Analysis and results: Looking back in L2 text production

1 Data analysis

All data were analysed in Bayesian mixed-effects models using the R package brms (Bürkner, 2017b, 2017a) and the probabilistic programming language Stan (Carpenter et al., 2016; Hoffman & Gelman, 2014). All models were fitted with random intercepts for participants and by-participant slope adjustments for Language (text written in L1 or L2). As predictors we used combinations of Language (levels: L1, L2), Transitions location (levels: before sentence, before word, within word), Lookback (levels: lookback, no lookback), Edit (levels: editing, no editing) and their interactions. Edit and Lookback will be used as predictor variables in some analysis; yet both are observed and will be analysed separately. Analyses on lookbacks and transition durations will focus on transition events that terminated in an ongoing writing event, not in a editing operation.

We report the most probable posterior (i.e. inferred) parameter value \(\beta\) as well as the interval around \(\beta\) that contains 95% of the posterior probability mass; 95% probability intervals (henceforth, PI).

We calculated the statistical support for the effects of interest and the support for the alternative hypothesis over the null hypothesis. This evidence was obtained using Bayes Factors (henceforth, BF\(_{10}\)) calculated using the Savage-Dickey method (see, e.g., Dickey et al., 1970; Wagenmakers et al., 2010). A BF\(_{10}\) larger than 5 indicate moderate and larger than 10 strong evidence for a statistically meaningful effect compared to the null hypothesis (see, e.g., Baguley, 2012; Jeffreys, 1961; Lee & Wagenmakers, 2014). For example BF\(_{10}\)=2 reflect that the alternative hypothesis is two times more likely than the null hypothesis given the evidence. Priors for all effects were weakly informative. We used these weakly informative priors favoring the null hypothesis over the alternative hypothesis for the slope parameters because BFs are sensitive to the distribution of the prior. Thus, our priors are not favoring the alternative hypothesis.

Transition duration and total-fixation duration were analysed in mixture models as described in Roeser et al. (2021). This is important for both variables to distinguish between relatively normal durations and those that results from processing difficulty are higher levels of activation. For the fixation duration this is related to different types of activities such as re-reading of a single word to refresh the writer’s memory and more demanding events such as reading entire sentences to ensure text coherence. For transition durations we must distinguish between normal transition durations associated with a smooth flow of activation from higher levels of activation into finger movements and transitions that were inhibited by, for example, difficulty to retrieve correct spelling or the right word. Rather than imposing threshold values to distinguish between simple and more demanding events, mixture models model the data as a combination of two processes using a mixing weight to capture the probability of processing difficulty occurring. The random-effects structure of the mixture models is identical to the mixed-effects models. In addition, mixture models included by-participant mixing proportions allowing to capture individual differences in typing style (Roeser et al., 2021).

Data were analysed in Bayesian mixed effects models (Gelman et al., 2014; McElreath, 2016). The R (R Core Team, 2020) package rstan (Stan Development Team, n.d.) was used to interface with the probabilistic programming language Stan (Carpenter et al., 2016) which was used to implement all models. Models were fitted with weakly informative priors (see McElreath, 2016), and run with at least 10,000 iterations on 6 chains with a warm-up of 5,000 iterations and no thinning. Model convergence was confirmed by the Rubin-Gelman statistic (\(\hat{R}\) = 1) (Gelman & Rubin, 1992) and inspection of the Markov chain Monte Carlo chains.

2 Text data

2.1 Sample

Data come from 39 participants of which each produced a text in their L1 (English) or their L2 (Spanish).

Table 2.1: Total number of events for analysis by language.
Participant id L1 L2
1 2938 2105
2 3542 2028
3 2041 985
4 2361 1973
5 4009 2012
6 721 858
7 1988 1164
8 2477 1394
9 2523 1037
10 3572 1956
11 3141 1310
12 5561 1783
13 3413 2256
14 3266 1692
15 3550 1114
16 3594 1496
17 3722 2559
18 1582 1266
19 5246 2593
20 3042 2267
21 2772 1629
22 2960 2468
23 6164 3661
24 2116 1108
25 5900 2095
26 3522 1230
27 6024 4520
28 4366 2315
29 3204 2023
30 5228 3360
31 4807 3316
32 2257 1606
33 2854 809
34 3703 1090
35 4292 2782
36 7241 2790
37 4467 1177
38 2385 1957
39 1528 946

2.2 Editing frequency

# Run model
#source("../scripts/brms_editing_frequency_simple.R")

# Load model posterior
fit <- readRDS(file = "../stanout/editing_frequency.rda")
fit$formula
edits | trials(total) ~ 1 + condition + (Lang | SubNo)

condition was coded with main effects of Language (levels: L1, L2), Transition location 1 (levels: before sentence, before word), Transition location 2 (levels: before word / sentence, within word) and all two and three-way interactions by-Transition location (i.e. no interactions that involve both Transition location 1 and Transition location 2). An analysis of editing frequency separated by Lookbacks can be found in the appendix.

fit$family

Family: binomial 
Link function: logit
Table 2.2: Editing frequency effects on logit scale.
Predictor Estimate with 95% PI \(BF_{10}\)
Main effects
Language (L1, L2) -1.47 [-1.76 – -1.17] > 100
Location 1 (before sentence, before word) 1.7 [1.49 – 1.9] > 100
Location 2 (before word / sentence, within word) 3.82 [3.59 – 4.03] > 100
Two-way interactions
Language : Location 1 0.15 [-0.06 – 0.35] 0.58
Language : Location 2 -0.3 [-0.51 – -0.09] 8.53
Note:
Colon indicates interactions. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis
\label{fig:editingfig1}Estimated cell means for editing frequency with 95\% PIs (probability intervals).

Figure 2.1: Estimated cell means for editing frequency with 95% PIs (probability intervals).

Table 2.3: Editing frequency. Cell means for L1 and L2 in proportion and language difference on logit scale both shown with 95% PIs in brackets.
Transition location L1 L2 Language effect \(BF_{10}\)
within word 0.05 [0.05 – 0.06] 0.07 [0.07 – 0.08] -0.41 [-0.5, -0.33] >100
before word 0.07 [0.07 – 0.08] 0.13 [0.12 – 0.14] -0.65 [-0.75, -0.55] >100
before sentence 0.17 [0.15 – 0.2] 0.25 [0.21 – 0.28] -0.57 [-0.78, -0.36] >100
Note:
PIs are probability intervals. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.

2.3 Number of transitions

# Run model
#source("../scripts/brms_transition_counts.R")
# Load model posterior
fit <- readRDS(file = "../stanout/transition_counts.rda")
fit$formula
n_transitions ~ 1 + condition + (Lang | SubNo)

condition was coded with main effects of Language (levels: L1, L2), Edit (levels: editing, no editing), Transition location 1 (levels: before sentence, before word), Transition location 2 (levels: before word / sentence, within word) and all two and three-way interactions by-Transition location (i.e. no interactions that involve both Transition location 1 and Transition location 2). An analysis of transition counts separated by Lookback can be found in the appendix.

fit$family

Family: negbinomial 
Link function: log

In addition we fitted a binomial model to account for the overall number of transitions by participant and language.

# Run model
#source("../scripts/brms_transition_binomial.R")
# Load model posterior
fit_binom <- readRDS(file = "../stanout/transition_binomial.rda")
fit_binom$formula
n_transitions | trials(total) ~ condition + (Lang | SubNo) 
fit_binom$family

Family: binomial 
Link function: logit
Table 2.4: Transition-count coefficients.
log number of transitions
logit prop. of transitions
Predictor Estimate with 95% PI \(BF_{10}\) Estimate with 95% PI \(BF_{10}\)
Main effects
Language (L1, L2) 1.91 [1.26 – 2.52] > 100 -1.42 [-1.6 – -1.25] > 100
Location 1 (before sentence, before word) -9.59 [-9.92 – -9.27] > 100 -9.51 [-9.68 – -9.34] > 100
Location 2 (before word / sentence, within word) -18.14 [-18.63 – -17.64] > 100 -23.17 [-23.37 – -22.98] > 100
Edit (edit, no edit) -11.84 [-12.21 – -11.46] > 100 -14.48 [-14.65 – -14.3] > 100
Two-way interactions
Language : Location 1 -0.47 [-0.8 – -0.15] 10.42 -0.44 [-0.61 – -0.27] > 100
Language : Location 2 -0.43 [-0.89 – 0.03] 1.23 -0.4 [-0.59 – -0.2] > 100
Language : Edit -1.49 [-1.85 – -1.12] > 100 -1.44 [-1.62 – -1.27] > 100
Location 1 : Edit 2.12 [1.8 – 2.44] > 100 2.78 [2.61 – 2.95] > 100
Location 2 : Edit 4.15 [3.68 – 4.61] > 100 9.53 [9.33 – 9.72] > 100
Three-way interactions
Location 1 : Edit : Language 0.17 [-0.15 – 0.49] 0.28 0.3 [0.12 – 0.47] 50.35
Location 2 : Edit : Language -0.3 [-0.77 – 0.15] 0.52 -0.14 [-0.33 – 0.05] 0.53
Note:
Colon indicates interactions. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis
\label{fig:transfig}Estimated cell means for transition counts with 95\% PIs (probability intervals).

Figure 2.2: Estimated cell means for transition counts with 95% PIs (probability intervals).

Table 2.5: Transition counts and percentages. Cell means for L1 and L2 as counts and percentages, and their respective language differences on log scale. 95% PIs in brackets.
Number of transitions
% transitions
Transition location L1 L2 Language effect \(BF_{10}\) L1 L2 Language effect \(BF_{10}\)
Editing
within word 129 [109 – 152] 103 [88 – 121] 0.22 [0.07 – 0.38] 3.63 4.1 [4 – 4.2] 5.9 [5.7 – 6.1] -0.4 [-0.43 – -0.35] > 100
before word 50 [42 – 59] 45 [38 – 53] 0.11 [-0.06 – 0.27] 0.19 1.6 [1.5 – 1.7] 2.6 [2.5 – 2.7] -0.49 [-0.56 – -0.43] > 100
before sentence 6 [5 – 7] 6 [5 – 8] -0.07 [-0.3 – 0.17] 0.14 0.2 [0.2 – 0.2] 0.4 [0.3 – 0.4] -0.66 [-0.83 – -0.48] > 100
Writing
within word 2,427 [2,051 – 2,867] 1,303 [1,117 – 1,522] 0.62 [0.47 – 0.77] > 100 74 [74 – 75] 74 [73 – 74] 0.04 [0.02 – 0.06] 4.76
before word 625 [528 – 737] 293 [251 – 342] 0.76 [0.6 – 0.9] > 100 19 [19 – 19] 17 [16 – 17] 0.17 [0.14 – 0.19] > 100
before sentence 27 [22 – 32] 17 [14 – 20] 0.45 [0.27 – 0.63] > 100 0.8 [0.8 – 0.9] 0.9 [0.9 – 1] -0.15 [-0.25 – -0.06] 10.34
Note:
PIs are probability intervals. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.

3 Transition duration

This analysis focuses on transition duration collapsing both lookback and non-looksback transitions. Transition duration shorter than (or equal to) 50 msecs (M=2.13%, SE=0.29%) or longer than (or equal to) 30 secs (M=0.05%, SE=0.01%) were removed from the analysis.

Selected a random subset of 200 observations per participant, Language, Location, and Editing to reduce the time and computational power necessary to run the statistical models. This mean, all before-sentence transitions were included in the analysis, and a sub-sample of before-word transitions (M=45.3%, SE=7.97%) and within-word transitions (M=11.9%, SE=5.19%).

Table 3.1: Mixture model results of the transition duration with predictor estimates for the distribution of slowdown for long transitions (on log scale) and the probability of long transitions (on logit scale). Estimates are shown with 95% PI.
Slowdown for long transitions
Probability of long transitions
Predictor Estimate \(BF_{10}\) Estimate \(BF_{10}\)
Main effects
Language (L1, L2) -0.38 [-0.48 – -0.26] > 100 -1.49 [-1.89 – -1.1] > 100
Location 1 (before sentence, before word) 1.07 [0.99 – 1.16] > 100 0.93 [0.47 – 1.41] > 100
Location 2 (before word / sentence, within word) 1.1 [0.9 – 1.25] > 100 3.48 [3.03 – 3.94] > 100
Two-way interactions
Language : Location 1 0.15 [-0.02 – 0.31] 0.39 0.51 [-0.43 – 1.42] 0.86
Language : Location 2 0.09 [-0.18 – 0.41] 0.16 0.63 [-0.23 – 1.48] 1.22
Note:
Colon indicates interactions. PI is the probability interval. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.
Estimated cell means of transition duration with 95\% PIs (probability intervals). Shown are the estimates for the mixture component of long long transition durations in msecs (on log scale) and the probability of long transition durations.

Figure 3.1: Estimated cell means of transition duration with 95% PIs (probability intervals). Shown are the estimates for the mixture component of long long transition durations in msecs (on log scale) and the probability of long transition durations.

Table 3.2: Transition duration. Cell means for L1 and L2 with values in msecs for durations and proportion for probability of long transition durations. Language difference are shown on log scale (for transition durations) and logit scale for probability of long transition durations. 95% PIs in brackets.
Transition location L1 L2 Language effect \(BF_{10}\)
Short transitions
overall 153 [142 – 163] 153 [142 – 163]
Long transitions
before sentence 1,366 [1,199 – 1,549] 1,912 [1,669 – 2,180] -0.34 [-0.49 – -0.18] > 100
before word 433 [400 – 468] 703 [652 – 756] -0.48 [-0.54 – -0.43] > 100
within word 271 [204 – 379] 370 [333 – 413] -0.32 [-0.58 – -0.02] 1.09
Probability of long transitions
before sentence .78 [.70 – .85] .94 [.90 – .97] -1.44 [-2.19 – -0.73] > 100
before word .53 [.43 – .62] .88 [.84 – .92] -1.95 [-2.53 – -1.38] > 100
within word .08 [.04 – .14] .20 [.14 – .27] -1.07 [-1.79 – -0.36] 32.56
Note:
PIs are probability intervals. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.
Mixture model plot for before-word transition durations. Shown are the distributions for both mixture components (short and long transition durations) and their respective weighting for every language group.

Figure 3.2: Mixture model plot for before-word transition durations. Shown are the distributions for both mixture components (short and long transition durations) and their respective weighting for every language group.

Model comparisons between the mixture model and a uni-modal unequal variance linear mixed effects model.

Table 3.3: Predictive performance was estimated as the expected log predictive density (\(\widehat{elpd}\)) (Vehtari et al., 2015, 2017). A negative difference \(\Delta\widehat{elpd}\) denotes lower predictive performance compared to the model with the highest predictive performance (top row). Standard error is shown in parentheses.
Model \(\Delta\widehat{elpd}\) \(\widehat{elpd}\)
Mixture model 0 (0) -215,705 (313)
LMM (unequal variance) -2,970 (98) -218,675 (331)
Note. LMM = linear mixed-effects model.

4 Reading during writing

4.1 Lookback probability

# Run model
#source("../scripts/brms_lookback.R")

# Load posterior
fit <- readRDS("../stanout/lookback.rda")
fit$formula
lookbacks | trials(total) ~ condition + (Lang | SubNo)

condition was coded with main effects of Language (levels: L1, L2), Transition location 1 (levels: before sentence, before word), Transition location 2 (levels: before word / sentence, within word) and all two-way interactions by-Transition location (i.e. no interactions that involve both Transition location 1 and Transition location 2).

fit$family

Family: binomial 
Link function: logit
Table 4.1: Lookback probability on logit scale.
Predictor Estimate with 95% PI \(BF_{10}\)
Main effects
Language (L1, L2) -1.37 [-1.98 – -0.71] > 100
Location 1 (before sentence, before word) 4 [3.77 – 4.22] > 100
Location 2 (before word / sentence, within word) 5.31 [5.15 – 5.49] > 100
Two-way interactions
Language : Location 1 0.86 [0.64 – 1.08] > 100
Language : Location 2 0.02 [-0.15 – 0.19] 0.18
Note:
Colon indicates interactions. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis
\label{fig:lookbackfig}Estimated cell means for lookback probability with 95\% PIs (probability intervals).

Figure 4.1: Estimated cell means for lookback probability with 95% PIs (probability intervals).

Table 4.2: Lookback probability. Cell means for L1 and L2 are proportions, and language difference on logit scale. 95% PIs in brackets.
Transition location L1 L2 Language effect \(BF_{10}\)
within word 0.003 [0.002 – 0.004] 0.008 [0.006 – 0.012] -1.1 [-1.38, -0.84] > 100
before word 0.041 [0.033 – 0.052] 0.108 [0.083 – 0.149] -1.05 [-1.29, -0.82] > 100
before sentence 0.342 [0.289 – 0.408] 0.396 [0.32 – 0.492] -0.24 [-0.54, 0.06] 0.52
Note:
PIs are probability intervals. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.

4.2 Total lookback duration (constrained)

Short lookback durations are distributed around a mean of 821 msecs (95% PI: [739, 908]).

Table 4.3: Mixture model results of the total-fixation duration with the predictor value estimates for the distribution long fixations (on log scale) and the probability of long fixations (on logit scale). Estimates are shown with 95% PI.
Slowdow for long lookbacks
Probability of long lookbacks
Predictor Estimate \(BF_{10}\) Estimate \(BF_{10}\)
Main effects
Language (L1, L2) -0.25 [-0.41 – -0.08] 4.27 0.25 [-0.51 – 1.02] 0.45
Location 1 (before sentence, before word) 0.33 [0.11 – 0.55] 7.22 -1.46 [-2.17 – -0.82] > 100
Location 2 (before word / sentence, within word) 0.8 [0.57 – 0.98] > 100 -1.05 [-2.13 – -0.02] 4.8
Two-way interactions
Language : Location 1 -0.31 [-0.66 – 0.04] 0.77 -0.36 [-1.52 – 0.84] 0.73
Language : Location 2 -0.3 [-0.68 – 0.12] 0.82 -1 [-3.08 – 1] 1.71
Note:
Colon indicates interactions. PI is the probability interval. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.
Estimated cell means of total-fixation durations during lookbacks with 95\% PIs (probability intervals). Shown are the estimates for each mixture components (short and long fixations) and the probability of long fixations

Figure 4.2: Estimated cell means of total-fixation durations during lookbacks with 95% PIs (probability intervals). Shown are the estimates for each mixture components (short and long fixations) and the probability of long fixations

Table 4.4: Total lookback duration of lookbacks. Cell means for L1 and L2 with values in msecs for durations and proportion for probability of long fixations. Language difference are shown on log scale (for fixation durations) and logit scale for probability of long fixations. 95% PIs in brackets.
Transition location L1 L2 Language effect \(BF_{10}\)
Short transitions
overall 820 [739 – 908] 820 [739 – 908]
Slowdown for long fixations
before sentence 2,577 [2,084 – 3,190] 3,500 [2,802 – 4,342] -0.31 [-0.57, -0.04] 1.77
before word 2,160 [1,640 – 2,857] 2,146 [1,809 – 2,558] 0 [-0.23, 0.23] 0.12
within word 913 [777 – 1,306] 1,437 [1,103 – 1,936] -0.45 [-0.79, -0.06] 5.07
Probability of long fixations
before sentence .75 [.61 – .87] .81 [.69 – .92] 0.4 [-0.59, 1.45] 0.66
before word .37 [.24 – .53] .56 [.44 – .67] 0.76 [0.18, 1.33] 8.09
within word .44 [.10 – .81] .34 [.15 – .59] -0.42 [-2.41, 1.54] 0.96
Note:
PIs are probability intervals. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.

Model comparisons between the mixture model and a uni-modal unequal variance linear mixed effects model.

Table 4.5: Predictive performance was estimated as the expected log predictive density (\(\widehat{elpd}\)) (Vehtari et al., 2015, 2017). A negative difference \(\Delta\widehat{elpd}\) denotes lower predictive performance compared to the model with the highest predictive performance (top row). Standard error is shown in parentheses.
Model \(\Delta\widehat{elpd}\) \(\widehat{elpd}\)
Mixture model 0 (0) -29,371 (81)
LMM (unequal variance) -85 (15) -29,456 (81)
Note. LMM = linear mixed-effects model.

4.3 Total lookback duration (unconstrained)

Table 4.6: Mixture model results of the total-fixation duration with the predictor estimates for the distribution of short and long fixations (on log scale) and the probability of long fixations (on logit scale). Estimates are shown with 95% PI.
Short fixations
Slowdow for long fixations
Probability of long fixations
Predictor Estimate \(BF_{10}\) Estimate \(BF_{10}\) Estimate \(BF_{10}\)
Main effects
Language (L1, L2) -0.2 [-0.31 – -0.07] 4.53 -0.08 [-0.31 – 0.15] 0.15 0.06 [-0.56 – 0.69] 0.32
Location 1 (before sentence, before word) 0.15 [-0.15 – 0.47] 0.17 0.37 [-0.02 – 0.7] 1.19 -1.02 [-2.7 – 0.57] 1.10
Location 2 (before word / sentence, within word) 0.14 [-0.03 – 0.32] 0.22 0.73 [0.38 – 1.05] 89.17 -1.46 [-2.67 – -0.28] 11.76
Two-way interactions
Language : Location 1 -0.14 [-0.45 – 0.23] 0.28 -0.27 [-0.69 – 0.15] 0.48 -0.62 [-1.8 – 0.65] 1.08
Language : Location 2 -0.03 [-0.25 – 0.17] 0.11 0.03 [-0.57 – 0.62] 0.30 -0.24 [-1.82 – 1.32] 0.80
Note:
Colon indicates interactions. PI is the probability interval. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.
Estimated cell means of total-fixation durations during lookbacks with 95\% PIs (probability intervals). Shown are the estimates for each mixture components (short and long fixations) and the probability of long fixations

Figure 4.3: Estimated cell means of total-fixation durations during lookbacks with 95% PIs (probability intervals). Shown are the estimates for each mixture components (short and long fixations) and the probability of long fixations

Table 4.7: Total-fixation duration of lookbacks. Cell means for L1 and L2 with values in msecs for durations and proportion for probability of long fixations. Language difference are shown on log scale (for fixation durations) and logit scale for probability of long fixations. 95% PIs in brackets.
Transition location L1 L2 Language effect \(BF_{10}\)
Short fixations
before sentence 884 [661 – 1,165] 1,167 [734 – 1,723] -0.26 [-0.55 – 0.1] 0.55
before word 812 [714 – 918] 914 [805 – 1,031] -0.12 [-0.21 – -0.03] 1.49
within word 726 [631 – 826] 899 [795 – 1,009] -0.21 [-0.34 – -0.09] 22.24
Slowdown for long fixations
before sentence 2,228 [1,212 – 3,772] 4,045 [1,903 – 8,120] -0.22 [-0.56 – 0.12] 0.42
before word 1,414 [896 – 2,228] 1,463 [1,049 – 2,026] 0.05 [-0.19 – 0.3] 0.13
within word 376 [40 – 959] 549 [115 – 1,322] -0.06 [-0.62 – 0.5] 0.27
Probability of long fixations
before sentence 0.36 [0.1 – 0.67] 0.4 [0.09 – 0.76] -0.17 [-1.11 – 0.91] 0.55
before word 0.65 [0.49 – 0.8] 0.55 [0.41 – 0.69] 0.45 [-0.2 – 1.11] 0.82
within word 0.77 [0.57 – 0.93] 0.79 [0.61 – 0.93] -0.1 [-1.57 – 1.35] 0.71
Note:
PIs are probability intervals. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.

Model comparisons between the constrained and the unconstrained mixture models and a uni-modal unequal variance linear mixed effects model (next section).

Table 4.8: Predictive performance was estimated as the expected log predictive density (\(\widehat{elpd}\)) (Vehtari et al., 2015, 2017). A negative difference \(\Delta\widehat{elpd}\) denotes lower predictive performance compared to the model with the highest predictive performance (top row). Standard error is shown in parentheses.
Model \(\Delta\widehat{elpd}\) \(\widehat{elpd}\)
Mixture model (constrained) 0 (0) -29,371 (81)
Mixture model (unconstrained) -1 (5) -29,372 (81)
LMM (unequal variance) -85 (15) -29,456 (81)
Note. LMM = linear mixed-effects model.

4.4 Total lookback duration (uni-modal unequal variance mixed-effects model)

Table 4.9: LMM (with unequal variance) results of the total-fixation duration with the predictor estimates (on log scale). Estimates are shown with 95% PI.
Predictor Estimate \(BF_{10}\)
Main effects
Language (L1, L2) -0.26 [-0.37 – -0.16] > 100
Language : Location 1 -0.13 [-0.31 – 0.05] 0.24
Language : Location 2 0.04 [-0.09 – 0.17] 0.08
Two-way interactions
Location 1 (before sentence, before word) 0.58 [0.49 – 0.67] > 100
Location 2 (before word / sentence, within word) 0.65 [0.58 – 0.72] > 100
Note:
Colon indicates interactions. PI is the probability interval. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.
Estimated cell means of total-fixation durations during lookbacks with 95\% PIs (probability intervals).

Figure 4.4: Estimated cell means of total-fixation durations during lookbacks with 95% PIs (probability intervals).

Table 4.10: LMM of total-fixation duration of lookbacks. Cell means for L1 and L2 with values in msecs. Language difference are shown on log scale. 95% PIs in brackets.
Transition location L1 L2 Language effect \(BF_{10}\)
before sentence 1,932 [1,677 – 2,216] 2,711 [2,309 – 3,162] -0.34 [-0.52 – -0.15] 49.37
before word 1,153 [1,037 – 1,274] 1,423 [1,281 – 1,575] -0.21 [-0.32 – -0.11] 51.64
within word 795 [707 – 891] 1,004 [898 – 1,119] -0.23 [-0.36 – -0.11] 31.74
Note:
PIs are probability intervals. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.

4.5 Number of words fixated

# Run model
#source("../scripts/brms_nwordsfixated.R")

# Load posterior
fit <- readRDS("../stanout/nwordsfixated.rda")
fit$formula
fix_nwords ~ 1 + condition + (Lang | SubNo)

condition was coded with main effects of Language (levels: L1, L2), Transition location 1 (levels: before sentence, before word), Transition location 2 (levels: before word / sentence, within word) and all two-way interactions by-Transition location (i.e. no interactions that involve both Transition location 1 and Transition location 2).

fit$family

Family: negbinomial 
Link function: log
Table 4.11: Number of words fixated during lookback on log scale.
Predictor Estimate with 95% PI \(BF_{10}\)
Main effects
Language (L1, L2) -0.32 [-0.63 – -0.01] 2.66
Location 1 (before sentence, before word) 1.57 [1.43 – 1.71] > 100
Location 2 (before word / sentence, within word) 3.81 [3.49 – 4.12] > 100
Two-way interactions
Language : Location 1 -0.22 [-0.36 – -0.08] 21.13
Language : Location 2 -0.26 [-0.57 – 0.06] 1.24
Note:
Colon indicates interactions. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis
\label{fig:nwordfixfig}Estimated cell means for number of words fixated during lookback with 95\% PIs (probability intervals).

Figure 4.5: Estimated cell means for number of words fixated during lookback with 95% PIs (probability intervals).

Table 4.12: Number of words fixated during lookback. Cell means for L1 and L2 as counts, and language difference on log scale. 95% PIs in brackets.
Transition location L1 L2 Language effect \(BF_{10}\)
within word 2.55 [2.24 – 2.88] 2.6 [2.31 – 2.94] -0.06 [-0.24 – 0.11] 0.23
before word 4.42 [4.07 – 4.78] 4.59 [4.18 – 5.05] -0.05 [-0.16 – 0.06] 0.15
before sentence 8.67 [7.82 – 9.6] 11.26 [9.95 – 12.73] -0.29 [-0.45 – -0.14] > 100
Note:
PIs are probability intervals. \(BF_{10}\) is the evidence in favour of the alternative hypothesis over the null hypothesis.

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