05 - Generalized Linear Models 广义线性模型
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Lecture 5 Generalized Linear Models
Outline
Introduction Theory of the Generalized Linear Models
Logistic Regression
Poisson and Negative Binomial Regression
Introduction
Review: How to compare treatments?
Usually an endpoint is compared across treatment groups,
while controlling for important predictors
• Example: Control for baseline measurements • Predictors could be continuous or categorical
Relapse
Remission
10 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Time
Introduction
RRMS study example – MRI scans
- T1 lesion - Combined unique active lesion (CUAL)
11 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Introduction
Introduction
AML study example – Data structure
PID: G: Patient identification Treatment group ACT = Drug A PBO = Placebo Time in months from last relapse YES / NO
If a patient had a more recent
relapse, he or she was more likely to have a relapse observed
Both groups have a similar
decreasing pattern
Patients receiving drug A seem
how can such endpoints be compared across treatment groups, while controlling for other predictors?
5 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
We can use the ANCOVA method, if
• The responses are normally distributed
• The responses depend on the continuous predictors linearly
Binary or count data are often observed in clinical trials,
to have lower relapse rate
Is the difference significant?
• Use logistic regression
9 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Patient identification Treatment group ACT = Drug B PBO = Placebo Baseline count of T1 lesions Number of CUALs at weerows of the data set are shown.
Introduction
AML study example – Background
Double-blind, parallel-group study to assess a new drug A
in patients with acute myelogenous leukemia (AML)
Introduction
RRMS study example – Disease background
Double-blind, parallel-group study to assess a new drug B in
patients with relapsing-remitting multiple sclerosis (RRMS)
RRMS study example – Study background
At the beginning of the study, the number of T1 lesions was
recorded as a baseline predictor
56 Patients randomly received either drug B or placebo
• 32 patients received drug B • 24 patients received placebo
After 24 weeks of treatment, the number of CUALs was
recorded for each patient as the response
recorded for each patient
• A binary response variable taking two values (“YES” or “NO”)
6 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
MS Patients suffer from neurological disability, including
numbness, vision problems, stiffness of joints
See lecture “Principles of Clinical Trials”
Increasing Disability
12 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Introduction
RRMS study example – Data structure
PID: G:
X: CUAL:
models
Be able to apply logistic, Poisson and negative binomial
regression models to real problems, and know how to interpret the fitted models
3 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Disease activity is measured by readings from MRI
(magnetic resonance imaging) scans
• Various types of lesions in the brain can be counted • Examples
Question: Is there any evidence that drug B has reduced
the number of CUALs?
• Need to consider baseline count of T1 lesions
13 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
X: Relapse:
* Only first 11 rows of the data set are shown.
Question: Is there any evidence that drug A has reduced
the relapse rate?
• Relapse rate is the probability of relapse occurrence
8 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Introduction
AML study example – Graphical data display
Sample proportion of relapse occurrence vs. time from last relapse
• Need to consider the baseline, i.e., time from last relapse
7 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Introduction
AML study example – Data structure
For the rest of the data set, the same calculation can be performed.
* Only first 23 rows of the data set are shown.
At the beginning of the trial, before each patient received
any treatment, time of remission in months from last relapse was taken as a baseline
• It is considered an important predictor in predicting how likely a relapse will occur
Summary
2 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Learning Objectives
Understand the basic theory of the generalized linear
Introduction Theory of Generalized Linear Model
Logistic Regression
Poisson and Negative Binomial Regression
Summary
4 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
102 Patients randomly received either drug A or placebo
• 52 patients received drug A; 50 patients received placebo
After 90 days, whether a relapse occurred or not was
Outline
Introduction Theory of the Generalized Linear Models
Logistic Regression
Poisson and Negative Binomial Regression
Introduction
Review: How to compare treatments?
Usually an endpoint is compared across treatment groups,
while controlling for important predictors
• Example: Control for baseline measurements • Predictors could be continuous or categorical
Relapse
Remission
10 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Time
Introduction
RRMS study example – MRI scans
- T1 lesion - Combined unique active lesion (CUAL)
11 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Introduction
Introduction
AML study example – Data structure
PID: G: Patient identification Treatment group ACT = Drug A PBO = Placebo Time in months from last relapse YES / NO
If a patient had a more recent
relapse, he or she was more likely to have a relapse observed
Both groups have a similar
decreasing pattern
Patients receiving drug A seem
how can such endpoints be compared across treatment groups, while controlling for other predictors?
5 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
We can use the ANCOVA method, if
• The responses are normally distributed
• The responses depend on the continuous predictors linearly
Binary or count data are often observed in clinical trials,
to have lower relapse rate
Is the difference significant?
• Use logistic regression
9 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Patient identification Treatment group ACT = Drug B PBO = Placebo Baseline count of T1 lesions Number of CUALs at weerows of the data set are shown.
Introduction
AML study example – Background
Double-blind, parallel-group study to assess a new drug A
in patients with acute myelogenous leukemia (AML)
Introduction
RRMS study example – Disease background
Double-blind, parallel-group study to assess a new drug B in
patients with relapsing-remitting multiple sclerosis (RRMS)
RRMS study example – Study background
At the beginning of the study, the number of T1 lesions was
recorded as a baseline predictor
56 Patients randomly received either drug B or placebo
• 32 patients received drug B • 24 patients received placebo
After 24 weeks of treatment, the number of CUALs was
recorded for each patient as the response
recorded for each patient
• A binary response variable taking two values (“YES” or “NO”)
6 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
MS Patients suffer from neurological disability, including
numbness, vision problems, stiffness of joints
See lecture “Principles of Clinical Trials”
Increasing Disability
12 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Introduction
RRMS study example – Data structure
PID: G:
X: CUAL:
models
Be able to apply logistic, Poisson and negative binomial
regression models to real problems, and know how to interpret the fitted models
3 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Disease activity is measured by readings from MRI
(magnetic resonance imaging) scans
• Various types of lesions in the brain can be counted • Examples
Question: Is there any evidence that drug B has reduced
the number of CUALs?
• Need to consider baseline count of T1 lesions
13 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
X: Relapse:
* Only first 11 rows of the data set are shown.
Question: Is there any evidence that drug A has reduced
the relapse rate?
• Relapse rate is the probability of relapse occurrence
8 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Introduction
AML study example – Graphical data display
Sample proportion of relapse occurrence vs. time from last relapse
• Need to consider the baseline, i.e., time from last relapse
7 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Introduction
AML study example – Data structure
For the rest of the data set, the same calculation can be performed.
* Only first 23 rows of the data set are shown.
At the beginning of the trial, before each patient received
any treatment, time of remission in months from last relapse was taken as a baseline
• It is considered an important predictor in predicting how likely a relapse will occur
Summary
2 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
Learning Objectives
Understand the basic theory of the generalized linear
Introduction Theory of Generalized Linear Model
Logistic Regression
Poisson and Negative Binomial Regression
Summary
4 | Basic Statistics in Clinical Trials | Generalized Linear Models | All Rights Reserved
102 Patients randomly received either drug A or placebo
• 52 patients received drug A; 50 patients received placebo
After 90 days, whether a relapse occurred or not was