More Statistics tutorial at Logistic Regression and the new:

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1 Logistic Regression and the new: Residual Logistic Regression 1

2 Outline 1. Logistic Regression 2. Confounding Variables 3. Controlling for Confounding Variables 4. Residual Linear Regression 5. Residual Logistic Regression 6. Examples 7. Discussion 8. Future Work 2

3 1. Logistic Regression Model In 1938, Ronald Fisher and Frank Yates suggested the logit link for regression with a binary response variable. ibl ln(odds of Y = 1 x) ( 1 ) ( 1 ) ln PY = x ln PY = x = = P( Y= 0 x) 1 PY ( = 1 x) ( x ) ( x) π = ln = β0+ β1x 1 π

4 A 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.)

5 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. PY ( = 1 x) ln 1 PY ( = 1 x ) This model can be rewritten as P ( Y = 1 x ) = exp( β + β 0 1x) 0 1x 1+ exp( β + β = β + β x ) 0 1 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) = β 0 + β 1 x

6 More on the properties of the logistic regression model In the simple logistic regression, the regression coefficient β 1 has the interpretation that it is the log of the odds ratio of a success event (Y=1) for a unit change in x. P( Y = 1 x + 1) ln P( Y = 0 x + 1) P( Y = 1 ln P( Y = 0 x) x) = [ β + β ( x + 1)] [ β + β x] = ] β 1 For multiple predictor variables, the logistic regression model is P( Y = 1 x, x,..., x ) = β + β x β k x = ) 1 2 k ln = ) P( Y 0 x1, x2,..., x k k

7 Logistic Regression, SAS Procedure 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 d d from data logit; set "c:\temp\hsb2"; honcomp = (write >= 60); run; proc logistic data= logit descending; di model honcomp = female read science; run; 7

8 Logistic Regression, SAS Output 8

9 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 9

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 10

11 Birth order Down Syndrome Maternal Age Confounding Maternal age is correlated with birth order and a risk factor for Down Syndrome, even if Birth order is low Alcohol Lung Cancer Smoking No Confounding Smoking is correlated with alcohol consumption and is a risk factor for Lung Cancer even for persons who don t drink alcohol 11

12 3. Controlling for Confounding In study designs Restriction Variables Random allocation of subjects to study groups to attempt to even out unknown confounders Matching subjects using potential confounders 12

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 13

14 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 i when excluding subjects Some strata might become too thin and thus insignificant i ifi creating also loss of information Regression methods, if well handled, can control lf for confounding factors 14

15 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 Y = as β 0 + follows β 1 X 1 + β 2 X β k X k + ε Residuals are 垐 calculated 垐 from this model, let ˆ = β β 1 1+ β β k k Y X X X 15

16 The residuals are Zero mean Homoscedasticity Normally distributed ( ) 0 Corr,, e i e j ej = Yj Y ) j i j with the following properties: This residual will be considered d the new dependant d variable. That is, the new model to be fitted is ) Y Y = α + β X + β X + + β X + ε 0 k+ 1 k+ 1 k+ 2 k t t which is equivalent to: ) Y = α + Y + β X + β X + + β X + ε 0 k+ 1 k+ 1 k+ 2 k t t 16

17 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 ibl is tosimply put all the n variables ibl in the same model as: π log = β 1 π 0 + β 1 X 1 + β 2 X β k X k β n X n 17

18 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 π as the probability of success where only confounding effect is analyzed π log 1 π = β + β X + β X β X k k 18

19 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 T = ) β 1 X 1 + ) β 2 X ) β k X k The new model to be fitted is π log = β0 + T + β 2 1 π k + 1X k βk + 2 X k β X n n 19

20 Method 2 Pearson residuals are calculated from the initial model using the Pearson residual ) (Hosmer and Y π Lemeshow, 1989) ) π Z = ) ) π [ 1 π ] where is the estimated probability of success ) k ) based on the confounding variables alone: ) e β 0 + β i X i i=1 π = ) k [ 0,1 ] 1+ e β 0 + i=1 ) β i X i The second level regression will use this residual 20 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 Z = β 0 + β k+1 X k+1 + β k+2 X k β n X n + ε 21

22 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 g Age Lwt Smk Ht Age Lwt Smk Ht

23 Potential ti confounding factor: Age Model for π (probability of low birth weight) Logisticregression i i π log = β 1 π 0 + β 1 age + β 2 lwt + β 3 smk + β 4 ht Residual logistic regression initial model Method 1 T = ) β 1 age Mthd2 Method π log = β 1 π 0 + β 1 age π log = β 1 π 0 + T + β 2 lwt + β 3 smk + β 4 ht Z = β 0 + β 2 lwt + β 3 smk + β 4 ht + ε 23

24 Results Variables Logistic Regression RLR Method1 Odds ratio P-value SE Odds ratio P-value SE lwt smk ht RLR Method 2 Variables P-value SE lwt Smk ht Conf. factors P-value Variables Log reg Ini model Age

25 Data: Alzheimer patients Example 2 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 Correlation matrix with Gender alpha=0.05 MMS PDS HDT

26 Potential ti confounding factors: Age, Gender Model for π (probability of declining) Logisticregression i i π log = β 1 π 0 + β 1 age + β 2 gender + β 3 mms + β 4 pds + β 5 hdt Residual logistic regression initial model Method 1 T = ) β 1 age + ) β 2 gender Mthd2 Method π log = β 1 π 0 + β 1 age + β 2 gender Z = β 0 + β 3 mms + β 4 pds + β 5 hdt + ε π log = β 1 π 0 + T + β 3 mms + β 4 pds + β 5 hdt 26

27 Results Variables Logistic Regression RLR Method1 Odds ratio P-value SE Odds ratio P-value SE mms pds hdt RLR Method 2 Conf. factors Variables P-value SE P-value Variables mms < Log reg Ini model pds < Age hdt Gender

28 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. 28

29 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. 29

30 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, Hosmer, D.W.; Jovanovic, B. and Lemeshow, S. Best Subsets Logistic Regression. Biometrics 45, Pergibon, D. Logistic Regression Diagnostics. The Annals of Statistics 19(4),