Ординатура / Офтальмология / Английские материалы / Principles Of Medical Statistics_Feinstein_2002

16.5.In the displays in Figure 16.12, each box shows a lower quartile, median, and upper quartile for the group. [The whiskers show the full range of data, i.e., from the 0th to the 100th percentile.] The actual values for the three quartile points of data can be estimated from the ordinates on the graph. Using this information for each group, prepare a quantile-quantile graph for the comparison of age in Africans and Caucasians.

16.6.Here is another “find-and-fix” set of exercises. From any published literature at your disposal, find an unsatisfactory graph, chart, or other visual display for two groups of data. (If you cannot find a drawing, a badly organized table can be substituted.) For each unsatisfactory display, indicate what is wrong and how you would improve it, sketching the improved arrangement. If possible, find such suboptimal displays for 2-group data that are

16.6.1.Binary

16.6.2.Dimensional

16.6.3.Ordinal

If you cannot find any “juicy” bad displays for these data, and are desperate, a particularly “good” display can be substituted.

16.7. In a scientifically famous legal case, called Daubert v. Merrell Dow Pharmaceuticals, Inc. (in which Bendectin was accused of causing birth defects), the U.S. Supreme Court reviewed a previous legal doctrine, called the Frye rule. The rule had been used by judges in lower courts to make decisions about admissible evidence in “toxic tort” cases involving the adverse effects of pharmaceutical substances or other products accused of “risk.” According to the Frye rule, the only admissible courtroom evidence should have been published in peer-reviewed journals; and any positive contentions about “risk” should have been accompanied by “statistically significant” evidence. The opponents of the Frye rule — who were the plaintiffs in the Daubert case — wanted a more lenient approach. It would allow “expert

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witnesses” to introduce unpublished data and also, when P values were not “statistically significant,” the potential magnitude of risks could be estimated from the upper ends of confidence intervals. Amicus curiae briefs were filed on both sides by a prominent array of institutions and individuals. The Frye rule was defended by the American College of Physicians, AMA, New England Journal of Medicine, a consortium of Nobel laureate scientists, and other persons (such as ARF) who argued that the dropping of standards would fill courtrooms with “junk science.” The Frye rule was opposed by some prominent scientists, statistical epidemiologists, and legal groups (mainly attorneys for plaintiffs) who argued that P values were too rigid a standard and that peer-reviewed publication was neither a guarantee of, nor a requirement for, good scientific work.

Briefly discuss what position you would take in this dispute. What position do you think was taken by the Supreme Court?

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17

Special Arrangements for Rates and Proportions

CONTENTS

17.1Ambiguous Concepts of a Rate

17.1.1Non-Biostatistical Rates

17.1.2Complexity in Biostatistical Rates

17.2Components of Probabilities for Risk

17.2.1Sources of Denominators

17.2.2Quasi-Proportions

17.2.3Inaccurate Numerators for Specific Diseases

17.2.4Quasi-Risks

17.2.5Bilateral Units of Observation

17.2.6Proportionate Mortality Rates

17.3Temporal Distinctions

17.3.1Cross-Sectional or Longitudinal Directions

17.3.2Durations

17.3.3Repetitions

17.4Issues in Eligibility

17.4.1Exclusions for New Development

17.4.2Inclusions for Repetitions

17.5The Odds Ratio

17.5.1Illustration of Numbers in a Cohort Study

17.5.2Illustration of Numbers in a Case-Control Study

17.5.3Ambiguities in Construction

17.5.4Incalculable Results

17.5.5Stochastic Appraisals

17.5.6Quantitative Appraisal of Odds Ratio

17.5.7Combination of Stratified Odds Ratios

17.5.8Scientific Hazards of Odds Ratios

17.6Specialized Terms for Contrasting Risks

17.6.1Glossary of Symbols and Terms

17.6.2Increments

17.6.3Ratios

17.6.4Cumulative Incidence and Incidence Density

17.6.5Proportionate Increments

17.7Advantages of Risk Ratios and Odds Ratios

17.8Adjustments for Numerators, Denominators, and Transfers

17.9Interdisciplinary Problems in Rates

References

Exercises

Several prominent ambiguities have already been noted when central indexes are chosen and contrasted for binary data. The central index itself is ambiguous, because information coded as 0, 0, 0, …, 1, 1, 1 can be cited with either of two complementary proportions, p or q. Thus, the same group might

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have a success rate of 29% or a failure rate of 71%. Another problem occurs for reporting changes. Choosing one central index rather than the other would not affect results when standard deviations are calculated as pq or when two-group increments are formed as p1 – p2 = q1 – q2 . The results can differ dramatically, however, if proportionate incremental changes are cited with either p or q in the denominator. Thus, a drop in the death rate from .09 to .07 could be cited proportionately as either a 22% decline in mortality [= (.07 − .09)/.09] or an equally correct 2% rise in survival [= (.93 − .91)/.91].

Beyond these ambiguities, however, the complexities of binary proportions create several profound scientific and statistical problems that are discussed in this chapter. The problems arise from the custom of calling the proportions “rates,” from their different sources of numerators and denominators, from their static or dynamic temporal constructions, from the uniqueness of the odds ratio as a statistical index of contrast, and from the special epidemiological jargon used for comparisons of rates.

17.1 Ambiguous Concepts of a Rate

Like probability, the idea of a rate is a fundamental concept in statistics; and like probability, rate is difficult to define. As noted in Chapter 6, probabilities can be subjective or objective, isolated or cumulative, empirical or theoretical, frequentist or Bayesian. Rates have many more sources of diversity, inconsistency, and confusion.

17.1.1Non-Biostatistical Rates

In nonmedical branches of science and daily life, a rate is often constructed as a quotient of different components, cited in the different units of measurement. Velocity (or speed) is a rate in which distance traveled, divided by duration of time, is listed in expressions such as km per hr. This usage suggests that a rate reflects something occurring over a period of time. The rate of an automobile’s gasoline consumption, however, is cited without reference to time, in such terms as miles per gallon.

Although the foregoing examples suggest that a rate is a quotient of two different constituents, such quotients also receive other names. When a quantity of calcium is divided by a quantity of fluid, the quotient is called a concentration, expressed in such units as mg/dl. A quantity of red blood cells divided by a quantity of fluid, however, is not called a concentration. It is called a count if the cells are enumerated and a hematocrit if their packed volume is measured. If the units of measurement are similar, but the measured entities are different, yet another word may be used. The quotient formed when a concentration of serum albumin is divided by a concentration of serum globulin is called a ratio and is unit-free. Many unit-free quotients, however, are not called ratios. They are called rates. Thus, a quotient of two quantities of money, such as the amount of interest for the principal of a bank loan, is called an interest rate. It is expressed as a unit-free proportion (or percentage). Although many proportions are called percentages as well as rates, other names are sometimes used. A baseball player’s proportion of number of hits per number of times at bat is called an average.

In maintaining the inconsistencies just noted for the world beyond, the rates examined in biostatistics may or may not be proportions; the proportions may or may not be rates; and the ratios may be either or neither.

17.1.2Complexity in Biostatistical Rates

Valiant attempts1 are sometimes made to define a rate as “a measure of change in one quantity (y) per unit of another quantity x … [which] is usually time.” Although a desirable standard, this definition is constantly ignored in the literature of medicine and public health. For practical purposes, a rate is almost anything that the user of the term wants it to be and that editors are willing to publish.

Any rate can readily be described (rather than defined) as a quotient of two quantities, a/n, in which a numerator, a, is divided by a denominator, n. In biostatistics, these quantities are usually frequency counts rather than measured dimensions. The quantities come from enumerations of either groups of

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people or events that occur for people, not from dimensional amounts of such entities as distance, gasoline, calcium, or fluid. The inconsistencies and ambiguities of biostatistical rates (and non-rates) arise from differences in what goes into the denominator, what goes into the numerator, and even what goes into the virgule, which is the “ /” mark between denominator and numerator.

Regardless of what the a/n quotient is called, it is always a proportion if the number counted in the numerator is a specific subset of the entity identified and counted in the denominator. If a denominator group of 60 people contains 20 men and 40 women, the proportion, not the rate, of men in the group is .33 = 20/60. If the numerator for those people, however, represents an event — such as death, success, or occurrence of a disease — the analogous proportion is often called a rate. Thus, if 20 of the group of 60 people die, have successful outcomes after treatment, or develop streptococcal infections, the enumerated 20 people are still a subset of the denominator of 60, and the quotient of .33 = 20/60 is still a proportion; but it will often be called, respectively, a death (or mortality) rate, a success rate, or an attack rate.

The next two main sections discuss the complexities produced by the different components and nomenclature used when rates express probabilities of risk, and when they denote different directions and durations of time. The main problems in these complexities are scientific: the diverse terms may be difficult to understand or remember, and the enumerations may have dubious accuracy.

17.2 Components of Probabilities for Risk

The rate with which a particular event occurs in a group of people is often called a probability or risk for occurrence of the

event. This structure, which uses the quotient of two frequency counts to form a rate of risk, is unique in medical biostatistics.

The weatherman may say there is an 18% chance or “risk” of rain, but does not derive the forecast from enumerations. In biostatistics, however, if 10 members of a group of 2000 people die in the next year, the proportion, .005 (= 10/2000), is often called a mortality rate and is further cited

as the probability or risk of death for that group. This rate or risk, like all frequentist probabilities, is constructed as a proportion. The denominator contains the enumerated people who are candidates for the risk of death; the numerator contains the count of those who died; and the rate is the proportion or relative frequency of death. The subset structure of a proportion is shown in Figure 17.1.

The basic idea is familiar and easy to understand, but clinicians and public-health epidemiologists produce confusion by using the same name — mortality rate — for death rates calculated from substantially different components in the numerators and denominators. Furthermore, when the necessary components are not specifically counted, the calculations may produce quasi-proportions and quasi-risks that are statistically reasonable, but scientifically inaccurate.

17.2.1Sources of Denominators

The basic source of different rates is the group of people who constitute the denominator. In publichealth statistics, this group is the general population of a community or region. At any particular point in time, some of the people in that region are sick, but most are healthy. In clinical work, however, the denominator group is usually a collection or “case series” of patients with a particular clinical ailment. The different proportions of subsequent death in these two different denominator groups, however, are often given the same name: mortality rate.

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Suppose a community contains 4000 people of

whom 80 are known to have lung cancer, and suppose 20 of those patients die in the ensuing

year. The rate of death for lung cancer in that year will often be expressed as a mortality rate of .005 (= 20/4000) in public-health reports and as a mortality rate of .25 (= 20/80) in clinical reports. The distinctive sources that produce these disparate mortality rates for the same dis-

ease in the same year are shown in Figure 17.2. With more precise citation, the result for the pub- lic-health denominator is often called an annual cause-specific mortality rate, whereas the result for the clinical denominator may be called a oneyear mortality rate.

To avoid the ambiguity of using mortality rate for both types of denominators, the term case-fatality rate has been proposed as a substitute name for the clinical citation. Clinicians do not like to give up

mortality rate, however, because they often refer to its complementary proportion, survival rate. Thus, the one-year survival rate for lung cancer in the foregoing group would be expressed as .75 = 60/80.

This term creates no conflict in public-health statistics, which seldom refer to a survival rate.

The public-health data sometimes contain demographic calculations for rates that reflect survival, but the results are called “longevity” or “life expectancy,” not “survival.” Accordingly, the idea of a death rate is regularly expressed by public-health investigators as mortality rate for a regional denominator, and by clinical investigators as the complementary proportion, survival rate, for a clinical denominator. The two citations of death rates, however, are not at all complementary. In the foregoing example, the public-health mortality rate is .005, but the clinical survival rate is .75. Confusion often occurs when clinical investigators ignore the distinctions and refer to their clinical results as a mortality rate of .25.

17.2.2Quasi-Proportions

A different type of problem arises when the public-health rates are regarded as proportions and used to express probabilities or risks. Because of inevitable difficulties in getting the data, these rates are really quasi-proportions; and their accuracy may sometimes be uncertain or dubious.

The challenge of finding and counting everyone to enumerate the entire human denominator of a large region or community is a daunting task. When this task is done every ten years, the census bureau is highly gratified if the enumeration seems reasonably complete. Aside from the problem of being completely counted, however, the denominator in a highly mobile society is constantly changing as various people move into or out of the region and other people die or are born. Because of depletions by deaths and out-migration and augmentations by births and in-migration, the denominators for intercensal years (such as 1991, 1992, …, 1999) depend on estimates derived from the counts obtained at the most recent census year (such as 1990). For inter-censal years, the basic decennial census results are altered according to counts of intervening annual births and deaths in the region, and then modified from educated guesses about annual out-migration and in-migration.

These estimates, without which no denominators would be available, make the regional mortality rates become quasi-proportions, rather than true proportions. The proportions are “quasi” because the counts of deaths, cancers, or other events in the numerators are not specific subsets of the people who were individually identified and counted in the denominators. The rates appear to be relative frequencies for a set and subset, but they are really a quotient of frequencies from two different sources, for groups counted in two different ways. The numerators come from an actual count of death certificates submitted to a regional agency, such as a state health department. These numerator counts can be accepted as complete, because a dead body cannot legally be disposed until a death certificate is prepared. The denominators for each of the nine inter-censal years, however, are not actual counts. They come from whatever formulas were used to estimate the augmentations and depletions that occurred after the actual counts in the census year.

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17.2.2.1 Problems of Inand Out-Migration —

Under-enumeration of a region is a striking problem, whose magnitude is difficult to determine. For various social, cultural, or other reasons, the census takers can seldom count everyone who lives in the region.2 The counted regional population in Figure 17.3 is therefore shown as a smaller subset of the actual regional

population. With geographic migration both into

and out of a region, some or many of the people originally enumerated in the

denominators may later move elsewhere. The rectangle counted as the regional

population in Figure 17.3 can be later depleted by a non-counted number of

out-migrants while being simultaneously augmented by a non-counted number of in-migrants. If the two migrating groups are similar in both numbers and composition (such as age-sex distribution), the counted denominator will be essentially

unchanged. Otherwise, the actual denominator will be different from the estimated denominator.

For the numerators, all of the deaths are usually counted completely in industrially developed countries. As shown in Figure 17.4, however, the deaths may not always be associated with the denominators from which they emerged. The originally enumerated people who remain and die in the region will be suitably managed. The out-migrants who die elsewhere, however, may be counted in the denominator of Region A but in the numerator of Region B. The in-migrants who die in Region A may be counted in its numerator, while appearing in the denominator of Region B.

17.2.2.2 Problems of Under-Enumeration — With suitable monitoring, data, and adjustments, the migratory phenomena may not produce major inaccuracies in the results. The main problem will then be the basic errors caused by census under-enumeration. For example, in many regions of the United States, the African-American or Hispanic population is regularly undercounted. The community leaders may then complain that the falsely low count has deprived their group of suitable political representation. A fascinating legal battle may then occur as the regional district, supported by an array of prominent statisticians, sues the government, supported by a different array of prominent statisticians, asking that the census be adjusted for the undercount.3,4

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During the legal and political skirmishes, however, an unrecognized biostatistical problem is that the falsely low denominators, when divided into the accurate numerators obtained from death certificates, will also lead to falsely elevated mortality rates. Thus, if 500 correctly counted deaths occur among 100,000 people erroneously counted as 90,000, the true death rate is .005, or 5 per thousand; but it will proportionately increase by 12% when reported as 5.6 per thousand.

17.2.3Inaccurate Numerators for Specific Diseases

If the numerators consist of all deaths and the appropriate denominators are suitably counted or estimated, the regional death rates may be reasonably accurate. Life insurance companies have managed to develop and maintain a profitable enterprise by using total death rates to anticipate longevity and set suitable costs for the premiums charged in policies.

For more than a century, however, death certificates have been used for much more than counts of total mortality. From the different diagnoses that may be listed on each death certificate, a single disease has been selected as the sole “cause of death”; the counts of these “causes” are then used as numerators for calculations called cause-specific mortality rates. These rates have been used to represent the frequency of occurrence, or “incidence rates,” of those diseases.

The errors produced by this peculiar custom have received thorough discussion elsewhere5,6 and are too extensive for more than a brief outline of all their lamentable consequences. The rates of “incidence” are fundamentally erroneous because they reflect death, not occurrence of disease; the rates are inevitably too low because only one of the patient’s many identified diseases is counted; the numerators contain counts of diagnoses rather than diseases and will vary with changes in diagnostic concepts and technol - ogy; the reported diagnoses may depend on inconsistent idiosyncrasies in the physicians who fill out the death certificates; and when one of the cited diseases is chosen as the cause of death, the selection criteria will vary both from one decade to the next and with inconsistent application during the coding process at official agencies, such as a Bureau of Vital Statistics or (in the U.S.) the National Center for Health Statistics.

17.2.4Quasi-Risks

Aside from the problem of accurate counts for numerators and denominators, the regional rates are seldom accurate as probability estimates of risk. The problem here arises from the concept of risk itself. Individual persons are “at risk” for a particular event if they are suitably susceptible to that event and if it has not yet occurred. Risk of death is easy to identify because any living person has that risk. If the numerator event, however, is a “causal disease,” such as cancer of the uterus or gallbladder, rather than death itself, any woman who has had a hysterectomy, or anyone with a previous cholecystectomy, is not at risk. Furthermore, someone in whom the numerator event has already occurred is also not at risk of developing it. Thus, a woman who is now pregnant is not at risk for becoming pregnant, and the development of coronary heart disease is not a risk for someone who already has angina pectoris.

When a regional population is entered into a denominator for calculating a rate that will represent a risk, however, the counted group will include not only the people who are truly at risk, but also those who are not at risk either because they are not susceptible to the numerator event or because they already have it. To identify the three separate categories of people — the at-risk, no-risk, and already affected groups — is seldom possible or feasible, however, in large communities. Consequently, because the denominator will almost inevitably be too large, the “risks” calculated for the community or region will be too small. The magnitude of error in these quasi-risks will depend on the relative proportions of the three types of constituent groups in the denominator. For some diseases — such as the cancers, infarctions, and other chronic diseases often found unexpectedly at necropsy after having been clinically “silent” and unsuspected or undiagnosed during life7,8 — the errors can be substantial.

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17.2.5Bilateral Units of Observation

A distinctly clinical problem in identifying constituents of the denominator arises when therapy is aimed at preventing or treating conjunctivitis, glaucoma, otitis media, diabetic neuropathy, or any other poten - tially bilateral disease for which a person may have two anatomic candidates, rather than one. In this circumstance, the main issue is whether to count persons or anatomic entities. If we study 24 persons, do we count the denominator group as 24 persons or as 48 eyes and/or ears?

The answer to this question may vary from one study to another and will often depend on whether the therapy is prophylactic or remedial. With prophylactic therapy, we probably want to count susceptible people who may develop the outcome event on one anatomic side or the other (or both). For remedial therapy, however, we may want to determine what happens to each treated entity. Sometimes a patient with similar involvement on both sides might even be used as a “self-control” for comparing two treatments.

17.2.6Proportionate Mortality Rates

Another potentially confusing entity is the proportionate mortality rate, symbolized as PMR. This proportion is almost never used in clinical work, but often appears in public-health or occupational epidemiology. The denominator for the PMR proportions is the collection of all deaths in a group or region during a particular time interval. The numerators are the counts of deaths ascribed to individual diseases. Thus, at the end of a year in a region with a total of 400 deaths, of which 50 are attributed to coronary heart disease, the PMR for coronary disease will be 50/400 = .125.

These calculations avoid the problem of erroneous counts in the denominator population, but magnify all the difficulty of getting accurate classifications and numbers for the “cause of death” cited in the numerators. Furthermore, the PMR values, although true proportions, cannot be used as probabilities for risk. They describe constituents in the univariate nominal spectrum of deaths, not the risk in living people.

The PMR has often been used in occupational epidemiology to examine death rates for workers in different industries. Getting the true death rate for an industry would involve the arduous task of identifying all the “exposed” employees in the denominator and following them as a cohort thereafter to discern the numerator fatalities. Instead, the deaths alone can be determined relatively easily (often from claims filed for “benefits”); and sometimes occupation is also listed on the death certificate. With either approach for relating deaths and occupation, a PMR can be constructed from death data alone, without the need for cohort studies.

Beyond the scientific difficulties of the PMR tactic, however, an additional mathematical problem is produced because the total of all proportionate mortality rates must add up to 1 (or 100%). Therefore, if the death rate declines for one disease but remains stable for a second disease, the PMR may rise for the second disease. Thus, if infectious diseases are substantially eliminated or reduced as causes of death, the PMR results can increase for chronic diseases, such as arteriosclerosis and cancer, even though the latter diseases may be occurring with unchanged lethality or even less often than before.

17.3 Temporal Distinctions

All of the problems just cited arise from counting the numerators and denominators in a rate such as a/n. A quite different problem is due to time. For each person enumerated in the denominator, the numerator event may have been counted at the same time, at a later time, or more than once. This distinction gives the virgule (“/”) a special temporal role beyond merely denoting division of a numerator by a denominator to form a/n. The virgule can imply a direction, duration, or repetition of events for an interval of time.

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17.3.1Cross-Sectional or Longitudinal Directions

The group of people in the denominator can be observed in either a static or forward direction. In static or cross-sectional observation, the condition counted in the numerator is observed at essentially the same time as the denominator condition. Thus, if we can simultaneously count the population of a region and the corresponding number of people with AIDS, we can calculate a “point prevalence” for AIDS in that region. In clinical work, group A streptococcal infection has a cross-sectional prevalence of 15% if we find 30 such infections in a survey examination of 200 schoolchildren.

In longitudinal observation, the numerator condition (or event) occurs afterward, at a serial time later than the original denominator condition. (The word longitudinal is itself a peculiar title. The name has no more of a forward connotation than latitudinal, but probably became popular because of its resemblance to “longevity,” which implies a forward direction of follow-up.)

A major distinction between cross-sectional and longitudinal data is the number of examinations required for each person. In a cross-sectional study, all the information can be obtained in essentially a single examination, which identifies the person’s age, sex, clinical state, or other descriptions of the condition counted in the denominator. The same examination can also identify the existence of a streptococcal infection, dietary habit, or previous exposure to a risk factor that may be counted in the numerator. To note the incidence of a longitudinal event, however, each person under observation must be examined on at least two occasions. If the 200 school children are examined repeatedly or at least one more time during the next year, and if 50 of them are found to have developed new streptococcal infections, the attack rate or incidence rate per person is 25% (= 50/200).

17.3.1.1Incidence vs. Prevalence — The difference between incidence and prevalence is a key distinction in epidemiologic nomenclature. Prevalence refers to what is there now; incidence refers to what happens later. Deaths are always an incidence event, but the occurrence of a disease may represent prevalence or incidence, according to when and how the occurrence was discovered. If a woman is suitably examined today and found to be free of breast cancer, its discovery two years from now is an incidence event. If she received no previous examinations the breast cancer discovered two years from now is usually called incidence because it seems to be “new,” but it may also be a revealed prevalence that would have been discovered had she been suitably examined today.

Clinicians constantly misuse the word incidence in referring to such prevalence events as the proportions of women, antecedent myocardial infarction, or college graduates observed in a counted group of people. The distinction is particularly important for the case-control studies discussed in Section 17.5.2, where all of the retrospective phenomena represent prevalence, not incidence. [If you accept the idea that rates involve a change over time, incidence rate is a satisfactory term, but prevalence rate is not, because prevalence does not involve a change over time.]

17.3.1.2Spectral Proportions — Clinicians also often misuse the idea of incidence when reporting proportions for the univariate age or sex spectrum of a disease. For example, suppose the spectrum of a clinical group of patients with omphalosis contains 70% men, and also 5% children, 20% young adults, 50% middle-aged adults, and 25% elderly adults. These results will regularly be reported with the comment that omphalosis has a higher “incidence” in men than in women, and has its highest “incidence” in middle-aged adults. Because the denominator of these univariate proportions is the number of patients with omphalosis, the results represent neither incidence nor prevalence in a group at risk. They show proportions in the univariate spectrum of distribution for each variable in the group of patients collected at whatever medical setting was the source of the research.

17.3.2Durations

Because all incidence events involve observation during a duration of time, a temporal period becomes the third component of an incidence rate. As a longitudinal quotient, a/n has two components, but in strict symbolism, it is a/n/t, where t is the interval of time. The extra symbol for t is commonly omitted, however, because the results are usually cited in such phrases as annual mortality rate or 3-year survival

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