Fwd: asymptotic standard error of lambda

[Douglas Bonett]

---------- Forwarded message ----------
From: Douglas Bonett <address@hidden>
Date: Mon, May 5, 2014 at 10:31 PM
Subject: Re: asymptotic standard error of lambda
To: John Darrington <address@hidden>

For lambda C|R, its variance can be expressed as

(N – A)(A + B – 2C)/(N – B)^3

where N is the total sample size, B is the largest column total, A is the sum across rows of the largest frequencies within each row. The C term is the hardest to explain in words – it is the summation of the largest frequency in each row for only those rows where the largest row frequency is in the same column as the largest column total.  It is easier to show it in an example.

Here is a 2x3 table from Bishop, Fienberg & Holland (page 388):

 

       c1     c2       c3

r1    225     53      206

r2      3      1       12

      228     54      218

 

A = 225 + 12 = 237

B = 228

C = 225 (since 225 is the only row maximum that occurs in the first column)

VAR[lambda(R|C)] = (500 – 237)(237 + 228 – 2*225)/(500 – 228)^3 = .000196

ASE1 = sqrt(.000196) = .014

SPSS also gives ASE1 = .014

 

On Mon, May 5, 2014 at 1:24 PM, John Darrington <address@hidden> wrote:

On Mon, May 05, 2014 at 11:27:40AM -0700, Ben Pfaff wrote:
     On Mon, May 5, 2014 at 10:51 AM, John Darrington
     <address@hidden> wrote:
     > On Mon, May 05, 2014 at 08:09:16AM -0700, Ben Pfaff wrote:
     >      I'm sure there is an error in our implementation.  NaN is coming from
     >      the square root of a negative number, as you said.
     >
     >      I made another mistake below.  PSPP actually calculates ASE0 correctly
     >      for asymmetric lambda (lambda divided by ASE0 is what's displayed as
     >      "Approx. T", which matches that calculated by SPSS for asymmetric
     >      lambda).  It's ASE1, displayed as "Asymp. Std. Error", that PSPP gets
     >      wrong.
     >
     > Ahh. I was calculating ASE0.
     >
     > ASE1 like you say seems wierd and results in an imaginary number.  I can only imagine
     > that this is a mistake in the SPSS documentation.  Unfortunately I haven't been able
     > to find any other references on how to calculate this value.
     >
     > Another issue: if we have T, we should be able to calculate the significance.  We just
     > need to know the degrees of freedom.  I wonder how these are calculated?
     >
     > Unfortunately the litereature on these values seems to be scarce.

     https://v8doc.sas.com/sashtml/stat/chap28/sect20.htm has a different formula,
     but I don't understand how to interpret r_i|l_i = l.

The text below it says:
 Also, let li be the unique value of j such that ri=nij, and let l be the unique value of j such that r = n·j.

I interpret this to mean that r_i is summed for all i where the condition l_i == l is true.

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