Generalized Linear Model
Simple Linear Regression
What if the assumption is violated?
Binary Outcome
Patient | Smoking (1 case 0 control) | Lung Cancer (1 case 0 control) |
1 | 1 | 1 |
2 | 0 | 1 |
3 | 1 | 0 |
… | … | … |
Count Data RNA -seq
Count Data RNA -seq
Count Data
| Sample 1 (Cancer) | Sample 2 (Cancer) | Sample 3 (Normal) | Sample 4 (Normal) |
Gene A | 100 | 150 | 30 | 40 |
What is common?
Linear regression
Logistic regression
Poisson regression
Three Components of Generalized Linear Model
Exponential Family Distribution (Canonical Form)
Estimation
Estimation
Score Function of Canonical Exponential Family�
Newton’s Method (single parameter)
Newton’s Method (Generalization to Vector)
Observed Information (single parameter)
Observed Information of Canonical Exponential Family
Fisher’s Information
Newton’s Method with Fisher scoring�(Iteratively Reweighted Least Square)
Likelihood ratio test�
Logistic Regression
Wald test
Wald Test for R’s glm
Rao’s Score test
Rao’s Score test
Comparison
| Likelihood Ratio Test | Wald | Score |
Basic idea | Compare maximized likelihood of reduced and full model | Check how far the MLE is from the null | Check slope of log-likelihood at null |
Computation | Both reduced and full model | Full model only | Reduced model only |
Large sample | | | |
Small sample | Your homework | ||