# Abstraction for Statistical Models

StatsAPI.jl defines an abstract type StatisticalModel, and an abstract subtype RegressionModel. They are both extended by StatsBase, and documented here.

Particularly, instances of StatisticalModel implement the following methods.

StatsAPI.adjr2Function
adjr2(model::StatisticalModel)
adjr²(model::StatisticalModel)

For linear models, the adjusted R² is defined as $1 - (1 - (1-R^2)(n-1)/(n-p))$, with $R^2$ the coefficient of determination, $n$ the number of observations, and $p$ the number of coefficients (including the intercept). This definition is generally known as the Wherry Formula I.

adjr2(model::StatisticalModel, variant::Symbol)
adjr²(model::StatisticalModel, variant::Symbol)

Adjusted pseudo-coefficient of determination (adjusted pseudo R-squared). For nonlinear models, one of the several pseudo R² definitions must be chosen via variant. The only currently supported variants are :MacFadden, defined as $1 - (\log (L) - k)/\log (L0)$ and :devianceratio, defined as $1 - (D/(n-k))/(D_0/(n-1))$. In these formulas, $L$ is the likelihood of the model, $L0$ that of the null model (the model including only the intercept), $D$ is the deviance of the model, $D_0$ is the deviance of the null model, $n$ is the number of observations (given by nobs) and $k$ is the number of consumed degrees of freedom of the model (as returned by dof).

StatsAPI.aicFunction
aic(model::StatisticalModel)

Akaike's Information Criterion, defined as $-2 \log L + 2k$, with $L$ the likelihood of the model, and k its number of consumed degrees of freedom (as returned by dof).

StatsAPI.aiccFunction
aicc(model::StatisticalModel)

Corrected Akaike's Information Criterion for small sample sizes (Hurvich and Tsai 1989), defined as $-2 \log L + 2k + 2k(k-1)/(n-k-1)$, with $L$ the likelihood of the model, $k$ its number of consumed degrees of freedom (as returned by dof), and $n$ the number of observations (as returned by nobs).

StatsAPI.bicFunction
bic(model::StatisticalModel)

Bayesian Information Criterion, defined as $-2 \log L + k \log n$, with $L$ the likelihood of the model, $k$ its number of consumed degrees of freedom (as returned by dof), and $n$ the number of observations (as returned by nobs).

StatsAPI.coeftableFunction
coeftable(model::StatisticalModel; level::Real=0.95)

Return a table with coefficients and related statistics of the model. level determines the level for confidence intervals (by default, 95%).

The returned CoefTable object implements the Tables.jl interface, and can be converted e.g. to a DataFrame via using DataFrames; DataFrame(coeftable(model)).

StatsAPI.confintFunction
confint(model::StatisticalModel; level::Real=0.95)

Compute confidence intervals for coefficients, with confidence level level (by default 95%).

StatsAPI.devianceFunction
deviance(model::StatisticalModel)

Return the deviance of the model relative to a reference, which is usually when applicable the saturated model. It is equal, up to a constant, to $-2 \log L$, with $L$ the likelihood of the model.

StatsAPI.dofFunction
dof(model::StatisticalModel)

Return the number of degrees of freedom consumed in the model, including when applicable the intercept and the distribution's dispersion parameter.

StatsAPI.informationmatrixFunction
informationmatrix(model::StatisticalModel; expected::Bool = true)

Return the information matrix of the model. By default the Fisher information matrix is returned, while the observed information matrix can be requested with expected = false.

StatsAPI.loglikelihoodFunction
loglikelihood(model::StatisticalModel)
loglikelihood(model::StatisticalModel, observation)

Return the log-likelihood of the model.

With an observation argument, return the contribution of observation to the log-likelihood of model.

If observation is a Colon, return a vector of each observation's contribution to the log-likelihood of the model. In other words, this is the vector of the pointwise log-likelihood contributions.

In general, sum(loglikehood(model, :)) == loglikelihood(model).

StatsAPI.nobsFunction
nobs(model::StatisticalModel)

Return the number of independent observations on which the model was fitted. Be careful when using this information, as the definition of an independent observation may vary depending on the model, on the format used to pass the data, on the sampling plan (if specified), etc.

StatsAPI.nulldevianceFunction
nulldeviance(model::StatisticalModel)

Return the deviance of the null model, obtained by dropping all independent variables present in model.

If model includes an intercept, the null model is the one with only the intercept; otherwise, it is the one without any predictor (not even the intercept).

StatsAPI.nullloglikelihoodFunction
nullloglikelihood(model::StatisticalModel)

Return the log-likelihood of the null model, obtained by dropping all independent variables present in model.

If model includes an intercept, the null model is the one with only the intercept; otherwise, it is the one without any predictor (not even the intercept).

StatsAPI.r2Function
r2(model::StatisticalModel)
r²(model::StatisticalModel)

Coefficient of determination (R-squared).

For a linear model, the R² is defined as $ESS/TSS$, with $ESS$ the explained sum of squares and $TSS$ the total sum of squares.

r2(model::StatisticalModel, variant::Symbol)
r²(model::StatisticalModel, variant::Symbol)

Pseudo-coefficient of determination (pseudo R-squared).

For nonlinear models, one of several pseudo R² definitions must be chosen via variant. Supported variants are:

• :MacFadden (a.k.a. likelihood ratio index), defined as $1 - \log (L)/\log (L_0)$;
• :CoxSnell, defined as $1 - (L_0/L)^{2/n}$;
• :Nagelkerke, defined as $(1 - (L_0/L)^{2/n})/(1 - L_0^{2/n})$.
• :devianceratio, defined as $1 - D/D_0$.

In the above formulas, $L$ is the likelihood of the model, $L_0$ is the likelihood of the null model (the model with only an intercept), $D$ is the deviance of the model (from the saturated model), $D_0$ is the deviance of the null model, $n$ is the number of observations (given by nobs).

The Cox-Snell and the deviance ratio variants both match the classical definition of R² for linear models.

StatsAPI.rssFunction
rss(model::StatisticalModel)

Return the residual sum of squares of the model.

StatsAPI.scoreFunction
score(model::StatisticalModel)

Return the score of the model, that is the gradient of the log-likelihood with respect to the coefficients.

StatsAPI.stderrorFunction
stderror(model::StatisticalModel)

Return the standard errors for the coefficients of the model.

StatsAPI.vcovFunction
vcov(model::StatisticalModel)

Return the variance-covariance matrix for the coefficients of the model.

RegressionModel extends StatisticalModel by implementing the following additional methods.

StatsAPI.crossmodelmatrixFunction
crossmodelmatrix(model::RegressionModel)

Return X'X where X is the model matrix of model. This function will return a pre-computed matrix stored in model if possible.

StatsAPI.leverageFunction
leverage(model::RegressionModel)

Return the diagonal of the projection matrix of the model.

StatsAPI.responseFunction
response(model::RegressionModel)

Return the model response (a.k.a. the dependent variable).

StatsAPI.responsenameFunction
responsename(model::RegressionModel)

Return the name of the model response (a.k.a. the dependent variable).

StatsAPI.predictFunction
predict(model::RegressionModel, [newX])

Form the predicted response of model. An object with new covariate values newX can be supplied, which should have the same type and structure as that used to fit model; e.g. for a GLM it would generally be a DataFrame with the same variable names as the original predictors.

An exception type is provided to signal convergence failures during model estimation:

StatsBase.ConvergenceExceptionType
ConvergenceException(iters::Int, lastchange::Real=NaN, tol::Real=NaN)

The fitting procedure failed to converge in iters number of iterations, i.e. the lastchange between the cost of the final and penultimate iteration was greater than specified tolerance tol.

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