Title: Logistic Regression and the new: Residual Logistic Regression
1Logistic Regression and the newResidual
Logistic Regression
- F. Berenice Baez-Revueltas
- Wei Zhu
2Outline
- Logistic Regression
- Confounding Variables
- Controlling for Confounding Variables
- Residual Linear Regression
- Residual Logistic Regression
- Examples
- Discussion
- Future Work
31. Logistic Regression Model
- In 1938, Ronald Fisher and Frank Yates
suggested the logit link for regression with a
binary response variable.
4A popular model for categorical response variable
- Logistic regression model 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 regression model 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.)
5More on the rationale of the logistic regression
model
- Consider a binary response variable Y0 or 1and a
single predictor variable x. We want to model
E(Yx) P(Y1x) as a function of x. The logistic
regression model expresses the logistic transform
of P(Y1x) as a linear function of the
predictor. -
- This model can be rewritten as
- E(Yx) P(Y1 x) 1 P(Y0x) 0 P(Y1x) is
bounded between 0 and 1 for all values of x. The
following linear model may violate this condition
sometimes -
- P(Y1x)
6More 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
(Y1) for a unit change in x. -
- For multiple predictor variables, the logistic
regression model is -
7Logistic 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/defa
ult.htm. - data logit
- set "c\temp\hsb2"
- honcomp (write gt 60)
- run
- proc logistic data logit descending
- model honcomp female read science
- run
8Logistic Regression, SAS Output
92. 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
10- 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
11Confounding
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 dont drink alcohol
123. 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
13- 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
14Pros 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
154. Residual Linear Regression
- Consider a dependant variable Y and a set of n
independent covariates, from which the first k
(kltn) of them are potential confounding factors - Initial model treating only the confounding
variables as follows - Residuals are calculated from this model, let
16- 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
17The Usual Logistic Regression Approach to
Control for Confounders
- Consider a binary outcome Y and n covariates
where the first k (kltn) 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
185. Residual Logistic Regression
- Each subject has a binary outcome Y
- Consider n covariates, where the first k (kltn)
are potential confounding factors - Initial model with as the probability of
success where only confounding effect is analyzed
19Method 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
20Method 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.
21- 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
226. Example 1
- Data Low Birth Weight
- Dow. Indicator of birth weight less than 2.5 Kg
- Age Mothers age in years
- Lwt Mothers weight in pounds
- Smk Smoking status during pregnancy
- Ht History of hypertension
Age Lwt Smk Ht
Age 1.0000 0.1738 -0.0444 -0.0158
Lwt 1.0000 -0.0408 0.2369
Smk 1.0000 0.0134
Ht 1.0000
Correlation matrix with alpha0.05
23- Potential confounding factor Age
- Model for (probability of low birth weight)
- Logistic regression
- Residual logistic regression
- initial model
- Method 1
-
- Method 2
24Results
Variables Logistic Regression Logistic Regression Logistic Regression RLR Method1 RLR Method1 RLR Method1
Variables Odds ratio P-value SE Odds ratio P-value SE
lwt 0.988 0.060 0.0064 0.989 0.078 0.0065
smk 3.480 0.001 0.3576 3.455 0.001 0.3687
ht 3.395 0.053 0.6322 3.317 0.059 0.6342
RLR Method 2
Conf. factors
Variables P-value SE
lwt 0.077 0.0024
Smk 0.000 0.1534
ht 0.042 0.3094
Variables P-value P-value
Variables Log reg Ini model
Age 0.055 0.027
25Example 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
-
Age Gender MMS PDS HDT
Age 1.0000 0.0413 -0.2120 0.3327 0.9679
Gender 1.0000 -0.1074 0.2020 -0.1839
MMS 1.0000 0.3784 -0.1839
PDS 1.0000 0.0110
HDT 1.0000
Correlation matrix with alpha0.05
26- Potential confounding factors Age, Gender
- Model for (probability of declining)
- Logistic regression
- Residual logistic regression
- initial model
- Method 1
-
- Method 2
27Results
Variables Logistic Regression Logistic Regression Logistic Regression RLR Method1 RLR Method1 RLR Method1
Variables Odds ratio P-value SE Odds ratio P-value SE
mms 0.717 0.023 0.1451 0.720 0.023 0.1443
pds 1.691 0.001 0.1629 1.674 0.001 0.1565
hdt 1.018 0.643 0.0380 1.018 0.644 0.0377
RLR Method 2
Conf. factors
Variables P-value P-value
Variables Log reg Ini model
Age 0.004 0.000
Gender 0.935 0.551
Variables P-value SE
mms lt0.001 0.0915
pds lt0.001 0.0935
hdt 0.061 0.0273
287. 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.
298. 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.
30Selected 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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