# Function to implement an adaptive two-stage test There are four key quantities for the specification of an adaptive two-stage test: the overall test level $alpha$, stopping bounds $alpha1 <= alpha0$ and the local level $alpha2$ of the test after the second stage. These quantities are interrelated through the overall level condition. The function `tsT` calculates any of these quantities based on the others. ```r tsT(typ, a = NA, a0 = NA, a1 = NA, a2 = NA) ``` ## Arguments - `typ`: type of test: `"b"` for Bauer and Koehne (1994), `"l"` for Lehmacher and Wassmer (1999), `"v"` for Vandemeulebroecke (2006) and `"h"` for the horizontal conditional error function - `a`: $alpha$, the overall test level - `a0`: $alpha0$, the futility stopping bound - `a1`: $alpha1$, the efficacy stopping bound and local level of the test after the first stage - `a2`: $alpha2$, the local level of the test after the second stage ## Details An adaptive two-stage test can be viewed as a family of decreasing functions $f[c](p1)$ in the unit square. Each of these functions is a conditional error function, specifying the type I error conditional on the p-value $p1$ of the first stage. For example, $f[c](p1) = min(1, c/p1)$ corresponds to Fisher's combination test (Bauer and Koehne, 1994). Based on this function family, the test can be put into practice by specifying the desired overall level $alpha$, stopping bounds $alpha1 <= alpha0$ and a parameter $alpha2$. After computing $p1$, the test stops with or without rejection of the null hypothesis if $p1 <= alpha1$ or $p1 > alpha0$, respectively. Otherwise, the null hypothesis is rejected if and only if $p2 <= f[c](p1)$ holds for the p-value $p2$ of the second stage, where $c$ is such that the local level of this latter test is $alpha2$ (e.g., $c = c(alpha2) = exp(-chi2_{4,alpha2}/2)$ for Fisher's combination test). The four parameters $alpha$, $alpha0$, $alpha1$ and $alpha2$ are interdependent: they must satisfy the level condition $$ \alpha_1 + \int_{\alpha_1}^{\alpha_0} cef_{\alpha_2}(p_1) d p_1 = \alpha,alpha1 + int_{alpha1}^{alpha0} cef_{alpha2}(p1) d p1 = alpha, $$ where $cef_{alpha2}$ is the conditional error function (of a specified family) with parameter $alpha2$. For example, this conditon translates to $$ \alpha = \alpha_1 + c(\alpha_2) * (\log(\alpha_0) - \log(\alpha_1))alpha = alpha1 + c(alpha2) * (log(alpha0) - log(alpha1)) $$ for Fisher's combination test (assuming that $c(alpha2) < alpha1$; Bauer and Koehne, 1994). The function `tsT` calculates any of the four parameters based on the remaining ones. Currently, this is implemented for the following four tests: Bauer and Koehne (1994), Lehmacher and Wassmer (1999), Vandemeulebroecke (2006), and the horizontal conditional error function. ## Returns If three of the four quantities $alpha$, $alpha0$, $alpha1$ and $alpha2$ are provided, `tsT` returns the fourth. If only $alpha$ and $alpha0$ are provided, `tsT` returns $alpha1$ under the condition $alpha1 = alpha2$ (the so-called "Pocock-type"). If the choice of arguments is not allowed (e.g., $alpha0 < alpha1$) or when a test cannot be constructed with this choice of arguments (e.g., $alpha0 = 1$ and $alpha < alpha2$), `tsT` returns `NA`. IMPORTANT: When the result is (theoretically) not unique, `tsT` returns the maximal $alpha1$, maximal $alpha2$ or minimal $alpha0$. In all cases, `tsT` returns the result for the test specified by `typ`. ## References Bauer, P., Koehne, K. (1994). Evaluation of experiments with adaptive interim analyses. **Biometrics** 50, 1029-1041. Lehmacher, W., Wassmer, G. (1999). Adaptive sample size calculations in group sequential trials. **Biometrics** 55, 1286-1290. Vandemeulebroecke, M. (2006). An investigation of two-stage tests. **Statistica Sinica** 16, 933-951. Vandemeulebroecke, M. (2008). Group sequential and adaptive designs - a review of basic concepts and points of discussion. **Biometrical Journal** 50, 541-557. ## Author(s) Marc Vandemeulebroecke ## Note The argument `typ`, and either exactly three of $alpha$, $alpha0$, $alpha1$ and $alpha2$, or only $alpha$ and $alpha0$, must be provided to `tsT`. ## See Also `adaptTest` package description ## Examples ```r ## Example from Bauer and Koehne (1994): full level after final stage, alpha0 = 0.5 alpha <- 0.1 alpha2 <- 0.1 alpha0 <- 0.5 alpha1 <- tsT(typ="b", a=alpha, a0=alpha0, a2=alpha2) plotCEF(typ="b", a2=alpha2, add=FALSE) plotBounds(alpha1, alpha0) ## See how similar Lehmacher and Wassmer (1999) and Vandemeulebroecke (2006) are alpha <- 0.1 alpha1 <- 0.05 alpha0 <- 0.5 alpha2l <- tsT(typ="l", a=alpha, a0=alpha0, a1=alpha1) alpha2v <- tsT(typ="v", a=alpha, a0=alpha0, a1=alpha1) plotCEF(typ="l", a2=alpha2l, add=FALSE) plotCEF(typ="v", a2=alpha2v, col="red") plotBounds(alpha1, alpha0) ## A remark about numerics tsT(typ="b", a=0.1, a1=0.05, a0=0.5) tsT(typ="b", a=0.1, a2=0.104877, a0=0.5) tsT(typ="b", a=0.1, a2=tsT(typ="b", a=0.1, a1=0.05, a0=0.5), a0=0.5) ## An example of non-uniqueness: the maximal alpha1 is returned; any ## smaller value would also be valid alpha <- 0.05 alpha0 <- 1 alpha2 <- 0.05 alpha1 <- tsT(typ="b", a=alpha, a0=alpha0, a2=alpha2) tsT(typ="b", a0=alpha0, a1=alpha1, a2=alpha2) tsT(typ="b", a0=alpha0, a1=alpha1/2, a2=alpha2) ```

tsT function