Linear Regression and Correlation

1
Linear Regression and Correlation

• Explanatory and Response Variables are Numeric
• Relationship between the mean of the response
  variable and the level of the explanatory
  variable assumed to be approximately linear
  (straight line)
• Model

• b1 gt 0 ? Positive Association
• b1 lt 0 ? Negative Association
• b1 0 ? No Association

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Least Squares Estimation of b0, b1

• b0 ? Mean response when x0 (y-intercept)
• b1 ? Change in mean response when x increases by
  1 unit (slope)
• b0, b1 are unknown parameters (like m)
• b0b1x ?? Mean response when explanatory
  variable takes on the value x
• Goal Choose values (estimates) that minimize the
  sum of squared errors (SSE) of observed values to
  the straight-line

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Example - Pharmacodynamics of LSD

• Response (y) - Math score (mean among 5
  volunteers)
• Predictor (x) - LSD tissue concentration (mean
  of 5 volunteers)
• Raw Data and scatterplot of Score vs LSD
  concentration

Source Wagner, et al (1968)
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Least Squares Computations
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Example - Pharmacodynamics of LSD
(Column totals given in bottom row of table)
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SPSS Output and Plot of Equation
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Inference Concerning the Slope (b1)

• Parameter Slope in the population model (b1)
• Estimator Least squares estimate
• Estimated standard error
• Methods of making inference regarding population
• Hypothesis tests (2-sided or 1-sided)
• Confidence Intervals

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Hypothesis Test for b1

• 1-sided Test
• H0 b1 0
• HA b1 gt 0 or
• HA- b1 lt 0

• 2-Sided Test
• H0 b1 0
• HA b1 ? 0

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(1-a)100 Confidence Interval for b1

• Conclude positive association if entire interval
  above 0
• Conclude negative association if entire interval
  below 0
• Cannot conclude an association if interval
  contains 0
• Conclusion based on interval is same as 2-sided
  hypothesis test

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Example - Pharmacodynamics of LSD

• Testing H0 b1 0 vs HA b1 ? 0

• 95 Confidence Interval for b1

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Confidence Interval for Mean When xx

• Mean Response at a specific level x is
• Estimated Mean response and standard error
  (replacing unknown b0 and b1 with estimates)
• Confidence Interval for Mean Response

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Prediction Interval of Future Response _at_ xx

• Response at a specific level x is
• Estimated response and standard error (replacing
  unknown b0 and b1 with estimates)
• Prediction Interval for Future Response

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Correlation Coefficient

• Measures the strength of the linear association
  between two variables
• Takes on the same sign as the slope estimate from
  the linear regression
• Not effected by linear transformations of y or x
• Does not distinguish between dependent and
  independent variable (e.g. height and weight)
• Population Parameter - r
• Pearsons Correlation Coefficient

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Correlation Coefficient

• Values close to 1 in absolute value ? strong
  linear association, positive or negative from
  sign
• Values close to 0 imply little or no association
• If data contain outliers (are non-normal),
  Spearmans coefficient of correlation can be
  computed based on the ranks of the x and y values
• Test of H0r 0 is equivalent to test of
  H0b10
• Coefficient of Determination (r2) - Proportion
  of variation in y explained by the regression
  on x

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Example - Pharmacodynamics of LSD
Syy
SSE
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Example - SPSS OutputPearsons and Spearmans
Measures
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Analysis of Variance in Regression

• Goal Partition the total variation in y into
  variation explained by x and random variation

• These three sums of squares and degrees of
  freedom are
• Total (SST) DFT n-1
• Error (SSE) DFE n-2
• Model (SSM) DFM 1

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Analysis of Variance in Regression

• Analysis of Variance - F-test
• H0 b1 0 HA b1 ?? 0

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Example - Pharmacodynamics of LSD

• Total Sum of squares

• Error Sum of squares

• Model Sum of Squares

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Example - Pharmacodynamics of LSD

• Analysis of Variance - F-test
• H0 b1 0 HA b1 ?? 0

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Example - SPSS Output
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Multiple Regression

• Numeric Response variable (Y)
• p Numeric predictor variables
• Model
• Y b0 b1x1 ??? bpxp e
• Partial Regression Coefficients bi ? effect (on
  the mean response) of increasing the ith
  predictor variable by 1 unit, holding all other
  predictors constant

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

• Response Height at Early adolescence (n 250
  cases)
• Predictors (p6 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

25
Analysis of Variance

• Direct extension to ANOVA based on simple linear
  regression
• Only adjustments are to degrees of freedom
• DFM p DFE n-p-1

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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???bp0 (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 p6 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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Testing for the Overall Model - F-test

• Tests whether any of the explanatory variables
  are associated with the response
• H0 b1???bp0 (None of Xs associated with Y)
• HA Not all bi 0

The P-value is based on the F-distribution with p
numerator and (n-p-1) denominator degrees of
freedom
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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 p-g predictors
  (including possibly cross-product terms) can be
  dropped from a model that contains the remaining
  g predictors. H0 bg1bp 0
• Complete Model Contains all k predictors
• Reduced Model Eliminates the predictors from H0
• Fit both models, obtaining the Error sum of
  squares for each (or R2 from each)

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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 p-g and
n-p-1 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 k levels, need to
  create k-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
  for which all k-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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Analysis of Covariance

• Combination of 1-Way ANOVA and Linear Regression
• Goal Comparing numeric responses among k groups,
  adjusting for numeric concomitant variable(s),
  referred to as Covariate(s)
• Clinical trial applications Response is Post-Trt
  score, covariate is Pre-Trt score
• Epidemiological applications Outcomes compared
  across exposure conditions, adjusted for other
  risk factors (age, smoking status, sex,...)

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Nonlinear Regression

• Theory often leads to nonlinear relations between
  variables. Examples
• 1-compartment PK model with 1st-order absorption
  and elimination
• Sigmoid-Emax S-shaped PD model

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Example - P24 Antigens and AZT

• Goal Model time course of P24 antigen levels
  after oral administration of zidovudine
• Model fit individually in 40 HIV patients

• where
• E(t) is the antigen level at time t
• E0 is the initial level
• A is the coefficient of reduction of P24 antigen
• kout is the rate constant of decrease of P24
  antigen

Source Sasomsin, et al (2002)
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Example - P24 Antigens and AZT

• Among the 40 individuals who the model was fit,
  the means and standard deviations of the PK
  parameters are given below

• Fitted Model for the mean subject

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Example - P24 Antigens and AZT
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Example - MK639 in HIV Patients

• Response Y log10(RNA change)
• Predictor x MK639 AUC0-6h
• Model Sigmoid-Emax

• where
• b0 is the maximum effect (limit as x??)
• b1 is the x level producing 50 of maximum
  effect
• b2 is a parameter effecting the shape of the
  function

Source Stein, et al (1996)
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Example - MK639 in HIV Patients

• Data on n 5 subjects in a Phase 1 trial

• Model fit using SPSS (estimates slightly
  different from notes, which used SAS)

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Example - MK639 in HIV Patients
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Data Sources

• Wagner, J.G., G.K. Aghajanian, and O.H. Bing
  (1968). Correlation of Performance Test Scores
  with Tissue Concentration of Lysergic Acid
  Diethylamide in Human Subjects, Clinical
  Pharmacology and Therapeutics, 9635-638.