lecture 7 case-control june 20

8/12/2019 Lecture 7 Case-Control June 20

1/71

Case-Control Studies


2/71

Case Control Studies

Readings

Fletcher, chapter 10

Walker, chapter 6 [Case-Control Studies] from

Observation and Inference, 1991 [course pack]


3/71

Objectives

Students will be able to:

1. Define the term case-control study

2. Explain the relationship between case-controland cohort studies

3. Understand the difference between

cumulative incidence and incidence density designs

Case-Control Studies - Slide 1


4/71

Objectives

4. Calculate parameters which may be validly obtainedfrom case-control studies, namely:

a. Odds parameters:

- odds of exposure in cases

- odds of exposure in controls

- odds ratio

b. Risk parameters:

- approximation of relative risk

- attributable fraction

c. Incidence rate parameters:

- incidence rate ratio

- attributable fraction among the exposed

- attributable fraction for the population



5/71

Objectives

5. Indicate situations in which case-control studies

permit estimation of rate differences between

exposure groups

6. Highlight advantages and disadvantages ofcase-control studies, including key biases

7. List possible sources of controls in

case-control studies

8. Identify biases which may result from

different types of control selection

Case-Control Studies- Slide 3


6/71

Case-Control Studies

Fletcher, p. 213:

Patients who have the disease and a group of

otherwise similar people who do not have the

disease are selected. The researchers then look

backward in time to determine the frequency of

exposure in the two groups.

In other words, a study population is first assembled

based on a determination as to whether subjects

have or have not developed an outcome of interest.

Subjects (or person-time) are then classified as to

whether an exposure of interest took place.

Data on other variables (e.g. potential confounders)

is also obtained.

Case Control Studies- Slide 4


7/71

Walker, 1991:

Case-control studies constitute the

major advance in epidemiologic methods

of our time

Classic example:

Doll & Hill, relationship between lung cancer

and cigarette smoking (1950)



8/71

Advantages

Useful for study of conditions that are rareand/or characterized by a long latency

between exposure(s) and outcomes of interest.

May be useful in evaluating the impact of

multiple types of exposure.

Disadvantages

May be particularly vulnerable to biases arising from

selection of subjects (most often of the control group),

and measurement (estimation) of exposure



9/71

In case-control studies, data about exposure status is

calculated after first determining outcome status.

However, subjects may be recruited prospectively

(concurrently), e.g.:

- All persons aged 30-50 who are diagnosed withhypertension on the island of Montreal during 2005,

within 2 weeks of diagnosis.

- Controls recruited among persons of the same age

who are newly diagnosed with appendicitis inMontreal during the same time period.



10/71

Often, outcome status is already available for all subjects

(historical) at the time of initiation, e.g.:

- During 2005, a researcher identifies all women

aged 40-50 who were diagnosed with breast cancer

on the island of Montreal in 2004.

- In 2005, she recruits a control group among

women of the same age who had negative

screening mammograms in Montreal in 2004.

Case Control Studies - Slide 8


11/71

Note that the terms

prospective and retrospective

are not very useful

with respect to case-control studies,

since data about exposure statusis always retrospective (by definition).

Case-Control Studies - Slide9


12/71

Cohort and Case-Control Studies

Every case control study corresponds to an underlying cohort study,

which is (ordinarily) hypothetical.

Example (from Doll & Hill, 1950):_____________________________________________________

Women diagnosed with lung cancer vs other diseases

at 20 London hospitals

Smokers Non-Smokers Total

Lung cancer cases 41 19 60No lung cancer (controls) 28 32 60Total 69 51 120

_________________________________________________________

Crudeodds ratio = odds of exposure in cases/odds of exposure in controls

= (a/b)/(c/d)

= ad/bc = (41x32) / (19x28) = 2.5



13/71

In the corresponding cohort study,

women from the same geographic area

would be recruited and classified as to

smoking status, then followed for the

development vs non-development of lung cancer.



14/71


Assuming all cases of lung cancer during the period of interest

were detected,

one possible 2x2 tablewould be


Lung cancer 41 19 60

No lung cancer (controls) 859 981 1,840

Total 900 1000 1,900OR = 2.5

but it could also be:

Smokers Non-Smokers TotalLung cancer 41 19 60

No lung cancer (controls) 70 81 151

Total 111 100 211OR = 2.5


15/71

The cases diagnosed and included, and the

controls sampled, relate to the exposure experienceof an underlying source population.

In each scenario, the estimated odds of cigarette

smoking among cases are 2.5 times thoseamong controls.

In each scenario, all cases of lung cancer were

included. The size of the source population

(and hence the number of non-cases) was varied.



16/71

Cumulative incidence case-control studies

Goal is toderive estimate of relative risks

(relative cumulative incidences)

of outcomes among

exposed vs. unexposed

Design:

- Cases are ascertained during a definedobservation period

- Controls are persons who did not become casesduring the period of observation.

- The underlying cohort is a fixed one(not open or dynamic).



17/71

Doll and Hill, 1950

Assume that the source population was as follows:

900 smokers & 1000 non smokers - followed 5 yearsThen the 2x2 table would be:


Cancer + 41 19 60

Cancer - 859 981 1,840

Total 900 1,000 2,000

________________________________________________


Risk of cancer in smokers: 41/900 = 0.046

Risk of cancer in non smokers: 19/1000 = 0.019

Risk ratio: 0.046/0.019 = 2.4

Odds of smoking in women with cancer: 41/19 = 2.2

Odds of smoking in women without cancer: 859/981 = 0.88

Odds ratio = 2.5


18/71

In the corresponding case control study we take 100% of cases, but

sample the controls (60/1840 or 3.3% of all potential controls - those

who happened to be admitted to hospital for some other reason).

Hence the new table is:

Smokers Non smokers Total

Cancer + 100% x 41 = 41 100% x 19 = 19 60

Cancer - 3.3% x 859 = 28 3.3% x 981 = 32 60

Total 69 51 120_________________________________________________________


Risk of cancer in smokers: 41/69 = 0.59 INVALID

Risk of cancer in non smokers: 19/51 = 0.37 INVALID

The risk ratio from this 2x2 table is also invalid

Odds of smoking among cases: 41/19 = 2.2 (as before)

Odds of smoking among controls: 28/32 = 0.88 (as before)

Odds ratio: 2.2/0.88 = 2.5 (as before)


19/71

General Form: Cumulative incidence case-control studies

exposure + exposure -

outcome + a b | total cases

outcome - c d | total controls___________ _____________ |

total exposed total unexposed | total subjects

Odds of exposure in cases = a/b

Odds of exposure in controls = c/d

Odds ratio = odds of exposure in cases = a/b = ad______________________ ___ __odds of exposure in controls c/d bc

but:

Odds of disease among exposed = a/c

Odds of disease among unexposed = b/d

Odds ratio = odds of disease among exposed = a/c = ad___________________________ ___ __odds of disease among unexposed b/d bc



20/71

Risk parameter estimationin cumulative incidence case-control studies:

Recall that relative risk = risk of disease in exposed______________________risk of disease in unexposed

From our 2x2 table, this is: a/(a+c) = a(b+d)_______ ______

b/(b+d) b(a+c)

If the disease is rare,

then a


21/71

In a case-control study, it is then possible to estimate

the attributable risk (fraction) among the exposed,

even if the risk for the population is unknown.

In a cohort study, the attributable risk fraction is:

Rexp

- Runexp__________Rexp

= (Rexp/Runexp) - (Runexp/Runexp)_______________________Rexp/Runexp

= RR-1_____RR

In a case-control study, this is estimated by (OR-1)/OR



22/71

Hence, from Doll and Hill (1950),

the estimated fraction of

lung cancer among female smokers

which is attributable to smoking is:

2.5 -1 = 0.6 or 60%______

2.5



23/71

Incidence Density Case-Control Studies

The incidence density case-control study involves theimplicit comparison of the person-timeexperience

of cases and controls with respect to the exposure(s)

of interest.

The absolute quantity of person-time sampled - and

hence the sampling fraction - is unknown. This is

analogous to the situation with respect to persons in a

cumulative incidence case-control study.



24/71

Hence the underlying (hypothetical) cohort is an openor dynamic one.

Persons considered controls at one point in timemay then become cases; they can then appear twicein the 2x2 table.

For this cohort, the general form of the 2x2 table is:


exposure + exposure -outcome + a bperson-time Pe Po

Where Pe = person-time among exposed

Po = person-time among unexposed

IRe = a/Pe and IRo= b/Po

IRR = aPo____

bPe


25/71

Suppose that all cases are counted, but the

controls are sampled with respect to person-time,

with sampling fraction f generating the incidencedensity case-control study.


Then the 2x2 table is:

exposure + exposure -outcome + a b

outcome - c = fPe d = fPo

Then OR = ad = afPo

= aPo___ _____ ____

bc bfPe bPe

which is equivalent to the IRR above.


26/71

Note that this formulation does not involve any

assumptions about disease rarity.

It requires that the likelihood of being sampled from the

source population of person-timevaries as

a proportion of the person-time potentially contributed

by each individual.

For example:

A potential control subject who was absent from

the geographic area of interest during most of the

accrual period should have less chance of being selected

than a potential subject who was present throughout.

As with the cumulative incidence design, validity hinges

on the assumption that f (the sampling fraction)

does not vary with exposurestatus.



27/71

An example of an incidence density case-control study:

A researcher wishes to evaluate the associationbetween the use of nonsteroidal anti-inflammatory

drugs (NSAIDS) and ventricular tachycardia (VT)

In an open cohort study lasting 2 years,

subjects are recruited and classified as toexposure status (NSAID use), then followed for

development of VT

In principle, it is possible to document periods

of exposure and non-exposure for individuals,

e.g. months on/off medication, as long as

exposure is somehow reassessed



28/71

Then for the cohort,

incidence rates and an incidence rate ratio can be calculated for

the exposed vs unexposed person-time experience, e.g.

NSAID No NSAID Total

VT, cases 80 40 120

Person-years 800 1200 2000

Incidence 0.1/p-y 0.033/p-y 0.06/p-y

The estimated incidence rate ratio is:

80/800_______

40/1200

= 3

So, assuming no confounding, we estimate that the

incidence of ventricular tachycardia among NSAID users

is 3 times that among non-users



29/71

Suppose we instead devise a case-control study.

Here, cases will be defined by a first diagnosisof VT at Montreal hospitals, and

controls will be recruited among persons whovisit the eye clinics of the same hospitals:

both over a 2-year accrual period.

They will be compared with respect to use of

NSAIDS within the last 24 hours prior to presentation.

If sampling is done correctly (e.g. the probability

of selection is unrelated to NSAID use) thenthe controls should represent theperson-time experience of the source population



30/71

If a possible control spent half the accrual period

on NSAIDS, and half off, he has a 50% chance

of contributing to the exposed group and a50% chance of contributing to the unexposed group

This individual will contribute one or the other,

depending on the date of the visit chosen as control;

but in a larger group of people,

the control days sampled will reflect the proportion

of exposed person-time

A person can be a control early in the accrual period

and a case later

In principle, a single person can also be sampled

repeatedly as a control if the time window for

exposure definition is short (more complicated in

terms of analysis)



31/71

Suppose that the case-control study includes all cases which

would have been detected with the open cohort design.

Two controls are recruited per case. This (unbeknownstto the researchers) corresponds to a sampling fraction

for controls of 0.12 person-day sampled per person-year

of follow-up that would have occurred in the open cohort.


Then the 2x2 table is:

NSAID No NSAID Total

VT, cases 80 40 120

No VT(controls) 800*0.12 1200*0.12 2000*0.12

= 96 = 144 = 240_____________________________________________

Total 176 184 360

OR = (80x144)/(40x96) = 3.0 same as earlier IRR


32/71

Another example of an incidence density design:

Bronchodilators are used for the treatment of asthma

There is concern that overuse may be associated with

an increased risk of adverse events, including death

Side effects can include arrhythmias, which may lead

to sudden death

Suissa et al conducted a case-control study using

the Saskatchewan health insurance database

They identified 30 persons prescribed anti-asthma

medications who died of cardiovascular events,

rather than of asthma; the date of death was

termed the index date



33/71

4080 control dayswere then sampled randomly

from the 574,103 person-months of follow-upfor the entire asthmatic group; each such day

was also an index date

Cases and controls were then compared as to

use of theophylline and beta-agonists during the

3 months preceding the index date

These were the main exposures of concern



34/71

Questions for discussion:

Why do you think the researchers chose

this study design?

What would have been the corresponding

cohort study?



35/71

With respect to the relationship between theophylline use and

sudden cardiac death, the authors found the following:

Theophylline in last 3 months

Yes No | Total

Cardiac Death Yes 17 13 | 30

No 956 3124 | 4080

Note that numbers in table refer toperson-days (not to persons)

OR (crude) = ad = 17 x 3124 = 4.3 (2.1 - 8.8)__ ________bc 13 x 956

IRR (crude) = 4.3 (2.1 - 8.8)



36/71

The odds of recent theophylline use among persons

aged 5-54 years prescribed anti-asthma drugswho died of cardiovascular events were

4.3 times those among other persons in the same age

range who were also prescribed anti-asthma drugs,

but did not die.

Asthmatics aged 5-54 who are prescribed theophylline

have an estimated 4.3 fold increase in incidence of

fatal cardiovascular events, compared with

asthmatics who are not prescribed theophylline.



37/71

As with the cumulative incidence design, an attributable ratefraction can be estimated for exposed persons:

It is: Ie-Io, where Ie= incidence among exposed and____

Ie Io= incidence among the unexposed

= IRR - 1 = OR - 1______ _____

IRR OR

For the Saskatchewan study, the estimated attributable

rate fraction among asthmatics who were prescribed

theophylline is:4.3 - 1 = 0.77______4.3

Among asthmatics aged 5-54 prescribed theophylline,

an estimated 77% of fatal cardiovascular events

were related to its prescription.



38/71

It is also possible to estimate the attributable rate fraction

for the entire population (PAR%)

In a cohort study, this is simply

It- Io, where It= incidence among the total population_____It Io= incidence among the unexposed

For the corresponding incidence density case-control study,the population attributable ratefraction is

IRR - 1 x proportion of cases who were exposed,____IRR

estimated as OR - 1 x a_____ ____OR a+b

Similar parameters involving riskcan be generated for

the cumulative incidence design


C C t l St di Slid 37


39/71

For the Saskatchewan study, recall the 2 x 2 table

Theophylline in last 3 months

Yes No | Total

Cardiac death Yes 17 13 | 30

No 956 3124 | 4080

OR = 4.3

Pexp |case = 17/30 = 0.57

then PAR fraction = OR -1 x Pexp |case_____OR

= 4.3 - 1 x 0.57 = 0.44______

4.3

Among Saskatchewan asthmatics aged 5-54, an estimated

44% of cardiovascular deaths relate to theophylline prescriptions.



40/71

Attributable rates (rate difference)

The absolute rate difference (i.e., the absolute

rate of disease attributable to exposure) is Ie- Io

Data from a standard case-control study alone

cannot validly be used to estimate

absolute rates of disease.

Even if case ascertainment is complete,

the controls represent an unknown and

arbitrary fraction of the true person-time at risk.

Hence the rate difference cannot be estimated.



41/71

However, incidence rates can be estimated if there is

additional knowledge about the amount of person-time at risk

Exposure

(+) (-)

Disease (+) a b

Disease (-) c = f x x Pt d = f x (1- ) x Pt

Then Ie = a = a_____ ___________x Pt [c/(c+d)] x Pt

Then Io = b = b_________ ___________(1- ) x P

t

[d/(c+d)] x Pt

and the rate difference is Ie-Io

where = proportion of person-time which is exposed



42/71

Example:

In this nested case-control study,

the researchers knew that in the source cohort

(Saskatchewan asthmatics aged 5-54), there were

47,842 person-years at risk during the study period

The 2x2 table was:Theophylline in last 3 months

Yes No | Total

Cardiac death Yes 17 13 | 30

No 956 3124 | 4080



43/71

Then the estimated incidence of cardiac death in asthmatics

prescribed theophylline (Ie) is:

a = 17 = 0.0015 per person-year___________ ________________[c/(c+d)] x Pt 956/4080 x 47,842

And in asthmatics who were not prescribed theophylline theestimated incidence (I

o) is:

b = 13 = 0.00035 per person-year___________ _________________[d/(c+d)] x Pt 3124/4080 x 47,842

The estimated rate difference is therefore

0.0015-0.00035 = 0.00115 per person-year.

Note that the IRR computed as Ie/Io remains 4.3


Case Control Studies Slide 42


44/71

Ieand Io may also be estimated if It is known for the source population

Recall that It = (Iex ) + [Iox (1- )]

But Ie = Iox OR

Then It = Io[(OR x ) + (1- )]

So Io = It = It______________ ________________________

(OR x ) + (1- ) {OR x [c/(c+d)]} + [d/(c+d)]

Then use Ie = Iox OR

Then RD = Ie- Ioas usual [= Io(OR-1)]




45/71

Example:

The total incidence (It) of cardiovascular death

in the Saskatchewan cohort was

30 deaths/47,842 person-years

= 0.00063 per person-year.

Then Io= 0.00063 = 0.00035___________________________

[4.3 x (956/4080)] + (3124/4080)

and Ie= 0.00036 x 4.3 = 0.0015

RD = 0.0015 - 0.00035 = 0.00115


l di l d


46/71

Additional points

Corresponding estimates of attributable risks and

risk differences can be made for cumulative incidence

case-control studies, if the corresponding additional data

is available

Estimates of absolute risks/incidence rates and

risk/rate differences can be made only if thetotal amount of persons/person-time at risk is known,

or at least one absolute risk/incidence rate is known

(i.e. for the total population, the exposed, or

the unexposed)

Nested case-control studies are a special type of study

where cases and controls are explicitly drawn from

a defined larger cohort (as in the Saskatchewan

asthma study)




47/71


Case-Control Studies: Strengths and Limitations

Advantages of case-control studies:

Efficiency - much less expensive/intensivethan cohort studies.

Very useful for outcomes that are rare

or occur after a long latency period.

Most outcomes are relatively rare overshort-term follow-up.

Permit evaluation of multiple exposures.

Can rapidly accrue person-time experience.

Avoid losses to follow-up inherent in cohort studies.



48/71

Disadvantages

Not useful/efficient for very rare exposures(may not be present in either cases or controls).

Cannot directly compute incidence rates.

Cannot usually evaluate more than one outcome.

Temporality may be lost or distorted.

Potential for considerable bias, i.e. loss of validity.

Bias relates to:

- Measurement of exposure status

- Selection of subjects (usually controls)




49/71

With respect to measurement,

exposure ascertainment must be consistent

for cases and controls.

There may be potential for misclassification of

exposure in relation to disease status




50/71

Example 1

Differential recall of exposures among casesvs controls

e.g. medication use and congenital malformations

- particularly if mothers attuned to

study hypothesis.

If cases more likely to recall exposure,

results will be biased toward a

positive association between exposure and outcome.

The more objective the source of exposure data,

the better.


Case-Control Studies Slide 49


51/71

Example 2

Different sources of information about exposure

e.g. family members asked about

alcohol consumption of persons

who died of gastric cancer,

vs Direct questioning of control subjects.

If family members tend to underestimate cases

alcohol consumption, results will be biased

against finding a positive association between

alcohol and gastric cancer.


C C l S di Slid 50


52/71

Example 3

Exposure status changes as a consequence ofthe outcome

e.g. patients with symptoms of lung cancer

stop smoking

If patients with newly diagnosed lung cancer arecompared to controls with respect to current

or recent smoking, results may be biased, i.e.,

the association between smoking and lung cancer

will be underestimated.

Data collection must reflect relevant person-time

experience and temporality of exposure and outcome.




53/71

Association may also be missed

if the exposure of interest is poorly documented

(an example of non-differential misclassification)

Example: mesothelioma

It can be caused by brief, intense exposures

to asbestos, with a very long latency period(>30 years).

In a case control study,

both cases and controls may recall such exposures

very poorly, thereby leading to an underestimate

of the true association.




54/71

Control selection in case-control studies

Recall that the validity of case-control studies

hinges on the assumption that the

sampling fraction for cases (which may be 100%)

and that for controls (usually unknown)

does not vary by exposure status.

In other words, controls should represent the

source population from which the cases arose,

with respect to exposure experience.




55/71

Example 1

A researcher wishes to test the hypothesis thatuse of nonsteroidal anti-inflammatory drugs (NSAIDs)

is associated with development of gastric cancer.

She plans a case-control study comparing gastric cancer

patients (cases) with patients seen at the same hospitalfor peptic ulcer disease (controls).

- NSAID use is a known risk factor for ulcers.

What will be the effect on her findings:

a) if NSAID use is truly a risk factor for gastric cancer?

b) if NSAID use is truly unassociated with gastric cancer?



56/71

Hence, controls should not differ systematicallyfrom the population of interest

with respect to exposure experience.

Sometimes the bias may be less obvious,

i.e. unrelated to explicit criteria for

control selection.




57/71

Example 2

A researcher wishes to evaluate the association between

cellular phone use and brain tumoursusing a case-control design.

Cases are recruited from the brain tumour clinic at theRoyal General Hospital, a neurosurgery referral centre.

Controls are recruited from the family medicine clinicat the same hospital. This clinic primarily serves alow-income population from the area adjacent tothe hospital.

This control group is less likely than the general population

to own cellular phones.

Result:

The study will be biased toward detecting anassociation between brain tumours and cell phone use.




58/71

Controls should be at risk for developing the outcome of interest

- otherwise they do not contribute useful data to the study

(inefficient)

- inclusion of individuals not at risk may

also distort the results if the reason they are not at risk

relates to the exposure under study. This may not be obvious.

Example:

Sleep apnea (exposure) and risk of traffic accidents (outcome)

Cases: Drivers involved in car accidents.

Including non-drivers in the control group would be

a waste of time

- it could bias the results if

persons with severe apnea have chosen not to drive

and are over-represented in the control group.




59/71

Controls should be persons who,

had they developed the outcome of interest,

would have had the same opportunity asthe actual cases to be included as such.

Similarly, cases should have

had the same opportunity as actual controls

to be included, had they

not developed the outcome of interest.

If this is not the case, controls may not properly

represent the source population.

e.g., study of brain tumours and cell phone use

discussed above




60/71

Types of controls in case-control studies

1. Population Controls

Suitable if cases are a representative sample

(or all cases) arising from a well-defined

source population.

Controls are then randomly sampledfrom the same population.

With the incidence-density design,the probability of being sampled should

vary with an individuals person-time at risk.

Often, it is not easy to define the

precise source population.

Case Co t o Stud es S de 58



61/71

2. Neighbourhood Controls

May match controls to individual cases

with respect to neighbourhood of residence.

If cases are from a hospital, their neighbours

may or may not be equally likely to betreated at the same hospital should

they develop the disease in question.

Example:

A hospital which caters to a particular group

within society.




62/71

3. Family members or friends as controls

May share exposure characteristics with cases

as opposed to broader source population

(e.g. tobacco and alcohol use, dietary intake,

use of household products).

This can obscure relevant associations.

Depends on information provided by cases;

investigator loses control over factors leading

to selection.

Cases friends may overlap, leading to

disproportionate probabilities of selection

of certain individuals as controls.




63/71

4. Hospital/clinic based controls

Often used when cases accrued at specific

hospital(s)/clinic(s).

Controls are recruited among persons seen

at the same hospitals/clinics forother reasons or conditions.

To avoid bias, the basis for control selection

cannot be related to the exposure under study.

The incidence of the control condition(s)

determines the sampling fraction.




64/71

Example:A researcher wishes to examine the relationship

between anti-hypertensive medication use

and car accidents.

What will happen if controls are recruited

in the cardiology clinic?




65/71

The best hospital controls are

persons with acute conditions that

consistently require hospital care but

are not related to the exposure of interest.

Example:

In a case control study of smoking as a

risk factor for colon cancer, a researcher

recruits controls who undergo appendectomy,

prostatectomy, or hysterectomy atthe same hospital as the cases.


Supplemental Material- Slide 1


66/71

Derivation of formula - Part 1

For the cohort study, the 2 x2 table is:

exposed unexposed total

Cases a b a + b

Person-time Pe Po Pe+ Po= Pt

IRR = Ie = a/Pe = aPo___ _____ ____Io b/Po bPe

= a(Pt - Pe) = a (1 - Pe/Pt)_______ _________

bPe b (Pe/Pt)

= a x (1-)_ ____b

Where = Pe/Pt = the proportion of person-years withexposure among total person-yearsin the source population


68/71

Derivation of formula - Part 2

if = proportion of person-years with exposure

then 1-= proportion of person-years without exposure

and It = Ie

+ Io(1-

)

i.e. a weighted average of incidence rates

among exposed and unexposed persons


S l l i l Sl d


69/71

Then the PAR fraction is:

It- Io = (Ie) + [(Io(1- )] - Io_____ ____________________

It (Ie) + [Io(1- )]

= (Ie/Io) + (1- ) (Io/Io) - Io/Io______________________________________

(Ie/Io) + (Io/Io) (1- )

= (IRR) + 1 - - 1________________

(IRR) + 1 -

= (IRR - 1)____________

(IRR - 1) + 1




70/71

Derivation - Part 3

= IRR - 1_____________IRR + (1/ ) - 1

= IRR - 1____________IRR + ( 1- )______

= IRR - 1______________IRR + IRR (1- )_________

IRR ()

pp

Supplemental Material - Slide 6


71/71

Substituting equation 1 for IRR, this is

IRR - 1____________________________IRR + IRR (1- ) () (1 - Pexp |case)_______________________

() (Pexp |case) (1- )

= IRR - 1__________________

IRR + IRR (1-Pexp |case)_____________

Pexp case

= IRR - 1______________________________

IRR (Pexp |case) + IRR - IRR (Pexp |case)______________________________

Pexp case

= IRR - 1 x Pexp |case = OR -1 x Pexp |case______ ____

IRR OR


lecture 7 case-control june 20

Documents