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Epidemiologic measures: Incidence & prevalence

Victor J. Schoenbach, PhD home page

Department of Epidemiology�Gillings School of Global Public Health�University of North Carolina at Chapel Hill

www.unc.edu/epid600/

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Incidence and prevalence

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Principles of Epidemiology for Public Health (EPID600)�

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Famous last words

5/20/2002

Incidence and prevalence

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Quotations that demonstrate the value of humility about predicting the future�(authenticity not established)

Courtesy of Suzanne Cloutier, 11/17/1998

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“Louis Pasteur's theory of germs is ridiculous fiction.”

- Pierre Pachet, Professor of Physiology at Toulouse, 1872

5/20/2002

Incidence and prevalence

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FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future

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“This `telephone´ has too many shortcomings to be seriously considered as a means of communication. The device is inherently of no value to us.”

- Western Union internal memo, 1876

(Source: 2000 National Ernst & Young Entrepreneur of the Year Awards special insert in USA Today, 2/11/2000, p9B)

5/20/2002

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FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future

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“Everything that can be invented has been invented.”

- Charles H. Duell, Commissioner, US Patent Office,1899

5/20/2002

Incidence and prevalence

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FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future

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“The wireless music box has no imaginable commercial value. Who would pay for a message sent to nobody in particular?”

- David Sarnoff's associates in response to his urgings for investment in radio in the 1920s.

9/11/2005

Incidence and prevalence

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FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future

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“My thesis in this lecture is that macroeconomics . . . has succeeded: Its central problem of depression-prevention has been solved, for all practical purposes, and has in fact been solved for many decades.”

- Robert E. Lucas, Jr., American Economics Association Presidential Address, January 10, 2003�http://home.uchicago.edu/~sogrodow/homepage/paddress03.pdf

1/25/2011

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FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future

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The population perspective requires measuring disease in populations

Science is built on classification and measurement.
Reality is infinitely detailed, infinitely complex.
Classification and measurement seek to capture the essential attributes.

Public health is distinguished by its population perspective, in contrast to the individual perspective of clinical medicine. Epidemiology is a basic science of population health, and in order to study population health we need to construct measures for that purpose.

Science is built on classification and measurement. Reality is infinitely detailed, infinitely complex, and it is simply not possible to work with all of that detail. Thus, every step in the process of understanding reality involves extraction and abstraction of elements we regard as meaningful. For example, although you are reading words on paper or a computer screen, your eyes are receiving a great variety of visual stimuli. Your retina is passing some but not all of these on to your visual cortex, the part of your brain responsible for vision. The visual cortex in turn passes a representation of what it has received on to the areas of your brain where cognition takes place. The visual cortex has “interpreted” the image that it received from the retina. This representation is then “interpreted” by the frontal cortex, so that you recognize and attach meanings to the letters, words, punctuation, etc. There may be imperfections on the paper or dust on the screen. Some of the letters may be slightly distorted. The intensity and sharpness of the print or image will vary. But the brain can ignore all of these “irrelevant” attributes since it has learned which attributes convey meaning and which do not.

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Deriving meaning from stimuli

Vase or faces?

Which line is longer?

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Measurement “captures” the phenomenon

Classification and measurement are based on:

Objective of the classification
Conceptual model (understanding of the phenomenon)
Availability of data (technology)

When they work well, measurement and classification “capture” the phenomenon by identifying and/or quantifying the essential attributes or characteristics within the totality.

There is no single classification that is the “right one” for all purposes. Rather, the appropriate classification and measurement for a particular phenomenon are based on three factors: 1) our objective in creating the classification or measurement, 2) the conceptual model within which we are operating, and 3) the availability of data, which in turn is determined by the technology we can apply. These three considerations – objective, conceptual model, and availability of data – will arise each time we need to create, select, or evaluate a measure or classification scheme.

We’re all familiar with the mean, or average, as a summary measure of a characteristic in a population. The focus of this lecture is the measurement of disease frequency, or more generally, of the frequency of an event, condition (e.g., fitness, immunity, longevity), or characteristic (e.g., right-handedness, usual diet, or exposure to environmental pollution). So the objective is to measure frequency of something in a population. Our conceptual model and the availability of data will depend on the specific characteristic or condition of interest.

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5/20/2002

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An example population (N=200)

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O

How can we quantify disease in populations?

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O

How can we quantify disease in populations?

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O

How can we quantify disease in populations?

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How can we quantify disease in populations?

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O

How can we quantify disease in populations?

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O

How can we quantify the frequency?

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Rate of occurrence of new cases�per unit time (e.g., 1 per month)

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1 new case in month 1

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O

1 new case in month 2

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5/20/2002

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









O

1 new case in month 3, for a total of 3 cases

22 of 80

5/20/2002

Incidence and prevalence

22

















O

2 new cases in month 4

23 of 80

5/20/2002

Incidence and prevalence

23

















O

1 new case in month 5 (total=6)

24 of 80

5/25/2011

Incidence and prevalence

24

















O

1 case in month 6

25 of 80

5/20/2002

Incidence and prevalence

25

















O

1 new case in month 7

26 of 80

5/20/2002

Incidence and prevalence

26

















O

2 new cases in month 8

27 of 80

5/20/2002

Incidence and prevalence

27

















O

2 cases in month 9

28 of 80

1/9/2007

Incidence and prevalence

28

















O

Rate of occurrence of new cases during 9 months: 1 case/month to 2 cases/month

29 of 80

1/9/2007

Incidence and prevalence

29

Number of cases depends on length of interval

Divide by length of time interval, so can compare across intervals

Number of new cases�Rate of new cases = –––––––––––––––––� Time interval

= 12 cases / 9 months = 1.33 cases / month �

One familiar way is to divide the number of cases by the length of the time interval. That gives us an average rate of new cases. With 12 cases in 9 months, that rate is 1.33 cases/month.

We measure the speed of a vehicle we are driving not by the odometer alone (distance traveled) but by the distance divided by the time it took us to cover that distance. In more time we expect to cover more distance, so the distance alone is not a good measure of the speed of the car (it would be an adequate measure if we always drove for the same length of time, but that is not how driving works). So we measure a car’s average speed as the ratio of the distance traveled to the length of time we have been driving. Similarly, we can measure the average rate of occurrence of new cases as the ratio of the number of cases that have occurred to the length of time during which they occurred. In this case, that rate is 12 cases / 9 months, or 1.33 cases/month.

However, does this rate meet our needs for a measure of disease occurrence in populations?

Not really. The population we observed had 200 people in it. What if we had been observing a larger population under exactly the same conditions. If the “force of morbidity” were the same in the larger population, would we not expect to have observed more cases per unit time? If so, then this rate is not quite what we need.

30 of 80

1/25/2011

Incidence and prevalence

30

Number of cases depends on population size

So, divide by population and time:

Number of new cases�Incidence rate = ––––––––––––––––––

Population-time

Since the number of cases generally depends on the size of the population, it makes sense to divide not only by the length of the time interval but also by the size of the population at risk – the number of people who were available to develop a disease and the time period for which they were at risk. We refer to the ratio of the number of cases to the combination of population and time as the incidence rate. The incidence rate serves to quantify the frequency of disease occurrence in a population per person and per unit time.

As we have derived it here, the incidence rate is an average rate, in the sense that we have ignored the possibility that the rate may have been changing from month to month. For the present we are assuming that the fluctuations we observed are not meaningful for our purposes. We are assuming that there is some underlying constant rate of disease occurrence, and our average rate provides an estimate of that constant rate. But the rate could change over time, for example, by season.

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31

How to estimate population-time?

Population at risk: the people eligible to become a case and to be counted as one.

In this example that population declines as each case occurs.

So estimate population-time as . . .

32 of 80

1/25/2011

Incidence and prevalence

32

Population-time =

Method 1: Add up the time that each person is at risk

Method 2: Add up the population at risk during each time segment

Method 3: Multiply the average size of the population at risk by the length of the time interval

In practice, we usually estimate the total population time at risk in one of three ways:

Method 1: Add up each person’s time. If we know the amount of time that each member of the population was at risk, we can take the sum of those times, e.g., 3 months for person A + 2 months for person B + 2.5 months for person C, etc..

Method 2: Add each time’s people. If we know the size of the population at risk during each small segment of time, then we can add up these sizes, e.g., 500 people in year 1 + 700 people in year 2 + 600 people in year 3.

Method 3: Multiply average time by average number of people. If we do not have such detailed information but we know the average size of the population, we can estimate population time as the average size of the population at risk multiplied by the number of time units in the risk interval, e.g., 600 people x 3 months.

33 of 80

1/9/2007

Incidence and prevalence

33

Estimating population-time - method 2

Total population-time over 9 months =

200 + 199 + 198 + 197 + 195 + 194 + 193 + 192 + 190

= 1,758 person-months

= 146.5 person-years

However, cases are not at risk for a full month.

This slide illustrates Method 2 with our little population example. As a first approximation to the total population-time over 9 months, we could simply add up the number of people at risk (200+199+198+197+195+194+193+192+190), which is 1,758 person-months, or 146.5 person-years. This procedure assumes that once a person becomes a case s/he is no longer at risk to become another case), since we have removed that person from the population at risk for subsequent months. That is a reasonable assumption for diseases like H1N1 influenza or coronary heart disease, for example. The disease is regarded as having occurred one time, even if different clinical manifestations and even recurrences occur at various times thereafter. One shortcoming of the above calculation, though, is that it ignores the fact that a person who becomes a case is no longer at risk for the rest of that month, either.

34 of 80

1/9/2007

Incidence and prevalence

34

Estimating population-time - method 2 - better

Total population-time over 9 months =

199.5 + 198.5 + 197.5 + 196 + 194.5 + 193.5 + 192.5 + 191 + 189

= 1,752 person-months

= 146 person-years

assuming that cases develop, on average, in the middle of the month

35 of 80

1/9/2007

Incidence and prevalence

35

Estimating population-time - method 3

Average size of the population at risk during the 9 months = 195.3 (1,758 / 9) or approximately: (200 + 188) /2 = 194

Population-time = 195.3 x 9 months or (approximately) 194 x 9 months

= 1,746 person-months

= 145.5 person-years

36 of 80

1/9/2007

Incidence and prevalence

36

Equivalent to - method 3

Take initial size of population at risk and reduce it for time the people were not at risk due to acquiring the disease:

200 - 12/2 = 194 (approximately)

Population-time = 194 x 9 months

= 1,746 person-months

1/9/2007

Incidence and prevalence

43

















O

6 / 200 = 0.03 = 3% = 30 / 1,000 in 5 months

And in month 5 one more new case occurred, so that at the end of 5 months the proportion of the population that was affected was 6 cases in 200 people, or 0.03. If we prefer not to use decimals, or at least would rather have fewer decimal digits, we can express this proportion as a percentage (3%), which is the same as “per 100”, or we can express the proportion as a number per 1,000, as 30 per 1,000. These expressions are simply alternate ways of expressing the same value, and it is largely a matter of personal preference (or sometimes convention) which variant to use.

It is important, though, that we indicate the location of the decimal point, so that if we wish to write 0.03 as 30, we had better write it as 30 per 1,000 or 30/1,000. It is also important that we indicate the time interval. After all, this proportion was different after each month passed, so if we don’t specify “5 months” then the 0.03 is quite ambiguous.

So when we report incidence, “time is of the essence. For an incidence rate, it is essential to state the unit of time. For an incidence proportion, it is essential to give the length of the time interval.

44 of 80

1/25/2011

Incidence and prevalence

44

Incidence proportion (“cumulative incidence”)

Number of new cases�5-month CI = –––––––––––––––––––� Population at risk

Incidence proportion estimates risk.

The measure we have just derived is called incidence proportion or cumulative incidence. It’s formula is simply the number of new cases (regardless of what happened to them after they developed the disease) divided by the size of the population at risk. Here, we do not have to divide by the time interval, since we are not defining a rate of occurrence per unit time, but rather a summation of what has happened over a time interval. As noted, we do need to specify the length of the time interval.

The term “risk” is typically used to refer to the probability that an (adverse) event will occur. The average risk for a member of a cohort during a specified period if time is conveniently estimated by the incidence proportion (IP) for that cohort, assuming that the outcome is known for all cohort members at the end of the period. Because the average risk and IP have the same numerical value, the terms are often used interchangeably.

Since the IP measures the accumulation of cases over time, it is directly related to the rate at which cases are occurring. If the incidence rate (IR) is constant and each cohort member can become a case only once (or we define a “case” as the first event for a person), then the IP is approximately equal to the IR multiplied by the length of the time period. The approximation is very close when the IP is small (e.g., less than 0.10). When IP is not small, then the proportion of the cohort still at risk declines noticeably as cases occur. If IR is constant, then ln(1 – IP) = –IR*t, or IP = 1 – exp(-IR*t).

45 of 80

1/25/2011

Incidence and prevalence

45

Incidence rate versus incidence proportion

Incidence rate measures how rapidly cases are occurring.
Incidence proportion is cumulative.
When care only about the “bottom line” (i.e., what has happened by the end of given period): incidence proportion (CI).

46 of 80

1/25/2011

Incidence and prevalence

46

Incidence rate versus incidence proportion

If risk period is long (e.g., cancer), we usually observe only a portion.
To compare results from studies with different length of follow-up, use incidence rate (IR)
If risk period is short, we usually observe all of it and can use incidence proportion.

This distinction suggests some ways for choosing between ID and CI. If the “risk period” – the period during which the population under study remains at risk – is long (e.g., adult cancers), then most studies will cover only a part of the risk period, and the length of the follow-up period will vary from study to study. It will therefore be much easier to compare incidence rates across different studies than it will be to compare incidence proportions, because the longer the follow-up period, the larger the incidence proportion (cumulative incidence) will tend to be.

In contrast, if the risk period is short (e.g., food poisoning after a contaminated meal), we are often more interested in comparing incidence proportions for groups of people according to the food they ate than in comparing their rates of disease per hour. Among other things, the rate will change during the time since exposure, so that an average will be a very inadequate summary. In fact, if we lengthen the follow-up time to include a longer period even though the number of new cases has become very small, the incidence rate may become tiny. The cumulative incidence, however, will change little.

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Incidence rate versus incidence proportion�(rare disease, IR = 0.005 / month)

1/25/2011

Incidence and prevalence

47

(see spreadsheet at epidemiolog.net/studymat/)

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Incidence rate versus incidence proportion� (common disease, IR = 0.1 / month)

2/7/2012

Incidence and prevalence

48

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5/20/2002

Incidence and prevalence

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Case fatality rate

“Case fatality rate” (but it’s really a proportion)

= proportion of cases who die� (in a specified time interval)

Like a “cumulative incidence of death” in cases�[ “incidence rate of death” in cases = “termination rate” = 1/(average survival time)]

Incidence rate and incidence proportion are the two principal concepts that epidemiologists use to describe and analyze the frequency of occurrence of a disease or other health-related condition or characteristic in a population. Much of epidemiology deals with deaths, though, since counts of deaths are often more accurate than counts of cases of a disease that no one is paying special attention to. The relation between incidence (events) and mortality (deaths) depends on how often and/or rapidly the disease is fatal. One measure of this relation is the proportion of cases who die in a given time, the case fatality rate (it’s really a case fatality proportion, but proportions are often called “rates” in epidemiology for sociological reasons undoubtedly).

Mathematically, the case fatality rate is completely analogous to a “cumulative incidence of death” where the population at risk consists of existing cases. There is also an analog to an “incidence rate of death” among cases, which is called the termination rate. Conveniently, the reciprocal of the termination rate is average survival time of an existing case.

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1/30/2007

Incidence and prevalence

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Mortality rate

Number of deaths�Mortality rate = ––––––––––––––––––––––––––––� Population at risk × Time interval

Number of deaths�Annual mortality rate = ––––––––––––––––––––––� Mid-year population (x 1 yr)

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6/6/2002

Incidence and prevalence

51

Mortality rate (more notes)

Number of deaths�Mortality rate = ––––––––––––––––––––––––––––� Population at risk × Time interval

Number of deaths�Annual mortality rate = ––––––––––––––––––� Mid-year population

A mortality rate is essentially an “incidence density” of death in some population. Thus, the numerator consists of deaths (i.e., events) and denominator consists of population-time. Since the most common type of mortality rate is an annual mortality rate, expressed per 1,000 person-years, the unit of time is often omitted. However, this is ambiguous, since any annual rate can also be expressed per month, per week, etc.

By convention, an estimate of the mid-year population is used as an estimate of the average population during the year. If the size of the population is not changing during the year, it makes no difference whether the population size from the beginning, middle, or end of the year is used – they will all be the same. If the population is steadily increasing or steadily decreasing, then the mid-year population will provide a better approximation to the average number of people at risk during the year than will other convenient choices.

Since multiplying the mid-year population by the interval length of one-year does not change its numerical value, the formula is written with just the mid-year population in the denominator. However, if the number of deaths is small one may want to compute an annual average over several years to obtain a more precise estimate. In that case the numerator will contain the total number of deaths occurring during the several-year period, and the denominator will contain an estimate of the entire population time, such as the mid-period population estimate multiplied by the number of years.

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5/20/2002

Incidence and prevalence

52

Mortality rates versus incidence rates

Mortality data are more generally available
Fatality reflects many factors, so mortality rates may not be a good surrogate of incidence rates
Death certificate cause of death not always accurate or useful

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O

What is the prevalence? (9 / 197)

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5/20/2002

Incidence and prevalence

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Fine points . . .

Who is “at risk”?

Endometrial cancer? Prostate cancer? Breast cancer?
Only women who have not had a hysterectomy?

“Could” develop the condition + “would” be counted.

Let’s consider several fine points.

First, who comprises the population at risk? What about a disease that occurs only in women or only in men? What is the population at risk for developing or having prostate cancer? Clearly, men. What about the population at risk for developing breast cancer? Women – and men, though the occurrence of breast cancer is so much less frequent in men that we would generally want to compute separate measures of breast cancer for women and men, rather than an average for both sexes combined.

What about endometrial cancer? Clearly women, but if they have had a hysterectomy, then they are not at risk of this disease. So ideally we would use in our denominator for incidence or prevalence only women with a uterus. Of course, we often don’t know how many women in a population have had a hysterectomy. Here, the third consideration we listed in the beginning – availability of data – may force us to include women in the denominator even if they have had a hysterectomy and are no longer biologically “at risk”. The treatment of women who have had a hysterectomy in the course of being treated for endometrial cancer depends on the study question and whether we can know this information.

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5/20/2002

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More fine points

Age?
Immunity?
Genetically susceptible?

What about differences in genetic susceptibility. After all, the differences we just discussed for women and men were certainly genetically based What do we do about people who due to other aspects of their genetic make-up are not in fact at risk to develop the disease? In principle we would want to omit them from the denominator. That is what we do, after all, when we omit women from the denominator for prostate cancer or when we calculate separate breast cancer incidences for women and men. But most of the time we do not have information on genetic susceptibility, either in terms of being “at risk” or “not at risk” or in terms of different levels of risk. So the “availability of data” consideration generally forces us to include all people whom we believe might be susceptible in the denominator.

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5/20/2002

Incidence and prevalence

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More fine points . . .

How do we measure time?

Are 10 people followed for 10 years the same as 100 people followed for 1 year?
Aging of the cohort? Secular changes?

Here is another fine point. How do we measure time when we construct an incidence rate? Are 10 people followed for 10 years always equivalent to 100 people followed for 1 year? Not necessarily.

It is possible that the level of risk changes during the time a person is being followed, for example, because the person is getting older or becoming increasingly susceptible. If we know that people with different lengths of follow-up are not equivalent, then we would not want to simply average their disease experience together. Instead, we would analyze the incidence separately for each duration of follow-up.

Also, if the incidence is changing over calendar time, then rather than estimate an overall incidence rate (essentially, an average), we might prefer to estimate an incidence rate for each part of the time period.

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9/22/2005, 9/8/2008

Incidence and prevalence

62

Fine points . . .

Importance of stating units and scaling unless they are clear from the context

e.g., 120 per 100,000 person-years = 10 per 100,000 person-months
Hazards from lack of clarity

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“You can never, never take anything for granted.”

Noel Hinners, vice president for flight systems at Lockheed Martin Astronautics in Denver, concerning the loss of the Martian Climate Orbiter due to the Lockheed Martin spacecraft team’s having reported measurements in English units whiles the orbiter’s navigation team at the Jet Propulsion Laboratory (JPL) in Pasadena, California assumed the measurements were in metric units.

1/30/2004

Incidence and prevalence

63

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5/20/2002

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Relation of incidence and prevalence

Prevalence depends on incidence
Higher incidence leads to higher prevalence if duration of cases does not change.
Limitation of the bathtub analogy – flow rate needs to be expressed relative to the size of the source
Introducing a new analogy . . .

In order for a case to exist, it must first occur. So in any population there must be some relation between incidence and prevalence. In general, the greater the incidence, the greater the prevalence. However, the prevalence also depends on the duration of the disease, since cures, deaths, and out-migration all remove existing cases from the population. In addition, existing cases may migrate into the population, so it is possible for prevalence to increase even without an increase in incidence.

A traditional diagram (I’ve used one for two decades, and Leon Gordis has one in his textbook) for illustrating the relation among incidence, prevalence, mortality, cure, and migration is a bathtub. However, the bathtub analogy is deficient in that the rate of flow at the faucet (representing incidence) is not proportional to the (unseen) size of the reservoir (representing the population at risk).

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9/23/2002

Incidence and prevalence

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5/20/2002

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Population at risk

Existing cases

Deaths, cures, etc.

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1/25/2011

Incidence and prevalence

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Incidence, prevalence, duration of hospitalization

Remote community of 101,000 people

One hospital, patient census = 1,000

Steady state

500 admissions per week

Prevalence = 1,000/101,000 = 9.9/1,000

IR = 500/100,000 = 5/1,000/week

Duration Prevalence / IR = 2 weeks

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1/25/2011

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Relation of incidence and prevalence

Under somewhat special conditions,� Prevalence odds = incidence × duration� Prevalence incidence × duration

(see spreadsheet at www.epidemiolog.net/studymat/)

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5/20/2002

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Standardization

When objective is comparability, need to adjust for different distributions of other determinants
Strategy:

Analyze within each subgroup (stratum)
Take a weighted average across strata
Use same weights for all populations

(See the Evolving Text on www.epidemiolog.net)

As mentioned last week, populations are diverse (different from each other) and heterogeneous (different within themselves). Yet we typically want to make comparisons across populations. Some of the differences between populations are of interest to us, but others are actually a distraction. For example, if we want to compare the death rates across several populations, differences in their age structure (which strongly influences death rates) could well interfere with the objective of our comparison. Developing countries, for example, tend to have much younger age distributions than countries that have been industrialized for many years. Because of their different age distributions, the death rate for a developing country may well be lower than that for an industrialized country, even if people of the same age have a lower death rate in the industrialized country.

Therefore, when there are differences between populations that may distort or cloud our comparison of interest, we often adjust that comparison to take account of the population differences that are not of interest at the moment. One method of doing this, widely used for presenting death rates, is called standardization. The strategy for standardization is basically straightforward: divide each population into subgroups defined by the factor(s) to be adjusted for. Estimate the death rate separately for each subgroup. Then take a weighted average across all of the subgroups. The key is to use the same set of weights for each population.

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8/17/2009

Incidence and prevalence

70

Familiar example of weighted averages

Liters of petrol per kilometer - differs for Interstate (0.050 LpK) and non-Interstate (0.100 LpK) driving.
To compare different cars, can:

Compare them for each type of driving separately (stratified analysis)
Average for each car, using one set of weights (e.g., 80% Interstate, 20% non-Interstate)

E.g. = 0.80 x 0.050 LpK + 0.20 x 0.100 LpK = 0.060 LpK

For the standardization to be straightforward, however, one does need to remember how weighted averages work. Here is a familiar example. Suppose you want to compare the fuel efficiency of two cars. Of course, any car will get better fuel efficiency in Interstate driving (e.g., 0.050 LpK) than other driving (e.g., 0.100 LpK). So if we want to compare fuel efficiency from information on kilometers driven and fuel consumed, we would make separate comparisons for Interstate and non-Interstate driving.

If we wanted to have a single number for each car, we could construct a weighted average of liters of petrol per kilometer driven (LpK) for each type of driving. [I used to use an example based on miles per gallon, but an EPID600 student pointed out to me that my algebra was incorrect, so I had to change to gallons per mile. Since we’re now a school of global public health, I thought I should switch to metric units. That is the reason the example uses liters (L) of petrol per kilometer (Km).]

So to summarize fuel efficiency by a single number for each vehicle, we construct a weighted average that combines fuel consumption in highway driving with fuel consumption in city driving.

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8/17/2009

Incidence and prevalence

71

Comparing a Suburu and a Mazda

Juan drives a Suburu 800 km on Interstate highways and 200 km on other roads. His car uses 0.050 LpK on Interstates and 0.100 LpK on other roads, for a total of 60 liters of petrol, an average of 0.060 LpK (60 L / 1000 km). His overall LpK can be expressed as a weighted average:

(800/1000) x 0.050 LpK + (200/1000) x 0.100 LpK

= 0.80 x 0.050 LpK + 0.20 x 0.100 LpK = 0.060 LpK

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8/17/2009

Incidence and prevalence

72

Comparing a Suburu and a Mazda

Shizu drives her Mazda on a different route, with only 200 km on Interstate and 800 km on other roads. She uses 0.045 lpk on Interstate highways and 0.080 LpK on non-Interstate. She uses a total of 73 liters, or 0.073 LpK. Her overall LpK can be expressed as a weighted average:

(200/1,000) x 0.045 LpK + (800/1,000) x 0.080 LpK

= 0.20 x 0.045 LpK + 0.80 x 0.080 LpK =0.073 LpK

Shizu’s overall fuel efficiency is also a weighted average, but in this case she drove a larger proportion of her trip on non-Interstate roads. Shizu drives her Mazda for 200 km on Interstate and 800 km on other roads. She uses 0.045 LpK in Interstate driving and 0.080 LpK on non-Interstate. So her total fuel consumption is 73 liters, or 0.073 LpK. In spite of the fact that her car was more efficient in Interstate driving and also in non-Interstate driving, Shizu used more fuel for the same trip.

To understand how this can be, let’s express Shizu’s overall LpK as a weighted average:

(200/1,000) x 0.045 LpK + (800/1,000) x 0.080 LpK

= 0.20 x 0.045 LpK + 0.80 x 0.080 LpK =0.073 LpK

This expression makes clear that although the numbers for LpK are smaller for Shizu’s Mazda, she is driving more of her distance at her lower fuel efficiency.

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8/17/2009

Incidence and prevalence

73

How can we compare their fuel efficiency?

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8/17/2009

Incidence and prevalence

74

Total fuel efficiency is not comparable because weights are different

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8/17/2009

Incidence and prevalence

75

By adopting a “standard” set of weights we can compare fairly

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8/17/2009

Incidence and prevalence

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Comparing a Suburu and a Mazda

Juan’s Suburu:

= 0.60 x 0.050 LpK + 0.40 x 0.100 LpK =0.070 LpK

Shizu’s Mazda:

= 0.60 x 0.045 LpK + 0.40 x 0.080 LpK =0.059 LpK

The choice of weights may often affect the results of the comparison.

This slide illustrates the computation of the standardized average liters per kilometer. It is clear that the choice of weights will affect the comparison, even if we use the same weights for both cars. The higher the weight for highway driving, the more the overall average will move in favor of Shizu’s Mazda. The higher the weight for city driving, the more the overall average will shift in favor of Juan’s Suburu.

Which weights should one use? As with other decisions about measures, the choice depends on our objective, our conceptual understanding of the phenomenon, and the availability of data. Convention also has a role. If we want to compare our results to those that others have published, using the same set of weights (often referred to as the “standard population”) as did the other studies improves the comparability across studies. If the data are sparse, we may want to avoid assigning large weights to imprecise estimates. This issue and related topics are covered in standard textbooks as well as in the Evolving Text.

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“I'm just glad it'll be Clark Gable who's falling on his face and not Gary Cooper.”

- Gary Cooper on his decision not to take the leading role in “Gone With The Wind”

5/20/2002

Incidence and prevalence

77

FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future

78 of 80

“A cookie store is a bad idea. Besides, the market research reports say America likes crispy cookies, not soft and chewy cookies like you make.”

- Response to Debbi Fields' idea of starting Mrs. Fields' Cookies.

5/20/2002

Incidence and prevalence

78

FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future

79 of 80

“Computers in the future may weigh no more than 1.5 tons.”

- Popular Mechanics, forecasting the relentless march of science, 1949

5/20/2002

Incidence and prevalence

79

FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future

80 of 80

“I think there is a world market for maybe five computers.”

-Thomas Watson, chairman of IBM, 1943

5/20/2002

Incidence and prevalence

80

FAMOUS LAST WORDS: quotations that demonstrate the value of humility in predicting the future