The first set of acoustic analyses focused on the pretest-posttest differences in the performance of the intervention and the control groups. A series of linear mixed- effects models were fitted for the analysis of duration, intensity, and pitch measures. A recommended method for fitting a linear mixed-effects model in the case of a confirmatory research study is to create the maximal model that takes the study design and the research questions into consideration (Barr, 2013; Barr et al., 2013; Cunnings & Finlayson, 2015). Following this recommendation, the formula below was used when fitting the maximal model for the analysis of the proportional difference in duration:
maximal model for duration: lmer (duration ~ condition * time + (1 + time | learner) + (1 + time | word))
In this formula, the outcome variable duration appears on the left side of the tilde operator (~), and on the right side of the tilde are condition and time, which are the fixed effects. The star sign between condition and time allows the model to
account for the interaction between two fixed effects. It should be noted that condition and time are crossed effects as both experimental conditions have data points that correspond to the two levels under time, which are pretest and posttest. Because the main goal of the study was to observe changes in acoustic measures over time, the interaction between time and condition as fixed effects was the main interest of the analysis. The second part of the formula shows that the variables learner and word are entered into the model as a random effect with their own intercept. The two random effects are partially crossed as half of the learners in both experimental conditions produced one half of the target words and the other half in both conditions produced the other half of the target words. As can be seen in the formula, time was also used in the second part along with the two random effects. This allows the model to create the random slopes of learner-by-time and word-by-time as it was discussed earlier in Chapter 6.7.2.
However, it was not possible to use the maximal model for the analysis of syllable duration because the model failed to converge; and as a result, adjustments needed to be made. Convergence issues are not uncommon when fitting mixed-effects models and they usually occur due to model complexity. Model complexity is a relative term and it is directly related to the amount of data used for model estimation. When the model is too complex with a number of random intercepts and random slopes, the model will not be able to estimate the correlations between the random intercepts and the random slopes as it will run out of degrees of freedom. It is possible to handle this issue by simplifying the model. Among the various possibilities for
model simplification, there are two common methods. One method is identifying and removing the random effect that has the lowest variance. This method works well in the case of explanatory data analysis, a case where researchers try to discover patterns in the data, sometimes using a method of trial and error. However, in the current situation, the mixed-effects model is based on a particular research design and it serves the purpose of confirmatory analysis. Since both random effects are crucial in terms of generalizing findings and the issue of language as a fixed-effect fallacy as it was discussed in Chapter 6.7.2, removing the random effect learner or word was kept as a last resort. As an alternative option, Cunnings and Finlayson (2015) recommend simplifying the model by removing the correlation between a random intercept and a random slope. They explain that when the correlation parameter is present, the model tries to account for the possibility that a higher intercept may also have a higher slope. To give an example, a learner who produces stressed syllables with a higher average duration than other learners may exhibit higher gains in duration on the posttest. In the same way, a word that has been produced with a longer second syllable on average may display higher gains in duration on the posttest. In the current model, the syntax 1 + time generates a random intercept and a random slope as well as the interaction between the two. As there were two random effects in the model, and hence two correlations, as a first step the model was simplified by removing the correlation parameter from the random effect word using the following expression: (1|word) + (0 + time | word). The complete version of second model can be seen below.
second model for duration: lmer (duration ~ condition * time + (1 + time | learner) + (1|word) + (0 + time | word)
However, the second model also failed to converge, and therefore, the model was simplified further by removing the correlation parameter from the random effect learner. The third model converged and no error messages were received. Thus, the third model, which is provided below, was used for the analysis of duration measures.
third model for duration: lmer (duration ~ condition * time + (1| learner) + (0 + time | learner) + (1|word) + (0 + time | word))
For the analysis of syllable intensity, once again the maximal model was fitted similar to the process that was followed for the analysis of syllable duration:
maximal model for intensity: lmer (intensity ~ condition * time + (1 + time | learner) + (1 + time | word))
The maximal model fitted for the analysis of intensity failed to converge, necessitating the use of the model simplification methods that were followed for the analysis of duration. After removing the correlation parameter from the random effect word, the model still failed the converge. Thus, the model was simplified further by removing the correlation parameter from the random effect learner. The third model converged successfully and was used for the analysis of intensity (see Appendix F).
third model for intensity: lmer (intensity ~ condition * time + (1| learner) + (0 + time | learner) + (1|word) + (0 + time | word))
measures. Once again, the maximal model was fitted for the linear mixed-effects analysis of pitch measures. Unlike duration and intensity measures, the maximal modal converged without any errors in the case of pitch measure (see Appendix F).
maximal model for pitch: lmer (pitch ~ condition * time + (1 + time | learner) + (1 + time | word))