In reply to Query 2, if I want to know whether mobility or age predicts
autism better (in the same direction), then I would usually compare the
2 Somers' D parameters, rather than the 2 Kendall's tau-a parameters.
The difference will be in the same firection. However, I think that
Somers' D is a better predictor performance indicator, because in this
case it is expressed on a scale from -1 for the best possible negative
predictor of autism to +1 for the best possible positive predictor of
autism, given the number of pairs of subjects whose autism level is
equal.
However, in reply to Query 1, the parameter measured by lincom is not an
age-adjusted Somers' D. It is a difference between 2 Somers' D
parameters. If you want an age-adjusted Somers' D of mobility with
respect to autism, then group age into groups and estimate a
within-strata Somers' D of mobility with respect to autism, using the
wstrata() option of somersd. For instance, to group age into 10 deciles,
you might type
xtile agegp=age, nquantiles(10)
tabulate agegp autism
somersd autism mobility, transf(z) tdist wstrata(age)
and somersd will give you a confidence interval for Somers' D of
mobility with respect to autism, restricted to comparisons within the
same age group. This Somers' D is equal to the difference between the
probability that a randomly-chosen autistic has a higher mobility than a
randomly-chosen non-autistic and the probability that a randomly-chosen
non-autistic has a higher mobility than a randomly-chosen autistic,
assuming that the autistic and the non-autistic are in the same age
group. (The tabulate command checks that each age group contains both
autistics and non-autistics.)
The wstrata() option can also use strata defined by multiple grouping
variables, eg age group and gender. The groups may defined either using
one variable or using a propensity score defined using multiple
variables. Examples are discussed in the manuals somersd.pdf and
censlope.pdf (distributed with the somersd package), and Newson (2006a),
Newson (2006b) and Newson (2006c), drafts of which can be downloaded
from my website if unfortunately you do not have access to The Stata
Journal (which of course everybody should subscribe to).
References
Newson R. 2006a. On the central role of Somers' D. Presented at the 12th
UK Stata User Meeting, 11-12 September, 2006.
Newson R. 2006b. Confidence intervals for rank statistics: Somers' D and
extensions. The Stata Journal 6(3): 309-334.
Newson R. 2006c. Confidence intervals for rank statistics: Percentile
slopes, differences, and ratios. The Stata Journal 6(4): 497-520.
Roger Newson
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton campus
Room 33, Emmanuel Kaye Building
1B Manresa Road
London SW3 6LR
UNITED KINGDOM
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: [email protected]
www.imperial.ac.uk/nhli/r.newson/
Opinions expressed are those of the author, not of the institution.
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Deidra Young
Sent: 06 December 2006 16:31
To: [email protected]
Subject: Re: st: probability and z-statistic
Dear Roger,
1. If I wanted to conduct the somersd test and adjust for age, do I
then
need to include a lincom command which provides the age adjusted
confidence
interval (see below)? [I referred to p. 10 of params.pdf]
2. In this case, is it more appropriate to chose somersd, rather than
tau-a
as the test of association (for an ordered categorical ?
. somersd autism mobility age, tr(z) tdist
. lincom (mobility-age)/2
Results:
. somersd autism mobility age, tr(z) tdist
Somers' D with variable: autism
Transformation: Fisher's z
Valid observations: 305
Degrees of freedom: 304
Symmetric 95% CI for transformed Somers' D
------------------------------------------------------------------------
----
| Jackknife
autism | Coef. Std. Err. t P>|t| [95% Conf.
Interval]
-------------+----------------------------------------------------------
----
mobility | -.3910524 .0899135 -4.35 0.000 -.5679839 -.2141209
Age | .0271106 .0782949 0.35 0.729 -.1269579 .1811791
------------------------------------------------------------------------
----
Asymmetric 95% CI for untransformed Somers' D
Somers_D Minimum Maximum
Mobility -.37226715 -.51387713 -.21090749
age .02710392 -.12628019 .1792223
. lincom (mobility - age)/2
(1) .5 mobility - .5 age = 0
------------------------------------------------------------------------
----
autism | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------
----
(1) | -.2090815 .0567701 -3.68 0.000 -.3207935 -.0973694
------------------------------------------------------------------------
----
Regards,
Deidra
On 6/12/06 9:04 PM, "Newson, Roger B" <[email protected]>
> Briefly, Kendall's tau-a is the difference between the probability of
> concordance and discordance, where "concordance" is the event that the
> larger of 2 X-values is associated with the larger of the 2
> corresponding Y-values, and discordance is the event that the larger
> X-value is associated with the smaller Y-value. Somers' D of Y with
> respect to X, denoted D_YX, is the difference between the 2
> corresponding CONDITIONAL probabilities, assuming that the 2 X-values
> are ordered (instead of being tied). And Kendall's tau-b is the
quantity
>
> taub_XY = sign(taua_XY) * sqrt(D_XY * D_YX)
>
> or (in other words) the common sign of D_XY and D_YX multiplied by the
> geometric mean of their 2 absolute values.
>
> I personally am more keen on having confidence intervals for Somers' D
> and Kendall's tau-a because they can be interpreted in words as
> differences between probabilities, which you cannot do with Kendall's
> tau-b. There are a large number of downloadable articles,
> pre-publication drafts and presentations on my website (see my
signature
> below) about the "Kendall family" of rank parameters, which are all
> defined in terms of Kendall's tau-a and also include median slopes,
> differences and ratios.
>
> I hope this helps.
>
> Best wishes
>
> Roger
>
>
> Roger Newson
> Lecturer in Medical Statistics
> Respiratory Epidemiology and Public Health Group
> National Heart and Lung Institute
> Imperial College London
> Royal Brompton campus
> Room 33, Emmanuel Kaye Building
> 1B Manresa Road
> London SW3 6LR
> UNITED KINGDOM
> Tel: +44 (0)20 7352 8121 ext 3381
> Fax: +44 (0)20 7351 8322
> Email: [email protected]
> www.imperial.ac.uk/nhli/r.newson/
>
> Opinions expressed are those of the author, not of the institution.
>
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]] On Behalf Of Joseph
> Coveney
> Sent: 06 December 2006 05:18
> To: Statalist
> Subject: Re: st: probability and z-statistic
>
> Deidra Young wrote:
>
> Hi Roger,
>
> Only one thing I don't quite follow...
> The tab command will produce Kendall's tau-b and approximate SE.
> However,
> Sommers' D produces tau-a only.
>
> How do tau-a and tau-b differ?
>
>
------------------------------------------------------------------------
> --------
>
> You'll find a description of each on Roger's website. Take a look at
> Section 2 on Page 2 of
>
> www.imperial.ac.uk/nhli/r.newson/papers/params.pdf
>
> Joseph Coveney
>
>
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