Constant in Cox Regresion

[Eric Black]

I have two questions on Cox Regression/general survival analysis.  They are
pretty general since I am just at the beginning of that.

1. We are trying to predict survival for new customers, i.e. model survival
with existing customers based on various covariates and then apply that to
new customers given their values on those covariates.
When I run the Cox regression, how is the hazard variable being calculated
that I can save? (  /SAVE= HAZARD)?
The online manual states
hi(t)=[h0(t)]e(b0+b1xi1+...+bpxip) but what is that constant b0? To the best
of my knowledge, there isn't a constant in Cox regression, is there?

2. If we have customers on annual contracts (but the implicit assumption
that they can drop out early by paying an Early Termination Fee (ETF)) - can
anybody point me in the direction of how to model monthly survival rates?
In other words, we have small churn rates for months 1-11/13-23 and then a
big drop in months 12, 24 etc.

Thanks kindly

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[Hector Maletta]

The constant is included in the baseline hazard rate h0(t). You may verify
this by formulating the Cox function with a constant, and solving. The
function with a constant would be =h0(t) x e^(b0+b1X1+b2X2 ...) = h0(t) x
e^b0 x e^(b1X1+b2X2 ...) = h*0(t) x e^(b1X1+b2X2 ...) where h*0(t)=h0(t) x
e^b0. The algorithm actually computes h*0(t), not the theoretical h0(t).

Hector

-----Mensaje original-----
De: SPSSX(r) Discussion [mailto:[hidden email]] En nombre de Matt
Black
Enviado el: Monday, January 21, 2013 10:05
Para: [hidden email]
Asunto: Constant in Cox Regresion

I have two questions on Cox Regression/general survival analysis.  They are
pretty general since I am just at the beginning of that.

1. We are trying to predict survival for new customers, i.e. model survival
with existing customers based on various covariates and then apply that to
new customers given their values on those covariates.
When I run the Cox regression, how is the hazard variable being calculated
that I can save? (  /SAVE= HAZARD)?
The online manual states
hi(t)=[h0(t)]e(b0+b1xi1+...+bpxip) but what is that constant b0? To the best
of my knowledge, there isn't a constant in Cox regression, is there?

2. If we have customers on annual contracts (but the implicit assumption
that they can drop out early by paying an Early Termination Fee (ETF)) - can
anybody point me in the direction of how to model monthly survival rates?
In other words, we have small churn rates for months 1-11/13-23 and then a
big drop in months 12, 24 etc.

Thanks kindly

=====================
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[hidden email] (not to SPSSX-L), with no body text except the
command. To leave the list, send the command SIGNOFF SPSSX-L For a list of
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=====================
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[Alex Reutter]

Hi Matt,

Assuming you're looking at http://pic.dhe.ibm.com/infocenter/spssstat/v21r0m0/topic/com.ibm.spss.statistics.cs/coxreg_model_prop_haz.htm, you're right that b0 should not be there; that's a bug in the documentation.

Thanks,
Alex




From:        Matt Black <[hidden email]>
To:        [hidden email],
Date:        01/21/2013 07:07 AM
Subject:        Constant in Cox Regresion
Sent by:        "SPSSX(r) Discussion" <[hidden email]>

---

I have two questions on Cox Regression/general survival analysis.  They are
pretty general since I am just at the beginning of that.

1. We are trying to predict survival for new customers, i.e. model survival
with existing customers based on various covariates and then apply that to
new customers given their values on those covariates.
When I run the Cox regression, how is the hazard variable being calculated
that I can save? (  /SAVE= HAZARD)?
The online manual states
hi(t)=[h0(t)]e(b0+b1xi1+...+bpxip) but what is that constant b0? To the best
of my knowledge, there isn't a constant in Cox regression, is there?

2. If we have customers on annual contracts (but the implicit assumption
that they can drop out early by paying an Early Termination Fee (ETF)) - can
anybody point me in the direction of how to model monthly survival rates?
In other words, we have small churn rates for months 1-11/13-23 and then a
big drop in months 12, 24 etc.

Thanks kindly

=====================
To manage your subscription to SPSSX-L, send a message to
[hidden email] (not to SPSSX-L), with no body text except the
command. To leave the list, send the command
SIGNOFF SPSSX-L
For a list of commands to manage subscriptions, send the command
INFO REFCARD

[Maguin, Eugene]

Matt,

I want be cautious in my response. I understand your question. If your plot the hazard, let's say by month, the hazard for the contract end months leaps upward from small values for the prior 11 months. I was interested recently in predicting person level survival time given a regression equation and a covariate values set but found it impossible to do so. It is possible to compute a median survival time and put a confidence interval around that value. I had planned to look at the median lifetimes for specific covariate combinations as a way of getting to the same thing you are interested in but didn't. Perhaps other, more survival-experienced, readers will comment on this strategy.

That said, and as I understand your problem, I wonder if a discrete time model or even a multinomial logistic model might be as, or more, useful because it sounds like you have three groups of people: people who drop in months 1-11, people who drop in month 12 and people who don't drop.

Gene Maguin


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of Matt Black
Sent: Monday, January 21, 2013 8:05 AM
To: [hidden email]
Subject: Constant in Cox Regresion

I have two questions on Cox Regression/general survival analysis.  They are pretty general since I am just at the beginning of that.

1. We are trying to predict survival for new customers, i.e. model survival with existing customers based on various covariates and then apply that to new customers given their values on those covariates.
When I run the Cox regression, how is the hazard variable being calculated that I can save? (  /SAVE= HAZARD)?
The online manual states
hi(t)=[h0(t)]e(b0+b1xi1+...+bpxip) but what is that constant b0? To the best of my knowledge, there isn't a constant in Cox regression, is there?

2. If we have customers on annual contracts (but the implicit assumption that they can drop out early by paying an Early Termination Fee (ETF)) - can anybody point me in the direction of how to model monthly survival rates?
In other words, we have small churn rates for months 1-11/13-23 and then a big drop in months 12, 24 etc.

Thanks kindly

=====================
To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD

=====================
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[Hector Maletta]

Gene,
The prediction might be at "person level" (e.g. survival after surgery) but
the outcome is a probability. Probability, when empirically determined on
the basis of frequencies of events, is not (strictly speaking) an attribute
of individuals. Each individual either dies or survives after a given time
since surgery. His or her "probability" is either zero or one. It is the
population of subjects (or the theoretical "representative individual" as a
member of that population) who are affected by the probability. Not the
median, though, nor the mode, but the mean, since the probability in
question is computed as an arithmetic mean of many zeroes and ones.

Now, a probability, computed on the basis of frequencies, may be INTERPRETED
as describing an individual propensity, e.g. the propensity to survive for
five years after undergoing a certain kind of surgery (say, heart
transplant). But it would be difficult to find a realistic interpretation of
this "propensity" not related to the population feature elicited in the
analysis (i.e. the relative frequency of survivors). This is, besides, not a
matter of statistical estimation but a matter of substantive interpretation,
and one having many complex philosophical underpinnings into which I will
not venture now.

Hector

-----Mensaje original-----
De: SPSSX(r) Discussion [mailto:[hidden email]] En nombre de
Maguin, Eugene
Enviado el: Monday, January 21, 2013 12:06
Para: [hidden email]
Asunto: Re: Constant in Cox Regresion

Matt,

I want be cautious in my response. I understand your question. If your plot
the hazard, let's say by month, the hazard for the contract end months leaps
upward from small values for the prior 11 months. I was interested recently
in predicting person level survival time given a regression equation and a
covariate values set but found it impossible to do so. It is possible to
compute a median survival time and put a confidence interval around that
value. I had planned to look at the median lifetimes for specific covariate
combinations as a way of getting to the same thing you are interested in but
didn't. Perhaps other, more survival-experienced, readers will comment on
this strategy.

That said, and as I understand your problem, I wonder if a discrete time
model or even a multinomial logistic model might be as, or more, useful
because it sounds like you have three groups of people: people who drop in
months 1-11, people who drop in month 12 and people who don't drop.

Gene Maguin


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Matt Black
Sent: Monday, January 21, 2013 8:05 AM
To: [hidden email]
Subject: Constant in Cox Regresion

I have two questions on Cox Regression/general survival analysis.  They are
pretty general since I am just at the beginning of that.

1. We are trying to predict survival for new customers, i.e. model survival
with existing customers based on various covariates and then apply that to
new customers given their values on those covariates.
When I run the Cox regression, how is the hazard variable being calculated
that I can save? (  /SAVE= HAZARD)?
The online manual states
hi(t)=[h0(t)]e(b0+b1xi1+...+bpxip) but what is that constant b0? To the best
of my knowledge, there isn't a constant in Cox regression, is there?

2. If we have customers on annual contracts (but the implicit assumption
that they can drop out early by paying an Early Termination Fee (ETF)) - can
anybody point me in the direction of how to model monthly survival rates?
In other words, we have small churn rates for months 1-11/13-23 and then a
big drop in months 12, 24 etc.

Thanks kindly

=====================
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[hidden email] (not to SPSSX-L), with no body text except the
command. To leave the list, send the command SIGNOFF SPSSX-L For a list of
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[Eric Black]

Thanks Alex, Gene and Hector

To address your questions/suggestions:
1. Yes, it was that part of the documentation where I was confused.
2. It looks like you can export the baseline hazard rate for each point in
time (/OUTFILE=TABLE) and with that you can model survival rates for new
cases given their covariates.  That in turn addresses the issue I had

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