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st: RE: Somers D
The choice between Somers' D and logistic regression depends whether you
wanted to know a difference between probabilities (which is what Somers'
D is) or to know about odds and odds ratios. If I am analysing a
case-control study, then I use logistic regression. This is because, in
a case-control study, the odds ratio is an estimate for the population
relative risk, and it is not possible to estimate absolute risks and
their differences, which we would have preferred to know, but which
would require a more expensive cohort study in order to estimate them.
However, if I want to measure the tendency of autistics to have a higher
ordinal disability score than non-autistics by comparing a sample of
autistics with a sample of non-autistics, then I want to know the
Somers' D of disability score with respect to autism, which is a
difference between the probability that a random autistic is more
disabled than a random non-autistic and the probability that a random
non-autistic is more disabled than a random autistic. I would consider
this difference between probabilities to be easier to explain than the
odds ratios in an ordinal logistic regression model.
I often use both Somers' D and logistic regression in an analysis,
because I use logistic regression to define a propensity score and use
this propensity score to stratify by propensity groups using the
wstrata() option. Newson (2006a) and Newson (2006b) gives an example of
this.
I hope this helps.
Best wishes
Roger
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: 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: 07 December 2006 15:56
To: [email protected]
Subject: st: Somers D
Thanks so much for pointing me in the Somers D collection of papers and
software!
The comments from statalist were very useful, especially as I had not
thought of using the wstrata option. I already had an age group
variable,
however it was not equally weighted as in your example. I had planned an
adjustment to logistic regression using the continuous variable age.
This
has the downside of assuming that age is linear (not usually a correct
assumption). Also, age group can be unequally distributed when created
manually. So this method of creating a new equal decile age group is
very
useful.
How does the use of Somers D compare to a logistic regression? Is it
always
better to use a rank, ordinal approach such as Somers D and tau-a when
trying to assess the significance of ordinal categorical variables? Or
is
it useful to also examine the logistic regressions (using xi to identify
ordinal/categorical independent variables)?
Regards,
Deidra
On 7/12/06 4:12 AM, "Newson, Roger B" <[email protected]>
> 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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