# Create a Link for BTSR models Given the name of a link, this function returns a link function, an inverse link function, the derivative $d\eta/d\mu$ and the derivative $d\mu/d\eta$. ```r link.btsr(link = "linear") ``` ## Arguments - `link`: character; one of `"linear"`,`"polynomial"`, `"logit"`, `"log"`, `"loglog"`, `"cloglog"`, `"SIP"`. See Details below. Default is `link = "linear"`. ## Returns An object of class `"link-btsr"`, a list with components - **linkfun**: Link function `function(mu)` $g(\mu)$ - **linkinv**: Inverse link function `function(eta)` $g^{-1}(eta)$ - **linkdif**: Derivative `function(mu)` $d\eta/d\mu$ - **mu.eta**: Derivative `function(eta)` $d\mu/d\eta$ - **name**: a name to be used for the link ## Details The available links are: - **`linear`:**: $g(x) = ax$, for $a$ real. - **`polynomial`:**: $g(x) = ax^b$, for $a$ and $b$ real. - **`logit`:**: $g(x) = \log((x-l)/(u-x))$ - **`log`:**: $g(x) = \log(x - l)$ - **`loglog`:**: $g(x) = \log(-\log((x-l)/(u - l)))$ - **`cloglog`:**: $g(x) = \log(-\log(1-(x-l)/(u - l)))$ - **`SIP` (Shifted Inverse Power):**: $g(x) = 1/(a+x)^b$, with $a \in \{0,1\}$ and $b$ real. Here $l$ and $u$ denote the lower and upper bounds of $x$. The linear link is a particular case of the polynomial link. It was kept for compatibility with older versions of the package. The parameters $a$, $b$, $l$, and $u$ are configured using the `configs.linkg` list, which can include the following elements * `ctt`: A constant multiplier for the link function (default: 1). * `power`: The power parameter for polynomial and SIP links (default: 1). * `lower`: The lower bound for `mu` (default: 0). * `upper`: The upper bound for `mu` (default: 1). This list must be passed to the functions created by `link.btsr`, when invoking them. ## Examples ```r #--------------------------------------------- # 0 < y < 1 and linear link #--------------------------------------------- mylink <- link.btsr("linear") y <- 0.8 a <- 3.4 gy <- a * y # comparing the expected result with the output of the function: mylink$linkfun(mu = y, configs.linkg = list(ctt = a)) gy mylink$linkinv(eta = gy, configs.linkg = list(ctt = a)) y mylink$diflink(mu = y, configs.linkg = list(ctt = a)) a mylink$mu.eta(eta = gy, configs.linkg = list(ctt = a)) 1 / a # same function, different ctt: mylink$linkfun(mu = y, configs.linkg = list(ctt = a + 1)) #--------------------------------------------- # For linear link bounds have no effect #--------------------------------------------- mylink <- link.btsr("linear") y <- 0.8 a <- 3.4 gy <- a * y mylink$linkfun(mu = y, configs.linkg = list(ctt = a, lower = 1, upper = 2)) mylink$linkfun(mu = y, configs.linkg = list(ctt = a)) # same result gy mylink$linkinv(eta = gy, configs.linkg = list(ctt = a, lower = 1, upper = 2)) y mylink$diflink(mu = y, configs.linkg = list(ctt = a, lower = 1, upper = 2)) a mylink$mu.eta(eta = gy, configs.linkg = list(ctt = a, lower = 1, upper = 2)) 1 / a #--------------------------------------------- # 0 < y < 1 and logit link #--------------------------------------------- mylink <- link.btsr("logit") y <- 0.8 gy <- log(y / (1 - y)) ginv <- 1 / (1 + exp(-gy)) mylink$linkfun(mu = y) gy mylink$linkinv(eta = gy) ginv mylink$diflink(mu = y) 1 / (y - y**2) mylink$mu.eta(eta = gy) ginv - ginv**2 #--------------------------------------------- # 1 < y < 2 and logit link #--------------------------------------------- mylink <- link.btsr("logit") a <- 1 b <- 2 y <- 1.8 gy <- log((y - a) / (b - y)) ginv <- b / (1 + exp(-gy)) + a / (1 + exp(gy)) mylink$linkfun(mu = y, configs.linkg = list(lower = 1, upper = 2)) gy mylink$linkinv(eta = gy, configs.linkg = list(lower = 1, upper = 2)) ginv mylink$diflink(mu = y, configs.linkg = list(lower = 1, upper = 2)) (b - a) / ((a + b) * y - y**2 - a * b) mylink$mu.eta(eta = gy, configs.linkg = list(lower = 1, upper = 2)) ((a + b) * ginv - ginv**2 - a * b) / (b - a) ```

link.btsr function