Study 1. Expectations shift in anticipation of uncertain outcomes

A growing body of social psychology literature suggests that individuals tend to pessimistically lower their expectations as the reveal of important news (e.g., result of a cancer screen) draws near (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. Intuitively, bracing pessimistically should reduce 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 as the grade reveal drew closer. To accomplish this, we took the difference between participants predictions at T1 and T2:

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

Bracing during anticipation is predominantly pessimistic

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 2.55 percentage points in anticipation of viewing their grade.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -2.55144 0.191167 2.217632 -13.34666 0.0036706

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 experience taking exams and thus maximal uncertainty surrounding grades.

Unsigned bracing (magnitude)

Below, 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 confirm our hypothesis, as the magnitude of bracing is greatest at the first exam, and decreases linearly over the remaining exams in the semester (B = -0.47 [0.09], p < 0.0001***). 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.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 5.6686512 0.2724260 24.69207 20.808038 0e+00
exam -0.4655993 0.0930255 720.02557 -5.005074 7e-07

Signed bracing

Looking at the trend of signed bracing over time, it appears that on average, participants brace most pessimistically at the first exam (average bracing at exam 1 = -3.20 [0.32]), and the magnitude of pessimsitic bracing decreases over time, by approximately 0.27 points with each additional exam.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -3.2010392 0.3193852 2544.455 -10.022504 0.0000000
exam 0.2665492 0.1110968 2119.082 2.399253 0.0165145

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.

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.

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

Does bracing impact learning?

Given that bracing inherently shifts PEs and theoretically changes their emotional impact, we hypothesized that bracing would impact the manner in which students learn from grade PEs to more accurately predict future exam grades.

Expectation accuracy (unsigned PE)

The model below tests whether bracing impacted the outcome of learning, that is, expectation accuracy. Here, to test whether students who braced prior to viewing their grades were more or less accurate after they updated their expectations for the next exam, we regressed expectation accuracy at the next exam onto bracing at the current exam.

Results indicate that regardless of the PE, pessimistic bracing is associated with reduced accuracy in expectations at the next exam. This suggests that bracing pessimistically negatively impacts PE learning, presumably by changing the way students update their expectations after both positive and negative PEs. Model parameters suggest that a one-point increase in pessimisic bracing predicts a 0.067 point decrease in expectation accuracy.

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: next_uPE_1 ~ pred_adj * pe_1 + exam + (1 | cohort_class/pt_id)
##    Data: df.retest
## 
## REML criterion at convergence: 10464.9
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.8286 -0.7187 -0.1803  0.4991  4.8372 
## 
## Random effects:
##  Groups             Name        Variance Std.Dev.
##  pt_id:cohort_class (Intercept)  7.162   2.676   
##  cohort_class       (Intercept)  1.277   1.130   
##  Residual                       48.335   6.952   
## Number of obs: 1526, groups:  pt_id:cohort_class, 696; cohort_class, 3
## 
## Fixed effects:
##                 Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)    1.042e+01  8.058e-01  3.601e+00  12.933 0.000379 ***
## pred_adj      -6.672e-02  2.979e-02  1.518e+03  -2.240 0.025261 *  
## pe_1          -1.693e-02  1.635e-02  1.517e+03  -1.035 0.300657    
## exam          -7.062e-01  1.987e-01  1.071e+03  -3.555 0.000395 ***
## pred_adj:pe_1  4.050e-04  1.401e-03  1.514e+03   0.289 0.772616    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) prd_dj pe_1   exam  
## pred_adj     0.104                     
## pe_1        -0.004 -0.378              
## exam        -0.508 -0.029 -0.047       
## pred_dj:p_1 -0.094 -0.032  0.152  0.058

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 10.4209143 0.8057695 3.601145 12.9328725 0.0003791
pred_adj -0.0667171 0.0297896 1518.444728 -2.2396085 0.0252606
pe_1 -0.0169318 0.0163532 1517.333592 -1.0353798 0.3006565
exam -0.7062048 0.1986596 1071.498911 -3.5548491 0.0003947
pred_adj:pe_1 0.0004050 0.0014013 1513.649534 0.2890065 0.7726160

Expectation updating

We followed up to test whether bracing modulates the manner in which students update their expectations after experiencing PEs.

Here, results indicate that bracing does indeed moderate expectation updating, such that following PEs between 0 and -15, pessimistic bracing predicts more negative updates to expectations, but smaller negative updates following negative PEs that are larger than -15. Following positive, PEs, pessimistic bracing predicts a reduced rate of updating. Taking with preceding results, this suggests that bracing impedes expectation accuracy by changing the way participants update their expectations after experiencing PEs. In the case of pessimistic bracing, this appears to impair expectation accuracy regardless of the size and magnitude of PE that one experiences.

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: 
## next_update_1 ~ pred_adj * pe_1 + exam + delta_grade + (1 | cohort_class/pt_id)
##    Data: df.retest[which(df.retest$pred_adj < 0), ]
## 
## REML criterion at convergence: 4860.3
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.3183 -0.5894 -0.0200  0.5858  3.7287 
## 
## Random effects:
##  Groups             Name        Variance Std.Dev.
##  pt_id:cohort_class (Intercept)   0.000   0.000  
##  cohort_class       (Intercept)   1.804   1.343  
##  Residual                       113.008  10.631  
## Number of obs: 640, groups:  pt_id:cohort_class, 459; cohort_class, 3
## 
## Fixed effects:
##                 Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)    -0.938939   1.519241  14.234631  -0.618    0.546    
## pred_adj        0.083595   0.097502 633.604369   0.857    0.392    
## pe_1            0.583826   0.061366 633.476840   9.514   <2e-16 ***
## exam            0.033829   0.473838 463.268110   0.071    0.943    
## delta_grade     0.653440   0.027597 590.990293  23.678   <2e-16 ***
## pred_adj:pe_1   0.002501   0.005599 633.690806   0.447    0.655    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) prd_dj pe_1   exam   dlt_gr
## pred_adj     0.492                            
## pe_1         0.261  0.387                     
## exam        -0.672 -0.079 -0.070              
## delta_grade  0.060  0.056  0.232 -0.026       
## pred_dj:p_1  0.261  0.632  0.761 -0.011 -0.033
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see ?isSingular

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -0.9389388 1.5192405 14.23463 -0.6180317 0.5463163
pred_adj 0.0835950 0.0975017 633.60437 0.8573696 0.3915648
pe_1 0.5838260 0.0613661 633.47684 9.5138132 0.0000000
exam 0.0338290 0.4738381 463.26811 0.0713936 0.9431153
delta_grade 0.6534398 0.0275973 590.99029 23.6776590 0.0000000
pred_adj:pe_1 0.0025014 0.0055992 633.69081 0.4467483 0.6552095

————————————————————————

Study 2. Bracing changes in the face of additional contextual information

Preceding results appear to indicate that bracing is a response to uncertainty in exam grade expectations. This is supported by the finding that participants brace less as they gain more experience taking exams and learn to more accurately predict their exam grades. As shown above, bracing to manage uncertainty impairs participants’ ability to accurately predict future exam grades, presumably because it changes the way participants update their expectations after PEs.

To follow up on these findings, we conducted an identical study with one difference: before asking students to report their T2 exam grade expectations (i.e., immediately before viewing their grades), we displayed the average exam grade for the entire chemistry class. This additional contextual information should in theory inform the way students changed their expectations before viewing their grades.

Here, we were interested to see if and how participants change their expectations when the uncertainty that presumably drove bracing in Study 1 is supplanted by new contextual information. Whereas bracing in Study 1 theoretically represents a response to uncertainty in the absence of new information, presenting the average exam grade to participants in Study 2 allows us to evaluate how bracing behavior changes when new, potentially helpful information is available.

Key questions are as follows:

Is bracing exclusively a response to a lack of information (i.e., uncertainty), or do participants also brace when we provide them with contextual information that might inform their expectations?

If participants still brace after viewing the average exam grade, does their bracing still negatively impact the accuracy of their expectations at the next exam? Or does the average grade lead to well-informed bracing, and thus better accuracy at the next exam?

Bracing is still predominantly pessimistic

Results suggest that like in Study 1, bracing in Study 2 was primarily pessimistic. However, pessimistic bracing in Study 2 across all exams was greater in magnitude, such that on average, participants reduced their expectations by -4.24 points - nearly half a letter grade - after viewing the average exam grade in their chemistry class.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -4.23967 0.3158388 213.3124 -13.42352 0

Bracing still decreases in magnitude over time, but remains negative at the final exam

Similar to in Study 1, bracing in Study 2 was also predominantly pessimistic and decreased progressively in magnitude over time. Whereas in Study 1, average bracing was -3.2 at the first exam, and decreased by 0.27 points with each additional exam, bracing in Study 2 was more pessimistic at the first exam (average bracing at exam 1 = -5.53 [0.71]) and decreased by 0.44 points with each additional exam. The predicted value of bracing at the final exam in Study 1 was -1.85 points, versus -3.32 points in Study 2, indicating that participants continued to brace pessimistically throughout the semester when they viewed the average grade in their class.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -5.5293699 0.7073443 1057.887 -7.817084 0.000000
exam 0.4444827 0.2182001 867.416 2.037042 0.041949

Estimate Std. Error df t value Pr(>|t|)
(Intercept) -5.5293699 0.7073443 1057.887 -7.817084 0.000000
exam 0.4444827 0.2182001 867.416 2.037042 0.041949

Bracing in Study 2 does not impair learning outcomes

In contrast to the effects of bracing on learning outcomes (i.e., expectation accuracy) in Study 1, bracing in Study 2 did not impede participants’ ability to accurately predict subsequent exam grades. This suggests that bracing in Study 2, while more pessimistic, was better informed relative to Study 1, presumably due to the provision of informative contextual information in the form of the class average.

In conclusion, bracing in response to uncertainty impedes accurate PE learning, whereas bracing in the face of information that contextualizes an uncertain outcome is similarly pessimistic but does not hinder PE learning. This suggests that bracing has different effects on learning, depending on whether it is well-informed (i.e., in the context of new information), or whether it is a putatively uninformed, emotion-driven response to uncertainty.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 11.7370461 0.8933918 704.9739 13.1376247 0.0000000
brace 0.0263219 0.0407796 693.5331 0.6454679 0.5188375
pe -0.0520084 0.0248355 683.0683 -2.0941117 0.0366180
exam -0.4479898 0.3030138 536.4154 -1.4784468 0.1398752
brace:pe 0.0014844 0.0011977 704.2225 1.2393038 0.2156459

————————————————————————

Follow up: Relationships between bracing and momentary affect

Does pessimistic bracing buffer reactivity to negative PEs?

Prior work frames bracing as an affect management strategy, whereby pessimistic bracing theoretically dampens one’s emotional response to a negative prediction error by allowing them to “prepare for the worst,” or by mathematically increasing the odds of a positive PE, which should intuitively yield a positive emotional reaction.

In Study 1, we found that bracing varied with the degree of uncertainty in exam grades over the semester, indicating that bracing may itself be an emotion-driven response to this uncertainty. Yet, it is unclear whether bracing serves to buffer latter emotional responses, as extant theory would suggest.

Below, we tested whether pessimistic bracing in Study 1 truly mitigated the affective impact of negative prediction errors. Assuming bracing buffers negative emotional responses to negative PEs, we should see reduced negative affect in participants who braced pessimistically and received negative PEs relative to pariticpants who received equivalent PEs but did not brace or braced less.

The following GAM was fit to momentary affective reactivity (negative affect; NA) data for participants who braced pessimistically:

brm(NA_score ~ t2(pe_1, by = pred_adj) + bin + NA_nondenseBaseline_overall + (1 | pt_id)

Here, a nonlinear interaction between PE and bracing was specified as the focal predictor of momentary NA, and time and baseline NA were specified as covariates.

The nonlinear relationship between momentary NA and the PE*bracing interaction revealed that pessismistic bracing did not buffer NA reactivity, but instead increased NA reactivity to negative PEs. This suggests that in Study 1, pessimistic bracing does not constitute an effective affect management strategy, as existing theory suggests. Rather, participants who braced pessimistically and received negative PEs were more reactive to those PEs than participants who braced less pessimistically.

Is bracing an emotion-driven response to uncertainty?

Given that bracing does not appear to buffer affective reactivity to negative PEs, we also tested whether bracing might instead constitute an emotion-driven response to uncertainty, as preceding associations between bracing and uncertainty might suggest.

Assuming bracing behavior is emotion-driven, we might expect it to be correlated with participants’ NA in anticipation of receiving their grades - that is, in the moments leading up to when they made their second predictions.

Here, a model was specified to test whether bracing is related to anticipatory NA. Given that anticipatory NA is driven in part by participants’ PEs at the preceding exam (i.e., negative PEs at the preceding exam predict more anticipatory NA at the next), an interaction was specified between anticipatory NA (baseline-corrected) and previous exam PE.

Results suggest that participants who experienced negative PEs on the preceding exam and reported heightened anticipatory NA braced the most pessimistically. This suggests that bracing not only scales with the degree of uncertainty in exam grades, but also with momentary NA and the extent to which a previous grade violated one’s expectation. In conclusion, bracing in the absence of new information in Study 1 appears to be an emotion-driven response to uncertainty, and does not effectively buffer affective reactivity to PEs.

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: pred_adj ~ anticipation_NA_mean_bc * pe_1_lag1 + (1 | pt_id)
##    Data: df.retest
## 
## REML criterion at convergence: 9585.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.6443 -0.4725  0.2190  0.3751  7.5871 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  pt_id    (Intercept)  2.855   1.690   
##  Residual             33.072   5.751   
## Number of obs: 1491, groups:  pt_id, 565
## 
## Fixed effects:
##                                     Estimate Std. Error         df t value
## (Intercept)                       -1.895e+00  1.786e-01  4.167e+02 -10.614
## anticipation_NA_mean_bc           -2.114e-02  1.071e-02  1.485e+03  -1.973
## pe_1_lag1                          1.151e-02  1.319e-02  1.478e+03   0.873
## anticipation_NA_mean_bc:pe_1_lag1  2.199e-03  8.171e-04  1.460e+03   2.692
##                                   Pr(>|t|)    
## (Intercept)                        < 2e-16 ***
## anticipation_NA_mean_bc            0.04863 *  
## pe_1_lag1                          0.38282    
## anticipation_NA_mean_bc:pe_1_lag1  0.00719 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) an_NA__ p_1_l1
## antcpt_NA__ -0.325               
## pe_1_lag1   -0.025  0.067        
## a_NA__:_1_1  0.063 -0.029  -0.401

————————————————————————

New/Updated Figures and Analyses (3/2)

Does bracing impact learning?

The figures below depict predictions from the models we reviewed together on Monday, 2/28. I’ve made a few changes to these: - PE is now on the x-axis, instead of bracing - Added marginal histograms - Limited range of axes to include predictions at values of the predictors for which we have an appropriate number of observations.

Expectation accuracy (unsigned PE)

The model below tests whether bracing impacted the outcome of learning, that is, expectation accuracy.

There is a significant main effect of bracing on expectation accuracy at the next exam, such that a one unit decrease in expectations due to bracing pessimistically predicts a 0.067 unit increase in absolute error at the next exam.

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: next_uPE_1 ~ pred_adj * pe_1 + exam + (1 | cohort_class/pt_id)
##    Data: df.retest
## 
## REML criterion at convergence: 10464.9
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.8286 -0.7187 -0.1803  0.4991  4.8372 
## 
## Random effects:
##  Groups             Name        Variance Std.Dev.
##  pt_id:cohort_class (Intercept)  7.162   2.676   
##  cohort_class       (Intercept)  1.277   1.130   
##  Residual                       48.335   6.952   
## Number of obs: 1526, groups:  pt_id:cohort_class, 696; cohort_class, 3
## 
## Fixed effects:
##                 Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)    1.042e+01  8.058e-01  3.601e+00  12.933 0.000379 ***
## pred_adj      -6.672e-02  2.979e-02  1.518e+03  -2.240 0.025261 *  
## pe_1          -1.693e-02  1.635e-02  1.517e+03  -1.035 0.300657    
## exam          -7.062e-01  1.987e-01  1.071e+03  -3.555 0.000395 ***
## pred_adj:pe_1  4.050e-04  1.401e-03  1.514e+03   0.289 0.772616    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) prd_dj pe_1   exam  
## pred_adj     0.104                     
## pe_1        -0.004 -0.378              
## exam        -0.508 -0.029 -0.047       
## pred_dj:p_1 -0.094 -0.032  0.152  0.058

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 10.4209143 0.8057695 3.601145 12.9328725 0.0003791
pred_adj -0.0667171 0.0297896 1518.444728 -2.2396085 0.0252606
pe_1 -0.0169318 0.0163532 1517.333592 -1.0353798 0.3006565
exam -0.7062048 0.1986596 1071.498911 -3.5548491 0.0003947
pred_adj:pe_1 0.0004050 0.0014013 1513.649534 0.2890065 0.7726160

Signed PE (NEW - 3/2)

Per our discussion on 2/28, I added a figure here to show how bracing impacts signed PEs at the next exam, in addition to unsigned PEs (accuracy) above.

There is a significant main effect of bracing on PE, such that a one unit decrease in expectations due to pessimistic bracing predicts a .11 unit increase in PEs at the next exam, controlling for all other predictors.

Here, the interaction between current exam PE and bracing is significant, and the moderating effect of bracing appears concentrated around positive PEs. Negative PEs predict similar PEs at the next exam, regardless of how one braced. On the other hand, PEs are more likely to be positive at the next exam if one braced pessimistically and their PE was negative. In contrast, positive bracing and positive PEs predict a PE at the next exam that is close to zero.

Putting this together, the way one braces at the current exam is positively related to the way they update their expectations for the next exam, but primarily if a positive PE occurred.

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: next_PE_1 ~ pred_adj * pe_1 + exam + (1 | cohort_class/pt_id)
##    Data: df.retest
## 
## REML criterion at convergence: 11823.7
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.9814 -0.5201  0.0232  0.5784  3.5689 
## 
## Random effects:
##  Groups             Name        Variance Std.Dev.
##  pt_id:cohort_class (Intercept)  38.4394  6.2000 
##  cohort_class       (Intercept)   0.6589  0.8117 
##  Residual                       103.2492 10.1612 
## Number of obs: 1526, groups:  pt_id:cohort_class, 696; cohort_class, 3
## 
## Fixed effects:
##                 Estimate Std. Error         df t value Pr(>|t|)  
## (Intercept)    1.371e-01  8.650e-01  7.965e+00   0.158   0.8780  
## pred_adj      -1.141e-01  4.634e-02  1.463e+03  -2.463   0.0139 *
## pe_1           4.015e-02  2.547e-02  1.459e+03   1.577   0.1151  
## exam           1.976e-01  2.952e-01  7.131e+02   0.669   0.5034  
## pred_adj:pe_1 -4.884e-03  2.173e-03  1.427e+03  -2.248   0.0247 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) prd_dj pe_1   exam  
## pred_adj     0.154                     
## pe_1        -0.008 -0.385              
## exam        -0.703 -0.031 -0.050       
## pred_dj:p_1 -0.138 -0.044  0.147  0.061

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 0.1370687 0.8649592 7.964867 0.1584685 0.8780319
pred_adj -0.1141200 0.0463371 1463.043336 -2.4628197 0.0138994
pe_1 0.0401540 0.0254653 1459.331491 1.5768142 0.1150550
exam 0.1976091 0.2951755 713.086711 0.6694632 0.5034167
pred_adj:pe_1 -0.0048845 0.0021728 1426.759788 -2.2479935 0.0247288

Expectation updating

Following up on the preceding result, there is indeed a significant main effect of bracing on updating. There is also a significant interaction between PE and bracing, such that more pessimistic bracing is associated with a lower rate of expectation updating, with differences in predicted updates concentrated around positive PEs. If one braced pessimistically and experienced a positive PE, the model predicts that they would make a smaller update relative to someone who did not brace, or braced optimistically.

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: 
## next_update_1 ~ pred_adj * pe_1 + exam + delta_grade + (1 | cohort_class/pt_id)
##    Data: df.retest
## 
## REML criterion at convergence: 11566.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.0438 -0.5607 -0.0143  0.5837  3.6950 
## 
## Random effects:
##  Groups             Name        Variance Std.Dev.
##  pt_id:cohort_class (Intercept)   7.3763  2.7159 
##  cohort_class       (Intercept)   0.2569  0.5069 
##  Residual                       105.8687 10.2892 
## Number of obs: 1526, groups:  pt_id:cohort_class, 696; cohort_class, 3
## 
## Fixed effects:
##                 Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)    4.150e-01  7.406e-01  1.450e+01   0.560    0.584    
## pred_adj       2.258e-01  4.305e-02  1.504e+03   5.244  1.8e-07 ***
## pe_1           6.751e-01  2.526e-02  1.222e+03  26.732  < 2e-16 ***
## exam          -2.255e-01  2.869e-01  5.111e+02  -0.786    0.432    
## delta_grade    7.112e-01  1.762e-02  9.474e+02  40.360  < 2e-16 ***
## pred_adj:pe_1  8.108e-03  2.028e-03  1.520e+03   3.998  6.7e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) prd_dj pe_1   exam   dlt_gr
## pred_adj     0.165                            
## pe_1        -0.022 -0.390                     
## exam        -0.810 -0.031 -0.046              
## delta_grade -0.046 -0.119  0.382  0.012       
## pred_dj:p_1 -0.143 -0.013  0.097  0.062 -0.122

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 0.4149539 0.7405965 14.49532 0.5602969 0.5838335
pred_adj 0.2257703 0.0430543 1504.21199 5.2438549 0.0000002
pe_1 0.6751433 0.0252560 1222.13418 26.7319510 0.0000000
exam -0.2255173 0.2868553 511.06817 -0.7861711 0.4321316
delta_grade 0.7111983 0.0176212 947.43656 40.3603628 0.0000000
pred_adj:pe_1 0.0081076 0.0020279 1519.69766 3.9979805 0.0000670

PE 1 vs PE 2 accuracy

Study 1

In study 1, when we didn’t display the average grade before students made their second predictions, the difference in accuracy between prediction 1 and prediction 2 is trending. Bracing appears to result in a slight, non-significant improvement in accuracy within an exam.

Below is a summary of a simple model testing for differences in magnitude between unsigned PE 1 and unsigned PE 2. I’ve also included a plot depicting these differences across exams 1-5.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 10.3479674 0.4758621 2.022302 21.745725 0.0019970
variableunsigned_pe_2 -0.3279687 0.1904082 4192.468459 -1.722451 0.0850616

Study 2 (displaying average grade before prediction)

Surprisingly, bracing in study 2 also did not cause improvement or any significant change in prediction accuracy.

Estimate Std. Error df t value Pr(>|t|)
(Intercept) 12.276950 0.4478639 362.1192 27.4122343 0.0000000
variableunsigned_pe_2 -0.141383 0.3811594 1827.9237 -0.3709288 0.7107336

Raw data figures

Below, I pulled the average updates, accuracy, and signed PE values at different levels of PE and bracing. Error bars represent SE of the mean. These are alternatives for the model-predicted results (aka ghost plots).

Average Update (Next Exam)

From the raw data averages plotted here, we can see the main effects of PE, bracing, and how they interact.

Regardless of PE valence, students who brace positively (i.e., optimistically; increase their expectations) appear more likely to increase their expectations at the next exam. The largest differences are in the case of PEs that were close to zero (-1:1). In the zero PE case, average updates for individuals who braced negatively or not at all are negative, while the average updates for those who braced positively are positive.

Average Accuracy (Next Exam)

Differences predicted by bracing within each level of PE don’t look that robust, but there is an apparent interaction between PE and bracing, such that positive bracing predicts poorer accuracy in the case of negative PEs, and improved accuracy in the case of positive PEs.

Average PE (Next Exam)

Looks like people who received negative or neutral PEs were less likely to brace positively given the size of the standard errors here, but interestingly, positive bracing in the case of a PE of zero is linked to negative PEs at the next exam.