For example, I'm doing some survival analysis and want to fit a model:
fit1 <- coxph(formula = Surv(week, arrest)~., data = rossi)
I know that the "~." is a shorthand way to include all covariates in the fit. However, there are instances like
fit1 <- coxph(formula = Surv(week, arrest)~1, data = rossi)
where "~1" is used instead, though I'm not really sure what the 1 does. What's the difference between these two things, if any?
I'm not sure what you're focused on but as you said, ~. means model using every variable.
And, ~1 means using no variable. Only constant coefficients will be included in your model.
Using GlobalDeviance::Rossi dataset, there exists very obvious difference between two model.
coxph(formula = Surv(week, arrest) ~ ., data = Rossi)
coef exp(coef) se(coef) z p
fin -3.832e+01 2.285e-17 4.471e+01 -0.857 0.39140
age 6.526e-01 1.920e+00 1.129e-01 5.778 7.56e-09
race 2.640e+01 2.914e+11 1.160e+00 22.756 < 2e-16
wexp -1.341e+01 1.503e-06 1.155e+00 -11.610 < 2e-16
mar -2.926e+01 1.965e-13 3.204e+02 -0.091 0.92723
paro -7.623e+00 4.888e-04 1.002e+00 -7.610 2.74e-14
prio -5.554e-01 5.739e-01 3.594e-01 -1.545 0.12224
educ 4.775e+00 1.186e+02 1.002e+00 4.768 1.86e-06
emp1 2.831e+01 1.977e+12 1.002e+00 28.262 < 2e-16
emp2 -1.586e+00 2.048e-01 1.002e+00 -1.583 0.11349
emp3 1.206e+01 1.724e+05 1.002e+00 12.036 < 2e-16
emp4 2.157e+01 2.341e+09 1.002e+00 21.536 < 2e-16
emp5 -3.970e+01 5.763e-18 1.002e+00 -39.625 < 2e-16
emp6 -4.786e+00 8.350e-03 1.155e+00 -4.144 3.42e-05
emp7 1.026e+01 2.856e+04 1.002e+00 10.242 < 2e-16
emp8 2.427e+01 3.462e+10 1.002e+00 24.225 < 2e-16
emp9 -2.961e+01 1.376e-13 1.002e+00 -29.562 < 2e-16
emp10 6.266e+00 5.264e+02 1.002e+00 6.255 3.98e-10
emp11 1.036e+01 3.142e+04 1.002e+00 10.337 < 2e-16
emp12 -5.055e+00 6.375e-03 1.002e+00 -5.046 4.50e-07
emp13 -1.462e+01 4.452e-07 1.002e+00 -14.599 < 2e-16
emp14 4.907e+00 1.352e+02 1.002e+00 4.898 9.67e-07
emp15 -2.577e+01 6.427e-12 1.002e+00 -25.725 < 2e-16
emp16 6.024e+00 4.132e+02 1.002e+00 6.013 1.82e-09
emp17 -1.644e+01 7.234e-08 1.002e+00 -16.413 < 2e-16
emp18 3.171e+01 5.900e+13 1.002e+00 31.652 < 2e-16
emp19 1.364e+01 8.397e+05 1.002e+00 13.617 < 2e-16
emp20 -1.646e+01 7.093e-08 1.002e+00 -16.432 < 2e-16
emp21 -1.895e-02 9.812e-01 1.002e+00 -0.019 0.98491
emp22 -1.249e+01 3.755e-06 1.002e+00 -12.470 < 2e-16
emp23 -3.892e+01 1.252e-17 1.170e+00 -33.252 < 2e-16
emp24 6.774e+01 2.621e+29 1.002e+00 67.618 < 2e-16
emp25 -4.181e+00 1.529e-02 1.002e+00 -4.173 3.00e-05
emp26 2.336e+01 1.396e+10 1.002e+00 23.318 < 2e-16
emp27 -7.105e+01 1.388e-31 1.002e+00 -70.926 < 2e-16
emp28 8.149e+00 3.462e+03 1.002e+00 8.135 4.12e-16
emp29 -2.850e+00 5.785e-02 1.002e+00 -2.845 0.00444
emp30 2.439e+01 3.910e+10 1.002e+00 24.346 < 2e-16
emp31 -3.805e+01 2.990e-17 1.002e+00 -37.981 < 2e-16
emp32 3.836e+01 4.575e+16 1.002e+00 38.294 < 2e-16
emp33 -2.804e+00 6.056e-02 1.002e+00 -2.799 0.00512
emp34 2.236e+01 5.162e+09 1.170e+00 19.109 < 2e-16
emp35 -8.057e+00 3.167e-04 1.170e+00 -6.884 5.80e-12
emp36 -3.150e+01 2.078e-14 1.171e+00 -26.914 < 2e-16
emp37 4.106e+01 6.825e+17 1.171e+00 35.081 < 2e-16
emp38 3.176e+00 2.394e+01 1.170e+00 2.713 0.00667
emp39 -2.956e+01 1.453e-13 1.170e+00 -25.254 < 2e-16
emp40 -7.740e+00 4.349e-04 1.002e+00 -7.727 1.10e-14
emp41 2.148e+01 2.129e+09 1.002e+00 21.441 < 2e-16
emp42 -2.305e+01 9.733e-11 1.171e+00 -19.694 < 2e-16
emp43 7.306e+00 1.489e+03 1.170e+00 6.242 4.32e-10
emp44 1.620e+01 1.087e+07 1.170e+00 13.842 < 2e-16
emp45 4.468e+01 2.535e+19 1.170e+00 38.174 < 2e-16
emp46 -5.447e+01 2.204e-24 1.170e+00 -46.542 < 2e-16
emp47 1.878e+01 1.426e+08 1.171e+00 16.040 < 2e-16
emp48 -1.159e+01 9.228e-06 1.171e+00 -9.904 < 2e-16
emp49 -2.961e+01 1.382e-13 1.171e+00 -25.296 < 2e-16
emp50 3.624e+01 5.468e+15 1.171e+00 30.958 < 2e-16
emp51 -1.418e-01 8.678e-01 1.171e+00 -0.121 0.90357
emp52 -1.442e+01 5.448e-07 1.171e+00 -12.321 < 2e-16
n.work.weeks 0.000e+00 1.000e+00 2.341e-02 0.000 1.00000
Likelihood ratio test=34.75 on 61 df, p=0.9973
n= 322, number of events= 4
(110 observations deleted due to missingness)
.
coxph(formula = Surv(week, arrest)~1, data = Rossi)
Null model
log likelihood= -675.3806
n= 432
The difference may be more easier to understand with fitting linear regression.
lm(Species ~ ., data = iris)
Call:
lm(formula = Species ~ ., data = iris)
Coefficients:
(Intercept) Sepal.Length Sepal.Width Petal.Length Petal.Width
1.18650 -0.11191 -0.04008 0.22865 0.60925
lm(Species ~ 1, data = iris)
Coefficients:
(Intercept)
2