# st: Hierarchical ordinal logistic regression model for diagnosticmeta-analysis using gllamm

 From "Ben Dwamena" <[email protected]> To <[email protected]> Subject st: Hierarchical ordinal logistic regression model for diagnosticmeta-analysis using gllamm Date Thu, 02 Dec 2004 12:26:18 -0500

```Described below is a multilevel model basd on ordinal regression for
diagnostic meta-analysis (SROC) for which codes are available for SAS
and WinBUGS and wanted to now how this may be modeled using gllamm?

I know how  to model the expression logit n=thetai+alphai*disij
However, I am not sure how to include the scale parameter so that the
above is multiplied by  exp(betai*disij ) .

HSROC MODEL
LEVEL 1
For each study (i), the number testing positive is assumed to follow a
binomial distribution
yij ~B(nij,, alphaij)

where    j=1 represents diseased group; j=2 represents non-diseased
group; nij  represents the number in group; nij  represents the
probability of a positive test     result in group j

The model is based on the ordinal logistic regression proposed by
McCullagh and takes the form:  logit (nij) =(thetai+alphai*disij)*
exp(-beta*disij)
where disij  represents the "true" disease status (coded as -0.5
for the non-diseased and 0.5 for the diseased).
thetai (threshold parameter) and  alphaI (accuracy measure) are modeled
as random effects while beta(modeling dependence of accuracy on
threshold) is a fixed effect.

When beta= 0, the model reduces to a logistic regression model and
thetai is estimated by (logit(tpri) + logit(fpri))/2 ( = Si/2)
alphai is estimated by logit (tpri) -logit (fpri) ( = Di)

Study level covariates may be added to explore associations with
threshold and/or accuracy and/or SROC shape

LEVEL 2
The random effects are assumed to be independent and normally
distributed:

thetai ~ N(omega, tau-squared ); alphai ~ N(lamda, tau-squared )

The SROC curve is computed using
E (tpr) = invlogit [logit (fpr) exp (- beta+ lamda* exp (-0.5 lamda]
for chosen values of fpr

When beta= 0, theta provides a global estimate of the expected test
accuracy (lnDOR) and the resulting SROC is symmetric.

The expected tpr is given by 1/[1+exp(-(omega+0.5*lamda)*exp(-
0.5*beta))]
The expected fpr is given by 1/[1+exp(-(omega-0.5*lamda)*exp(-
0.5*beta))]

How may this be modeled using gllamm?
I know how  to model the expression logit n=thetai+alphai*disij
However, I am not sure how to include the scale parameter so thatthe
above is multiplied by  exp(betai*disij ) .

Thanks
Ben Dwamena

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