# Compute likelihood ratio tests. Functions to compute likelihood ratio tests of two nested copula models with the same marginal models, and of marginal parameters in the same copula model but with different nested marginal models where one or more parameters are set to 0 (Yilmaz & Lawless, 2011). ```r lrt_copula(minlogl_null = NULL, minlogl_altern = NULL) lrt_param(minlogl_null = NULL, minlogl_altern = NULL, df = NULL) ``` ## Arguments - `minlogl_null`: Minus log-likelihood of the null model, which is nested within the larger alternative model `minlogl_altern`. - `minlogl_altern`: Minus log-likelihood of the alternative model, which contains the null model `minlogl_null`. - `df`: Degrees of freedom in the likelihood ratio test of marginal parameters in the same copula model, i.e. the number of parameters that are in the alternative model but that are not contained in the smaller nested null model. ## Returns List including the $\chi^2$ value and p-value of the likelihood ratio test. ## Details References: Yilmaz YE, Lawless JF (2011). Likelihood ratio procedures and tests of fit in parametric and semiparametric copula models with censored data. Lifetime Data Analysis, 17(3): 386-408. ## Examples ```r # Example 1: Test whether 2-parameter copula model has a better # model fit compared to Clayton copula (no). set.seed(10) genodata <- generate_genodata(n_SNV = 20, n_ind = 100) phenodata <- generate_phenodata_2_copula(genodata = genodata$SNV1, MAF_cutoff = 1, prop_causal = 1, tau = 0.2, b1 = 0.3, b2 = 0.3) predictors <- data.frame(X1 = phenodata$X1, X2 = phenodata$X2, SNV = genodata$SNV1) estimates_c <- get_estimates_naive(Y1 = phenodata$Y1, Y2 = phenodata$Y2, predictors_Y1 = predictors, predictors_Y2 = predictors, copula_param = "phi") minusloglik_Clayton <- minusloglik(Y1 = phenodata$Y1, Y2 = phenodata$Y2, predictors_Y1 = predictors, predictors_Y2 = predictors, parameters = estimates_c, copula = "Clayton") estimates_2p <- get_estimates_naive(Y1 = phenodata$Y1, Y2 = phenodata$Y2, predictors_Y1 = predictors, predictors_Y2 = predictors, copula_param = "both") minusloglik_2param <- minusloglik(Y1 = phenodata$Y1, Y2 = phenodata$Y2, predictors_Y1 = predictors, predictors_Y2 = predictors, parameters = estimates_2p, copula = "2param") lrt_copula(minusloglik_Clayton, minusloglik_2param) # Example 2: Test marginal parameters (alternative model has better fit). set.seed(10) genodata <- generate_genodata(n_SNV = 20, n_ind = 100) phenodata <- generate_phenodata_2_copula(genodata = genodata$SNV1, MAF_cutoff = 1, prop_causal = 1, tau = 0.2, b1 = 2, b2 = 2) predictors_1 <- data.frame(X1 = phenodata$X1, X2 = phenodata$X2) estimates_1 <- get_estimates_naive(Y1 = phenodata$Y1, Y2 = phenodata$Y2, predictors_Y1 = predictors_1, predictors_Y2 = predictors_1, copula_param = "phi") minusloglik_1 <- minusloglik(Y1 = phenodata$Y1, Y2 = phenodata$Y2, predictors_Y1 = predictors_1, predictors_Y2 = predictors_1, parameters = estimates_1, copula = "Clayton") predictors_2 <- data.frame(X1 = phenodata$X1, X2 = phenodata$X2, SNV = genodata$SNV1) estimates_2 <- get_estimates_naive(Y1 = phenodata$Y1, Y2 = phenodata$Y2, predictors_Y1 = predictors_2, predictors_Y2 = predictors_2, copula_param = "phi") minusloglik_2 <- minusloglik(Y1 = phenodata$Y1, Y2 = phenodata$Y2, predictors_Y1 = predictors_2, predictors_Y2 = predictors_2, parameters = estimates_2, copula = "Clayton") lrt_param(minusloglik_1, minusloglik_2, df=2) ```

lrt function