Notation & Formulas for HS161

Measures and Notation

Epidemiologic terminology is far from uniform. The chart below will help clarify some of the key terms used in the course and the text.

Measure	Verbal Description	Related Terms
Prevalence	Proportion of people with disease at a point in time.	Prevalence "rate" (misnomer)
Incidence	Risk or rate (see below) of new occurrences.	Many forms of incidence exist. The two main categories of incidence are "risks" and "rates" (see below).
Risk	Number of disease onsets divided by the number at risk at beginning of follow-up.	Incidence Proportion

Cumulative Incidence

Probability of Disease
Rate	Number of disease onsets divided by sum or person- time at risk.	Incidence Rate

Incidence Density

Central Rate

Hazard Rate

Force of morbidity
Relative Risk	Ratio of two risks (or rates).	Almost anything that ends in "ratio" (e.g., rate ratio, risk ratio, prevalence ratio, cumulative incidence ratio, incidence density ratio, standardized mortality ratio, hazard ratio, etc.)
Risk Difference	Difference of two risks (or rates).	Almost anything that ends in "difference" (e.g., incidence difference, rate difference, cumulative incidence difference, incidence density difference, etc.)
Attributable Fraction, Population	Expected % reduction in number cases following elimination of the exposure in population.	Etiologic Fraction, Population

Attributable Fraction, Exposed Cases	Expected % reduction in number cases following elimination of the exposure in exposed case.	Etiologic Fraction, Exposed Cases

Odds ratio	Ratio of two odds, especially useful in case-control studies.	Two forms exist: a disease odds ratio and exposure odds ratio.

Chapter 6: Incidence and Prevalence

Basic Formulas

where represents the average ("central") population size and Dt represents the duration of observations.

Examples of Specific Rates

Chapter 7: Adjusted Rates

Notation: Capital letters (e.g., Ni) denote values from the reference population. Small letters (e.g., ni) denote values from the study population.

Direct Adjustment

where aR(direct) represents the directly adjusted rate, Ni represents the number of people in strata i in the reference population (there are k strata), and ri represents the rate of disease in strata i of the study population.

Indirect Adjustment

Expected number of cases ( )

where Ri represents the rate in the ith stratum of the reference population and ni represents the number of people in the ith strata of the study population. The product Rini is the expected number of cases in the ith stratum of the study population ( i).

Standardized Mortality Ratio

where x represents the observed number of cases in the study population and represents the expected number of cases as calculated above.

Indirectly Adjusted Rate

Chapter 8: Measures of Association

Dichotomize the population into two groups: an "exposed" group and an "unexposed" group. In fact, the exposure may represent any trait, intervention, or environmental exposure in the usual sense. Then, let:

R1	represent the risk or rate of disease in exposed group

R0	represent the risk or rate of disease in the unexposed group

R	represent the risk or rate of disease in the population as a whole.

Risk Difference

The risk difference is the difference between two risks:

For example, if the risk in the exposed group was 2 per 1000 and the risk in the unexposed group was 1 per 1000, the risk difference would be 1 per 1000, suggesting an additional cases per every 1000 exposures.

The risk difference quantifies the effect of the exposure in absolute terms. That is, it quantifies the excess number of cases per m exposures.

Relative Risk

The relative risk is the ratio of two risks:

For example, if the risk in the exposed group is 2 per 1000 and the risk in the unexposed group is 1 per 1000, then the relative risk is 2.

The relative risk indicates the direction and strength of the association. Notice that when R1 = R0, RR = 1, indicating no association between the exposure and disease. Relative risks greater than 1 indicate a positive association, and relative risks less than 1 indicate a negative association. The strength of the association is quantified by the size of the relative risk; the relative risk can be viewed as a risk multiplier. It also measures the effect of the exposure in relative terms.

If we define the relative effect of the exposure as the absolute effect (i.e., risk difference) compared to the baseline risk, then "the relative effect of the exposure" = . That is, the "relative effect of the exposure" = RR - 1. Therefore, "relative effect" has two components: the relative risk (proper) and the constant "-1." This allows us to say that a relative risk of 2 suggest the effect of the exposure is to increase risk by 2 - 1 = 1 = 100%. This may seem awkward at first, but it is really no different than any other relative comparison. For example, we say that $2 is 100% more than $1.

Attributable Fraction

Attributable fractions allow for the quantification of the impact of the exposure on either a population basis or case basis.

The attributable fraction in the population is:

This quantifies the expected proportional reduction in risk if the exposure were eliminated from the population. For example, a population attributable fraction of 50% suggests that eliminating the exposure from the population would eliminate half the cases.

The attributable fraction in exposed cases is:

This quantifies the expected proportional reduction in cases had the exposed cases not been exposed (it is a liability measure). For example, an attributable fraction in exposed cases of 75% would suggest that three-quarters of the exposed cases would have been avoided had they not been exposed.

Chapter 9: Case-Control Studies

Notation:

Exposed	Not Exposed
Cases	a	b	m1
Controls	c	d	m0
n1	n0	n

The case-control method precludes the direct estimation of risk, but allows the estimation of relative risk in the form on an odds ratio.

The odds ratio is

This statistic can be shown to be equivalent to a rate ratio from a cohort study when proper (density) sampling is used. Therefore, an odds ratio of 1 indicates no association between the exposure and disease, an odds ratio of 2 indicates a doubling of the rate, and so on. For example, as case-control study with the following data:

Exposed+ Exposed-

Case 647 2

Cntl 622 27

has OR =(647)(27)/(622)(2) = 14.0. This indicates that the exposed group has a rate of disease that is 14 times that of the unexposed group (equivalently, a 1300% increase in risk).

Attributable fractions in exposed cases can be determined from case-control studies as:

For example, when the OR = 14.0, AFe = (14.0 – 1)/(14.0) = .929.

The attributable fraction in the population is

where p0 represent the exposure proportion in controls, which is equal to p0 = c / m0. For the above data, p0 = 622 / 649 = .9584 and AFp = [(.9584)(14.0 – 1)] / [(.958)(14.0 – 1) + 1] = .926.

Chapter 4: Reproducibility and Validity

Reproducibility Statistics

Rater B
Rater A	+	−
+	a	b	p1
−	c	d	q1
p2	q2	N

Overall agreement = (a + d) / N

Agreement in Subjects w/ At Least One Positive Diagnosis = a / (a + b + c)

Kappa

Iinterpreting : 0 indicates random agreement; < .4 represents poor agreement; .4 ≤ < .7 represents moderate agreement; > .7 represents excellent agreement; = 1 indicates perfect agreement.

Validity Statistics

Disease +	Disease −
Test +	True Positives (TP)	False Positive (FP)	n1
Test −	False Negative (FN)	True Negative (TN)	n2
m1	m2	N

SENsitivity = TP / m1	Note: TP = (SEN)(m1)

SPECificity = TN / m2	Note: TN = (SPEC)(m2)

PVP = (TP) / n1	Bayesian:

PVN = (TN) / n2	Baysian:

P = m1 / N	where P represents the true prevalence of disease. Note: m1 = (P)(N)

P* = n1 / N	where P* represents the apparent prevalence of disease