On Fri, 30 Jun 2000, dennis roberts wrote:
> interesting but ... 3 questions:
>
> 1. how can the r squared for the best model be 100% when, the errors
> are not all 0s?
R-sq is not 100% exactly, it is reported as 100.0%.
Examining the SS reported shows that R-sq = 3361.7/3361.9 = 99.994%,
less than 100.000% but equal to 100.0% (to one decimal place).
> 2. we are talking about a model that goes from an r squared of 99.5%
> ... to (nearly) 100% ... is this important?
In the second step, adding the quadratic term accounts for 90% (well,
89.67%) of the residual variance from the linear model. In the third
step, adding the cubic term accounts for about 90% (89.47%) of the
residual variance from the quadratic model. Is it important to be able
to account for 90% of the remaining variance by a single predictor?
> 3. while there is a dinky gain in r squared ... it comes in relation to
> using a scale that is not as understandable as age ... it is the square
> of age ... or the cube of age ... is this gain worth the transposition
> of a scale in 1 year increments ... to something like squares or cubes
> of age increments?
I'm not altogether sure that I would characterize the explaining of 99%
(98.91%, if one can believe 4 digits' precision) of the residual variance
as "a dinky gain".
I do not follow the "not understandable as age" part. The fitted
function gives an explicit estimate of height (in cm) as a function of
age (in, presumably, years). To one decimal place, these estimates are:
yrs 2 3 4 5 6 7 8 9 10 11
cm 86.8 95.4 103.1 110.2 116.7 122.7 128.5 134.2 139.8 145.6
delta 8.6 7.7 7.1 6.5 6.0 5.8 5.7 5.6 5.8
The annual increment in predicted height is shown in the line labelled
"delta". We see that the fitted RATE of growth diminishes from the
initial value, levels off about age 9 or 10, and increases a little in
the last year for which data are supplied. This, one supposes, is a part
of what the residual plots were trying to tell us. (Something like this
will have been visible in the raw data as well, but I neglected to copy
that information to a place where I can easily retrieve it right now.)
> i would say that in this case ... using a much more complicated model .
> does not add to the clarity of the prediction problem ...
I wouldn't call it _much_ more complicated. Agreed, with a linear
approximation, the annual increments are constant (6.37 cm per year);
and agreed, this is simpler than increments that change from year to
year, from a value 20% higher in the first year to values about 10%
lower in the last four years. On the other hand, it is hard to believe
that a constant increment per year adequately describes the true
relationship between age and height for human female children.
Just one more example of the eternal trade-offs between "reality" and
spurious simplicity.
-- Don.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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