# Simulate Data from Generalized Linear Models Simulate data from linear and generalized linear models. Only the first covariate truely affects the response variable with coefficient equal to `lambda`. ```r simglm(family=c("binomial", "gaussian", "poisson","Gamma"),lambda=3,n=50,p=3) ``` ## Arguments - `family`: the family of the distribution. - `lambda`: size of the coefficient of the first covariate. - `n`: the sample size. - `p`: the number of covarites. ## Details The first covariate takes 1 in half of the observations, and 0 or -1 in the other half. When `lambda` gets larger, it is supposed to easier to predict the response variable. ## Returns Returned values include `yx` and `beta`. - **yx**: a data frame including the response `y` and covariates `x.1`, `x.2`, and so on. - **beta**: true values of the regression coefficients. ## References Zhang, D. (2017). A coefficient of determination for generalized linear models. **The American Statistician**, 71(4): 310-316. ## Author(s) Dabao Zhang, Department of Epidemiology and Biostatistics, University of California, Irvine ## See Also `rsq, rsq.partial, pcor`. ## Examples ```r # Poisson Models sdata <- simglm(family="poisson",lambda=4) fitf <- glm(y~x.1+x.2+x.3,family=poisson,data=sdata$yx) rsq(fitf) # type='v' fitr <- glm(y~x.2+x.3,family=poisson,data=sdata$yx) rsq(fitr) # type='v' rsq(fitr,type='kl') rsq(fitr,type='lr') rsq(fitr,type='n') pcor(fitr) # type='v' pcor(fitr,type='kl') pcor(fitr,type='lr') pcor(fitr,type='n') # Gamma models with shape=100 n <- 50 sdata <- simglm(family="Gamma",lambda=4,n=n) fitf <- glm(y~x.1+x.2+x.3,family=Gamma,data=sdata$yx) rsq(fitf) # type='v' rsq.partial(fitf) # type='v' fitr <- glm(y~x.2,family=Gamma,data=sdata$yx) rsq(fitr) # type='v' rsq(fitr,type='kl') rsq(fitr,type='lr') rsq(fitr,type='n') # Likelihood-ratio-based R-squared y <- sdata$yx$y yhatr <- fitr$fitted.values fit0 <- update(fitr,.~1) yhat0 <- fit0$fitted.values llr <- sum(log(dgamma(y,shape=100,scale=yhatr/100))) ll0 <- sum(log(dgamma(y,shape=100,scale=yhat0/100))) # Likelihood-ratio-based R-squared 1-exp(-2*(llr-ll0)/n) # Corrected likelihood-ratio-based R-squared (1-exp(-2*(llr-ll0)/n))/(1-exp(2*ll0/n)) ```

simglm function