When there is a continuous distribution of bio-marker values with no obvious modal values, then a good alternative to using cut-points to derive discrete subgroups of patients with different outcomes, is to consider a biomarker as a continuous rather than a dichotomous variable in multivariate analyses. For example, Chapman et al32
reported the multivariate results obtained by use of continuous hormonal receptor factors, followed by multivariate examination of various cut-points. They showed there was no best cut-point for estrogen receptor (ER) or for the progesterone receptor (PgR) in breast cancer patients and suggested that they be considered as continuous rather than dichotomous (negative, positive) variables for prognosis. Thompson et al33
report the percent risk of prostate cancer as a continuous dependent variable and prostate specific antigen (PSA) as a continuous independent variable, in contrast to the previous discrete risk groups with cut-points of 4.0 ng/mL and 10 ng/mL.
If two cut-points are more informative than one cut-point, then this raises the question whether more cut-points, or different cut-points would be an improvement. That is, should one search among a variety of possible cut-points until the difference between groups of patients becomes statistically significant with the minimum p-value? Altman et al34
have shown that seeking an optimal cut-point among several cut-points causes an inflation of type I error rate, and that this requires corrections for multiple testing; they also advocate against use of a data set to determine a factor cut-point, followed by use of the cut-point in the same data to determine its (multivariate) significance.
However, if the distribution of the values of a biomarker appears by inspection and/or outcome to have subgroups, e.g. bimodal, i.e. Mobbs et al35
or trimodal, then it is reasonable to consider multivariate investigations aimed at definition of appropriate cut-points, i.e. Chapman et al.32
This requires that the number of subgroups be chosen, e.g. 2 or 3, so that the patients will be aggregated to minimize overlap between subgroups.
Alternatively, a population of patients who have a continuous distribution of values of a quantitative biomarker can be separated into two groups by specifying a neutral to investigation cut-point of its distribution. For instance, one cut-point at the mean value would separate patients into two groups, one with a higher and the other with a lower value of the biomarker. Or, the population can be separated by two cut-points into three groups, low, medium, and high values of the biomarker (). These groups are not necessarily intrinsically different biological classes,36
but they may be informative. These new discrete groups can be compared to patient outcome; for one cut-point, the patients might be described graphically as healthy or sick, good or poor survival, and for two cut-points good, moderate, and poor survival, using univariate Kaplan-Meier or cumulative incidence,37
or multivariately with Cox or log-normal survivor plots.38
Recently, Royston et al39
demonstrated that a visual display of length of survival accomplished with a log-normal distribution of survival times would complement a Kaplan-Meier plot.
Figure 1 Cut-points of a continuous distribution of breast cancer patients. Eighty patients with in situ duct carcinoma of the breast were ranked by a canonical variable derived by weighting 39 nuclear features of 200 cells per patient.25 Left panel: there is (more ...)
There are several caveats to the application of cut-points. The first is in the use of visual subjective estimates of values rather than of objective measurements. In some cases qualitative information is combined with quantitative information to derive a score. The score is then separated into discrete classes. For example, grading of invasive carcinoma of the breast40
by the Nottingham histologic score, combines visual estimates of degree of acinus formation (1, 2, 3) and nuclear atypia (1, 2, 3) with numerical count of mitoses(converted to score 1, 2, 3) to create a total score which determines the grade (Total score of 3, 4, or 5 = grade 1, 6 or 7 = grade 2, and 8 or 9 = grade 3.). Two cut-points are then used to give three discrete nuclear grades.
In a series of patients with in situ
duct carcinoma of the breast, objective quantitative image analysis measurements of 39 nuclear features including size, shape, texture, and stain intensity were combined in a dimension reduction method, Fisher discriminant analysis, to derive a single value, the canonical variable, for each of 80 patients.25
(left) This result indicates that there is a continuous distribution of values. (middle) shows two groups of patients resulting from one cut-point, and (right) shows three groups of patients resulting from two cut-points. Since the distribution of nuclear values is continuous without obvious groups, the number of cut-points and position of the cut-points are arbitrary and not derived from distinct subpopulations. The two or three groups are not intrinsic biological classes.36
Although cut-points may not provide classes of patients with distinctly different nuclei, the grouping (binning) of patients can be useful. For instance, it was determined that one cut-point at the mean resulted in two groups of patents that differed in recurrence of ductal carcinoma in situ
and in development of invasive cancer.25,41
A notable example of the use of two cut-points to derive three informative groups of patients is given by Camp et al.42
The amount of p53 staining in breast cancer biopsies was determined by immunohistochemistry. Two cut-points were determined by a method that allows the effects of scanning many cut-points and simultaneously visualizing the histogram of the distribution of p53 staining and of the Kaplan-Meier survival plots. The surprising result was that the survival was not proportional to the amount of p53, but rather that patients with intermediate values of p53 survived better than patients with low and high values of p53. We have confirmed this non-linear dependence of survival on p53.43
In addition we have shown that two groups of patients, produced by one cut-point at the mean, do not differ in survival.
In addition to grouping patients by cut-points of the amounts of proteins measured by immunohistochemistry, patients have been grouped by cut-points of the amounts of messenger RNA measured in gene expression microarrays. A multivariable method, Logical Analysis of Data (LAD) has been used to group breast cancer patients for good and poor prognosis,44
and for diagnosis of lymphoma patients as having diffuse large B-cell lymphoma or follicular lymphoma45
based on gene expression data. LAD is a method that uses combinatorics and optimization to derive one or more cut-points for each of many individual variables.46
LAD has recently been extended to prognosis of censored survival data.47
If two cut-points are more informative than one cut-point, then these results raise the question whether more cut-points, or different cut-points would be an improvement, which brings us back to the beginning of this discussion that it could be appropriate to examine the effects of continuous factors without categorization, as continuous factors in multivariate modeling.
In summary, while there is frequently an impetus to assign cut-point(s) at an early investigational point to facilitate scientific or medical application, this assignment may be detrimental to progress if it is not robust over a broad range of data. Good work-ups of a continuous factor’s multivariate effects on outcome, with repeated testing of (externally) generated hypothesized cut-points will avoid confounding of apparent effect that may arise from cut-points applied inappropriately in some future contexts.