Calculates the MLE of a heteroscedastic linear model (HLM) with possibly right-censored responses. See Details.
hlm(formula, data, subset, weights, na.action, control, model = TRUE, qr = TRUE, x = FALSE, y = FALSE, offset)
| formula | An object of class |
|---|---|
| data | An optional data frame, list or environment containing the variables in the model. If not found in |
| subset | An optional vector specifying a subset of observations to be used in the fitting process. |
| weights | Currently ignored. |
| na.action | A function which indicates what should happen when the data contain |
| control | List of parameters to control the fitting process. See Details. |
| model, x, y, qr | Logical values indicating which data elements should be returned in the output. These correspond to the response, the covariance matrices, the |
| offset | Currently ignored, as are |
An object of class hlm with the following elements:
coefficientsA list with elements beta and gamma containing the coefficient MLEs.
loglikThe loglikelihood at the MLE.
df.residualThe residual degrees of freedom.
iterThe number of iterations used in the fitting algorithm.
callThe matched call.
termsA list with elements X and Z giving the terms object for mean and variance models.
yIf requested, the response used.
xIf requested, a list with elements X and Z giving the mean and variance model matrices uses.
modelIf requested, the model.frame used.
na.action(Where relevant) information returned by model.frame on the special handling of NAs.
qrIf requested, the QR decomposition of the observed Fisher information matrix of size (p+q) x (p+q).
The heteroscedastic linear model (HLM) is defined as $$ y_i \mid \boldsymbol{x}_i, \boldsymbol{z}_i \stackrel{\mathrm{ind}}{\sim} \mathcal N\big(\boldsymbol{x}_i'\boldsymbol{\beta}, \exp(\boldsymbol{z}_i'\boldsymbol{\gamma})\big), $$ where for each subject \(i\), \(y_i\) is the response, and \(\boldsymbol{x}_i \in \mathbb{R}^p\) and \(\boldsymbol{z}_i \in \mathbb{R}^q\) are mean and variance covariate vectors, respectively.
The formula term is specified as e.g.,
y ~ x1 + x2 | z1 + z2
where the vertical bar separates mean and variance components. If no bar is provided, an intercept variance term of the form y ~ x1 + x2 | 1 is assumed, corresponding to the usual linear model (but with a different parametrization).
Right censoring of observations is supported by specifying the response as a two-column matrix, where the first column is the response and the second column is a censoring status indicator with 0: right-censored and 1: uncensored.
Fitting the hlm model is done by blockwise coordinate ascent, alternatively maximing the mean parameters by weighted least-squares, and the variance parameters either via Fisher scoring or Iteratively Reweighted Least Squares. When there is right-censoring, these maximization steps are embedded within an Expectation-Conditional-Maximization algorithm.
Warning: At present hlm cannot handle pure LM or LVLM models. For datasets without censoring, please use the low-level functions lm_fit and lvlm_fit instead.
Wang, Y., You, T., and Lysy, M. "A heteroscedastic accelerated failure time model for survival analysis" (2019): https://arxiv.org/abs/1508.05137.
Current methods for hlm objects are: print, nobs, vcov, and summary.
#> time status rx sex age #> Min. : 23.0 Mode :logical Obs :305 female:428 Min. :18.00 #> 1st Qu.: 809.8 FALSE:458 Lev :294 male :460 1st Qu.:53.00 #> Median :1983.0 TRUE :430 Lev+5FU:289 Median :61.00 #> Mean :1674.8 Mean :59.81 #> 3rd Qu.:2378.5 3rd Qu.:69.00 #> Max. :3329.0 Max. :85.00 #> obstruct perfor adhere nodes differ extent #> FALSE:717 FALSE:861 FALSE:760 Min. : 0.000 well : 90 submuc : 19 #> TRUE :171 TRUE : 27 TRUE :128 1st Qu.: 1.000 moder:653 muscle :102 #> Median : 2.000 poor :145 serosa :730 #> Mean : 3.663 cstruct: 37 #> 3rd Qu.: 5.000 #> Max. :33.000 #> surg node4 #> short:650 FALSE:653 #> long :238 TRUE :235 #> #> #> #># fit model without censoring (i.e., ignoring it) M <- hlm(log(time) ~ . | rx, data = colond) M # brief display#> #> Call: #> hlm(formula = log(time) ~ . | rx, data = colond) #> #> Mean Coefficients: #> (Intercept) statusTRUE rxLev rxLev+5FU sexmale #> 8.094110 -1.088684 -0.060590 -0.038627 -0.015568 #> age obstructTRUE perforTRUE adhereTRUE nodes #> -0.003779 -0.166131 0.138035 0.037240 -0.004408 #> differmoder differpoor extentmuscle extentserosa extentcstruct #> 0.035946 -0.245295 0.106529 -0.008666 -0.175723 #> surglong node4TRUE #> -0.022991 -0.192685 #> #> Variance Coefficients: #> (Intercept) rxLev rxLev+5FU #> -1.3556 0.1286 0.2252 #>#> #> Call: #> hlm(formula = log(time) ~ . | rx, data = colond) #> #> Mean Coefficients: #> Estimate Std. Error z value Pr(>|z|) #> (Intercept) 8.094110 0.161689 50.060 < 2e-16 *** #> statusTRUE -1.088684 0.038777 -28.076 < 2e-16 *** #> rxLev -0.060590 0.043158 -1.404 0.160344 #> rxLev+5FU -0.038627 0.044734 -0.863 0.387877 #> sexmale -0.015568 0.036382 -0.428 0.668729 #> age -0.003779 0.001540 -2.454 0.014133 * #> obstructTRUE -0.166131 0.046563 -3.568 0.000360 *** #> perforTRUE 0.138035 0.106750 1.293 0.195988 #> adhereTRUE 0.037240 0.053404 0.697 0.485598 #> nodes -0.004408 0.007998 -0.551 0.581559 #> differmoder 0.035946 0.061278 0.587 0.557474 #> differpoor -0.245295 0.074141 -3.309 0.000938 *** #> extentmuscle 0.106529 0.136155 0.782 0.433972 #> extentserosa -0.008666 0.127635 -0.068 0.945870 #> extentcstruct -0.175723 0.155631 -1.129 0.258856 #> surglong -0.022991 0.041165 -0.559 0.576497 #> node4TRUE -0.192685 0.064219 -3.000 0.002696 ** #> --- #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #> #> Variance Coefficients: #> Estimate Std. Error z value Pr(>|z|) #> (Intercept) -1.35562 0.08281 -16.369 <2e-16 *** #> rxLev 0.12858 0.11916 1.079 0.2806 #> rxLev+5FU 0.22519 0.11917 1.890 0.0588 . #> --- #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #> #> AIC: 1523.1 #> Number of algorithm iterations: 3 #>#> X_(Intercept) X_statusTRUE X_rxLev X_rxLev+5FU #> X_(Intercept) 2.614343e-02 -1.236219e-04 -7.016475e-04 -9.930187e-04 #> X_statusTRUE -1.236219e-04 1.503654e-03 4.158904e-05 1.646970e-04 #> X_rxLev -7.016475e-04 4.158904e-05 1.862588e-03 8.539018e-04 #> X_rxLev+5FU -9.930187e-04 1.646970e-04 8.539018e-04 2.001132e-03 #> X_sexmale -3.902654e-04 -5.854324e-06 -4.277758e-05 7.981676e-05 #> X_age -1.359035e-04 -3.029764e-06 -1.240753e-06 -1.143409e-06 #> X_obstructTRUE -5.105193e-04 -8.094224e-05 1.895699e-05 5.228386e-05 #> X_perforTRUE -2.162797e-04 -3.920988e-05 -5.966730e-05 -2.353477e-05 #> X_adhereTRUE 3.884689e-04 -1.284340e-04 -3.724624e-05 1.795973e-05 #> X_nodes -9.465147e-05 -3.577784e-05 5.730984e-06 7.867816e-06 #> X_differmoder -2.834128e-03 8.348546e-05 1.300987e-04 -1.094200e-05 #> X_differpoor -2.870937e-03 2.958005e-05 1.807907e-04 -4.871304e-05 #> X_extentmuscle -1.431536e-02 -1.288739e-04 -2.137776e-04 1.013180e-04 #> X_extentserosa -1.428071e-02 -3.128462e-04 -2.374501e-04 3.055481e-05 #> X_extentcstruct -1.422080e-02 -5.106152e-04 -5.690122e-05 1.842411e-04 #> X_surglong -2.531197e-04 -1.127285e-04 6.644265e-05 7.051479e-05 #> X_node4TRUE 3.245213e-04 -2.418950e-04 -5.740609e-05 -3.192449e-05 #> Z_(Intercept) 5.202376e-04 -2.757366e-04 -2.178431e-05 -2.412023e-05 #> Z_rxLev -1.298183e-03 4.185820e-04 1.850457e-05 2.813008e-05 #> Z_rxLev+5FU -2.778045e-04 4.213863e-04 4.810837e-05 4.549348e-05 #> X_sexmale X_age X_obstructTRUE X_perforTRUE #> X_(Intercept) -3.902654e-04 -1.359035e-04 -5.105193e-04 -2.162797e-04 #> X_statusTRUE -5.854324e-06 -3.029764e-06 -8.094224e-05 -3.920988e-05 #> X_rxLev -4.277758e-05 -1.240753e-06 1.895699e-05 -5.966730e-05 #> X_rxLev+5FU 7.981676e-05 -1.143409e-06 5.228386e-05 -2.353477e-05 #> X_sexmale 1.323672e-03 -4.805577e-07 5.528224e-05 -2.451072e-05 #> X_age -4.805577e-07 2.371536e-06 7.368944e-06 4.214665e-06 #> X_obstructTRUE 5.528224e-05 7.368944e-06 2.168143e-03 -3.382309e-04 #> X_perforTRUE -2.451072e-05 4.214665e-06 -3.382309e-04 1.139560e-02 #> X_adhereTRUE 6.221910e-05 -5.096654e-06 -6.876429e-05 -7.775554e-04 #> X_nodes -8.525867e-06 6.353176e-08 2.061360e-06 -2.904181e-05 #> X_differmoder -9.868294e-05 2.005042e-06 6.362023e-05 1.130515e-04 #> X_differpoor -1.002048e-04 3.447152e-06 5.410667e-05 1.299652e-04 #> X_extentmuscle -2.412944e-04 -9.503254e-06 -3.179455e-04 -1.081152e-06 #> X_extentserosa -2.060822e-04 -8.247074e-06 -4.644602e-04 -1.967325e-04 #> X_extentcstruct -2.509378e-04 -5.128729e-06 -2.488893e-04 -5.355568e-04 #> X_surglong -6.928519e-06 -1.139398e-06 -1.119633e-06 -9.760759e-05 #> X_node4TRUE 1.325594e-04 7.054099e-06 9.660433e-05 2.508340e-04 #> Z_(Intercept) -2.537538e-05 -2.048667e-06 -2.550134e-04 5.276219e-05 #> Z_rxLev 1.393548e-04 9.534626e-06 3.258801e-04 -2.553656e-04 #> Z_rxLev+5FU -6.379916e-05 -3.404992e-06 4.520195e-04 9.767024e-05 #> X_adhereTRUE X_nodes X_differmoder X_differpoor #> X_(Intercept) 3.884689e-04 -9.465147e-05 -2.834128e-03 -2.870937e-03 #> X_statusTRUE -1.284340e-04 -3.577784e-05 8.348546e-05 2.958005e-05 #> X_rxLev -3.724624e-05 5.730984e-06 1.300987e-04 1.807907e-04 #> X_rxLev+5FU 1.795973e-05 7.867816e-06 -1.094200e-05 -4.871304e-05 #> X_sexmale 6.221910e-05 -8.525867e-06 -9.868294e-05 -1.002048e-04 #> X_age -5.096654e-06 6.353176e-08 2.005042e-06 3.447152e-06 #> X_obstructTRUE -6.876429e-05 2.061360e-06 6.362023e-05 5.410667e-05 #> X_perforTRUE -7.775554e-04 -2.904181e-05 1.130515e-04 1.299652e-04 #> X_adhereTRUE 2.851956e-03 1.860939e-05 -1.326077e-04 -3.113655e-04 #> X_nodes 1.860939e-05 6.396250e-05 -2.364810e-05 -3.245073e-05 #> X_differmoder -1.326077e-04 -2.364810e-05 3.755038e-03 3.371084e-03 #> X_differpoor -3.113655e-04 -3.245073e-05 3.371084e-03 5.496859e-03 #> X_extentmuscle -2.825091e-04 9.943178e-06 -4.928944e-04 -4.002767e-04 #> X_extentserosa -2.986773e-04 -9.469006e-06 -5.216127e-04 -4.974208e-04 #> X_extentcstruct -1.112823e-03 -3.428818e-05 -8.631957e-04 -8.665510e-04 #> X_surglong -1.110779e-05 8.451058e-06 -1.828495e-05 -3.591306e-05 #> X_node4TRUE -5.245026e-05 -3.774189e-04 -3.792445e-05 -2.576355e-04 #> Z_(Intercept) 2.319267e-04 7.360723e-05 -3.092939e-04 -5.327932e-04 #> Z_rxLev -5.844850e-04 -1.163334e-04 5.057801e-04 8.722203e-04 #> Z_rxLev+5FU -1.180054e-04 -1.078147e-04 4.357857e-04 7.497150e-04 #> X_extentmuscle X_extentserosa X_extentcstruct X_surglong #> X_(Intercept) -1.431536e-02 -1.428071e-02 -1.422080e-02 -2.531197e-04 #> X_statusTRUE -1.288739e-04 -3.128462e-04 -5.106152e-04 -1.127285e-04 #> X_rxLev -2.137776e-04 -2.374501e-04 -5.690122e-05 6.644265e-05 #> X_rxLev+5FU 1.013180e-04 3.055481e-05 1.842411e-04 7.051479e-05 #> X_sexmale -2.412944e-04 -2.060822e-04 -2.509378e-04 -6.928519e-06 #> X_age -9.503254e-06 -8.247074e-06 -5.128729e-06 -1.139398e-06 #> X_obstructTRUE -3.179455e-04 -4.644602e-04 -2.488893e-04 -1.119633e-06 #> X_perforTRUE -1.081152e-06 -1.967325e-04 -5.355568e-04 -9.760759e-05 #> X_adhereTRUE -2.825091e-04 -2.986773e-04 -1.112823e-03 -1.110779e-05 #> X_nodes 9.943178e-06 -9.469006e-06 -3.428818e-05 8.451058e-06 #> X_differmoder -4.928944e-04 -5.216127e-04 -8.631957e-04 -1.828495e-05 #> X_differpoor -4.002767e-04 -4.974208e-04 -8.665510e-04 -3.591306e-05 #> X_extentmuscle 1.853814e-02 1.577224e-02 1.584670e-02 -1.571652e-04 #> X_extentserosa 1.577224e-02 1.629062e-02 1.599654e-02 -1.470173e-04 #> X_extentcstruct 1.584670e-02 1.599654e-02 2.422107e-02 7.357173e-05 #> X_surglong -1.571652e-04 -1.470173e-04 7.357173e-05 1.694574e-03 #> X_node4TRUE -4.199103e-04 -3.476679e-04 -3.515506e-05 3.515357e-05 #> Z_(Intercept) 1.160999e-04 -6.989647e-05 2.097538e-04 -2.815588e-04 #> Z_rxLev -5.439295e-05 2.101769e-04 -5.446882e-04 3.651163e-04 #> Z_rxLev+5FU -3.013872e-04 9.459765e-07 -9.036372e-05 4.936661e-04 #> X_node4TRUE Z_(Intercept) Z_rxLev Z_rxLev+5FU #> X_(Intercept) 3.245213e-04 5.202376e-04 -1.298183e-03 -2.778045e-04 #> X_statusTRUE -2.418950e-04 -2.757366e-04 4.185820e-04 4.213863e-04 #> X_rxLev -5.740609e-05 -2.178431e-05 1.850457e-05 4.810837e-05 #> X_rxLev+5FU -3.192449e-05 -2.412023e-05 2.813008e-05 4.549348e-05 #> X_sexmale 1.325594e-04 -2.537538e-05 1.393548e-04 -6.379916e-05 #> X_age 7.054099e-06 -2.048667e-06 9.534626e-06 -3.404992e-06 #> X_obstructTRUE 9.660433e-05 -2.550134e-04 3.258801e-04 4.520195e-04 #> X_perforTRUE 2.508340e-04 5.276219e-05 -2.553656e-04 9.767024e-05 #> X_adhereTRUE -5.245026e-05 2.319267e-04 -5.844850e-04 -1.180054e-04 #> X_nodes -3.774189e-04 7.360723e-05 -1.163334e-04 -1.078147e-04 #> X_differmoder -3.792445e-05 -3.092939e-04 5.057801e-04 4.357857e-04 #> X_differpoor -2.576355e-04 -5.327932e-04 8.722203e-04 7.497150e-04 #> X_extentmuscle -4.199103e-04 1.160999e-04 -5.439295e-05 -3.013872e-04 #> X_extentserosa -3.476679e-04 -6.989647e-05 2.101769e-04 9.459765e-07 #> X_extentcstruct -3.515506e-05 2.097538e-04 -5.446882e-04 -9.036372e-05 #> X_surglong 3.515357e-05 -2.815588e-04 3.651163e-04 4.936661e-04 #> X_node4TRUE 4.124080e-03 -2.503804e-04 5.181461e-04 2.421917e-04 #> Z_(Intercept) -2.503804e-04 6.858312e-03 -7.034739e-03 -6.996389e-03 #> Z_rxLev 5.181461e-04 -7.034739e-03 1.419812e-02 7.171536e-03 #> Z_rxLev+5FU 2.421917e-04 -6.996389e-03 7.171536e-03 1.420178e-02#> #> Call: #> hlm(formula = cbind(log(time), status) ~ . | rx, data = colond) #> #> Mean Coefficients: #> (Intercept) rxLev rxLev+5FU sexmale age #> 9.49981 0.07233 0.46443 -0.03225 -0.01051 #> obstructTRUE perforTRUE adhereTRUE nodes differmoder #> -0.35492 0.06564 -0.15956 -0.05284 0.21673 #> differpoor extentmuscle extentserosa extentcstruct surglong #> -0.23659 -0.34714 -0.90472 -1.36850 -0.25503 #> node4TRUE #> -0.64961 #> #> Variance Coefficients: #> (Intercept) rxLev rxLev+5FU #> 0.1716 0.3003 0.6502 #>