Method

Semester-long ecological momentary assessment paradigm. In a sample of 873 students enrolled in university chemistry classes, we used cell phone-based ecological momentary assessment (EMA) to measure expected grades and emotions surrounding four major exams, the average of which comprised their final grade. Shortly after completing each exam and several days prior to the release of grades, students were prompted via EMA to report the grade they expected to receive on that exam. Taking participants’ actual exam grades, we computed exam grade prediction errors (PEs) as the difference between self-reported grade expectations and actual grades on each exam.

After each exam, participants were subjected to an uncertain waiting period of 3-4 days before grades were released. In the hours prior to the release of exam grades, we induced a state of anticipation by notifying participants via SMS that grades would be released in 2.25 hours. For the duration of this anticipation period, participants received EMA surveys of momentary positive and negative affect at 45-minute intervals until grades were ready to be released. Before students viewed their exam grades, they were prompted once more via EMA to report the grade they expected to receive. To measure emotional responses to exam grades, we yoked a set of EMA surveys of PA and NA to the precise moment when participants received their exam grades.


Glossary of terms

  • Prediction error (PE): the difference between the expected value and actual value of a given outcome. In this context, PEs occur when a student over- or underestimates their actual exam grade:

\[PE_t = Grade_t - Expectation_t\]

  • Expectation precision: the magnitude of the difference between an expected and actual outcome (i.e., PE). Operationalized as abs(Prediction Error).

    • Change in expectation precision: the numeric difference between expectation precision on successive exams. A positive value indicates a decrease in precision, whereas a negative value indicates an increase in expectation precision between exams.

  • Expectation updating: the numeric difference between successive exam grade predictions. A positive values indicates that expectations increased from one exam to the next, while a negative value indicates that expectations decreased from one exam to the next.

  • Bracing: While waiting for an outcome to be revealed, individuals’ expectations often change in a predictable manner. Prior research suggests that in the immediate anticipation of an uncertain outcome, individuals frequently lower their expectations, likely in an attempt to “brace” themselves for a surprising negative outcome. In the context of academic exams, bracing is defined as the difference between a participant’s initial exam grade expectation (T1 prediction), reported immediately after taking an exam, and a second expectation (T2 prediction), reported immediately prior to viewing their exam grade (typically 3-4 days later). Positive (i.e., optimistic) bracing indicates that one increased their expectations, whereas negative (i.e., pessimistic) bracing indicates that one lowered their expectations in anticipation of their grade.

    • Note: Unlike expectation updating, bracing represents change in one’s expectation for a single outcome, while expectation updating represents change in expectations between two distinct, consecutive outcomes.


Results

Section 1. Prediction Errors promote learning

Learning is the ability to utilize information from prior events to predict more precisely what will happen in future. Thus, learning enables us to update our expectations so we are better prepared for, and less surprised by, the future. Surprise can be quantified as the difference between what happened and what one expected to happen. This difference, termed prediction error (PE), is a driver of learning. PEs represent the extent to which the outcome was better or worse than one’s expectation. Individuals learn from PEs and update expectations accordingly to more precisely predict what will happen in the future.

Here, we tested whether students learned from grade PEs to more precisely predict their grades on future exams during a semester. We also hypothesized that updates to exam grade predictions would be impacted by their PEs on the preceding exam (because individuals can update their expectations without becoming more accurate). In other words, they would learn from previous misestimations of performance and would become more accurate in their future grade predictions. By updating expectations in line with PEs, we hypothesized that students would reduce the magnitude of PEs (i.e., surprise) over the semester.

Expectation precision improves over time

Given that learning should result in more precise expectations over time, we tested whether participants’ more precisely predicted their grades over the course of the semester. As noted above, expectation precision was operationalized as the absolute value of each exam grade PE (i.e., unsigned PEs). An unsigned PE trend towards zero would indicate that students had learned to more precisely predict their actual exam grades.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 13.322310 0.3942875 2320.508 33.788312 0
exam -1.216115 0.1485863 1876.373 -8.184574 0

Across exams, students become more precise in their predictions. The trend of increasing precision in expectations over time indicates learning.

Prediction errors lead to shifts in expectations for uncertain events

To test for evidence of PE-driven learning, we specified a multilevel linear model in which changes in expectations between exams were regressed onto PEs from the preceding exam.

Unlike traditional reinforcement learning paradigms where the value of an uncertain outcome is static over time, students’ grades tend to fluctuate between exams. Therefore, changes in expectations are inherently influenced by changes in actual grades between exams. For the purposes of the present study, we were not interested in capturing changes in actual grades, but rather, changes in students’ tendency to over- or underestimate their actual grades. Therefore, to test for evidence of PE-driven learning while controlling for wide changes in actual grades, we specified a linear model in which changes in expectations between exams are regressed onto the linear combination of prior exam PE and the change in actual grades between exams.

No covariates

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -0.2940914 1.1158783 2.225759 -0.2635515 0.8145504
pe_1 0.3092869 0.0307785 1521.165518 10.0487819 0.0000000

Results indicate that participants update their expectations in line with previous PEs.

Controlling for change in grade

In traditional reinforcement learning paradigms, individuals must learn the value of an outcome through repeated experience. Yet, in these studies, the values of outcomes typically do not change during learning (e.g., a stimulus reliably predicts a reward of $5). Given that participants’ exam grades naturally fluctuate over time, it is reasonable to assume that expectation updates will vary with changes in grades between exams.

To test whether PE-learning is present when controlling for shifts in grades between exams, we ran the same linear mixed effects model with change in grade (delta grade) included as a covariate:

\[\Delta Grade_t = Grade_{t+1} - Grade_t\]

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -0.5440164 0.5712813 1.861833 -0.9522742 0.4477797
pe_1 0.6994605 0.0233719 1202.604369 29.9274577 0.0000000
delta_grade 0.7255268 0.0176048 978.913734 41.2119837 0.0000000

Results indicate that participants update their expectations in line with previous PEs. By controlling for changes in grades between exams, we confirmed that changes in expectations are primarily driven by PEs.

Section 2. Expectations shift in anticipation of uncertain outcomes

A separate social psychology literature suggests that individuals tend to lower their expectations (become pessimistic) as important news (e.g., result of a cancer screen) is about to be revealed (Sweeny, 2018). Similarly, setting “defensively pessimistic” expectations appears to help individuals to cope with anxiety regarding future outcomes (Norem & Cantor, 1986). Such coping strategies during which individuals “brace” for potentially negative outcomes may be used to mitigate their emotional impact. Reducing one’s expectations reduces the likelihood of a negative prediction error, and thus, the likelihood of an intense negative emotional reaction (Rutledge et al., 2014; others)

Because we collected students’ grade expectations at two separate timepoints: T1, immediately after the exam and T2, immediately before grades were released, we were able to assess whether participants braced (i.e., became more pessimistic in their grade expectations) as the grade reveal drew closer. To accomplish this, we simply took the difference between participants predictions at T1 and T2:

\[Bracing = Prediction_{T2} - Prediction_{T1}\]

Bracing during anticipation is predominantly pessimistic

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -2.754523 0.3878612 2.203013 -7.101826 0.014764

As students get closer to seeing their actual exam grades, they tend to brace pessimistically (i.e.,they reduce their expectations). On average, participants reduced their expectations by 2.75 percentage points in anticipation of viewing their grade.

Magnitude of bracing decreases over time, as experience accrues

Current theory suggests that bracing is a strategy for managing uncertainty. As participants become more experienced with taking exams and estimating their performance, we assumed that uncertainty surrounding exam grades would decrease. Thus, we hypothesized that participants would brace less over time, and that participants would brace the most at the first exam of the semester, when they have no prior exam experiences to refer to.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 5.815667 0.4505570 4.272861 12.907730 0.0001389
exam -0.437696 0.0933692 2000.175738 -4.687799 0.0000029

Here, the absolute value of bracing, that is the number of points by which a participant shifted their expectation during the anticipation period, is regressed onto the sequence of exams throughout the semester.

Results indicated that the magnitude of bracing is greatest at the first exam of the semester and decreases over subsequent exams, which confirms our hypotheses. This suggests that as uncertainty about the course decreases (participants become more experienced with exams, gain information about their performance, and learn how to accurately predict their performance), bracing becomes less prominent. Bracing thus varies as a function of familiarity of the contingencies of given events.

Magnitude of bracing is negatively associated with confidence in expectations

Given the apparent link between uncertainty and bracing, we tested whether a related construct, confidence in expectations, predicted the manner in which participants braced for uncertain exam grades.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -4.0775207 0.4142750 455.2873 -9.842547 0e+00
conf_1 0.0303342 0.0061338 1848.3152 4.945417 8e-07

Individuals who report being less confident in their exam grade expectations show a greater propensity to brace pessimistically. While greater confidence is associated with less bracing, models suggest that participants who reported being maximally confident in their initial grade expectations still reduced their expectations by approximately one point in anticipation of receiving their grade.

Does bracing impact expectation updating and precision?

Given that bracing inherently shifts PEs, which in turn drive expectation updating and improvements in precision, we tested whether bracing impacts expectation updates and precision - our two proxies for learning.

Expectation updating

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -0.0790365 0.3116282 245.1685 -0.2536243 0.7999984
pe_1 0.6710442 0.0251928 1212.8495 26.6363034 0.0000000
pred_adj 0.2256462 0.0430358 1505.1398 5.2432209 0.0000002
delta_grade 0.7104078 0.0175570 1030.3165 40.4630259 0.0000000
pe_1:pred_adj 0.0081077 0.0020234 1520.6860 4.0069164 0.0000645

Results indicate that bracing significantly predicts expectation updating, such that pessimistic bracing is associated with reductions in future expectations, whereas optimistic bracing is associated increases in future expectations.

Expectation precision

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 9.1834584 0.2302158 555.8429 39.8906491 0.0000000
pe_1 -0.0147977 0.0163689 1518.5680 -0.9040148 0.3661309
pred_adj -0.0719847 0.0300186 1520.4785 -2.3980051 0.0166046
pe_1:pred_adj 0.0006640 0.0014077 1514.3152 0.4717120 0.6372003

Results indicate that pessimistic bracing is associated with reduced precision in expectations at the next exam.

Section 3. Individual differences in neuroticism explain variability in PE learning

Neuroticism is associated with reduced expectation precision

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 9.0429252 0.5372501 702.8216 16.83187 0.000000
neurot 0.0909969 0.0324782 702.0408 2.80179 0.005222

Despite a lack of differences in grades between high- and low-neuroticism participants (b = -0.121, SE = 0.0893, p = 0.177), neuroticism is associated with reduced precision in exam grade expectations.

Neuroticism is linked to more capricious updating following PEs

Using b-splines to account for potential nonlinearities in the effect of PEs on updating, we found that elevated neuroticism is linked to a greater rate of expectation updating following PEs.

Neuroticism is linked to reduced precision following small PEs

Results indicate that relative to low-neuroticism participants, high-neuroticism participants learn sub-optimally from small PEs. Here, predicted values above zero indicate that expectations became less precise at the next exam – that is, PEs increased in magnitude. Conversely, high-neuroticism participants report more precise expectations following large positive and negative PEs relative to low-neuroticism participants.

In conjunction with the finding that neurotic participants update their expectations at a greater rate following PEs, the above result suggests that high-neuroticism participants “over-learn” from small PEs. Put differently, high-neuroticism participants seem to over-correct their expectations, and thus become less precise in their expectations after small PEs.


Section 4. Additional findings to discuss


Connecting emotion, bracing, and neuroticism to variability in learning

In the three sections above, the findings on bracing (Section 2) seem tangential and do not fit cleanly into my working model of how neuroticism impacts learning. Moreover, emotion is largely absent from these results.

To keep this project tight and cohesive, one option is to remove the section on bracing altogether and focus strictly on learning and individual differences.

Conversely, another solution might be to keep the section on bracing and present additional results linking bracing, neuroticism, emotion, and learning. Below are some additional findings that might reconcile these missing links.


Proposed model

Summary:

  • Neuroticism predicts variability in anticipatory negative emotion (submodel A).
  • Bracing varies as a function of the interaction between previous PE and anticipatory NA (submodel B).
  • Bracing shifts PEs, which may impact learning. Indeed, as shown in section 2, bracing predicts differences in both expectation updating and precision.
  • Updating and precision are predicted by the 3-way interaction between Neuroticism, PE, and bracing (submodel C).


New Results

Neuroticism predicts variability in anticipatory negative emotion

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: anticipation_NA_mean_bc ~ neurot + (1 + neurot | cohort_class/id)
##    Data: df.new[which(df.new$exam < 5), ]
## 
## REML criterion at convergence: 21320.9
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.0285 -0.5946 -0.0616  0.5709  4.0370 
## 
## Random effects:
##  Groups          Name        Variance  Std.Dev.  Corr 
##  id:cohort_class (Intercept) 1.444e+01  3.800600      
##                  neurot      1.409e-05  0.003754 0.96 
##  cohort_class    (Intercept) 1.251e-01  0.353737      
##                  neurot      1.917e-02  0.138438 -1.00
##  Residual                    2.050e+02 14.317628      
## Number of obs: 2591, groups:  id:cohort_class, 855; cohort_class, 3
## 
## Fixed effects:
##             Estimate Std. Error       df t value Pr(>|t|)    
## (Intercept)  5.76586    0.95166 14.01668   6.059 2.93e-05 ***
## neurot      -0.02563    0.09856  1.82581  -0.260    0.821    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##        (Intr)
## neurot -0.705
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see ?isSingular

Results indicate that neuroticism is associated with significantly elevated negative emotion in the immediate anticipation of viewing exam grades.

Bracing varies by anticipatory negative emotion and previous PEs (2-way interaction)

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -1.5245989 0.4039856 756.8603 -3.773895 0.0001733
anticipation_NA_mean -0.0102218 0.0077547 884.3970 -1.318143 0.1877972
pe_1_lag1 -0.0399793 0.0316224 1343.7154 -1.264269 0.2063526
anticipation_NA_mean:pe_1_lag1 0.0012997 0.0005722 1345.8668 2.271497 0.0232742

Here, results suggest that bracing is driven by the interaction between anticipatory negative emotion and one’s PE on the preceding exam, such that participants who feel worse exhibit similar levels of bracing regardless of their previous PE. Conversely, participants who reported lower negative emotion in the anticipation period braced more pessimistically if they had negative PEs on the previous exam.

Updating and precision are predicted by the 3-way interaction between neuroticism, PE, and bracing

Updating

Here, we assessed whether individual differences in neuroticism explain differences in the way participants update their expectations following bracing-shifted PEs (i.e., PEs emerging from T2 predictions).

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 1.1757798 0.9016125 223.6693 1.3040855 0.1935449
pe_2 0.6413445 0.0721695 1309.5140 8.8866413 0.0000000
pred_adj -0.3712160 0.1250098 1500.6545 -2.9694949 0.0030302
neurot -0.0659304 0.0537740 209.7113 -1.2260633 0.2215498
delta_grade 0.7122334 0.0176886 1035.6230 40.2650816 0.0000000
pe_2:pred_adj -0.0186382 0.0085384 1490.5951 -2.1828700 0.0292010
pe_2:neurot -0.0001989 0.0041585 1298.7698 -0.0478207 0.9618665
pred_adj:neurot 0.0157353 0.0070508 1503.8623 2.2316989 0.0257817
pe_2:pred_adj:neurot 0.0009828 0.0004797 1480.6223 2.0488846 0.0406493

Results indicate that participants with elevated neuroticism make similar updates to expectations following PEs, regardless of whether their PE was shifted by bracing. On the other hand, expectation updates for participants with lower neuroticism scores vary more notably as a function of bracing, such that low-neuroticism participants make the largest positive updates to expectations when they braced pessimistically, but were positively surprised by their grades.

Precision

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -0.8491033 0.8075039 1507 -1.0515161 0.2931902
pe_2 0.1341594 0.0649543 1507 2.0654422 0.0390518
pred_adj 0.2277843 0.1143775 1507 1.9915126 0.0466048
neurot -0.0299490 0.0480666 1507 -0.6230719 0.5333315
pe_2:pred_adj -0.0040968 0.0078409 1507 -0.5224876 0.6014077
pe_2:neurot -0.0084829 0.0038009 1507 -2.2317978 0.0257749
pred_adj:neurot -0.0135148 0.0064721 1507 -2.0881779 0.0369493
pe_2:pred_adj:neurot -0.0013116 0.0004409 1507 -2.9749759 0.0029767

In contrast to participants with lower neuroticism scores, high-neuroticism participants become appreciably less precise in their expectations after bracing optimistically and experiencing a negative PE.


Follow-up analyses (discussed over Zoom)

Do learning effects vary as a function of neuroticism?

Testing the correlation between neuroticism and random effects of PE in learning models

Outcome variable: Change in expectation (updating between exams)

Model formula: next_update_short ~ pe_2 + delta_grade + pred_adj + (1 + pe_2 | id)

Correlation between PE random effects and neuroticism scores: -0.0068086

No apparent association between random effects of PE on change in expectation (updating) and neuroticism scores.

Expectation precision over time

## [1] 0.001334528

Model formula: unsigned_pe_1 ~ exam + (1 + exam | id)

Correlation between random effects of time (exam index) and neuroticism scores: 0.0013345

No apparent association between random effects of time on expectation precision and neuroticism scores.

Change in expectation precision (between exams)

## [1] -0.02301128

Model formula: **delta_uPE_1 ~ pe_1*grade + (1 + pe_1 | id)**

Correlation between PE random effects and neuroticism scores: -0.0230113

No apparent association between random effects of PE on change in expectation precision and neuroticism scores.


Does bracing vary as a function of neuroticism?

Testing the correlation between neuroticism and random effects in models predicting bracing

Bracing ~ Time

## [1] -0.001293687

Model formula: Bracing ~ exam + pe_1_lag1 + grade_lag1 + (1 + exam | id)

Correlation between random effects of time (exam) and neuroticism scores: -0.0012937

No apparent association between random effects of time on bracing and neuroticism scores.

Bracing ~ Previous Grade PE

## [1] 0.006165037

Model formula: Bracing ~ pe_1_lag1 + grade_lag1 + exam + (1 + pe_1_lag1 | id)

Correlation between random effects of previous PE and neuroticism scores: 0.006165

No apparent association between random effects of previous PE on bracing and neuroticism scores.

Bracing ~ Previous Grade (outcome)

## [1] -0.008721068

Model formula: Bracing ~ grade_lag1 + pe_1_lag1 + exam + (1 + grade_lag1 | id)

Correlation between random effects of previous grade and neuroticism scores: -0.0087211

No apparent association between random effects of previous grade on bracing and neuroticism scores.

Bracing ~ Anticipatory Negative Emotion

## [1] -0.03655439

Model formula: Bracing ~ anticipation_NA_mean + (1 + anticipation_NA_mean | id)

Correlation between random effects of anticipatory negative emotion and neuroticism scores: -0.0365544

No apparent association between random effects of anticipatory negative emotion on bracing and neuroticism scores.


does the effect of bracing on learning vary as a function of neuroticism?

Probing the Bracing * Neuroticism interaction, while controlling for PE and outcome

Outcome Variable: Change in expectations (updating between exams)

Model Summary

Model formula: **next_update_short ~ pe_2 + pred_adj*neurot + delta_grade + (1 | id)**

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 1.3671245 0.9030704 221.0832 1.513863 0.1314889
pe_2 0.6500468 0.0252343 1104.9309 25.760447 0.0000000
pred_adj -0.4406831 0.1203841 1488.2754 -3.660642 0.0002604
neurot -0.0779760 0.0538258 206.5056 -1.448674 0.1489454
delta_grade 0.7154252 0.0176325 1015.7440 40.574301 0.0000000
pred_adj:neurot 0.0195252 0.0068070 1508.9597 2.868417 0.0041827
Predicted effects

Results indicate that high-neuroticism participants update their expectations in accordance with PEs that are shifted by bracing (i.e., PEs stemming from their second predictions (T2) made during the anticipation period), regardless of how they braced in that T2 prediction. Conversely, low-neuroticism participants’ updates to expectations depend on how they braced in their T2 predictions

This result suggests that low-neuroticism participants make updates in accordance with their T1 PEs, emerging from their initial exam grade predictions, and discount PEs that result from bracing when updating their expectations (T2 PEs). On the next tab (labeled Follow-Up Test), the same model is tested, but with T1 PE as the focal predictor variable instead of T2 PE. If the above interpretation stands, we should see more variability in updating as a function of bracing, specifically in high-neuroticism participants.

Follow-up Test

Model formula: **next_update_1 ~ pe_1 + pred_adj*neurot + delta_grade + (1 | id)**

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: next_update_1 ~ pe_1 + pred_adj * neurot + delta_grade + (1 |      id)
##    Data: df.new
## 
## REML criterion at convergence: 11494.2
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.0918 -0.5554 -0.0158  0.6025  3.5304 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  id       (Intercept)   4.753   2.18   
##  Residual             109.106  10.45   
## Number of obs: 1515, groups:  id, 690
## 
## Fixed effects:
##                   Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)      1.367e+00  9.031e-01  2.211e+02   1.514  0.13149    
## pe_1             6.500e-01  2.523e-02  1.105e+03  25.760  < 2e-16 ***
## pred_adj        -9.073e-02  1.196e-01  1.491e+03  -0.759  0.44816    
## neurot          -7.798e-02  5.383e-02  2.065e+02  -1.449  0.14895    
## delta_grade      7.154e-01  1.763e-02  1.016e+03  40.574  < 2e-16 ***
## pred_adj:neurot  1.953e-02  6.807e-03  1.509e+03   2.868  0.00418 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) pe_1   prd_dj neurot dlt_gr
## pe_1         0.045                            
## pred_adj     0.348 -0.074                     
## neurot      -0.942 -0.094 -0.304              
## delta_grade  0.010  0.406  0.013 -0.043       
## pred_dj:nrt -0.326 -0.071 -0.932  0.322 -0.059

As anticipated, low-neuroticism subs update their expectations in accordance with PEs stemming from their initial predictions (T1 PE), whereas high-neuroticism subs update their expectations in accordance with the PEs that are shifted by bracing (T2 PE)

Change in expectation precision (between exams)

Model Summary

Model formula: **delta_uPE_1 ~ pred_adj*neurot + grade + (1 | id)**

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: delta_uPE_1 ~ pred_adj * neurot + grade + (1 | id)
##    Data: df.new
## 
## REML criterion at convergence: 11325.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.8235 -0.5868 -0.0004  0.5728  4.1582 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  id       (Intercept)   0       0.0    
##  Residual             102      10.1    
## Number of obs: 1515, groups:  id, 690
## 
## Fixed effects:
##                   Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)     -5.409e+00  1.482e+00  1.510e+03  -3.651 0.000270 ***
## pred_adj         9.737e-02  1.128e-01  1.510e+03   0.863 0.388287    
## neurot          -8.842e-03  4.921e-02  1.510e+03  -0.180 0.857425    
## grade            5.819e-02  1.578e-02  1.510e+03   3.689 0.000234 ***
## pred_adj:neurot -1.419e-02  6.426e-03  1.510e+03  -2.208 0.027413 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) prd_dj neurot grade 
## pred_adj     0.253                     
## neurot      -0.576 -0.324              
## grade       -0.829 -0.061  0.060       
## pred_dj:nrt -0.209 -0.942  0.326  0.029
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see ?isSingular
Predicted linear effects

Results indicate that broadly, improvements in expectation precision between exams vary as a function of bracing, and vary more widely in participants with higher neuroticism scores. Interpreting the predicted linear effects, after bracing pessimistically, high-neuroticism subs appear to become less precise in their next predictions. Conversely, after bracing optimistically, high-neuroticism subs become more precise in their next predictions.

Note: There is very little data at the more extreme levels of bracing (i.e., > 10 points in either direction), so these predicted effects should be interpreted with caution.

On the next tab (Predicted Nonlinear Effects), I test the same model, but with splines to allow for nonlinearities in the Neuroticism * Bracing interaction.

Here, we can see that most of the differentiation in the effects of neuroticism in this interaction appears on the optimistic side, where participants are increasing their expectations from T1 to T2. However, these differences are highly uncertain.

These results raise an interesting question. We know that high neuroticism is associated with more pessimistic expectations overall. Keeping this in mind, the prediction that high-neuroticism participants become more precise after optimistic bracing leads me to wonder whether optimistic bracing corrects neurotic participants’ general tendency to set more pessimistic expectations, and thus makes resultant PEs more informative for optimal learning.

I’m currently following up on this question by testing models that split optimistic and pessimistic bracing and estimate effects separately for both directions of bracing. This is more of an exploratory analysis, but I will fill you in if anything interesting emerges.