The following formula for the tetrachoric correlation is from Glass &
Hopkins (1996, p. 137), full reference given below. They just give
the formula without citing any other works. I'm wondering if this
formula is still in use. Thanks.

YY*NN - YN*NY
r_tet = -----------------
ord(1)*ord(2)*N^2

where YY = count in the Yes-Yes cell
NN = count in the No-No cell
NY = count in the No-Yes cell
YN = count in the Yes-No cell
p(1) = p(YES) for variable 1
p(2) = p(YES) for variable 2
z(1) = z that cuts off p(1) of the area in tail of standard
normal dist.
z(2) = z that cuts off p(2) of the area in tail of standard
normal dist.
ord(1) = ordinate of the standard normal distribution at z(1)
ord(2) = ordinate of the standard normal distribution at z(2)

EXAMPLE

V2
V1 YES NO TOTAL
YES 110 10 120
NO 90 190 280
TOTAL 200 200 400

YY = 110, NN = 190, YN = 10, NY = 90, N = 400
p(1) = 120/400 = 0.3
p(2) = 200/400 = 0.5
z(1) = + or - 0.5244
z(2) = 0
ord(1) = 0.3477
ord(2) = 0.3989

110*190 - 10*90
r_tet = ------------------- = 0.901
0.3477*0.3989*400^2

Reference

Glass GV, Hopkins KD (1996). Statistical Methods in Education and
Psychology (3rd Ed.). Boston, MA: Allyn and Bacon. ISBN: 0-205-14212-5

--
Bruce Weaver
bwea... at (no spam) lakeheadu.cawww.angelfire.com/wv/bwhomedir
"When all else fails, RTFM."

On Jun 26, 10:33 am, Bruce Weaver <bwea... at (no spam) lakeheadu.ca> wrote:



The following formula for the tetrachoric correlation is from Glass &
Hopkins (1996, p. 137), full reference given below. They just give
the formula without citing any other works. I'm wondering if this
formula is still in use. Thanks.

YY*NN - YN*NY
r_tet = -----------------
ord(1)*ord(2)*N^2

where YY = count in the Yes-Yes cell
NN = count in the No-No cell
NY = count in the No-Yes cell
YN = count in the Yes-No cell
p(1) = p(YES) for variable 1
p(2) = p(YES) for variable 2
z(1) = z that cuts off p(1) of the area in tail of standard
normal dist.
z(2) = z that cuts off p(2) of the area in tail of standard
normal dist.
ord(1) = ordinate of the standard normal distribution at z(1)
ord(2) = ordinate of the standard normal distribution at z(2)

EXAMPLE

V2
V1 YES NO TOTAL
YES 110 10 120
NO 90 190 280
TOTAL 200 200 400

YY = 110, NN = 190, YN = 10, NY = 90, N = 400
p(1) = 120/400 = 0.3
p(2) = 200/400 = 0.5
z(1) = + or - 0.5244
z(2) = 0
ord(1) = 0.3477
ord(2) = 0.3989

110*190 - 10*90
r_tet = ------------------- = 0.901
0.3477*0.3989*400^2

Reference

Glass GV, Hopkins KD (1996). Statistical Methods in Education and
Psychology (3rd Ed.). Boston, MA: Allyn and Bacon. ISBN: 0-205-14212-5

--
Bruce Weaver
bwea... at (no spam) lakeheadu.cawww.angelfire.com/wv/bwhomedir
"When all else fails, RTFM."

I've never seen that particular approximation before. It's not one of
the six that Castellan (Psychometrika, 1966) reports on, and I thought
that since then just about everyone had abandoned approximations in
favor of exact answers (which in this case is .820).

On Jun 26, 6:13 pm, Ray Koopman <koop... at (no spam) sfu.ca> wrote:
On Jun 26, 10:33 am, Bruce Weaver <bwea... at (no spam) lakeheadu.ca> wrote:

The following formula for the tetrachoric correlation is from Glass &
Hopkins (1996, p. 137), full reference given below. They just give
the formula without citing any other works. I'm wondering if this
formula is still in use. Thanks.

YY*NN - YN*NY
r_tet = -----------------
ord(1)*ord(2)*N^2

where YY = count in the Yes-Yes cell
NN = count in the No-No cell
NY = count in the No-Yes cell
YN = count in the Yes-No cell
p(1) = p(YES) for variable 1
p(2) = p(YES) for variable 2
z(1) = z that cuts off p(1) of the area in tail of standard
normal dist.
z(2) = z that cuts off p(2) of the area in tail of standard
normal dist.
ord(1) = ordinate of the standard normal distribution at z(1)
ord(2) = ordinate of the standard normal distribution at z(2)

EXAMPLE

V2
V1 YES NO TOTAL
YES 110 10 120
NO 90 190 280
TOTAL 200 200 400

YY = 110, NN = 190, YN = 10, NY = 90, N = 400
p(1) = 120/400 = 0.3
p(2) = 200/400 = 0.5
z(1) = + or - 0.5244
z(2) = 0
ord(1) = 0.3477
ord(2) = 0.3989

110*190 - 10*90
r_tet = ------------------- = 0.901
0.3477*0.3989*400^2

Reference

Glass GV, Hopkins KD (1996). Statistical Methods in Education and
Psychology (3rd Ed.). Boston, MA: Allyn and Bacon. ISBN: 0-205-14212-5

--
Bruce Weaver
bwea... at (no spam) lakeheadu.cawww.angelfire.com/wv/bwhomedir
"When all else fails, RTFM."

I've never seen that particular approximation before. It's not one of
the six that Castellan (Psychometrika, 1966) reports on, and I thought
that since then just about everyone had abandoned approximations in
favor of exact answers (which in this case is .820).

Ray, the Glass & Hopkins equation I cited is identical to equation 6
in Castellan's paper, but Castellan calls the result "epsilon". And
if I follow, he then plugs that value of epsilon into equation 7, and
solves for the tetrachoric correlation.

How did you obtain the "exact" solution?

Thanks,
Bruce

--
Bruce Weaver
bwea... at (no spam) lakeheadu.cawww.angelfire.com/wv/bwhomedir
"When all else fails, RTFM."

On Jun 27, 7:16 am, Bruce Weaver <bwea... at (no spam) lakeheadu.ca> wrote:



Ray, the Glass & Hopkins equation I cited is identical to equation 6
in Castellan's paper, but Castellan calls the result "epsilon". And
if I follow, he then plugs that value of epsilon into equation 7, and
solves for the tetrachoric correlation.

But Glass & Hopkins didn't solve Castellan's (7) for rtet.
They just gave epsilon and called it rtet!

How did you obtain the "exact" solution?

The major (IMHO) contribution of Pearson's 1901 paper was to
show that pxy - px*py = integral[0...rtet: f(x,y,r) dr], where
pxy = the observed joint proportion of successes,
px & py are the observed marginal proportions,
x = F^(px), y = F^(py),
F^(p) is the univariate standard normal inverse cdf, and
f(x,y,r) is the bivariate standard normal density.

I solve that equation numerically (using Mathematica), with the
upper bound of integration as the unknown. There are some special
cases that simplify a little, and it turns out to be better to
change r -> sin(t) and integrate with respect to t to eliminate
the division by sqrt(1-r^2) in f, but otherwise it's relatively
straightforward number crunching.