# Post-Hoc Maximum Likelihood Refitting of a Conditional Graphical Lasso ‘cggm’ is used to perform post-hoc maximum likelihood refitting of a selected conditional graphical lasso model with censored and/or missing values. ```r cggm(object, GoF = AIC, lambda.id, rho.id, tp.min = 1.0E-6, ntp = 100L, maxit.em = 1.0E+4, thr.em = 1.0E-3, maxit.bcd = 1.0E+5, thr.bcd = 1.0E-4, trace = 0L, ...) ``` ## Arguments - `object`: an object of S3 class ‘cglasso’ , that is, the output of the function `cglasso`. - `GoF`: a valid goodness-of-fit function, such as `AIC.cglasso` or `BIC.cglasso`, or an object of class ‘GoF’ . - `lambda.id`: an optional integer used to identify a $lambda$-value stored in `object`. See section Details for more details. - `rho.id`: an optional integer used to identify a $rho$-value stored in `object`. See section Details for more details. - `tp.min`: the smallest $lambda$ and $rho$ value. See section Details for more details. - `ntp`: integer; number of $lambda$- and $rho$-values used to compute the coefficients path. See section Details for more details. - `maxit.em`: maximum number of iterations of the EM algorithm. Default is `1.0E+4`. - `thr.em`: threshold for the convergence of the EM algorithm. Default value is `1.0E-4`. - `maxit.bcd`: maximum number of iterations of the glasso algorithm. Default is `1.0E+5`. - `thr.bcd`: threshold for the convergence of the glasso algorithm. Default is `1.0E-4`. - `trace`: integer for printing information out as iterations proceed: `trace = 0` no information is printed out on screen; `trace = 1` minimal information is printed; `trace = 2` detailed information is printed on screen. - ``...``: the **penalty** parameter passed to the goodness-of-fit function (argument ‘GoF’ ). ## Details The model-fitting function `cggm` is used to obtain the maximum likelihood estimates of a censored Gaussian graphical model whose structure was found by penalised cglasso inference. That is, given a fitted cglasso model, identified along the path via a goodness-of-fit function (passed by `GoF`) or via the optional arguments `lambda.id` and `rho.id`), `cggm` computes the maximum likelihood estimates of the parameters with zero constraints associated to the given structure. Maximum likelihood estimates are computed using `cglasso` as workhorse function, that is, `cggm` fits a sequence of cglasso models, with length equal to `ntp`, reducing $lambda$ and $rho$ until they are equal to the value ‘min(tp.min, lambda.min, rho.min)’ , where `lambda.min` and `rho.min` are the smallest $lambda$- and $rho$-values stored in `object`. Maximum likelihood estimates are obtained from the last fitted cglasso model. The model-fitting function `cggm` returns an object inheriting the S3 class ‘cglasso’ , for which all printing and plotting functions, designed for a `cglasso` object, can be used (see section ‘See Also’ ). Function `ShowStructure` can be used to show the structure of the package. NOTE: if the user tries a problem when the sample size is not large enough, then increase the number of fitted cglasso models (argument `ntp`). ## Returns `cggm` returns an object of S3 class ‘cggm’ , i.e., a list containing the following components: - **call**: the call that produced this object. - **Yipt**: an array storing the working response matrix . - **B**: the maximum likelihood estimate of the regression coefficient matrix. - **mu**: the fitted expected values. - **R**: the working residuals matrix. - **S**: the working empirical covariance matrix . - **Sgm**: the maximum likelihood estimate of the covariance matrix. - **Tht**: the maximum likelihood estimate of the precision matrix. - **dfB**: the number of estimated non-zero regression coefficients. Only for internal purpose. - **dfTht**: the number of estimated non-zero (off-diagonal) partial correlation coefficients. Only for internal purpose. - **InfoStructure**: a named list whose elements are used to store the information about the estimated networks. Only for internal purpose. - **nit**: the number of EM steps. - **Z**: the ‘datacggm’ object used to compute the censored graphical lasso estimator. - **nlambda**: Only for internal purpose. - **lambda**: the $lambda$-value of the selected cglasso model. - **nrho**: Only for internal purpose. - **rho**: the $rho$-value of the selected cglasso model. - **maxit.em**: maximum number of iterations of the EM algorithm. - **thr.em**: threshold for the convergence of the EM algorithm. - **maxit.bcd**: maximum number of iterations of the glasso algorithm. - **thr.bcd**: threshold for the convergence of the glasso algorithm. - **conv**: a description of the error that has occurred. - **subrout**: the name of the Fortran subroutine where the error has occurred (for internal debug only). - **trace**: the integer used for printing information on screen. - **nobs**: the sample size - **nresp**: the number of response variables used to fit the model. - **npred**: the number of predictors used to fit the model. ## Author(s) Luigi Augugliaro (luigi.augugliaro@unipa.it ) ## See Also `cglasso`, `coef.cglasso`, `fitted.cglasso`, `residuals.cglasso`, `predict.cggm`, `impute`, `AIC.cglasso`, `BIC.cglasso`, `summary.cglasso`, `to_graph`, `plot.cglasso2igraph` and `ShowStructure`. ## Examples ```r set.seed(123) # Y ~ N(XB, Sigma) and # 1. probability of left/right censored values equal to 0.05 # 2. probability of missing-at-random values equal to 0.05 n <- 100L p <- 3L q <- 2L b0 <- runif(p) B <- matrix(runif(q * p), nrow = q, ncol = p) X <- matrix(rnorm(n * q), nrow = n, ncol = q) rho <- 0.3 Sigma <- outer(1L:p, 1L:p, function(i, j) rho^abs(i - j)) Z <- rcggm(n = n, b0 = b0, X = X, B = B, Sigma = Sigma, probl = 0.05, probr = 0.5, probna = 0.05) out <- cglasso(. ~ ., data = Z) # MLE of the censored Gaussian graphical model identified by 'BIC' out.mle <- cggm(out, GoF = BIC) out.mle # accessor functions coef(out.mle, drop = TRUE) fitted(out.mle, drop = TRUE) residuals(out.mle, type = "working", drop = TRUE) impute(out.mle, type = "both") # goodness-of-fit functions AIC(out.mle) BIC(out.mle) summary(out.mle) # network analysis out.graph <- plot(out.mle) out.graph ```

cggm function