Numerical integration with an application in sample size reestimation

We introduce a new integrate() function for Mata that evaluates single-dimensional integrals. This function uses three different Gaussian quadrature algorithms: Gauss–Hermite and Gauss–Laguerre for indefinite integrals; and Gauss–Legendre for definite integrals. The algorithms were implemented using the methods of Golub and Welsch (Mathematics of Computation, 1968). The user can specify any integrand by defining a new function in the Mata language. The integrand function is allowed to have two arguments: the first is the variable of integration, and the second is a real scalar. Thus the integrate() function can be used in combination with optimise() to solve for the value of x in an integral f(u,x) du = 0.05. Such calculations are used in the sample size re-estimation methodology introduced by Li, Shih, and Xie (Biostatistics, 2002). We apply these methods to a clinical trial where a single interim analysis is carried out, and the analysis is used to reevaluate the sample size.