1. Confounding and Interaction: Part III
When Evaluating Association Between an Exposure and an Outcome, the Possible Effects of a 3rd Variable are:
Intermediary Variable
Effect Modifier
Confounder
No Effect
Using Stratification to Form “Adjusted” Summary Estimates to Evaluate Presence of Confounding
Concept of weighted average
Woolf’s Method
Mantel-Haenszel Method
Avoid statistical testing
Handling more than one potential confounder
Limitations of Stratification to Adjust for Confounding
the motivation for multivariable regression
Any questions from last week? If not, here is our roadmap for today. Two weeks ago, we defined and discussed confounding. Last week, we defined and discussed interaction, also known as effect modification. Today, we will integrate all of this material by reviewing the possible effects of a third variable when evaluating the association between an exposure and a disease - in other words, that a third variable can either be an intermediary variable, serve as an effect modifier, be a confounder, or have no effect. We will spend the majority of today talking about how stratification can be used to to form “adjusted” summary effect estimates in order to assess for and deal with the presence of confounding. To do this, we will review the concept of weighted averages and look at specific techniques known as Woolf’s method and the Mantel Haenszel method. In particular, we will describe, unlike what we did when we were evaluating for interaction, that there is no role for statistical testing when evaluating for confounding. We will also talk about the possible approaches for when there is more than one potential confounder to contend with.
Finally, although stratification is a very useful technique for evaluating for interaction and confounding, it does have its limitations that we will discuss. These limitations served as the driving impetus for the development of multivariable regression - a topic we will discuss in the next two sessions with Mitch Katz, starting next week.

2. When Assessing the Association Between an Exposure and a Disease, What are the Possible Effects of a Third Variable?
No Effect
Ok, for purposes of review, when we are assessing the association between an exposure and a disease, what are the possible roles of a third variable? A third variable could be acting as a confounder, or it could be an an intermediary variable depending upon how you are conceptualizing system, or it could be an effect modifier. Or, it could be none of these and simply have no effect. + C
Intermediary
Variable I
Effect Modifier _
Interaction:
MODIFIES THE EFFECT OF THE EXPOSURE
Confounding:
ANOTHER PATHWAY TO
GET TO THE DISEASE D

3. What are the Possible Effects of a 3rd Variable?
Intermediary Variable
Effect Modifier (interaction)
Confounder
No Effect
Intermediary Variable (conceptual decision)?
So, there are four possibilities: the third variable can be an intermediary variable, an effect modifier, a confounder, or have no effect.
Let me give you some context here. If you are working in a system where the effects of a third variable are already well known, then there is no question of what the third variable is. The issue is that you just need to crank through the numbers to see the particular magnitude of effect that the third variable has in your sample.
More typically, however, you are doing an analysis where you are not sure of the effect of the third variable. All that you know is that the third variable is not an intermediary in the system, at least the way you have conceptualized it. If the third variable is being conceptualized as an intermediary variable, then you should stop right there and do nothing else. In other words, just report the crude estimate. But assuming that the third variable is not being conceptualized as an intermediary variable, the more the typical situation is that you have suspicions that the third variable could be a confounder because, for example, it is known to be associated with the disease in question. You are also interested in whether the third variable is an effect modifier because this will giver a richer understanding of the system.
So, functionally, what you do first is to look for effect modification. If this is present, then you should report the association between the exposure and disease in terms of stratum-specific estimates based on the effect modifier. If no effect modification is present, you then look for confounding. If you deem that confounding is present, you will want to report an adjusted measure of association, one that is no longer confounded and summarizes the stratum-specific measures. If no confounding is present, you are left with the conclusion that the third variable had no effect. In that case, report the crude estimate as the final measure of association between exposure and disease.
Report Crude Estimate no
yes
Effect Modifier? no
yes
Report stratum-specific estimates
Confounder?
No Effect: Report Crude Estimate
Report “adjusted” summary estimate no
yes

4. Interaction with a Third Variable
Crude
RR crude = 1.7
Stratified
Heavy Caffeine Use
No Caffeine Use
Remember this example from a study of the effects of smoking and caffeine use in the occurrence of delayed pregnancies among women hoping to conceive. The principal exposure in question is smoking. We were interested in the effects of a third variable, caffeine use.
First of, in the way were conceptualizing the system, caffiene is not an intermediary variable. I guess we could have considered caffeine use an intermediary variable if we thought that the only way that smoking caused delayed conception was thru caffeine use, but this is not our intent. Instead, we are interested in knowing whether smoking has any direct biolological pathways in preventing conception.
So, we stratified by caffeine use and remember that we saw interaction.
It is not appropriate to try to summarize these two effects, 2.4 and 0.7, into one overall number and so instead we would report the two stratum-specific estimates separately. Stop here. End of story.
RRcaffeine use = 0.7
RRno caffeine use = 2.4

5. Is Interaction Present?
Does the relationship between the exposure and the outcome vary meaningfully (in a clinical/biologic sense) across strata of the third variable?
Does an average (adjusted) effect (formed by averaging the strata formed on the basis of the third variable) reasonably represent all strata?
if yes, go on to form an average (adjusted) measure
if no, stop - this is interaction; report stratum-specific estimates
Again, the real issue here when contemplating whether you believe interaction is occurring is whether there is a biologically meaningful difference across the strata. Stated another way, does an average of all the strata reasonably represent all of the strata. If no, this is interaction and stop right there. If yes, then go on to actually form this average using techniques we will discuss right now.

6. Declare vs Ignore Interaction?
Remember last week, we went through many examples of when we should declare versus ignore nteraction. We talked about how you need to take both the absolute magnitude of the differences in the point estimates between strata and the statistical test for interaction into account. The difference between a RR of 2.3 and 2.6, for example, is trivial and in general not worth shouting about regardless of the statistical test, whereas the difference between 2.0 and 20 usually is worth declaring but this does depend upon biol.plausibility especially when p values get high.
Any questions on this?

7. No Effect of Third Variable
Crude
OR crude = 21.0
(95% CI: )
Stratified
So, assuming you have not detected any interaction, what should you do next? Remember, the effect of matches on the association between smoking and lung cancer. There was no interaction and furthermore, matches had no effect on the association. In this case, we would report the crude estimate, only right?. Why not report the average of the stratum-specific estimates - after all the average is 21, the same as the crude? Well, one answer is that it is too much work. The second answer is that most of the time when you stratify, you pay a little price in terms of statistical precision. In other words, the confidence interval of the crude estimate will be narrower than the CI of the adjusted measure - they both will have 21 as their pt estimate but the crude assocation will be more precise. As you can see, the 95% confidence interval for the crude estimate is 16.4 to 26.9 compared to 14.2 to 31.1, which is determined by techniques we are going to describe later today. So, the point is, don’t adjust unless you have to.
Matches Present
Matches Absent
ORmatches = 21.0
OR no matches = 21.0
OR adj = 21.0
(95% CI: )

8. Confounding by Third Variable
Crude
OR crude = 8.8
Stratified
Smokers
Non-Smokers
How about our initial example looking at the association between matches and lung cancer. Here, after stratification on smoking: smoking present and smoking absent, we first saw no evidence of interaction, but more importantly, in each of the stratum, we found no effect of matches. In other words, smoking was confounding the relationship between matches and lung cancer. Here we would report the adjusted estimate, an average of the two stratum-specific estimates, in this case the average of 1.0 and 1.0 is 1.0.
ORsmokers = 1.0
OR non-smokers = 1.0
OR adj = 1.0

9. Forming an Adjusted Summary Estimate
Crude
OR crude = 3.5
Stratified
Age < 35
Age > 35
But what about a situation like this, looking at our example from last week evaluating the association between spermicide use and Down’s syndrome with the third variable being age. Remember, there is a fairly marked differences in the OR in the two stratum but look at the sample sizes. Some of the cells are rather small and therefore we know these numbers are not very statistically precise. When we looked at the test of homogeneity, we saw the the p value was 0.xx. We decided to ignore interaction and move on to see if there was confounding present. How should we do this?
ORage <35 = 3.4
ORage >35 = 5.7
Test of homogeneity: p = 0.71

10. How do we decide on a weight?
Assuming Interaction is not Present, Form a Summary of the Unconfounded Stratum-Specific Estimates
Right. We need to assign a weight to each stratum and then perform a weighted average.
Construct a weighted average
Assign weights to the individual strata
Summary Estimate = Weighted Average of the stratum-specific estimates
a simple mean is a weighted average where the weights are equal to 1
which weights to use depends on type of effect estimate desired (OR, RR, RD) and characteristics of the data
e.g.
Woolf’s method
Mantel-Haenszel method
To do this, we will form a weighted average. In other words, we will assign weights to the various strata and then take an average of the strata using these weights. Mathematically, it looks like this. We take the measure of association in each stratum and multipy by the weight and then divide by the total weight. A simple mean is an example of a weighted average where all the weights are 1 (Consider writing on the board: the average of 2, 4, 6, and 7 equal 1(2) + 1(4) + 1(6) + 1(7) / 4*1).
Which weight we should use depends upon a lot of factors, like the measure of association you are calculating and the nature of the data. Some methods that we will discuss are called Woolf’s method and the Method of M-H.
Hopefully the concept of a weighted average is understood by everyone. A simple mean is in fact a weighted average where the weights equal one. To get the average height of everyone in class, we add up everyone’s height and divide by the number of persons contributing. The weight is one.
How do we decide on a weight?
The second approach to getting a summary estimate is actually the one used by multivariable modeling approaches and we will touch on this briefly today. It is called the maximum likelihood approach

11. Forming a Summary Estimate for Stratified Data
After we have formed our strata and gotten rid of confounding, how do we summarize what the unconfounded estimates from the two or more strata are telling us. In the examples of last week, the measures of association from the different strata were identical. This is seldom the case.
Goal:
Create a summary “adjusted” estimate for the relationship in question while adjusting for the potential confounder
e.g.:
Case-control study of post-exposure AZT use in preventing HIV seroconversion after needlestick (NEJM 1997)
A more realistic is described in the Rothman chapter regarding the question of whether spermicide use might cause Down’s
To illustrate, lets work through another neat example. This is a case control study looking at the efficacy of AZT in preventing HIV seroconversion after a needlestick in health care workers. So, the exposure in question is the use of AZT and the outcome is the occurrence of HIV. In the crude analysis, the OR was 0.61, which by the way was not statistically significant, ie no strong evidence of a benefit from AZT. Are there any potential confounders we should be concerned about (without looking ahead in your handout?)
Crude
ORcrude =0.61
(95% CI: )
The goal is to combine or average the results from the different strata into one summary estimate. Any thoughts on how to do this?

12. Post-exposure prophylaxis with AZT after a needlestick
AZT Use
Right, severity of exposure. It is likely that the health care workers who took AZT were also the ones who had the most severe exposures (ie deep wound, big inoculum, end-stage AIDS patient source), the same exposures that were associated with a greater probability of HIV transmission.
Severity of Exposure
HIV

13. Forming a Summary Estimate for Stratified Data
After we have formed our strata and gotten rid of confounding, how do we summarize what the unconfounded estimates from the two or more strata are telling us. In the examples of last week, the measures of association from the different strata were identical. This is seldom the case.
Potential confounder: severity of exposure
Crude
A more realistic is described in the Rothman chapter regarding the question of whether spermicide use might cause Down’s
So, when the authors stratified by the severity of the needlestick., look at the stratum specific estimates; they are much lower. The first thing to decide is whether interaction is present. What do you think?
Well, this is a big difference in magnitude but note how few cases there are in the minor needlestick severity group . . .
ORcrude =0.61
Stratified
Minor Severity
Major Severity
OR = 0.0
OR = 0.35
The goal is to combine or average the results from the different strata into one summary estimate. Any thoughts on how to do this?

14. To stratify the subjects into those women with maternal age less than 35 and those with maternal age >= 35, you add a “by(matage) option. If you add a “, pool” option as I have here, the program will give you not only the default MH summary but also the Woolf estimate.
When we look at a statistical test of heterogeneity, we see a p value of 0.44 showing that chance could have easily caused this difference between strata. So, the authors did not decide to declare that there was any important interaction. That said, is there confounding here? To address this, we have to form a summary estimate of the two strata.
Finally, you are already familiar with this command but for sake of comparison let’s look at the summary estimate as obtained by logistic regression which as you know uses the MLE approach. As you can see, the MH estimate is essentially identical to the MLE in this problem.

15. Forming a Summary Estimate for Stratified Data
After we have formed our strata and gotten rid of confounding, how do we summarize what the unconfounded estimates from the two or more strata are telling us. In the examples of last week, the measures of association from the different strata were identical. This is seldom the case.
Goal:
Create a summary “adjusted” estimate for the relationship in question while adjusting for the potential confounder
e.g.:
AZT use, severity of needlestick and HIV seroconversion after needlestick (NEJM 1997)
A more realistic is described in the Rothman chapter regarding the question of whether spermicide use might cause Down’s
So, how are we going to summarize these two strata. How would you weight these strata? Would you give them equal weight?
Weight according to sample size? No. of cases? Variance?
Crude
ORcrude =0.61
Stratified
Minor Severity
Major Severity
OR = 0.0
OR = 0.35
The goal is to combine or average the results from the different strata into one summary estimate. Any thoughts on how to do this?

16. Summary Estimators: Woolf’s Method
One of the first approaches developed for forming summaryl adjusted estimates was Woolf’s method:
aka Directly pooled or precision estimator
Woolf’s estimate for odds ratio
where wi
wi is the inverse of the variance of the stratum-specific log(odds ratio)
Well, variance is the most intuitive approach and in fact is the basis of the one of the techniques developed called Woolf’s method. In this method, we actually initially work with log of the odds ratio. The weight is not the variance of the log odds ratio per se but it is the inverse of the variance. This makes sense, right because the bigger the variance, the inverse of the variance is small and thus a smaller weight. The smaller the variance, ie the more confident that you have nailed down the estimate, the inverse is larger - ie more weight. After you have taken the weighted average on the log scale, at the end you exponentiate this to get back to the native scale.
This is the inverse of the variance of the log odds ratio. This makes sense the more precise strata have the smallest variances and the inverse of a small number is a large number

17. Calculating a Summary Effect Using the Woolf Estimator
After we have formed our strata and gotten rid of confounding, how do we summarize what the unconfounded estimates from the two or more strata are telling us. In the examples of last week, the measures of association from the different strata were identical. This is seldom the case.
e.g. AZT use, severity of needlestick, and HIV
Crude
ORcrude =0.61
A more realistic is described in the Rothman chapter regarding the question of whether spermicide use might cause Down’s
Stratified
Minor Severity
Major Severity
When we actually crank through the numbers, assign each stratum a weight based on the inverse of its variance and take a weighted average, we uncover a problem with Woolf’s methods. It cannot handle cells that have zeros in them because a) you cannot take a log of 0; and b) you cannot divide by zero.
OR = 0.0
OR = 0.35
The goal is to combine or average the results from the different strata into one summary estimate. Any thoughts on how to do this?

18. Summary Estimators: Woolf’s Method
I discuss this approach first not only because it was one of the first proposed but also because it is the most conceptually straightforward.
Conceptually straightforward although computationally messy
Best when:
number of strata is small
sample size within each strata is large
Cannot be calculated when any cell in any stratum is zero because log(0) is undefined
1/2 cell corrections have been suggested but are subject to bias
Formulae for Woolf’s summary estimates for other measures (RR, RD, AR) available in texts and software documentation
sensitive to small strata, cells with “0”
computationally messy
It seems the most reasonable to assign each stratum according to how sure you are of the inference and the variance of the estimate is the best measure we have for this.
In the days before computers, this was considered computationally messy such that other easier methods were sought
So, Woolf’s methods is conceptually very straightforward and has its uses, especially when the number of strata are small and the sample size within each strata is large but does not work when there are zeros in cells. You can use substitute 0.5 in these cells to get answers but this is an imperfect solution.

19. Summary Estimators: Mantel-Haenszel
A more robust approach is the Mantel-Haenszel method
Summary Estimators: Mantel-Haenszel
Again, using the same cell definitions, the M-H estimate for the summary OR is the sum of a times d divided by T divided by the sum of . . .
Mantel-Haenszel estimate for odds ratios
ORMH =
wi =
wi is inverse of the variance of the stratum-specific odds ratio under the null hypothesis (OR =1)
Given this problem with cells of zero, a second method, the Mantel-Haenszel method is more widely used. Its formula is shown here. Remember, when the cells are set up like this, ad/bc is the odds ratio. So, here the weight is bc/N which happens to be the inverse of the variance of each stratum under the conditions when the OR=1. This is one of those things you are going to have to accept without proof. So,the weight is again related to the variance, but a special form of the variance, that is, when there is no association.
Note: need to find out more about bc/N to give students a more intuitive feel as to why it is a good weighting scheme
If we decompose this slightly, we can see that the weight is for each stratum is actually b times c divided by T. This is actually the inverse of the . . .
And the same logic as before, strata with the smallest variance get the most weight

20. Summary Estimators: Mantel-Haenszel
The MH is the most commonly used estimator.
Mantel-Haenszel estimate for odds ratios
resistant to the effects of large numbers of strata with few observations
resistant to cells with a value of “0”
computationally easy
most commonly used
It is fairly resistant (ie it doesn’t blow up) . . .
The Mantel-Haenszel technique, although not as straightforward conceptually as the Woolf technique, has many advantages. It is relatively resistant to large numbers of strata and can handle cells with zeros. It also is computationally very easy, although this is not an issue these days. For all these reasons, it is the most common approach to forming adjusted summary estimates.
Although really not a factor in the computer era, the computation of the MH estimator is a breeze.
More importantly is that the M-H closely approximates the MLE estimate which is generally regarded as the most accurate.

21. Calculating a Summary Effect Using the Mantel-Haenszel Estimator
After we have formed our strata and gotten rid of confounding, how do we summarize what the unconfounded estimates from the two or more strata are telling us. In the examples of last week, the measures of association from the different strata were identical. This is seldom the case.
e.g. AZT use, severity of needlestick, and HIV
ORMH =
Crude
ORcrude =0.61
A more realistic is described in the Rothman chapter regarding the question of whether spermicide use might cause Down’s
Stratified
Minor Severity
Major Severity
Let’s work our example and obtain an summary estimate for the association between AZT use and HIV seroconversion after adjusting for severity of needlestick.
What do we end up with? This is an example of negative confounding. The crude estimate was 0.6 and not even statistically significant. After adjustment for severity of needlestick the OR is now an impressive There is a 70% reduction in odds of HIV seroconversion associated with using AZT.
I really like this example. First, it illustrates how an elegant observational study design determined an extremely important biologic inference, one that could not be determined with a randomized experiment, because as you know health care workers refused to be randomized when such a study was started. Second, it illustrates how the authors would have blown making this inference if they had not paid attention to measuring and adjusting for important confounders.
OR = 0.0
OR = 0.35
The goal is to combine or average the results from the different strata into one summary estimate. Any thoughts on how to do this?

22. Calculating a Summary Effect in Stata
How can we make our lives a lot easier and implement all of this on the computer?
The epitab command - Tables for Epidemiologists is quite a little handy command. Has anyone used it ?
epitab command - Tables for epidemiologists
see Reference manual A-G
To produce crude estimates and 2 x 2 tables:
For cross-sectional or cohort studies:
cs variablecase variable exposed
For case-control studies:
cc variablecase variableexposed
To stratify by a third variable:
cs varcase varexposed, by(varthird variable)
cc varcase varexposed, by(varthird variable)
Default summary estimator is Mantel-Haenszel
, pool will also produce Woolf’s method
How do we do this in Stata? You know the commands already. Remember, for a cross-sectional or cohort study, to get the crude measure of association it is “cs followed by the outcome variable, followed by the exposure variable”. For a case-control study, it is “cc followed by outcome variable followed by exposure variable”
To stratify by a third variable, you add a comma and a “by(name of third variable)”. The default summary estimator is Mantel-Haenszel, but if you add a “, pool” you will also get the summary estimate using the Woolf method.

23. Calculating a Summary Effect Using the Mantel-Haenszel Estimator
After we have formed our strata and gotten rid of confounding, how do we summarize what the unconfounded estimates from the two or more strata are telling us. In the examples of last week, the measures of association from the different strata were identical. This is seldom the case.
e.g. AZT use, severity of needlestick, and HIV
. cc HIV AZTuse,by(severity) pool
severity | OR [95% Conf. Interval] M-H Weight
minor |
major |
Crude |
Pooled (direct) |
M-H combined |
Test of homogeneity (B-D) chi2(1) = Pr>chi2 =
Test that combined OR = 1:
Mantel-Haenszel chi2(1) =
Pr>chi2 =
Crude
ORcrude =0.61
A more realistic is described in the Rothman chapter regarding the question of whether spermicide use might cause Down’s
Stratified
Minor Severity
Major Severity
So, on the bottom is the Stata command. Cc outcome predictor, by(third variable) and then we added the pool option.
Here is the crude estimate again of 0.6.
Here are the stratum-specific estimates: 0 and 0.35
Here is the Woolf estimate, that Stata calls Pooled or Direct: Note it is undefined because Stata had the same problem we had when dividing by zero.
Here is the Mantel-Haenszel adjusted measure: 0.30 just like we got by hand on the prior slide. Note that the confidence interval is also provided for the MH-adjusted odds ratio and it is 0.12 to 0.79, thus not crossing 1. There is also a hypothesis being tested as well for the MH-adjusted OR and it is that the adjusted OR is equal to 1. This follows a chi-square distribution, regardless of the number of strata. Here the chi-square statistic is 6.06 with the corresponding p value of
OR = 0.0
OR = 0.35
The goal is to combine or average the results from the different strata into one summary estimate. Any thoughts on how to do this?

24. Calculating a Summary Effect Using the Mantel-Haenszel Estimator
In addition to the odds ratio, Mantel-Haenszel estimators are also available in Stata for:
risk ratio
“cs varcase varexposed, by(varthird variable)”
or st stir
rate ratio
“ir varcase varexposed vartime, by(varthird variable)”
or st strate
In addition to the adjusted summary estimate for the odds ratio, Mantel and Haenszel also have developed techniques for forming adjusted risk ratios and rate ratios. Again, the epitab command in Stata, using cs for risk ratio and ir for rate ratios can be used.

25. Mantel-Haenszel Confidence Interval and Hypothesis Testing
It is worth pointing that Stata is working hard for you behind the scenes in the Mantel-Haenszel confidence interval formation and hypothesis testing. Whereas the point estimate for the adjusted measure of association is easy to calculate, we just did it, the standard error, shown here,is not. But luckily Stata calculates all of this for you. And remember, the standard formula for a 95% confidence interval is the point estimate plus or minus 1.96 times the standard error.
For the hypothesis test of adjusted measure equal to one, this is the formula and it follows a chi-square distribution with 1 degree of freedom.

26. Mantel-Haenszel Techniques
Mantel-Haenszel estimators
Mantel-Haenszel chi-square statistic
Mantel’s test for trend (dose-response)
It is also worth pointing that in addition to the Mantel-Haenszel estimators we are discussing, you’ll see other techniques named after Mantel and/or Haenszel and it is worth keeping it straight which is which. We actually described the M-H chi-square statistic in the last slide and there is also something known as the Mantel test for trend , for which there’s a nice discussion in the Appendix of our green text book. . .

27. Assessment of Confounding: Interpretation of Summary Estimate
If the summary estimate, here a M-H OR estimator of 3.8
Compare “adjusted” summary estimate to crude estimate
e.g. compare ORMH (= 0.30 in the example) to ORcrude (= 0.61 in the example)
If “adjusted” measure “differs meaningfully” from crude estimate, then confounding is present
e.g., does ORMH = 0.30 “differ meaningfully” from ORcrude = 0.61?
What does “differs meaningfully” mean?
a matter of judgement based on biologic/clinical sense rather than on a statistical test
no one correct answer
10% change often used
your threshold needs to be stated a priori and included in your methods section
So, now that we have determined an adjusted summary estimate, what are we going to do with it? We now need to compare it to the crude estimate to determine if confounding is indeed present. If the adjusted estimate is meaningfully different than the crude, we then conclude that confounding is present.
So, of course, your next question is what does “differ meaningfully” mean? Is this the same arbitrary process we used when looking at interaction. Not really, because here we are guarding against bias whereas in the case of interaction we weren’t trying to prevent a bias but instead we’re giving a more detailed explanation of the system. So, already you know we have to be more conservative. Most importantly, this is this decision is based on your biological sense of the system under study - and this is not a statistical issue, as I will discuss in greater length in a few minutes. There is no one correct answer but if forced me to give you an answer I would say differences of 10% or greater in the measure of association would typically be considered meaningful. But, this is the kind of thing you want to be intellectually honest about and you should always state upfront before you embark upon an analysis what difference you are going to consider big enough.
So, its in the hands of the researcher

28. Summary Effect in Stata -example
e.g. Spermicide use, maternal age and Down’s
With this in mind, let’s consider an example using
Crude
OR = 3.5
Age < 35
Age > 35
Stratified
Let’s look at another example. Remember our question looking at the effect of spermicide use and the development of Down’s Syndrome. Remember after we stratified by maternal age, we saw these two stratum specific estimates: 3.4 and 5.7, but the p value for the test of homogeneity was 0.71 and so we were willing to pass on interaction and now we need to assess for confounding.
Here the crude estimate is 3.5 The adjusted OR by Woolf method is 3.82 and adjusted OR by M-H is Confounding or not? Well, this is a close call, right below the 10% cut-off, I guess. The point estimate has changed after adjustment from 3.5 to 3.8 but this is actually fairly trivial. You would probably be fine if concluded there was no meaningful confounding by age and you went with the OR of 3.5 as your final answer Part of the answer is in your feeling or knowledge about age and your certainty about its role as a confounder. On the one hand, if you are absolutely certain that age is associated with both spermicide use and Down’s, then it should probably be adjusted for and you should accept the adjusted OR as the right answer. But here, whereas we might be pretty sure that age is associated with Down’s, how certain are we that age is associated with Spermicide use. Maybe it is in general, but maybe it actually isn’t in your population. If you’re saying, why quibble - just adjust and accept the adjusted measure of association, the problem with this is when you adjust you add a certain amount of statistical imprecision. Note the CI for the crude vs adjusted; the adjusteds are slightly wider. So, there is no reason to adjust and take the statistical penalty unless you really need to.
OR = 3.4
OR = 5.7
Should we pool these?
Is there confounding present?

29. Presence or Absence of Confounding by a Third Variable?
Let’s go through a few more numerical examples.
Pretend this is the crude estimate, these are the stratum-specific estimates and this is the adjusted estimate, the weighted average of the stratum specific estimates. If the crude is 4.0 and the adjusted is 2, you would always report the adjusted estimate. If the crude is 0.2 and the adjusted is 0.8, I would definitely adjust.
However, if the crude is 4.0 and the adjusted is 4.1, I would not adjust. I would ignore this in favor of the crude estimate. Likewise, if the crude is 1.9 and the adjusted is 1.8, I would also probably ignore.

30. Statistical Testing for Confounding is Inappropriate
Testing for statistically significant differences between crude and adjusted measures is inappropriate
e.g. when examining an association for which a factor is a known confounder (say age in the association between HTN and CAD)
if the study has a small sample size, even large differences between crude and adjusted measures will not be statistically different
yet, we know confounding is present
therefore, the difference between crude and adjusted measures cannot be ignored as merely chance and must be reported as confounding
Many of you are perhaps saying why don’t we just apply a statistical test to see if the difference between the crude and adjusted is significant. If there is anything I want you to leave today with is that this is a bad idea.
Say you are working with a variable that you know is a confounder, like age in the evaluation of the relationship between hypertension and coronary artery disease. What if you had a small sample size? Then even large differences between the crude and adjusted measures of association might not be statistically significant. What are you going to do? Throw out age and just go with crude estimate of the relationship between hypertension and CAD? Of course, you won’t do this because you know that confounding is present! You cannot ignore differences between crude and adjusted measures just because they are not statistically significant. In other words, when protecting against bias, you have to do whatever you can regardless of statistical significance. We have to live with what we see as differences between crude and adjusted regardless of the statistics.

31. Statistical Testing for Confounding is Inappropriate
Furthermore, with large sample sizes, even factors which truly are not confounders can appear to cause confounding that is “statistically significant”
e.g., study of sunlight exposure and melanoma
prior knowledge: no relationship between gum chewing and melanoma
data: gum chewing is assoc. with sunlight exposure and with melanoma and adjusted measure of association is statistically different than the crude association?
should gum chewing be controlled for?
To resolve this paradox, only adjust for factors for which you have biologic rationale (i.e., some prior probability)
As a corollary to this, what if we had a large sample size study. In this case, factors, which truly are not confounders may appear to cause confounding which, if you performed a statistical test, is statistically significant.
Consider a study of sunlight exposure and melanoma. We have no prior knowledge that gum chewing is associated with melanoma (and for that matter I don’t think it is associated with sunlight exposure). But what if we did our study and then naively started to ask whether every variable that was collected, including gum chewing, was a confounder in the sunlight-melanoma association. What if we found that gum chew was associated with both sunlight exposure and with melanoma, and that it indeed result in a different adjusted measure of association than the crude association? Are we forced to adjust for gum chewing? The answer is that we don’t have to adjust for it if we have don’t believe that is biologically a confounder. In other words, we really never should have looked at gum chewing in the first place as a potential confounder if we did not have some biological rationale or some prior probability. In other words, we cannot let statistical tests guide us.
So, although you are forced to live with the result of the adjusted measure if it is meaningfully different than the crude, you don’t have to adjust for any or old thing that you don’t have some prior prob for.

32. Stratification - Effect of Excessive Correlation Between Exposure & Confounder
After we have formed our strata and gotten rid of confounding, how do we summarize what the unconfounded estimates from the two or more strata are telling us. In the examples of last week, the measures of association from the different strata were identical. This is seldom the case.
e.g. race/SES; income/education; no. of sexual partners/no. of anal intercourse partners
aka collinearity
precludes ability to adjust
A more realistic is described in the Rothman chapter regarding the question of whether spermicide use might cause Down’s
Crude
RRcrude =8.0
Let’s talk about a few more aspects of stratification to control for confounding. We covered this in the problem set this week. What happens if there is a tremendous correlation between the exposure in question and the potential confounder? Like what we be the case if race was the principal exposure and SES the confounder, or income as the exposure and education as the confounder, or no. of sexual partners and no. of anal intercourse partners.
As an example, let’s look at a hypothetical study of the association between income and acquisition of HIV. If we wanted to control for education what would happen? Well, if income and education are very tightly linked,the following would occur. In our low education stratum, we would mainly be limited to low income persons - there would no Hi income persons to serve as the comparison group. This stratum would have an undefined risk ratio - ie we can’t actually control for education. The statisticians call this collinearity and it has lots of bad effects. The effects are even more nefarious in multivariable regression models where the models may appear to work, but they cause standard errors and confidence intervals to become enormous. There is no real solution to this other than that you cannot adjust for these kinds of variables unless you have enormous sample sizes.
Stratified
Low Education
High Education
RR = undefined
RR = 12.5
The goal is to combine or average the results from the different strata into one summary estimate. Any thoughts on how to do this?

33. When More than One Additional Variable is Present
Crude
Stratified
white smokers
Finally, all of our examples so far today have dealt with situations when there is only one “third variable” - one potential effect modfier, one potential confounder. What if, as shown here in the example of chlamydia and CAD if there were more than one additional variable present?
black smokers
latino smokers
white non-smokers
black non-smokers
latino non-smokers

34. The Need for Evaluation of Joint Confounding
Variables that evaluated alone show no confounding may show confounding when evaluated jointly
Crude
Stratified
by Factor 1
by Factor 2
by Factor 1 & 2
The examples I have shown thus far have just one potential confounder to worry about. What should we do when more than . . .
In this example, the crude estimate is identical to the stratum specific measures when the 2 other variables are looked at separately.
There are a couple of approaches here. You could look at each third variable one variable at a time but this is a dangerous practice because it ignores the joint effects of the third and fourth variables. Consider this case control study where the crude OR is 2.2 and there are two additional variables, typically we call these covariates: factor 1 and factor 2. If we just stratified by factor 1, we would see no evidence of interaction or confounding, right? If we just stratified by factor 2, we would also see no evidence of interaction or confounding. If you stopped here, you might be tempted to say that there is no effect of factors 1 and 2 in the relationship between the primary exposure and the outcome.
But if we formed four strata based on the combinations of the two potential confounders, we would see the following. Each of the four strata shows no effect. In other words, when we looked at the joint effects of factors 1 and 2, confounding is present - there is no longer any association between the exposure and the disease.

35. Approaches for When More than One Potential Confounder is Present
This introduces the whole topic of
Backward versus forward confounder evaluation strategies
relevant both for stratification and especially multivariable modeling
Backwards Strategy
initially evaluate all potential confounders together (look for joint confounding)
conceptually preferred because in nature variables are all present and act together
Procedure:
with all potential confounders considered, form adjusted estimate
one variable can then be dropped and the adjusted estimate is re-calculated (adjusted for remaining variables)
if the dropping of the first variable results in an inconsequential change, it can be eliminated
procedure continues until no more variables can be dropped
Problem:
with many potential confounders, cells become very sparse and strata very imprecise
I know you are learning a bit about this in biostatistics. Which is preferable -backward or forwards?
So, this gets at the question of what approach you should use when there are multiple potential confounders. This is relevant not only for stratification, our topic of discussion today, but also for when you do multivariable regression.
We won’t have time to cover this in detail, but this to get you started with the some of the vocabulary In a backwards strategy, you start by initially looking at all the potential confounders jointly; ie look for joint confounding. This really is conceptually preferred because in nature these variables are all occuring together not in isolation.
Procedurally, we would form mutually exclusive and exhaustive strata based on all the potential confounders. Assuming interaction was not present, we would, just like we did when there was only one potential confounder, form an adjusted summary measure by averaging the different strata. We then drop one of the variables and recalculate an adjusted estimate. If the dropping of the variable results in no consequential change in the measure of association then it can be eliminated from further consideration. If dropping the variable does result in a meaningful change, then it cannot be dropped. Instead, another variable and you recalculate. The procedure continues until no more variables can be dropped. For example, using the last slide, you would adjust for the two factors jointly and get an OR adjusted of If you then dropped one of the two variables and got an adjusted estimate you would see a very large change and there conclude you could not drop that variable. You would then try to drop the other variable and see the same result. You would conclude that you have to keep both variables for adjustment.
The problem, however, with this is depending upon how many potential confounders there are and how many levels there are for each confounder the number of strata you’d need could be very large, and the cells in the strata very small rendering the weighting procedures unusable.
In fact, you may not even be able to get off the ground because the initial stratification is just too thin

36. Approaches for When More than One Potential Confounder is Present
In the forward selection approach, you start with . . .
Forward Strategy
start with the variable that has the biggest “change-in-estimate” impact
then add the variable with the second biggest impact
keep this variable if its presence meaningfully changes the adjusted estimate
procedure continues until no other added variable has an important impact
Advantage:
avoids the initial sparse cell problem of backwards approach
Problem:
does not evaluate joint confounding effects of many variables
In contrast to the backwards strategy is what is known as a forward strategy. This is where you start with just one potential confounder and then add variables one at a time keeping them only if they result in meaningful changes. This procedure has some advantages like that it gets away from the problem of having many sparse cells when looking at joint confounding of many variables but it has a big disadvantage in that it really does not fully look at the effects of joint confounding.
Procedurally, you would first form an adjusted measure of association using the confounder that gives the greatest change-in-estimate. Then, you would consider another variable and form an adjusted measure based on stratifying on the first variable and the second variable. If the addition of the second variable results in a meaningfuly change compared to the first variable, you would keep this variable. If not, drop it and consider the remaining variables. This continues until no other added varialbe has an important impact.

37. Stratification to Reduce Confounding
Although you are all now learning about the wonderful world of multivariable modeling, I would encourage you to examine your data whenever you can with stratification because it is the most native way to see your data and the easiest to explain your data to others
Advantages
straightforward to implement and comprehend
easy to evaluate interaction
Limitations
Looks at only one exposure-disease assoc. at a time
Requires continuous variables to be discretized
loses information; possibly results in “residual confounding”
Deteriorates with multiple confounders
e.g. suppose 4 confounders with 3 levels
3x3x3x3=81 strata needed
unless huge sample, many cells have “0”’s and strata have undefined effect measures
Solution:
Mathematical modeling (multivariable regression)
e.g.
linear regression
logistic regression
proportional hazards regression
It does, however, have its limitations which is principally that it breaks down with multiple confounders
Finally, although we have spent the last several sessions focusing on stratification as a very straightforward approach to evaluate interaction and confounding, stratification does have its limitations.
First, we can only look at one exposure-disease association at a time. Each time you want to use your same data to look at the association of another exposure and the disease under study, you have to re-format your 2x2 tables.
Second, what do we do if we have continuous variables as our exposure or potential coufounders - something like age, for example? To use stratification, we have to break these continous variables into categories in order to get them into our contingency tables. This unfortunately is not the richest use of continuous data.
But finally the biggest limitation of stratification, that we have already touched upon, is that it really deteriorates with multiple potential confounders. Suppose there are 4 potential confounders present each with 3 levels. If you wanted to look for the presence of joint confounding by these 4 pot CF’s you would have to form 81 strata and unless you had an enormous sample size, many of these strata would have unusable data and you could not perform adjustment.
The solution to all of these limitations lies in the use of mathematical models also known as multivariable regression. Mitch Katz will give you a conceptual approach to these in the next three sessions of this course and then you will learn the technical aspects of how to do these in the Biostatistics course starting in Jan.
These approaches are the topics of Mitch Katz’s upcoming sessions and your Thursday sessions.