A new version of the AUXAL program is immediately available for purchase.

Auxal4 uses the same empirical Bayes estimation procedure as AUXAL3, but has new features and improvements that extend the functionality of the program and make it easier to use. The accuracy of the estimated population covariance matrix of the model parameters has been improved. Although archival of data from the Fels longitudinal study continues as the source of the default prior distribution means and covariance matrices for the several models, the more accurately estimated covariance matrices provide the priors of Auxal4. When the user wishes to replace of the priors in the current job by those based on other longitudinal data, the steps involved have been simplified (see MEAN and COVARIANCE commands below).

The evaluated heights of the structural average curve that appear at the end of the summary output listing now include standard errors of height at any given age. They facilitate statistical analysis of group comparisons of structural average curves. The standard errors are computed from the estimated population covariance matrix of the parameters and the derivatives of the structural average curve with respect to the parameters at any given age point (see Rao, C. R. (2002). Linear Statistical Analysis and Its Applications, 2nd edition, paperback, New York: Wiley, pp.386-389). If the cases in the current analysis are drawn from the same population as the assumed prior distribution, the default covariance matrix of the prior is used in these computations. If the size of the current sample is large enough to justify large sample assumptions, the population covariance matrix estimated in the current job may be used in place of the default (see TECHNICAL command below).

Facilities have been added for cross-sectional analysis of mixed longitudinal data. They include standard errors for average height in given age intervals, and a provision for evaluating so-called plausible values of height at any given age that permit conventional multivariate analysis of group differences in mixed longitudinal data. The standard errors are exact when each case is represented by only one observed height in each interval. Otherwise it is conservative: additional observations within the interval that information, but it is difficult to evaluate the reduction in the standard error because of the observations are correlated.

**Changed commands, options, and keywords**

**MEAN and COVARIANCE commands**

** New option: IMPORT**

These commands allow the user to replace the prior mean and covariance matrix with others more relevant to the current analysis. If new versions of the population mean and covariance matrix have been estimated by the program and saved using the MEAN keyword of the SAVE command, the appearance of the new option in these commands automatically extracts the mean and/or covariance matrix from an existing file created by the SAVE command. In the absence of the IMPORT option, the mean or covariance matrix will be read from existing files containing the parameter values in the standard order described in the AUXAL manual.

**ERROR command**

Setting larger error standard deviations (SD) for purposes of suppressing failures of the maximum posterior eye estimation procedure from converging is no longer required globally. It is now applied only to those cases that do not converge in the iterative estimation of the model parameters for the case. When this occurs, the program automatically attempts re-estimation up to five times with increasing values of the error SD. If convergence is then obtained, the last value of the SD appears in the case processing list. These adjustments typically reduce the number of failed convergences. (Failures of convergence are also fewer when autocorrelation of the residuals across ages is neglected by invoking and the TECHNICAL option UNCOR.

The ERROR command is no longer needed, but is still operative.

**TECHNICAL command**

** New option: SAMPLE**

If this option is present, the program will base the standard errors of the structural average curve at any given age on the population covariance matrix of the model parameters estimated in the current job. Otherwise, the default prior distribution is the source of the covariance matrix in the computations.

** New keyword: PLSEVAL = ***t*

If the EVALUATE key word of the PROCEDURE command is present, and the evaluated heights of the cases at successive ages are saved using the EVAL keyword of the SAVE command, the evaluated heights are converted into plausible values on the assumption that the measurement error distribution at each age has mean zero and standard deviation equal to the square root of the error variance for the individual case. A random deviant from this distribution is added to each of the evaluated heights. Group differences of cross-sectional average growth can be analyzed by multivariate analysis of variance using these values as data. Their sampling variance includes the effects of sampling the cases as well as that of those of measurement error and the equation error (see also SAVE command).

The quantity *t* is the seed of the random number generator—any integer greater than 1 and less than 2147483647.

** New keyword: ONLY = ***u*

This keyword allows the program to compute the standard errors of a structural average curves directly from user-supplied values of the parameters means (possibly those from analyses by other investigators). The parameters must be input by use of the MEAN command in the standard order for the model in question. The quantity *u* is the number of cases in the sample from which the putative mean was obtained. The program must be then executed in a dummy job of least a few cases. The plot of the curve follows as usual. The default prior covariance matrix is used in these calculations.

**SAVE command**

** Keyword: COVARIANCE**

This keyword was not implemented in AUXAL3. It is now operative.

** Option: HGTROW**

If case heights or plausible value heights are evaluated, this option lists the output in rows of space-delimited values; otherwise, the values will be listed in a single column.

## Adjusting priors for different populations

If measurements of a large sample of *N* cases from a suitable longitudinal growth study are available, the population mean covariance matrix of the model parameters can be estimated from the MAP estimate and posterior covariance matrix for each case.

Because the population mean and covariance matrix are required in the prior distribution for estimating the parameter means and covariances, a “boot strap” procedure is required in their use. Initially, one starts with the existing AUXAL priors for the BTT, JPA2 or Jenss-Bayley models. These priors are based on USA data. Provided the number of well-spaced data points per case exceeds, say, 20, a pass through the cases with this provisional prior will give a good approximation to the population quantities. The revised prior can be saved to an external file via the SAVE command

`>SAVE means=’priors.par’;`

The file **priors.par **contains the estimated population mean and population covariance matrix of the model parameters.

A second or third pass, each time substituting the resulting provisional prior will yield a sufficiently accurate estimation of the population mean and covariance matrix for practical use in MAP estimate of the model parameters. The prior obtained from the previous run is read into the program via the commands

`>MEAN MALE FILE = ‘male.mea’;`

`>COVARIANCE MALE FILE = ‘male.cov’;`

See examples **exampl10.axl** and **exampl11.axl**.