summary.heft.RdThis function summarizes both the stepwise selection process of the
model fitting by heft, as well as the final model
that was selected using AIC/BIC.
heft object, typically the result of heft.
other arguments are ignored.
These function produce identical printed output. The main body is a table with six columns:
the first column is a possible number of knots for the fitted model;
the second column is 0 if the model was fitted during the addition stage and 1 if the model was fitted during the deletion stage;
the third column is the log-likelihood for the fit;
the fourth column is -2 * loglikelihood + penalty * (dimension),
which is the AIC criterion - heft selected the model with
the minimum value of AIC;
the fifth and sixth columns give the
endpoints of the interval of values of penalty that would yield the
model with the indicated number of knots. (NAs imply that the model is
not optimal for any choice of penalty.)
At the bottom of the table the number of knots corresponding to the selected model is reported, as are the value of penalty that was used and the coefficients of the log-based terms in the fitted model and their standard errors.
Charles Kooperberg, Charles J. Stone and Young K. Truong (1995). Hazard regression. Journal of the American Statistical Association, 90, 78-94.
Charles J. Stone, Mark Hansen, Charles Kooperberg, and Young K. Truong. The use of polynomial splines and their tensor products in extended linear modeling (with discussion) (1997). Annals of Statistics, 25, 1371–1470.
fit1 <- heft(testhare[,1], testhare[,2])
summary(fit1)
#> knots A(0)/D(1) loglik AIC minimum penalty maximum penalty
#> 3 1 -2954.28 5931.37 82.52 Inf
#> 4 0 -2913.02 5856.45 2.79 82.52
#> 5 0 -2912.87 5863.75 NA NA
#> 6 1 -2910.63 5866.87 NA NA
#> 7 1 -2910.36 5873.92 NA NA
#> 8 1 -2907.45 5875.71 2.64 2.79
#> 9 1 -2906.50 5881.40 NA NA
#> 10 1 -2905.58 5887.16 NA NA
#> 11 1 -2903.97 5891.54 NA NA
#> 12 1 -2902.83 5896.87 NA NA
#> 13 1 -2900.86 5900.53 1.68 2.64
#> 14 1 -2900.64 5907.69 NA NA
#> 15 1 -2899.18 5912.37 0.75 1.68
#> 16 1 -2899.17 5919.95 NA NA
#> 17 1 -2898.45 5926.11 NA NA
#> 18 1 -2898.40 5933.62 NA NA
#> 19 1 -2897.68 5939.78 0.37 0.75
#> 20 1 -2897.50 5947.01 0.20 0.37
#> 21 1 -2897.49 5954.60 NA NA
#> 22 1 -2897.30 5961.82 0.05 0.20
#> 23 0 -2897.28 5969.37 0.00 0.05
#> the present optimal number of knots is 4
#> penalty(AIC) was the default: BIC=log(samplesize): log( 2000 )= 7.6
#> theta SE t
#> left tail 1.01 0.09 11.01
#> right tail -1.00 NA NA
# modify tail behavior
fit2 <- heft(testhare[,1], testhare[,2], leftlog = FALSE, rightlog = FALSE,
leftlin = TRUE)
summary(fit2)
#> knots A(0)/D(1) loglik AIC minimum penalty maximum penalty
#> 2 0 -3008.39 6024.39 86.12 Inf
#> 3 0 -2995.90 6007.00 NA NA
#> 4 1 -2922.28 5867.36 13.67 86.12
#> 5 0 -2915.45 5861.29 5.92 13.67
#> 6 1 -2913.17 5864.34 NA NA
#> 7 0 -2912.23 5870.07 NA NA
#> 8 1 -2906.57 5866.35 2.81 5.92
#> 9 1 -2906.37 5873.54 NA NA
#> 10 1 -2903.76 5875.94 1.46 2.81
#> 11 1 -2903.40 5882.80 NA NA
#> 12 1 -2902.76 5889.12 NA NA
#> 13 1 -2902.33 5895.87 NA NA
#> 14 1 -2901.60 5902.00 NA NA
#> 15 1 -2901.05 5908.52 NA NA
#> 16 1 -2899.72 5913.45 NA NA
#> 17 1 -2899.31 5920.23 NA NA
#> 18 1 -2897.91 5925.04 0.85 1.46
#> 19 1 -2897.49 5931.79 0.23 0.85
#> 20 1 -2897.38 5939.17 0.05 0.23
#> 21 1 -2897.36 5946.74 NA NA
#> 22 1 -2897.33 5954.27 0.02 0.05
#> 23 0 -2897.32 5961.85 0.00 0.02
#> the present optimal number of knots is 5
#> penalty(AIC) was the default: BIC=log(samplesize): log( 2000 )= 7.6
#> theta SE t
#> left tail 0 NA NA
#> right tail 0 NA NA
fit3 <- heft(testhare[,1], testhare[,2], penalty = 0) # select largest model
summary(fit3)
#> knots A(0)/D(1) loglik AIC minimum penalty maximum penalty
#> 3 0 -2954.28 5908.56 82.52 Inf
#> 4 0 -2913.02 5826.05 2.48 82.52
#> 5 0 -2912.87 5825.75 NA NA
#> 6 0 -2912.36 5824.72 NA NA
#> 7 0 -2911.57 5823.14 NA NA
#> 8 0 -2909.73 5819.46 NA NA
#> 9 0 -2907.57 5815.14 NA NA
#> 10 0 -2906.92 5813.84 NA NA
#> 11 0 -2904.50 5809.00 NA NA
#> 12 0 -2903.10 5806.20 1.84 2.48
#> 13 0 -2902.18 5804.35 1.10 1.84
#> 14 0 -2901.63 5803.25 0.99 1.10
#> 15 0 -2901.24 5802.47 NA NA
#> 16 0 -2900.94 5801.89 NA NA
#> 17 0 -2900.62 5801.23 NA NA
#> 18 0 -2899.82 5799.64 NA NA
#> 19 0 -2899.18 5798.35 NA NA
#> 20 0 -2898.65 5797.29 0.91 0.99
#> 21 0 -2898.42 5796.85 NA NA
#> 22 0 -2897.94 5795.89 NA NA
#> 23 0 -2897.28 5794.55 0.00 0.91
#> the present optimal number of knots is 23
#> penalty(AIC) was 0 , the default (BIC) would have been 7.6
#> models with fewer than 3 knots can be fitted, but they are not optimal for the
#> present choice of penalty - choose penalty in heft larger to see these fits
#> theta SE t
#> left tail 0.20 0.53 0.38
#> right tail -0.66 0.55 1.20