Rhame and Sudderth paper

AMERICAN

Journal of Epidemiology
Formerly AMERICAN JOURNAL OF HYGIENE
© 1981 by The Johns Hopkins University School of Hygiene and Public Health

JANUARY, 1981

NO. 1

Reviews and Commentary
INCIDENCE AND PREVALENCE AS USED IN THE ANALYSIS
OF THE OCCURRENCE OF NOSOCOMIAL INFECTIONS
FRANK S. RHAME1 AND WILLIAM D. SUDDERTH1

Two recent articles (1, 2) have helped
to clarify the proper usage of the terms
incidence and prevalence. However,
neither paper considered their use in
hospital epidemiology. In the analysis of
the occurrence of nosocomial infections
these terms are used somewhat differently. This paper will discuss the differences and present their rationale. We
will then present the mathematical interrelationship of the prevalence rate and
incidence rate of nosocomial infection.
Finally, we will discuss the practical difficulties and pitfalls in compiling these
rates and applying the interrelationship
formula.

number of infections acquired during a
given month by the number of patients
discharged (or admitted) during that
month,
/ = incidence rate of nosocomial
infections for month A

number of infections
acquired in month A
(1)
number of patients
discharged in month A
This parameter is called the infection
rate, incidence rate, or, less properly,
the incidence. Purists may object to this
usage of the word incidence on at least
three grounds. The most trivial objection
INCIDENCE RATE
is that the numerator and denominator
The occurrence of nosocomial infections do not have the same dimension. The
is most often computed by dividing the numerator is the number of infections;
the denominator the number of discharged
patients.
On these grounds, equation 1
1
Department of Medicine and Department of
should
be
called a ratio. The reason all
Laboratory Medicine and Pathology, University of
Minnesota Medical School; University of Minnesota infections are tallied, rather than infected
Hospitals and Clinics.
patients, is that two patients acquiring
Reprint requests to Dr. Rhame, Infection Control
Program, University of Minnesota Hospitals and one infection each is just as undesirable
Clinics, Box 421 Mayo, 420 Delaware Street SE, as one patient acquiring two infections.
Minneapolis, MN 55455.
Furthermore, if the number of infected
' Department of Statistics, University of Minnepatients
were used, an anomaly would
sota. Research supported by National Science Founarise. If a patient acquired infections in
dation Grant MCS 77-28424.
The authors thank Drs. Robert W. Haley, David H. two different months, he/she would be
Culver, and John V. Bennett for their helpful
counted twice. If both infections arose
comments.

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VOL. 113

RHAME AND SUDDERTH

The final objection to equation 1 arises
because, as used in community epidemiology, incidence rates have a time
unit in the denominator (3). For instance,
the current incidence rate of measles is
about 1.1 cases per 100,000 children per
month. In a steady-state hospital, the
magnitude of the nosocomial infection incidence rate, as computed by equation 1,
is independent of the survey interval.
The sense of this parameter is, however,
highly analogous to a true incidence rate.
The hospital situation is approached differently because of differences in the relative lengths of the survey interval and
the sojourn of the population-at-risk in
the at-risk status. In community epidemiology the survey interval is usually
shorter than the average length of time
people are susceptible to the illness. For
instance, people are at risk of developing
measles for about 10 years and survey
intervals of the incidence rate of measles
are usually one month or one year. The
incidence rate of measles per one month
is roughly half that per two months. But
how would one compute the incidence
rate of measles per century? per millennium? To be meaningful, all the people at risk should be included in the
denominator which would then become
the number of people having lived during
the interval. In a steady-state universe,
this parameter would also become time
independent. It would be computed in the
same way as is the incidence rate defined
in equation 1. In the hospital the average
length of stay is about eight days and
survey intervals range from one month
to one year.
PREVALENCE RATE

The prevalence rate is determined at a
single point in time. The number of both
active and cured nosocomial infections
that are or have been present in patients
hospitalized on a given day is divided by
the number of patients present at the
time of the survey,

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during the same month he/she would be
counted only once.
A more fundamental problem with the
statistic defined by equation 1 is that
the numerator and the denominator are
not drawn from the same population. A
patient who becomes infected in one
month may not be discharged until a subsequent month. Such a patient would be
counted in the numerator and denominator of different months. A more rigorous
way to present incidence data would be
to define a cohort (e.g., patients admitted
in June) and follow them for instances of
nosocomial infection. The numerator of
equation 1 would become "instances of infection among June-admitted patients."
However, this is not the most useful statistic because outbreaks of infection are
more likely to occur at a point in time
than within an admission cohort. The
rationale for choosing total discharges as
the denominator in equation 1 is less
secure. The argument against omitting
the denominator altogether (producing a
true incidence) is that its use adjusts for
fluctuations in patient admissions and
permits interhospital comparisons. Although substantial variation in the total
hospital census is uncommon, equation 1
is also used to produce ward- and servicespecific infection rates. In smaller units,
census variation is more substantial. The
interhospital comparison argument is
weak. Meaningful interhospital comparisons are difficult because of differences
in hospital population, surveillance
methods and definitions of infection. Alternative denominators include patientdays and average daily census (each can
be simply computed from the other using
the number of days in the survey interval). Use of either does adjust for variations in census and mitigates the following anomaly. For chronic care facilities
and services with long durations of hospitalization, equation 1 produces high
rateB of infection even when few infections are occurring.

RATES OF NOSOCOMIAL INFECTION

RELATIONSHIP BETWEEN PREVALENCE
RATE AND INCIDENCE RATE

Published surveys of the occurrence of
ndsocomial infection (4-18) have presented both the prevalence and incidence
rates (table 1). The incidence rate is more
readily conceptualized. When expressed
per 100 patients, it is slightly more (because of multiple infections in occasional
patients) than the percentage of patients
who acquire an infection during their
hospitalization. However, the prevalence
rate is often determined because it doesn't
require the sustained effort needed to produce incidence data. Either appears acceptable in meeting accreditation requirements for surveillance. Since both rates
are in use and the easier to obtain is the
less desirable, a formula expressing their
interrelationship would be useful.
The relationship between P and / is
LA

(2)

LN - INT
where LA is the average length of stay

of all patients, LN is the average length
of stay of patients who acquire one or
more nosocomial infections and INT is
the average interval between admission
and onset of the first nosocomial infection
for those patients who acquire one or more
nosocomial infections.
The derivation of equation 2 is presented in the appendix for two stochastic
models. For both models it is assumed
that infections occur independently so
that the chance of one patient becoming
infected is not dependent on whether or
not other patients get infected. To the
extent that epidemics and clusters of infection occur, this assumption is unjustified. However, since the bulk of nosocomial
infections are endemic, the assumed independence seems close to the truth. Furthermore, the analysis done in the appendix requires only such approximate independence or sufficiently small correlations
between the occurrence of infection in
different patients.
In the first model it is also assumed,
as a first approximation, that patients
never acquire more than one nosocomial
infection. In the second model, which
allows for multiple infections, it is assumed that, for patients who suffered at
least one infection, the probability of
each subsequent infection does not depend
on the number of prior infections. That
is, after acquiring a first infection, there
is a probability (q in the appendix) of acquiring a second infection. It is assumed
that patients acquiring a second infection
have the same probability of acquiring a
third, and so forth. This assumption is
unwarranted insofar as patients who have
contracted two infections are likely to be
more susceptible to future infection than
patients who have acquired one infection.
Nevertheless, in two of the three incidence studies presenting multiplicity data
(table 2), the frequencies do not deviate
greatly from what would be expected
under the assumption in question. The
deviation that is present is toward in-

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P = prevalence rate of
nosocomial infections
number of infections (active
and cured) having occurred
in patients hospitalized at the
time of the survey
number of patients present
at the time of the survey
Both active and cured infections are included because of the difficulty in deciding the point at which an infection becomes cured. Some surveys have presented
the number of infected patients in the
numerator and separate data on multiplicity of infection. In actual practice, the
"single point in time" is usually taken
to be the interval (generally several hours)
during which the survey team visits the
ward. Since very few patients acquire a
nosocomial infection during their first
few hours of hospitalization, this comes
very close to a true point prevalence.

(18)

Britt et al.

(9)

(16)
(17)

Adler et al.
Moody and Burke

Mulholland et al.

(12)
(13)
(14)
(15)

(11)

John

Prevalence surveys
Kislak et al.
Barrett et al.
Adler and Schulman
Edwards

(10)

(5)
(6)
(7)
(8)
(9)

(4)

(Reference no.)

McNamara et al.
Eickhoffet al.
Thoburn et al.
Gardner and Carles
Mulholland et al.
Wenzel et al.

Incidence surveys
Roy et al.

Authors

TABLE 1

Boston City
Boston City
Grady Memorial
Public Health
Service
Boston City
Latter Day
Saints
18 intermountain
U. Pennsylvania

Moncrief Army

Location

Boston, MA
Salt Lake City,
UT
Rocky Mountain
States
Philadelphia, PA

NY

Boston, MA
Boston, MA
Atlanta, GA
Staten Island,

Baltimore, MD
Boston, MA
Philadelphia, PA
Charlottes ville,
VA
Fort Jackson, SC

USA

Lexington, KY

Toronto, Ontario

Hospital

Hospital for
Sick Children
U. Kentucky
6 community
Johns Hopkins
Children's
U. Pennsylvania
U. Virginia

Name

'65-Aug. '65
'65-Dec. '66
'65-March '66
'70-Dec. '71
'71-Sept. '72
'72-Aug. '73

7.6

14.4

June '73

9.2

15.0

8.4

15.4
19.7
13.0

1.5

6.7

16.7

6.1
3.5
4.0
4.6

6.5

Infections per
100 patients

Oct. '72-Feb. '73

Jan. '70
Jan. '71

Feb. '67
Oct. '67
Feb. '68

Jan. '64

July '74-Dec. '75

June
July
Oct.
March
Oct.
Sept.

Jan. '59-Dec. '59

Survey date

Published hospital-wide surveys of the occurrence of all types of nosocomial infection

oxfordjournals.org at CDC Public Health Library & Information Center on February 23, 2011
X

tB

sa

H

0.016
0.118
0.161

{0.037}

{0.273}t
{0.333}
{0.062}

0.032
0.264
0.461
0.340

920
55
31
15
1

1,253
141
26
1

972
76
8
2
1
0.059
0.082
0.126

12,209

(8)

87,708

(*)

Incidence surveys

17,836

(4)

{0.059}

0.135
0.136

108
16
1

928

(12)

{0.308}

0.120
0.153

72
9
4

709

0.072

{0.026}

0.085

37
1

525

(18)

{0.083}

44
4

566

(17)

Prevalence surveys
(16)

• Unpublished data observed in a continuation of the study described by Eickhoff et al. (6). The infections occurred between July 1968 and June
1969.
t Frequencies in braces were computed from <10 infections.

Observed conditional frequencies
»1 infection
*2 infections given 1 infection
» 3 infections given 2 infections
s>4 infections given 3 infections
»5 infections given 4 infections

Total patients studied
No. of patients with
1 infection
2 infections
3 infections
4 infections
5 infections

TABLE 2

Multiplicity of infection; conditional frequencies of subsequent infections

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RHAME AND SUDDERTH

the prevalence rate compared to the incidence rate. Specifically, (LN - INT)
must be greater than LA. That is, the
length of time patients are hospitalized
after acquiring nosocomial infection must
be longer than the average length of
admission.
In the epidemiology of community acquired disease, the relationship of incidence and prevalence rates is described by
Prevalence Rate =
Incidence Rate x Duration of
Dlness of Active Infection.
Equation 2, rearranged, is similar
P = -^— • (LN - INT).
LA

P is the prevalence of both actively infected and cured patients; (LN - INT) is
the duration of that combined condition.
The "true" incidence rate (3) as used in
community epidemiology is analogous to
the term I/LA. For instance, in a hospital
with LA = eight days and a nosocomial
infection incidence rate of five infections
per 100 discharged patients, the "true" inIn the prevalence surveys (table 2), the cidence rate of nosocomial infections
assumption seems to be quite consistent would be 0.625 infections per 100 patient
with the data partly because very few days. When LA is known, data from surpatients were observed to acquire two in- veys using patient-day denominators may
fections and even fewer to acquire three. be converted to the incidence rate defined
In any case, the defects in equation 2 in equation 2.
caused by the assumption are minor.
APPLICATION WITHIN THE HOSPITAL
With an / of 0.05, which is typical, the
Both incidence and prevalence surveys
contributions of infections after the second
fail to identify inadequately documented
are quite small.
The only additional assumption is that or unrecognized infections. Beyond this,
the expected intervals between infections the two surveillance methods tend to obare identical. Since fourth infections are tain different biased samples of the infecrare enough to be negligible, the sub- tions which can be documented by a comstance of the assumption is that the av- plete, post discharge chart review. Prevaerage interval between the first and sec- lence surveys tend to miss infections with
ond infection is the same as that between a greater lag between onset and documenthe second and third infection. While we tation. A patient with fever occurring on
know of no data bearing on this point, the day before the survey who has a blood
culture drawn which does not turn posiwe also know of no better presumption.
Examination of equation 2 allows a tive until the day after the survey will not
more precise statement of the basis of the be recorded as infected. They are also
generally observed greater magnitude of biased toward patients who have longer

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creasing likelihood of infection as more
infections occur. Some of this increase in
the occurrence of multiple infections is
probably artifactual. Incidence surveillance is usually done by reviewing a subset of the charts of currently hospitalized
patients. Various strategies may be used
to choose the charts of patients with increased likelihood of having had a nosocomial infection. For instance, surveillance personnel may examine the charts
of patients who are febrile and/or have
had cultures obtained. This sample is
probably biased toward multiple infection
patients. Furthermore, subsequent infections are more likely to be identified by
surveillance personnel than first infections because the patient's chart must be
carefully examined in the course of determining and assessing the data bearing
on the first infection. In support of this, the
conditional frequencies are more nearly
similar for the survey of Roy et al. (4)
which involved a uniform chart review.
Otherwise, these models are very general.
Specifically, no presumptions about dayspecific infection rates are required.

RATES OF NOSOCOMIAL INFECTION

may not be exact because the sample of
patients present may be biased toward a
longer staying subgroup of patients acquiring a nosocomial infection. LN would
be artifactually higher; bias in INT in
such a sample is unknown. A more rigorous
INT and LN can be obtained by surveying a cohort of patients (say, those admitted
during several randomly selected days).
In the future, published surveys presenting prevalence data should be accompanied by the simply obtainable data
which will enable the conversion to incidence data. LN and INT should be determined by the methods described in the
previous paragraph. LA may be approximated from the average daily census and
the average daily admissions for the
month during which the survey was performed. Obtaining the average length of
stay for the patients present on the day of
the survey would probably not be a good
approximation of LA; this patient sample
is biased toward longer staying patients.
Note: Since the submission of this
manuscript, we have become aware of a
paper (21) describing the relationship
between incidence and prevalence.
REFERENCES

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than average durations of stay. Incidence
surveillance is usually carried out so that
infections can be identified soon after
they occur rather than in record rooms on
the charts of discharged patients. Generally, surveillance personnel review only
a selected portion of charts of currently
hospitalized patients. Patients are selected using clues (e.g., those patients
from whom a culture is obtained) which
yield a population with a high probability
of being infected. This method generally
detects those infections which tend to produce the clues which lead to chart review.
To the extent that prevalence and incidence surveys identify different biased
samples of all infected patients, equation
2 is invalid.
LA is readily available in most hospitals. It is usually computed by dividing
the average daily census by the average
daily admissions. Unfortunately, this
computation produces a systematic exaggeration of LA because most hospitals
generate an "average daily census" rather
than an "average instantaneous census."
In the former are included all patients
hospitalized during the midnight-to-midnight 24-hour period, even those discharged just after the day begins or admitted just before midnight. In a hospital
with a true LA of eight days, where all
admissions and discharges occur at noon
and the average daily census includes all
patients hospitalized from midnight, the
computed LA would be nine days (12.5
per cent high). This problem may be partially compensated for by including all
the patients in the prevalence who are on
the ward at any time of the day of the
survey.
The average length of stay of patients
acquiring at least one nosocomial infection (LN) and their interval from admission to onset of infection QNT) may be
closely approximated during a prevalence
survey by tallying INT and attaching to
the charts of infected patients a form to
be completed with the discharge date and
mailed to survey personnel. This method

1. Elandt-Johnson RC. Definition of rates: some remarks on their use and misuse. Am J Epidemiol
1975;102:267-71.
2. Friedman GD. Medical usage and abusage:
"prevalence" and "incidence." Ann Intern Med
1976;84:502-4.
3. MacMahon B, Pugh TF. Epidemiology. Boston:
Little Brown and Co, 1970.
4. Roy TE, McDonald S, Patrick ML, et al. A survey of hospital infection in a pediatric hospital.
Part I. Description of hospital, organization of
survey, population studied and some general
findingB. Can Med Assoc J 1962;87:531-8.
5. McNamara MJ, Hill MC, Balows A, et al. A
study of the bacteriologic patterns of hospital
infections. Ann Intern Med 1967;66:480-8.
6. Eickhoff TC, Brachman PS, Bennett JV, et al.
Surveillance of nosocomial infections in community hospitals. I. Surveillance methods, effectiveness and initial results. J Infect Dis 1969;
120:305-16.
7. Thoburn R, Fekety FR Jr, Cluff LE, et al. Infections acquired by hospitalized patients: an
analysis of the overall problem. Arch Intern
Med 1968:121:1-10.

8

RHAME AND SUDDERTH
15. Edwards LD. Infections and use of antimicrobials in an 800-bed hospital. Public Health Rep
1969;84:461-7.
16. Adler JL, Burke JP, Finland M. Infection and
antibiotic use at Boston City Hospital, January,
1970. Arch Intern Med 1971;127:460-5.
17. Moody ML, Burke JP. Infections and antibiotic
use in a large private hospital, January, 1971.
Arch Intern Med 1972;130:261-6.
18. Britt MR, Burke JP, Nordquist AG, et al. Infection control in small hospitals. Prevalence surveys in 18 institutions. JAMA 1976^236:1700-3.
19. Feller W. Introduction to probability theory
and its applications. Volume 2. New York: John
Wiley & Sons, 1971.
20. Chung KL. A course in probability theory. New
York: Harcourt, Brace and World, 1968.
21. Freeman J, Hutchison GB. Prevalence, incidence and duration. Am J Epidemiol 1980;
112:707-23.

APPENDIX: TWO STOCHASTIC MODELS FOR NOSOCOMIAL INFECTIONS IN A LARGE HOSPITAL

The epidemiologic assumptions underlying these models and the definitions of P and
/ have been discussed in the main text. The simpler model is based on the presumption
that patients do not acquire more than one nosocomial infection.
Let Xu X2, . . . Xn, . . . be random variables corresponding to the time spent in the
hospital by successive occupants of a single hospital bed. The XK'B are assumed to be
independent and have the same distribution.
Let i = the probability that the nth patient in the bed contracts an infection. (3)
i is assumed to be the same for all patients. Let ZJ? be a variable corresponding to the
time spent in the hospital by the rath patient prior to the occurrence of any infection
and let Z\ be the time spent by the patient after an infection occurs. Z\ = 0 if patient n
is never infected.
Then Xn = Z° + Zl
(4)
Now take FJ} to be the event that therathpatient does not become infected and Fl to
be the event that one infection is acquired. By equation 4, the following relationship
between expected lengths of stay may be expressed
E(Xn \Fi) = E(Z°\Fl) + E(Zl \Fl).
(5)
Next suppose that the process corresponding to a single bed is examined after it has
been in operation for a large time t, and define
p = probability that the patient in the bed at time t has been infected.
Then, under very mild assumptions on the distribution of X which doubtless hold in
practice, the renewal theorem (see reference 19, Theorems XI.1.1 and XI.1.2 and
Example XI. 8a) implies that
expected time in hospital for any patient after being infected
expected time in hospital for any patient

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8. Gardner P, Carles DG. Infections acquired in a
pediatric hospital. J Pediatr 1972;81:1205-10.
9. Mulholland SG, Creed J, Dieraug LA, et al.
Analysis and significance of nosocomial infection rates. Ann Surg 1974;180:827-30.
10. Wenzel RP, Ostennan CA, Hunting KJ, et al.
Hospital-acquired infections. I. Surveillance at
a university hospital. Am J Epidemiol 1976;
103:251-60.
11. John JF Jr. Nosocomial infection rates at a general Army hospital. Am J Surg 1977;134:381-4.
12. Kislak JW, Eickhoff TC, Finland M. Hospitalacquired infections and antibiotic usage in the
Boston City Hospital—January, 1964. N Engl J
Med 1964^271:834-5.
13. Barrett FF, Casey JI, Finland M. Infections and
antibiotic use among patients at Boston City
Hospital, February, 1967. N Engl J Med 1968;
278:5-8.
14. Adler JL, Schulman JA. Nosocomial infection
and antibiotic usage at Grady Memorial Hospital: a prevalence survey. South Med J 1970;
63:102-5.

RATES OF NOSOCOMIAL INFECTION

(6

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Solving equation 5 for E(Zi \F£), substituting in equation 6, and rearranging yields
EX
l=P
(7
' E(X.\Fi) Now assume that the hospital contains a large number of beds. By definition, P calculated at time t is just the proportion of patients in the hospital who have been
infected. Assume also that the random variables associated with different beds are
independent or at least have small correlation. Then, by the law of large numbers,

P*p.
(8)
It is also true that if / is the incidence rate for a month beginning at time t, then
/ == i.
(9)
To verify equation 9, calculate as follows:
number of infections in the given month
r _
number of patients admitted in the given month
____
expected number of infections
expected number of patients admitted
length of the month a
EYm
'*
length of the month _
EX,
*
(10)

EYm

where Y\ is the time of dismissal of the first infected patient and where Ym is the time
from the dismissal of the (m - l)st infected patient until the time of dismissal of the
mth infected patient. The successive lines in equation 10 are by definition, by the law
of large numbers, by the renewal theorem, and obvious (Q is the number of beds in the
hospital), respectively. If S is the number of patients up to and including the first to be
infected, then S is a geometric random variable with expectation i~'. Y! can be written
in the form

y, = I xn.
n - 1

By an equation due to Wald (see reference 20, Theorem 5.5.3),
EY{ = (ES) (EX,) = »-' EXX.
(11)
Because the Xn's and the Y"m's all have the same probability distribution, equation 11
may be generalized to

10

RHAME AND SUDDERTH

By equations 10 and 12, equation 9 is established. By the law of large numbers again,
the expected values in equation 7 are approximately equal to the respective average
values. Thus by equations 7, 8, and 9
/-P.

LN -INT

Xn =Zi

+ Zl+Z*+

- ••

where Zf} is defined as before, and, for k 3= 1, ZJ is the time spent by the nth patient
from the onset of the Jfeth infection until the onset of the k + 1st or dismissal from the
hospital. ZJj is set equal to zero if there is no &th infection. Take FJ to be the event that
the nth patient develops exactly k infections. Generalizing equation 5, the relationship
between expected lengths of stay, given that FJ occurs, may be expressed
E(Xn\K) =E(ZS\FS) +E(Zi\F*) + • • • +E(Z*\F*).
Now let ik = probability that any patient contracts exactly k infections, and pt =
probability that the patient in bed at time t has contracted k infections. Again, by the
renewal theorem, for t sufficiently large,
expected time in hospital for any patient
__ during which the patient has exactly k infections
expected time in hospital for any patient

(13)

EXn

Equation 13 may be simplified by the assumptions discussed in the text of the article.
The probability q of developing one subsequent infection, given that at least one infection has occurred, is assumed to be independent of the number of prior infections. This
assumption gives a geometric distribution for the ik's except possibly at the first
term to.
ifc = g*- 1 i,forft = 1,2,....

(14)

It is also assumed that the expected time to a subsequent infection after a first or subsequent infection is a constant C regardless of the number of prior infections
E(Z{ \F*) = C for all n and 1 =£/ =£*.

(15)

Let AH be the event that the nth patient has at least one infection and let
Tn=Zl+Z*+

•••

be the total length of stay after the first infection occurs. Then
Xn = Z% + Tn and, consequently,
E(Xn\AJ = E(Z°\An) + E(Tn\An).

(16)

(17)

Using equations 14, 15, and 16, one can calculate the expected value of Tn given An.
x

E(Tn\AH)=

I
k - l

E(.T,\n)P(F*B\An)

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In the second, more complex model, patients may acquire any number of infections.
Corresponding to equation 4, there is now the equality

RATES OF NOSOCOMIAL INFECTION

11

30

= X (hC) (qk " » (1 - q))
k = 1

c
1-9
Using equation 13 for k 3=1 and equations 14, 15, and 18

(18)

(EZJ (1 iJS(Tn\An)

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