fixed estimate can be elicited by the command, ranef(myresult), after the model has run. Note that we used an identity matrix as our covariance structure (pdIdent), but other options may be more suitable for different situations (see , for more details). The argument start = c(g = 0.1,r = 1,0,0,0,….)specifies starting values for each predictor for each condition or grouping variable. The first two statements, g = 0.1 and r = 1, specify the starting values of each predictor for the first group, whereas the numbers in the subsequent list specify variation around the initial numbers for the other groups, if these predictors are allowed to vary across groups. Frankly, this can be the trickiest and most counterintuitive part of the analysis. If, as in the example above, the model allows both predictors to vary across three groups (fixed = c(g + r ~ group)), then there are six starting values in this list because there are three g estimates (one for each group) and three r estimates (one for each group). Therefore, the full argument is start = c(g = 0.1,r = 1,0,0,0,0). If the model allows only the g predictor to vary across the three groups (fixed = c(g ~ group, r ~ 1)), then there are four starting values in this list because there are three g estimates (one for each group) and one r estimate (the same for each group). If the model fails to converge, then different starting values can be used (e.g., a “0” can be replaced with small positive or negative values). Finally, the argument verbose=TRUE provides information for each iteration of the algorithm and can be deleted in order to speed up the computation. The detailed description of this and other commands in the nlme library is available at Once the model has converged on a solution, the following command can be used to display the results: summary(myresult)
selection process. The absence of a significant effect for a fixed predictor indicates that it has little effect on the predictive ability of the model. The absence of a significant effect for a predictor that is allowed to vary across the training groups indicates that the best-fitting predictors do not, in fact, differ significantly across training groups. The initial model always included generalization, g, and relational disparity, r, as predictors because our and others’ empirical data strongly suggested that both generalized associative strength and relational disparity play an important role in determining an organism’s behavior in the transposition task. We next established whether g and r produced the best fit when allowed to vary across training groups or when fixed. Finally, we added the fixed familiarity predictor, f, and if it improved the fit, then we allowed it to vary across conditions to find the best-fitting model. Model selection for Experiment 1: Small Circle condition Model 1: Two predictors (r and g), both vary AIC = -304.17 Gradient: F (2, 199) = 0.12, p = 0.89; no evidence that strength of g varies across groups Relational disparity: F (2, 199) = 29.31, p < 0.0001; significant difference in strength of r across groups Model 2: Two predictors, r varies, g is fixed AIC = -307.79 Gradient: F (1, 201) = 44.15, p < 0.0001; significantly improves explanatory power and should be retained Relational disparity: F (2, 201) = 38.89, p < 0.0001; same as above Model 3: Three predictors, r varies, g and f are fixed
AIC = -368.43 Gradient: F (1, 200) = 47.21, p < 0.0001; same as above Relational disparity: F (2, 200) = 18.43, p < 0.0001; same as above Familiarity: F (1, 200) = 48.35, p < 0.0001; significantly improves explanatory power, but it is not yet clear whether its strength should vary across groups Model 4: Three predictors, r and f vary, g is fixed AIC = -372.66 Gradient: F (1, 198) = 49.61, p < 0.0001; same as above Relational disparity: F (2, 198) = 17.17, p < 0.0001; same as above Familiarity: F (2, 198) = 4.27, p = 0.015; significant difference in strength of f across groups Final model: All three predictors included, strength of g is constant, strengths of r and f vary across groups Model selection for Experiment 1: Large Circle condition Model 1: Two predictors (r and g), both vary AIC = 4.07 Gradient: F (2, 199) = 1.52, p = 0.22; no evidence that strength of g varies across groups Relational disparity: F (2, 199) = 5.73, p = 0.004; significant difference in strength of r across groups Model 2: Two predictors, r varies, g is fixed AIC = 3.71 Gradient: F (1, 201) = 14.45, p = 0.0002; significantly improves explanatory power and should be retained as a fixed factor