12 Four Common Models

We can extend the ideas from the last chapter on logistic regression to a larger class of probability models.

Choose a distribution.
Choose the parameters to model as functions of covariates and those to model as fixed. Typically, this is a parameter that largely determines the location of the distribution, but we can also model the dispersion or multiple parameters at once.
Choose an inverse link function that maps the unbounded linear predictor \(X_i \beta \in \mathbb R\) onto the parameter space of the modeled parameters.
Fit the model using maximum likelihood. Use Fisher information to estimate the variance. Use invariance property and delta method to obtain quantities of interest and estimate their variance.

12.1 Summary of four common models

Just as an example of the kinds of models we can develop using this approach, here are four common models.

Normal linear: common for unbounded outcomes (without severe outliers)
Logit: common for binary outcomes 3: Negative binomial: common four count data (allows for overdispersion, unlike the Poisson) 4: Weibull: common for time-to-event or duration data, though we have many options here and Weibull might not be the best.

12.1.1 Normal Linear

Component	Expression
Outcome	Continuous
Model	\(y_i \sim \mathcal{N}(\mu_i, \sigma^2)\), where \(\mu_i = X_i \beta\)
Expected value	\(\hat{\mu} = X_c \hat{\beta}\) for chosen row \(X_c\)
First difference	\(\hat{\Delta} = \hat{\mu}_{hi} - \hat{\mu}_{lo} = X_{hi} \hat{\beta} - X_{lo} \hat{\beta}\) for chosen rows \(X_{hi}\) and \(X_{lo}\)
Function	`lm()` or `glm(..., family = gaussian)`
Parameterization notes	Parameterized as expected
Alternatives	\(t\) regression

12.1.2 Logit

Component	Expression
Outcome	Binary
Model	\(y_i \sim \text{Bernoulli}(\pi_i)\), where \(\pi_i = \operatorname{logit}^{-1}(X_i \beta)\)
Expected value	\(\hat{\mu} = \hat{\pi} = \operatorname{logit}^{-1}(X_c \hat{\beta})\)
First difference	\(\hat{\Delta} = \hat{\pi}_{hi} - \hat{\pi}_{lo} = \operatorname{logit}^{-1}(X_{hi} \hat{\beta}) - \operatorname{logit}^{-1}(X_{lo} \hat{\beta})\)
Function	`glm(..., family = binomial)`
Parameterization notes	Parameterized as expected
Alternatives	Probit

12.1.3 Negative Binomial

Component	Expression
Outcome	Count (non-negative integers)
Model	\(y_i \sim \text{NegBin}(\mu_i, \theta)\), with \(\log \mu_i = X_i \beta\) and \(\operatorname{Var}(y_i) = \mu_i + \mu_i^2/\theta\)
Expected value	\(\hat{\mu} = \exp(X_c \hat{\beta})\)
First difference	\(\hat{\Delta} = \hat{\mu}_{hi} - \hat{\mu}_{lo} = \exp(X_{hi} \hat{\beta}) - \exp(X_{lo} \hat{\beta})\)
Function	`MASS::glm.nb()` for fitting models; `dnbinom()` for densities
Parameterization notes	Uses the mean–dispersion form: regression is parameterized in terms of the mean \(\mu_i\) with log link. The dispersion parameter \(\theta\) (`size` in R) controls overdispersion. In `dnbinom()` you can use either `(size, prob)` or `(size, mu)`; `glm.nb()` uses the mean–dispersion form \((\mu_i, \theta)\).
Alternatives	Poisson regression (but variance equals mean), zero-inflated variants

12.1.4 Weibull

It’s harder to choose a default model for duration data. We’ve got lots of common options and folks tend to use a semiparametric approach for these data. But to pick a decent default (for both data analysis and an example to learn from), the Weibull model is a good choice.

Component	Expression
Outcome	Survival or duration time (positive continuous)
Model	\(y_i \sim \text{Weibull}(\lambda_i, k)\), with scale parameter \(\lambda_i = \exp(X_i \beta)\) and shape parameter \(k\) (constant across units)
Expected value	\(\hat{\mu} = \hat{\lambda} \, \Gamma(1 + 1/k)\) for chosen row \(X_c\) (the Gamma function \(\Gamma(\cdot)\) adjusts the mean away from the scale \(\lambda\) depending on shape \(k\))
First difference	\(\hat{\Delta} = \hat{\mu}_{hi} - \hat{\mu}_{lo} = \hat{\lambda}_{hi}\,\Gamma(1+1/k) - \hat{\lambda}_{lo}\,\Gamma(1+1/k)\)
Function	`survreg(..., dist = "weibull")` in survival package for fitting models; `dweibull()` for densities; note differences in parameterization, though.
Parameterization notes	The base R density function `dweibull(x, shape, scale)` uses the standard shape–scale parameterization. However, `survreg()` uses parameterization in terms of \(\lambda\) and \(\sigma\), where `shape` \(= \frac{1}{\sigma}\) and `scale` \(= \exp(X_i \beta)\). The two forms are equivalent after reparameterization. To clarify, the log-likelihood used by `survreg()` is below.
Alternatives	Exponential (special case of Weibull with \(k = 1\)); log-normal; gamma, generalized gamma, gompertz, semiparametric Cox proportional hazards

weibull_ll <- function(theta, y, X) {
  # extract parameters for ease of reading
  beta <- theta[1:ncol(X)]  # coefs are first ncol(X) values of theta
  sigma <- theta[ncol(X) + 1]  # sigma is the last value, after coefs
  
  # linear predictor
  eta <- X %*% beta
  # transform parameters
  k <- 1 / sigma               # Weibull shape
  lambda <- exp(eta)           # Weibull scale

  ll <- sum(dweibull(y, shape = k, scale = lambda, log = TRUE))
  return(ll)
}