how to deal with extreme values of longitudinal biomarker? My dataset includes repeated measured longitudinal biomarker values on cancer patients. For example, every patient would take CEA (Carcinoembryonic antigen) test every 8 weeks. Now the problem is that the range of CEA value [0.4,10060] (ng/mL) is not so "friendly" when I conduct subsequent analysis like fitting COX model. Most values lie in the range [0,200], but some patients have dramatically change like from 2000 to 10000.
I don't want to exclude these information as outliers, since over 10 patients have this kind of huge CEA values. And I'm afraid that dividing them into categorical covariates would lose information. And I tried log transformation and min-max transformation, they did not work well.
In medical sense, healthy people's CEA values are usually <5 ng/mL. So the change from 3 to 10 may be more meaningful than the change from 2000 to 3000. I want to find a way to transform data without losing a lot of information, so that the CEA value changes could reflect biological changes inside patients' body. I would be grateful if someone could give me some advice.
 A: First, remember that a Cox model with a time-dependent covariate assumes that the instantaneous value of the covariate at any time determines the hazard at that time. Use your knowledge of the subject matter to consider whether that really is an appropriate way to model the relationship between CEA levels and outcome. With this longitudinal study some average over N previous measurements, or a sudden difference in CEA levels between sequential measurements (perhaps with some lag in time; representing a time of potential tumor recurrence or expansion) might be expected to provide a better relationship with outcome. (Such alternate types of modeling might minimize or remove the extreme-value problem.)
Even if you proceed with CEA levels as you have started to, no simple transformation might work adequately and you will have to use a more flexible model for CEA to capture its relationship with outcome. The danger is that the more flexibly you model CEA, the more degrees of freedom you use up and the more events you need to avoid overfitting or losing power to detect real relationships in a Cox model. So the best solution might depend on the total number of events.
One approach would be to model CEA as a spline function, with the regression itself providing the coefficient values for the terms introduced into the model. See this page for a description of two general types of splines for modeling. Restricted cubic splines are generally a good choice, implemented in standard statistical software packages. Given the wide range of CEA values, this might work better on log-transformed rather than raw values as restricted cubic spline fits are linear beyond the two outermost knots. You choose the flexibility of the fit by the number of knots and, if desired, their locations along the CEA axis to handle values at which the relationship with outcome is expected to change (e.g., placing one at the value generally recognized to distinguish normal patients from those with tumors). The default tends to be to choose knots at particular quantiles of the distribution.
It's possible that in this particular situation the usual cautions against categorizing a continuous predictor are a bit too conservative. If, say, everyone with CEA > 20 ng/ml has an aggressive tumor, that once you are known to have an aggressive tumor then the actual CEA level doesn't matter, and CEA values below that level don't reliably distinguish individuals who are presently free of cancer from those who have aggressive tumors, then dichotomization might not be too bad. Slightly less Procrustean than a hard cutoff would be an arctangent transformation, for which you could choose the center (say at the clinically accepted, if misguided, cutoff value) and the slope to estimate the probability of serious disease as a function of CEA levels near that center. (The arctangent transformation effectively assumes that all cases with extreme high values have the same relationship to outcome regardless of CEA level.) If you have enough data, though, a well thought-through spline model should demonstrate that type of relationship directly. 
