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SESSION 2

MEASURES OF EFFECT

Objectives

After this session students should be able to:

• define, calculate and understand the application of measures of association between riskfactors and disease based on ratio of measures of frequency (risk ratio, rate ratio, oddsratio)

• define, calculate and understand the application of measures of association between riskfactors and disease based on the difference between measures of frequency (risk differenceand rate difference)

• select which of the measures (the ratio or the difference) is appropriate for specific situations

• understand and apply critically the concept of strata specific rates, comparisons withinstratum and weighted summary measures

• define, calculate and understand the application of measures of vaccine efficacy

See Hennekens & Buring Chapter 4.

Ratios and differences

Measures of effect are used to compare the frequency of disease in specified populations. Whenone of the populations is exposed to a risk factor and the other is not, measures of effect can beused to study associations between frequency of disease and the risk factor. They reflect theincrease in frequency of disease in one population in comparison with another, treated as thebaseline. Incidences can be compared by estimating their ratios or their differences. The choicebetween a ratio measure or a difference measure should be based on our understanding of themechanism by which a risk factor increases the incidence of disease: if we expect it to multiply thebaseline incidence by a factor, then the best estimate of its effect is a ratio measure; if it is expectedto add to the baseline incidence then the best measure of its effect is the risk difference. Alternativelythe use of either measure may be chosen depending on need, so having a rate difference will help inplanning health service needs.

Ratio measures

The most commonly used measure of effect is the ratio of incidence rates — the rate ratio (RR). Theratio estimates the magnitude of the effect of a risk factor on incidence of disease. The ratio is ameasure of the strength of the association between a risk factor and a disease.

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It is possible to compare any type of measure of frequency in two populations, and so one can havethe rate ratio, the risk ratio, the odds ratio (OR), the prevalence ratio. Because risks and rates tendto be numerically similar in rare diseases, rate ratios and risk ratios tend also to be numerically verysimilar, and often are used interchangeably, and called either name, or called the relative risk.

Examples of ratio measures of effect are:

◊ Smokers are 10 times more likely to develop lung cancer than non- smokers (RR=10).

◊ The mortality rate for stomach ulcers in males in the UK was 96 per million population peryear in 1950 and 31 per million in 1980. This cause of death was 3 times more common in1950 than in 1980. The mortality rate ratio 1950/1980 is three.

◊ In a study in Pelotas, Brazil, the risk of low birth weight was 2 times higher among babiesborn into low income families compared to high income families, 1.6 higher among thoseborn to mothers who smoked compared to those who did not smoke and 1.5 higher tothose born to mothers aged 20 years or less, compared to mothers aged 21 years and over.

Ratios: if we define

Rate in exposed (r1) and rate unexposed (r0); we calculate the rate ratio (RR) as:

RR = r1/r0

For example, a study found that menopausal status increases the rate of coronary heart disease(CHD) in women, based on the data below:

Table 1

Person years at risk Cases Rates per 1000 person years

Post menopause 6848 26 3.8

Pre menopause 8384 6 0.7

According to this table, treating menopause as the exposure, the rate ratio of CHD associated withmenopause is:

RR = r1/r0 = 3.8/0.7 = 5.43

This is interpreted as: ‘post-menopausal women have a rate of CHD roughly five and a half timesthat of pre-menopausal women’.

Risk ratios and odds ratios can also be calculated:

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Defining risk in exposed r1 and risk in unexposed r0, and defining odds of disease in exposed o1 (= r1/(1-r1)) and odds of disease in unexposed o0 (= r0/(1-r0)), we can calculate:

Risk Ratio (RR) = r1/r0 and Odds Ratio (OR) = o1/o0

In general, the risk ratio, the rate ratio and the odds ratio are numerically similar if the disease is rareand often the terms are used interchangeably.

Issues in the use of ratio measures

Effect and impact

Ratio measures are a very good way to estimate the strength of the association between a risk factorand a disease but not a good way to estimate how much of the disease present in a population wascaused by the risk factor. This measure of disease impact depends on the distribution of the riskfactor in the population. For example, although repeated x-rays during pregnancy are known tocause leukaemia in childhood, this causes a very small fraction of all leukaemia in childhood.

Causation

Measures of effect are used in analytical studies which are studies of causality of disease.

Cases and non cases

Measures of effect, like those of frequency and those of impact, assume a dichotomy between casesand non cases, and so are perhaps less useful in the studies of diseases that constitute the extremerange of normal values (for example hypertension). Correlation or regression coefficients may bebetter measures of effect when both the outcome and the risk factor are continuous (e.g. to estimatethe effect of exercise on obesity).

Estimating measures of effect for more than one level of exposure

In most situations we are interested in the ratio between the incidence in two groups, one exposedand one unexposed to the risk factor. It is possible to estimate the effect of different levels ofexposure, calculating, for example, the relative risk of lung cancer in different levels of smoking:number of cigarettes smoked a day. There may also be different strata of exposure that are notordered, for example the risk of tuberculosis in different ethnic groups. When calculating ratiomeasures with more than 2 levels of exposure, the measure of frequency (say, the rate) in each levelis divided by the measure of frequency in the baseline level.

For example, if we define as rate in exposed to the first level (r1), to the second level (r2) and rateunexposed (r0), the rate ratio in the first level of exposure (RR1) is

RR1 = r1/r0

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and in the second level of exposure (RR2):

RR2 = r2/r0

For example, a study of pellagra among persons of different income in 7 villages in the United Statesof America in 1916 found:

Table 2

Income per capita Person years Cases Rate per 1,000 person years

r0 > US$20 1027 4 3.9

r1 US$20 to US$12 1821 37 20.3

r2 < US$12 1312 56 42.7

In this example, using as baseline those with an income of over US$20 per capita a month, wecalculate the rate ratio of pellagra associated with having an income per capita between US$12 andUS$20:

RR=r1/r0 = 20.3/3.9 = 5.2

and for those with income under US$12 as

RR=r2/r0 = 42.7/3.9 = 10.9

The interpretation is that in this study, when compared to those in the higher income group, those inthe intermediate income group had 5 times as high a rate of pellagra, and those in the lower incomegroup, 11 times.

Selecting the baseline group

When studying the effect of a risk factor on an outcome, it is necessary to define a reference, orbaseline, group. Often there is a natural choice for the baseline group (for example never smokers ifthe exposure is smoking). By convention the base line stratum is usually defined as that with thelowest risks (but not in the study of protective measures), or, for statistical reasons, the stratum withthe largest number of subjects may be chosen.

Stratum specific ratios

When both exposed and unexposed populations can be divided into strata according to anothervariable of interest - say age - it is possible to calculate stratum specific ratios. So for examplewhen looking at the association between smoking and coronary heart disease (CHD) the ratio insmokers and non smokers was estimated separately for 6 age groups.

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Table 3 Death rate from CHD in smokers and non-smokers by age

Age Smokers rate(per 1,000)

Non-smokers rate(per 1,000)

Rate ratio

35-44 0.61 0.11 5.5

45-54 2.40 1.12 2.1

55-64 7.20 4.90 1.5

65-74 14.69 10.83 1.4

75-84 19.18 21.20 0.9

85+ 39.52 35.93 1.1

All ages 4.29 3.30 1.3

The rate ratio decreases with age, suggesting that the effect of smoking on the rate of CHD is higherin the younger age groups.

Summary measures

There are ways to combine estimated stratum-specific rate ratios to produce summary rate ratios.These usually give weights to strata. For example, a summary measure may give weights accordingto the size of the strata in the study population, the size of the strata in the target population, etc.Examples of summary measures are SMRs (standardized mortality ratios), Mantel-Haenszelsummary relative risk etc. These are discussed further in the session on Controlling for Confounding.

Difference measures

Difference measures estimate the excess risk caused by exposure in the exposed group. These aresometimes discussed under measures of effect and sometimes under measures of impact.

Risk or Rate difference (RD): ‘the absolute difference between two risks (or rates)’ (Last 1988).This is calculated by subtracting the frequency estimate for the reference group from the comparableestimate for the exposed group. Thus the risk difference is calculated by subtracting the risk in theexposed (r1) from the risk in the unexposed (r0)) and the rate difference by subtracting the rate inexposed (r1) from the rate in the unexposed (r0)).

In most situations, where disease is not very common, risk differences and rate differences will benumerically similar. The risk or rate difference has also been called the attributable risk or excessrisk (even though you may be working with rates!):

Risk difference = risk in exposed - risk in unexposed, or (r1 - r0)Rate difference = rate in exposed - rate in unexposed, or (r1 - r0)

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For example, if the risk of HIV infection among children born to HIV infected mothers who do notbreast feed is 150/1000, and 280/1000 among those who breast feed, the risk difference is130/1000 (130=280-150). The interpretation - always assuming causality - is that breast feedingwas responsible for the infection of 130/1000 of the children who were born to, and breastfed by,HIV infected mothers.

Risk (or rate) difference percent (RD%) also called attributable fraction for exposed, oretiologic fraction among the exposed:

Always assuming a causal association, this represents the proportion of cases in the exposed groupthat were caused by the exposure. It is estimated by dividing the risk (or rate) difference by the risk(or rate) among exposed:

RD% = (r1-r0)/r1

In the same example as above:

RD% = (28/1000 -15/1000) / 28/1000 = 0.46 (or 46%)

The interpretation of this is that breast feeding in that study was responsible for 46% of the infectionamong those children born to HIV infected mothers who breast fed.

Notice that measures of effect do not take into account the rate in the whole population, and areindependent of the frequency of exposure in the population. Measures of impact, to be discussednext session, estimate the risk or rate in the population attributable to the exposure, and aredependent on how widespread the exposure is in that specific population.

Vaccine efficacy

Vaccine efficacy: percentage reduction in incidence among vaccinated attributable to vaccination,assuming that the vaccine is the cause of this reduction. Essentially similar to the rate differencepercent. If r0 = rate in unvaccinated, r1 = rate in vaccinated,

VE = (r0-r1)/r0 or VE= 1-(r1/r0)

The last formula is the same as 1 - the rate ratio (of vaccinated to unvaccinated): vaccine efficacycan be calculated if the rate ratio is known even if the rates among vaccinated and unvaccinated arenot known, for example from a case-control study.

For example, in a study of the protection of hepatitis vaccine, the rate of developing hepatitis wasestimated among those receiving vaccine and receiving placebo:

Table 4

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Cases Person-years Rate per 100 person-years

Placebo 56 443.9 13

Vaccine 11 482.1 2

Vaccine efficacy is VE = (r0-r1)/r0 or (13 - 2)/13 = 85%

Or if all we knew was the rate ratio (of vaccinated over unvaccinated)

VE= 1-(r1/r0)r1/r0 = (2/13) = 0.151-(r1/r0) = (1-0.15) = 0.85 (or 85%).

It is possible to calculate vaccine efficacy when data are available about the vaccine coverage in thepopulation and among cases. Defining pN as proportion vaccinated in the population and pc asproportion vaccinated among cases:

VE = (pN-pc)/pN(1-pc)

For example, if 90% of two-year-olds are vaccinated against measles and 70% of cases in the sameage group are vaccinated, the vaccine efficacy can be roughly estimated to be

pN = 0.9; pC = 0.7;VE = [(0.9 - 0.7)] / [0.9x(1 - 0.7)] = 74%

(This assumes that all the difference in incidence between vaccinated and unvaccinated was causedby vaccine — which is very rarely the case!)

References

Anderson R, May R. Infectious diseases in humans: dynamics and control. OxfordScience Publications, 1992. Chapters 1 to 5.

Kleinbaum DG et al. Epidemiologic research. Van Nostrand Cia. New York, 1982.Chapter 8.

Last JM. A dictionary of epidemiology. Oxford University Press. 1988.

Orenstein WA, Bernier RH, Dondero TJ et al. Field evaluation of vaccine efficacy. BullWHO 1985; 65: 1055-68.

Smith PG, Rodrigues LC, Fine PM. Assessment of the protective effect of vaccines againstcommon diseases using the case-control and cohort studies. Int J Epidemiol 1984; 13:87-93.

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Walter SD and Holford TR. Additive, multiplicative and other models for disease risk. Am JEpidemiol 1978; 108 (5) 341-346.