Page 102 - 20dynamics of cancer

PROGRESSION DYNAMICS 87

THE IMPORTANCE OF COMPARATIVE HYPOTHESES
A mathematical analysis for the age of cancer onset depends on sev-
eral parameters. Those parameters might include the number of stages
in progression, the somatic mutation rate that moves a tissue from one
stage to the next, the number of cells in the tissue, and the precancerous
rate of cell division. Given values for those parameters, the mathemati-
cal model generates an age-specific incidence curve.
A mathematical model may be used in two different ways: fit or com-
parison.
A fit chooses values for all parameters that minimize the distance be-
tween the predicted and observed age-specific incidence curves. A good
fit provides a close match between prediction and observation. A good
fit also uses realistic values for parameters such as rates of mutation
and cell division.
A comparison sets an explicit hypothesis: as a parameter changes,
the model predicts a particular direction of change for the age-specific
incidence curve. For example, an inherited mutation may reduce by one
the number of stages that must be passed during progression. Mathe-
matical models predict that fewer stages cause the incidence curve to
have a lower slope and to shift to earlier ages (higher intercept). I will
show data that support this comparative prediction.

FITTING
One can fit theory to observation, but the match usually arises be-
cause a model with several parameters creates a flexible manifold that
conforms to the data. Even when one constrains parameter estimates
to realistic values, an incorrect model with several parameters often has
great flexibility to conform to the shape of the data. A fit is achieved so
easily that such a model, fitting widely and well, actually explains very
little. As Dyson (2004) tells it:
In desperation I asked Fermi whether he was not impressed by
the agreement between our calculated numbers and his measured
numbers. He replied, “How many arbitrary parameters did you
use for your calculations?” I thought for a moment about our cut-
off procedures and said, “Four.” He said, “I remember my friend
Johnny von Neumann used to say, with four parameters I can fit
an elephant, and with five I can make him wiggle his trunk.” With
that, the conversation was over.