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Lecture 11: Regression methods and adjustment for confounding
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Outline
1. methods of confounding adjustment in data analysis
a. stratification
b. MLR (linear, log, poisson, cox)
2. Alternative approaches for control of confounding
a. mendelian randomization & instrumental variables
b. propensity scores
3. incomplete adjustment for confounding
stratification (adjustment)
1. examines both confounding and EM; controls for its effect
2. heterogeneity
a. Confounding: strata ORs different from crude; association is confounded
b. interaction: strata ROs disimilar, assess magnitude, maybe random due to small sample
3. if homogeneity
a. confounding: strat ORs similar to 1, crude explained by confouding effect, report single estimate across strata
b. interaction: strata ORs similar, no interaction, calc weighted OR
simple linear regression
1. assess assoc. between two continuous varibales
2. best fit line (least squares method)
3. parameters
a. intercept: value of y when x is 0
b. slope: mean change in Y per unit change in independent variable
4. limitations
a. curvilinear might explain relationship better
b. other variables might complete model
pearson correlation coefficient
1. range from -1 to 1
2. strength assoc. b/t two variables linear relationship
multiple regression (adjustment; general)
1. adjustment when outcome (Y) is continuous
2. X variables can be either cont. or cat.
assumptions MLR
1. linearity of assoc
2. independence
3. constant variance
4. residual confounding, estimate Bs constant across al llevels
MLR and interaction
1. use interaction term (product)
logistic regression
1. binary outcome, binomial distribution
2. log(p/1-p) = log(odds) = bo+b1x
Assumption logistic regression
1. linear relationship on log odds scale
a. not apply to categorized variables; homogeneous within categories
interpretation of logistic regression coefficient
1. for smoking, b(smoking) is the increase of __ in log odds of Y if move from non smoking to smoking, adjusted for other Y if smoking exposure, adjusting for other variables in model
interpretation of logistic categorical preditors
1. adjusted RO for Y (#), the ratio of Y incidence odds for individuals with X to the odds of those not X
Log regression in cohort or X-sect
1. only for rare outcomes
a.if freq. high, estimate biased (exaggerated)
Conditional log regression for Matched CC
1. takes into account pairing/matching
2. interpretation same as normal but "Adjusted" for variables and matching
Conditional log regression for Matched nested CC study
1. analyzed as "matched CC study"
2. analogous to Cox PH regression
Goodness of fit for logistic regression
1. deviance and LRTs
2. Pseudo R^2
poisson regression (log linear model)
1.outcomes of interest are rates (rate ratios)
a. rare diseases in large populations
log(rate) = bo+b1x1
a. can separate person time from the count using offset in equation, log(person time)
2. mean = variance
3. exponentiate to get rate ratio
Poisson regression coefficient interpretation and rate ratio
1. bx is the increase of # in log rate of Y if move classification of x vs non-x, adjusting for other variable in the model
2. estimate adjusted rate ratio of Y for those E vs. non E = RR =1.67
cox PH regression model
1. based on time to event (or survival) data, models hazard
Cox assumptions
1. proportionality
a. constant hazard ratio at any point during follow up
b. if violated, stratify according to f/u time
Cox hazard ratio interprestion
1. adjusting for other covariates, an additional unit of X increases the hazard of Y by a factor of # (1.015 = 1.5%)
approaches to modeling non-linear relationships w/ linear regression models
1. use indicator (dummy) variables
2. instrumental variable method
3. mendelian randomization
4. propensity score method
indicator (dummy) variables
1. regression coefficient for each dummy variable represents average difference in Y b/t individuals in dummy category and reference category
a. splits up non linear line to portions, more or less linear portions
2. intercept interpreted as average value of Y among individuals in ref category
Instrumental variable method
1. build an instrument variable
a. must be causally assoc with E
b. affects outcome
c. not associated with any confounders between E and O
2. must regress E on instrument to obtain predicted variables (E as function of instrument variable)
3. regress predicted values on outcome
limitations of instrumental variable method
1. if instrument fails to mimic randomization -> non valid causal inference. this happens when
a. instrument has weak assoc. w/ E
b. relationship b/t instrument and O confounded
Mendelian randomization and limitations
1. when instrument is genetic polymorphism, affects the presence/level of E (phenotype)
a. true causal E to disease assoc. b/c genes randomly segregated
2. limitations
a. genetic trait must be strongly assoc w/ E
b. might be confounded by linkage disequilibrium
Propensity score method
1. predicted probability of E in individuals based on set of relevant characteristics
2. can be used to control for confounding in Obs cohort studies by
a. propensity score matching
b. used in MLR
c. stratification on propensity scores
d. weighting by inverse propensity score
propensity score matching
1. conditional reg performed on new dataset including E and NE indiviudals match w/ respect to score
2. matched groups similar w/ respect to overall set of known covariates, approach could be though of as simulated randomization
propensity score method steps
1. model used to calculate predictive probability of E as function of set of possible confounders
2. scores the applied to generate new dataset of matched E and NE based on scores to control for score as covariate in reg model
disadvatage to propensity score methods
1. lose useful info about predictors
2. subject to residual confounding
residual confounding
1. incomplete adjustment
a. improper def. of categories
b. adjustment imperfect surrogate (does it rep. the E?)
c. confounders not included in model
d. misclass.
overadjustment (overmatching)
1. adjustment carried out for variable either on causal path or strongly related to E or O
2. obscure true effect or create effect that doesnt exist
model for xsect and reported MoA
1. direct /indirect adjustment, MH method, Log reg
2. report rate ratio, ORs
model for CC and reported MoA
1. MH or Log reg
2. report ORs
model for cohort and reported MoA
1. Direct/indirect adjust for cum. incid. ; RR/standardized RRs
2. MH and log reg ; ORs
3. Cox (time to event data) ; HR
4. MH or poisson (IR) ; RR
nested CCon, CCoh
1. conditional reg, Cox model ; HRs
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