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From |
"Rosy Reynolds" <[email protected]> |

To |
"statalist" <[email protected]> |

Subject |
st: reverse prediction - confidence interval for x at given y in nonlinear model |

Date |
Thu, 25 Oct 2007 10:50:50 +0100 |

Dear Statalist,

Sigmoid models are customary in pharmacodynamics (dose-response studies).

According to custom, I am using a 4-parameter logistic (sigmoid Emax) model. This is very easily done with -nl- as Stata has this model already built in.

The model is y= b0 + b1/(1 + exp(-b2*(x-b3))) + error

and the coefficients can be interpreted as

b0 = baseline outcome

b1 = Emax i.e. largest change from baseline

b2 = Hill or slope coefficient

b3 = ED50 i.e. value of x (dose) required to produce half-maximal effect, that is x required for y=b0 + b1 / 2

As the ED50 is a parameter of the model, -nl- reports it with a standard error and confidence interval.

What I would like to do is obtain estimates with standard errors and confidence intervals for other similar measures e.g. the ED90, the dose required for 90% of maximal effect.

In general, how could I calculate an estimate and confidence interval for the x required to achieve any given value of y?

I will be grateful for any guidance anyone can give me.

I hope I'm not missing something very obvious.

best wishes

Rosy Reynolds

BSAC Resistance Surveillance Co-ordinator

www.bsacsurv.org

Below are simulated data and output from -nl- .

In this case, by design, half maximal effect is at y=6, and the ED50 value of x to produce that is 5.

-nl- estimates the ED50 (b3) nicely as 4.9 with CI [4.7, 5.1]

I would like to obtain an estimate of x for y=9, for example, with a CI.

* Simulate data with fairly small errors.

. clear

. set obs 101

obs was 0, now 101

. gen x=(_n-1)/10

. local b0 2 // Yo

. local b1 8 // Emax

. local b2 1 // Hill coefficient

. local b3 5 // ED50

. gen y= `b0' + `b1'/(1 + exp(-`b2'*(x-`b3'))) + 0.5*invnormal(uniform())

*Fit model

. nl log4: y x // Stata built-in 4-parameter logistic

(obs = 101)

Iteration 0: residual SS = 37.0629

Iteration 1: residual SS = 25.18804

Iteration 2: residual SS = 22.87

Iteration 3: residual SS = 22.86999

Iteration 4: residual SS = 22.86999

Iteration 5: residual SS = 22.86999

Source | SS df MS

-------------+------------------------------ Number of obs = 101

Model | 1019.30494 3 339.768313 R-squared = 0.9781

Residual | 22.8699928 97 .235773122 Adj R-squared = 0.9774

-------------+------------------------------ Root MSE = .4855647

Total | 1042.17493 100 10.4217493 Res. dev. = 136.6108

4-parameter logistic function, y = b0 + b1/(1 + exp(-b2*(x - b3)))

------------------------------------------------------------------------------

y | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

/b0 | 1.853862 .1387566 13.36 0.000 1.578469 2.129256

/b1 | 8.259274 .2210395 37.37 0.000 7.820572 8.697977

/b2 | .9559409 .0661914 14.44 0.000 .8245693 1.087312

/b3 | 4.879619 .0739367 66.00 0.000 4.732876 5.026363

------------------------------------------------------------------------------

Parameter b0 taken as constant term in model & ANOVA table

*

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