Matching

Objectives Discuss methods of matching Discuss advantages and disadvantages of matching Discuss applications of matching Confounding residual Overmatching

Recommended reading: Szklo and Nieto pgs: Further reading: See Rothman and Greenland. Modern Epidemiology

What is matching? The process of making a study group and a comparison group comparable with respect to extraneous factors (Last JM. A Dictionary of Epidemiology. 3rd Ed. New York, NY: Oxford University Press; 1995) In case-control studies, we match to make cases and controls as similar as possible with regard to potentially important confounding factors

Why we do matching?

Strategies for control of confounding 1. Random allocation of exposure 2. Restriction 3. Matching - Partial restriction 4. Stratification 5. Modeling

Types of matching Individual (paired) matching: for each case, one (or more) controls with the relevant characteristics matching the case are chosen –For continuous variables such as age or weight, controls may be selected if they are within a specified range of the control value Example: Age ± 2 years Example: Weight ± 5 Kg

Caliper matching: | X 1- X 0 | <= e | X 1- X 0 | = 0 (exact)

Nearest available matching no fixed tolerance

Stratified matching, for categorical variables Bias reduction N. Of strata % Bias reduction

Frequency matching: Controls are selected such that the distribution of the relevant characteristic in the controls is similar to the distribution in the cases Example: If 30% of cases are smokers, then select a control group in such a way that 30% of controls are smokers

Mean matching

When matching on a variety of characteristics, including continuous variables, it may be very difficult to individually match on all characteristics Minimum Euclidean distance: identify the individual who is the closest match with regard to all of the variables. –We usually do this using mathematical modeling techniques

Discriminant matching

Advantages of Matching May be the best way to control for a strong confounder when there is little overlap of the confounder between the cases and controls –Example: If the cases tend to be older (CHD, prostate cancer) and a random sample of controls would result in a much younger control group, then there may not be much overlap of age between cases and controls –This lack of overlap makes adjustment for confounding difficult. Why? When the confounder is strong, matching increases the efficiency of the study (by decreasing the width of the confidence intervals around an estimate)

Matching can be a useful method of sampling controls when cases and controls are identified from a reference population for which there is no available sampling frame (list). –Example: Reference population is patients at a clinic or hospital

Age Blood Pressure × × × ×‌ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ‌ × × ×‌‌‌ × × × × ‌ × × ‌ × × × × × × × × × × × × × × × × × × × × ‌ × × × × × × × × × × × × × × × × × × × × × ‌ × × × × × × × × × × × × × × × × × × × × × × × × × × × ×‌ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ‌ × × ×‌‌‌ × × × × ‌ × × ‌ × × × × × × × × × × × × × × × × × × × × ‌ × × × × × × × × × × × × × × × × × × × × × ‌ × × × × × × × × × × × × × × × × × × × × × × × × Non- Smokers Smokers

Age Blood Pressure Y NS = a NS + bX NS Y S = a S + bX S Non- Smokers Smokers

Y S = a S + bX S Y NS = a NS + bX NS (Y S – Y NS ) = a S - a NS + b (X S - X NS )

Age Blood Pressure Y NS = a NS + bX NS Y S = a S + bX S Non- Smokers Smokers

Age Blood Pressure Y S - Y NS Non- Smokers Smokers

Age Blood Pressure a S - a NS Non- Smokers Smokers

Age Blood Pressure b (X S - X NS ) Non- Smokers Smokers

How Matching Can Reduce Confounding Bias?

Age Blood Pressure Non- Smokers Smokers

Age Blood Pressure b (X S - X NS ) = 0 Non- Smokers Smokers

Disadvantages of Matching It may difficult (and expensive) to identify a matched control When you match on a characteristic, you create an equal distribution in the cases and controls. Therefore, you cannot examine the association between the matched characteristic and the outcome

Factors influencing efficiency of matching in Bias Reduction 1.The difference of mean of matching factor between groups.

Age Blood Pressure Non- Smokers Smokers

Age Blood Pressure Non- Smokers Smokers

1.The difference of mean of matching factor between groups. Factors influencing efficiency of matching in Bias Reduction 2. The ratio of the population variances.

Age Blood Pressure Non- Smokers Smokers

Age Blood Pressure Non- Smokers Smokers

1.The difference of mean of matching factor between groups. Factors influencing efficiency of matching in Bias Reduction 2. The ratio of the population variances. 3. The size of the control sample from which the investigator forms a comparison group.

Age Blood Pressure Non- Smokers Smokers

Age Blood Pressure Non- Smokers Smokers

1.The difference of mean of matching factor between groups. Factors influencing efficiency of matching in Bias Reduction 2. The ratio of the population variances. 3. The size of the control sample from which the investigator forms a comparison group.

Disadvantages of matching (cont.) You must account for matching in the data analysis You may create groups that are no longer representative of the reference population, thus decreasing your ability to generalize your findings

Overmatching Overmatching occurs when you match on a variable that is strongly correlated with the exposure of interest By setting the distribution of the matching variable to be equal between cases and controls, you are effectively setting the distribution of the exposure variable to be equal between cases and controls In doing so, you will be unable to detect a difference in exposure between cases and controls

Hypotethical target population of 2 million people FemalesMales Exposed Unexposed Exposed Unexposed No. cases in 1 year Total 4, , , , ,000 1-year risk Crude risk ratio = ( )/1,000,000 (50+90)/1,000,000 = 33

Expected results of a matched 1-year cohort study of 200,000 subjects drawn from the target population FemalesMales Exposed Unexposed Exposed Unexposed No. cases in 1 year Total Relative Risk Crude risk ratio =

Expected results of a matched 1-year cohort study of 200,000 subjects drawn from the target population FemalesMales Exposed Unexposed Exposed Unexposed No. cases in 1 year Total , , ,000 1 Relative Risk10 Crude risk ratio = (450+10)/100,000 (45+1)/100,000 = 10 10

Expected results of a case-control study matched on sex when the source of subjects is the same target population FemalesMales Exposed Unexposed Exposed Unexposed Cases (4740) Controls (4740) Approximate expected OR Approximate expected Crude OR = =

Expected results of a case-control study matched on sex when the source of subjects is the same target population FemalesMales Exposed Unexposed Exposed Unexposed Cases (4740) Controls (4740) Approximate expected OR 10 Approximate expected Crude OR ( )( ) ( )(50+90) = 5 10 =

Matching may induce selection bias in case-control studies

Costs of Matching Loss of ability to study the effects of matching factor on outcome. Possible expense entailed in the process of choosing control subjects. Effects of matching on study efficiency. - size efficiency vs.. cost efficiency.

Overmatching Matching that harms statistical efficiency. - e.g: case-control matching on a variable associated with exposure but not disease (not confounder). Matching that harms validity. - e.g: matching on an intermediate variable between exposure and disease (causal pathway). Matching that harms cost efficiency. Overmatching includes three forms:

Conclusion Conventional matching is rarely the optimal stratified design.

Residual confounding Occurs when you categorize continuous variables Ex. Create age categories for matching For each case between 20 and 25, select a control who is also between 20 and 25 Now your cases and controls are comparable with respect to age right?

Are these two groups really comparable?

Age Blood Pressure Non- Smokers Smokers

Age Blood Pressure Non- Smokers Smokers

Y S = a S + b X 2 S Y NS = a NS + b X 2 NS (Y S - Y NS ) = a S - a NS + b (X 2 S - X 2 NS ) If X S = X NS then b (X 2 S - X 2 NS ) may be unequal to zero.

موردشاهدموردشاهد مواجهه‌يافته7550مواجهه‌يافته3080 مواجهه‌نيافته2550مواجهه‌نيافته1080 جمع100 جمع40160 نسبت‌شانس = 0/3نسبت شانس = 0/3 فاصله‌ اطمينان 95 درصد:‌7/5 – 6/1فاصله اطمينان 95 درصد: 1/7 – 3/1