This is a wrapper function to estimate the average distance into an age interval lived by those dying within that age interval. It calls 4 different methods: "keyfitz", "schoen", "midpoint" or "preston". These generally use formulas proposed by their respective namesakes, although all have been modified by the current author in minor ways, usually in order to provide values for the final ages, which are left as NAs using certain methods. The Preston method is called so not because he explicitly invented it, but rather because it follows a series of rules of thumbs drawn from other sources and described so well in Preston et al (2001). See the individual ax estimation functions to see the details of the various methods. a0 is handled using a variant of the Andreev-Kingkade method.
axEstimate(Mx, n, axsmooth = TRUE, method = "keyfitz", sex, verbose)a numeric vector of the age-specific central death rates, calculated as D(x)/N(x) (deaths/exposure)
a numeric vector of age interval widths.
logical. default = TRUE. Should the a(x) values be calculated from a smoothed M(x) series? In this case, the M(x) series is smoothed within the function for a(x) estimation, but the smoothed M(x) function that was used is not returned. In general, it is better to smooth the M(x) function prior to putting it in this function, because the loess smoother used here has no weights or offset. If this is not possible, loess M(x) smoothing still produces more consistent and less erratic a(x) estimates.
either "keyfitz", "schoen", "preston" or "midpoint". Default = "keyfitz", although this is not recommended for abridged ages. See comparisons in examples below.
"female" or "male". default "female". This is only used by the "preston" method and need no be specified for any other method.
logical, default = TRUE. Should informative but possibly annoying messages be returned when the function does something that you might want to know about?
ax, a numeric vector of a(x) values.
This function is a wrapper, and it is called by the lifetable function LT(), although it can be used independently as well. For fuller explanations, see the descriptions and code of the various methods. Formulas are available in the referenced works.
Chiang C.L.(1968) Introduction to Stochastic Processes in Biostatistics. New York: Wiley.
Coale Anseley and Paul Demeny, with B Vaughan (1983). Regional Model Life Tables and Stable Populations. New York Academic Press.
Keyfitz, Nathan (1966) A Life Table that Agrees with the Data. Journal of the American Statistical Association, 61 (314):305-12. (as described on page 44-45 of Preston et al (2001). Demography: Measuring and Modelling Population Processes. Blackwell Publishing)
Schoen R. (1978) Calculating lifetables by estimating Chiang\'s a from observed rates. Demography 15: 625-35.
Andreev, Evgueni M and Kingkade, Ward W (2011) Average age at death in infancy and infant mortality level: reconsidering the Coale-Demeny formulas at current levels of low mortality. MPIDR Working Paper WP-2011-016.
This function dispatches to one of four different a(x) estimation functions axMidpoint for the "midpoint" method, axSchoen for the "schoen" method, axPreston for the "preston" method and axKeyfitz for the "keyfitz" method. Look to these pages for specifics. Compare using the examples below. This function is called by LT, a single decrement lifetable function.