help ie_rate


ie_rate -- intrinsic estimator for age, period, cohort (APC) applications

Syntax ie_rate depvar [indepvars] [if] [in] [weight] [,options]

options Description ------------------------------------------------------------------------- irr eform results offset() log of exposure for rate parameterization scale(dev) calculate std. errors using deviance-based mse Description

ie_rate computes coefficients from an APC analysis characterized by a design matrix that is not full rank due to the perfect linear dependence between age, period, and cohort. However, unlike apc_ie, ie_rate does not require strict linear dependence to estimate the APC model. However, it does require care in inputting the proper APC design matrix (described below).

The dependent variable for ie_rate is assumed to be in the form of counts. An offset (log exposure) should be specified in order to estimate a rate model. If an offset is not specified, it defaults to 0 resulting in a standard Poisson regression. Initial values are obtained using ie_reg, which is a companion routine for linear regression on the log rates.

In order to obtain APC estimates normalized according to conventional applications (i.e., apc_ie), it is necessary that an ANOVA design matrix be constructed using the last factor level of each APC factor as the reference as shown below.

After estimation the user may request the a display of the full set of ANOVA normalized estimates, including those pertaining to the reference categories, which are obtained using ie_norm, which is a modified version of Ben Jann's devcon utility with exactly the same syntax. As mentioned above, it is important to code the APC design with the last category as reference. Additionally, the model must include a constant term.

Example code for ANOVA design coding

qui tab age, gen(a) scal arow = r(r) qui tab period, gen(p) scal prow = r(r) qui gen cohort = period - age qui tab cohort, gen(c) scal crow = r(r)

* construct ANOVA normalization using last category as reference

forval i = 1/`=arow' { gen aC`i' = a`i' - a`=arow' }

forval i = 1/`=prow' { gen pC`i' = p`i' - p`=prow' } forval i = 1/`=crow' { gen cC`i' = c`i' - c`=crow' }


APC rate model

. ie_rate d aC1-aC6 pC1-pC3 cC1-cC9, offset(logn)

fully normalized solution

. ie_norm, groups(aC1-aC7, pC1-pC4, cC1-cC10)

Estimation Details

ie_rate uses a Newton-Raphson algorithm and employes the method outlined in Fu (2000). Starting values are obtained from a loglinear regression using the empirical rates via ie_reg, which may also be used as a standalone program for a loglinear analysis of empirical rates.

Saved Results

ie_rate saves the following in e():

Scalars e(N) number of cells e(ll) deviance e(k) number of estimated parameters Macros e(depvar) name of dependent variable

Matrices e(Coef) 1 X K vector of estimates e(Var) K X K variance-covariance matrix

Functions e(sample) marks estimation sample


Wenjiang J. Fu (2000). "Ridge Estimator in Singular Design with Application to Age-Period-Cohort Analysis of Disease Rates, Communications in Statistics - Theory and Methods, 29:2, 263-278

Daniel A. Powers (2012). "Black-White Differences in Maternal Age, Maternal Birth Cohort, and Period Effects on Infant Mortality in the U.S. (1983-2002)." Presented at the annual meetings of the Population Research Association of America, San Francisco, CA, May 4 2012.


Daniel A. Powers, University of Texas at Austin,

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