Multiple Regression

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Chapter 12

• Multiple Regression

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Multiple Regression

• Numeric Response variable (y)
• k Numeric predictor variables (k lt n)
• Model
• Y b0 b1x1 ??? bkxk e
• Partial Regression Coefficients bi ? effect (on
  the mean response) of increasing the ith
  predictor variable by 1 unit, holding all other
  predictors constant
• Model Assumptions (Involving Error terms e )
• Normally distributed with mean 0
• Constant Variance s2
• Independent (Problematic when data are series in
  time/space)

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Example - Effect of Birth weight on Body Size in
Early Adolescence

• Response Height at Early adolescence (n 250
  cases)
• Predictors (k6 explanatory variables)
• Adolescent Age (x1, in years -- 11-14)
• Tanner stage (x2, units not given)
• Gender (x31 if male, 0 if female)
• Gestational age (x4, in weeks at birth)
• Birth length (x5, units not given)
• Birthweight Group (x61,...,6 lt1500g (1),
  1500-1999g(2), 2000-2499g(3), 2500-2999g(4),
  3000-3499g(5), gt3500g(6))

Source Falkner, et al (2004)
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Least Squares Estimation

• Population Model for mean response

• Least Squares Fitted (predicted) equation,
  minimizing SSE

• All statistical software packages/spreadsheets
  can compute least squares estimates and their
  standard errors

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Analysis of Variance

• Direct extension to ANOVA based on simple linear
  regression
• Only adjustments are to degrees of freedom
• DFR k DFE n-(k1)

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Testing for the Overall Model - F-test

• Tests whether any of the explanatory variables
  are associated with the response
• H0 b1???bk0 (None of the xs associated with
  y)
• HA Not all bi 0

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Example - Effect of Birth weight on Body Size in
Early Adolescence

• Authors did not print ANOVA, but did provide
  following
• n250 k6 R20.26
• H0 b1???b60 HA Not all bi 0

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Testing Individual Partial Coefficients - t-tests

• Wish to determine whether the response is
  associated with a single explanatory variable,
  after controlling for the others
• H0 bi 0 HA bi ? 0 (2-sided
  alternative)

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Example - Effect of Birth weight on Body Size in
Early Adolescence
Controlling for all other predictors, adolescent
age, Tanner stage, and Birth length are
associated with adolescent height measurement
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Comparing Regression Models

• Conflicting Goals Explaining variation in Y
  while keeping model as simple as possible
  (parsimony)
• We can test whether a subset of k-g predictors
  (including possibly cross-product terms) can be
  dropped from a model that contains the remaining
  g predictors. H0 bg1bk 0
• Complete Model Contains all k predictors
• Reduced Model Eliminates the predictors from H0
• Fit both models, obtaining sums of squares for
  each (or R2 from each)
• Complete SSRc , SSEc (Rc2)
• Reduced SSRr , SSEr (Rr2)

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Comparing Regression Models

• H0 bg1bp 0 (After removing the effects of
  X1,,Xg, none of other predictors are associated
  with Y)
• Ha H0 is false

P-value based on F-distribution with k-g and
n-(k1) d.f.
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Models with Dummy Variables

• Some models have both numeric and categorical
  explanatory variables (Recall gender in example)
• If a categorical variable has m levels, need to
  create m-1 dummy variables that take on the
  values 1 if the level of interest is present, 0
  otherwise.
• The baseline level of the categorical variable is
  the one for which all m-1 dummy variables are set
  to 0
• The regression coefficient corresponding to a
  dummy variable is the difference between the mean
  for that level and the mean for baseline group,
  controlling for all numeric predictors

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Example - Deep Cervical Infections

• Subjects - Patients with deep neck infections
• Response (Y) - Length of Stay in hospital
• Predictors (One numeric, 11 Dichotomous)
• Age (x1)
• Gender (x21 if female, 0 if male)
• Fever (x31 if Body Temp gt 38C, 0 if not)
• Neck swelling (x41 if Present, 0 if absent)
• Neck Pain (x51 if Present, 0 if absent)
• Trismus (x61 if Present, 0 if absent)
• Underlying Disease (x71 if Present, 0 if absent)
• Respiration Difficulty (x81 if Present, 0 if
  absent)
• Complication (x91 if Present, 0 if absent)
• WBC gt 15000/mm3 (x101 if Present, 0 if absent)
• CRP gt 100mg/ml (x111 if Present, 0 if absent)

Source Wang, et al (2003)
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Example - Weather and Spinal Patients

• Subjects - Visitors to National Spinal Network in
  23 cities Completing SF-36 Form
• Response - Physical Function subscale (1 of 10
  reported)
• Predictors
• Patients age (x1)
• Gender (x21 if female, 0 if male)
• High temperature on day of visit (x3)
• Low temperature on day of visit (x4)
• Dew point (x5)
• Wet bulb (x6)
• Total precipitation (x7)
• Barometric Pressure (x7)
• Length of sunlight (x8)
• Moon Phase (new, wax crescent, 1st Qtr, wax
  gibbous, full moon, wan gibbous, last Qtr, wan
  crescent, presumably had 8-17 dummy variables)

Source Glaser, et al (2004)
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Modeling Interactions

• Statistical Interaction When the effect of one
  predictor (on the response) depends on the level
  of other predictors.
• Can be modeled (and thus tested) with
  cross-product terms (case of 2 predictors)
• E(Y) a b1X1 b2X2 b3X1X2
• X20 ? E(Y) a b1X1
• X210 ? E(Y) a b1X1 10b2 10b3X1
• (a 10b2)
  (b1 10b3)X1
• The effect of increasing X1 by 1 on E(Y) depends
  on level of X2, unless b30 (t-test)

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Logistic Regression

• Logistic Regression - Binary Response variable
  and numeric and/or categorical explanatory
  variable(s)
• Goal Model the probability of a particular
  outcome as a function of the predictor
  variable(s)
• Problem Probabilities are bounded between 0 and 1

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Logistic Regression with 1 Predictor

• Response - Presence/Absence of characteristic
• Predictor - Numeric variable observed for each
  case
• Model - p (x) ? Probability of presence at
  predictor level x

• b 0 ? P(Presence) is the same at each level
  of x
• b gt 0 ? P(Presence) increases as x increases
• b lt 0 ? P(Presence) decreases as x increases

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Logistic Regression with 1 Predictor

• b0, b1 are unknown parameters and must be
  estimated using statistical software such as
  SPSS, SAS, or STATA
• Primary interest in estimating and testing
  hypotheses regarding b1
• Large-Sample test (Wald Test) (Some software
  runs z-test)
• H0 b1 0 HA b1 ? 0

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Example - Rizatriptan for Migraine

• Response - Complete Pain Relief at 2 hours
  (Yes/No)
• Predictor - Dose (mg) Placebo (0),2.5,5,10

Source Gijsmant, et al (1997)
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Example - Rizatriptan for Migraine (SPSS)
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Odds Ratio

• Interpretation of Regression Coefficient (b1)
• In linear regression, the slope coefficient is
  the change in the mean response as x increases by
  1 unit
• In logistic regression, we can show that

• Thus eb1 represents the change in the odds of
  the outcome (multiplicatively) by increasing x by
  1 unit
• If b10, the odds (and probability) are equal at
  all x levels (eb11)
• If b1gt0 , the odds (and probability) increase as
  x increases (eb1gt1)
• If b1lt 0 , the odds (and probability) decrease
  as x increases (eb1lt1)

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95 Confidence Interval for Odds Ratio

• Step 1 Construct a 95 CI for b

• Step 2 Raise e 2.718 to the lower and upper
  bounds of the CI

• If entire interval is above 1, conclude positive
  association
• If entire interval is below 1, conclude negative
  association
• If interval contains 1, cannot conclude there is
  an association

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Example - Rizatriptan for Migraine

• 95 CI for b1

• 95 CI for population odds ratio

• Conclude positive association between dose and
  probability of complete relief

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Multiple Logistic Regression

• Extension to more than one predictor variable
  (either numeric or dummy variables).
• With p predictors, the model is written

• Adjusted Odds ratio for raising xi by 1 unit,
  holding all other predictors constant

• Inferences on bi and ORi are conducted as was
  described above for the case with a single
  predictor

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Example - ED in Older Dutch Men

• Response Presence/Absence of ED (n1688)
• Predictors (k12)
• Age stratum (50-54, 55-59, 60-64, 65-69, 70-78)
• Smoking status (Nonsmoker, Smoker)
• BMI stratum (lt25, 25-30, gt30)
• Lower urinary tract symptoms (None, Mild,
  Moderate, Severe)
• Under treatment for cardiac symptoms (No, Yes)
• Under treatment for COPD (No, Yes)
• Baseline group for dummy variables

Source Blanker, et al (2001)
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Example - ED in Older Dutch Men

• Interpretations Risk of ED appears to be
• Increasing with age, BMI, and LUTS strata
• Higher among smokers
• Higher among men being treated for cardiac or
  COPD