The GENMOD Procedure

Lagrange Multiplier Statistics

When you select the NOINT or NOSCALE option, restrictions are placed on the intercept or scale parameters. Lagrange multiplier, or score, statistics are computed in these cases. These statistics assess the validity of the restrictions, and they are computed as

chi squared equals StartFraction s squared Over upper V EndFraction

where s is the component of the score vector evaluated at the restricted maximum corresponding to the restricted parameter and upper V equals bold upper I 11 minus bold upper I 12 bold upper I 22 Superscript negative 1 Baseline bold upper I 21. The matrix bold upper I is the information matrix, 1 refers to the restricted parameter, and 2 refers to the rest of the parameters.

Under regularity conditions, this statistic has an asymptotic chi-square distribution with one degree of freedom, and p-values are computed based on this limiting distribution.

If you set k = 0 in a negative binomial model, s is the score statistic of Cameron and Trivedi (1998) for testing for overdispersion in a Poisson model against alternatives of the form upper V left-parenthesis mu right-parenthesis equals mu plus k mu squared.

See Rao (1973, p. 417) for more details.

Last updated: December 09, 2022