# Generate simulated data Generate data for simulations under the generalized linear model and Cox model. ```r gen.data( n, p, k = NULL, rho = 0, family = c("gaussian", "binomial", "poisson", "cox"), beta = NULL, cortype = 1, snr = 10, censoring = TRUE, c = 1, scal, sigma = 1, seed = 1 ) ``` ## Arguments - `n`: The number of observations. - `p`: The number of predictors of interest. - `k`: The number of nonzero coefficients in the underlying regression model. Can be omitted if `beta` is supplied. - `rho`: A parameter used to characterize the pairwise correlation in predictors. Default is `0`. - `family`: The distribution of the simulated data. `"gaussian"` for gaussian data.`"binomial"` for binary data. `"poisson"` for count data. `"cox"` for survival data. - `beta`: The coefficient values in the underlying regression model. - `cortype`: The correlation structure. `cortype = 1` denotes the exponential structure, where the covariance matrix has $(i,j)$ entry equals $rho^{|i-j|}$. codecortype = 2 denotes the constant structure, where the $(i,j)$ entry of covariance matrix is $rho$ for every $i \neq j$ and 1 elsewhere. `cortype = 3` denotes the moving average structure. Details can be found below. - `snr`: A numerical value controlling the signal-to-noise ratio (SNR). The SNR is defined as as the variance of $x\beta$ divided by the variance of a gaussian noise: $\frac{Var(x\beta)}{\sigma^2}$. The gaussian noise $\epsilon$ is set with mean 0 and variance. The noise is added to the linear predictor $\eta$ = $x\beta$. Default is `snr = 10`. This option is invalid for `cortype = 3`. - `censoring`: Whether data is censored or not. Valid only for `family = "cox"`. Default is `TRUE`. - `c`: The censoring rate. Default is `1`. - `scal`: A parameter in generating survival time based on the Weibull distribution. Only used for the "`cox`" family. - `sigma`: A parameter used to control the signal-to-noise ratio. For linear regression, it is the error variance $\sigma^2$. For logistic regression and Cox's model, the larger the value of sigma, the higher the signal-to-noise ratio. Valid only for `cortype = 3`. - `seed`: seed to be used in generating the random numbers. ## Returns - **x**: Design matrix of predictors. - **y**: Response variable. - **Tbeta**: The coefficients used in the underlying regression model. ## Details We generate an $n \times p$ random Gaussian matrix $X$ with mean 0 and a covariance matrix with an exponential structure or a constant structure. For the exponential structure, the covariance matrix has $(i,j)$ entry equals $rho^{|i-j|}$. For the constant structure, the $(i,j)$ entry of the covariance matrix is $rho$ for every c("$i\n$", "$\\neq j$") and 1 elsewhere. For the moving average structure, For the design matrix $X$, we first generate an $n \times p$ random Gaussian matrix $\bar{X}$ whose entries are i.i.d. $\sim N(0,1)$ and then normalize its columns to the $\sqrt n$ length. Then the design matrix $X$ is generated with $X_j = \bar{X}_j + \rho(\bar{X}_{j+1}+\bar{X}_{j-1})$ for $j=2,\dots,p-1$. For `family = "gaussian"` , the data model is $$ Y = X \beta +\epsilon. $$ The underlying regression coefficient $\beta$ has uniform distribution [m, 100m], $m=5 \sqrt{2log(p)/n}.$ For `family= "binomial"`, the data model is $$ Prob(Y = 1) = \exp(X\beta + \epsilon)/(1 + \exp(X \beta + \epsilon)). $$ The underlying regression coefficient $\beta$ has uniform distribution [2m, 10m], $m = 5\sigma \sqrt{2log(p)/n}.$ For `family = "poisson"` , the data is modeled to have an exponential distribution: $$ Y = Exp(\exp(X \beta +\epsilon)). $$ For `family = "cox"`, the data model is $$ T = (-\log(S(t))/\exp(X \beta))^{1/scal}. $$ The centering time is generated from uniform distribution $[0, c]$, then we define the censor status as $\delta = I\{T \leq C\}, R = min\{T, C\}$. The underlying regression coefficient $\beta$ has uniform distribution [2m, 10m], $m = 5\sigma \sqrt{2log(p)/n}.$ In the above models, c("$\\epsilon \\sim N(0,\n$", "$\\sigma^2 ),$") where $\sigma^2$ is determined by the `snr`. ## Examples ```r n <- 200 p <- 20 k <- 5 rho <- 0.4 SNR <- 10 cortype <- 1 seed <- 10 Data <- gen.data(n, p, k, rho, family = "gaussian", cortype = cortype, snr = SNR, seed = seed) x <- Data$x[1:140, ] y <- Data$y[1:140] x_new <- Data$x[141:200, ] y_new <- Data$y[141:200] lambda.list <- exp(seq(log(5), log(0.1), length.out = 10)) lm.bsrr <- bsrr(x, y, method = "pgsection") ``` ## See Also `bsrr`, `predict.bsrr`. ## Author(s) Liyuan Hu, Kangkang Jiang, Yanhang Zhang, Jin Zhu, Canhong Wen and Xueqin Wang.

gen.data function