SAMPLING AND BIOSTATISTICS Statistical Inference on Associated Fertility Life Table Parameters Using Jackknife Technique: Computational Aspects ALINE DE H. N. MAIA, ALFREDO J. B. LUIZ, AND CLAYTON CAMPANHOLA Embrapa Meio Ambiente, Rodovia SP-340, km 127,5, Caixa Postal 69, CEP 13.820 Ð 000, Jaguariu ´ na, Sa ˜ o Paulo, Brazil J. Econ. Entomol. 93(2): 511Ð518 (2000) ABSTRACT Knowledge of population growth potential is crucial for studying population dynamics and for establishing management tactics for pest control. Estimation of population growth can be achieved with fertility life tables because they synthesize data on reproduction and mortality of a population. The Þve main parameters associated with a fertility life table are as follows: (1) the net reproductive rate (Ro), (2) the intrinsic rate of increase (r m ), 3) the mean generation time (T), (4) the doubling time (Dt), and (5) the Þnite rate of increase (). Jackknife and bootstrap techniques are used to calculate the variance of the r m estimate, which can be extended to the other parameters of life tables. Those methods are computer-intensive, their application requires the development of efÞcient algorithms, and their implementation is based on a programming language that encompasses quickness and reliability. The objectives of this article are to discuss statistical and computational aspects related to estimation of life table parameters and to present a SAS program that uses jackknife to estimate parameters for fertility life tables. The SAS program presented here allows the calculation of conÞdence intervals for all estimated parameters, as well as provides one-sided and two-sided t-tests to perform pairwise or multiple comparison between groups, with their respective P values. KEY WORDS fertility life tables, intrinsic rate of increase, jackknife technique, computer program, SAS program FERTILITY LIFE TABLES are appropriate to study the dy- namics of animal populations, especially arthropods, as an intermediate process for estimating parameters related to the population growth potential, also called demographic parameters. Methods for construction, description, and analysis of life tables for animal pop- ulations can be found in Deevey (1947), Birch (1948), and Southwood (1978). Examples of practical use of life tables are abundant in the literature (Elliot et al. 1988, Kieckhefer and Elliott 1989, Michels and Behle 1989, Vargas and Carey 1990, Fernandez-Casalderrey 1992, Zeng et al. 1993, Brodsgaard 1994, Hance et al. 1994, Chakraborty et al. 1996, Sharma et al. 1997). In addition, the population growth potential of insects and mites can be used as an indicator in studies that aim to assess environmental effects of agricultural technologies and practices (Stark and Wennergren 1995, Nascimento et al. 1998). For instance, in the risk assessment of biocontrol agents, it is important to know their possible negative effects on beneÞcial ar- thropods population dynamics. The parameters usually estimated from fertility life tables are the net reproductive rate (Ro); the intrinsic rate of increase (r m ); the mean generation time (T); the doubling time (Dt), and the Þnite rate of increase (). To compare life table parameters of different groups using statistical tests, it is necessary to have information on the uncertainty degree associated with its estimates, expressed as their variances. Information on variances for estimates coming from individual observations like longevity, immature development time, or number of eggs per female are easily obtained by calculating the variability among observed values from individuals of the same group, the so-called in- ternal or within group variance. However, the same procedure cannot be applied to synthetic parameters like Ro, r m , T, Dt, and , which summarize information on immature development, reproduction, and survival into a single statistic. In that case, variances can be calculated by the Monte Carlo methods, such as boot- strap, jackknife, and randomization tests. Detailed in- formation on those methods applied to biological sci- ences can be found in Manly (1991). Generally, estimates of associated life table parameters are re- ported in the literature without any measure of un- certainty. In some cases, inappropriate methods like analysis of variance and multiple range tests are em- ployed (Vargas and Nishida 1985, Zeng et al. 1993). Only a few authors (Elliot et al. 1988, Kieckhefer and Elliott 1989, Smith 1992, Smith 1993) reported stan- dard error for the r m estimate, using the jackknife method applied to life table analysis proposed by Meyer et al. (1986). The r m estimation using the iterative method is a computer-intensive technique itself. In addition, an- other computer intensive technique, the jackknife, is required to assess r m estimate uncertainty. In this way, to jointly use both techniques it is necessary to de- velop effective algorithms and to implement them using a computer language which provides speed and reliability for the estimation process. 0022-0493/00/0511Ð0518$02.00/0  2000 Entomological Society of America