Logistic and Nonlinear Regression
• Logistic Regression - Dichotomous Response
variable and numeric and/or categorical
explanatory variable(s)
– Goal: Model the probability of a particular as a function
of the predictor variable(s)
– Problem: Probabilities are bounded between 0 and 1
• Nonlinear Regression: Numeric response and
explanatory variables, with non-straight line
relationship
– Biological (including PK/PD) models often based on
known theoretical shape with unknown parameters
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
  x
e
p ( x) 
  x
1 e
•  = 0  P(Presence) is the same at each level of x
•  > 0  P(Presence) increases as x increases
•  < 0  P(Presence) decreases as x increases
Logistic Regression with 1 Predictor
 ,  are unknown parameters and must be
estimated using statistical software such as SPSS,
SAS, or STATA
· Primary interest in estimating and testing
hypotheses regarding 
· Large-Sample test (Wald Test):
· H0:  = 0
HA:   0
2
T .S . : X obs
2
R.R. : X obs
 ^
 
 ^
 ^

 
  2 ,1






2
2
P  val : P (  2  X obs
)
Example - Rizatriptan for Migraine
• Response - Complete Pain Relief at 2 hours (Yes/No)
• Predictor - Dose (mg): Placebo (0),2.5,5,10
Dose
0
2.5
5
10
Source: Gijsmant, et al (1997)
# Patients
67
75
130
145
# Relieved
2
7
29
40
% Relieved
3.0
9.3
22.3
27.6
Example - Rizatriptan for Migraine (SPSS)
t
d
B
p
.
a
ig
E
S
D
5
7
9
1
0
0
a
1
C
0
5
6
1
0
3
a
V
2.490 0.165x
e
p ( x) 
 2.490 0.165x
1 e
^
H0 :   0 H A :   0
2
2
T .S . : X obs
 0.165 

  19.819
 0.037 
2
RR : X obs
  .205,1  3.84
P  val : .000
Odds Ratio
• Interpretation of Regression Coefficient ():
– 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:
odds( x  1)
 e
odds( x)

p ( x) 
 odds( x) 

1  p ( x) 

• Thus e represents the change in the odds of the outcome
(multiplicatively) by increasing x by 1 unit
• If  = 0, the odds and probability are the same at all x levels (e=1)
• If  > 0 , the odds and probability increase as x increases (e>1)
• If  < 0 , the odds and probability decrease as x increases (e<1)
95% Confidence Interval for Odds Ratio
• Step 1: Construct a 95% CI for  :
^
^
^
^
^
^
^ 
    1.96   ,   1.96   


^
  1.96  
^
• Step 2: Raise e = 2.718 to the lower and upper bounds of the CI:
 e  1.96  , e  1.96  




^
^ ^
^
^ ^
• 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  :
^
^
  0.165
   0.037
^
95% CI : 0.165  1.96(0.037)  (0.0925 , 0.2375)
• 95% CI for population odds ratio:
e
0.0925

, e0.2375  (1.10 , 1.27)
• 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:
   x   x
p p
e 11
p
   x   p x p
1 e 1 1
• Adjusted Odds ratio for raising xi by 1 unit, holding
all other predictors constant:
ORi  e  i
• Inferences on i 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 (n=1688)
• Predictors: (p=12)
–
–
–
–
Age stratum (50-54*, 55-59, 60-64, 65-69, 70-78)
Smoking status (Nonsmoker*, Smoker)
BMI stratum (<25*, 25-30, >30)
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
Predictor
Age 55-59 (vs 50-54)
Age 60-64 (vs 50-54)
Age 65-69 (vs 50-54)
Age 70-78 (vs 50-54)
Smoker (vs nonsmoker)
BMI 25-30 (vs <25)
BMI >30 (vs <25)
LUTS Mild (vs None)
LUTS Moderate (vs None)
LUTS Severe (vs None)
Cardiac symptoms (Yes vs No)
COPD (Yes vs No)
b
0.83
1.53
2.19
2.66
0.47
0.41
1.10
0.59
1.22
2.01
0.92
0.64
sb
0.42
0.40
0.40
0.41
0.19
0.21
0.29
0.41
0.45
0.56
0.26
0.28
Adjusted OR (95% CI)
2.3 (1.0 – 5.2)
4.6 (2.1 – 10.1)
8.9 (4.1 – 19.5)
14.3 (6.4 – 32.1)
1.6 (1.1 – 2.3)
1.5 (1.0 – 2.3)
3.0 (1.7 – 5.4)
1.8 (0.8 – 4.3)
3.4 (1.4 – 8.4)
7.5 (2.5 – 22.5)
2.5 (1.5 – 4.3)
1.9 (1.1 – 3.6)
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
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
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:
E (t )  E0 (1  A)e  kout t  E0 A
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)
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:
Parameter
E0
A
kout
Mean
472.1
0.28
0.27
Std Dev
408.8
0.21
0.16
• Fitted Model for the “mean subject”
E (t )  472.1(1  0.28)e 0.27t  (472.1)(0.28)
Example - P24 Antigens and AZT
Example - MK639 in HIV+ Patients
• Response: Y = log10(RNA change)
• Predictor: x = MK639 AUC0-6h
• Model: Sigmoid-Emax:
0 x 
Y 

x  1
2
2
2
• where:
• 0 is the maximum effect (limit as x)
• 1 is the x level producing 50% of maximum effect
• 2 is a parameter effecting the shape of the function
Source: Stein, et al (1996)
Example - MK639 in HIV+ Patients
• Data on n = 5 subjects in a Phase 1 trial:
Subject
1
2
3
4
5
log RNA change (Y) MK639 AUC0-6h (x)
0.000
10576.9
0.167
13942.3
1.524
18235.3
3.205
19607.8
3.518
22317.1
• Model fit using SPSS (estimates slightly different from notes,
which used SAS)
35.60
3.52 x
Y  35.60
x
 18374.39
^
Example - MK639 in HIV+ Patients