The Performance of Multilevel Growth Curve Models Under an Autoregressive Moving Average Process
Daniel L. Murphy & Keenan A. Pituch
The present study have systematically evaluated the use of an incorrect serial correlation specification when another type of serial correlation underlies the data and examined the performance of growth curve models when an unstructured (UN) Level-1 covariance matrix is used to model serial correlation processes.
It accomplished that by fitting alternative within-participant covariance matrixes to multilevel linear growth curve models when a first-order ARMA process is present in the data.
The linear growth model was reviewed in detail, in which the problems of misspecifying serial correlation were brought out. Some alternatives methods to Modeling Serial Correlation: Ignoring Serial Correlation: Variance Components (VC), Stationary Autoregressive Models [AR(1)], Moving Average Models [MA(1)], Autoregressive Moving Average Models [ARMA(1, 1)], Unstructured Models (UN), Flexibility of the ARMA(1, 1) Approach. Each model has its merit and disadvantage.
Research questions:
The performance of a two-level linear growth model when an ARMA(1, 1) process is present in the data and the data are modeled as VC, AR(1), ARMA(1, 1), and UN were examined.
Simulation study:
Monte Carlo methods were used to generate data with SAS/IML andanalyzed with the SAS PROC MIXED procedure.
The data were generated by a 2 (autoregressive parameter) ×2 (moving average parameter) × 2 (sample size) × 2 (series length) factorial design having 16 cells. A repeated measures factor with four levels, was
crossed with all study factors. Thus, the study design had 64 cells. For each cell, a total of 10,000 data sets were generated.
Relative Bias, Type I error rates, fit indexes RLRT, AIC, AICC, and BIC were for each simulated data set analyzed under the four covariance-structure specifications. The effect sizes were calculated.
Most of the results support the previous research. With the exception of elevated Type I error rates that occurred under some conditions, the best performance was obtained by use of an unstructured covariance matrix at the first level of the growth curve model.
Comments:
1 it is a long paper. The research procedure is described very concrete. The reasons about why the author choose one variable or do what is explained very clear, which makes the paper very long.
2 In formulation 2b (p258), the coefficient B00 before x1i should be replaced by B01 which represents the fixed effect of explanatory variable x1 on an individual’s starting value.
3 I am a little confused. Since the Σ was correctly specified as ARMA(1, 1), why the estimation of the variance component is not better than the situation when Σ was misspecified as AR(1), and misspecified as VC.（Table3）
4 It seems the author choose the simulation variable is the same with the former research, so that it can have good support from other paper. If he includes more situations rather than just cite the former researchers’, the research may have more information about the results.