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 4: Additional Topics in Regression

- John McGready, PhD, MSAssociate Scientist, Biostatistics

Bloomberg School of Public Health

So in this section we'll give two more examples of propensity score methods and

Â propensity score adjustment.

Â Focusing on two articles from the scientific literature.

Â So hopefully this will reinforce the concept of propensity scores and

Â propensity score adjustment with these examples.

Â So the first example we're going to look at was an article from the journal

Â the American medical association in 2010.

Â Pneumococcal vaccination and risk of acute myocardial infarction in men.

Â So they give it from the abstract here.

Â They say multiple studies have shown that preventing influenza by

Â vaccine reduces the risk of vascular events.

Â However, the effect of the pneumo,

Â pneumococcal polysaccharride vaccine on vascular events remains controversial.

Â So, their objective is to examine the association between this vaccination and

Â the risk of acute MI and stroke as well among men.

Â And, they used a prospective cohort study of Kaiser Permanente Northern and

Â Southern Californian health plans.

Â With ov, over 84,000 participants between 45 and

Â 69 years from the California Men's Health Study.

Â Who were recruited between January 2002-December 2003, and

Â they were followed up until December 31st, 2007.

Â And the cohort was similar to the population of health plan members and

Â men who responded to a general health survey in California,

Â on important demographic and clinical characteristics.

Â Demographic and

Â detailed lifestyle characteristics were collected from surveys.

Â And the vaccination records were

Â obtained from the Kaiser Immunization Tracking System.

Â So their main outcome measure here was the incidence of acute myocardial infarction

Â and stroke during the followup period in men.

Â WHo had no history of such conditions.

Â And they go on to detail the results, the unadjusted incident rates.

Â And then my amongst those who received the vaccine, and then those who didn't.

Â And then they say here.

Â With the propensity score adjustment, we found no evidence for

Â an association between pneumococal vaccination and reduced risk of acute MI.

Â And their adjusted hazard ratio was 1.09, so slightly higher estimated risk.

Â But the 95% confidence interval went from .98 to 1.21, so

Â the result was not statistically significant.

Â And they also do the same thing for the outcome of stroke.

Â And the adjust hazard ratio of stroke comparing those who got the vaccine

Â to those who did not was 1.14 with a confidence interval of 1 to 1.31.

Â So in both situations they found an increased risk.

Â Amongst those who got the vaccine, but it was not statistically significant.

Â They go on to say the inverse association was not,

Â also not found of men of different age and risk groups.

Â In other words, there was no apparent effect modification.

Â And, so they go on to conclude, among a cohort of men aged 45 years or older.

Â Recipi, receipt of the pnumococcal vaccine was not associated with

Â subsequent reduced risk of acute miocardio infarction and stroke.

Â So let's look at how theyd did this.

Â Let's first look at their methods section.

Â And just for

Â some, we'll read, I'll read this to you because we'll hear some things we've

Â covered over the entire course before we get into the propensity scores part.

Â But they say the association between vaccination status and

Â patient characteristics was assessed either using the chi-squared test for

Â categorical factors or the Keruskal-Wallis for continuous factors.

Â The Kruskal-Wallis test is a, a slightly different version of

Â the tee test teranova but it's testing the same idea.

Â That the centers of the distributions being compared are the same.

Â The association between vaccination and myocardial infarction or

Â stroke was assessed using the Cox proportional hazards regression model,

Â both in bivariate and multivariate models.

Â The latter, to adjust for the propensity score of vaccination with

Â a pneumococcal vaccine estimated from a logistic regression model.

Â The number of pneumococcal vaccines received was included in the models as

Â a time varied covariant which means they allowed this.

Â Measured to change the information unless they had multiple measures for

Â each person over the followup period.

Â The number of vaccines received before the study entry was included as

Â the initial number of vaccines, and the each vaccination during the followup

Â period resulted in the increment in the number vaccines administered.

Â Followup started at the time of the base line survey.

Â And ended at the time of acute MI or stroke diagnosis, so

Â if they had the event termination of health plan or membership.

Â So they were censored before having the even, death, or

Â December 31, 2007, the end of the study, whichever came first.

Â Mainly reported having a previous acute MI or

Â stroke at baseline were excluded from the analysis.

Â So let's talk about, let's read how they did the propensity scores and

Â then we'll get into the results.

Â They say the propensity score was created using

Â a logistic regression model predicting the probability of receiving at

Â least one dose of pneumococcal vaccine during the study period.

Â Quintiles the propensity score were used to adjust for

Â the likelihood of vaccination in the models of MI and stroke outcomes.

Â So what they did was estimated the probability of being vaccinated given

Â a bunch of patient characteristics and then adjusted for these.

Â Adjusted with this, split it into quintiles, five categories,

Â in the Cox model relating myocardial infarction to vaccination status.

Â So they go on to give a whole list of the variables they used to

Â estimate the propensity scores.

Â And these include race, ethnicity, region, household income, etc.

Â So they had a lot of potential confounders here that they

Â rolled into a single propensity score by doing that logistic aggression.

Â And then they in turn used the propensity score to adjust for

Â these distributions of these confounders that may differ between the vaccinated and

Â non-vaccinated group in the Cox regression.

Â Looking at the ri, log hazard of myocardial infractio, farction or stroke.

Â So then they go on, I just want to note this because they're actually getting into

Â the idea of effect modification here.

Â To examine whether the association between pneumococcal vaccination, acute MI,

Â and stroke would vary among participants of different ages, risk groups.

Â And number of influenza vaccination records, adjusted hazard ratios and

Â 95% confidence intervals estimated by Cox proportional hazards regressions were

Â presented separately for various subgroups.

Â And we'll show that this means they had to re,

Â replicate their analyses that they had done on everyone.

Â Slightly differently when they were scoring effect modification.

Â So when they were to look and see whether there was effect modification by age, for

Â example, they had to fit a Cox model that included predictors

Â of, vaccination status, call it X1.

Â Age, or some categorization of age, and then an interaction.

Â Between, vaccination status and age and

Â they actually ultimately dichotomized age for this analysis.

Â So age, or x two, was a binary above or below a threshold.

Â And then they had to adjust for propensity score quintiles.

Â But they would have had to re-run for this analysis the propensity scores without

Â using age as one of the factors they use to estimate the propensity score.

Â Similarly when they did it for the three presumable high risk groups.

Â Separately for current smokers, patients with history of diabetes, and

Â pasties, patients with history of hypertension, and one low risk group.

Â They had four different groups.

Â And they had to exclude things like smoking and

Â history of diabetes in the propensity score computation to do this analysis.

Â Since the risk groups that they need to include is predictors on their own, so

Â they could do the interaction, could be included in the model.

Â So here is what they got, here are the results they got.

Â I am only showing you for the myocardial outcomes, myocardial infarction outcomes,

Â but they had another table for stroke.

Â They give the hazard ratio 95% confidence interval unadjusted, and

Â then adjusted with the propensity scores.

Â So what they've actually found is that unadjusted those men who

Â actual received the vaccine had a higher risk of acute myocardial infarction and

Â it was statistically significant.

Â After adjustment, there was a slightly elevated risk among men in the sample but

Â it was not statistically significant.

Â As evidenced by this confidence interval containing one and the p value for

Â the association of 0.13.

Â They went on to show separate estimates of this association by age.

Â So here they dichotomized ages.

Â I was referring to for men less than 65.

Â And greater than 65 and

Â they found an elevated risk of myocardial infarction for those men who

Â were vaccinated in the younger group, but no association in the older group.

Â So there's some effect modification perhaps.

Â Those who were the effect actually presented itself,

Â the effect was harmful in terms of the myocardial infarction outcome.

Â And, they go on to do this again for the risk groups that we talked about, and

Â then the doses of influenza vaccine they had to see if

Â there was any evidence of effect modification.

Â And, they didn't find any subgroups with these different investigations where

Â the vaccine was protective against the outcome of myocardial infarction.

Â So, just to reiterate their results and what they said in their dis,

Â conclusions was that, with adjustment for propensity scores.

Â We found no evidence for association between pneumococcal vaccination and

Â reduced risk of myocardial infarction.

Â And this in addition part, they just go on to say.

Â That in addition there was no evidence of

Â effect modification such that the vaccine was protective for some subgroups.

Â Here's another example where propensity scores are used for adjustment.

Â This is from the British medit, Medical Journal, a study on mortality and

Â anti-psychotic drug use, or prescription in the elderly,.

Â So the abstract here says the TM objective is to

Â assess the risks of mortality associated with use of

Â individual anti psychotic drugs in elderly residents in nursing homes.

Â And this is actually done even despite the fact it was published in

Â the British Medical Journal, it was done in the U.S..

Â It was a population based cohort study with link data from Medicaid,

Â Medicare, something called the minimum data set in the national death index, and

Â a national assessment of nursing home quality.

Â And the setting was nursing homes in the United States.

Â And they had 75,000 plus participants.

Â Who were new users of anti psychotic drugs, and previously used them and

Â they had, six different drugs haloperidol, et cetera.

Â And they all participants were age greater than 65, greater than or

Â equal to 65 were eligible for Medicaid and lived in a nursing home.

Â And, what they used, their mean outcome measure,

Â they used Cox proportional hazard models to compare the 180 day risks of all-cause.

Â And cause-specific mortality by individual drugs after the start of the drug use,

Â with propensity score adjustment to control for potential confounders.

Â And they highlighted the drugs where they found differences.

Â And the results they say compared with risperidone users of haloperidol had

Â an increased risk of mortality, hazard ratio 2.07 95% confidence interval.

Â Not 1.89 to 2.26, and users of quetiapine a decreased risk compared

Â to risperidone and has a ratio of 0.81 with a significant confidence interval.

Â They go on to say the effects were strongest after the start of the treatment

Â and that implies to me that they, they didn't observe, or

Â somehow estimated, or allowed for non-proportional hazards.

Â But there was no mention of that in the article.

Â So, maybe I'm misinterpreting that.

Â But, the associations remain after adjustment for

Â dose, and were seen for all causes of death.

Â And they go on then in their conclusions to say,

Â though these findings cannot prove causality.

Â And we cannot rule out the possibility of residual confounding,

Â confounders that they didn't control for with their propensity score approach.

Â They provide more evidence of the risk of using these drugs in older patients,

Â reinforcing that the concept that they should not be used in the absence of,

Â in clear need.

Â The data suggests that the risk of mortality with these drugs is

Â generally increased with higher doses and

Â seems to be highest for haloperidol, and least for quetiapine.

Â So what they had to do here is they had 6 different antipsychotic drugs,

Â all comparing them to risperidone.

Â So they had to estimate separate propensity scores for

Â comparing or estimating the probability of using each of

Â these drugs versus risperidone, that reference drug.

Â So they talk about though the set up here,

Â they say we compare distributions of sociodemic graphic, clinical and

Â use characteristics among participants who started taking.

Â Different antipsychotics and calculated mortality rates during follow-up.

Â We censored, this is important, follow-up time at the time of discontinuation of

Â treatment, augmentation, or switch to a different drug.

Â And admission to the hospital for

Â ten days or more as treatment status is unknown during in-patient days.

Â So they go on to say, we fitted proportional hazards models for

Â pairwise comparisons against risperidone, unadjusted.

Â Then adjusted for age, sex and calendar years.

Â And then adjusted for multiple variables.

Â So they did separate Cox regressions.

Â Where they, their exposure or

Â their predictor variable was the drug they were looking at, was it wrong?

Â And if it was the drug they were looking at versus the common reference

Â of risperdone.

Â And what they had to do when they did the propensity scores for

Â each of these is that they had to estimate them separately.

Â For the probability of being on each of these drugs compared to risperidone.

Â In multi varied analysis we use propensity score adjustment to

Â balance potential confounders.

Â Propensity scores were derived from predicted probabilities of

Â the started treatment estimated.

Â In logistic models that contain all covariance above.

Â So I, there's a previous section that listen the entire list of these.

Â Cox models were stratified across tenths of the propensity score.

Â This means they created ten categories or no categories for the propensity scores.

Â The first decile, the second decile, and adjusted for

Â that, using the categorical indicators.

Â In addition we plotted multivariable adjusted Kaplan-Meier curves for

Â survival as a function of the duration of the use of the index antipsychotic drug

Â using what's called the inverse probability treatment weight.

Â That's another way of saying they estimated adjusted survival curves from

Â the Cox regressions.

Â It went on saying something co,

Â in confirmatory analysis we hit we hit high dimensional propensity scores.

Â So, they did one more layer of adjustment with another set of

Â propensity score that included more confounders.

Â Just to see if, by adjusting further, the results were robust and

Â similar to what they saw with their original adjustment.

Â So here's what we got.

Â The hazard ratios for death in elderly people in nursing homes within 180 days

Â of the start of the treatment with various antipsychotic drugs.

Â And what they show is the ha, hazard ratios comparing the relative hazard of

Â mortality across the follow-up period.

Â For each of these five treatments, all each compared to risperidone.

Â So for haloperidol the unadjusted association was

Â a hazard ratio of 2.42 and it was statistically significant.

Â After they adjusted for

Â age, sex and calendar year, the association was about the same.

Â When they adjusted for the propensity scores it was.

Â There was still over a two fold increase in the hazard.

Â And it was statistically significant but

Â the estimate and confidence intervals shifted down slightly.

Â And when they did this, high dimensional propensity scores, something where

Â they've added even more covariants in estimating the propensity score.

Â They got very similar results,

Â a slightly attenuated estimate in confidence interval.

Â But again, and

Â even in this best case scenario, an estimated 81% increase in mortality.

Â With a confidence interval 1.65 to 1.98 for haloperidol compared to ziprasidone.

Â And then they showed the results for the other drugs as well.

Â And after adjustment,

Â the associations that were significant at the original adjustment with the original

Â propensity scores were the difference between Haliperadol and Risperadone.

Â And the difference between Queitiapine and Risperadol.

Â The other results were not statistically significant.

Â And as a sensitivity analysis to adding even more potential confounders,

Â they showed that this significance held.

Â And similar magnitude and

Â direction of association with the additional adjustment.

Â So, anyway, hopefully this is giving you some sense of how propensity scores

Â are used to adjust with a large number of covariance when there's one.

Â Predictor outcome interest of association.

Â These authors were not concerned with the relative impact of

Â these other things on mortality.

Â And didn't want to water down the precision of these estimates between

Â competing antipsychotic drugs over the followup period.

Â By including multiple Xs for each of the potential confounders, so

Â they roll them all into one propensity score and adjust it with that.

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