Wald tables for the mean and precision (or dispersion) coefficients, information criteria, a pseudo \(R^2\), the censoring counts and a summary of the randomized quantile residuals.
Usage
# S3 method for class 'brs'
summary(object, ...)Value
A list of class "summary.brs" with coefficients
(tables mean and precision with columns Estimate,
Std. Error, z value, Pr(>|z|)), residuals
(RQR), loglik, AIC, BIC, df, nobs,
pseudo.r2, censoring (counts by type), link,
link_phi, repar, convergence and iterations.
Details
For each coefficient \(\hat\theta_j\), the standard error is
\(SE_j = \sqrt{[(-H)^{-1}]_{jj}}\) from vcov.brs, the Wald
statistic is \(z_j = \hat\theta_j / SE_j\) and the two-sided p-value is
\(2\Phi(-|z_j|)\) (Lopes, 2023, "Inferencia"). The test is on the link
scale (\(H_0: \theta_j = 0\)). When \(-H\) is singular, or its inverse
has negative variances, the affected standard errors, statistics and
p-values are NA (see 'Fit diagnostics' in brs).
\(\mathrm{AIC} = -2\ell + 2k\) and \(\mathrm{BIC} = -2\ell + k\log n\),
with \(\ell\) the maximised log-likelihood, \(k\) the number of
coefficients and \(n\) the number of observations. The pseudo
\(R^2\) is the squared correlation between the fitted linear predictor
of the mean and \(g_1(y_i)\) at the cell centres yt (Ferrari and
Cribari-Neto, 2004); under repar = 0 both sides are on the logit
scale. It uses the cell centres, so it is rough when most observations are
censored (the print says so).
The randomized quantile residuals (residuals.brs,
type = "rqr") are drawn without changing the caller's RNG state.
References
Lopes, J. E. (2023). Modelos de regressao beta para dados de escala. Master's dissertation, Universidade Federal do Parana, Curitiba. URI: https://hdl.handle.net/1884/86624.
Ferrari, S. L. P., and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799–815. doi:10.1080/0266476042000214501
Examples
set.seed(2023)
d <- data.frame(time = factor(rep(c("6h", "12h", "24h"), each = 60),
levels = c("6h", "12h", "24h")))
shp <- brs_repar(mu = plogis(-1.3 + c(0, 0.75, 0.3)[d$time]), phi = 0.3)
d$y <- round(10 * rbeta(nrow(d), shp$shape1, shp$shape2))
fit <- brs(y ~ time, data = d, ncuts = 10)
s <- summary(fit)
s
#>
#> Call:
#> brs(formula = y ~ time, data = d, ncuts = 10)
#>
#> Quantile residuals:
#> Min 1Q Median 3Q Max
#> -2.3714 -0.7045 -0.0964 0.6605 2.6798
#>
#> Coefficients (mean model with logit link):
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) -1.3079 0.1628 -8.036 9.31e-16 ***
#> time12h 0.5245 0.2156 2.433 0.0150 *
#> time24h 0.4992 0.2151 2.321 0.0203 *
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Phi coefficients (precision model with logit link):
#> Estimate Std. Error z value Pr(>|z|)
#> (phi) -0.8393 0.1103 -7.61 2.73e-14 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> ---
#> Log-likelihood: -377.6945 on 4 Df | AIC: 763.3891 | BIC: 776.1609
#> Pseudo R-squared: 0.0537 (midpoint approx.; interpret with caution for heavily censored data)
#> Number of iterations: 26 (BFGS)
#> Censoring: 141 interval | 37 left | 2 right
#>
s$coefficients$mean
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) -1.3079316 0.1627663 -8.035643 9.308940e-16
#> time12h 0.5245223 0.2155579 2.433324 1.496090e-02
#> time24h 0.4992069 0.2150869 2.320955 2.028928e-02
c(AIC = s$AIC, BIC = s$BIC, pseudo_R2 = s$pseudo.r2)
#> AIC BIC pseudo_R2
#> 763.38908156 776.16090897 0.05368292