Splines and Knots, again

[Bob Schacht-3]

I'm trying to get my mind around splines and knots, having read the helpful
reference provided by Anita Kooij

>More detail and graphical illustrations:
>
>page 16-22 of chapter 2 at
>https://openaccess.leidenuniv.nl/dspace/handle/1887/12386 (CATPCA) . . .


I thought I understood what knots were: I have many questions scored on a
Likert scale from 1-5, which looked to me like 4 knots, no?

On my CATPCA Analysis command, for each of these variables, I specify only
LEVEL = SPORD, leaving the default Degree = 2. But from this, CATPCA
decided that
>a. Optimal Scaling Level: Spline Ordinal (Degree 2, Interior Knots 2).

The Online help explains "INKNOT" as "The number of interior knots. The
minimum is 0, and the maximum is the number of categories of the variable
minus 2.

The number of Categories is 5, so ISTM INKNOT should be 5-2=3, right? So
why does CATPCA think that INKNOT should be 2?

And why doesn't CATPCA provide "rotated" factors? Or is that what the
"Component loadings" are?

Thanks in advance,
Bob Schacht

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[Kooij, A.J. van der]

Answers below.
 
>I thought I understood what knots were: I have many questions scored on a
>Likert scale from 1-5, which looked to me like 4 knots, no?

A spline always has 2 boundary knots, in your case on category 1 and on 5. So the maximum number of interior knots that can be choosen is number of categories minus 2. The interior knots divide the range of the variable into intervals. Separate transformation curves (piecewise polynomials) are fitted within the intervals. These separate curves join togehter at the interior knots. The degree of the spline determines the smoothness of the joining. The number of interior knots defines the number of intervals (pieces) and thus the smoothness of the total transformation curve.


>On my CATPCA Analysis command, for each of these variables, I specify only
>LEVEL = SPORD, leaving the default Degree = 2. But from this, CATPCA
>decided that
>a. Optimal Scaling Level: Spline Ordinal (Degree 2, Interior Knots 2).
>The Online help explains "INKNOT" as "The number of interior knots. The
>minimum is 0, and the maximum is the number of categories of the variable
>minus 2.
>The number of Categories is 5, so ISTM INKNOT should be 5-2=3, right? So
>why does CATPCA think that INKNOT should be 2?

Because you have specified the default of 2. The maximum that you can specify is 3. With 5 categories you can specify 1, 2, or 3 interior knots.


>And why doesn't CATPCA provide "rotated" factors? Or is that what the
>"Component loadings" are?

Yes, but unrotated. If you want rotation, save the transformed variables and use as input to FACTOR with PCA extraxtion. The results of FACTOR on the transformed variables is the same as CATPCA results, but with FACTOR you can choose a rotation option.

Thanks in advance,
Bob Schacht

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