Confounding, Matching, and Related Analysis Issues Kevin Schwartzman MD Lecture 8a June 22, 2005.

Confounding, Matching, and Related Analysis Issues

Kevin Schwartzman MDLecture 8a

June 22, 2005

Readings

• Fletcher, chapter 1

• Hennekens and Buring, Epidemiology in Medicine, 1987: Chapter 12, Analysis of Epidemiologic Studies: Evaluating the Role of Confounding [course pack]

Confounding, Matching & Related Analysis Issues

Confounding, Matching & Related Analysis Issues - Slide 1

Objectives

Students will be able to:

1. Define confounding

2. Explain what must be true of a confounding variable

3. Describe design strategies for control of confounding

a. Restriction

b. Randomization, including stratified design

c. Matching, including different matching

schemes

Objectives

4. Describe analytic strategies for control of confounding

a. Stratified analyses

b. Standardization

c. Calculation of pooled effect estimates: the example of the Mantel-Haenszel odds ratio

d. The special case of matched pair case-control studies

e. Multivariate analyses

5. Identify advantages and disadvantages of matching

6. Define and identify effect modification


Confounding

• Refers to

distortion of the true underlying relationship (or lack thereof)

between an exposure and an outcome of interest,

because of the influence of a third factor(a “confounder” or a “confounding variable”)

• At the design phase,

confounding is potential; its true presence or absence is assessed through appropriate data analyses


Confounding Variables

A variable is said to be a confounder if:

- it is associated with the exposure of interest

- it is an independent risk factor for the outcome of interest

- it is not an intermediate along the causal pathway from exposure to outcome

Exposure Confounder

Outcome



Case-Control Study

Coffee No Coffee

Lung cancer 260 80

No lung cancer 190 150

Odds ratio = (260 x 150) / (80 x 190) = 2.6

Case-Control Study

Smokers

CoffeeNo

Coffee

LungCancer

250 50

NoLungCancer

150 30

OR = (250 x 30)/(50 x 150) = 1

Non-Smokers

CoffeeNo

Coffee

LungCancer

10 30

NoLungCancer

40 120

OR = (10 x 120)/(30 x 40) = 1


Smoking as Confounder

Smoking was associated with coffee drinking

- 400/450 coffee drinkers were smokers, vs 80/230 non-coffee drinkers

Smoking is an independent risk factor for lung cancer

- here, OR = (300 x 160)/(40 x 180) = 6.7

By separating the group intosmokers and non-smokers, and examining the relationship between coffee and lung cancer within each subgroup, confounding by smoking was eliminated


Smoking as Confounder

There was no independent association of coffee drinking with lung cancer (odds ratio within both smoking subgroups or strata was 1)

The apparent relationship was due entirely to confounding by smoking

Confounding can also reduce, eliminate, exaggerate, or even change the directionof true underlying associations

The presence of confounding can be assessed by comparing crude and adjusted effect estimates(some investigators use 10% “rule of thumb”)


Design Strategies to Control Confounding

First of all, any potential confounder must be measured appropriately

Simplest strategy (in terms of design) is restriction,to eliminate variation in potential confounder

If there is no variation in the potential confounder, it cannot influence the outcome

Example: restriction of the lung cancer-coffee study to smokers only

However, in this particular case, there could still be residual variation in smoking which could influence outcome


Randomization

Goal is to distribute potential confounders equally between study groups

Again, if there is no variationin a potential confounder, it cannot be responsible for differences in outcome

Smaller sample sizes may lead toimbalance between groups with respect to potential confounders, simply by chance


Randomization

Stratified randomization (often combined with blocked randomization): promotes equal distribution of treatment groups across strata of variable(s) of interest e.g. gender, age, study centre

Number of strata limited by logistical constraints

All reports of randomized studies include a table for assessing the adequacy of randomization

As soon as analysis is limited to subgroups, the control of confounding disappears e.g. compliance bias (healthy behaviours etc.)


Matching

Matching is an element of observational study design, introduced to help control potential confounders

it involves selection of a comparison group that is forced to resemble the index group with respect to the distribution of one or more potential confounders

in case-control studies selection of control group(matched to cases with respect to potentialconfounders)

in cohort studies selection of unexposed group (matched to exposed with respect to potentialconfounders)


Subjects can be matched for continuous covariates (e.g. age) or categorical covariates (e.g. sex, HIV serology, etc.)

Matching may be done at the level of the individual or of the group

In a case-control study, individual matching meansthat each case is separately matched toone or more control(s) according tothe matching factor(s)

Matching or variable ratio may be fixed(e.g. 1 case:1 control, 1:2, etc.)


• We will primarily discuss matchingin case-control studies

• For categorical covariates, individual matching means that for each case, the control subject(s) is/are drawn from the same category, e.g. male controls for male subjects

• Continuous covariates may also be “categorized”, e.g. age divided into categorical ranges: 20-39, 40-59, 60-79, etc.


Continuous variables may be matched by

a) Caliper matching: a rule by which values are considered sufficiently close

Matching done on sex plus age within 3 years

Potential controls:men aged 28, 35, 39, 49, 57women aged 31, 34, 43

Case 1: 31 y.o. male matched to 28 y.o. male

Case 2: 38 y.o. female no match found

case discarded

or additional controls identified


Continuous variables may be matched by

b) Nearest available matching- controls are selected based on

the closest value of the matching factor

In above example, the match for the 38 y.o. female case would be a 34 y.o. female control

Advantage: less restrictive, more efficient

Disadvantage: Subjects may be less well matched if the distribution of the matching variable is quite different between cases and controls


Example:

cases of a disease which affects primarily elderly persons

Controls drawn from the general population with matching based on nearest age may be considerably younger, on average, depending on the number of potential controls identified.

- the same may occur when continuous variables are categorized into wide ranges

- the impact of the study will depend on the nature of the relationship between the matching factor,the exposure, and the outcome of interest


Group level matching

Cases are stratified according to the matching factor, and then

controls are selected to match the grouping of cases

a) Stratified sampling:

The levels of the covariate in whichsampling occurs are defined.

Then preset numbers of cases and controlsare drawn from each stratum, with a consistent matching ratio


Example of stratified sampling:

Case-control study examining coffee intake and lung cancer


CoffeeStratum Yes No Total

Current smokers Lung cancer 100

No cancer 200

Former smokers Lung cancer 100 preset

No cancer 200

Never smokers Lung cancer 200

No cancer 400

Note that within each stratum,

There are 2 controls fo every lung cancer case

b) Frequency matching

There is also a constant proportion of controls to cases,

but the distribution of cases is not fixedaccording to the matching factor.

However, controls are forced to have the same distribution of the matching factor as do the cases.

The distribution of the matching factors is therefore representative of that among

the population that gave rise to cases.


Example of frequency matching: Coffee intake and lung cancer


Coffee

Stratum Yes No Total

Current smokers Lung cancer 140

No cancer 280 not

Former smokers Lung cancer 220 preset

No cancer 440

Never smokers Lung cancer 40

No cancer 80

- here the number of cases in each smoking stratumreflects the distribution of smoking behaviour among lung cancer cases

- the matching ratio is 2 controls per case throughout

Analysis of case-control studies with matching:

- Always requires stratification by the matching factor (or the multivariate equivalent - conditional logistic regression).

- The crude odds ratio will be biased toward the null value.

- This is because matching forces the cases and controls to be more alike with respect to the exposure of interest than would ordinarily be the case.


Hypothetical example: ObesityYes No Total

Smokers Heart disease 480 20 | 500No heart disease 420 80 | 500_________________________________

Total 900 100 | 1000_________________________________

OR = 4.6

ObesityYes No Total

Non-smokers Heart disease 8 42 | 50No heart disease 2 48 | 50_________________________________

Totals 10 90 | 100_________________________________

OR = 4.6


Crude analysis of same data

ObesityYes No Total

Heart disease 488 62 | 550No heart disease 422 128 | 550____________________Totals 910 190 | 1100

OR crude = 2.4

Despite matching, the underlying association betweensmoking (confounder) and obesity (exposure) remains:smokers were much more likely than non-smokers to be obese.

However, matching on smoking behaviour made cases and controls more similar with respect to obesity, therebyleading to underestimation of the odds ratio.

Stratified analysis corrects this problem.


Matching in cohort studies - does not lead to inappropriate crude risk/rate ratio estimates

e.g. cohort study of obesity and heart disease

ObesityYes No Total

Smokers Heart disease 460 100No heart disease 540 900_______________________________________

Total 1000 1000 2000_______________________________________RR = 4.6

ObesityYes No Total

Non-smokers Heart disease 46 10No heart disease 954 990_______________________________________

Total 1000 1000 2000_______________________________________RR = 4.6


Crude analysis

Coffee Yes No Totals

Smokers Lung cancer 506 110 | 616No cancer 1494 1890 | 3384___________________________________Totals 2000 2000 | 4000

RR = 4.6

Here the crude RR is the same as within the individual strata.

This is because matching eliminates the association between smoking (confounder) and coffee drinking (the exposure studied).


Stratified Analysis

If effect estimates are identical across strata, then it is easy to report a single summary estimate (e.g. odds ratio)

More often, they are not precisely identical, which may reflect random error/imprecision (e.g. small strata), residual confounding, ortruly different effects (effect modification)

Effect modification will be described separately


Combining Effects from Strata

• Can take some type of weighted average

• One approach is to use weights which reflect the distribution of the stratification variable in the population of interest

• For example, age-specific risk ratios could be combined using a weighted average that accounts for the age distribution of the general population

• This is an example of standardization: the effect is adjusted to reflect a standard age distribution

• This does not assume that the effects are homogeneous

• The most heavily weighted strata may not have much information


Mantel-Haenszel Odds Ratio

• An odds ratio that reflects pooling of effects across strata, to summarize the overall association between exposure and outcome, while adjusting for the effect of the confounder of concern

• Pooling assumes that the effect is homogeneous, andvariation reflects random error

• Is a weighted average of odds ratio estimates across strata

• Weights reflect quantity of information in each stratum, expressed as bc/T where b and c are exposed controls and unexposed cases within the stratum, and T is total subjects within the stratum

• Note this differs from standardization using “external” weights


Mantel-Haenszel Odds Ratio

OR MH = Σ[(bc/T) x ad/bc] = Σ(ad/T)__________________ _________

Σ(bc/T) Σ(bc/T)

For the case-control study of obesity and heart disease, this would be:

(480 x 80)/1000 + (8 x 48)/100__________________________(20 x 420)/1000 + (42 x 2)/100

= (38.4 + 3.84)/(8.4 + 0.84) = 4.6


Analysis of matched pair data in case control studies

• can be thought of as a special case of stratified analysis

• each matched pair constitutes a single stratum with 2 subjects

• only informative strata are those where exposure status of case and control are discordant


Recall Mantel-Haenszel OR estimates

OR MH = ( ad/T)_______

( bc/T)

Concordant strata: E + E -

D + 1 0D - 1 0

or E + E -

D + 0 1D - 0 1

ad = 0, bc = 0


The pairs can be grouped as follows:

Case

Control Exposed Unexposed

Exposed r s

Unexposed t u

Then OR MH = t/s

i.e. N(case exposed, control unexposed)_____________________________N(case unexposed, control exposed)

where N refers to number of pairs


Example:

Marrie et al conducted a study evaluating the relationship between certain infections (the exposure)and the subsequent development of multiple sclerosis (the outcome). Data was taken from a general practicedatabase.

Cases and controls were matched on age ( 2 years),sex, physician practice, and date seen.

Imagine a 1:1 design (in fact it was 1:4, on average).


Hypothetical data

MS (cases)

No MS (controls) Infection No infection

Infection 30 5

No infection 20 170

OR = 20/5 = 4


Suppose the key confounder is physician practice

- the physicians most likely to see and diagnose infectionsmay also be those most likely to pursue and establish the diagnosis of multiple sclerosis

Unmatched analysisMS No MS

Infection 50 35No infection 175 190

Crude OR = (190x50) / (175x35) = 1.6

As before, the unstratified analysis yields an OR estimatebiased toward the null.

As before, this is because the matching forces the controlsto “resemble” the cases with respect to the distribution of exposure in the crude analysis.


Multivariate Analysis

Has become the standard approach for identifying and accounting for confounding

Complex process: computer essentially solves multiple equations to identify “best guess” effect estimatewhile holding other covariates constant, e.g. effect of obesitywhile holding smoking behaviour, sex, diabetes constant

Mathematically breaks the data down into numerous strata

Examples:logistic regression for binary outcome data (very frequent),Cox proportional hazards modelling for incidence data,Poisson model for count data


Rationale for Matching

• Matching can be considered a form of partial restriction: the controls are restricted so as to resemble the cases with respect to some factor(s).

• The main purpose of matching is to improve statistical efficiency (precision).

• In principle, stratified analysis alone (including multivariate techniques) should be sufficient to deal with the confounderin question.

• However, matching may be needed to ensure that all strata are sufficiently informative.


Example:

An investigator wishes to investigate a possibleassociation between use of calcium channel blockers (drugs used for blood pressure and heart disease) and Alzheimer’s disease.

Age is obviously a key confounder: increasing age is associated with use of the drugs in question and with the onset of Alzheimer’s disease

Unmatched controls drawn from the general populationwill be younger and hence less likely to be using calcium channel blockers, leading the crude analysis to overestimate any potential association


• This can be handled through stratified analysis by age (e.g. various age categories)

• If unmatched general population controls are used, there may be few controls in the oldest age strata, leading to imprecise OR estimates in those strata (wide confidence intervals)

• Matching ensures sufficient numbers of subjects foreach level of the matching variable(s) - in this case, age

• Matched cohort studies are also more efficiently analyzedusing stratification by the matching factor(s)


Advantages of matching

1. Promotes efficiency, as discussed above.

Studies are most efficient when the the ratio of index to referent subjects (e.g. cases:controls) is constant across the different strata of a confounder.

2. Very useful in situations where the confounder is difficult to quantify or control, making stratification impossible.

Classic example: using sibling controls.


Disadvantages of matching

1. Practical - may be cumbersome, expensive, time consuming.

Depending on the circumstances, index subjects may be dropped if no matching referent subjects are found loss of data.

Also very onerous when many matching factors are used.

2. The effect of the matching factor on the outcome of interest cannot be evaluated.

3. Potential for overmatching.


Overmatching

Refers in general to situations where matching interferes with the logistics, statistical efficiency, or scientific validity of a study.

1. Overmatching as a cause of logistical inefficiency

matching on many factors, or on factors that are difficult to match, adds to the expense and difficulty of study conduct

difficulty with matching may lead to loss of cases as well as of potential controls (in case-control studies)


2. Overmatching as a cause of reduced statistical efficiency

occurs when matching factor is not a true confounder,e.g. associated with exposure but not with outcome

simplest example is with matched pair case-control design

if cases and controls made more similar with respect to exposure frequency, then there will be many uninformative pairs

these do not contribute to the odds ratio estimate and are essentially “ wasted”

conversely with fewer discordant pairs, the precision of the odds ratio estimate is reduced

the same holds true for other matching ratios


• With weak confounders (e.g. limited effect on outcome) the loss of statistical efficiency may outweigh any apparent benefits of matching

• Recall that stratified analysis and multivariate techniqueswill still account for potential confounders in the absence of matching


3. Overmatching as a cause of biased effect estimates

Occurs when matching factor is:

a) produced by exposure and related to disease (e.g. an intermediate in pathway)

or

b) produced by disease and related to exposure


Effect Modification

• Effect modification refers to the situation

where the biologic effect of exposure on outcome differs according to some additional factor,

e.g. different influence of smoking on development of COPD in men and women

• Also known as interaction

• In stratified analysis, will see differentexposure-outcome relationships within different strata, e.g. different odds ratios, rate ratios, etc.


• In the absence of confounding, the overall effect estimatewill simply be an average of the stratum-specific estimates,weighted by the size of the strata e.g. males and females

• Effect modification is NOT the same as confounding- It refers to biologic variation in an effect,

not artefactual distortion of results because of inadequate design or analysis

• Effect modification should be noted and reported, rather than “controlled” through design and analysis strategies

• Effect modification is relevant to randomized trials as well as observational studies


• Effect modification is only evident from stratified analysis, with stratification by the factor(s) of interest

• Analyses/effect estimates restricted to specific strata (e.g. women, young adults) have less precisionand statistical power than the study as a whole

• If investigators wish to detect and document effect modification, they need to ensure the necessary sample sizes


Confounding, Matching, and Related Analysis Issues Kevin Schwartzman MD Lecture 8a June 22, 2005.

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Transcript of Confounding, Matching, and Related Analysis Issues Kevin Schwartzman MD Lecture 8a June 22, 2005.