Abstraction for Statistical Models

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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)

Adjusted coefficient of determination (adjusted R-squared).

For linear models, the adjusted R² is defined as , with the coefficient of determination, the number of observations, and 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 and :devianceratio, defined as . In these formulas, is the likelihood of the model, that of the null model (the model including only the intercept), is the deviance of the model, is the deviance of the null model, is the number of observations (given by nobs) and 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 , with 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 , with the likelihood of the model, its number of consumed degrees of freedom (as returned by dof), and the number of observations (as returned by nobs).

# StatsAPI.bicFunction

bic(model::StatisticalModel)

Bayesian Information Criterion, defined as , with the likelihood of the model, its number of consumed degrees of freedom (as returned by dof), and the number of observations (as returned by nobs).

# StatsAPI.coefFunction

coef(model::StatisticalModel)

Return the coefficients of the model.

# StatsAPI.coefnamesFunction

coefnames(model::StatisticalModel)

Return the names of the coefficients.

# 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 , with 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.fitFunction

Fit a statistical model.

# StatsAPI.fit!Function

Fit a statistical model in-place.

# 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.isfittedFunction

isfitted(model::StatisticalModel)

Indicate whether the model has been fitted.

# StatsAPI.islinearFunction

islinear(model::StatisticalModel)

Indicate whether the model is linear.

# 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.mssFunction

mss(model::StatisticalModel)

Return the model sum of squares.

# 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 , with the explained sum of squares and 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 ;

  • :CoxSnell, defined as ;

  • :Nagelkerke, defined as .

  • :devianceratio, defined as .

In the above formulas, is the likelihood of the model, is the likelihood of the null model (the model with only an intercept), is the deviance of the model (from the saturated model), is the deviance of the null model, 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.

# StatsAPI.weightsFunction

weights(model::StatisticalModel)

Return the weights used in 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.dof_residualFunction

dof_residual(model::RegressionModel)

Return the residual degrees of freedom of the model.

# StatsAPI.fittedFunction

fitted(model::RegressionModel)

Return the fitted values of the model.

# StatsAPI.leverageFunction

leverage(model::RegressionModel)

Return the diagonal of the projection matrix of the model.

# StatsAPI.cooksdistanceFunction

cooksdistance(model::RegressionModel)

Compute Cook’s distance for each observation in linear model model, giving an estimate of the influence of each data point.

# StatsAPI.meanresponseFunction

meanresponse(model::RegressionModel)

Return the mean of the response.

# StatsAPI.modelmatrixFunction

modelmatrix(model::RegressionModel)

Return the model matrix (a.k.a. the design matrix).

# 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.

# StatsAPI.predict!Function

predict!

In-place version of predict.

# StatsAPI.residualsFunction

residuals(model::RegressionModel)

Return the residuals of the model.

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.