Use of the multinomial jackknife and bootstrap in generalized nonlinear canonical correlation analysis

The estimation of mean and standard errors of the eigenvalues and category quantifications in generalized non-linear canonical correlation analysis (OVERALS) is discussed. Starting points are the delta method equations. The jackknife and bootstrap methods are compared for providing finite difference approximations to the derivatives. Examining the basic properties of the jackknife method indicates that the vector of profile proportions is perturbed by leaving out single observations. The grid of perturbed values is used to estimate relevant derivatives. Bootstrapping means resampling with replacement from the original sample. Both procedures, bootstrapping and jackknifing, are used to compute pseudo-value means and standard errors for four different data sets: (1) the characteristics of 36 kinds of marine mammals; (2) data describing the attributes of 47 countries; (3) data from a study of school choice for 520 children leaving elementary school; and (4) a sample of 4,863 secondary students from the Second International Mathematics Study. For the small data sets the jackknife and bootstrap were used; for the larger sets, Monte Carlo versions of both were used. The jackknife method appeared less imprecise than the bootstrap method, and jackknife approximations were less stable for smaller samples. It is concluded that the bootstrap method performed better than did the jackknife method. For large samples, the bootstrap procedure works quite well for computing confidence intervals, and eigenvalues computed from OVERALS seem quite stable.