On Sun, 4 Mar 2001, Philip Cozzolino wrote in part:
> However, after the cubic non-significant finding, the 4th and 5th
> order trends are significant.
>
> Intuitively, it seems that if there is no cubic trend of significance,
> there will not be any higher order trend, but this is relatively new
> to me.
Your intuition is, in this case, incorrect. The five
trends are mutually independent in the sense that any combination of them
may be operating. (I am for the moment accepting the implied premise
that a power function of the IV is a reasonable function to try to fit to
your data. In most instances I know of, this is not "really" the case,
and the power function is more usefully thought of as an approximation
to whatever the "real" functionality is.) This may be seen by
considering the following relationships between Y and X (think of them as
DV and IV if you wish):
I. + * *
- * *
Y -
- * *
-
+ * *
-
- * *
- *
-
+---------+---------+---------+---------+---------+- X
II. + *
- * * *
-
Y - * * *
-
+ * * *
-
- * * *
-
- * *
+---------+---------+---------+---------+---------+- X
In I. above, the linear trend is approximately zero, and the quadratic
component of X accounts for nearly all the variation in Y. A "rule"
that claimed "If the linear trend is insignificant there can be no
significant quadratic trend" is clearly false in this case.
In II. above, both the linear and quadratic components of trend are
virtually zero -- certainly insignificant -- and the cubic component
accounts for nearly all the varition in Y. Similar situations can be
imagined, where only the quartic, or only the quintic, or only the
linear, quadratic, and quartic, or any other arbitrary combination of
the basic trends are significant, and other components are not.
If you are carrying out your trend analysis by using orthogonal
polynomials (as you probably should be), try constructing the model
derived from your linear + quadratic fit only, and plot those as
predicted values against X; then construct the model derived from linear
+ quadratic + quartic + quintic, and plot those predicted values against
X. You may find it illuminating also to plot the residuals in each case
against X, especially if you force the same vertical scale on the two
sets of residuals.
I note in passing that you haven't stated how much of the variance of Y
is accounted for by each of the significant components, nor how much
residual variance there is after each component is entered. That also
might be illuminating.
-- DFB.
----------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 (603) 535-2597
Department of Mathematics, Boston University [EMAIL PROTECTED]
111 Cummington Street, room 261, Boston, MA 02215 (617) 353-5288
184 Nashua Road, Bedford, NH 03110 (603) 471-7128
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