Logistic Regression and Confounding

8/13/2019 Logistic Regression and Confounding

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Advanced Data Analysis:

Methods to Control for Confounding(Matching and Logistic Regression)


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Goals Understand the issue of confounding in

statistical analysis

Learn how to use matching and logisticregression to control for confounding


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Confounding Example: people in a gastrointestinal outbreak

Mostly members of the same dinner club BUT many clubmembers also went to a city-wide food festival

Food handling practices in the dinner club might be blamedfor the outbreak when food eaten at the festival was thecause

Membership in the dinner club could be a confounderof therelationship between attendance at the food festival andillness

Analyzing the data to account for both dinner clubmembership and food festival attendance could helpdetermine which event was truly associated with theoutcome


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Confounding Gastrointestinal outbreak (continued)

Stratification methods could be used to calculate

the risk of illness due to the food festival for thosein the dinner club vs. those not in the dinner club

If attending the food festival was a significant riskfactor for illness in both groups, then the festival

would be implicated because illness occurredwhether or not people were members of thedinner club


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Confounding What if there are multiple factors that might be

confounding the exposure-disease relationship? Using our previous example, what if we had to stratify by

membership in the dinner club andby health status? Orstratify by other potential confounders (age, occupation,income, etc.)?

Trying to stratify by all of these layers becomes difficult

At this point more advanced methods are needed: Logistic regressioncontrols for many potential

confounders at one time

Matchingwhen incorporated correctly into the studydesign, reduces confounding before analysis begins


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Confounding Confounders In field epidemiology, we commonly compare two

groups by using measures of association: Risk ratio (RR) in cohort studies

Odds ratio (OR) in case-control studies

May have multiple exposures significantly associatedwith disease or no exposures associated In these cases you need to explore whether a confounder is

present making it appear that exposures are associated with

the disease (when they really are not) or making it appearthat no association exists (when there really is one)


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Confounders A confounderis a variable that distorts the risk ratio

or odds ratio of an exposure leading to an outcome Confounding is a form of bias that can result in a distortion

in the measure of association between an exposure anddisease

Confounding must be eliminated for accurate results (1)

Confounding can occur in an observationalepidemiologic study whenever two groups are

compared to each other Confounding is a mixing of effects when the groups are

compared (exposure-disease relationship can be affected byfactors other than the relationship)


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Common Confounders Common confounders include age,

socioeconomic status and gender.

Examples: Children born later in the birth order are more

likely to have Downs syndrome. Does birth order cause Downs syndrome?

Norelationship is confounded by mothers age, older

women are more likely to have children with Downs Mothers age confounds the association between birth

order and Downs syndrome: appears there is anassociation when there is not (2)


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Common Confounders--Examples Womens use of hormone replacement therapy (HRT)

and risk of cardiovascular disease Some studies suggest an association, others do not

Women of higher socio-economic status (SES) are morelikely to be able to afford HRT

Women of lower SES are at higher risk of cardiovasculardisease

Differences in SES may thus confound the relationship

between HRT and cardiovascular disease Need to control for SES among study participants (3)


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Common Confounders--Examples Hypothetical outbreak of gastroenteritis at a

restaurant Study shows women were at much greater risk of

the disease than men Association is confounded by eating salad

women were much more likely to order salad thanmen

Salad was contaminated with disease-causing

agent Relationship between gender and disease was

confounded by salad consumption (which was thetrue cause of the outbreak)


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Controlling for Confounding To control for confounding you must take the confounding

variable out of the picture There are 3 ways to do this:

Restrict the analysisanalyze the exposure-disease relationship

only among those at one level of the confounding variable Example: look at the relationship between HRT and cardiovascular

disease ONLY among women of high SES

Stratifyanalyze the exposure-disease relationship separately forall levels of the confounding variable

Example: look at the relationship between HRT and cardiovasculardisease separately among women of high SES and low SES

Conduct logistic regressionregression puts all the variables into amathematical model

Makes it easy to account for multiple confounders that need to becontrolled


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Controlling for Confounding:

Stratification Stratification can be used to separate the

effects of exposures and confounders Example: tuberculosis (TB) outbreak among

homeless men Homeless shelter and soup kitchen implicated as

the place of transmission Men likely to spend time in both places

To determine which site is most likely, couldexamine the association between the homelessshelter and TB among men who did NOT go to thesoup kitchen and among men who DID go to thesoup kitchen


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Stratification--Example After conducting a case-control study, overall

data show the following:

Cases Controls Total

Cookies 37 21 58

No Cookies 13 29 42

Total 100

Cookie Exposu re

OR = (37x29)/(21x13) = 3.93; 95% CI, 1.69 9.15

p= 0.001*


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Stratification--Example

Data continued..


Punch 40 20 60

No Punch 10 30 40

Total 100

Punch Exposu re

OR = (40x30)/(20x10) = 6.00; 95% CI, 2.83 12.71

p= 0.0004*


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Both cookies and punch have a high odds ratio forillness & a confidence interval that does not include 1 OR (cookies) = 3.93; 95% CI, 1.699.15, p= 0.001*

OR (punch) = 6.00; 95% CI, 2.8312.71, p= 0.0004* To stratify by punch exposure, we want to know:

Among those who did notdrink punch, what is the oddsratio for the association between cookies and illness?

Among those who diddrink punch, what is the odds ratio forthe association between cookies and illness?

If cookies are the culprit, there should be an associationbetween cookies and illness, regardless of whether anyonedrank punch


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Stratification of the cookie association bypunch exposure:


Cookies 35 17 52

No Cookies 5 3 8

Total 60

Did have punch

OR = (35x3)/(17x5) = 1.3; 95% CI, 0.17 7.22

p= 1.0*


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Stratification of the cookie association by punch

exposure:


Cookies 2 4 6

No Cookies 8 26 34

Total 40

Did not have punch

OR = (2x26)/(4x8) = 1.63; 95% CI, 0.12 13.86

p= 0.63*


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To stratify by cookie exposure, we want toknow: Among those who did noteat cookies, what is the

odds ratio for the association between punch andillness?

Among those who dideat cookies, what is theodds ratio for the association between punch and

illness? If punch is the culprit, there should be an

association between punch and illness, regardlessof whether anyone ate cookies


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Stratification of the punch association bycookie exposure:


Punch 35 17 52

No Punch 2 4 6

Total 58

Did have cookies

OR = (35x4)/(17x2) = 4.12; 95% CI, 0.52 48.47

p= 0.18*


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Stratification of the punch association bycookie exposure:


Punch 5 3 8

No Punch 8 26 34

Total 42

Did not h ave cook ies

OR = (5x26)/(3x8) = 5.42; 95% CI, < 0.80 40.95

p= 0.08*


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Stratification

Stratification allows us to examine two riskfactors independently of each other

In our cookies and punch example we cansee that cookies were not really a risk factorindependent of punch (stratified ORs 1)

Punch remained a potential risk factorindependent of cookies (large ORs and p-values close to significant)


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More on Stratification

Mantel-Haenszel odds ratio Method of controlling for confounding using stratified

analysis

Takes an association, stratifies it by a potential confounderand then combines these by averaging them into oneestimate that is controlled for the stratifying variable

Cookies and punch example: 2 stratum-specific estimates of the association between

punch and illness (ORs of 4.1 and 5.4) More convenient to have only one estimatecan average

two estimates into a pooled or common odds ratio


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Stratification andEffect Measure Modifiers

Effect measure modification One stratum shows no association (OR 1) while another

stratum does have an association No confounding third variable present, rather, need to

identify and present estimates separately for each level orstratum

Example: if gender is an effect measure modifier, youshould give 2 odds or risk ratios, 1 for men and 1 forwomen

You identify effect measure modification bystratification (same technique used to identifyconfounding) but you are looking for the measure ofeffect to be different between the 2 or more strata


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Effect Measure Modifiers--Examples

Among the elderly, gender is an effect modifier of theassociation between nutritional intake and osteoporosis Nutritional intake (calcium) is associated with osteoporosis

among women Among men this association is not so strong because mens

bone mineral content is not affected as much by nutritionalintake

In developing countries, sanitation is an effect modifier ofthe association between breastfeeding and infantmortality In unsanitary conditions, breastfeeding has a strong effect in

reducing infant mortality In cleaner conditions infant mortality is not very different

between breastfed and bottle-fed infants


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Matching

Matching can reduce confounding

In case-control studies cases are matched to

controls on desired characteristics In cohort studies unexposed persons are matched

to exposed persons on desired characteristics

You must account for matching when

analyzing matched data Important that the matched variables not be

exposures of interest


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Matching--Example

Hypothetical study where students in a high schoolhave reported a strange smell and sudden illness Test the association between smelling an unusual odor and

a set of symptoms Match cases and controls on gender, grade and hallway

Precedents for outbreaks of illness related to unusual odors inbuildings, possibly psychogenic (ie. illness spread by panicrather than true cause)

Women are more reactive in this situation, grade level controls

for age (different ages may react differently) and matching onhallway controls for actual odor observed (different locationsmay produce different odors)


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Matching--Example

Cells e and h are concordant cells because the case and thecontrol have the same exposure status

Cells f and g are discordant because the case and control have adifferent exposure status

Only the discordant cells give us useful data to contrast theexposure between cases and controls

Controls

Cases

Exposed Not Exposed Total

Exposed e f e + f

Not Exposed g h g + h

Total e + g f + h

With matched case-control pairs, a 2x2 table is set up to examine pairs

Table 1: Analysis of matched pairs for a case control study


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Matching--Example

A chi-square for matched data (McNemarschi-square) can be calculated using a

statistical computing program Calculation examines discordant pairs and results

in a McNemar chi-square value and p-value

If the p-value


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Matching--Example

A table of discordant pairs can also be usedto calculate a measure of association

Controls

Cases

Smell No Smell Total

Smell 6 12 18

No Smell 4 5 9

Total 10 17

Table 2: Sample data for sudden illness in a high school.Controls matched to cases on gender, grade, and hallway in the school


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Matching--Example

Calculating the odds ratio:

OR = (# pairs with exposed cases and unexposed cases)

(# pairs with unexposed cases and exposed controls)

= f / g = 12/4 = 3.0

Interpretation: The odds of having a sudden onset of nausea, vomiting, or

fainting if students smelled an unusual odor in the school

were 3.0 times the odds of having a sudden onset of thesesymptoms if students did not smell an unusual odor in theschool, controlling for gender, grade, and location in theschool.


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Matching

An important note about matching: Once you have matched on a variable, you

cannot use that variable as a risk factor inyour analysis

Cases and controls will have the exactsame matched variables so they are

useless as risk factors Do not match on any variable you suspect

might be a risk factor


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An Introduction to LogisticRegression

Logistic regression is a mathematicalprocess that results in an odds ratio

Logistic regression can control fornumerous confounders

The odds ratio produced by logistic

regression is known as the adjustedodds ratio because its value has beenadjusted for the confounders


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An Introduction to LogisticRegression

Logistic regression uses an equation called alogit functionto calculate the odds ratio

Using our earlier punch and cookies example,we suspect one of these food items isconfounding the other

Variables would be:

SICK (value is 1 if ill, 0 if not ill) PUNCH (1 if drank punch, 0 if did not drink punch)

COOKIES (1 if ate cookies, 0 if did not eatcookies)


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Logistic Regression--Example

General equation is:

Logit (OUTCOME) = EXPOSURE + CONFOUNDER1

+ CONFOUNDER2 + CONFOUNDER3 + (etc) For our example:

Outcome = variable SICK

Exposure = variable PUNCH

Confounder = variable COOKIES

Equation is: Logit (SICK) = PUNCH + COOKIES


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Logistic Regression:Important Points

Each variable on the right side of the equation iscontrolling for all the other variables on the right sideof the equation If you are not sure whether one of several variables is a

confounder, you can examine them all at the same time

Two important warnings: Do not put too many variables in the equation (a loose rule

of thumb is you can add one variable for every 25observations)

You cannot control for confounders you did not measure(Example: if a childs attendance at a particular daycare wasa confounder of the SICK-PUNCH relationship, but you donot have data on childrens daycare attendance, you cannotcontrol for it.)


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Logistic Regression

For many investigations you may not need to uselogistic regression

Logistic regression is helpful in managing

confounding variables, useful with large datasets andin studies designed to establish risk factors forchronic conditions, cancer cluster investigations orother situations with numerous confounding factors

Many software packages can simplify data analysisusing logistic regression SAS, SPSS, STATA and Epi Info are a few examples


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Logistic Regression:Software Packages

Common software packages used for data analysis,including logistic regression* SASCary, NC http://www.sas.com/index.html

SPSSChicago, IL http://www.spss.com/ STATACollege Station, TX http://www.stata.com

Epi InfoAtlanta, GA http://www.cdc.gov/EpiInfo/

EpisheetBoston, MAhttp://members.aol.com/krothman/modepi.htm

(Episheet cannot do logistic regression but is useful forsimpler analyses, e.g., 2x2 tables and stratified analyses.)

*This is not a comprehensive list, and UNC does not specifically

endorse any particular software package.
http://www.sas.com/index.htmlhttp://www.spss.com/http://www.stata.com/http://www.cdc.gov/EpiInfo/http://members.aol.com/krothman/modepi.htmhttp://members.aol.com/krothman/modepi.htmhttp://www.cdc.gov/EpiInfo/http://www.stata.com/http://www.spss.com/http://www.sas.com/index.html


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Logistic Regression--Examples

Wedding Reception, 1997 (4)

Guests complained of a diarrheal illness diagnosed

as cyclosporiasis Univariate analysis (using 2x2 tables) showed

eating raspberries was the exposure most stronglyassociated with risk for illness

Multivariate logistic regression showed sameresults

Investigators determined raspberries had not beenwashed


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Logistic Regression--Examples

Assessing the relationship between obesity andconcern about food security (5) Washington State Dept. of Health analyzed data from the

1995-99 Behavioral Risk Factor Surveillance System A variable indicating concern about food security was

analyzed using a logistic regression model with income andeducation as potential confounders

Persons who reported being concerned about food securitywere more likely to be obese than those who did not report

such concerns (adjusted OR = 1.29, 95% CI: 1.04-1.83)


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Matching & Conditional LogisticRegression--Examples

Foodborne SalmonellaNewport outbreak, 2002 (6) Affected 47 people from 5 different states

Case-control study carried out, controls matched by age-

group Logistic regression conducted to control for confounders

Cases were more likely than controls to have eaten groundbeef (MOR = 2.3, 95% CI: 0.9-5.7) and more likely to haveeaten raw or undercooked ground beef (MOR = 50.9, 95%CI: 5.3-489.0)

No specific contamination event identified but public healthalert issued to remind consumers about safe food-handlingpractices


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Matching & Conditional LogisticRegression--Examples

Outbreak of typhoid fever in Tajikistan, 1996-97 (7) 10,000 people affected in outbreak, case-control study conducted Cases were culture positive for the organism (Salmonella serotype

Typhi)

Using 2x2 tables, illness was associated with: Drinking unboiled water in the 30 days before onset (MOR = 6.5, 95%

CI: 3.0-24.0) Using drinking water from a tap outside the home (MOR = 9.1, 95%

CI: 1.6-82.0) Eating food from a street vendor (MOR = 2.9, 95% CI: 1.4-7.2)

When all variables were included in conditional logistic regression,

only drinking unboiled water (MOR = 9.6, 95% CI: 2.7-334.0) andobtaining water from an outside tap (MOR = 16.7, 95% CI: 2.0-138.0) were significantly associated with illness

Routinely boiling drinking water was protective (MOR = 0.2, 95% CI:0.05-0.5)


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Conclusion

Controlling for confounding can be doneusing matched study design and logistic

regression While complicated, with practice these

methods can be as easy to use as 2x2

tables


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References1. Gregg MB. Field Epidemiology. 2nd ed. New York, NY: Oxford

University Press; 2002.2. Hecht CA, Hook EB. Rates of Down syndrome at livebirth by one-year

maternal age intervals in studies with apparent close to completeascertainment in populations of European origin: a proposed revisedrate schedule for use in genetic and prenatal screening.Am J Med

Genet.1996;62:376-385.3. Humphrey LL, Nelson HD, Chan BKS, Nygren P, Allan J, Teutsch S.

Relationship between hormone replacement therapy, socioeconomicstatus, and coronary heart disease. JAMA. 2003;289:45.

4. Centers for Disease Control and Prevention. Update: Outbreaks ofCyclosporiasis -- United States, 1997. MMWR Morb Mort Wkly Rep.1997;46:461-462. Available at: http://www.cdc.gov/mmwr/PDF/wk/mm4621.pdf.Accessed December 12, 2006.

5. Centers for Disease Control and Prevention. Self-reported concernabout food security associated with obesity --- Washington, 19951999. MMWR Morb Mort Wkly Rep.2003;52:840-842. Available at:http://www.cdc.gov/mmwr/preview/mmwrhtml/mm5235a3.htm.Accessed December 12, 2006.
http://www.cdc.gov/mmwr/PDF/wk/mm4621.pdfhttp://www.cdc.gov/mmwr/PDF/wk/mm4621.pdfhttp://www.cdc.gov/mmwr/PDF/wk/mm4621.pdfhttp://www.cdc.gov/mmwr/PDF/wk/mm4621.pdfhttp://www.cdc.gov/mmwr/PDF/wk/mm4621.pdfhttp://www.cdc.gov/mmwr/preview/mmwrhtml/mm5235a3.htmhttp://www.cdc.gov/mmwr/preview/mmwrhtml/mm5235a3.htmhttp://www.cdc.gov/mmwr/preview/mmwrhtml/mm5235a3.htmhttp://www.cdc.gov/mmwr/preview/mmwrhtml/mm5235a3.htmhttp://www.cdc.gov/mmwr/preview/mmwrhtml/mm5235a3.htmhttp://www.cdc.gov/mmwr/PDF/wk/mm4621.pdfhttp://www.cdc.gov/mmwr/PDF/wk/mm4621.pdf

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Logistic Regression and Confounding