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A collapsed coding technique for analyzing longitudinal data
Consider a model with "n" fixed blocks (blocks may be a convenience sample), "r + 1" fixed treatments (within blocks) and no block by treatment interactions. When "n" is large enough, and data are missing at random, an ordinary least squares (OLS) solution is infeasible due to the numerous parameter estimates required. If only treatment parameter estimates are desired, one may use an indirect estimation technique developed by Harvey (1970) and used in the SAS ABSORB procedure (SAS, 1992). This technique includes blocks in the model, but produces only treatment parameter estimates. Collapsed coding extends Harvey's technique. In collapsed coding, the standard design matrix of "n + r" columns, ordinarily employed to obtain an OLS solution, is "collapsed" into a design matrix of "1 + r" columns, using a pair of orthogonal projection matrices. Since the collapsed coding design matrix's column size is invariant to "n", a model of any size "n" can be fit. Using the collapsed design matrix, one may obtain fitted and imputed values, OLS model and error sums of squares, treatment parameter estimates, and hypothesis tests. For models fit noniteratively, one can obtain standardized residuals. SAS ABSORB cannot do these things, nor can OLS for large enough "n". The collapsed coding technique also extends to two models which are variations of the basic model. Neither model can be fit using SAS ABSORB, or by OLS for large "n". The first model contains between block or group variables; the second contains block variables, two treatments in a crossed design, and block by treatment interaction terms. For the first model, all collapsed coding estimates are identical to standard OLS results. The second model is fit iteratively. Treatment parameter estimates, fitted and imputed values, are identical to OLS results. Standardized residuals, and the estimated covariance matrix for treatment parameters, differ from the usual OLS results. Finally, the problem of non-estimable model parameters, caused by particular patterns of missing data, is considered for the second model. The iterative collapsed coding solution incorporates constraints on non-estimable model parameters to solve the OLS equations. Two ways of identifying non-estimable treatment parameters are presented.
Fitzgerald, Gordon Abbott, "A collapsed coding technique for analyzing longitudinal data" (1994). Doctoral Dissertations Available from Proquest. AAI9510468.