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H(t)
cumulative hazard is a non-decreasing & unbounded function on [0; inf.)
integral of non-negative function
Survival & Hazard
t–> inf. S(t) –> 0 and H(t) –> inf.
indv. can life infinitely long but under taking S(t) as R.V has to take an infinite number (everyone dies at some point) and thus S(t) –> 0
ni
number at risk (i.e alive) just before ti (KM = LT)
di
number of deaths (events) at ti
number of death in interval (tí; tí+1)
ci
number of censored in interval (ti;ti+1)
number of censored in interval (tí ;tí +1)
ní
ni - 1/2 * ci
actuarial notation
exposed to risk only half of the time
why taking Logs
distribution of ratio is typically skewed
to approximate the distribution of ratio with a normal ditribution more observations are needed and thus it´s preferred to perform inference on the logarithm of the ratio
remove/reduce skewness (adjust for skewness)
specially because lamda, phi and S(ti) > 0
Cox-proportional model
form of baseline hazard is not specified (non-parametric) and only the ratios of the hazard are parametrised
We prefer estimate from this model as it makes less assumption
Exponential - hazards are constant
Weibull - hazards can be in/decreasing with time
Xij
individual (xi) with specific indicator (xj = 0 or 1)
Shape of S.T
positively skewed longer tail to the right median < mean no upper-bound non-symmetric (thus can´t assume that data have normal distribution)
KM
Product limit estimator
can be obtained as limit of actuarial LT estimator as the nr. of intervals increase (width –> 0)
Normal Version
computational harder
can be used in a one-side version which allows to explore the direction of the hypothetical deviation from homogeneity
Chi-Square Version
easier to compute
easily generalised to more than two groups
ignores “the sign” of the possible deviation
One-sided Version
motivated by the nature of the data
E.g. - new treatment be tested if it has positive effect (how large the deviation is)
E.g. - new technology maybe expected to improve the durability of technical devices
Why do we need to modify S.T Intervals
constraint face when C.I for S(t) using Gw formula due to symmetry of formula
Thus for small/large t (ST resp.) the lower/upper bound may go outside (0,1)
Addition - underestimation of Var in the tails of S. distribution
S.T intervals - solution
pragmatic - replace by 0 or 1
alternative - transform S(t) to value in (-inf, inf) and obtain C.I for transformed value
which is then transformed back to give C.I for S(t) itself
limitations that apply to the use of Cox´s formula for the partial LH
there should be no ties in the non-censored pooled data
Using Life-Table Estimator
we have interval censoring
appropriate to use when data can be reported as total from fixed intervals
Remember to use ni´ to compute S(t) but ni to know ni´
(include censoring within it)
Weighted LRT
It seems logical to give more importance to the deviation at earlier stage of experiment where the population is greater than later in time (data set decreases and people die)
Chi-Square Statistic
The decision rule - knowing that X2 is X(1) 2 distributed
X2 > k1(100a) then reject H0 , where a is a chosen significance level and X1(100*a) is the corresponding percentage point of the X(1) 2 distribution.
