2. Outline. Logistic RegressionConfounding VariablesControlling for Confounding VariablesResidual Linear RegressionResidual Logistic RegressionExamplesDiscussionFuture Work. 1. Logistic Regression Model. . In 1938, Ronald Fisher and Frank Yates suggested the logit link for regression with a binary response variable..
Logistic Regression and the new: Residual Logistic Regression
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Logistic Regression and the new:Residual Logistic Regression
F. Berenice Baez-Revueltas
Controlling for Confounding Variables
Residual Linear Regression
Residual Logistic Regression
In 1938, Ronald Fisher and Frank Yates suggested the logit link for regression with a binary response variable.
1. Logistic Regression Model
A popular model for categorical response variable
Logistic regressionmodel is the most popular model for binary data.
Logistic regression model is generally used to study the relationship between a binary response variable and a group of predictors (can be either continuous or categorical).
Y = 1 (true, success, YES, etc.) or
Y = 0 ( false, failure, NO, etc.)
Logistic regressionmodel can be extended to model a categorical response variable with more than two categories. The resulting model is sometimes referred to as the multinomial logistic regression model (in contrast to the ‘binomial’ logistic regression for a binary response variable.)
More on the rationale of the logistic regression model
Consider a binary response variable Y=0 or 1and a single predictor variable x. We want to model E(Y|x) =P(Y=1|x) as a function of x. The logistic regression model expresses the logistic transform of P(Y=1|x) as a linear function of the predictor.
This model can be rewritten as
E(Y|x)= P(Y=1| x) *1 + P(Y=0|x) * 0 = P(Y=1|x) is bounded between 0 and 1 for all values of x. The following linear model may violate this condition sometimes:
More on the properties of the logistic regression model
In the simple logistic regression, the regression coefficient has the interpretation that it is the log of the odds ratio of a success event (Y=1) for a unit change in x.
For multiple predictor variables, the logistic regression model is
This page shows an example of logistic regression with footnotes explaining the output. The data were collected on 200 high school students, with measurements on various tests, including science, math, reading and social studies. The response variable is high writing test score (honcomp), where a writing score greater than or equal to 60 is considered high, and less than 60 considered low; from which we explore its relationship with gender (female), reading test score (read), and science test score (science). The dataset used in this page can be downloaded from http://www.ats.ucla.edu/stat/sas/webbooks/reg/default.htm.
honcomp = (write >= 60);
proc logistic data= logit descending;
model honcomp = female read science;
Logistic Regression, SAS Output
2. Confounding Variables
Correlated with both the dependent and independent variables
Represent major threat to the validity of inferences on cause and effect
Add to multicollinearity
Can lead to over or underestimation of an effect, it can even change the direction of the conclusion
They add error in the interpretation of what may be an accurate measurement
For a variable to be a confounder it needs to have
Relationship with the exposure
Relationship with the outcome even in the absence of the exposure (not an intermediary)
Not on the causal pathway
Uneven distribution in comparison groups
Maternal age is correlated with birth order and a risk factor for Down Syndrome, even if Birth order is low
Smoking is correlated with alcohol consumption and is a risk factor for Lung Cancer even for persons who don’t drink alcohol
3. Controlling for Confounding Variables
In study designs
Random allocation of subjects to study groups to attempt to even out unknown confounders
Matching subjects using potential confounders
In data analysis
Stratified analysis using Mantel Haenszel method to adjust for confounders
Restriction (is still possible but it means to throw data away)
Model fitting using regression techniques
Pros and Cons of Controlling Methods
Matching methods call for subjects with exactly the same characteristics
Risk of over or under matching
Cohort studies can lead to too much loss of information when excluding subjects
Some strata might become too thin and thus insignificant creating also loss of information
Regression methods, if well handled, can control for confounding factors
4. Residual Linear Regression
Consider a dependant variable Y and a set of n independent covariates, from which the first k(k<n) of them are potential confounding factors
Initial model treating only the confounding variables as follows
Residuals are calculated from this model, let
The residuals are with the following properties:
This residual will be considered the new dependant variable. That is, the new model to be fitted is
which is equivalent to:
The Usual Logistic Regression Approach to ‘Control for’ Confounders
Consider a binary outcome Y and n covariates where the first k(k<n) of them being potential confounding factors
The usual way to ‘control for’ these confounding variables is to simply put all the n variables in the same model as:
5. Residual Logistic Regression
Each subject has a binary outcome Y
Consider n covariates, where the first k(k<n) are potential confounding factors
Initial model with as the probability of success where only confounding effect is analyzed
The confounding variables effect is retained and plugged in to the second level regression model along with the variables of interest following the residual linear regression approach.
That is, let
The new model to be fitted is
Pearson residuals are calculated from the initial model using the Pearson residual (Hosmer and Lemeshow, 1989)
where is the estimated probability of success based on the confounding variables alone:
The second level regression will use this residual as the new dependant variable.
Therefore the new dependant variable is Z, and because it is not dichotomous anymore we can apply a multiple linear regression model to analyze the effect of the rest of the covariates.
The new model to be fitted is a linear regression model
6. Example 1
Data: Low Birth Weight
Dow. Indicator of birth weight less than 2.5 Kg
Age: Mother’s age in years
Lwt: Mother’s weight in pounds
Smk: Smoking status during pregnancy
Ht: History of hypertension
Correlation matrix with alpha=0.05
Potential confounding factor: Age
Model for (probability of low birth weight)
Residual logistic regression
RLR Method 2
Data: Alzheimer patients
Decline: Whether the subjects cognitive capabilities deteriorates or not
Age: Subjects age
Gender: Subjects gender
MMS: Mini Mental Score
PDS: Psychometric deterioration scale
HDT: Depression scale
Correlation matrix with alpha=0.05
Potential confounding factors: Age, Gender
Model for (probability of declining)
Residual logistic regression
RLR Method 2
The usual logistic regression is not designed to control for confounding factors and there is a risk for multicollinearity.
Method 1 is designed to control for confounding factors; however, from the given examples we can see Method 1 yields similar results to the usual logistic regression approach
Method 2 appears to be more accurate with some SE significantly reduced and thus the p-values for some regressors are significantly smaller. However it will not yield the odds ratios as Method 1 can.
8. Future Work
We will further examine the assumptions behind Method 2 to understand why it sometimes yields more significant results.
We will also study residual longitudinal data analysis, including the survival analysis, where one or more time dependant variable(s) will be taken into account.
Menard, S. Applied Logistic Regression Analysis. Series: Quantitative Applications in the Social Sciences. Sage University Series
Lemeshow, S; Teres, D.; Avrunin, J.S. and Pastides, H. Predicting the Outcome of Intensive Care Unit Patients. Journal of the American Statistical Association 83, 348-356
Hosmer, D.W.; Jovanovic, B. and Lemeshow, S. Best Subsets Logistic Regression. Biometrics 45, 1265-1270. 1989.
Pergibon, D. Logistic Regression Diagnostics. The Annals of Statistics 19(4), 705-724. 1981.