Hi everyone,

Many thanks to everyone who replied to my questions regarding Cox
regression on SPSS(see below)

Basically here is the general consensus:

To check the proportional hazards assumption.....if the hazard ratio for
two patients changes through time, the proportional hazard assumption is
invalid.  So if we have x_1 as 'sex' (male=1, female=0), x_2 as
'treatment' (yes=1, 0=no) and x_3 as 'history' (yes=1, no=0) then for a
male and a female with treatment = 1 and history = 1 we would have

h(t) = h_o(t) * e^(B_1 + B_2 + B_3)    for the male

h(t) = h_o(t) * e^(B_2 + B_3) for the female

The baseline h_o(t) is the same for both males and females, the ratio of
their estimated hazard rates across all time points is e^B_1 i.e. for
the proportional hazards assumption to hold, the ratio of the estimated
hazard across time is a constant.  Parmar and Machin give a nice example
in their book "Survival Analysis" which shows a log(-log(Survival))plot
to demonstrate that the proportional hazards assumption holds. They have
produced lines for each of the possible combinations of the levels of
their two variables.....x1=0,x2=0 and x1=1,x2=0;  x1=0,x2=1 and
x1=1,x2=1.  They say that the parallel lines indicate approximately
proportional hazards in the 4 patient groups.

In SPSS we check that the proportional hazards assumption is true using
the "plots" button in the cox regression option and choosing a log(-log)
plot for the 'separate lines' option.

Now to assess  if a time dependent variable should be added... say, in
my example above, the plot which tested the proportional assumption for
'treatment' was non parallel / or crossed and hence you thought that
'treatment' was time dependent you could fit the model with terms 

h(t) = h_o(t) * e^(B_1*sex + B_2*treat + B_2*treat*t_cov + B_3*history)

After fitting this model (using the TIME program in Cox regression) you
found that the null hypothesis treat*T_cov=0 was rejected then the
hazards were not proportional for this variable and treatment is indeed
time dependent and the time dependent term treat*t_cov should be
included in the model.

Stratification is a totally different ball game.....say we stratify by
'sex', we get two different baseline functions...one for male and one
for female.  A plot ,log(-log(survival)) against t ,for males and a
similar plot for females (where both male and female patients share the
same values for 'history' and 'treatment') will reveal if the data
should be stratified.  Now say if the lines are non parallel, indicating
that stratification is required.....we can split the data and we create
one model for males and one model for females (i.e. different betas for
males and females.

You may still test within each stratum whether hazards non-parallel
(i.e. whether betas are different for OTHER covariates, not for the
stratification variable), and then you may decide to use time-dep
covariates to account for that. It may possibly be that the time
covariate is significant in one stratum and not in the other (say, time
interacts with X for males, but not for females), but there is doubt
whether this will happen frequently. In any case, if you find betas
should be different for different strata, you should split the dataset
and run Cox regression separately for each stratum.

Now the answers to my other questions (in '>'):

>
> I find the SPSS manual a little confusing as regards how to establish

> if a specific effect is constant over time (i.e whether a time 
> dependent covariate is in existence).  The example in the manual I am 
> reading finds that the data should be stratified by treatment (where 
> treatment takes the values 0 or 1).  It then tries a model
>
> H(t) = h_o(t) e^ (B_1 *treat + B_2 *treat*t_cov)
>
> The manual says  that "whenever you want to test that hazards are 
> proportional for different strata, you incorporate the
> time-by- stratification-variable interaction.  If the coefficient for 
> this term is significant then the hazards are not proportional."  
> Could anyone explain what this means please?

The t_cov is a time dependent covariate you may create with the TIME
PROGRAM command before starting COX REG proper. For the exercise
described in the manual, you do not stratify, but use the stratifying
variable (e.g.
treatment) as another covariate, multiplied by t_cov in this case, to
see whether the effects of time for each treatment describe a different
curve. Time is basically your time variable (e.g. time after surgery),
but it could also be some other function of time, such as log time,
squared time or whatever.
You can also treat the t_cov time-dependent covariate by itself, to see
whether hazards are proportional along time, or the interaction of t_cov
with any of your covariates to see whether time interacts with that
covariate.
In effect, time-dependent covariates are the mechanism through which you
analyze non-proportional hazards in general.


> Finally some books state that the log(-log) plots should be against t;

> some say against log of t.  SPSS plots against t.does anyone know the 
> reason for the descrepency?

It depends on theory. The effect of time elapsed on the hazard may be
constant or decreasing. For instance, the risk of dying for a newborn is
greater in the first few days, and then decreases; so increasing time by
one day has not the same effect when the child is one day old or 30 days
old.
But perhaps increasing time by 20% has the same effect at all times, and
in this case log time is the appropriate measure. Unlike infant
mortality per unit time, which is a decreasing function of time after
birth, the risk of being mugged in the street can reasonably be
conceived of as a constant function of time elapsed since the start of
the study, since that hazard is probably the same at all times per unit
of time.

**************************************
Finally, some references for the assessment of goodness of fit of the
Cox regression model:

"Regression modelling strategies" by Frank Harrell Jr.

Explained randomness in proportional hazards models O'Quigley J, Xu RH,
Stare J STATISTICS IN MEDICINE
24 (3): 479-489 FEB 15 2005

If you are looking for something like R^2 you might find the web site of
the biometricians at the medical university of Vienna helpful
http://www.meduniwien.ac.at/imc/biometrie/index_en.htm
Click on the link to "publications", then for example on the link to the
paper by Schemper and Henderson in Biometrics 56.
Follow the link "back to programs" to find SAS, R and SPLUS programs to
compute the measures of predictive accuracy and explained variation
explained in the aforementioned paper. 

Once again, thanks for all your help,
Kim


-----Original Message-----
From: K F Pearce 
Sent: 30 August 2005 10:03
To: [log in to unmask]
Subject: Cox regression: time dependent covariates: SPSS plots

Hello everyone,

I would like to ask a few questions about Cox regression and how to
assess its assumptions on SPSS.  Perhaps someone who has carried out Cox
regression can help.

Firstly some background....

1) Background about the Proportional Hazard Assumption

The cumulative hazard function is 

h(t) = h_o(t) * e^(beta_1 x1 + b_2 x2 +.....)

the baseline hazard is h_o(t).

Cumulative Survival function is S(t)=exp(- h(t))

The model is called the "proportional hazards model" because for two
patients, the ratio of their hazards will be constant for all time
points. 

For example if you had patients with the same age with presence of
characteristic A but different stages of disease, the ratio of the
estimated hazard rates across all time points is constant at  e^beta
where the regression coefficient is for the case with stage coded as 1.


According to Paramar and Machin ("Survival Analysis") we could plot the
log(-log) value of the survival function against (log of) time for the
distinct covariate patterns we are dealing with to assess if the
proportional hazard assumption holds.  

So, if we focus on 'stage', we could check the proportional hazard
assumption was true for this variable in SPSS by using the "plots"
option in SPSS to plot separate log(-log) survival curves/"lines"  for
pattern 1) where stage=0 and pattern 2) where stage=1.   We could do
similar plots to check the proportional hazard assumption for presence
and absence of 'characteristic A'.  Parallel lines indicate proportional
hazards (Paramar and Machin, p140).  Note that the same baseline
function is used to generate the different lines.

2) Background about stratification

We can also establish if the model should be stratified in SPSS by
splitting the data into strata to generate several separate hazard
baseline functions, one for each stratum.

One set of coefficients is generated regardless of stratum.  The value
of the  hazard functions in both strata are calculated using the same
set of variables e.g. if the data was stratified by 'sex',  the hazard
function for those with characteristic A and characteristic B would be
generated for both males and females over all time points.

Again, we examine SPSS's  'log minus log' against t  plot to see if the
ratio of the hazard functions for the two patient groups is constant
over time.  Parallel lines signify that this is true.  If this is the
case, then the variable used to form the strata ('sex' in our example)
can be used in the model and a common baseline hazard function can be
estimated for all of the groups.

Questions:

I find the SPSS manual a little confusing as regards how to establish
if a specific effect is constant over time (i.e whether a time dependent
covariate is in existence).  The example in the manual I am reading
finds that the data should be stratified by treatment (where treatment
takes the values 0 or 1).  It then tries a model 

H(t) = h_o(t) e^ (B_1 *treat + B_2 *treat*t_cov)

The manual says  that "whenever you want to test that hazards are
proportional for different strata, you incorporate the
time-by-stratification-variable interaction.  If the coefficient for
this term is significant then the hazards are not proportional."  Could
anyone explain what this means please?

Also, I'd like to know if we can assess if a time dependent covariate
should be added by looking at plots?  If so, which plots?  I would say
that the plots described above (in 1) to assess the proportional hazards
assumption would be the ones to look at as these make use of the index,
beta_x.  Non parallel lines would indicate that a predictor depends on
time. Do you agree?

Finally some books state that the log(-log) plots should be against t;
some say against log of t.  SPSS plots against t...does anyone know the
reason for the descrepency?

Many thanks again,
All the Best,
Kim.