graphs and CI's

[brouwerl]

I have analyzed survival with time dependence, two age classes and one individual covariate. The individual cov affects survival, but there is no interaction with year. As my data set consists of 20 years i would like to make a graph of the effect of the individual covariate for an average year. How can I do this and how can I calculate the CI's?

I read the appendix about the delta method for calculating the SE's, can I use these for calculating CI's? Why are the SE's not assymetric?

Thanks,
Lyanne

[Doherty]

I have analyzed survival with time dependence, two age classes and one individual covariate. The individual cov affects survival, but there is no interaction with year. As my data set consists of 20 years i would like to make a graph of the effect of the individual covariate for an average year. How can I do this and how can I calculate the CI's?

I read the appendix about the delta method for calculating the SE's, can I use these for calculating CI's? Why are the SE's not assymetric?

Thanks,
Lyanne

Dear Lyanne,

You would like to make a graph with survival on the y axis and the covariate on the x-axis with the prediction equation being for an "average" year.

The simplest model for an "average" year would be to take the beta's from the "dot" model for survival - at least in terms of years (i.e., no temporal variation). This would equate to a weighted-average over time.

You would export the beta values from this model and then back transfrom them through the appropriate link function (you probably used the default logit link). The help file in Program Mark details how to do this, as does Chapter 7 (?) in Evan and Gary's manual.

To put error bars around the line you would need to calculate these using the methods in the Appendix 2 of the manual.

Why the SEs are asymetric you ask? I guess I am not sure when a SE would be asymetric... but I think I am following your line of thinking. You probably want to take +- 1.96 x SE to get the CI and are wondering why the CI doesn't seem to be symetric in the output? I beleive that the calculation is done on the logit scale and then back-transformed. So the CI is symetric on the logit scale, but does not look that way after transformation.

Evan hounds me to do my part in answering some of these questions. I hope this does not lead you astray.

Sincerley,

Paul

[brouwerl]

Dear Paul,

i thought of this indeed, run a model without time, but then the beta of the individual covariate will also change?

Lyanne

[jlaake]

On the surface your question appears to be fairly simple but you need to think carefully about what you are doing here and without more details it is not trivial to answer it because it will depend on what the covariate is and whether it is varying by year. For example, I'll describe an example I have where we mark young of the year each year. The average first year survival varies tremendously from year to year but first year survival also depends on the animal's mass at the time of marking and the slope of that relationship doesn't change with time similar to the example you described. The model is equivalent to regression with a constant slope but annually varying intercept. Now here is the tricky part. The average mass of the young varies from year to year. In a good year the survival of young at an average mass of 15kg can be considerably higher than the survival of averge mass of 30kg in a bad year because the intercept is varying. The slope parameter of mass is constant but with an annual intercept it only describes the relative survival differences between individuals in the same cohort. If you were to plot the survival curve for each year as a function of mass for the range of values encountered in each year, sometimes the curve would be nearly flat (high intercept) and at other times quite steep (low intercept). It is linear on the logit scale but is not linear on the survival scale. Thinking off the top of my head and not knowing the details of your study, you may want to use the average over years and plot the survival as a function of your covariate using that value and then show the plot with the smallest and largest annual betas. However, that only captures the possible range of relationships and doesn't capture the variation in the covariate values as a function of year. Not certain if you can accomplish that with MARK but you can easily do it in a plotting package or Excel.

[jlaake]

ps

How you average betas by year will depend on the way you created the design matrix. If you used an identity matrix for time, it is straightforward. If you used treatment contrast in which beta0 was year 1 etc then you need to add beta0 to the beta for years 2..20 etc.

Hope that makes sense. --jeff