Advanced Data Analysis: Methods to Control for Confounding (Matching and Logistic Regression)

1
Advanced Data AnalysisMethods to Control for
Confounding(Matching and Logistic Regression)
2
Goals

• Understand the issue of confounding in
  statistical analysis
• Learn how to use matching and logistic regression
  to control for confounding

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Confounding

• Example people in a gastrointestinal outbreak
• Mostly members of the same dinner club BUT many
  club members also went to a city-wide food
  festival
• Food handling practices in the dinner club might
  be blamed for the outbreak when food eaten at the
  festival was the cause
• Membership in the dinner club could be a
  confounder of the relationship between attendance
  at the food festival and illness
• Analyzing the data to account for both dinner
  club membership and food festival attendance
  could help determine which event was truly
  associated with the outcome

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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
  those in the dinner club vs. those not in the
  dinner club
• If attending the food festival was a significant
  risk factor for illness in both groups, then the
  festival would be implicated because illness
  occurred whether or not people were members of
  the dinner 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 and by
  health status? Or stratify 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 regression controls for many potential
  confounders at one time
• Matching when incorporated correctly into the
  study design, 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
  associated with 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 appear that no
  association exists (when there really is one)

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Confounders

• A confounder is 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 and disease
• Confounding must be eliminated for accurate
  results (1)
• Confounding can occur in an observational
  epidemiologic 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 by factors 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
  an association 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
  more likely to be able to afford HRT
• Women of lower SES are at higher risk of
  cardiovascular disease
• 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 saladwomen
  were much more likely to order salad than men
• Salad was contaminated with disease-causing agent
• Relationship between gender and disease was
  confounded by salad consumption (which was the
  true cause of the outbreak)

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Characteristics of Confounders

• Confounders must have two key characteristics
• A confounder must be associated with the disease
  being studied
• A confounder must be associated with the exposure
  being studied

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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 for all levels of the
  confounding variable
• Example look at the relationship between HRT and
  cardiovascular disease separately among women of
  high SES and low SES
• Conduct logistic regressionregression puts all
  the variables into a mathematical model
• Makes it easy to account for multiple confounders
  that need to be controlled

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Controlling for ConfoundingStratification

• 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, could
  examine the association between the homeless
  shelter and TB among men who did NOT go to the
  soup kitchen and among men who DID go to the soup
  kitchen

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Stratification--Example

• Outbreak at a reception, cookies and punch have
  both been implicated
• Suspicion that one food item is confounding the
  other
• Cannot tease out the effects without stratifying
  because many people consumed both cookies and
  punch

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Stratification--Example

• After conducting a case-control study, overall
  data show the following

Cookie Exposure

• OR (37x29)/(21x13) 3.93 95 CI, 1.69 9.15
• p 0.001  

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Stratification--Example

• Data continued..

Punch Exposure

• OR (40x30)/(20x10) 6.00 95 CI, 2.83 12.71
• p 0.0004

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Stratification--Example

• Both cookies and punch have a high odds ratio for
  illness a confidence interval that does not
  include 1
• OR (cookies) 3.93 95 CI, 1.69 9.15, p
  0.001
• OR (punch) 6.00 95 CI, 2.83 12.71, p
  0.0004
• To stratify by punch exposure, we want to know
• Among those who did not drink punch, what is the
  odds ratio for the association between cookies
  and illness?
• Among those who did drink punch, what is the odds
  ratio for the association between cookies and
  illness?
• If cookies are the culprit, there should be an
  association between cookies and illness,
  regardless of whether anyone drank punch

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Stratification--Example

• Stratification of the cookie association by punch
  exposure

Did have punch

• OR (35x3)/(17x5) 1.3 95 CI, 0.17 7.22
• p 1.0 

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Stratification--Example

• Stratification of the cookie association by punch
  exposure

Did not have punch

• OR (2x26)/(4x8) 1.63 95 CI, 0.12 13.86
• p 0.63 

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Stratification--Example

• To stratify by cookie exposure, we want to know
• Among those who did not eat cookies, what is the
  odds ratio for the association between punch and
  illness?
• Among those who did eat cookies, what is the odds
  ratio for the association between punch and
  illness?
• If punch is the culprit, there should be an
  association between punch and illness, regardless
  of whether anyone ate cookies

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Stratification--Example

• Stratification of the punch association by cookie
  exposure

Did have cookies

• OR (35x4)/(17x2) 4.12 95 CI, 0.52 48.47
• p 0.18

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Stratification--Example

• Stratification of the punch association by cookie
  exposure

Did not have cookies

• OR (5x26)/(3x8) 5.42 95 CI, lt 0.80 40.95
• p 0.08

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Stratification

• Stratification allows us to examine two risk
  factors independently of each other
• In our cookies and punch example we can see that
  cookies were not really a risk factor independent
  of punch (stratified ORs 1)
• Punch remained a potential risk factor
  independent 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 confounder and then combines these by
  averaging them into one estimate 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 and Effect 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 or stratum
• Example if gender is an effect measure modifier,
  you should give 2 odds or risk ratios, 1 for men
  and 1 for women
• You identify effect measure modification by
  stratification (same technique used to identify
  confounding) but you are looking for the measure
  of effect 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 the association 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 nutritional intake
• In developing countries, sanitation is an effect
  modifier of the association between breastfeeding
  and infant mortality
• 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
  school have 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 in buildings, possibly psychogenic
  (ie. illness spread by panic rather than true
  cause)
• Women are more reactive in this situation, grade
  level controls for age (different ages may react
  differently) and matching on hallway controls for
  actual odor observed (different locations may
  produce different odors)

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Matching--Example

• 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

• Cells e and h are concordant cells because the
  case and the control have the same exposure
  status
• Cells f and g are discordant because the case and
  control have a different exposure status
• Only the discordant cells give us useful data to
  contrast the exposure between cases and controls

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Matching--Example

• A chi-square for matched data (McNemars
  chi-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 lt0.05, you can conclude that there
  is a statistically significant difference in
  exposure between cases and controls

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Matching--Example

• A table of discordant pairs can also be used to
  calculate a measure of association

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 these symptoms
  if students did not smell an unusual odor in the
  school, controlling for gender, grade, and
  location in the school.

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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 in your
  analysis
• Cases and controls will have the exact same
  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 Logistic Regression

• Logistic regression is a mathematical process
  that results in an odds ratio
• Logistic regression can control for numerous
  confounders
• The odds ratio produced by logistic regression is
  known as the adjusted odds ratio because its
  value has been adjusted for the confounders

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An Introduction to Logistic Regression

• Outcome variable (sick or not sick) and exposure
  variable (exposed or not exposed) must both be
  dichotomous
• Other variables (the confounders) can be
  dichotomous, categorical, or continuous

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An Introduction to Logistic Regression

• Logistic regression uses an equation called a
  logit function to calculate the odds ratio
• Using our earlier punch and cookies example, we
  suspect one of these food items is confounding
  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 eat
  cookies)

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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--Example

• Computer uses the math behind logistic regression
  to give the results as odds ratios
• Each variable on the right side will have its own
  odds ratio
• Odds ratio for PUNCH would be the odds of
  becoming ill if punch was consumed compared to
  the odds of becoming ill if punch was not
  consumed, controlling for COOKIES
• Odds ratio for COOKIES is the odds of becoming
  ill if cookies were consumed compared to the odds
  of becoming ill if cookies were not consumed,
  controlling for PUNCH

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Logistic Regression Important Points

• Each variable on the right side of the equation
  is controlling for all the other variables on the
  right side of 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 25 observations)
• You cannot control for confounders you did not
  measure (Example if a childs attendance at a
  particular daycare was a confounder of the
  SICK-PUNCH relationship, but you do not have data
  on childrens daycare attendance, you cannot
  control for it.)

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Logistic Regression Matching

• Logistic regression can also account for matching
  in the data analysis
• Known as conditional logistic regression
• Computer calculates odds ratios similar to
  McNemars test but the results are conditioned
  on the matching variables
• Can be done using Epi Info
• Interpretation of matched odds ratios (MORs)
  using conditional logistic regression is the same
  as interpretation of MORs calculated from tables

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Logistic Regression

• For many investigations you may not need to use
  logistic regression
• Logistic regression is helpful in managing
  confounding variables, useful with large datasets
  and in studies designed to establish risk factors
  for chronic conditions, cancer cluster
  investigations or other situations with numerous
  confounding factors
• Many software packages can simplify data analysis
  using 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
• SAS Cary, NC http//www.sas.com/index.html
• SPSS Chicago, IL http//www.spss.com/
• STATA College Station, TX http//www.stata.com
• Epi Info Atlanta, GA http//www.cdc.gov/EpiInfo
  /
• Episheet Boston, MA http//members.aol.com/kro
  thman/modepi.htm
• (Episheet cannot do logistic regression but is
  useful for simpler analyses, e.g., 2x2 tables and
  stratified analyses.)
• This is not a comprehensive list, and UNC does
  not specifically
• endorse any particular software package.

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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 strongly
  associated with risk for illness
• Multivariate logistic regression showed same
  results
• Investigators determined raspberries had not been
  washed

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Logistic Regression--Examples

• Assessing the relationship between obesity and
  concern 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 and education as potential
  confounders
• Persons who reported being concerned about food
  security were 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 Logistic
Regression--Examples

• Foodborne Salmonella Newport 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 ground beef (MOR 2.3, 95 CI 0.9-5.7)
  and more likely to have eaten raw or undercooked
  ground beef (MOR 50.9, 95 CI 5.3-489.0)
• No specific contamination event identified but
  public health alert issued to remind consumers
  about safe food-handling practices

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Matching Conditional Logistic
Regression--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) and obtaining
  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 done using
  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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References

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  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 complete
  ascertainment in populations of European origin
  a proposed revised rate schedule for use in
  genetic and prenatal screening. Am J Med Genet.
  199662376-385.
• 3. Humphrey LL, Nelson HD, Chan BKS, Nygren P,
  Allan J, Teutsch S. Relationship between hormone
  replacement therapy, socioeconomic status, and
  coronary heart disease. JAMA. 200328945. 
• 4. Centers for Disease Control and Prevention.
  Update Outbreaks of Cyclosporiasis -- United
  States, 1997. MMWR Morb Mort Wkly Rep.
  199746461-462. Available at http//www.cdc.gov/
  mmwr/PDF/ wk/mm4621.pdf. Accessed December 12,
  2006.
• 5. Centers for Disease Control and Prevention.
  Self-reported concern about food security
  associated with obesity --- Washington,
  19951999. MMWR Morb Mort Wkly Rep.
  200352840-842. Available at http//www.cdc.gov/
  mmwr/preview/mmwrhtml/mm5235a3.htm. Accessed
  December 12, 2006.