For all intepretations, “predicts” really means “predicts on average”.

# Linear Regression

$\mathbb{E}(Y|X) = \beta_0 + \beta_1X$

• $$\beta_0$$ is the average value of $$Y$$ when $$X = 0$$.
• $$\beta_1$$: A 1 increase in $$X$$ predicts a change of $$\beta_1$$ in $$Y$$.

For log-transforms, see Log Transform Interpretation. The tldr is

• If $$Y$$ is log-transformed, a 1 increase in $$X$$ predicts a $$\exp(\beta_1)$$% change in $$Y$$.
• If $$X$$ is log-transformed, a $$r$$% increase in $$X$$ predicts a $$\log(1 + r/100)\beta_1$$ change in $$Y$$.
• If $$X$$ and $$Y$$ are log-transformed, a $$r$$% increase in $$X$$ predicts a $$(1 + r/100)^{\beta_1}$$ change in $$Y$$.

# Logistic Regression

Let $$p = \mathbb{P}(Y = 1)$$.

$\textrm{logit}(p) = \log\left(\frac{p}{1 - p}\right) = \beta_0 + \beta_1X$

• $$\beta_0$$ is the log odds when $$X = 0$$.
• $$\beta_1$$: A 1 increase in $$X$$ predicts a change of $$\beta_1$$ in the log odds
• $$\exp(\beta_0)$$ is the odds ratio when $$X = 0$$.
• $$\exp(\beta_1)$$: A 1 increase in $$X$$ predicts a change of $$\exp(\beta_1)$$ in the odds ratio.

## Odds ratio and log odds

The odds ratio is

$\frac{p}{1-p}$

• The ratio of successes to failures.
• The number of successes we expect per failure. (E.g. if $$p = .6$$, then the odds ratio is 1.5, so we expect 1.5 successes for every failure.)

The log odds is

$\log\left(\frac{p}{1-p}\right)$

and should not be interpreted, other than that the coefficients are linear.

# Poisson Regression

$\log(\mathbb{E}(Y|X)) = \beta_0 + \beta_1X$

where $$Y$$ is a count.

• Interpretations are similar to a log transformed $$Y$$ in linear regression. A 1 increase in $$X$$ predicts a $$\exp(\beta_1)$$% change in $$Y$$.

## Negative Binomial Regression

Same interpretation as Poisson.