# ML fit of hyperbolic or sigmoidal Emax models to continuous/binary dose response data. Calls Newton-Raphson optimizers, nls and nlm, for a hyperbolic or sigmoidal Emax model. Different intercepts for multiple protocol-data are supported. For binary data, the Emax model is on the logit scale. ```r fitEmax(y,dose,iparm,xparm,modType=4, prot=rep(1,length(y)),count=rep(1,length(y)),xbase=NULL, binary=FALSE,diagnostics=TRUE,msSat=NULL, pboAdj=rep(FALSE,max(prot)),optObj=TRUE) ``` ## Arguments - `y`: Outcome for each patient. Missing `Y` values are are not permitted. Dose/protocol group means for grouped continuous data. For binary data, `y` must be 0/1 and counts must be supplied for each 0/1 value. - `dose`: Dose for each patient. - `iparm`: Optional starting values for the Newton-Raphson algorithm. The order of the variables is (log(ED50),Emax,E0) or (log(ED50),lambda,Emax,E0). Note the transformation of ED50. If there is more than one protocol, the E0 is automatically duplicated. - `xparm`: Optional starting values for the baseline covariate slopes (if any). `xparm` must be specified when `iparm` and `xbase` are specified. `startEmax` is used to obtain starting values if no starting values are specified. - `modType`: modType=3 (default) for the 3-parameter hyperbolic Emax model. modType=4 for the 4-parameter sigmoidal Emax model. - `prot`: Protocol (group) membership used to create multiple intercepts. The default is a single protocol. - `count`: Counts for the number of patients when the `Y` are dose continuous group means or binary 0/1 values. Default is 1 (ungrouped data). - `xbase`: A matrix of baseline covariates with rows corresponding to `y` that enter as linear additive predictors. The baseline covariates must be centered about their (protocol-specific) means. `xbase` does not include an intercept or protocol indicators. Covariates cannnot be specified with PBO adjusted or aggregated input. - `diagnostics`: Print trace information per iteration and any error messages from the optimizing methods. Printing can be suppressed for use in simulation studies. - `binary`: When `TRUE`, the `y` are assumed to be coded `0/1`, and the the means reported are proportions. The Emax model is specified on the logit scale, and proportions are estimated from the model by back-transformation. - `msSat`: If continuous `Y` are dose/protocol group means rather than individual measurements, the within group variance, `msSat`, should be supplied. This variance is the mean square from the model saturated in dose and protocol. It is used for goodness-of-fit (GOF) testing, and to improve the residual variance estimate for the Emax model. If it is not supplied, statistics needed for GOF will not be available, and the residual SD (and associated SE) will have low degrees of freedom. - `pboAdj`: For published data with only pbo-adjusted dose group means and SEs, the model is fit without an intercept(s). If initial parameters are supplied, the intercept (E0) should be assigned `0`. A zero for the placebo mean should not be included in `Y`. This option is not available for binary data. Potential correlation between between placebo-adjusted means is ignored. - `optObj`: Include the output object from the R optimization code in the `fitEmax` output. ## Details Fits the 3- or 4- Emax model using `nls`. A newton-raphson algorithm is tried first followed by a partial linear optimatization if needed. Binary data are fit using `nlm`. ## Returns A list assigned class "fitEmax" with: - **fit**: The parameter estimates and their variance-covariance matrix. - **y, dose, modType, prot, count, binary, pboAdj**: Input values. - **gofTest**: Goodness of fit p-value based on likelihood ratio comparison of the model to a saturated fit. - **nll**: `-2*loglikelihood` for the Emax model and the saturated model. Residual sums of squares are returned for continuous data models. These statistics can be used to construct other tests using multiple calls to `fitEmax` (e.g., 3 vs 4 parameter Emax models, or a common intercept model across protocols). - **df**: Residual degrees of freedom for the Emax model and the saturated model. - **optobj**: When requested, the fit object returned by the R optimation functions. ## Author(s) Neal Thomas ## See Also `nls`, `nlm`, `nllogis`, `predict.fitEmax`, `plot.fitEmax`, `coef.fitEmax` ## Examples ```r ## the example changes the random number seed doselev<-c(0,5,25,50,100,350) n<-c(78,81,81,81,77,80) ### population parameters for simulation e0<-2.465375 ed50<-67.481113 dtarget<-100 diftarget<-9.032497 emax<-solveEmax(diftarget,dtarget,log(ed50),1,e0) sdy<-8.0 pop<-c(log(ed50),emax,e0) dose<-rep(doselev,n) meanlev<-emaxfun(dose,pop) y<-rnorm(sum(n),meanlev,sdy) testout<-fitEmax(y,dose,modType=4) ```

fitEmax function