Remember

Question 1

H(t)

Answer 1

cumulative hazard is a non-decreasing & unbounded function on [0; inf.)
integral of non-negative function

Question 2

Survival & Hazard

Answer 2

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

Question 3

ni

Answer 3

number at risk (i.e alive) just before ti (KM = LT)

Question 4

di

Answer 4

number of deaths (events) at ti

number of death in interval (tí; tí+1)

Question 5

ci

Answer 5

number of censored in interval (ti;ti+1)

number of censored in interval (tí ;tí +1)

Question 6

ní

Answer 6

ni - 1/2 * ci
actuarial notation
exposed to risk only half of the time

Question 7

why taking Logs

Answer 7

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

Question 8

Cox-proportional model

Answer 8

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

Question 9

Xij

Answer 9

individual (xi) with specific indicator (xj = 0 or 1)

Question 10

Shape of S.T

Answer 10

positively skewed 
longer tail to the right
median < mean 
no upper-bound 
non-symmetric (thus can´t assume that data have normal distribution)

Question 11

KM

Answer 11

Product limit estimator

can be obtained as limit of actuarial LT estimator as the nr. of intervals increase (width –> 0)

Question 12

Normal Version

Answer 12

computational harder

can be used in a one-side version which allows to explore the direction of the hypothetical deviation from homogeneity

Question 13

Chi-Square Version

Answer 13

easier to compute
easily generalised to more than two groups
ignores “the sign” of the possible deviation

Question 14

One-sided Version

Answer 14

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

Question 15

Why do we need to modify S.T Intervals

Answer 15

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

Question 16

S.T intervals - solution

Answer 16

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

Question 17

limitations that apply to the use of Cox´s formula for the partial LH

Answer 17

there should be no ties in the non-censored pooled data

Question 18

Using Life-Table Estimator

Answer 18

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)

Question 19

Weighted LRT

Answer 19

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)

Question 20

Chi-Square Statistic

Answer 20

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.