Testing dense and anticipatory affect in prediction updating models

1. Testing Competing Model Specifications

1.1 Distribution of updating response variable

Model formula:

## next_update_short ~ pe_2 + exam_f + (1 | class/id)

Comparison:

• Gaussian outcome distribution
• Asymmetric double exponential (asymmetric Laplace) distribution

1.1.1 Posterior predictive check: Gaussian response distribution

1.1.2 Posterior predictive check: Asymmetric Laplace response distribution

1.1.3 Comparing response distributions (LOO / WAIC)

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_delta_laplace	0.000	0.000	-6268.070	37.787	14.613	0.223	12536.14	75.575
md_update_delta_gaussian	-74.936	17.965	-6343.006	41.093	9.355	0.758	12686.01	82.186

NOTE: Asymmetric Laplace distribution yields optimal fit for prediction update outcome variable.

1.2 Including grade covariate in updating model

##                             elpd_diff se_diff   elpd_loo  se_elpd_loo p_loo    
## md_update_delta_grade           0.000     0.000 -5864.865    36.201     323.737
## md_update_delta_gradePE_int    -3.299     3.114 -5868.163    36.285     317.769
## md_update_delta_noGrade      -403.245    31.873 -6268.109    37.694      14.169
##                             se_p_loo  looic     se_looic 
## md_update_delta_grade          13.219 11729.729    72.401
## md_update_delta_gradePE_int    13.207 11736.327    72.571
## md_update_delta_noGrade         0.226 12536.218    75.388

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_delta_grade	0.000	0.000	-5846.952	36.376	305.825	12.891	11693.90	72.751
md_update_delta_gradePE_int	-4.398	2.971	-5851.350	36.430	300.956	12.861	11702.70	72.860
md_update_delta_noGrade	-421.122	32.100	-6268.074	37.693	14.133	0.225	12536.15	75.387

NOTE: Without affect in model, Grade covariate improves model fit. Interaction between Grade x PE yields slightly poorer fit, but WAIC difference is not interpretable.

1.3 Random effects structure

1.3.1 Random effects levels

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
Raneff.id.class.cohort	0.000	0.000	-5822.739	35.866	326.093	12.725	11645.48	71.731
Raneff.id.class	-22.574	10.607	-5845.313	36.244	303.438	12.897	11690.63	72.488
Raneff.id	-38.892	11.367	-5861.632	36.375	294.413	12.552	11723.26	72.750

NOTE: Maximal nested random effects structure (Cohort / Class / ID) yields optimal model fit.

1.3.2 Random effects slopes

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_grade_id_class_cohort_re_pe_grade_GRADEINT	0.000	0.000	-5506.731	35.036	255.796	7.881	11013.46	70.071
md_grade_id_class_cohort_re_pe_grade	-58.603	8.661	-5565.335	34.352	334.582	8.917	11130.67	68.704
md_grade_id_class_cohort_re_pe	-284.469	19.346	-5791.200	35.763	370.089	13.988	11582.40	71.525
md_grade_id_class_cohort_re_pe_GRADEINT	-292.579	19.076	-5799.311	35.685	357.094	13.539	11598.62	71.371
Raneff.id.class.cohort	-316.008	20.101	-5822.739	35.866	326.093	12.725	11645.48	71.731
md_id_class_cohort_re_pe	-691.544	35.246	-6198.275	37.760	167.023	5.152	12396.55	75.521

NOTE: Including random slopes for each predictor (1 + pe_2 + grade | Cohort / Class / ID) yields optimal model fit. With maximal random effects, prediction updating is best explained by the Grade x PE interaction.

1.4 Nonlinear PE effects

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_grade_id_class_cohort_re_pe_grade_GRADEINT	0.000	0.000	-5506.731	35.036	255.796	7.881	11013.46	70.071
md_update_delta_grade.id.class.nlPExGrade	-77.675	11.726	-5584.406	35.531	337.161	12.020	11168.81	71.061
md_update_delta_grade.id.class.nlPE.rePE_grade	-81.649	11.852	-5588.381	35.209	354.969	11.616	11176.76	70.419
md_update_delta_grade.id.class.nlPE_noGrade	-681.186	34.822	-6187.918	37.687	163.231	5.137	12375.83	75.373

NOTE: Model with linear Grade x PE interaction fits the data better than models with nonlinear PE effects.

1.5 Excluding high-grade observations

Parameter estimates

NOTE: Here, Updating ~ PE parameter estimates are displayed for models excluding grades above 90%, 80%, and outside the 25th and 75th percentiles for grades, respectively. Point estimates for Updating ~ PE effect increase with exclusion of grades above 80% and 90%, though credible intervals widen with the exclusion of these observations.

Model comparison: grades < 90%

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_updating_truncated_grades_90_wGradeInt	0.000	0.000	-4824.476	30.610	286.377	11.341	9648.951	61.219
md_updating_truncated_grades_90_wGrade	-0.029	1.406	-4824.504	30.692	288.626	11.502	9649.009	61.384
md_updating_truncated_grades_90	-317.471	27.205	-5141.947	31.796	24.879	0.440	10283.893	63.592

NOTE: When excluding observations with exam grades above 90%, a model with a linear Grade x PE interaction still fits the data best, but does not differ appreciably in WAIC relative to a model with only Grade and PE main effects.

Model comparison: grades < 80%

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_updating_truncated_grades_80_wGrade	0.000	0.00	-3514.616	25.437	251.363	10.076	7029.232	50.875
md_updating_truncated_grades_80_wGradeInt	-7.794	1.55	-3522.410	25.390	251.154	10.014	7044.820	50.780
md_updating_truncated_grades_80	-244.814	21.87	-3759.430	26.084	30.618	0.539	7518.861	52.168

NOTE: When excluding observations with exam grades above 80%, the model with a linear Grade x PE interaction no longer fits the data best. Instead, a model with Grade and PE main effects yields a superior model fit. While AICs do not differ substantially between the interactive and main effects model, the elpd_diff between both models is interpretable and favors the main effect model. Thus, it appears that the Grade x PE interaction improves model fit for high-grade observations where subjects reach a ceiling in grade prediction updates.

1.6 Updating without “bad” prediction data

Here, subjects who reported exam grade expectations of 50 for both their first and second predictions for any exam are excluded prior to model fitting. This represents the most stringent exclusion criteria to remove so-called “bad prediction” subjects. Below, updating models are refit following the application of this maximally stringent exclusion criterion to see A) if best-fitting models shift with this subset of participants, and B) if parameter estimates for the Updating ~ PE effect change appreciably.

Prediction 1 data

NOTE: 5.93% of Prediction 1 data is considered “bad data” by this standard.

Prediction 2 data

NOTE: 7.25% of Prediction 1 data is considered “bad data” by this standard.

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
updating_rm_bad_preds_gradeInt.brm	0.000	0.000	-4918.892	33.820	291.174	8.329	9837.785	67.640
updating_rm_bad_preds_grade.brm	-72.901	6.887	-4991.794	33.798	324.452	10.525	9983.587	67.595
updating_rm_bad_preds.brm	-584.054	32.689	-5502.946	36.918	140.381	4.591	11005.893	73.837

NOTE: 10.88% of subjects are excluded after applying the maximally stringent exclusion criterion. Regardless of whether these so-called “bad prediction” subjects are included or excluded, a model with a linear PE x Grade interaction fits the data best.

1.6.1 Conditional effects with and without bad prediction data

All prediction data

Removing “bad prediction” observations

NOTE: Grade x PE interaction is significantly different from zero in models both including and excluding “bad prediction” subjects. Credible intervals change appreciably after removing “bad prediction” subjects such that certainty in effects.

2. Dense Sampling Affect as Moderator of Updating Effect

2.1 Testing Negative Affect x PE Interaction

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_na5h_int	0.000	0.000	-6141.510	37.039	19.790	0.405	12283.02	74.077
md_update_DL_na5h_noInt	-2.057	3.173	-6143.566	37.184	17.778	0.359	12287.13	74.369

NOTE: In a model without grade covariate or random slopes, including the PE * NA interaction improves model fit, but not appreciably. WAIC estimates overlap, and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	1.181	8.680	-11.749	34.574	1.011	245	64
pe_2	0.307	0.033	0.233	0.372	1.006	325	56
NA_score_5h_mean_bc	0.170	0.022	0.109	0.211	1.010	249	45
exam_f	-0.573	0.329	-1.346	0.035	1.012	238	62
pe_2:NA_score_5h_mean_bc	0.004	0.001	0.001	0.007	1.004	980	1224

NOTE: Interaction effect is relatively weak, but shows that NA primarily differentiates updating following higher, positive PEs. Higher mean NA during the dense sampling period predicts a steeper updating effect.

Negative Affect x PE interaction excluding high grades

Summary: Grades below 90%

setwd("//datastore01.psy.miami.edu/Groups/AHeller_Lab/Undergrad/WVillano/BRMS_updating/MKD_models")
md_update_DL_na5h_int_g90 <- brm(next_update_short ~ pe_2*NA_score_5h_mean_bc + exam_f + (1 | cohort / class / id), 
                             data = df.new[which(df.new$grade <= 90),],
                             family = asym_laplace,
                             chains = 2,
                             cores = 4, 
                             file = "brm_update_DL_na5h_int_g90")
md_update_DL_na5h_int_g90$loo <- loo(md_update_DL_na5h_int_g90)
md_update_DL_na5h_int_g90 <- add_criterion(md_update_DL_na5h_int_g90, criterion = "waic")

kable(summary(md_update_DL_na5h_int_g90)$fixed, digits = 3)

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	4.281	6.832	-8.721	18.779	1.006	145	60
pe_2	0.451	0.040	0.372	0.523	1.003	569	1188
NA_score_5h_mean_bc	0.163	0.024	0.118	0.209	1.000	882	1368
exam_f	-0.664	0.361	-1.337	0.062	1.001	1327	1131
pe_2:NA_score_5h_mean_bc	0.001	0.002	-0.002	0.005	1.000	1528	1518

Summary: Grades below 80%

setwd("//datastore01.psy.miami.edu/Groups/AHeller_Lab/Undergrad/WVillano/BRMS_updating/MKD_models")
md_update_DL_na5h_int_g80 <- brm(next_update_short ~ pe_2*NA_score_5h_mean_bc + exam_f + (1 | cohort / class / id), 
                             data = df.new[which(df.new$grade <= 80),],
                             family = asym_laplace,
                             chains = 2,
                             cores = 4, 
                             file = "brm_update_DL_na5h_int_g80")
md_update_DL_na5h_int_g80$loo <- loo(md_update_DL_na5h_int_g80)
md_update_DL_na5h_int_g80 <- add_criterion(md_update_DL_na5h_int_g80, criterion = "waic")

kable(summary(md_update_DL_na5h_int_g80)$fixed, digits = 3)

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	5.885	6.572	-7.341	19.901	1.002	1134	1145
pe_2	0.526	0.054	0.418	0.632	1.003	1898	1176
NA_score_5h_mean_bc	0.118	0.033	0.054	0.187	1.004	2498	1287
exam_f	-0.641	0.554	-1.697	0.479	1.001	2567	1634
pe_2:NA_score_5h_mean_bc	0.001	0.002	-0.003	0.006	1.001	2165	1236

Comparison: PE x NA estimates

NOTE: PE x NA parameter estimate is only significantly different from zero when all grades are included.

Negative Affect x PE adding Grade

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
NA * PE + Grade + RE(PE + Grade)	0.000	0.000	-5572.478	34.959	339.613	11.602	11144.96	69.918
NA * PE + Grade + RE(PE)	-207.461	16.521	-5779.939	35.757	354.045	13.388	11559.88	71.514
NA * PE + Grade	-224.797	17.334	-5797.275	35.789	318.582	12.373	11594.55	71.578
NA * PE * Grade + RE(PE + Grade)	-280.211	18.617	-5852.688	34.700	108.145	5.416	11705.38	69.400
NA * PE * Grade	-287.956	18.942	-5860.434	34.389	51.864	2.055	11720.87	68.778
NA * PE * Grade + RE(PE)	-306.790	19.961	-5879.268	35.480	107.064	6.117	11758.54	70.960

NOTE: Model with PE x NA interaction and random slopes for PE and Grade fits the data best.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	49.694	8.308	32.568	64.291	1.314	6	30
pe_2	0.341	0.557	-1.067	1.140	1.120	12	26
NA_score_5h_mean_bc	0.006	0.014	-0.018	0.032	1.316	6	62
grade	-0.529	0.107	-0.724	-0.272	1.086	14	29
exam_f	0.107	0.161	-0.217	0.468	1.123	19	34
pe_2:NA_score_5h_mean_bc	0.003	0.001	0.001	0.005	1.220	7	81

NOTE: PE x NA interaction is weak but significantly different from zero.

2.2 Positive Affect x PE Interaction

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_pa5h_int	0.000	0.000	-6189.882	38.411	18.876	0.403	12379.76	76.822
md_update_DL_pa5h_noInt	-0.084	2.136	-6189.966	38.493	17.243	0.369	12379.93	76.985

NOTE: In a model without grade covariate or random slopes, including the PE * PA interaction improves model fit, but not appreciably. WAIC estimates overlap, and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-0.983	5.877	-12.858	10.944	1.001	789	780
pe_2	0.195	0.036	0.123	0.264	1.001	1326	1235
PA_score_5h_mean_bc	-0.133	0.026	-0.182	-0.084	1.002	1673	1524
exam_f	-0.460	0.318	-1.107	0.137	1.002	2691	1370
pe_2:PA_score_5h_mean_bc	-0.003	0.002	-0.007	0.001	1.001	2418	1573

NOTE: While including the PE * PA interaction improves model fit, the interaction is not significantly different from zero. Nonetheless, the interaction effect is plotted above.

Positive Affect x PE interaction excluding high grades

Summary: Grades below 90%

setwd("//datastore01.psy.miami.edu/Groups/AHeller_Lab/Undergrad/WVillano/BRMS_updating/MKD_models")
md_update_DL_pa5h_int_g90 <- brm(next_update_short ~ pe_2*PA_score_5h_mean_bc + exam_f + (1 | cohort / class / id), 
                             data = df.new[which(df.new$grade <= 90),],
                             family = asym_laplace,
                             chains = 2,
                             cores = 4, 
                             file = "brm_update_DL_pa5h_int_g90")
md_update_DL_pa5h_int_g90$loo <- loo(md_update_DL_pa5h_int_g90)
md_update_DL_pa5h_int_g90 <- add_criterion(md_update_DL_pa5h_int_g90, criterion = "waic")

kable(summary(md_update_DL_pa5h_int_g90)$fixed, digits = 3)

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	3.570	6.135	-8.968	16.081	1.003	666	865
pe_2	0.462	0.037	0.390	0.535	1.001	1489	1463
PA_score_5h_mean_bc	-0.177	0.027	-0.230	-0.120	1.000	1850	995
exam_f	-0.886	0.394	-1.633	-0.097	1.001	2093	1202
pe_2:PA_score_5h_mean_bc	-0.001	0.002	-0.005	0.003	1.000	2202	1531

Summary: Grades below 80%

setwd("//datastore01.psy.miami.edu/Groups/AHeller_Lab/Undergrad/WVillano/BRMS_updating/MKD_models")
md_update_DL_pa5h_int_g80 <- brm(next_update_short ~ pe_2*PA_score_5h_mean_bc + exam_f + (1 | cohort / class / id), 
                             data = df.new[which(df.new$grade <= 80),],
                             family = asym_laplace,
                             chains = 2,
                             cores = 4, 
                             file = "brm_update_DL_pa5h_int_g80")
md_update_DL_pa5h_int_g80$loo <- loo(md_update_DL_pa5h_int_g80)
md_update_DL_pa5h_int_g80 <- add_criterion(md_update_DL_pa5h_int_g80, criterion = "waic")

kable(summary(md_update_DL_pa5h_int_g80)$fixed, digits = 3)

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	6.044	6.738	-7.348	20.122	1.002	923	873
pe_2	0.528	0.053	0.425	0.627	1.000	2090	1641
PA_score_5h_mean_bc	-0.101	0.037	-0.173	-0.029	1.002	2221	1358
exam_f	-0.622	0.525	-1.624	0.437	1.001	2875	1300
pe_2:PA_score_5h_mean_bc	0.000	0.003	-0.006	0.006	1.000	2060	1535

Comparison: PE x PA estimates

NOTE: Parameter estimates for PE x PA interaction are not significantly different from zero regardless of which grades are excluded. However, point estimate for interaction effect is highest for grades < 80%.

2.3 High vs. low arousal affect items

2.3.1 Low arousal Negative Affect x PE Interaction

Tired

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_tired5h_noInt	0.000	0.000	-6199.771	38.746	13.704	0.360	12399.54	77.491
md_update_DL_tired5h_int	-1.003	0.649	-6200.774	38.675	14.906	0.373	12401.55	77.349

NOTE: In updating model with tired emotion item, model with only main effects of PE and tired fits the data best. WAICs overlap, but elpd_diff is moderately interpretable and indicates that main effect model yields optimal fit.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-1.778	5.648	-12.245	10.393	1.002	756	930
pe_2	0.089	0.033	0.031	0.159	1.001	1105	1504
Tired_5h_mean_bc	0.032	0.017	0.001	0.068	1.001	1397	1062
exam_f	-0.322	0.264	-0.837	0.165	1.000	1822	1608

NOTE: Effect of PE on Updating, conditioned on tired scores from the dense sampling period shows that updating is primarily occurring in the negative direction after negative PEs.

Sad

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_sad5h_int	0.000	0.000	-6147.893	36.907	19.925	0.388	12295.79	73.814
md_update_DL_sad5h_noInt	-3.391	3.531	-6151.284	37.073	18.975	0.358	12302.57	74.145

NOTE: In updating model with sad emotion item, model with PE x tired interaction fits the data best, though WAICs overlap and elpd_diff is not entirely interpretable given the wide margin of error.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-0.644	5.621	-11.995	11.232	1.005	570	558
pe_2	0.296	0.034	0.228	0.365	1.001	1581	1438
Sad_5h_mean_bc	0.142	0.015	0.112	0.173	1.000	1549	1507
exam_f	-0.595	0.315	-1.256	0.014	1.004	2007	1017
pe_2:Sad_5h_mean_bc	0.003	0.001	0.001	0.006	1.002	2047	1667

NOTE: Interaction between PE x sad is significantly different from zero and indicates that increased sad scores during the dense sampling period are associated with steeper updating slopes. Sad scores seem to differentiate updating effects most substantially after higher, more positive PEs.

2.3.2 High arousal Negative Affect x PE Interaction

Irritable

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_irritable5h_int	0.00	0.000	-6164.877	36.905	20.141	0.405	12329.75	73.811
md_update_DL_irritable5h_noInt	-4.57	4.283	-6169.447	36.986	19.464	0.368	12338.90	73.971

NOTE: In updating model with irritable emotion item, model with PE x irritable interaction fits the data best, though WAICs overlap and elpd_diff may not be reliable to interpret.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-0.518	5.376	-11.049	10.831	1.002	880	889
pe_2	0.235	0.033	0.172	0.300	1.001	1942	1663
Irritable_5h_mean_bc	0.101	0.018	0.069	0.137	1.004	1635	1267
exam_f	-0.677	0.317	-1.292	-0.055	1.000	2309	1440
pe_2:Irritable_5h_mean_bc	0.005	0.001	0.002	0.007	1.005	2171	1459

NOTE: Interaction between PE x Irritable is significantly different from zero and indicates that increased irritable scores during the dense sampling period are associated with steeper updating slopes. Irritable scores seem to differentiate updating effects most substantially after higher, more positive PEs.

Stressed

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_stress5h_noInt	0.000	0.000	-6158.761	37.232	16.554	0.306	12317.52	74.463
md_update_DL_stress5h_int	-1.877	1.368	-6160.638	37.123	19.725	0.382	12321.28	74.247

NOTE: In updating model with stressed emotion item, model with only main effects of PE and stressed fits the data best, though WAICs overlap and elpd_diff may not be reliable to interpret.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	1.273	5.412	-9.737	12.269	1.004	773	827
pe_2	0.299	0.031	0.237	0.358	1.001	1751	1233
Stressed_5h_mean_bc	0.134	0.016	0.102	0.165	1.000	1706	1537
exam_f	-0.849	0.314	-1.493	-0.246	0.999	2642	1336

NOTE: Effect of PE on Updating, conditioned on stressed scores from the dense sampling period shows a symmetrical updating effect over positive and negative PEs of similar magnitude.

Anxious

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_anx5h_noInt	0.00	0.000	-6157.911	37.389	17.328	0.349	12315.82	74.778
md_update_DL_anx5h_int	-0.64	2.534	-6158.551	37.336	20.385	0.407	12317.10	74.673

NOTE: In updating model with anxious emotion item, model with only main effects of PE and anxious fits the data best, though WAICs overlap, and elpd_diff is not interpretable given the wide margin of error.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	0.683	6.010	-11.719	13.505	1.000	471	717
pe_2	0.280	0.029	0.222	0.338	1.000	1413	1422
Anxious_5h_mean_bc	0.132	0.017	0.100	0.164	0.999	1306	1358
exam_f	-0.814	0.311	-1.451	-0.243	1.003	1627	1245

NOTE: Effect of PE on Updating, conditioned on anxious scores from the dense sampling period shows a symmetrical updating effect over positive and negative PEs of similar magnitude.

2.3.3 Low arousal Positive Affect x PE Interaction

Relaxed

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_relaxed5h_int	0.000	0.000	-6164.434	37.109	21.493	0.403	12328.87	74.217
md_update_DL_relaxed5h_noInt	-2.334	3.509	-6166.768	37.093	20.051	0.371	12333.54	74.185

NOTE: In updating model with relaxed emotion item, model with PE x relaxed interaction fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	0.636	5.513	-11.071	11.409	1.012	614	704
pe_2	0.284	0.031	0.225	0.342	1.000	2085	1300
Relaxed_5h_mean_bc	-0.110	0.019	-0.148	-0.073	0.999	1930	1593
exam_f	-0.857	0.341	-1.528	-0.182	1.001	1811	1150
pe_2:Relaxed_5h_mean_bc	-0.004	0.002	-0.007	-0.001	1.000	2379	1416

NOTE: Interaction between PE x Relaxed is significantly different from zero and indicates that lower relaxed scores during the dense sampling period are associated with steeper updating slopes. Relaxed scores seem to differentiate updating effects most substantially after higher, more positive PEs.

Content

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_content5h_int	0.000	0.000	-6136.147	36.919	19.634	0.389	12272.29	73.839
md_update_DL_content5h_noInt	-0.117	2.091	-6136.264	37.000	17.817	0.361	12272.53	74.000

NOTE: In updating model with content emotion item, model with PE x content fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	0.267	5.656	-12.499	11.185	1.001	567	674
pe_2	0.357	0.035	0.288	0.425	1.000	1657	1283
Content_5h_mean_bc	-0.171	0.017	-0.206	-0.137	1.000	1591	1374
exam_f	-0.588	0.317	-1.231	0.036	1.000	2177	1300
pe_2:Content_5h_mean_bc	-0.002	0.001	-0.005	0.000	1.000	2430	1419

NOTE: Parameter estimate for PE x content interaction is not significantly different from zero, but the interaction effect is plotted above nonetheless.

2.3.4 Ambiguous arousal Positive Affect x PE Interaction

Happy

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_happy5h_int	0.000	0.000	-6145.467	36.839	21.224	0.453	12290.93	73.679
md_update_DL_happy5h_noInt	-0.139	2.674	-6145.606	36.944	18.743	0.399	12291.21	73.888

NOTE: In updating model with happy emotion item, model with PE x happy interaction fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	0.617	5.846	-10.830	13.368	1.004	499	561
pe_2	0.345	0.033	0.282	0.408	1.000	1463	1348
Happy_5h_mean_bc	-0.155	0.018	-0.189	-0.121	1.000	1273	1374
exam_f	-0.687	0.300	-1.271	-0.082	1.000	1794	1134
pe_2:Happy_5h_mean_bc	-0.003	0.001	-0.005	0.000	1.001	1804	1317

NOTE: PE x Happy interaction estimate is not significantly different from zero.

2.3.5 High arousal Positive Affect x PE Interaction

Excited

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_DL_excited5h_int	0.000	0.000	-6169.744	36.899	23.296	0.458	12339.49	73.798
md_update_DL_excited5h_noInt	-0.311	2.196	-6170.055	37.011	21.811	0.426	12340.11	74.021

NOTE: In updating model with excited emotion item, model with PE x excited interaction effect fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	0.693	5.311	-9.764	12.072	1.003	909	897
pe_2	0.297	0.037	0.226	0.368	0.999	1663	1450
Excited_5h_mean_bc	-0.109	0.021	-0.150	-0.067	1.001	1647	1381
exam_f	-0.843	0.319	-1.453	-0.234	1.000	2768	1293
pe_2:Excited_5h_mean_bc	-0.003	0.001	-0.006	0.000	1.002	2555	1520

NOTE: Parameter estimate for PE x Excited interaction is not significantly different from zero.

3. Anticipation Affect as Moderator of Updating Effect

3.1 Testing Negative Affect x PE Interaction

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_na5h_int	0.000	0.000	-5677.465	35.511	20.697	0.359	11354.93	71.021
md_update_AL_na5h_noInt	-1.014	1.924	-5678.479	35.569	20.641	0.351	11356.96	71.137

NOTE: In a model without grade covariate or random slopes, including the PE * Anticipation NA interaction improves model fit, but not appreciably. WAIC estimates overlap, and elpd_diff may not be reliable to interpret.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-3.371	5.953	-15.859	8.701	1	692	731
pe_2	0.068	0.078	-0.079	0.220	1	2551	1640
anticipation_NA_mean	0.085	0.017	0.054	0.118	1	2602	1371
exam_f	-0.875	0.327	-1.519	-0.235	1	2664	1457
pe_2:anticipation_NA_mean	0.002	0.001	0.000	0.005	1	2300	1424

NOTE: Interaction effect is relatively weak, but is significantly different from zero. The plotted interaction effect shows that anticipation NA primarily differentiates updating following higher, positive PEs. Higher mean anticipation NA predicts a steeper updating effect.

3.1.1 Negative Affect x PE interaction excluding high grades

Summary: Grades below 90%

setwd("//datastore01.psy.miami.edu/Groups/AHeller_Lab/Undergrad/WVillano/BRMS_updating/MKD_models")
md_update_AL_na5h_int_g90 <- brm(next_update_short ~ pe_2*anticipation_NA_mean + exam_f + (1 | cohort / class / id), 
                             data = df.new[which(df.new$grade <= 90),],
                             family = asym_laplace,
                             chains = 2,
                             cores = 4, 
                             file = "brm_update_AL_na5h_int_g90", 
                             threads = threading(2, grainsize = 315),
                             backend = "cmdstanr")
md_update_AL_na5h_int_g90$loo <- loo(md_update_AL_na5h_int_g90)
md_update_AL_na5h_int_g90 <- add_criterion(md_update_AL_na5h_int_g90, criterion = "waic")

kable(summary(md_update_AL_na5h_int_g90)$fixed, digits = 3)

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	0.303	6.454	-12.528	13.528	1.001	669	904
pe_2	0.169	0.087	0.004	0.351	1.002	1886	1110
anticipation_NA_mean	0.096	0.019	0.059	0.133	1.001	2299	1332
exam_f	-0.926	0.399	-1.720	-0.134	1.001	1983	1279
pe_2:anticipation_NA_mean	0.003	0.002	0.000	0.006	1.004	1909	1307

Summary: Grades below 80%

setwd("//datastore01.psy.miami.edu/Groups/AHeller_Lab/Undergrad/WVillano/BRMS_updating/MKD_models")
md_update_AL_na5h_int_g80 <- brm(next_update_short ~ pe_2*anticipation_NA_mean + exam_f + (1 | cohort / class / id), 
                             data = df.new[which(df.new$grade <= 80),],
                             family = asym_laplace,
                             chains = 2,
                             cores = 4, 
                             file = "brm_update_AL_na5h_int_g80", 
                             threads = threading(2, grainsize = 315),
                             backend = "cmdstanr")
md_update_AL_na5h_int_g80$loo <- loo(md_update_AL_na5h_int_g80)
md_update_AL_na5h_int_g80 <- add_criterion(md_update_AL_na5h_int_g80, criterion = "waic")

kable(summary(md_update_AL_na5h_int_g80)$fixed, digits = 3)

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	1.595	6.885	-12.549	15.139	1.004	781	747
pe_2	0.236	0.117	0.016	0.468	1.001	1536	1489
anticipation_NA_mean	0.105	0.025	0.054	0.154	1.000	1706	1507
exam_f	-0.559	0.606	-1.712	0.653	0.999	2047	1527
pe_2:anticipation_NA_mean	0.003	0.002	0.000	0.007	1.001	1591	1657

Comparison: PE x NA estimates

NOTE: PE x Anticipation NA parameter estimate is most reliably different from zero when all grades are included.

3.1.2 Negative Affect x PE adding Grade

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
NA * PE + Grade + RE(PE + Grade)	0.000	0.000	-5128.628	32.999	326.275	9.549	10257.26	65.998
NA * PE + Grade + RE(PE)	-192.173	13.858	-5320.801	34.076	334.738	13.324	10641.60	68.152
NA * PE + Grade	-215.006	15.093	-5343.634	34.197	307.094	12.715	10687.27	68.395
NA * PE * Grade + RE(PE + Grade)	-338.263	19.778	-5466.891	33.398	79.482	5.866	10933.78	66.796
NA * PE * Grade + RE(PE)	-359.112	18.651	-5487.740	33.177	85.747	4.610	10975.48	66.354
NA * PE * Grade	-495.376	21.560	-5624.004	32.882	129.423	3.551	11248.01	65.764

NOTE: Model with PE x Anticipation NA interaction, grade covariate, and random slopes for PE and Grade fits the data best.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	49.339	8.487	32.967	67.078	1.042	46	115
pe_2	0.445	0.251	-0.119	0.926	1.023	74	75
anticipation_NA_mean	0.006	0.013	-0.019	0.033	1.142	11	141
grade	-0.578	0.104	-0.792	-0.395	1.022	70	155
exam_f	-0.017	0.206	-0.378	0.427	1.060	55	174
pe_2:anticipation_NA_mean	0.002	0.001	0.000	0.004	1.027	68	431

NOTE: PE x NA interaction is weak but significantly different from zero. Greater NA during the anticipation period predicts steeper prediction updating effects.

3.2 Positive Affect x PE Interaction

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_pa5h_noInt	0.000	0.000	-5686.421	35.502	18.668	0.336	11372.84	71.004
md_update_AL_pa5h_int	-1.071	1.539	-5687.492	35.422	20.838	0.385	11374.98	70.844

NOTE: In a model without grade covariate or random slopes, a model with only main effects of PE and Anticipation PA fits the data best, though WAIC estimates overlap and elpd_diff is not reliable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	3.899	5.886	-6.712	16.092	1.003	576	744
pe_2	0.170	0.032	0.108	0.235	1.000	2055	1210
exam_f	-0.862	0.353	-1.558	-0.147	1.000	1978	1408
anticipation_PA_mean	-0.069	0.020	-0.108	-0.028	1.002	2056	1437

NOTE: Main effect of PE conditioned on Anticipation PA shows a symmetrical updating effect.

3.3 High vs. low arousal affect items

3.3.1 Low arousal Negative Affect x PE Interaction

Tired

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_tired5h_noInt	0.000	0.00	-5687.996	35.485	19.130	0.347	11375.99	70.969
md_update_AL_tired5h_int	-1.165	0.19	-5689.161	35.458	20.317	0.396	11378.32	70.916

NOTE: In updating model with Anticipation tired emotion item, model with only main effects of PE and Anticipation tired fits the data best. WAICs overlap, but elpd_diff is interpretable and indicates that main effect model yields optimal fit.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-0.436	5.591	-11.988	10.732	1.001	794	654
pe_2	0.172	0.031	0.110	0.234	1.000	2625	1310
anticipation_Tired_mean_bc	0.055	0.019	0.019	0.091	1.000	2631	1235
exam_f	-0.810	0.367	-1.524	-0.110	1.002	2602	1127

NOTE: Effect of PE on Updating, conditioned on Anticipation tired scores shows a symmetrical updating effect.

Sad

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_sad5h_int	0.000	0.00	-5660.704	35.338	21.403	0.397	11321.41	70.675
md_update_AL_sad5h_noInt	-2.833	2.97	-5663.537	35.415	21.674	0.372	11327.07	70.830

NOTE: In updating model with Anticipation sad emotion item, model with PE x Anticipation sad interaction fits the data best, though WAICs overlap and elpd_diff is not reliably interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-0.339	5.629	-11.398	10.670	1.005	666	747
pe_2	0.258	0.035	0.191	0.327	1.001	2273	1320
anticipation_Sad_mean_bc	0.149	0.018	0.113	0.184	1.003	2158	1659
exam_f	-0.679	0.367	-1.373	0.033	1.000	2347	1553
pe_2:anticipation_Sad_mean_bc	0.003	0.001	0.001	0.006	1.003	1936	1234

NOTE: Interaction between PE x Anticipation sad is significantly different from zero and indicates that increased Anticipation sad scores are associated with steeper updating slopes. Anticipation Sad scores seem to differentiate updating effects most substantially after higher, more positive PEs.

3.3.2 High arousal Negative Affect x PE Interaction

Irritable

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_irritable5h_noInt	0.000	0.000	-5675.638	35.380	18.783	0.357	11351.28	70.760
md_update_AL_irritable5h_int	-1.121	1.933	-5676.759	35.279	21.596	0.457	11353.52	70.559

NOTE: In updating model with Anticipation irritable emotion item, model with only main effects of PE and Anticipation irritable fits the data best, though WAICs overlap and elpd_diff may not be reliable

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-0.694	5.485	-11.150	10.427	1.000	965	747
pe_2	0.225	0.031	0.161	0.291	1.003	2310	1291
anticipation_Irritable_mean_bc	0.119	0.021	0.078	0.159	1.002	2292	1425
exam_f	-0.546	0.347	-1.189	0.132	0.999	2850	1056

NOTE: Main effect of PE, conditioned on Anticipation irritable scores shows a symmetrical updating effect.

Stressed

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_stress5h_noInt	0.000	0.000	-5667.069	35.584	18.813	0.367	11334.14	71.168
md_update_AL_stress5h_int	-1.519	0.282	-5668.587	35.543	20.899	0.413	11337.17	71.085

NOTE: In updating model with Anticipation stressed emotion item, model with only main effects of PE and Anticipation stressed fits the data best. While WAICs overlap, elpd_diff is interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	1.757	5.178	-9.172	11.836	1.001	616	827
pe_2	0.263	0.032	0.201	0.325	1.004	1362	1339
anticipation_Stressed_mean_bc	0.142	0.019	0.103	0.179	1.003	1358	1233
exam_f	-0.963	0.342	-1.619	-0.269	1.000	2057	1426

NOTE: Effect of PE on Updating, conditioned on Anticipation stressed scores from the shows a symmetrical updating effect over positive and negative PEs of similar magnitude.

Anxious

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_anx5h_int	0.000	0.000	-5669.311	35.657	20.367	0.392	11338.62	71.313
md_update_AL_anx5h_noInt	-0.448	1.855	-5669.759	35.686	19.783	0.362	11339.52	71.373

NOTE: In updating model with Anticipation anxious emotion item, model with PE x Anticipation anxious fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	1.586	5.300	-10.191	13.039	1.003	771	695
pe_2	0.239	0.031	0.181	0.298	0.999	2275	1233
anticipation_Anxious_mean_bc	0.125	0.020	0.084	0.163	1.001	2073	1601
exam_f	-0.931	0.346	-1.614	-0.275	1.000	2412	1404
pe_2:anticipation_Anxious_mean_bc	0.002	0.002	-0.001	0.005	1.000	2472	1734

NOTE: Parameter estimate for PE x Anticipation anxious interaction effect is not significantly different from zero.

3.3.3 Low arousal Positive Affect x PE Interaction

Relaxed

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_relaxed5h_int	0.000	0.000	-5674.765	35.541	19.930	0.401	11349.53	71.082
md_update_AL_relaxed5h_noInt	-0.099	1.809	-5674.865	35.491	18.788	0.350	11349.73	70.982

NOTE: In updating model with Anticipation relaxed emotion item, model with PE x Anticipation relaxed interaction fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	1.273	5.927	-11.163	13.001	1.003	841	885
pe_2	0.237	0.031	0.179	0.297	1.003	2278	1394
anticipation_Relaxed_mean_bc	-0.119	0.022	-0.161	-0.075	1.003	2176	1392
exam_f	-0.863	0.360	-1.561	-0.170	1.000	2807	1305
pe_2:anticipation_Relaxed_mean_bc	-0.002	0.002	-0.006	0.001	1.003	2578	1494

NOTE: Interaction between PE x Anticipation Relaxed is not significantly different from zero.

Content

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_content5h_int	0.000	0.000	-5654.898	35.285	20.813	0.416	11309.80	70.570
md_update_AL_content5h_noInt	-0.105	1.742	-5655.003	35.302	19.856	0.370	11310.01	70.604

NOTE: In updating model with Anticipation content emotion item, model with PE x Anticipation content interaction fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	0.525	5.436	-11.129	12.064	1.002	825	824
pe_2	0.297	0.034	0.226	0.364	1.002	2070	1375
anticipation_Content_mean_bc	-0.166	0.020	-0.205	-0.129	1.001	1829	1241
exam_f	-0.587	0.360	-1.298	0.082	1.001	2673	1411
pe_2:anticipation_Content_mean_bc	-0.002	0.001	-0.005	0.001	1.005	2149	1307

NOTE: Parameter estimate for PE x Anticipation content interaction is not significantly different from zero.

3.3.4 Ambiguous arousal Positive Affect x PE Interaction

Happy

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_happy5h_noInt	0.000	0.000	-5660.538	35.437	19.171	0.399	11321.08	70.874
md_update_AL_happy5h_int	-0.098	1.382	-5660.637	35.415	19.979	0.419	11321.27	70.831

NOTE: In updating model with happy emotion item, model with only main effects for PE and Anticipation happy fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	-0.158	5.177	-10.729	9.794	1.000	927	969
pe_2	0.301	0.035	0.233	0.368	1.001	2282	1359
anticipation_Happy_mean_bc	-0.167	0.020	-0.204	-0.128	1.000	2082	1353
exam_f	-0.594	0.341	-1.286	0.073	1.002	2092	1355

NOTE: PE updating effect, conditioned on Anticipation happy scores shows a symmetrical effect over both positive and negative PEs.

3.3.5 High arousal Positive Affect x PE Interaction

Excited

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
md_update_AL_excited5h_noInt	0.000	0.000	-5679.996	35.424	21.682	0.400	11359.99	70.848
md_update_AL_excited5h_int	-1.279	1.397	-5681.275	35.361	24.115	0.488	11362.55	70.722

NOTE: In updating model with Anticipation excited emotion item, model with only PE and Anticipation excited interaction effect fits the data best, though WAICs overlap and elpd_diff is not interpretable.

Effects from winning model

	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Intercept	0.577	5.430	-9.939	12.622	1.000	988	917
pe_2	0.247	0.035	0.179	0.315	1.002	1434	1364
anticipation_Excited_mean_bc	-0.120	0.023	-0.163	-0.074	1.001	1773	1509
exam_f	-0.831	0.354	-1.534	-0.137	1.000	2554	1581

NOTE: Updating ~ PE effect, conditioned on Anticipation happy scores shows a symmetrical updating effect.

4. Mediation models

4.1 Simple vs. moderated mediation

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
mod_mediation_dense	0.000	0.000	-12054.55	48.846	513.425	13.018	24109.10	97.692
simple_mediation_dense	-558.759	31.695	-12613.31	51.755	456.686	13.846	25226.62	103.509

NOTE: Moderated mediation model, in which grade moderates the path from PE to Prediction Update fits the data better than the nested simple mediation model.

Bayesian R²

	Estimate	Est.Error	Q2.5	Q97.5
R2NAscore5hmeanbc	0.298	0.046	0.206	0.368
R2nextupdateshort	0.493	0.014	0.464	0.521

NOTE: Coefficient of determination for Prediction Update (yellow density) is pretty impressive and appears to be a reliable estimate. The R² distribution for NA mean (5h) indicates that approximately 25% of the variance in NA during the first 5 hours of dense sampling is explained by PE. Additional models (tested in section 4.2) will include Grade as a predictor of PE and as a moderator of the path from PE to Prediction Update.

Visualizing AB path

##    50%   2.5%  97.5% 
## -0.023 -0.270  0.291

Direct vs. total effects

NOTE: Zero appears to be a more credible estimate for the direct effect than the total effect through the mediator, Dense NA, with grade moderating the path from PE to Prediction Update.

4.2 Serial and Parallel Mediation

Model List

Name	Model.Type	Formula
mediation_test_1.1	mediator model	NA_score_5h_mean_bc ~ grade + pe_2 + (1 + grade + pe_2 | cohort/class/id)
mediation_test_1.1	outcome_model	next_update_short ~ pe_2 + NA_score_5h_mean_bc + (1 + pe_2 | cohort/class/id)
mediation_test_1.2	mediator model	NA_score_5h_mean_bc ~ grade + pe_2 + (1 + grade + pe_2 | cohort/class/id)
mediation_test_1.2	outcome_model	next_update_short ~ pe_2 * grade + NA_score_5h_mean_bc + (1 + grade + pe_2 | cohort/class/id)
mediation_test_1.3	mediator model	NA_score_5h_mean_bc ~ grade + pe_2 + (1 + grade + pe_2 | cohort/class/id)
mediation_test_1.3	outcome_model	next_update_short ~ pe_2 * NA_score_5h_mean_bc + (1 + pe_2 | cohort/class/id)
mediation_test_1.4	mediator model	NA_score_5h_mean_bc ~ grade + pe_2 + (1 + grade + pe_2 | cohort/class/id)
mediation_test_1.4	outcome_model	next_update_short ~ s(pe_2) + NA_score_5h_mean_bc + (1 + pe_2 | cohort/class/id)
mediation_test_1.5	mediator model [1]	NA_score_5h_mean_bc ~ grade + pe_2 + (1 + grade + pe_2 | cohort/class/id)
mediation_test_1.5	mediator model [2]	pe_2 ~ grade
mediation_test_1.5	outcome_model	next_update_short ~ pe_2 + NA_score_5h_mean_bc + (1 + pe_2 + NA_score_5h_mean_bc | cohort/class/id)

Model comparison

	elpd_diff	se_diff	elpd_waic	se_elpd_waic	p_waic	se_p_waic	waic	se_waic
mediation_test_1.2	0.000	0.000	-11901.55	49.941	525.451	13.761	23803.10	99.882
mediation_test_1.4	-579.759	31.891	-12481.31	52.128	378.719	12.442	24962.62	104.256
mediation_test_1.3	-586.797	32.482	-12488.35	52.143	378.235	12.334	24976.70	104.287
mediation_test_1.1	-589.850	32.349	-12491.40	52.205	379.583	12.456	24982.80	104.410
mediation_test_1.5	-6276.324	48.833	-18177.88	65.786	509.798	15.797	36355.75	131.572

NOTE: Moderated mediation model with the addition of a path from grade to Dense NA (5 hour) yields the best fit to the data relative to other models tested.

Bayesian R² (winning mediation model)

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