Can I draw 1-CIF (Cumulative Incidence Function) as a survival curve? I analyze survival data with competing risks. To me, the CIF is fine, but the researchers want to see how fast the survival falls, so they want the 1-CIF. They know it's NOT the Kaplan-Meier, it's a different analysis (cause specific, but with overall survival in it) than the cause-specific K-M, it's just a question for a different presentation.
Honestly, I couldn't find any examples of 1-CIF curves (all mention 1-KM = CIF when there's no competing risk). Will that be OK to draw a 1-CIF curve as a survival curve and mention in the plot title it's 1-CIF (so calculated differently, via Aalen-Johansen estimator; no covarites), just showing how fast people experience the event of interest?
EDIT: They want to see each cause-specific CIF. In the overall case it's not a problem, as CIF = sum of specific CIFs = 1-KM.
For the cause-specific CIF I cannot use the KM (would make it too high), but I can calculate CIF and THEN draw the 1-CIF to "mimic" the survival curve.
By competing risks I mean, death from many reasons, while the even of interest is some clinical event. The death(s) prevent the event to happen (while censoring does not).

This seems to be similar to the analysis in Section 2.3.2 of the R survival vignette, where individuals were evaluated until development of (a) plasma cell malignancy or (b) death, whichever came first, as competing risks. The standard plot for competing risks is, as you note, the cumulative probability of each type of event over time. The first figure in that section of the vignette is an example.
I know of no rule against plotting each of those cumulative probabilities as apparent survival curves as you propose, but that runs a risk of being misleading to those who don't already understand the issues thoroughly. It might be too tempting to interpret each of the individual "survival" curves as representing a "cure" model, as each curve will plateau at some finite apparent "survival." I would thus not recommend that for publication, only for internal use.
If you nevertheless do produce such a plot, I'd recommend including the survival curve representing occurrence of either event. That will help emphasize that those cause-specific "survival" curves represent alternate, mutually exclusive ways to be lost from the initial event/death-free state. The first plot in Section 2.3.2 of the survival vignette shows such a curve, along with the standard cumulative probabilities.
