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Chapter 12

- Multiple Regression

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)

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)

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

Analysis of Variance

- Direct extension to ANOVA based on simple linear

regression - Only adjustments are to degrees of freedom
- DFR k DFE n-(k1)

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

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

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)

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

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)

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.

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

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)

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)

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)

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

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

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

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)

Example - Rizatriptan for Migraine (SPSS)

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)

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

Example - Rizatriptan for Migraine

- 95 CI for b1

- 95 CI for population odds ratio

- Conclude positive association between dose and

probability of complete relief

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

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)

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

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