a low incidence if people live with it for a long time, such as diabetes. A disease can have a high incidence and low prevalence if it's short-lived or fatal, such as a severe acute infection.

The relationship worth remembering is that, roughly, prevalence is incidence multiplied by average duration. If a disease is prolonged by better treatment (people survive longer with it), prevalence goes up even if incidence stays the same.

Risk, relative risk and attributable risk

These come from cohort studies, where you follow exposed and unexposed groups forward.

Suppose 200 smokers and 800 non-smokers are followed for ten years. In the smokers, 40 develop a particular lung disease. In the non-smokers, 24 do.

• Risk (incidence) in the exposed: 40 out of 200 = 0.20, or 20%.
• Risk in the unexposed: 24 out of 800 = 0.03, or 3%.
• Relative risk (RR): risk in exposed divided by risk in unexposed = 0.20 / 0.03, about 6.7. Smokers have a risk roughly 6.7 times that of non-smokers.
• Attributable risk: the difference, 0.20 − 0.03 = 0.17, or 17%. This is the portion of the risk in smokers that's attributable to smoking.

Relative risk tells you the strength of an association. Attributable risk tells you how much disease could be prevented by removing the exposure. They answer different questions, and the exam knows it.

Odds ratio: the case-control measure

In a case-control study you start with people who have the disease (cases) and people who don't (controls), and ask about past exposure. You can't calculate incidence, because you haven't followed anyone, so you can't calculate relative risk directly. You calculate the odds ratio instead.

Using the two-by-two layout with exposure in the rows and case status in the columns: odds ratio = (a × d) / (b × c), sometimes remembered as the cross-product.

Example: among 50 cases, 30 had been exposed and 20 hadn't. Among 50 controls, 10 had been exposed and 40 hadn't. Then a = 30, b = 10, c = 20, d = 40. OR = (30 × 40) / (10 × 20) = 1200 / 200 = 6.

An odds ratio above 1 suggests the exposure is associated with the disease. When the disease is rare, the odds ratio approximates the relative risk, which is why case-control studies are such an efficient way to study rare diseases.

Choosing the right study design

Exam questions often give you a scenario and ask which design was used, or which is best. A few anchors help.

• Wants incidence or relative risk, and can follow people over time: cohort study.
• Rare disease, wants to explore many past exposures quickly: case-control study.
• Wants prevalence at one moment: cross-sectional study.
• Testing whether an intervention works: randomised controlled trial.
• Comparing whole populations, not individuals: ecological study.

Remember the direction: cohort studies go forward from exposure to outcome. Case-control studies go backward from outcome to exposure.

Measures of central tendency and spread

A few quick ideas that appear often.

The mean is the average. The median is the middle value when data is ordered. The mode is the most frequent value. In a skewed distribution the mean is pulled toward the tail, so the median is a better summary. In a normal distribution, all three coincide.

The standard deviation describes how spread out data are around the mean. In a normal distribution, about 68% of values lie within one SD of the mean, about 95% within two, and about 99.7% within three. Learn those three numbers, they're asked constantly.

A tiny example: if the mean systolic blood pressure in a group is 120 with a standard deviation of 10, then about 95% of the group would have a systolic pressure between 100 and 140.

Hypothesis testing in one paragraph

You begin with a null hypothesis, which says there is no difference. You collect data, and calculate a p-value: the probability of seeing a difference at least this big if the null hypothesis were true. If the p-value is below a chosen threshold, often 0.05, you reject the null.

Two errors are possible. Type I is rejecting a null hypothesis that is true, a false positive. Type II is failing to reject a null hypothesis that is false, a false negative. The probability of a Type I error is alpha. The probability of a Type II error is beta. Power is 1 − beta, the chance of detecting a real effect.

A small p-value doesn't tell you the effect is large or important. It tells you the data would be surprising if there were no effect.

Demography in a few lines

A few indicators come up again and again.

• Crude birth rate: births per 1000 mid-year population.
• Crude death rate: deaths per 1000 mid-year population. Affected by age structure.
• Infant mortality rate: infant deaths (under one year) per 1000 live births. Widely used as a sensitive indicator of community health.
• Maternal mortality ratio: maternal deaths per 100,000 live births.
• Life expectancy: average number of years a newborn is expected to live under current mortality patterns.

Note the denominators. Many questions are really asking whether you know which denominator goes with which rate.

How to practise this subject

Numbers need doing, not reading.

Draw the two-by-two table every time. Make it a reflex.

Make small numbers. If you can't follow an example with 1000 people, shrink it to 100. Easy numbers make the structure visible.

Build a one-page formula sheet and rewrite it from memory each week until you don't need to look.

Practise questions in batches of ten. PSM calculations improve quickly with repetition, because the same few structures recur.

Check denominators. Before you compute, say out loud what the denominator is. Many errors come from using the wrong one.

Keep programme details separate. National health programmes, targets and statistics change. Keep an updated sheet from official sources and revise it near the exam.

Standardisation in two paragraphs

Crude rates can mislead when populations differ in age structure. A town with many older people will have a higher crude death rate than a town with young people, even if its health care is better. Standardisation fixes this by applying each population's age-specific rates to a common standard population, producing a rate that can be fairly compared. Direct standardisation needs age-specific rates in the study population. Indirect standardisation, used when those rates are unreliable or unavailable, applies standard rates to the study population's age structure and gives a standardised mortality ratio.

If a question gives you two populations with very different age profiles and asks which has the better health, think of standardisation before you compare the crude rates.

Putting the formulas to work in one mixed example

Imagine a town of 50,000 people. In one year there are 1,000 births and 400 deaths, of which 30 are infants under one year old. Then the crude birth rate is 1,000 / 50,000 × 1,000, which is 20 per 1000. The crude death rate is 400 / 50,000 × 1,000, which is 8 per 1000. The infant mortality rate is 30 / 1,000 live births × 1,000, which is 30 per 1000 live births. Notice that the denominator for the infant mortality rate is live births, not the population.

Work through a few of these with your own numbers each week, and notice which denominator you've used each time. That habit alone prevents many lost marks.

A short summary

• Draw the two-by-two table. Sensitivity is about the diseased column, specificity the healthy one.
• Predictive values depend on prevalence; sensitivity and specificity don't.
• Prevalence is a snapshot, incidence is a rate of new cases; prevalence is roughly incidence times duration.
• Relative risk and attributable risk come from cohorts; the odds ratio comes from case-control studies.
• Match the design to the question: cohort forward, case-control backward, cross-sectional for prevalence.
• Within ±1, ±2 and ±3 SD of a normal mean lie about 68%, 95% and 99.7%.
• Type I is a false positive; Type II is a false negative; power is 1 − beta.

NextPrep's PSM questions and flashcards cover these ideas in small worked examples with a reason for every option. They are original practice material still awaiting independent expert review, so verify key facts and any programme details against official sources. Pick one of the calculations above and do it with your own small numbers. It takes two minutes, and it's the surest way to stop fearing the chapter.