A practical and example filled tour of simple and multiple regression techniques (linear, logistic, and Cox PH) for estimation, adjustment and prediction.

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Statistical Reasoning for Public Health 2: Regression Methods

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A practical and example filled tour of simple and multiple regression techniques (linear, logistic, and Cox PH) for estimation, adjustment and prediction.

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Module 2B: Effect Modification (Interaction

Effect modification (Interaction), unlike confounding, is a phenomenon of "nature" and cannot be controlled by study design choice. However, it can be investigated in a manner similar to that of confounding. This set of lectures will define and give examples of effect modification, and compare and contrast it with confounding.

- John McGready, PhD, MSAssociate Scientist, Biostatistics

Bloomberg School of Public Health

So in this section we'll continue our discussion of effect modification and

Â we'll look at several examples of studies where one of

Â the researcher questions involved was investigating effect modification.

Â So this lecture section will give more examples of effect modification and/or

Â the processes used where necessary to investigate effect modification.

Â So let's look at this first article from the American Journal of Epidemiology.

Â And the title of the article gives some hint as to where it

Â was going in terms of effect modification.

Â It says, Similar Relation of Age and

Â Height to Lung Function Among Whites, African Americans, and Hispanics.

Â So notice the talk about in this case.

Â If you read the article more thoroughly you'll see their outcomes of

Â interest have more to deal with lung function.

Â And some of their predictors of interest include age and height.

Â And what they conclude based on this article that,

Â is that the relationship between lung functioning variables and age and

Â height is statistically equivalent among Whites, African Americans, and Hispanics.

Â So that they can estimate one overall association that applies to

Â all three ethnic groups as opposed to there being effect modification,

Â which would necessitate separate measures of the associations between lung function,

Â age, and height.

Â For each of the three eth, ethnicities.

Â So lets look at what they say in the abstract.

Â They say, current guidelines recommend separate spirometry reference

Â equations for whites, African Americans, and

Â Mexican Americans, but the justification for this recommendation is controversial.

Â So what the authors were curious to see is if data they

Â collected supported this process of having separate estimate

Â reference equations relating lung functioning to age and height, etc.

Â For the separate ethnic groups.

Â In other words,

Â they are investigating whether there is indeed effect modification.

Â So what they say is the authors examine the statistical justification for race and

Â ethnic specific reference equations in adults in both

Â the Third National Health and Nutrition Examination Survey and

Â the Multi-Ethnic Study of Atherosclerosis Lung Study.

Â Spirometry was measuring followed,

Â measured following American Thoracic Society guidelines.

Â And then they go on to describe what they mean by statistical justification.

Â Statistical justification, and, for estimating separate associations was

Â defined as the presence of effect modification by race or

Â ethnicity among never smoking participants without respiratory disease or symptoms.

Â So they go on to say in the abstract, there was actually no evidence of

Â effect modification by race, ethnicity for forced expiratory volume in one second.

Â Forced vital capacity or the forced expiratory volume in one second

Â ratio compared to forced vital capacity in the three different ethnic groups.

Â So, they went on to do an analysis and use the statistical techniques to,

Â to test whether or not the relationship between these respiratory outcomes and

Â predictors such as age and height were statistically different among the three

Â ethnicity groups, and they found no evidence of a difference.

Â We'll see how to do such tasks shortly in the upcoming sections on

Â multiple regression.

Â But what they're concluding here is based on this updated data this more

Â modern data, there was no evidence to support the previously thought notion

Â that the relationship or the reference equations that related lung functioning,

Â the characteristics in such as the age and height needed to be different.

Â And there were associations for

Â white, Africa-American and Mexican-American men or women.

Â They did go on to say though that the mean lung function for a given age, gender and

Â height was the same for whites and

Â Mexican Americans, but was lower for African Americans.

Â So they did conclude that there were some overall differences between

Â the ethnicities after adjusting for age, gender, and

Â height, but then ultimately that the relationship between lung function and

Â age, gender, and height did not depend on what ethnicity the person is.

Â And this second article from the New Jer, England Journal Medicine their looking at

Â statins to prevent vascular vents in men and

Â women with elevated c-reactive protein.

Â So this is a randomized study where the researchers randomized 17,800 healthy,

Â in other words those without a history of cardiovascular disease, men and

Â women with non-elevated LDL cholesterol levels to

Â receive either 20 milligrams of statins daily, or a placebo.

Â And the subjects were followed for up to five years.

Â At the end of the follow-up period, the study results include the following.

Â Of the 8900 subjects randomized to the statins groups,

Â a 142 developed cardiovascular disease.

Â And of the 8900 subjects who were randomized to the placebo group,

Â 251 developed cardiovascular disease.

Â So the unadjusted incidence rate ratio here is very similar to

Â the straight comparison proportions, but the incidence rate ratio accounts for

Â potentially differing follow up periods for

Â each individual is the incidence rate ratio is 0.56 indicating that.

Â And this is unadjusted.

Â This compares the incidence rate cardiovascular disease development for

Â those who are randomized to receive statins to those who are randomized to

Â receive a placebo.

Â And this estimates a 44% reduced instance or risk of developing cardiovascular

Â disease in the follow up period in the statins group relative to the placebo.

Â And this result is statistically significant and

Â the 95% confidence interval goes from 0.46 to 0.69.

Â Now the authors did not go ahead and

Â report adjusted relative risk or incidence rate ratio.

Â Despite other characteristics that may be associated with

Â cardiovascular disease development.

Â Sex.

Â Age, and smoking.

Â But why do you think they didn't have to go ahead and

Â report an adjusted incidence rate ratio?

Â Well, the study was large and randomized.

Â So ostensibly, if they were to report the adjusted incidence rate ratio,

Â which should be similar,

Â if not identical, to this other adjusted incidence rate ratio, 0.56.

Â But because of randomization there was very little potential for any confounding.

Â However, the authors did investigate interactions or

Â effect modification between some of these characteristics and stats.

Â They said, well, the overall result that we just presented is not

Â confounded by distributional differences between the statin and

Â the placebo groups in these other measures.

Â However, it is possible to the association between cardiovascular disease and

Â statin use differs depending on the level of some of these other characteristics.

Â So, this is a very common type of table shown in

Â the results from randomized clinical trials.

Â Especially where they look at the association of interests separately for

Â different levels of other variables.

Â So for example, they go ahead and show the estimated instance rate

Â ratio of mortality for those on statins compared to those on

Â placebo among males only and they give the estimated as a ratio here.

Â Now that's confidence interval and females only.

Â And they give the estimated haz ratio here and it's confidence interval.

Â This vertical line here, this dotted vertical line is

Â the 0.56 that they estimated for the overall association.

Â And this solid line here is one, which would be the null value.

Â So we can see very quickly that the association is statistically significant

Â for both males and females, as neither confidence interval includes one, but

Â you can see the estimates are relatively close to one another, and

Â the confidence intervals overlap.

Â So this suggests strongly that the relationship between

Â cardiovascular disease development statins does not differ between males and females.

Â In other words the association is not modified by sex.

Â They do report something here that we haven't explored yet, but

Â we will when we get into multiple regression techniques.

Â There is a way to formally test whether the population level

Â associations between cardiovascular disease and

Â statins are statistically different for males and females.

Â The null is that they are not different and this P-value is quite high,

Â indicating the we would fail to reject a null, which is consistent with

Â the fact that the estimates were similar and the confidence intervals overlap.

Â They went in to do this type of analysis stratifying by age.

Â They wanted to see if there was a difference in the association for

Â younger people as defined by those less than equal 65 years and

Â those greater to equal 65 years.

Â And the estimates are different, as you can see, but the confidence intervals

Â overlap and the interaction was not statistically significant.

Â And they do this for several other characteristics.

Â So what they ultimately report, the ultimately did not find any evidence of

Â effect modification, even though they investigated it.

Â So they go on to actually report the results like this.

Â They say the rates of the primary end point were 0.77 and

Â 1.36 per 100 person-years of follow-up in the statins and

Â placebo group, respectively, with a hazard ratio for statins of 0.56.

Â In a 95% confidence interval 0.46 to 0.69,

Â this is what we showed when we started talking about this.

Â And a very small p-value.

Â However, they go on to say consistent effects were observed in

Â all subgroups evaluated.

Â So, what they're saying, in other words, is there was an overall association and

Â it did not, in their investigation,

Â appear to differ for different subgroups of the population.

Â So the message that they are giving is pretty clear that there's an overall

Â association of reduced cardiovascular risk associated with statin use.

Â And this relationship does not vary by sex or by age.

Â Or by any other characteristic they did a subgroup analysis on to look for

Â effect modification.

Â So they found no evidence of effect modification by any of

Â the factors that they examined in their study.

Â Here's another study.

Â Plasma Enterolignan Concentrations in

Â Colorectal Cancer Risk in a Nested Case-Control Study.

Â 'Kay. So this is a nested case-control study.

Â We haven't looked at many case-control studies but

Â we can still appreciate the associations.

Â So enterolignans and biphenolic compounds that

Â possess several biologic activities whereby they may influence carcinogenesis.

Â The authors investigated the association between plasma entero lignan and

Â enterodiol and colorectal cancer risk in a Dutch prospective study.

Â Among more than 35,000 participants age 20-59 years

Â 160 colorectal cancer cases were diagnosed after seven point years of follow up.

Â So they used this as their starting point, these 160 cases.

Â And they matched members in the cohort on frequency matching to the cases on age,

Â sex, and study center.

Â So they selected about double, two and a half times the number of controls.

Â Frequency match.

Â Not one to one matching.

Â But they took a, a control group that had similar characteristics in terms of

Â the age, sex, and sex distribution, and study center distribution as the cases.

Â Okay, so they actually show that plasma, enterodiol and enterolactone were

Â not associated with the risk of colorectal cancer after adjustment for

Â known colorectal cancer risk factors.

Â And so they estimated odds ratio comparing the highest quartile versus

Â the lowest quartile.

Â So they categorized the enterodiol levels into four quartiles.

Â The odds ratio is 1.11, and the results are not statistically significant.

Â And similarly they did this thing for the enterolactone quartiles.

Â And while they showed in the sample an elevated odds

Â of colorectal cancer in the highest quartile to the lowest, the results were

Â not statistically significant as the confidence interval includes one.

Â However, they go on to say, sex and body mass index

Â modified the relationship between plasma enterolactone and colorectal cancer risk.

Â Increased risks were observed among women and subjects with high body mass index.

Â So what they're saying is on the whole there was an association after accounting

Â for the sampling variability.

Â But in certain subgroups they found that there was an association of increased risk

Â associated with increased levels of these, of the plasma enterolactone.

Â And this was found among women, but necessarily among men.

Â And among those with a high body mass index, but

Â the association didn't hold up for other body mass index.

Â So what they're saying is that the effect of ent-,

Â enterolactones on their risk of colorectal cancer as

Â measured by the odds ratio was modified by these characteristics.

Â A different association existed for women than men, for example.

Â And let's look at one more example,

Â the association of race with age among survival patients undergoing dialysis.

Â And we looked at this several times with statistical reasoning one, but

Â we'll come back to it.

Â Now I'll just give you that context.

Â It says from the abstract here, many studies have reported that black

Â individuals undergoing dialysis survive longer than those who are white.

Â This observation is paradoxical given racial sparities in

Â access to inequality in care.

Â And is inconsistent with observed lower survival among black patients with

Â chronic kidney disease.

Â And one of the things they hypothesized was that age modified survival differences

Â by race.

Â So this goes on to talk about the study design this is just to

Â say they pulled a large number of medical records from the Center for Medicare and

Â Medicaid services forms.

Â And then coupled it with data from the United States Renal Data System.

Â And what they did in order to replicate previous studies was look at a Cox

Â proportional hazard model to estimate the association between mortality and

Â race and they actually went on to adjust for

Â a bunch of other characteristics that may differ.

Â Between the black and white subjects receiving dialysis including age, sex and

Â insurance type.

Â And we'll see how to adjust with multiple cox regression in our

Â section on multiple regression.

Â But they did this first for everyone.

Â And then they went on to actually look at the association.

Â Between mortality and

Â race adjusted for these other characteristics but separately by age.

Â And so they said to confirm whether the differences between age groups were

Â statistically significant an additional model was built.

Â And including interactions terms for each category and black race.

Â And again, we'll get how to do this in the multiple regression section.

Â But essentially what they said is they used an approach that allowed them to

Â estimate the separate relationship between mortality and

Â race, adjusted for other characteristics, for separate age groups.

Â To see if the association between mortality and

Â race differed by age of the subjects.

Â And so we've looked at this picture before.

Â But this is a close up of table 2.

Â As close as I can get it.

Â And these are the results from resulting analysis.

Â What they are presenting here is the adjusted relative hazard of mortality for

Â black patients to white patients in each of these age groups.

Â And so they adjust for

Â a bunch of other characteristics in each analysis that may differ between black and

Â white patients who are going on dialysis and may affect mortality.

Â But what you see here is that in the early ages, 18 to 30,

Â this dot here is the estimated hazard ratio and this is the confidence interval.

Â And this is actually all scaled, you may remember on the long scale.

Â So these intervals are symmetric and comparable in terms of the risk.

Â But what we see here is that younger ages the instance rate ratio.

Â The mortality for black to white patients is above one and

Â statistically significant.

Â So black patients on, on dialysis have a higher risk of

Â mortality compared to white patients in the 18 to 30-year-old age group.

Â And this goes, decreases, but persists to be higher and

Â statistically significant for blacks compared to

Â white patients who are 31 to 40 years old when receiving dialysis.

Â But after that age group, the trend goes the other direction,

Â the relationship between mortality and race changes the other direction.

Â After age 41 blacks consistently have a lower risk of

Â mortality compared to white patients after adjustment for other differences.

Â And so what the authors are showing here effectively is that

Â there's effect modification by age.

Â That the relationship between mortality and race, as defined by black or

Â white in dialysis patients is modified by age.

Â So in summary, we'll look at a few more examples here.

Â Effect modification occurs when the relationship between two quantities,

Â Y and X, depends on the level of a third quantity, Z.

Â And effect modification cannot be ascertained by comparing unadjusted or

Â crude associations and adjusted estimates adjusted fo Z.

Â We actually need to see separate estimates of the Y/X relationship for separate

Â levels of Z in order to ascertain whether that association is different or not.

Â We will show very shortly how to set this up in a regression context and

Â how to formally test whether at least some of the associations are different for

Â some of the levels of Z.

Â But, the fundamental idea holds that in order to investigate effect

Â modification the researcher has to consider doing so in advance and.

Â Allow for the estimation of the relationship of interest separately for

Â different levels of the potential effect modifier.

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