Download Modelling Binary and Categorical Data using NONMEM

PAGE 2005
Introduction to Categorical Data Analysis
Adrian Dunne
Department of Statistics and Actuarial Science,
Roinn na Staitisticí agus na hAchtúreolaíochta
University College Dublin
An Coláiste Ollscoile Baile Átha Cliath
Data Types
• Quantitative
– Continuous
– Discrete
Plasma drug conc., BP, Muscle Tension, Time
Number of blood cells, Number of heart attacks
• Categorical
– Nominal
– Ordinal
– Binary
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Religion, Nationality, Gender
Social class, Treatment outcome
Gender, Dead/Alive
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Example: Categorical Data
Population of patients require a
particular analgesic
Estimate population
proportions in each
category –
none, some, complete.
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Drug administered to
sample of patients
Each patient is assessed
for pain relief –
none, some, complete
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Binary Data
• Describe the two categories as
“Success” (S) and “Failure” (F).
• Code
Z = 0 for F
Z = 1 for S
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Binary Data
• Proportion of S in population.
• Randomly select a member of the
population – probability of S.
• Proportion = Probability
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Binary Data
• Z has the Bernoulli Distribution
Pr (Z = 1 ) = π
Prob
Pr (Z = 0 ) = 1 − π
Pr( Z = r ) = π
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r
of S
Prob
of F
(1 − π ) ( 1 − r )
r = 0 ,1
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Estimation: Method of Maximum
Likelihood
• Likelihood
n
L(π ) = ∏ π (1 − π )
zi
(1− zi )
i =1
• πˆ is the value of π that maximises the
likelihood
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Simple Example
10 observations:- 3S’s 7F’s
Simple model with no structure
Z ~ Bernoulli (π )
n
L(π ) = ∏ π (1 − π )
zi
(1− zi )
i =1
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Example: Likelihood
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Data Modelling
• Previous example had no structure in the
data.
• Consider the case where the subjects were
administered different doses of drug and the
response depends on dose.
• Another example would be where response
changes with time following drug
administration (PK-PD).
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Data Modelling
• Take account of the structure by recording
the values of covariates (x’s) for each
member of the sample e.g. dose, time.
• Then construct a model which describes
how the parameters depend on the
covariates.
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Modelling Binary Data
• We model π i e.g.
• However,
π i = f ( xi ,θ )
0 ≤ πi ≤1
• There is no guarantee that
0 ≤ f ( xi ,θˆ) ≤ 1
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Modelling Binary Data
Transform πi from (0,1) to (−∞,+∞) and
model the transformed value to ensure that
model predicted probabilities lie in (0,1).
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Transformations
• Logit
 πi 

logit (π i ) = log
1−π i 
• Probit
probit (π i ) = Φ −1 (π i )
• Log-log
log(− log(π i ))
• Complementary log-log
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log(− log(1 − π i ))
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Probit transformation
Normal
distribution
π
Probit
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Transformations
Logit
Log-log
Comp. Log-log
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Probit
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Logistic Regression Model
Now consider a model with structure
logit (π i ) = f ( xi ,θ )
Example
logit (π i ) = θ1 + θ 2 x1i + θ 3 x2i + ...
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Example: Bioassay
Beetle deaths following dosing with an
insecticide
Dose
0.0028
0.0056
0.0112
0.0225
0.0450
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# Exposed
40
40
40
40
40
# Dead
5
19
31
34
39
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Logit Model
Linear logistic model with log(dose)
zi ~ Bernoulli (π i )
log it (π i ) = θ1 + θ 2 log(dosei )
n
L(θ) = ∏ π i (1 − π i )
zi
(1− zi )
i =1
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Observed & Predicted Values
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Mixed Effects Modelling
• Modelling correlation between
responses/variation between groups.
– Groups of related items
– Repeated measures/Longitudinal data
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Example: Binary PK-PD response
• 10 subjects all received dose of 100 units.
• Bolus iv administration.
• Binary response (dry mouth) recorded for
each subject at times 0.5, 1, 2, 3, 5, 7, 9, 12,
15, 18, 24, 30 hours.
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PK-PD Model
Bolus (D) at t=0
Central Compartment
Cp(t)
K
k1e
Effect Compartment
Ce(t)
keo
Eff ( t ) = f ( C e ( t ))
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PK-PD Model
• Linear PD model
Eff (t ) = θ1 + θ 2Ce (t )
logit (π (t )) = θ1 + θ 2Ce (t )
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Example: Binary Population PK-PD model
• Variation between subjects
• Longitudinal (repeated measures) data –
observations on same subject are correlated
• Model intrasubject correlation and
intersubject variation using random effects
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Example: Binary Population PK-PD model
logit ( π i ( t j )) = θ 1 + θ 2 C e ( t j ) + η i
η i ~ N (0, Ω )
n
L(θ, Ω) = ∏
+∞ mi
∫ ∏ π (t )
i
j
zi
(1 − π i (t j ))
(1− zi )
f (ηi , φ )dηi
i =1 − ∞ j =1
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Observed & Predicted Values
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Latent Variable
• Consider again the insecticide bioassay
• Assume that each insect has an
(unobserved) tolerance ti which varies
randomly across the population of insects
ti ≤ d i ⇒ zi = 1
ti > d i ⇒ zi = 0
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Tolerance distribution
π i = Pr(ti ≤ d i )
πi
1−π i
di
zi = 1
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zi = 0
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Latent Variables
• di is known as the cut-point
• Here the latent variable is tolerance
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π i = Pr(ti ≤ d i )
πi =
di
∫
f (ti , β)dti
−∞
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exp((ti − β 0 ) / β1 )
f(ti , β) =
2
β1 (1 + exp((ti − β 0 ) / β1 ) )
logit (π i ) = θ1 + θ 2 d i
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exp((log(ti ) − β 0 ) / β1 )
f(ti , β) =
2
ti β1 (1 + exp((log(ti ) − β 0 ) / β1 ) )
logit (π i ) = θ1 + θ 2 log(d i )
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exp(−0.5((ti − β 0 ) / β1 ) )
f (ti , β) =
2π β1
2
probit (π i ) = θ1 + θ 2 d i
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exp(−0.5((log(ti ) − β 0 ) / β1 ) )
f (ti , β) =
ti 2π β1
2
probit (π i ) = θ1 + θ 2 log(d i )
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f (ti , β) =
exp((ti − β0 ) / β1 ) exp(− exp((ti − β0 ) / β1 ))
β1
log(-log(1- π i )) = θ1 + θ2di
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exp((log(ti ) − β0 ) / β1 ) exp(− exp((log(ti ) − β0 ) / β1 ))
f (ti , β) =
ti β1
log(-log(1- π i )) = θ1 + θ2 log(di )
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Ordinal Data
• Ordered categories e.g. severity of
symptoms, none, mild, moderate, severe.
• Ordered categories
• Probabilities
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Z = 1,2,..., K
π 1 , π 2 ,..., π K
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Latent Variable
α1
Z =1
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α2
Z =2
Z =3
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Cumulative Logits
• Cumulative probabilities
Fk = π 1 + π 2 + ... + π k
• Cumulative Logits
 Fk
Lk = logit ( Fk ) = log
 1 − Fk



k = 1,2,..., K − 1
• A model for Lk is a logit model for a binary
response.
• We need K-1 logit models
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Cumulative Logits
Based on K-1 dichotomizations.
–
–
–
–
(1) and (2 to K)
(1 and 2) and (3 to K)
(1 to 3) and (4 to K)
etc.
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Proportional Odds Model
• Covariate x influences all cumulative logits
equally
logit ( Fk ) = α k − f ( x, θ )
• Such a model is equivalent to x influencing
the location (but not the spread) of the
distribution of the latent variable.
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Proportional Hazards Model
• Covariate x influences all cumulative
complementary log-logs equally
log(− log(1 − Fk )) = α k − f ( x,θ )
• Such a model is equivalent to x influencing
the location (but not the spread) of the
distribution of the latent variable.
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