## Disease frequency

Prevalence = number of existing cases ÷ total population, at a point in time (point prevalence) or over a period (period prevalence)

A proportion, with no units. Measured by cross-sectional studies. Useful for planning services

Cumulative incidence (risk): new cases ÷ people at risk at the start, over a stated time (e.g. 2% over 5 years)
Incidence rate: new cases ÷ total person-time at risk (e.g. per 1000 person-years). It allows for people followed up for different lengths of time
Measured by cohort studies. Useful for studying causes

Prevalence ≈ incidence × average duration of disease (for a stable, uncommon disease)

Worked example: an incidence of 2 per 1000 per year and an average duration of 10 years give a prevalence of about 20 per 1000, or 2%
A treatment that keeps people alive without curing them (e.g. better glaucoma care) increases prevalence. A cure, or a rapidly fatal disease, lowers it

Mortality rate: deaths ÷ population per period. Case fatality: deaths ÷ people with the disease
Standardisation removes the effect of different age structures when comparing populations. Direct standardisation applies each population's age-specific rates to a standard population; indirect standardisation gives the standardised mortality (or morbidity) ratio (observed ÷ expected)

## The 2 × 2 table

Risk in the exposed = a ÷ (a + b); risk in the unexposed = c ÷ (c + d)
Relative risk (RR) = [a ÷ (a + b)] ÷ [c ÷ (c + d)]. Used in cohort studies and trials
Odds = events ÷ non-events. Odds ratio (OR) = (a ÷ b) ÷ (c ÷ d) = ad ÷ bc. The only measure possible in case-control studies; also given by logistic regression
RR or OR = 1: no association. More than 1: increased risk. Less than 1: reduced risk (protective). If the 95% confidence interval includes 1, the result is not statistically significant
The OR approximates the RR when the outcome is rare (under about 10%). For common outcomes the OR exaggerates the effect, moving further from 1 than the RR

## Worked example: a cohort study

1000 smokers and 1000 non-smokers are followed for 10 years. 30 smokers and 10 non-smokers develop late AMD

Risk in smokers = 30 ÷ 1000 = 3%; in non-smokers = 10 ÷ 1000 = 1%
RR = 3% ÷ 1% = 3.0
OR = (30 × 990) ÷ (970 × 10) = 3.06: close to the RR because AMD is rare in this sample
Attributable risk (risk difference) = 3% − 1% = 2%, or 20 extra cases per 1000 smokers over 10 years
Attributable fraction in the exposed = (RR − 1) ÷ RR = 2 ÷ 3 = 67% of AMD in smokers is due to smoking

## Worked example: a case-control study

100 patients with AMD and 100 controls. 60 cases and 30 controls have ever smoked

a = 60, b = 30 (exposed controls), c = 40, d = 70
OR = (60 × 70) ÷ (30 × 40) = 4200 ÷ 1200 = 3.5
The RR cannot be calculated, because the investigator chose how many cases and controls to include

## Measures of treatment effect

Control event rate (CER) and experimental event rate (EER): the proportion with the bad outcome in each arm
Absolute risk reduction (ARR) = CER − EER
Relative risk reduction (RRR) = ARR ÷ CER = 1 − RR
Number needed to treat (NNT) = 1 ÷ ARR, always rounded up. The number of patients treated for one to benefit, over the trial's time period
Number needed to harm (NNH) = 1 ÷ absolute risk increase of an adverse effect

Worked example	Calculation	Result
A trial of a glaucoma treatment over 5 years: 20% of controls and 12% of treated patients progress	CER = 0.20, EER = 0.12
Relative risk	0.12 ÷ 0.20	0.6
Absolute risk reduction	0.20 − 0.12	0.08 (8%)
Relative risk reduction	0.08 ÷ 0.20	40%
Number needed to treat	1 ÷ 0.08 = 12.5, rounded up	13 over 5 years
Odds ratio	(0.12 ÷ 0.88) ÷ (0.20 ÷ 0.80)	0.55

The RRR stays the same when baseline risk changes, but the ARR and NNT do not. The same 40% RRR in a population whose control risk is only 2% gives an ARR of 0.8% and an NNT of 125
Relative measures make treatments look more impressive; the absolute measures (ARR, NNT) are what matter to an individual patient
Case-control: OR. Cohort and RCT: RR (or OR). Survival analysis: hazard ratio (see Hypothesis Testing)

## Measures of burden and value

Disability-adjusted life year (DALY) = years of life lost (YLL) + years lived with disability (YLD). One DALY is one lost year of healthy life. Vision loss contributes almost entirely through YLD
Quality-adjusted life year (QALY): years of life multiplied by a utility from 0 (dead) to 1 (perfect health). One year at a utility of 0.7 is 0.7 QALYs
Incremental cost-effectiveness ratio (ICER) = difference in cost ÷ difference in QALYs between two treatments. Cataract surgery is among the most cost-effective procedures in medicine

Sensitivity, specificity and predictive values use a similar 2 × 2 table (see Diagnostic Tests and Screening)