ASA Connect

In a survival analysis setting using Cox regression, one often constructs plots of adjusted survival curves.   These plots involving setting some of the variables in the Cox regression model to their "average" value. For example, if the model includes the variables Treatment and Gender, Gender is set to its average value (e.g., proportion of Males in the data) so that we can plot the gender-adjusted survival curves for patients in the Novel Treatment and those in the Standard Treatment groups. 

Here is what I don't understand about this method: 

1. Does it assume that we are comparing the survival behaviour of patients with an "average" value for Gender across the two treatment groups?  What does it even mean for patients to have an "average" value for Gender? 

2. Does it assume that we are comparing the survival behaviour of patients placed on the two treatments by "averaging out" the effect of Gender?

3.  What if Gender had 3 levels (e.g., Male, Female, Undeclared)? If we code the Gender effect via two dummy variables (say, D1 and D2, with D1 comparing Female vs Male and D2 comparing Undeclared vs Male), do we set the values of D1 and D2 to the proportion of Females and Undeclared values in the data?  And how do we interpret the gender-adjusted curves (i.e., to what kind of patients do they apply)?

These may be "stupid questions", but I would appreciate any insights.

Thanks,

Isabella

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Isabella Ghement
Ghement Statistical Consulting Company Ltd.
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What statistical software allows or encourages you to 'adjust to' interpolations between the values of a categorical variable? Frank Harrell's R package 'rms' will typically choose modes of categorical variables as default 'adjust-to' values.

Sometimes in mathematics, the formalism seems to give rise to apparently meaningless construct that later turn out to be fantastically useful. The most obvious example of this is √(-1), but there are others like non-Euclidean geometries.

In statistics, on the other hand, I would suggest that this does not happen! Our formalisms might suggest seductive ideas we feel compelled to 'interpret', but we may be better off resisting such temptations. Just because binary variables can be 'coded' as being from {0,1}, and that set is an element of a larger set like [0,1] or even all of ℜ, this need not induce a search for individuals with a 'gender' of 0.5 or π or -14. Statistics is not physics.

Kind regards,

In a survival analysis setting using Cox regression, one often constructs plots of adjusted survival curves.   These plots involving setting some of the variables in the Cox regression model to their "average" value. For example, if the model includes the variables Treatment and Gender, Gender is set to its average value (e.g., proportion of Males in the data) so that we can plot the gender-adjusted survival curves for patients in the Novel Treatment and those in the Standard Treatment groups. 

Here is what I don't understand about this method: 

1. Does it assume that we are comparing the survival behaviour of patients with an "average" value for Gender across the two treatment groups?  What does it even mean for patients to have an "average" value for Gender? 

2. Does it assume that we are comparing the survival behaviour of patients placed on the two treatments by "averaging out" the effect of Gender?

3.  What if Gender had 3 levels (e.g., Male, Female, Undeclared)? If we code the Gender effect via two dummy variables (say, D1 and D2, with D1 comparing Female vs Male and D2 comparing Undeclared vs Male), do we set the values of D1 and D2 to the proportion of Females and Undeclared values in the data?  And how do we interpret the gender-adjusted curves (i.e., to what kind of patients do they apply)?

These may be "stupid questions", but I would appreciate any insights.

Thanks,

Isabella

Hi David, 

Thanks for your thoughtful response.   I agree with you that "averaging" binary variables doesn't make sense from an interpretation point of view (hence my original questions). I may be off about this, but I think the "effects" package in R uses this "averaging" argument to produce effect plots (not in a survival setting, but in a similar situation where some model variables have to be set to 'typical' values in order to visualize the effects of other variables in the model on the response). 

In my own setting, I actually have to produce adjusted survival curves based on N = 20 imputation data sets and a Cox regression model which includes 5 categorical predictors. I guess I could build adjusted survival curves for the levels of each predictor variable in turns while setting the values of the other predictors in the model to their modal values, as in the rms package. (The other option would be to define some 'prototypical' patients and compute adjusted survival curves for those. But there are so many options that it's hard to come up with a select few.) The adjusted survival curves would then be "averaged" across imputation data sets in an appropriate manner.

I just wanted to get a sense on how other people tend to go about a task like this and make sure that the end result is interpretable. 

All the best, 

Isabella

What statistical software allows or encourages you to 'adjust to' interpolations between the values of a categorical variable? Frank Harrell's R package 'rms' will typically choose modes of categorical variables as default 'adjust-to' values.

Sometimes in mathematics, the formalism seems to give rise to apparently meaningless construct that later turn out to be fantastically useful. The most obvious example of this is √(-1), but there are others like non-Euclidean geometries.

In statistics, on the other hand, I would suggest that this does not happen! Our formalisms might suggest seductive ideas we feel compelled to 'interpret', but we may be better off resisting such temptations. Just because binary variables can be 'coded' as being from {0,1}, and that set is an element of a larger set like [0,1] or even all of ℜ, this need not induce a search for individuals with a 'gender' of 0.5 or π or -14. Statistics is not physics.

Kind regards,

In a survival analysis setting using Cox regression, one often constructs plots of adjusted survival curves.   These plots involving setting some of the variables in the Cox regression model to their "average" value. For example, if the model includes the variables Treatment and Gender, Gender is set to its average value (e.g., proportion of Males in the data) so that we can plot the gender-adjusted survival curves for patients in the Novel Treatment and those in the Standard Treatment groups. 

Here is what I don't understand about this method: 

1. Does it assume that we are comparing the survival behaviour of patients with an "average" value for Gender across the two treatment groups?  What does it even mean for patients to have an "average" value for Gender? 

2. Does it assume that we are comparing the survival behaviour of patients placed on the two treatments by "averaging out" the effect of Gender?

3.  What if Gender had 3 levels (e.g., Male, Female, Undeclared)? If we code the Gender effect via two dummy variables (say, D1 and D2, with D1 comparing Female vs Male and D2 comparing Undeclared vs Male), do we set the values of D1 and D2 to the proportion of Females and Undeclared values in the data?  And how do we interpret the gender-adjusted curves (i.e., to what kind of patients do they apply)?

These may be "stupid questions", but I would appreciate any insights.

Thanks,

Isabella

In a survival analysis setting using Cox regression, one often constructs plots of adjusted survival curves.   These plots involving setting some of the variables in the Cox regression model to their "average" value. For example, if the model includes the variables Treatment and Gender, Gender is set to its average value (e.g., proportion of Males in the data) so that we can plot the gender-adjusted survival curves for patients in the Novel Treatment and those in the Standard Treatment groups. 

Here is what I don't understand about this method: 

1. Does it assume that we are comparing the survival behaviour of patients with an "average" value for Gender across the two treatment groups?  What does it even mean for patients to have an "average" value for Gender? 

2. Does it assume that we are comparing the survival behaviour of patients placed on the two treatments by "averaging out" the effect of Gender?

3.  What if Gender had 3 levels (e.g., Male, Female, Undeclared)? If we code the Gender effect via two dummy variables (say, D1 and D2, with D1 comparing Female vs Male and D2 comparing Undeclared vs Male), do we set the values of D1 and D2 to the proportion of Females and Undeclared values in the data?  And how do we interpret the gender-adjusted curves (i.e., to what kind of patients do they apply)?

These may be "stupid questions", but I would appreciate any insights.

Thanks,

Isabella

In a survival analysis setting using Cox regression, one often constructs plots of adjusted survival curves.   These plots involving setting some of the variables in the Cox regression model to their "average" value. For example, if the model includes the variables Treatment and Gender, Gender is set to its average value (e.g., proportion of Males in the data) so that we can plot the gender-adjusted survival curves for patients in the Novel Treatment and those in the Standard Treatment groups. 

Here is what I don't understand about this method: 

1. Does it assume that we are comparing the survival behaviour of patients with an "average" value for Gender across the two treatment groups?  What does it even mean for patients to have an "average" value for Gender? 

2. Does it assume that we are comparing the survival behaviour of patients placed on the two treatments by "averaging out" the effect of Gender?

3.  What if Gender had 3 levels (e.g., Male, Female, Undeclared)? If we code the Gender effect via two dummy variables (say, D1 and D2, with D1 comparing Female vs Male and D2 comparing Undeclared vs Male), do we set the values of D1 and D2 to the proportion of Females and Undeclared values in the data?  And how do we interpret the gender-adjusted curves (i.e., to what kind of patients do they apply)?

These may be "stupid questions", but I would appreciate any insights.

Thanks,

Isabella

Thank you very much to everyone who responded to my initial inquiry about adjusted survival curves.  

Terry, I finally had a chance to review your excellent CRAN vignette regarding adjusted survival curves and wanted to follow up with a few questions.  

Let's say I am interested in constructing an adjusted "conditional" survival curve using the methodology in the vignette.  The way I understand this methodology, it implies two steps: 1) model and 2) balance.  

For 1), a Cox proportional hazards regression model is fitted to the sample data. For 2), the model is used to obtain a survival curve for each individual in the sample (i.e., for each configuration of predictor variables represented in the sample) and then the adjusted "conditional" survival curve is obtained by performing a simple averaging of the obtained survival curves.  

Now, let's say that the Cox proportional hazards regression model includes 3 predictors (i.e., X1, X2 and X3) such that X1 and X2 are both categorical with two levels each (i.e., levels a and b for X1; levels A and B for X2) and X3 is continuous.  Let's also say that we are interested in a variation of the adjusted "conditional" survival curves that would enable us to visualize the effect of each of these predictor variables in turns on survival (rather than the combined effect of all three predictor variables on overall survival).   

For the effect of X1, we could separate our sample into two sub-samples:  i) subjects where X1 = a and ii) subjects where X1 = b.   We could then use the Cox proportional hazards regression model to separately construct survival curves for all configurations of values for X2 and X3 present in each of the two sub-samples.  Simple averaging of those survival curves across the subjects in each sub-sample would yield two adjusted "conditional" survival curves  - one for each level of X1.          

For the effect of X2, we would proceed in a similar fashion as described for X1, except that the two sub-samples would correspond to subjects where X2 = A and X2 = B, respectively, and the construction of survival curves would pertain to all configurations of values of X1 and X3 present in the two sub-samples. 

For the effect of X3, things are trickier if we only care about some pre-specified values of X3 (rather than all observed values of X3 present in the sample).  For example, if X3 stands for Age, the pre-specified values of Age might be 30, 40 and 40 years.  For a relatively small study, it's entirely possible that none of the subjects in the study actually have any of those ages (or some of those ages). So the question is: How would we proceed in this situation, especially when it comes to looking at the effects of X1 and X2 on survival?  

For this situation, could we use a slightly different reasoning where, instead of averaging survival curves over all subjects in the sample with various observed configurations of predictor variables, we would average survival curves over "prototypical" subjects with idealized configurations of predictor variables?  In the context of the example given, we could define the following "prototypical" subjects:

    X1 = a,  X2 = A, X3 = 30  (where X3 = Age)

    X1 = a,  X2 = B, X3 = 30

    X1 = b, X2 = A, X3 = 30

    X2 = b, X2 = B, X3 = 30

    X1 = a,  X2 = A, X3 = 40     

    etc. 

Each of these idealized configurations of predictor variables would yield a single survival curve.  Eliciting the effect of X1 would mean averaging the survival curves corresponding to subjects with (i) X1 = a for whom X2 can be either A or B and X3 can be either 30, 40, 50 and (ii) X1 = b for whom X2 can be either A or B and X3 can be either 30, 40, 50, etc.  This would yield to average survival curves across "prototypical" subjects with either X1 = a or X1 = b for whom X2 can be either A or B and X3 can be either 30, 40, 50 (rather than average survival curves in the entire cohort of patients with either X1 = a or X1 = b).  

Does this make sense?  And if it doesn't make sense, is there a better way to deal with adjusted survival curves in situations where one of the predictor variables is continuous and we are interested in learning more about its effect on survival by comparing adjusted survival curves at some of its pre-specified values?  

Thanks a lot,

Isabella  

In a survival analysis setting using Cox regression, one often constructs plots of adjusted survival curves.   These plots involving setting some of the variables in the Cox regression model to their "average" value. For example, if the model includes the variables Treatment and Gender, Gender is set to its average value (e.g., proportion of Males in the data) so that we can plot the gender-adjusted survival curves for patients in the Novel Treatment and those in the Standard Treatment groups. 

Here is what I don't understand about this method: 

1. Does it assume that we are comparing the survival behaviour of patients with an "average" value for Gender across the two treatment groups?  What does it even mean for patients to have an "average" value for Gender? 

2. Does it assume that we are comparing the survival behaviour of patients placed on the two treatments by "averaging out" the effect of Gender?

3.  What if Gender had 3 levels (e.g., Male, Female, Undeclared)? If we code the Gender effect via two dummy variables (say, D1 and D2, with D1 comparing Female vs Male and D2 comparing Undeclared vs Male), do we set the values of D1 and D2 to the proportion of Females and Undeclared values in the data?  And how do we interpret the gender-adjusted curves (i.e., to what kind of patients do they apply)?

These may be "stupid questions", but I would appreciate any insights.

Thanks,

Isabella

Steve Simon's answer pretty much nails it. The interpretation is not about an individual but about groups of individuals. Adjusting by gender (either 2 or 3 categories) would attempt to estimate survival behavior differences between the treatment groups where these groups have (hypothetically) similar gender breakdown (assuming that the form of the model, a linearized equation, is appropriate enough). It doesn't necessarily apply to a particular individual.

In a survival analysis setting using Cox regression, one often constructs plots of adjusted survival curves.   These plots involving setting some of the variables in the Cox regression model to their "average" value. For example, if the model includes the variables Treatment and Gender, Gender is set to its average value (e.g., proportion of Males in the data) so that we can plot the gender-adjusted survival curves for patients in the Novel Treatment and those in the Standard Treatment groups. 

Here is what I don't understand about this method: 

1. Does it assume that we are comparing the survival behaviour of patients with an "average" value for Gender across the two treatment groups?  What does it even mean for patients to have an "average" value for Gender? 

2. Does it assume that we are comparing the survival behaviour of patients placed on the two treatments by "averaging out" the effect of Gender?

3.  What if Gender had 3 levels (e.g., Male, Female, Undeclared)? If we code the Gender effect via two dummy variables (say, D1 and D2, with D1 comparing Female vs Male and D2 comparing Undeclared vs Male), do we set the values of D1 and D2 to the proportion of Females and Undeclared values in the data?  And how do we interpret the gender-adjusted curves (i.e., to what kind of patients do they apply)?

These may be "stupid questions", but I would appreciate any insights.

Thanks,

Isabella

In a survival analysis setting using Cox regression, one often constructs plots of adjusted survival curves.   These plots involving setting some of the variables in the Cox regression model to their "average" value. For example, if the model includes the variables Treatment and Gender, Gender is set to its average value (e.g., proportion of Males in the data) so that we can plot the gender-adjusted survival curves for patients in the Novel Treatment and those in the Standard Treatment groups. 

Here is what I don't understand about this method: 

1. Does it assume that we are comparing the survival behaviour of patients with an "average" value for Gender across the two treatment groups?  What does it even mean for patients to have an "average" value for Gender? 

2. Does it assume that we are comparing the survival behaviour of patients placed on the two treatments by "averaging out" the effect of Gender?

3.  What if Gender had 3 levels (e.g., Male, Female, Undeclared)? If we code the Gender effect via two dummy variables (say, D1 and D2, with D1 comparing Female vs Male and D2 comparing Undeclared vs Male), do we set the values of D1 and D2 to the proportion of Females and Undeclared values in the data?  And how do we interpret the gender-adjusted curves (i.e., to what kind of patients do they apply)?

These may be "stupid questions", but I would appreciate any insights.

Thanks,

Isabella