University of North Carolina, Department of Epidemiology
Fundamentals of Epidemiology (EPID 168)

I don't understand the odds ratio. (1998-10/06a)

Prevalence in a cohort study versus prevalence in the population (1998-10/06b)

Are specificity and sensitivity equally important in screening tests? (1998-09/15a)

I am still confused about the definitions of adjusted rates and specific rates and why one or the other is used. (1998-09/15b)

I'm not sure about the definition of prevalence and incidence as involving 'entities' and 'events' respectively (1998-09/15c)

I am confused about rates and proportions. (1998-09/15d)

1 - Do proportions have a time element to them? I understood them not to be associated with a time dimension, but in the "Evolving Text" you mentioned that prevalence was a proportion, even though it has a time period associated with it. (1998-09/15d1)

2 - Gordis (1996) says that rates cannot be the same as a proportion, it seems your evolving text is not in agreement with this. Do you know why? (1998-09/15d2)

3 - Are prevalence and incidence figures rates? Are they proportions? Is prevalence relative and incidence absolute? (1998-09/15d3)

I don't understand the odds ratio. (1998-10/06a)

- Odds ratios:

Of the various epidemiologic measures of association, the odds ratio (OR) seems to strike people as the most "mysterious". That may be due to it's tendency to appear in various contexts, various study designs, and have various meanings.

- When can one compute an odds ratio (OR)?

The OR is simply the ratio of two odds. Since an odds is simply the ratio of a probability to its inverse [i.e., p/(1-p)] an OR can be defined and computed anytime one can compare two probabilities". A probability can be defined even in a situation where it has no external meaning. For example, if the top row of a 2 x 2 tables has a 3 and a 5 in its cells, then I can say that the proportion 3/(3+5) = 3/8 is "the probability of choosing one of the 8 people at random and finding out that s/he is in the left-hand cell. Therefore, an odds ratio can be defined based on almost any two probabilities (the exception, of course, is where the denominator probability is zero.

- Advantages of the odds ratio:

Epidemiologists like to look at 2 x 2 tables. The OR is very handy for 2 x 2 tables, because it is easily calculated (ad/bc) and its value is the same whether you construct the table with the outcome variable in the rows or in the columns. Furthermore, the natural logarithm of the odds ("logit") has several nice properties. Consider that a probability can be only a real number between zero and one, inclusive. In contrast, an odds can be any positive real number. But the range of possible values for the logarithm is skewed, since the middle of the range of probabilities (0.5) becomes an odds of 50:50 or 1.0 which is at nearly the lower end of the range of values for odds. The logit fixes that problem by providing a range of ALL real numbers with its midpoint at zero. The logit's convenient mathematical properties make it more suited for mathematical modeling (e.g., logistic regression).

- Why would I want to use an odds ratio?

Epidemiologists often want to compare two incidences by dividing one by the other. This ratio is either a rate ratio (or IDR), when the incidences are rates, or a risk ratio (or CIR), when the incidences are proportions. In some cases, the odds ratio can provide an estimate of an incidence ratio. In other cases an odds ratio may be a useful measure of association even though incidences are available.

- Case-control study:

In a case-control study, incidences cannot be estimated without data other than that in the 2 x 2 table. The only ratio measure of association that is useful is the OR. Since the OR is numerically equivalent whether the odds are obtained from the rows or from the columns, we don't have to worry about the fact that in a case-control study with no extra information, only the exposure proportions, and not the disease proportions, are meaningful.

Depending upon the manner of selection of controls and the rarity of the disease outcome, the OR may provide an estimate of the CIR or the IDR. With small incidences (e.g., both under 10%), all three of these measures are numerically similar to one another. Under suitable circumstances, the OR can estimate the IDR even when the disease outcome is not rare.

- Cross-sectional study:

In a cross-sectional study, a prevalence odds ratio (POR) can be computed using the same ad/bc formula. The POR is a measure of the association between the two factors in the 2 x 2 table. If the duration of the condition is the same in both groups being compared, then the POR can estimate the IDR for the incidence rates in the two groups. If the disease outcome is rare, then of course OR, CIR, and IDR are all similar.

- Cohort study:

In a cohort study, incidences and incidence ratios can be estimated directly. Therefore, the OR is usually not of interest. However, logistic regression estimates OR's, rather than incidence ratios. So sometimes it's convenient to compute OR's in a cohort study in order to compare them to those estimated with the logistic regression models. And, of course, if the incidences are small, the OR will be about the same size as the CIR, anyway, so one could get by with just the OR.

- Mathematical relationship between the OR and CIR:

The null value for the OR, CIR, and IDR is 1.0. In a situation where the OR approximates the CIR, if the CIR is 1.0 the OR will be exactly 1.0. If the CIR is above or below 1.0, the OR will always even further above or below 1.0; the larger the incidences, the greater the separation between the two measures of association. This statement is based on the simple algebraic relationship betweem the OR and CIR:

Suppose that p1 is the disease probability (or incidence proportion) in the "exposed" group. Therefore, the incidence odds in this group is p1/(1-p1). Suppose that p0 and p0/(1-p0) are the corresponding values for the "unexposed" group. Then the risk ratio or cumulative incidence ratio (CIR) is simply p1/p0 and the odds ratio (OR) is

 p1 / (1-p1) p1 * (1-p0) p1 (1-p0) OR = --------------- = --------------- = -------- * --------- p0 / (1-p0) p0 * (1-p1) p0 (1-p1)

The expression on the right can also be written as:

 (1-p0) CIR * --------- (1-p1)

If the CIR=1, then p1=p0 so the OR will also = 1. If p1 and p0 are both very small, then 1-p1 and 1-p0 will both be very close to 1.0, so the OR will be close to the CIR. If p1 is greater than p0. then (1-p0) will be greater than (1-p1), and the OR will equal the CIR multiplied by a number that is greater than 1.0; conversely, if p1 is smaller than p0, (1-p0)/(1-p1) will be less than 1.0 and the OR will be smaller than the CIR.

Try some numbers in a spreadsheet to see how the CIR and OR compare across a range of circumstances.

Prevalence in a cohort study versus prevalence in the population (1998-10/06b)

Anytime one has a proportion, one can call it a "prevalence". To be meaningful as a prevalence, however, the numerator and denominator must have some external meaning. If a representative sample is obtained from a population, then the data in that sample can be used to obtain estimates of the population. In particular, prevalences computed from the sample estimate the corresponding prevalences in the population.

A cohort study may or may not constitute a representative sample of some population of interest. The Framingham and Evans County cohort studies are two examples of cohort studies that were constructed by drawing samples from their respective populations, so baseline prevalences provide estimates of prevalences in the population at that time.

After a period of time has elapsed, the cohort will usually become less like the population from which it was chosen, so subsequent cohort prevalences will not correspond so closely with population prevalences. Some cohort studies use volunteers recruited through mass media and referals. Prevalences can still be computed for such cohorts, but these prevalences do not estimate population prevalences (unless it so happens that people volunteered in proportion to their population proportions, an unlikely scenario).

Are specificity and sensitivity equally important in screening tests? (1998-09/15a)

The relative importance of sensitivity and specificity needs to be considered in a decision-making framework. We need to weigh the costs (of all kinds) or lack of sensitivity (missed cases) and lack of specificity (false alarms).

If a serious disease can be readily cured if it is detected in time, then we want to miss as few cases as possible; in other words, we want high sensitivity. On the other hand, if the cost of missing the disease is not great, either because we'll find it soon anyway or because the later treatment is only somewhat less effective, then it is not so critical if we miss some cases.

If a positive test means an expensive, painful, or otherwise burdensome diagnostic evaluation, then we want to have as few false positive tests as possible; in other words, we want high specificity. On the other hand, if a confirmatory test is not burdensome and there is no stigma attendant on a positive test result, then high specificity is not so critical.

If we cannot increase both sensitivity and specificity, but have to trade-off one for the other (e.g., by varying a cut-point), then considerations such as those above are important.

In applying these cost considerations, however, it is also important to consider the actual prevalence of the condition. If a condition is very rare, there are only a small number of cases to detect. Even a very high specificity could still mean many false positives. So even if a false positive does not have a high cost, there may be so many of them that we still need to avoid the situation.

If a condition is quite common, then even a relatively high sensitivity could still mean missing many cases. So even if the cost of a single missed case is not great, having a large number of them is a problem.

However, if we do not screen, we will not detect any cases early. If we screen with low sensitivity (e.g., 30%), at least we will detect 30% of cases, so some good will be done. On the other hand, if we screen with only modest specificity (e.g., 80%), we may have so many positive tests to work up we may exhaust our health care resources. In this sense, therefore, low sensitivity is not as critical as low specificity.

Similarly, a study of a disease will often be compromised more if most of the people in the case group do not have the disease (because of imperfect specificity) than if the case group is "pure" but many cases have been missed.

I am still confused about the definitions of adjusted rates and specific rates and why one or the other is used. (1998-09/15b)

IN A NUTSHELL:

A rate (or proportion, or other measure) can be crude, specific, or adjusted.

The concepts come into play in situations where a population is composed of subgroups that differ in some important way. A rate for an individual subgroup is "specific" (to that subgroup). A rate for the total population, without regard to the variation across subgroups, is "crude". A rate for the total population, but which makes an allowance for the variation across subgroups is "adjusted".

Crude rates are simple, but comparisons of crude rates have the potential to mislead when the populations being compared differ in some important way(s).

Specific rates are simple, but if there are many subgroups then it can be awkward to have to compare so many pairs of rates.

Adjusted rates enable us to summarize sets of specific rates so that we can report and compare the summaries.

If there is no heterogeneity within the population, then the crude rate is fine. If there is heterogeneity then we need to look at specific rates. If it would be useful to have a summary of the specific rates, then we compute an adjusted rate. If we want to compare heterogenous populations then we compare the specific rates across the populations. If the comparison indicates that it would be meaningful to present and compare summary measures, then we compute adjusted rates.

A SIMPLE (BUT WORDY) EXAMPLE:

I have been hired by a suburban school district to compare the quality of urban schools and suburban schools. I obtain the average scores on the most recent statewide test for all NC suburban schools and all NC urban schools. The average test score over all students attending suburban schools is greater than that for all students attending urban schools. I show the results to the school district, who proudly release them to the local press. So far, so good.

A reporter asks the superintendent whether the results show that suburban schools are better than urban schools. That raises the question of whether suburban schools can be fairly compared to urban schools, or are there differences that might be interfering with the comparison. Available data indicate that 60% of suburban students have college-educated parents, versus only 20% of urban students.

In general, students with college-educated parents score much higher on these tests. The comparison between the (crude) average scores for suburban and urban schools is "confounded" by differences in parental education.

I break down both groups according to parental education. I compare average scores for suburban students with college-educated parents to average scores for urban students with college-educated parents. I do the same for suburban and urban students without college-educated parents. In other words, I compare scores that are parental-education-specific, rather than crude.

The suburban students have higher average scores than rural students within each parental-education subgroup. I show the results to the superintendent and tell her/him that the higher average scores for suburban schools are not due to their having higher proportions of students with college-educated parents.

The reporter asks if, for the headline, I could provide two rates instead of four [1) suburban, college-educated; 2) suburban, not college-educated; 3) urban, college-educated; 4) urban, not college educated]. I give the reporter an adjusted (in this case, an adjusted score for suburban schools and an adjusted score for urban schools. I compute the adjusted (in this case, standardized) score by weighting the scores for students with college-educated parents by 30%, the national average across all school students, and weighting the scores for the students without college-educated parents by 70%. The reporter computes a ratio and writes

that suburban schools have 20% higher test scores than urban schools after allowing for differences in parental education.

The math teacher spots an arithmetic error. In fact, the urban students without college-educated parents scored higher than the suburban students without college-educated parents. So among students with college-educated parents, the suburban schools had higher scores, but among students without college-educated parents the urban schools had higher scores. The adjusted scores are now very similar for suburban and urban school students.

The reporter asks if that means that there is no difference between suburban and urban schools. I say "'yes' and 'no'" -- no difference in an overall comparison that takes account of differences in the proportion of students with college-educated parents, but differences within each subgroup of students. The superintendent gives me a sharp look. I shrug my shoulders and say "this just shows that we need to do more research".

IF YOU STILL WANT MORE!

"Crude" refers to an overall measure that ignores possible heterogeneity within the group.

"Specific" refers to a measure that applies to a subgroup regarded as homogenous with regard to some characteristic or characteristics (e.g., sex, age).

"Adjusted" refers to an overall measure that takes account of possible heterogeneity within the group to which it applies. An adjusted measure is, in most cases, a weighted average of a set of specific measures -- the measures for each of the subgroups that comprise the overall group.

("Standardized" means adjusted by the kind of weighted average procedure described in the Standardization chapter in the "Evolving Text".)

The simplest of these measures is the crude one, and it may be completely sufficient for our purposes. Problems arise when a population of interest is heterogeneous in some factor(s) that influences the rate or other measure we are examining. The problem is compounded when we want to compare the rate (as we almost always want to do at some point) to the rate for a population that has a different distribution of the relevant factor(s).

If the population is heterogenous we can subdivide it into subgroups that are homogenous (or more homogenous) with respect to the relevant factors. The rates for these subgroups are "specific" to them. If we have succeeded in forming homogenous groups, then we have all of the information that is available.

However, it can be inconvenient to work with a handful of rates rather than a single one. If we wish to have a single rate, then we compute an adjusted rate that in some way takes into account the variation across subgroups. The standardized rates we saw in the chapter represent one class of adjusted rates. Another class of adjusted rates are weighted averages in which the weights are the precision of the specific rate estimates. That is a subject for another day.

In exchange for the convenience of a single rate, though, we are (deliberately) masking the variation across subgroups. As long as we recognize that (and report variation if it is significant - not necessarily in the "statistical" usage of the word), then no harm is done.

I'm not sure about the definition of prevalence and incidence as involving 'entities' and 'events' respectively (1998-09/15c)

Prevalence (a "still life") refers to an existing state of affairs. Therefore, the "cases" for the numerator of a prevalence measure are "entities" -- things that exist.

Incidence (a "moving picture") refers to a process taking place over time. Therefore, the "cases" for the numerator in an incidence measure are "events" -- things that occur.

If among 500 people, 100 are bald, then the prevalence of baldness is 100/500 = 0.20 or 20%. Baldness is an entity -- the state of being bald -- something that exists.

If among 500 people with hair, 100 become bald during a five year period, then the t-year (cumulative) incidence of baldness is 100/500 = 0.20 or 20%. Here, baldness is an event -- becoming bald -- something that happened.

When I lost my hair, that was an event. Now I have no hair -- my bare head is an entity.

I am confused about rates and proportions. (1998-09/15d)

Your questions have puzzled many an epidemiology student (and faculty member) for years. Here is my understanding. It's possible that when you reach 268 you'll obtain a different perspective, since there is not unanimity on some things.

1 - Do proportions have a time element to them? I understood them not to be associated with a time dimension, but in the "Evolving Text" you mentioned that prevalence was a proportion, even though it has a time period associated with it. (1998-09/15d1)

As a mathematical entity, a proportion has neither time nor any other unit, it is simply a number between 0 and 1 (inclusive). In order to know what that number represents, though, we need some additional information (for example, it could be a prevalence, an incidence, an interest rate, a test score, etc.) Once I know that a proportion is a prevalence, I need to know what it is a prevalence of (i.e., who is going in the numerator), among whom (i.e., the PAR in the denominator), and if the phenomenon is something that may be changing over time I may need to know to what point in time the prevalence applies (e.g., May 1998). Sometimes the prevalence was obtained over a period of time, for logical or logistical reasons, but that just means a "fat" point. A single prevalence does not examine change during that time period.

2 - Gordis (1996) says that rates cannot be the same as a proportion, it seems your evolving text is not in agreement with this. Do you know why? (1998-09/15d2)

I don't recall the context for Gordis' statement, but some of these issues are definitional, and not everybody uses the same definitions! Regina Elandt-Johnson (formerly a professor in the BIOS department here) published an article in AJE which gave definitions of ratio, proportion, and rate. These definitions are fairly widely followed. I try to follow them too, but I'm also not inclined toward dogmatism. Generations of epidemiologists (probably including Gordis at one time, but that's just a guess) have employed terms like "prevalence rate", "case fatality rate", and so on. These terms refer to proportions but use the word "rate". In defense of this usage, Ollie Mietinnen (who is credited for developing or at least introducing into epidemiology many of the theoretical concepts we now use) once argued (in an unpublished document that I saw, though it may now be in his textbook) that a proportion was a rate because the denominator represented a "change in the population space" [or words to that effect]. For safety's sake (mine), I generally follow the usage of some authority figure, such as Elandt-Johnson or Rothman, and avoid correcting people who are my betters (or peers)!

3 - Are prevalence and incidence figures rates? Are they proportions? Is prevalence relative and incidence absolute? (1998-09/15d3)

Except for the Absolute, Everything is Relative! (that's an inside joke). In ordinary and officlal usage, prevalence is a proportion and generally not regarded as a rate (though the term "prevalence rate" is still in fairly wide usage). The terms "relative" and "absolute" aren't generally linked with the concept of prevalence. As a proportion, prevalence already expresses its numerator in relation to a denominator of the same type of entity (e.g., people, other animals, aliens, etc.).

In contrast, a rate (see Elandt-Johnson) can be either absolute (the denominator contains only the change in some quantity, usually time) or relative (the rate also expresses the numerator in relation to a denominator of its type). For example, if 50 UNC students pulled their hair out during the past five years, the average annual rate would be 10 hairless students/year. Between 1955 and 1959, inclusive, only 20 UNC students pulled their hair out, yielding an average annual rate of 4 per year. However, these ABSOLUTE rates (change in something / change in time) are difficult to interpret since the numerator must be examined in relation to the number of UNC students during the relevant time period. When we divide these rates by the average number of UNC students during each of these five-year periods, we may well find that the resulting RELATIVE rates are about the same for the two time periods. In contrast, the rate of rainfall (average inches per year) for two time periods is quite interpretable even though it is an absolute rate.

There are two varieties of incidence measures in wide use. Rothman and Greenland (1998) refer to them as "incidence proportion" and "incidence rate", which makes their mathematical form abundantly clear. I learned them as, respectively, "cumulative incidence" and "incidence density", which terms where then coming into style. There are situations where CI (incidence proportion) is preferred and others where ID (incidence rate) is preferred. There are also situations where it's not clear if one or the other is preferable. The preference is often related to whether it's more useful or meaningful to describe a period of time that has already ended (CI) versus one that is still going on (ID) and whether it matters when the event occurred (ID) versus just that the event has occurred (CI).

Victor_Schoenbach@unc.edu, public\faq.htm, last updated 9/15/1998, 10/3/2000