11.1

X:\606\May 2003\study guide\Chapter 11.doc

CHAPTER 11. Confounding

Objectives. Students should be able to:

a) Define: weighted average, arithmetic average

b) Define: confounding effect, confounding variable

c) Describe the structure of a crude relative risk in terms of

weighted averages

d) Define and distinguish strategies for the control of

confounding: direct and indirect standardization, Mantel-

Haenszel procedure, restriction, randomization, matching,

multivariate analysis

e) Distinguish between control strategies used in design and

those used in analysis

f) Define effect-modification; distinguish between confounding

and effect modification

Assignment: Confounding

Confounding 11.2

Outline

1. Weighted average

2. Confounding effect

3. Confounding variable

A risk factor for the disease under study

Associated with exposure

Not in the causal pathway

4. Weights in estimation of relative risk

5. Control of confounding: stratification

Direct standardization

Indirect standardization

Mantel-Haenszel procedure

6. Control of confounding: matching (a separate lecture)

7. Control of confounding: other strategies

Restriction

Randomization

Multivariate analysis

8. Overview of control strategies

9. Effect modification

Compulsory readings

Hennekens CH, Buring JE. Chapter 12.

Confounding 11.3

LECTURE NOTES ON CONFOUNDING

(prepared by Jean-François Boivin and William Hodge)

The design and conduct of epidemiologic studies can be affected

by three types of biases: selection bias, misclassification

bias, and confounding bias. Other types of biases may arise at

the level of analysis, for example through inappropriate

modelling assumptions. The subject of this section is

confounding.

WEIGHTED AVERAGE

Central to the understanding of confounding and how to control

it is the concept of weighted average. The arithmetic mean is

one particular type of mean, calculated in a specific way and

with specific weights (insert 1). To calculate the arithmetic

mean, the weights used are equal for each value being averaged.

Depending on the weights chosen, however, a mean may be very

different from the arithmetic mean. It will, however, always

fall within a range of values, the limits of which are the

maximum and minimum values in the set of numbers being averaged.

The concept of weighted average plays an essential role in the

understanding and controlling of confounding.

Confounding 11.4

CONFOUNDING EFFECT

A confounder is an extraneous variable that totally or partially

accounts for the apparent effect of the study exposure on the

outcome. It may even mask an underlying true association or

reverse it. Some examples will serve to illustrate these

possibilities. In table 1, the exposure variable is vitamin

deficiency and the outcome is depression. Is vitamin deficiency

a risk factor for depression? The overall relative risk is 2.5,

giving the erroneous impression that it is a risk factor.

However, if we stratify the results by age category, we see that

among old subjects the relative risk is 1.0 and among the young

it is also 1.0. Hence there is no relationship between vitamin

deficiency and depression. The confounding effect of age totally

accounts for the relationship between vitamin deficiency and

depression.

A confounding effect may partially account for the effect of the

exposure. In table 2, the overall relative risk of heart

disease in subjects with a red meat diet is 2.4, but when we

stratify the results by sex, the risk is 2.0 for each stratum.

Hence sex is partially responsible for the association between

red meat diet and heart disease.

In table 3, the relative risk of respiratory disease after

exposure to air pollution is 1.0, suggesting that there is no

Confounding 11.5

association between this exposure and the outcome. If we

stratify by smoking status, however, we see that the relative

risk in each category is actually 3.3. This represents an

example of a confounding effect masking a true association.

Finally in table 4, we see an example on how confounding can

reverse the true exposure-outcome relationship. Looking at the

effect of drug A relative to drug B on the risk of death, one

would conclude from the crude data that subjects exposed to drug

A have a 2.5-fold increase in the risk of dying relative to

drug B patients. However, if we stratify by asthma severity

status, we see that drug A patients are really protected from

dying relative to drug B patients.

We will now define the concept of confounding more formally.

CONFOUNDING VARIABLE

A confounding variable is a risk factor for the disease under

study and is also associated with the exposure under study.

Some refinements of this definition are needed. While the

confounder must be a risk factor for the outcome under study,

this relationship need not be causal. Causality is a complex

philosophical and scientific concept and will be addressed later

in this course. Age and sex are examples of variables which

often are confounders without being causal risk factors.

Confounding 11.6

A factor which meets the definition for confounding but is an

intermediary in the exposure-outcome pathway is not a

confounder. For example, if a high fat diet causes increased

cholesterolemia which in turn results in an increased risk of

myocardial infarction, the intermediary variable cholesterolemia

is not a confounder.

WEIGHTS IN THE ESTIMATION OF RELATIVE RISK

Return to table 1. The overall or crude relative risk of

depression was found to be 2.5, suggesting that vitamin

deficiency was a risk factor for depression. However, when we

stratify by age, we see that the relative risk in each stratum

is really 1.0, indicating that age confounds the relationship

between vitamin deficiency and depression. Why has this

happened? Table 5 shows data from table 1, from a different

perspective. We see that age is a risk factor for depression

(relative risk= 3.0). We also see that age is associated with

vitamin deficiency. That age is a risk factor for depression is

a fact that cannot be modified. However, the association

between age and vitamin deficiency is a design feature of the

study that can be manipulated by modifying the weights used.

Insert 2 demonstrates that subjects with vitamin deficiency and

those without vitamin deficiency receive different weights in

Confounding 11.7

the estimation of the crude relative risk. For subjects with

vitamin deficiency, the weights heavily favor the old age group

whereas for those without vitamin deficiency, the weights are

heavily tipped toward the young age group. Hence the crude

comparison of subjects with vitamin deficiency and subjects

without vitamin deficiency represents to a large extent the

comparison of old and of young subjects. In tables 6 and 7, we

changed the weights to an appropriate set of weights and the

confounding effect disappears. Choosing appropriate weights

represents one approach for controlling confounding, which leads

us into our next topic.

METHODS TO CONTROL CONFOUNDING

(A) STRATIFICATION

Stratification refers to a group of methods which yields a

summary measure of association which is an average of stratum-

specific values. If, for example, sex were a potential

confounding variable, the measure of association for males and

females could be calculated separately. By definition, each

stratum-specific estimate is now unconfounded by sex. One

option is to report the unconfounded measure of association

separately for each stratum. This, however, has the

disadvantage of being unparsimonious. Usually an overall pooled

measure of association, representing a weighted average of the

measure of association in each stratum, is calculated. The

Confounding 11.8

pooled measure of association will fall between the lower and

upper limits of the stratified measures of association. Table 8

demonstrates the relationship between the use of a drug and

development of a rash. There is strong confounding by sex as

the crude relative risk of 0.63 does not fall within the

stratum-specific relative risks of 2.0 and 1.5. By using a

weighting system which yields a weighted average of these two

values, the corrected or standardized relative risk will fall

between 1.5 and 2.0. Various stratification methods are

distinguished on the basis of how they determine weights used in

the analysis. The main stratification methods include direct

standardization, indirect standardization, and the Mantel-

Haenszel procedure.

Standardization

Both direct and indirect standardization consist of obtaining a

weighted average of stratum-specific risks, using within each

stratum the same weights for exposed and unexposed subjects.

Let us use the example from table 8 to work through these two

methods. Table 9 demonstrates the crude weights and risks for

this example. One can see from this table that the confounding

arises as a result of the extreme difference in the values of

the weights between the two groups. Drug A was predominantly

prescribed to males and drug B predominantly to females. In

table 9, these weights are changed and standard arbitrary

Confounding 11.9

weights are chosen. The summary unconfounded estimate then

becomes 1.57. In table 10, indirect standardization is

performed. Note that the method is identical to direct

standardization, except that the weights are not in this case

arbitrary: the population of exposed subjects (drug A) is chosen

to provide the weights. The indirectly standardized estimate is

1.8.

Mantel-Haenszel procedure

A common method used to obtain an unconfounded estimate of the

odds ratio is the Mantel-Haenszel procedure. The principle of

the Mantel-Haenszel procedure is to give the largest weights to

the stratum-specific estimates with the smallest variance. The

weights assigned to the stratum-specific values are therefore

inversely proportional to the variance of each estimate. Table

11 illustrates the use of the Mantel-Haenszel procedure in a

study of occupation and lung dysfunction. The crude odds ratio

is 0.54 but the stratum-specific odds ratios (stratified by age)

range from 1.42 to 2.7. Using the Mantel-Haenzel procedure, the

adjusted odds ratio is 2.2.

(B) MATCHING

The question of matching will be covered in a separate lecture.

Confounding 11.10

(C) RESTRICTION

In restriction, the investigator allows only subjects in one

category of the potential confounding variable to be included in

the study. For example, if the association between asbestos

exposure and lung cancer was to be studied, and one wanted to

control for cigarette smoking as an important confounding

variable, one could restrict the study to nonsmokers, or to

smokers. One disadvantage of restriction is that it may reduce

the number of subjects available for study.

(D) RANDOMIZATION

Randomization is a very important way to control confounding and

has unique advantages. In randomized studies, known confounders

are expected on average to be equally distributed between these

groups. It is this presumptive ability to control even unknown

confounders that makes randomization such an attractive way to

control confounding.

(E) MULTIVARIATE ANALYSIS

By far the most important and practical way to control

confounding in modern epidemiology is multivariate modelling.

This topic is the subject of several textbooks and specialized

courses and will not be developed here.

Confounding 11.11

OVERVIEW OF CONTROL STRATEGIES

Table 12 summarizes at which level of a study specific

strategies aimed at the control of confounding are used.

Randomization and restriction are used in the design phase while

stratification and multivariate analysis are used in the

analysis. Matching is used in the design phase and depending on

the study design may also need to be taken into account in the

analysis.

EFFECT MODIFICATION

Effect modification (synonym: interaction) is sometimes confused

with confounding. While confounding is a bias to be controlled

for, effect modification is a phenomenon to be assessed and

explored. Does the effect of diabetes mellitus on risk of

coronary heart disease differ across sex or age group? For

example, are diabetic males more likely to suffer from coronary

heart disease than diabetic females? Such relationships must be

studied to better understand this clinical and epidemiologic

question.

When a measure of association differs across strata, effect

modification exists. Effect modification may exist with or

without confounding. Table 13 gives examples. In the first

example, confounding exists because the crude relative risk of

0.63 does not fall between the stratum-specific relative risks

Confounding 11.12

of 1.5 and 2.0. Effect modification also exists: the stratum-

specific relative risks of 1.5 and 2.0 are different. In the

second example, confounding is not present as the crude relative

risk of 2.36 falls between the stratum-specific risks of 2.0 and

6.0; effect modification is certainly present as the stratum-

specific relative risks of 2.0 and 6.0 are very different. By

the same reasoning, in the third example, confounding exists but

effect modification does not. Finally, in the fourth example,

there is neither confounding nor effect modification. We have

assumed in all of these examples that the sample sizes were

large and therefore that these estimates of relative risk were

very precise.

Effect modification depends on the measure of association. Look

for example at the third example of table 12. Based on the

relative risk, there is no effect modification as the relative

risk in each stratum is 2.0. The risk difference in the first

stratum, however, is 0.1 (200/1000-20/200) and in the second

stratum it is 0.4 (80/100-800/2000). Hence on the risk

difference scale there is effect modification.

Confounding 11.13

INSERT 1 DEFINITION OF AVERAGE AND WEIGHTED AVERAGE

1. Mean: a value that lies within a range of values and is

computed according to a prescribed law (synonym: average)

(Webster's New Collegiate Dictionary, 1977).

2. Arithmetic mean: a value that is computed by dividing the

sum of a set of terms by the number of terms (Webster).

Algebraically:

3. Weighted mean (or weighted average): a value that is

computed by adding a set of terms weighted in such a way

that the sum of the weights is equal to one.

Algebraically:

The arithmetic mean is a special case of the weighted mean

where weights are equal for all i's. If one defines wi =

1/n in equation II above, one obtains:

( )n/x = n

x+...+x+x = x ofMean in

1=i

n21 Σ

wx = wx+...+wx+wx = x ofMean iin

1=inn2211 Σ

1 = w where in

1=iΣ

nx+...+x+x =

)n1(x+...+)

n1(x+)

n1(x = xofMean

n21

n21

Confounding 11.14

TABLE 1 CONFOUNDING TOTALLY ACCOUNTS FOR EXPOSURE EFFECT A COHORT STUDY OLD Vitamin deficiency + -

1.0 = 10060

20001200

÷

+ Depression

-

1200 60

800 40

2000 100

YOUNG Vitamin deficiency + -

1.0 = 1000200

10020

÷

+

Depression -

20 200

80 800

100 1000 ALL Vitamin deficiency + -

2.5 = 1100260

21001220

÷

+ Depression

-

1220 260 880 840

2100 1100

Confounding 11.15

TABLE 2 CONFOUNDING PARTIALLY ACCOUNTS FOR EXPOSURE EFFECT (source: Boivin JF, Wacholder S. Conditions for confounding of the risk ratio and of the odds ratio. American Journal of Epidemiology 1985; 121:152-158.)

A COHORT STUDY MALES Red meat diet + -

2.0 = 480200

16801400

÷

Heart + Disease -

1400 200

280 280

1680 480 FEMALES Red meat diet + -

2.0 = 24020

26444

÷ Heart + Disease -

44 20

220 220

264 240 ALL Red meat diet + -

2.4 = 720220

19441444

÷ Heart + Disease -

1444 220

500 500

1944 720

Confounding 11.16

TABLE 3 CONFOUNDING MASKS THE EFFECT OF EXPOSURE

A COHORT STUDY SMOKERS Air pollution + -

3.3 = 73367

10030

÷ Respiratory +

Disease -

30 67

70 666

100 733 NON-SMOKERS Air pollution + -

3.3 = 100

142614

÷ Respiratory +

Disease -

14 1

412 99

426 100 ALL Air pollution + -

1.0 = 83368

52644

÷ Respiratory +

Disease -

44 68

482 765 526 833

Confounding 11.17

TABLE 4 CONFOUNDING REVERSES THE EFFECT OF EXPOSURE

A COHORT STUDY SEVERE ASTHMA Drug A B

0.5 = 104

1000200

÷ +

Death -

200 4

800 6

1000 10 MILD ASTHMA Drug A B

0.5 = 100

4100

2÷

+ Death

-

2 4

98 96

100 100 ALL PATIENTS Drug A B

2.5 = 110

81100202

÷ +

Death -

202 8

898 102

1100 110

Confounding 11.18

TABLE 5 CONDITIONS FOR CONFOUNDING IN THE EXAMPLE OF VITAMIN DEFICIENCY AND DEPRESSION

(1) Age is associated with depression In subjects without vitamin deficiency: Old Young

3.0 = 1000200

10060

÷ +

Depression -

60 200

40 800

100 1000 In subjects with vitamin deficiency: Old Young

3.0 = 10020

20001200

÷ +

Depression -

1200 20

800 80

2000 100 (2) Age is associated with vitamin deficiency Old Young

200 = 10010010002000 =ratio Odds

××

+

Vitamin deficiency

-

2000 100

100 1000

Confounding 11.19

INSERT 2 EXPRESSING DATA FROM TABLE 1 IN TERMS OF WEIGHTS In tables 1 and 5, it can be seen that the stratum-specific risks of depression among subjects with vitamin deficiency are 0.60 (old) 0.20 (young). The crude risk, seen in table 1, is 1220/2100 = 0.58. This crude risk is a weighted average of 0.60 (old) and 0.20 (young) and the weights are 2000/2100 and 100/2100, respectively. Thus,

Similarly, the stratum-specific risks of depression among subjects without vitamin deficiency are 0.60 (old) and 0.20 (young) and the crude risk is 260/1100 = 0.24. The weights are 100/1100 and 1000/1100, respectively:

It can be seen that for each age group, a different weight is used in the subjects with vitamin deficiency and those without: Crude relative risk:

9 9 Old Young Relative risk=1 Relative risk = 1 Most of the information which goes into the crude relative risk is the risk of 0.60 in the exposed and of 0.20 in the unexposed.

0.58 = 2100

20+1200 = 2100100+(0.20)

21002000(0.60)

0.24 = 1100260 =

1100200+60 =

11001000+(0.20)

1100100(0.60)

2.5 =

11001000(0.20) +

1100100(0.60)

2100100(0.20) +

21002000(0.60)

Confounding 11.20

TABLE 6 VITAMIN DEFICIENCY AND DEPRESSION: SELECTING NEW WEIGHTS

A COHORT STUDY OLD Vitamin deficiency + -

+ Depression

-

60% 60%

40% 40%

N1 N2 YOUNG Vitamin deficiency + -

+ Depression

-

20% 20%

80% 80% N3 N4

Select N1, N2, N3, N4 such that For example: N1, N2, N3, N4 = 100

then . . . table 7

1 = NNNN

32

41

Confounding 11.21

TABLE 7 VITAMIN DEFICIENCY AND DEPRESSION: RESULTS WITH NEW WEIGHTS

OLD Vitamin deficiency + -

Relative risk = 1 +

Depression -

60 60

40 40

100 100 YOUNG + -

Relative risk = 1 +

Depression -

20 20

80 80

100 100 ALL + -

Relative risk = 1 +

Depression -

80 80

120 120

200 200 This represents the principle of stratified analysis. The existing weights are replaced by other more a weights.

Confounding 11.22

TABLE 8 CONFOUNDING REVERSES THE EFFECT OF EXPOSURE

A COHORT STUDY MALE Drug A B

2.0 = 20020

1000200

÷ +

Rash -

200 20

800 180

1000 200 FEMALE Drug A B

1.5 = 2000800

10060

÷ +

Rash -

60 800

40 1200

100 2000 ALL Drug A B

0.63 = 2200820

1100260

÷ +

Rash -

260 820

840 1380

1100 2200

Confounding 11.23

TABLE 9 DIRECT STANDARDIZATION Replace crude weights by appropriate weights (the standard weights). Using data from table 8:

Drug A Drug B

Risk Population Crude weight Risk Population Crude

weight

male 0.20 1000 11001000

male 0.10 200 2200200

female 0.60 100 1100100

female 0.40 2000 22002000

Replace the observed population distribution by a standard arbitrary one. The one constraint is that the population weights must be identical within sexes.

Drug A Drug B

Risk Standard population Weight Risk Standard

population Weight

male 0.20 1000 1000/3000 male 0.10 1000 1000/3000

female 0.60 2000 2000/3000 female 0.40 2000 2000/3000

0.47 = 30002000(0.60) +

30001000(0.20) 0.30 =

30002000(0.40) +

30001000(0.10)

Relative risk:0.47/0.30 = 1.57 (an average of stratum-specific

values of the relative risk; the stratum-specific values in table 8 were 2.0 and 1.5)

Confounding 11.24

TABLE 10 INDIRECT STANDARDIZATION Replace the crude weights by weights in the exposed group

Drug treatment A = Exposed B = Unexposed

Risk Crude

population Crude weights Risk

Standard population = Crude

population in A

New weights = Crude

weights in A

male 0.20 1000 1000/1100 male 0.10 1000 1000/1100

female 0.60 100 100/1100 female 0.40 100 100/1100

0.24 = 1100100(0.60) +

11001000(0.20) 0.13 =

1100100(0.40) +

11001000(0.10)

Relative risk: 0.24/0.13 = 1.8 (an average of the

stratum-specific values of the relative risk)

The relative risk is more specifically called 'indirectly standardized relative risk.'

Confounding 11.25

TABLE 11 MANTEL-HAENSZEL PROCEDURE Procedure to obtain a weighted average of odds ratios (applicable in cohort and case-control studies; the original paper dealt with case-control studies) Example: A cohort study of lung dysfunction in occupational groups Young OCCUPATION

Manufacture Service Dysfunction + 10 5

- 90 95

100 100

Middle age 20 15

80 85

100 100

Old

+ 80 600

- 20 400

100 1000

2.1 = 90 595 10 = ORy ×

×

)OR( Variance1 W = youngfor weight MH

yy ≈

1.42 = 80 1585 20 = ORm ×

×

)OR( Variance1 W = age middlefor weightsMH

mm ≈

2.7 = 20 600

400 80 = ORo ××

)OR( Variance1 W = oldfor weightsMH

oo ≈

Confounding 11.26

TABLE 11 (continued) All ages

110 620

190 580

300 1200

Reference: Mantel N, Haenszel W. Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute 1959;22:719-748.

0.54 = 190 620580 110 = OR

××

1 = W + W + W omy

= )OR(W + )OR(W + )OR(W = OR oommyyMH

2.2 = (2.6)W + (1.42)W + (2.1)W = omy

Confounding 11.27

TABLE 12 OVERVIEW OF CONTROL STRATEGIES FOR CONFOUNDING Strategy Level of

use

Randomization Design Restriction Design Stratification Analysis Multivariate analysis

Analysis (represents an extension of stratification in which stratum-specific estimates are influenced by values of other strata)

Matching Design and sometimes analysis, depending on the measure of association being estimated

Confounding 11.28

TABLE 13 EFFECT MODIFICATION AND CONFOUNDING 1. Effect modification, confounding

Stratum 1 Exp

Stratum 2 1 + 2

+ - + -

Dis + 200 20 60 800 260 820

- 800 180 40 1200 840 1380

RR = 2.0 RR = 1.5 RR = 0.63 2. Effect modification, no confounding

200 20 60 200 260 220

800 180 40 1800 840 1980

RR = 2.0 RR = 6.0 RR = 2.36 3. Confounding, no effect modification

200 20 80 800 280 820

800 180 20 1200 820 1380

RR = 2.0 RR = 2.0 RR = 0.68 4. No effect modification, no confounding

200 20 200 20 400 40

800 180 800 180 1600 360

RR = 2.0 RR = 2.0 RR = 2.0