# Calculates Relative Efficiency for Locally Optimal Designs Given a vector of initial estimates for the parameters, this function calculates the D-and PA- efficiency of a design $\xi_1$ with respect to a design $\xi_2$. Usually, $\xi_2$ is an optimal design. ```r leff( formula, predvars, parvars, family = gaussian(), inipars, type = c("D", "PA"), fimfunc = NULL, x2, w2, x1, w1, npar = length(inipars), prob = NULL ) ``` ## Arguments - `formula`: A linear or nonlinear model `formula`. A symbolic description of the model consists of predictors and the unknown model parameters. Will be coerced to a `formula` if necessary. - `predvars`: A vector of characters. Denotes the predictors in the `formula`. - `parvars`: A vector of characters. Denotes the unknown parameters in the `formula`. - `family`: A description of the response distribution and the link function to be used in the model. This can be a family function, a call to a family function or a character string naming the family. Every family function has a link argument allowing to specify the link function to be applied on the response variable. If not specified, default links are used. For details see `family`. By default, a linear gaussian model `gaussian()` is applied. - `inipars`: Vector. Initial values for the unknown parameters. It will be passed to the information matrix and also probability function. - `type`: A character. `"D"` denotes the D-efficiency and `"PA"` denotes the average P-efficiency. - `fimfunc`: A function. Returns the FIM as a `matrix`. Required when `formula` is missing. See 'Details' of `minimax`. - `x2`: Vector of design (support) points of the optimal design ($\xi_2$). Similar to `x1`. - `w2`: Vector of corresponding design weights for `x2`. - `x1`: Vector of design (support) points of $\xi_1$. See 'Details' of `leff`. - `w1`: Vector of corresponding design weights for `x1`. - `npar`: Number of model parameters. Used when `fimfunc` is given instead of `formula` to specify the number of model parameters. If not given, the sensitivity plot may be shifted below the y-axis. When `NULL`, it will be set here to `length(inipars)`. - `prob`: Either `formula` or a `function`. When function, its argument are `x` and `param`, and they are the same as the arguments in `fimfunc`. `prob` as a function takes the design points and vector of parameters and returns the probability of success at each design points. See 'Examples'. ## Returns A value between 0 and 1. ## Details For a known $\theta_0$, relative D-efficiency is $$ exp(\frac{log|M(\xi_1, \theta_0)| - log|M(\xi_2, \theta_0)|}{npar})exp{(log|M(\xi_1, \theta_0)| - log|M(\xi_2, \theta_0)|)/npar}. $$ The relative P-efficiency is $$ \exp(\log(\sum_{i=1}^k w_{1i}p(x_{1i}, \theta_0) - \log(\sum_{i=1}^k w_{2i}p(x{2_i}, \theta_0))exp(log (\sum w1_i p(x1_i, \theta_0) - log(\sum w2_i p(x2_i, \theta_0)), $$ where $x1$ and $w1$ are usually the support points and the corresponding weights of the optimal design, respectively. The argument `x1` is the vector of design points. For design points with more than one dimension (the models with more than one predictors), it is a concatenation of the design points, but dimension-wise . For example, let the model has three predictors $(I, S, Z)$. Then, a two-point optimal design has the following points: ${point1 = (I1, S1, Z1), point2 = (I2, S2, Z2)}$. Then, the argument `x1` is equal to `x = c(I1, I2, S1, S2, Z1, Z2)`. ## Examples ```r p <- c(1, -2, 1, -1) prior4.4 <- uniform(p -1.5, p + 1.5) formula4.4 <- ~exp(b0+b1*x1+b2*x2+b3*x1*x2)/(1+exp(b0+b1*x1+b2*x2+b3*x1*x2)) prob4.4 <- ~1-1/(1+exp(b0 + b1 * x1 + b2 * x2 + b3 * x1 * x2)) predvars4.4 <- c("x1", "x2") parvars4.4 <- c("b0", "b1", "b2", "b3") # Locally D-optimal design is as follows: ## weight and point of D-optimal design # Point1 Point2 Point3 Point4 # /1.00000 \ /-1.00000\ /0.06801 \ /1.00000 \ # \-1.00000/ \-1.00000/ \1.00000 / \1.00000 / # Weight1 Weight2 Weight3 Weight4 # 0.250 0.250 0.250 0.250 xopt_D <- c(1, -1, .0680, 1, -1, -1, 1, 1) wopt_D <- rep(.25, 4) # Let see if we use only three of the design points, what is the relative efficiency. leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(), x1 = c(1, -1, .0680, -1, -1, 1), w1 = c(.33, .33, .33), inipars = p, x2 = xopt_D, w2 = wopt_D) # Wow, it heavily drops! # Locally P-optimal design has only one support point and is -1 and 1 xopt_P <- c(-1, 1) wopt_P <- 1 # What is the relative P-efficiency of the D-optimal design with respect to P-optimal design? leff(formula = formula4.4, predvars = predvars4.4, parvars = parvars4.4, family = binomial(), x1 = xopt_D, w1 = wopt_D, inipars = p, type = "PA", prob = prob4.4, x2 = xopt_P, w2 = wopt_P) # .535 ``` ## References McGree, J. M., Eccleston, J. A., and Duffull, S. B. (2008). Compound optimal design criteria for nonlinear models. Journal of Biopharmaceutical Statistics, 18(4), 646-661.

leff function