Gottfried said:
>
>Here you focus the crux of the normal-distributed variables.
>If there is a reality with n1 and n2 as normal distributed causes and
>y as effect like y<-n1+n2, you have measured standardized z1 and z2 (not
knowing
>which represents y and which represents n1 of the model)
>then you will find about 5% cases with values >2 - in both items.
>Now how will you resample: for uniformity of x1 or uniformity of x2?
Bill responds:
First, when we use least square methods the estimates are driven by the
frequency of certain patterns in the data. Part of the pattern we seek in
corresponding regressions concerns the values of x1 and x2 are the extremes
of y, We look for correlation(and the polarization effect), since two
extremes of x1 and x2 lead to higher extremes of y, But these two extremes
are very rare because our design fails to sample at the extremes, then we
are simply loading the research against the detection of polarization, This
is an artifact,
If I am interested in discovering causes I will do as the experimentalists
do when they seek to isolate and control their measurements, Because I was
trainned as an experimental personality psychologist, this seems a natural
thing to do, It would also be the best strategy to a chemist or physicist,
Experimental scientists seek to understand causal connections in their
simplicity and then build up the complex model, Applied social scientists
tend to mistrust such systematic research and prefer to constrain their
modeling to the joint causes expressed in a particular sample of numerous
variables, But because the circumstances present myriad dimensional
possible interactions, we end up with the ambiguity that SEM researchers
face, The same data will support many models,
I would hypothesize that variables A and B are causes of C and then collect
a uniformly distributed array of both A and B, or better yet, do a factorial
sampling in which all levels of A and B are crossed with equal frequency per
cell, This is what I call a manifold in my papers, I would then see if A
and B cause C, using corresponding regressions or correlations, Thus I am
interested in the simple statement do A and B cause C. I am not
hypothesizing the complex nonlinear functions that are entailed by the
assumption that A and B are normally distributed,
>As the direction-coefficient depends much on the (rare) extremes, I expect
>the coefficient will show that variable z1 or z2 as independent, which
>you sample uniformly. In which of z1 or z2 you assume you were missing
>some values, if you have a normal distributed sample in z1 and z2?
>I have not tested this procedure, but I may modify a graph, that I
>have shown last year (with uniform causes), for normal variables.
Bill responded:
I am not sure what you are saying, But if you are suggesting that CR will
say the cause is which ever variable is sampled uniformly then you are
wrong, It is not the shape of the distribution that defines the cause, A
uniform distribution is a prerequisite for uniform cause, But uniformity
does not imply causation, Polarization of the correlations between x1 and
x2 across the ranges of the dependent variable (y) implies causation,
>
>Let z1 and z2 be instances for n1 and y in our sample, not knowing
>which z represents y. We will get a bivariate distribution like the
>following, where rare combinations are indicated with the period,
>medium frequent combinations with the star and most frequent
>combinations with a double-cross:
>
> .!.
> *!* * .
> * #!# * * .
> ---.-*-#-#!# # *--------- z1
> * * #!# *
> * *!* .
> .! z2
>
>You can produce a direction coefficient showing any variable as dependent
>just by resampling and weighting some cases.
>
>a)
>For instance to have uniformity in z2 and triangularity in z1 weight all
>cases to the frequency of the edges in 30 deg and 210 deg (extreme values
>of z1)
>
> !
> *!* * * * *
> * *!* * * *
> -----*-*-*!* * *--------- z1
> * * * *!* *
> * * * * *!*
> ! z2
>
>This shows z1 dependent (more complex distribution: triangular) and assumes
>a third (latent) cause z0, which can be retrieved as residual of a
>regression.
Bill responds, NO. No. No. It is not the triangularity as such that implies
the complexity of a dependent variable, It is the polarization of the causes
across the ranges of the dependent variable that implies greater complexity
in y, This is because y contains x1 and x2, not because y is triangular,
True y is triangular, but this another issue, This is why CR works even if
we feed it the uniform ranks of all the data AFTER the model is first
generated using uniform causes.
Gottfried continues:
>
>b)
>For instance to have uniformity in z1 and triangularity in z2 weight all
>cases to the frequency of the edges in 60 deg and 240 deg (extreme values
>of z2)
>
>
> ! *
> ! * *
> !* * *
> *!* * *
> * *!* * *
> -----*-*-*!* * *--------- z1
> * * *!* *
> * * *!*
> * * *! z2
> * *
> *
>This shows z2 dependent (more complex distribution: triangular) and assumes
>a third (latent) cause z0, which can be retrieved as residual of a
>regression.
>
>
>c)
>If you weight cases to have uniformity both in z1 and z2, then weight all
>cases to the frequency of the edges in 45,135,215,305 deg (extreme values
>of z1 AND z2),
>
> !
> * * *!* * *
> * * *!* * *
> * * *!* * *
> -----*-*-*!* * *--------- z1
> * * *!* * *
> * * *!* * *
> ! z2
>
>Here you will detect no causal relation between z1 and z2.
>
>These are sketches of ideal data. If errors are there we will randomly
>find some cases with extreme values in z1 as well as in z2. I expect
>a serious problem, how to decide which variable, z1 or z2, shall be
>brought to uniformity just by sampling. (This is all expectation, not
>really tested yet).
>
Bill responded:
Gottfried, I think you are still thinking that it is the shape of the
distribution that drives the CR inference, It is not the shape, though there
are shapes that are necessary to bring forth the logical relationships in
uniform causal patterns, The polarization is based on asymmetrical logical
relationships, This is why CR works even if we only send it ranks of the
causal data. But if we arbitrarily multiply the extremes of x1 and x2 AFTER
the causation (based on normal distributions) then the cause will still be
nonlinear, being based on the disjunctive (either x1 or x2 but not both).
Blowing up the extremes AFTER may make x1 and x2 appear to be uniform, but
during the causal process there is still very few cases or real extremes of
x1 and x2 combining, Without those extreme combinations, then the
polarization effect does not occur, regardless of how we might post hoc
transform the variables,
The up shot is that if we have a lot of data and want to pull the full
manifold of x1 and x2, including an equal number of cases in which x1 and
x2 are BOTH extreme together, then we are going to have to sample very large
data sets to being with, It would perhaps be smarter just to go out and
look for a set number of cases at each combination rather than using a
dragnet. This is what the chemist and physicist do, They do not sample
randomly from nature to find rare elements, They know where to look and
build their design intelligently, It is somewhat Romantic nonsense to
believe we can only study causes by sampling what nature throws at us as in
clumps,
Bill
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