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Logistic Regression and the new: Residual Logistic Regression

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Outline

- Logistic Regression
- Confounding Variables
- Controlling for Confounding Variables
- Residual Linear Regression
- Residual Logistic Regression
- Examples
- Discussion
- Future Work

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:

P(Y=1|x) =

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

Logistic Regression, SAS Procedure

- http://www.ats.ucla.edu/stat/sas/output/SAS_logit_output.htm
- Proc Logistic
- 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.
- data logit;
- set "c:\temp\hsb2";

honcomp = (write >= 60);

run;

proc logistic data= logit descending;

model honcomp = female read science;

run;

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

Exposure

Outcome

Third variable

Birth order

Down Syndrome

Maternal Age

Alcohol

Lung Cancer

Smoking

Confounding

Maternal age is correlated with birth order and a risk factor for Down Syndrome, even if Birth order is low

No Confounding

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
- Restriction
- 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
- Case-control studies
- Cohort studies
- 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
- Initial model treating only the confounding variables as follows
- Residuals are calculated from this model, let

The residuals are with the following properties:

- Zero mean
- Homoscedasticity
- Normally distributed
- ,

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
- 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
- Initial model with as the probability of success where only confounding effect is analyzed

Method 1

- 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

Method 2

- 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)
- Logistic regression
- Residual logistic regression

initial model

- Method 1
- Method 2

Example 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)
- Logistic regression
- Residual logistic regression

initial model

- Method 1
- Method 2

7. Discussion

- 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.

Selected References

- 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.

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