Logistic Regression

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Logistic Regression Clinical Prediction Rules

• DOC Research
• October 21, 2009
• Brian F. Gage, MD
• Thanks to Curtis A. Parvin, Ph.D

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Regression Relate 1 predictor (independent)
variables to an outcome dependent) variable

• Cox regression
• Binary outcome variable
• Used to quantify the time to event (the hazard)
• Assumptions
• 1. Proportional hazard (multiplicative risk) and
• 2. Non-informative censoring
• Logistic regression
• Binary outcome variable
• Quantify the relationship between the odds of the
  outcome occurring and the predictor variable(s)
• Ordinary linear regression
• Continuous outcome variable
• Determine the relationship between a continuous
  outcome variable and the predictor variable(s)

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Other Flavors of Logistic Regression

• Conditional Logistic Regression
• Matched pairs data (11, 1k, k1k2 matching)
• Ordinal Logistic Regression
• More than two ordered groups for outcome
• Multinomial Logistic Regression
• More than two unordered groups for outcome

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Example 1 Odds of CA After Background Exposure
to Radiation

• Study Question Does the background level of
  radiation fallout cause CA?
• Predictor Variable ?
• http//www.elementsdatabase.com/

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• Hypothesis Higher levels of radioactive
  strontium (Sr-90) in deciduous teeth lost in the
  1960s predicts subsequent CA over the next 40
  years.
• Subjects St. Louis children
• Statistical Analysis Could be Cox or logistic
  regression
• Study Design What do you recommend?
• Assume that you are collaborating w/ the tooth
  fairy
• Finding http//newsok.com/st.-louis-baby-teeth-y
  ield-new-findings-on-nuclear-fallout/article/feed/
  95578

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Example 2 Development of Angina. Logistic
Regression Michael P. LaValleyCirculation
20081172395-2399
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Angina Goodness of Fit Calibration
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Angina Discrimination

• C-statistic measures how well we can
  differentiate volunteers in the 2 groups
• Generally ranges from 0.5 (no discrimination
  better than chance) to 1.0 (perfect
  discrimination)
• 0.8 0.9 is excellent
• C 0.64 for this model

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Example 3 Odds of Major Bleeding Around Time of
NSTEMI

• Background Tx of MI often causes bleeding
• Significance Being able to predict major
  bleeding, could allow us to minimize that risk
• Hypothesis We could develop validate an
  accurate clinical prediction rule for bleeding
• Study Design Split-sample, retrospective cohort
• Subjects 89,134 participants in CRUSADE

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Split Samples 80 Derivation 20 Validation
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http//www.crusadebleedingscore.org
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Example 4 Relationship between gestational age
(GA) and whether an infant is breast feeding at
time of hospital discharge
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Ordinary Linear Regression
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Logistic Regression
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How do we get an S-shaped curve?

• Rather than using probability as our outcome
  variable, we use a transformation that is a
  function of probability
• We choose our transformation so that it ranges
  between (-8,8) as probability ranges between
  (0,1)
• We will use the logit transform
• Fitting a straight line using the logit transform
  as the outcome variable is called logistic
  regression
• After we estimate the straight line we can
  transform back to get our S-shaped curve

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Probability, Odds, and the Logit Transform

• Probability, P, ranges between 0 and 1
• Define Odds P/(1-P)
• Odds range between 0 and 8
• Note P Odds/(1Odds)
• The Logit transform is the logarithm of the Odds
• Logit log(Odds) logP/(1-P)
• Logit ranges between -8 and 8
• Note Odds eLogit
• Note P eLogit/(1eLogit)

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Log(Odds) -16.72 0.577GA
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Logistic Regression

• Model the logarithm of the odds of an outcome as
  a linear combination of predictor variables
• Logit log(Odds) abXcY. . .
• Estimate the coefficients a, b, c based on a
  random sample of subjects data
• Determine which of the predictors are good
• Assess model fit
• Use the model to predict future cases

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Logistic Regression Coefficients

• For a single predictor variable, logistic
  regression fits a straight line to the log of the
  Odds
• log(Odds) a bX
• b is the slope coefficient for X
• Each 1 unit change in X, changes the log(Odds) by
  b units

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Logistic Regression Coefficients

• b logOdds(X1) logOdds(X)
• Note log(A) log(B) log(A/B)
• b logOdds(X1)/Odds(X)
• Note Odds(X1)/Odds(X) is called an Odds ratio

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Odds and Odds Ratios

• Odds defines the probability that an event occurs
  divided by the probability that the event doesnt
  occur
• An Odds ratio is the ratio of two Odds
• An Odds ratio could represent the ratio of the
  odds in two different groups
• An Odds ratio could represent the ratio of the
  odds at two different values for a risk variable

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Breast Feeding Example
The Odds ratio for breast feeding at hospital
discharge for GA32 compared to GA28 is 4.0/0.5
8.0
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Logistic Regression Coefficients and Odds Ratios

• b logOdds(X1)/Odds(X)
• b estimates the log of the Odds ratio associated
  with a 1 unit increase in X
• eb estimates the the odds ratio for a 1 unit
  increase in X
• For the breast feeding example
• log(Odds) -16.72 0.577GA
• the odds of breast feeding at hospital discharge
  increase by a factor of e0.577 1.78 for each
  additional week of GA

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Logistic Regression Odds Ratios
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Logistic Regression When There is Only One Binary
Predictor

• This situation can be handled as a classic
  case-control study

Disease Cases Controls
Risk Yes a b Factor No c
d Odds Ratio (OR) a/c ad b/d bc
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The Real Strength of Logistic Regression is When
There are Multiple Predictor Variables

• The independent variables (predictors, risk
  factors) can be categorical or continuous
• Example TDx-FLM II and gestational age as
  predictors of risk for respiratory distress
  syndrome (RDS)
• TDx-FLM II measures mg surfactant/g of albumin in
  amniotic fluid

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The Data (some of it)
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Logistic Regression Parameter Estimates
--------------------------------------------------
---------------------------- rds
Coef. Std. Err. z Pgtz 95 Conf.
Interval ---------------------------------------
--------------------------------------
tdxflm -.1121656 .0163848 -7.11 0.000
-.1442792 -.0800520 ga -.3661113
.1192559 -2.58 0.010 -.5998486
-.1323740 _cons 15.68597 4.322678
3.63 0.000 7.213680 24.15827 -----------
--------------------------------------------------
-----------------
log(Odds) 15.69 - 0.112TDxFLM - 0.366GA
Odds Ratio for a 1 g/mg increase in TDxFLM
e-0.112 0.894 Odds Ratio for a 1 week increase
in GA e-0.366 0.693
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Using the Logistic Model to Predict Risk of RDS

• We can use the logistic model equation to
• Identify variables that are significant
  predictors
• calculate the absolute risk (probability) of RDS
  (may give biased estimates)
• calculate the relative risk (odds ratio) of RDS
• develop a classifier for diagnosing RDS

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Logistic Regression Parameter Estimates
Significant coefficients mean significantly
different from zero
--------------------------------------------------
---------------------------- rds
Coef. Std. Err. z Pgtz 95 Conf.
Interval ---------------------------------------
--------------------------------------
tdxflm -.1121656 .0163848 -7.11 0.000
-.1442792 -.0800520 ga -.3661113
.1192559 -2.58 0.010 -.5998486
-.1323740 _cons 15.68597 4.322678
3.63 0.000 7.213680 24.15827 -----------
--------------------------------------------------
-----------------
Significant Odds ratios mean significantly
different from one
--------------------------------------------------
---------------------------- rds Odds
Ratio Std. Err. z Pgtz 95 Conf.
Interval ---------------------------------------
--------------------------------------
tdxflm .893896 .0154269 -7.11 0.000
.8636601 .9241324 ga .6934256
.0871025 -2.58 0.010 .5227078
.8641434 -----------------------------------------
-------------------------------------
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Odds ratios for RDS relative to a TDX FLM II
ratio of 70 mg/g at 37 weeks gestational age
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Logistic Regression Predicted Probabilities and
Classification with 0.20 cutoff
TDxFLM GA RDS Logistic
P Classify 75 30 0
.0115517 0 TN 7 31 1
.9521286 1 TP 14.8 31
1 .8912354 1 TP 18.3
31 1 .8462539 1 TP 27
31 1 .6718219 1 TP
22 31 0 .7832782 1
FP 29 31 0 .6198854
1 FP 135 31 0
.0000095 0 TN 4 32
1 .9543484 1 TP 15
32 1 .8568574 1 TP 16.5
32 1 .8346432 1 TP
25 32 1 .6575863 1
TP 44.2 32 1 .1779585
0 FN 35.5 32 0
.3679177 1 FP 41 32
0 .2374989 1 FP 48
32 0 .1232235 0 TN 49
32 0 .1114575 0 TN
55.8 32 0 .0547323 0
TN 59 32 0 .0386864
0 TN 59 32 0
.0386864 0 TN
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Software Packages that perform Logistic Regression

• STATA
• SAS
• SPSS
• R
• JMP
• Others

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References

• Hosmer DW, Lemeshow S. Applied logistic
  regression, 2nd ed., New York, NY John Wiley
  Sons, 2000.
• Kleinbaum DG. Logistic regression a
  self-learning text. New York, NY
  Springer-Verlag, 1994.
• Bagley SC, White H, Golumb BA. Logistic
  regression in the medical literature standards
  for use and reporting, with particular attention
  to one medical domain. J Clin Epidemiol
  200154979-85.
• (http//www.sciencedirect.com/science/publications
  /journal)
• Ostir GV, Uchida T. Logistic regression a
  nontechnical review. Am J Phys Med Rehabil
  200079565-72.
• (pdf file available online through Ovid gateway)
• http//www.ioa.pdx.edu/newsom/pa551/lectur21.htm
• http//personal.ecu.edu/whiteheadj/data/logit/
• Parvin CA, Kaplan LA, Chapman JF, McManamon TG,
  Gronowski AM. Predicting respiratory distress
  syndrome using gestational age and fetal lung
  maturity by fluorescent polarization. Am J Obstet
  Gynecol 2005192199-207.

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Next Week

• Attend 4th Annual Research Symposium and Poster
  Session 1230-430
• Farrell Learning and Teaching Centers Connor
  Auditorium
• Lecture 440-545
• Read Hulley et al. Chapter 9
• Problem Set 4