1. Instructor Resource Chapter 15 Copyright © Scott B. Patten, 2015. Permission granted for classroom use with Epidemiology for Canadian Students: Principles, Methods & Critical Appraisal (Edmonton: Brush Education Inc. www.brusheducation.ca).

2. Chapter 15. Stratified analysis and regression modelling in analytical studies

3. Objectives Explain the key importance of analysis-stage techniques to control confounding. Describe the following aspects of stratified analysis: fit with noncontinuous variables, pooling estimates for better precision, identification of effect modification, detection and adjustment for confounding variables. List the steps of stratified analysis. Describe the advantages of regression modelling over stratified analysis. List the steps of regression modelling.

4. Why study these techniques? These techniques play a role in almost every analytical study, even in studies that have also used design-stage techniques to control confounding. If you want to understand most of the empirical papers published in the clinical and public health literature, you need to understand the goals and strategies of stratified analysis and regression modelling.

5. Review of design-stage strategies Randomization can control unmeasured confounders, a unique strength, but randomization is not usually feasible, except in treatment studies. Restriction detracts from representation of the population and nonrestricted variables are still potential confounders. Matching, similar to restriction, distorts population relationships, and nonmatched variables can still confound.

6. Stratified analysis Stratified analysis is straightforward for noncontinuous confounders (e.g., sex). Sex cannot confound an exposure-disease association within strata containing only men and only women. Confounding is an intermixing. Stratification is an unmixing.

7. Stratified analysis (continued) Stratified analysis has tw2 main goals: detection of confounding control of confounding Confounding is detected when the stratum-specific estimates differ from the nonstratified (crude) estimate. Confounding is controlled when adjusted estimates (e.g., stratum-specific estimates) are reported rather than the (confounded) crude association.

8. Stratified analysis and continuous variables It is possible to stratify on continuous variables (such as age) by creating categories from them (such as age ranges). If there are too many strata, the number of study participants in each stratum will be small and subject to random variation, making it very difficult to detect whether confounding has occurred. If strata are too large, a failure to fully adjust for the effect of a confounding variable leads to residual confounding. Where continuous variables are potential confounders, stratified analysis must walk a fine line between these 2 problems.

9. Stratum-specific estimates and pooling Stratum-specific estimates are adjusted for confounding. But because they are based on subsets of the larger sample, they are often imprecise. Precision is important, so techniques have been developed to pool these stratified estimates in a way that preserves control over confounding but increases their precision.

10. Stratum-specific estimates and pooling (continued) The classical parameter is the Mantel-Haenszel family of estimators. These pooled estimators can be easily calculated across a series of strata. The family includes pooled versions of the risk difference, risk ratio, rate ratio, and odds ratio.

11. Heterogeneity in stratified analysis Pooled estimates are only meaningful if the stratum-specific estimates comprise a single underlying effect. Such consistency of effect across strata is called homogeneity. A lack of homogeneity— heterogeneity—implies that there are different effects of exposure in different strata—in other words, effect modification. Effect modification makes pooling inappropriate.

12. Assessing heterogeneity One way to assess heterogeneity is to look at the stratum-specific estimates and “eyeball” how different they are. Confidence intervals of the stratum-specific estimates should also be considered. There is a Mantel-Haenszel test to assess heterogeneity, which quantifies whether the difference between stratum-specific estimates (or a greater difference) is more than could occur by chance, at a defined level of confidence.

13. The “steps” of stratified analysis First, you estimate a crude effect using a risk difference, risk ratio, odds ratio, hazard ratio, or other parameter. Next, you stratify and examine the association within strata formed by levels of the potential confounding variable. Then, you assess whether the resulting estimates are similar to each another (exhibit homogeneity). If the estimates are dissimilar (heterogenous), effect modification has been identified. In this case, confounding is NOT the issue.

14. The “steps” of stratified analysis (continued) If stratum-specific effects are similar (homogenous), you assess whether they differ from the crude estimate. If they do differ, then confounding has been demonstrated. At this stage, you should report an adjusted estimate of effect, such as 1 of the Mantel-Haenszel family of parameters. If there was no confounding, it is not necessary to report an adjusted estimate. The crude estimate is good enough.

15. Effect modification in stratified analysis The occurrence of effect modification precludes any assessment of confounding. Recall that confounding can be defined according to whether some form of adjustment alters the effect of exposure. If there is effect modification, there is no longer a single effect of exposure upon which to make such a determination. The demonstration of effect modification confirms that there is more than 1 effect of exposure.

16. Effect measure modification The determination of whether an effect is modified often depends on the measure of effect used in the analysis: a stratified analysis based on the risk difference might uncover heterogeneity where a ratio-based measure of association would not, and vice versa. For this reason, many epidemiologists prefer the term effect measure modification over the simpler, but less precise, effect modification.

17. Regression models Regression models deal more effectively with sparse-data problems than stratified analysis. They are more capable of simultaneously controlling for multiple confounding variables and joint confounding (where several variables act together to produce confounding). They also deal better with continuous variables, such as age, since these can be modelled directly, as continuous variables.

18. The “steps” of epidemiological regression modelling You usually begin by making an estimate of the crude measure of association. This is done using a statistical model that includes only 1 variable—the exposure. Such models are those presented in chapter 10—for example, a model of this type:

19. The “steps” of epidemiological regression modelling (continued) (where X e is a 0,1 indicator variable for exposure) This model describes a log odds of α for the nonexposed (where X e is 0) and a log odds of α + β for those exposed. In this type of model, β is a log odds ratio, so it can be used to calculate the crude estimate.

20. The “steps” of epidemiological regression modelling (continued) Next, you use a model to assess the possibility of effect measure modification. This is done using a model that has 3 terms: the exposure, the extraneous variable, and a term that represents exposure both to the exposure variable and the extraneous variable—something called an interaction term. The interaction term derives from the model being fit to a new variable created as the cross-product of the exposure and the extraneous variable.

21. The “steps” of epidemiological regression modelling (continued) Statistical software used to fit regression models will produce a P value for the interaction term. This is the equivalent of the Mantel-Haenszel test, which assesses heterogeneity. However, it is more flexible, as it can accommodate continuous variables.

22. The “steps” of epidemiological regression modelling (continued) If there is no evidence of effect measure modification (e.g., the interaction term is not important) the term can be removed. Now the model (which previously contained 3 terms: exposure, extraneous variable, and interaction term) just has 2 (exposure and extraneous variable). In such a model, the effect of exposure is adjusted for the potential confounding effects of the extraneous variable.

23. The “steps” of epidemiological regression modelling (continued) If the adjusted estimate of exposure differs from the crude, then confounding is demonstrated. If there is no confounding, the extraneous variable can be removed from the model and the crude effect can be reported. If there is confounding, the adjusted effect should be reported.

24. Critical appraisal and regression models In critical appraisal, you need to be able to understand analyses that present modelling results to discern whether the data collected in a study advance the state of knowledge about an exposure- disease association. The most common models encountered in the epidemiological literature are, like logistic regression, based on linear equations. Models use different transformations, so their parameters reflect different types of epidemiological estimates. A final note: when assessing analyses of models in critical appraisal, the first step is always to assess for effect measure modification.

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