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.

From the lesson

Module 2A: Confounding and Effect Modification (Interaction)

This module, along with module 2B introduces two key concepts in statistics/epidemiology, confounding and effect modification. A relation between an outcome and exposure of interested can be confounded if a another variable (or variables) is associated with both the outcome and the exposure. In such cases the crude outcome/exposure associate may over or under-estimate the association of interest. Confounding is an ever-present threat in non-randomized studies, but results of interest can be adjusted for potential confounders.

- John McGready, PhD, MSAssociate Scientist, Biostatistics

Bloomberg School of Public Health

Welcome back.

Â In this very short section, we're just going to give a little bit of insight as

Â to how adjusted estimates come about, the general idea behind the computations.

Â What we'll see, very shortly, is that multiple regression methods provide

Â a nice frame work for doing adjustments quickly and easily.

Â So, hopefully, upon completion of this short section you'll gain some insight,

Â conceptually, as to how adjusted estimates are computed.

Â So, let's first look at our fictitious study.

Â You'll recall that this was the fictitious study on a random sample from a population

Â of persons that were males, in a population of male and female adults.

Â And, there were 210 smokers and 240 non-smokers in this study sample.

Â And, the crude association between smoking, and

Â this not so rare disease outcome, again this is fictitious.

Â Was such that it was close to 1 relative risk was close to 1 but

Â in the sample smokers had a slightly lower risk of the disease than non smokers.

Â Then, when we actually broke things out specifically, we have sex,

Â we looked back behind the scenes, and

Â we had that sex was related to both the probability of having the disease.

Â Females were more likely to have the disease than males.

Â But, males were more likely to smoke than females.

Â So, sex was related to the outcome of disease in the predictor of smoking.

Â When we removed the variation in sex, between smoking and

Â non-smoking groups, i.e, we looked at the sex groups separately.

Â The relative risk among males of having the disease for

Â smokers compared to non-smokers was one point eight.

Â And, for females it was 1.5.

Â Both estimates are greater than one.

Â And, again, we're not considering statistical significance at this moment,

Â just using the estimates to illustrate the point.

Â So, how would we adjust for confounding?

Â Well, we, what we did here when z was categorical,

Â our potential confounder z of sex was categoricals.

Â We at, looked at the association between our outcome of disease and our predictor

Â of smoking separately by levels of that potential categorical confounder, sex.

Â So, our example of separate tables for males and

Â females is an example stratifying by a potential confounder.

Â And, what we could, we do, well we saw that the estimates,

Â the estimated relative risks were both greater than one by differing degrees, but

Â the difference in the estimates could be because of sampling variability.

Â Again, we're not at this point considering statistical significance for

Â this particular section, just talking about the overall concept.

Â Well, what could we do to take those sex specific estimates, and

Â aggregate them into one overall association between disease and

Â smoking that had been adjusted for sex.

Â Well, one way to do this would be to take a weighted average of

Â these stratum specific estimates, these sex specific estimates.

Â So, for example, to get a sex adjusted relative risk for

Â the smoking disease relationship, We could weight the sex

Â specific relative risk, for example, by the number of males and females.

Â And we could take a weighted average by taking the number of

Â males times the relative risk estimate for males plus the number of females times

Â the relative risk estimate for females, and divide it by the total sample size.

Â So, in this example there are 200 males, and

Â the relative risk of disease for smokers to non-smokers is 1.8.

Â There were 250 females, and the relative risk of disease for smokers to non-smokers

Â was 1.5, and the weighted average using that weighing scheme is 1.6.

Â So, this would be what we might call our sex adjusted relative risk of

Â disease and smoking.

Â There are better ways to do this, to take such a weighted average.

Â Instead of weighting by the sample size, we might be weight by the standard error

Â of the relative risk estimates, or the log relative risk estimates, and

Â do the weighted average on the log scale, and then exponentiate the results.

Â Bu,t this just illustrates the idea of stratifying by the potential confounder,

Â estimating the stata, stratam specific estimates of the outcome exposure

Â association, and then taking away the average across the strata.

Â We could also compute confidence intervals for these adjusted measures, but

Â we're going to save that until we get very shortly to multiple regression.

Â In this case, our outcome of disease was binary.

Â So, we could do a multiple logistic regression to relate the binary

Â outcome to smoking.

Â And, we'll see that this multiple logistic can be used to adjust that association for

Â other predictors.

Â And, this will be a very useful tool for performing adjustments, so

Â that we don't have to do this stratifying, averaging approach.

Â We've looked at the relationship between arm circumference and

Â height in the sample Nepalese children, less than a year old, and

Â we found that behind the scenes, not surprisingly, weight was

Â related to both the outcome of arm circumference and the predictor of height.

Â So, how could we go about adjusting this?

Â Well, weight is a little trickier as a potential confounder,

Â because it's measured on a continuum.

Â And, the adjusted results we presented in a previous lecture set were adjusted for

Â weight as a continuous variable.

Â But, here's the idea.

Â We could, behind the scenes, look at the relationship.

Â It's as if we were looking at the relationship between arm circumference and

Â height, for very tight weight ranges.

Â So, this is, you know, weight equal, or between 10 to 11 kilograms.

Â And, we do the same thing for arm circumference and height.

Â So, this is just trying to explain it conceptually for

Â the next weight group, between 11 and 12 kilograms, et cetera.

Â And, we could keep doing this for

Â small ranges of weight across the entire range of the sample.

Â And, what we'd get is we could estimate separate associations of the relationship

Â between arm circumference, get separate regression

Â lines of arm circumference and height for each of these small weight strata.

Â And, then what the algorithm for

Â presenting an over all weight adjusted association,

Â weight adjusted association, association between,

Â Arm circumference and height would involve taking the estimated regression slopes for

Â the regressions of arm circumference and height on each of these weight groups,

Â so I'll call that Beta One, weight one, Beta One, weight two.

Â And, we'd have multiple weight groups.

Â And, we can take a weighted average of these regression slopes to

Â get an overall adjusted regression slope of arm circumference and

Â height, after adjusting for weight.

Â This is just the idea behind the process.

Â This would not be feasible to do by hand.

Â And, this is where multiple regression, again, will be our, our saving grace,

Â because it will do this effortlessly and easily by the computer.

Â So, in summary, the adjusted association between an outcome y and a predictor x,

Â adjusted for a a single potential cofounder Z, can estimate, be estimated by

Â stratifying on Z which is actually hard to operationalize if Z is continuous.

Â When Z is binary, like sex, or

Â multi-categorical, the stratum are well defined.

Â But, if Z is continuous, we couldn't do this easily by hand,

Â unless we designated small ranges to enumerate the strata.

Â Then when, within each strata of Z,

Â we would estimate the Y X relationship, in whatever metric we were using,

Â whether it be a relative risk or a linear regression slope, etc.

Â >> and then we could take some sort of weighted average of all the Z

Â strata level specific Y/X associations.

Â So, we, across all stratum take our measure of association, and

Â average those across all the strata based on either the sample size of each strata,

Â or the standard error of the estimate in each strata, etc.

Â But, some weighting process that would give more weight to

Â those estimates informed by more information, or more precise information.

Â This idea can be generalized, estimating the adjusted association between Y and X,

Â adjusted for multiple potential confounders, Z1,

Â Z2...up to Z, however many potential confounders we have,

Â but obviously, that would be nearly impossible to do by hand.

Â Breaking our data up into groups stratified on multiple potential

Â confounders by all possible combinations of these multiple confounder values.

Â So this is where multiple regression methods are going to

Â make the adjustment process easy and straightforward.

Â But, at their core, this is essentially what they're doing behind the scenes with

Â some assumptions built in.

Â They're separating the data out into different strata based on the adjustment

Â variables, and then estimating the outcome exposure association, and,

Â then averaging across all those levels.

Â And, multiple regression can estimate multiple adjustment associations in

Â the context of one model.

Â So, when we'll see shortly when we expand what we

Â did in the first three lectures one of the natural ways to

Â interpret the results will be in terms of adjusted estimates

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