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Math Forum » Discussions » sci.math.* » sci.stat.math.independent

Topic: Interpretation of coefficients in multiple regressions which model
linear dependence on an IV
Replies: 146 Last Post: Dec 15, 2012 6:44 PM

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Ray Koopman

Posts: 3,345
Registered: 12/7/04

Re: Interpretation of coefficients in multiple regressions which
model linear dependence on an IV
Posted: Dec 1, 2012 4:46 PM

Plain Text

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On Dec 1, 11:00 am, Ray Koopman <koop...@sfu.ca> wrote:
> [...]
>
> There are two average slopes. Call them Av1 and Av2:
>
> dy/dx1 = a1 + a3*x2 ==> Av1 = a1 + a3*mean_x2
>
> dy/dx2 = a2 + a3*x1 ==> Av2 = a2 + a3*mean_x1
>
> To estimate the coefficients, give your regression program
> three predictors: x1, x2, and x3 = x1*x2. Then
>
> var[Av1] = var(a1) + 4*var(a3)*(mean_x2)^2 + 4*cov(a1,a3)*mean_x2,
>
> var[Av2] = var(a2) + 4*var(a3)*(mean_x1)^2 + 4*cov(a2,a3)*mean_x1,

Those are a good example of the sort of mistake it is easy to
make when cutting and pasting instead of typing from scratch.
The correct variances are

var[Av1] = var(a1) + var(a3)*(mean_x2)^2 + 2*cov(a1,a3)*mean_x2,

var[Av2] = var(a2) + var(a3)*(mean_x1)^2 + 2*cov(a2,a3)*mean_x1.

Also, there is a covariance that might be needed sometime:

cov[Av1,Av2] = cov[a1,a2] + var[a3]*mean_x1*mean_x2 +
cov[a1,a3]*mean_x1 + cov[a2,a3]*mean_x2.

Date

Subject

Author

11/21/12

Interpretation of coefficients in multiple regressions which model
linear dependence on an IV

Halitsky

11/21/12

The problematic regression is actually ln(c) on ( ln(u), ln(u^2) ),
not c on (u, u^2)

Halitsky

11/22/12

Re: The problematic regression is actually ln(c) on ( ln(u), ln(u^2)
), not c on (u, u^2)

Ray Koopman

11/22/12

Off-line Zip File with one Summ File and 12 Detl files for lnc on (lnu,(lnu)^2)

Halitsky

11/23/12

Re: Off-line Zip File with one Summ File and 12 Detl files for lnc on (lnu,(lnu)^2)

Ray Koopman

11/23/12

Re: Off-line Zip File with one Summ File and 12 Detl files for lnc on (lnu,(lnu)^2)

Halitsky

11/23/12

Complete "a1_N_1_S" zipfile with results from all 3 new regressions

Halitsky

11/24/12

Re: Complete "a1_N_1_S" zipfile with results from all 3 new regressions

Ray Koopman

11/24/12

Re: Complete "a1_N_1_S" zipfile with results from all 3 new regressions

Halitsky

11/24/12

You now have N_1_S, N_2_S, and N_3_S files for all folds

Halitsky

11/25/12

As per your suggestion in the other thread, scaled e on scaled u, c, L

Halitsky

11/26/12

Re: As per your suggestion in the other thread, scaled e on scaled u,
c, L

Ray Koopman

11/26/12

Re: Interpretation of coefficients in multiple regressions which
model linear dependence on an IV

Ray Koopman

11/26/12

Them there is some neat algebraic mechanics !

Halitsky

11/27/12

Re: Them there is some neat algebraic mechanics !

Ray Koopman

11/27/12

OK – I think I’m set, at least till we get to c
on (e, u, u*e).

Halitsky

11/27/12

Re: OK – I think I’m set, at least till we get t
o c on (e, u, u*e).

Ray Koopman

11/28/12

Re: OK – I think I’m set, at least till we get t
o c on (e, u, u*e).

Ray Koopman

11/28/12

Thanks for your review of Tables I/II from previous analysis

Halitsky

11/27/12

Holy Cow! Look at your "average a1" slope regressed on Len Int

Halitsky

11/27/12

Re: Holy Cow! Look at your "average a1" slope regressed on Len Int

Ray Koopman

11/27/12

Re: Holy Cow! Look at your "average a1" slope regressed on Len Int

Halitsky

11/27/12

Re: Holy Cow! Look at your "average a1" slope regressed on Len Int

Ray Koopman

11/27/12

Here's how I did logs ...

Halitsky

11/27/12

Please note that $u = u in last post (the $ prefix is from PERL - sorry).

Halitsky

11/27/12

Re: Here's how I did logs ...

Ray Koopman

11/28/12

Average slopes and means of u' for c on (u',u'^2) WITHOUT logs

Halitsky

11/28/12

Results (!!) on average slopes and means for a1_N_1_C (complement
instead of core subset)

Halitsky

11/28/12

Re: Results (!!) on average slopes and means for a1_N_1_C (complement
instead of core subset)

Ray Koopman

11/28/12

Finally! Pay-off for all that work I did with the "A" matrix returned
by Ivor Welch's module!

Halitsky

11/29/12

Average Slope SEs for a1_N_1_S and a1_N_1_C (and some questions
regarding them ...)

Halitsky

11/30/12

Re: Average Slope SEs for a1_N_1_S and a1_N_1_C (and some questions
regarding them ...)

Ray Koopman

12/2/12

Re: Average Slope SEs for a1_N_1_S and a1_N_1_C (and some questions
regarding them ...)

Ray Koopman

12/2/12

Re: Average Slope SEs for a1_N_1_S and a1_N_1_C (and some questions
regarding them ...)

Ray Koopman

12/2/12

Glad you brought up “singleton” length intervals
... been thinkin’ on ‘em also ...

Halitsky

12/2/12

Re: Glad you brought up “singleton” length inter
vals ... been thinkin’ on ‘em also ...

Ray Koopman

12/2/12

It's still 24...124 - don't know why I bothered to say "roughly
25...125" instead of "exactly "24...124"

Halitsky

12/2/12

You should probably clear your data deck and start fresh with the two
csv's I just mentioned in the last email

Halitsky

12/2/12

Re: Glad you brought up “singleton” length inter
vals ... been thinkin’ on ‘em also ...

Halitsky

12/2/12

One last thought: definitions for the third regression (will save a
complete re-run if I incorporate them now) ...

Halitsky

12/3/12

Number of Bonferroni entries for each singleton length is still 72 (duh!)

Halitsky

11/30/12

En passant question: What if a plot of slope CI’s
is lousy, but splits the “m’s” perfectly?

Halitsky

11/30/12

Re: En passant question: What if a plot of slope CI
’s is lousy, but splits the “m’s” perfectly?

Ray Koopman

12/1/12

I’m glad the perfect m split legitimately suggests
a subset effect; here’s why.

Halitsky

12/1/12

Re: I’m glad the perfect m split legitimately sugg
ests a subset effect; here’s why.

Ray Koopman

12/1/12

Re: I’m glad the perfect m split legitimately sugg
ests a subset effect; here’s why.

Halitsky

12/1/12

Slope and intercept for R'uq in the above example ...

Halitsky

12/1/12

Bonferroni tables for p’s from new 2-ways for Auq
per fold and length interval

Halitsky

12/1/12

Nope! 24-entry Bonferroni tables for (a1,a3) and (b1,b47) do NOT
improve results for a3 nor b47

Halitsky

12/5/12

I'm VERY glad you'll know how to answer this "perms and combs"
question !

Halitsky

12/5/12

“L-H Het” Table for Average Slopes Auq, Aubu, Au
bqu

Halitsky

12/5/12

In "L-H Het table", L-H Het for N1 Aubu should be 4, NOT 2

Halitsky

12/5/12

Holy Moly, were you right about covariances for Rub and Rubq !!!!

Halitsky

12/5/12

Re: Holy Moly, were you right about covariances for Rub and Rubq !!!!

Ray Koopman

12/6/12

So do we need to "Bonferroni-correct" in this case

Halitsky

12/7/12

Re: So do we need to "Bonferroni-correct" in this case

Ray Koopman

12/7/12

Response to your last of 12/7 at 12:17am

Halitsky

12/7/12

Re: Response to your last of 12/7 at 12:17am

Ray Koopman

12/7/12

Thanks for the guidance on how to evaluate the contribution of u^2 in
the second model.

Halitsky

12/7/12

Please ignore my first question about "estimated standard errpr" in
my last post !!!! Sorry !

Halitsky

12/7/12

The u^2 coefficient in c on (e,u,u*e,u^2) does NOT distinguish among
the four subset x MoSS roll-ups

Halitsky

12/7/12

Sorry! Those were the SE's in my last post, not the t's !

Halitsky

12/7/12

SE's and p's for four subset x MoSS roll-ups of u*e coefficient in c
= (u,e,u*e)

Halitsky

12/7/12

Re: SE's and p's for four subset x MoSS roll-ups of u*e coefficient
in c = (u,e,u*e)

Ray Koopman

12/7/12

I'm sorry Ray - excitement (probably unwarranted) has disconnected my
brain from my fingers ...

Halitsky

12/7/12

Must we say S,N instead of N,S if we've said "Subset x MoSS" (not
MoSS x Subset) ???

Halitsky

12/7/12

Re: Must we say S,N instead of N,S if we've said "Subset x MoSS" (not
MoSS x Subset) ???

Ray Koopman

12/7/12

Response to your last

Halitsky

12/8/12

Re: Response to your last

Ray Koopman

12/8/12

Re: Response to your last

Ray Koopman

12/8/12

I think I understand; if so, then here’s what I ex
pect you’ll agree I should do next

Halitsky

12/9/12

Thanks so much for the sample picture you sent off-line

Halitsky

12/8/12

One other thing - because we're using "c-average", not "c-simple",
"c" is no longer a pure count

Halitsky

12/8/12

One other possibly worthwhile observation regarding the term u*e in
the regression c on (e,u,u^e,u^2)

Halitsky

12/8/12

Typo's of u^e for u*e in previous post.

Halitsky

12/9/12

Could I impose on you for four more ordered p “ref
erence plots”?

Halitsky

12/9/12

Have sent off-line a PDF of the four plots themselves graphed all together.

gimpeltf@hotmail.com

12/9/12

I'm getting the hang of the plotting now - see PDF SNa1_1_for_Rubq
sent offline

Halitsky

12/9/12

Am resending the last PDF sent off-line, since I've now learned how
to highlight the line of interest against the random backdrop.

Halitsky

12/10/12

Re: Am resending the last PDF sent off-line, since I've now learned
how to highlight the line of interest against the random backdrop.

Ray Koopman

12/10/12

1) Just u*e and u^2(!!); 2) IOTs vs “proper” tes
ts

Halitsky

12/10/12

Re: 1) Just u*e and u^2(!!); 2) IOTs vs “proper”
tests

Ray Koopman

12/10/12

Response to your last re Q and p

Halitsky

12/10/12

Sorry! I meant set=2, not set =1 in last post ...

Halitsky

12/11/12

Re: Response to your last re Q and p

Ray Koopman

12/11/12

1) yes - I am using abs(t); 2) subtraction from 1

Halitsky

12/10/12

Results of p's obtained by referring Q’s to the ch
i-square distribution.

Halitsky

12/11/12

Correction to harmless "thought-typo" in last post

Halitsky

12/11/12

Another way to bring the other folds in might be via investigation of
your average slopes and covar vis a vis "hetness"

Halitsky

12/11/12

Re: Results of p's obtained by referring Q’s to th
e chi-square distribution.

Ray Koopman

12/11/12

OK then, how ‘bout “hetness”? Are you amenabl
e to its further investigation?

Halitsky

12/12/12

Re: OK then, how ‘bout “hetness”? Are you amen
able to its further investigation?

Ray Koopman

12/12/12

I need to correct an apparent miscommunication regar
ding derivation of het H’s and L’s

Halitsky

12/13/12

Re: I need to correct an apparent miscommunication r
egarding derivation of het H’s and L’s

Ray Koopman

12/13/12

The SE's are in the zipped files but here they are for your
convenience ....

Halitsky

12/13/12

Re: The SE's are in the zipped files but here they are for your
convenience ....

Ray Koopman

12/13/12

Re your question about "linearity of SE’s in lengt
h"

Halitsky

12/14/12

Re: Re your question about "linearity of SE’s in l
ength"

Ray Koopman

12/14/12

Your question re features of (L,Aubqe) plots

Halitsky

12/13/12

I think I may have found something relevant to Aubqe
“het-ness” and heteroscedasticity

Halitsky

12/13/12

Re: I think I may have found something relevant to A
ubqe “het-ness” and heteroscedasticity

Ray Koopman

12/14/12

Re your questions about the plots sent off-line (and the underlying
data posted here 12/13 at 10:33am)

Halitsky

12/14/12

Re: Re your questions about the plots sent off-line (and the
underlying data posted here 12/13 at 10:33am)

Ray Koopman

12/14/12

Thanks for the terminological/methodological corrections, and also
for the ref to gnuplot.

Halitsky

12/14/12

Re: Thanks for the terminological/methodological corrections, and
also for the ref to gnuplot.

Ray Koopman

12/14/12

Response to your last of 12/14 at 227pm re terminology and methodology.

Halitsky

12/14/12

Re linearity of the Axxxx SE plots – hold on to yo
ur hat

Halitsky

12/14/12

Re: Re linearity of the Axxxx SE plots – hold on t
o your hat

Ray Koopman

12/14/12

Thanks for doing those two plots - yes - we agree on what we're seeing

Halitsky

12/14/12

Re: Thanks for doing those two plots - yes - we agree on what we're seeing

Ray Koopman

12/15/12

Re: Thanks for doing those two plots - yes - we agree on what we're seeing

Ray Koopman

12/15/12

Re plot of SEP against L

Halitsky

12/15/12

Effect of multiplying SE by sqrt(N), as per your post of 12/14 at 10:34pm

Halitsky

12/15/12

Re: Effect of multiplying SE by sqrt(N), as per your post of 12/14 at 10:34pm

Ray Koopman

12/14/12

One other general question regarding scaling to [0,1].

Halitsky

12/14/12

Re: One other general question regarding scaling to [0,1].

Ray Koopman

12/14/12

Sorry - I will be typographically more careful re Aubqe in the future.

Halitsky

12/1/12

Re: Interpretation of coefficients in multiple regressions which
model linear dependence on an IV

Ray Koopman

12/1/12

Thanks for elucidation of 2nd new regression.

Halitsky

12/1/12

Re: Interpretation of coefficients in multiple regressions which
model linear dependence on an IV

Ray Koopman

12/1/12

Roger corrected defs; also, will add new cov, just in case it's
needed later

Halitsky

12/2/12

Re: Interpretation of coefficients in multiple regressions which
model linear dependence on an IV

Ray Koopman

12/2/12

1) thanks for the 3rd regression defs; 2) Yes - I see why the terms
aren't "symmetrical" in this case.

Halitsky

12/3/12

New copies of a1_N_1_C and a1_N_1_S with data for all three
regressions at each singleton length.

Halitsky

12/3/12

Since 3rd regression computation needs df = 5, am requiring 15
observations for any given length singleton in any cell

Halitsky

12/3/12

Have sent off-line all N_1 regression coefficient files and master N
per length index file for N1

Halitsky

12/3/12

Same as above post for f_N_2_ss

Halitsky

12/3/12

Same as above post for f_N_3_ss

Halitsky

12/3/12

Same as above post for f_R_1_ss

Halitsky

12/3/12

Same as above post for f_R_2_ss

Halitsky

12/3/12

Same as above post for f_R_3_ss

Halitsky

12/3/12

Re: Since 3rd regression computation needs df = 5, am requiring 15
observations for any given length singleton in any cell

Ray Koopman

12/4/12

Sparseness of b1 data ...

Halitsky

12/4/12

I realized I should clarify my 4-way b1 match table: it's AFTER
subtracting df of 3

Halitsky

12/4/12

Re: I realized I should clarify my 4-way b1 match table: it's AFTER
subtracting df of 3

Ray Koopman

12/4/12

No - the counts in the files themselves are all OK.

Halitsky

12/4/12

Re: Sparseness of b1 data ...

Ray Koopman

12/5/12

We cross posted, so I just saw your revised "counts" table after I
made my last two posts ...

Halitsky

12/4/12

Let me know if you're ready for some interesting data, or if you're
too busy analyzing

Halitsky

12/4/12

Re: Let me know if you're ready for some interesting data, or if
you're too busy analyzing

Ray Koopman

12/4/12

Please evaluate this "yield" table of method/subset avg slope 2-ways
per fold and len with p < .05

Halitsky

12/5/12

One other question about using Auq avg slope as a constant when
computing the other two regressions

Halitsky

12/5/12

Re: One other question about using Auq avg slope as a constant when
computing the other two regressions

Ray Koopman

12/5/12

Re: One other question about using Auq avg slope as a constant when
computing the other two regressions

Halitsky

12/4/12

Some of your counts apparently ARE off.

Halitsky

12/4/12

Sorry! those counts in my last post were for len 63 in b1 (forgot to
tell you the length!!!!)

Halitsky

12/4/12

Re: Since 3rd regression computation needs df = 5, am requiring 15
observations for any given length singleton in any cell

Ray Koopman

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