Using PROC GENMOD for Loglinear Smoothing

AND INTRODUCTION The goal of smoothing is to replace an observed frequency distribution with a distribution that preserves some features of the observed data without the irregularities that are attributable to sampling. The type of smoothing covered in this paper involves the fitting of loglinear, Poisson-based models to discrete distributions. Loglinear smoothing can preserve a variety of different features in observed data with a relatively small number of parameters. In this paper we use SAS/STAT® PROC GENMOD (SAS, 2002) to demonstrate the smoothing of univariate (one variable) and bivariate (two variables and one variable for separate subgroups) frequency distributions. For univariate distributions, we will produce smoothed distributions that preserve 1) the mean, 2) the mean and variance, 3) the mean, variance and skewness, and finally 4) the mean, variance, skewness and kurtosis in the observed distribution of one variable, X. For bivariate distributions, we will produce smoothed distributions that preserve three univariate moments in each of the marginal distributions of two variables, X and Y, as well as the correlation between X and Y. Finally, the incorporation of indicator functions is used to model overall and subset-specific features of distributions within the same overall model. LOGLINEAR SMOOTHING MODELS Assume we have a discrete random variable X with possible values x0,...,xJ , or xj, with j=0,...,J (the possible values), and a corresponding vector of observed frequencies n = (n0,...,nJ) t that sum to the total sample size, N. Under multinomial or Poisson distributional assumptions about n, the vector of the population probabilities p = (p0,...,pJ ) t is said to satisfy the following loglinear model: log ( ) e j j p u α = + + j b where the {pj} are assumed to be positive and sum to 1, bj is a row vector of known constants, is a vector of free parameters, uj is a known constant that specifies the distribution of the {pj} when the vector is set to zero, and is a normalizing constant that insures that the probabilities sum to one (Holland & Thayer, 1987; 2000). Throughout this paper u will be set to 0 so that the “null” model will be a uniform distribution where the frequencies for all j score values are equal to N/J. For the modeling of test score distributions, we write the loglinear model as: 1 log ( ) ( ) I i e j i j i p x α β