2024-06-05
Dist’n of Y | Typical uses | Link name | Link function | Common name |
---|---|---|---|---|
Normal | Linear-response data | Identity | \(g(\mu)=\mu\) | Linear regression |
Bernoulli / Binomial | outcome of single yes/no occurrence | Logit | \(g(\mu)=\text{logit}(\mu)\) | Logistic regression |
Poisson | count of occurrences in fixed amount of time/space | Log | \(g(\mu)=\log(\mu)\) | Poisson regression |
Bernoulli / Binomial | outcome of single yes/no occurrence | Log | \(g(\mu)=\log(\mu)\) | Log-binomial regression |
Multinomial | outcome of single occurence with K > 2 options, nominal | Logit | \(g(\mu)=\text{logit}(\mu)\) | Multinomial logistic regression |
Multinomial | outcome of single occurence with K > 2 options, ordinal | Logit | \(g(\mu)=\text{logit}(\mu)\) | Ordinal logistic regression |
\[ E(Y \mid X) = \mu = \beta_0 + \beta_1 X\]
\[ \text{logit}(\mu) = \beta_0 + \beta_1 X\]
\[ \log(\mu) = \beta_0 + \beta_1 X\]
YouTube video on R tutorial for Poisson Regression
\[ \log(\mu) = \beta_0 + \beta_1 X\]
Population models\[ \log \left(\dfrac{\mu_{\text{group } 2}}{\mu_{\text{group } 1}} \right) = \beta_0 + \beta_1 X\] \[ \log \left(\dfrac{\mu_{\text{group } 3}}{\mu_{\text{group } 1}} \right) = \beta_0 + \beta_1 X\]
Interpretations
YouTube video on R tutorial for Poisson Regression
\[ \log \left(\dfrac{P(Y \leq 1)}{P(Y > 1)} \right) = \beta_0 + \beta_1 X\] \[ \log \left(\dfrac{P(Y \leq k)}{P(Y > k)} \right) = \beta_0 + \beta_1 X\]
Dist’n of Y | Typical uses | Link name | Link function | Common name |
---|---|---|---|---|
Bernoulli / Binomial | outcome of single yes/no occurrence | Probit | \(g(\mu)=\Phi^{-1}(\mu)\) | Probit regression |
Bernoulli / Binomial | outcome of single yes/no occurrence | Complementary log-log | \(g(\mu)=\log(-\log(1-\mu))\) | Complementary log-log regression |
Multinomial | outcome of single occurence with K > 2 options, nominal | Probit | \(g(\mu)=\Phi^{-1}(\mu)\) | Multinomial probit regression |
Multinomial | outcome of single occurence with K > 2 options, ordinal | Probit | \(g(\mu)=\Phi^{-1}(\mu)\) | Ordered probit regression |
Lesson 17: Wrap-up and other regressions