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The proportion of questions each student answered correctly on the first try and the posterior mean of $\eta_T$ for each student from the 2PL and GRM models plotted against the student posterior means of $\eta_T$ from the Rasch model

The latent variable $\eta_T$ measures differences in implementation between students. Figure <ref>  compares estimates (posterior means) of  $\eta_T$ in the Rasch model to analogous estimates from the 2PL and GRM models, respectively, as well as to the
proportion of problems each student answered correctly on the first try,  $\hat{p}$.
The three latent variable models agree on  $\eta_T$ to a remarkable extent–the Pearson correlations between estimates across Rasch, 2PL, and GRM models are all above 0.98. The correspondence to the observed $\hat{p}$ is more moderate, though also high, with a correlation of around 0.86.
The posterior standard deviations of $\eta_T$ between the Rasch, 2PL, and GRM models were also highly correlated, with Pearson correlations of roughly 0.99.

Table <ref> shows the results of the submodel predicting $\eta_T$
as a function of Z-scored student-level covariates. The covariates–-including teacher indicators-–were, collectively, fairly predictive; in the 2PL and GRM models, they explained roughly 60% of the variance in $\eta_T$, while in the Rasch model, $R^2$ was roughly 80%. The reason $R^2$ differed between the Rasch model and the others is likely due to the fact that the variance of $\eta_T$ in the Rasch model is relatively unconstrained, whereas the
residual variance of $\eta_T$ was fixed at 1 in the 2PL and GRM models (this was necessary because $a_1$ was unconstrained).

Estimated coefficients and standard errors from models predicting $\eta_T$ as a function of covariates.

Rasch          2PL          GRM

Male     –0.07 *     (0.03)     –0.09     (0.05)     –0.06     (0.05)

Hispanic/Latino     –0.01     (0.03)     –0.01     (0.06)     0.01     (0.07)

Asian     0.09 *     (0.04)     0.1     (0.07)     0.09     (0.07)

Other     –0.02     (0.03)     –0.04     (0.06)     –0.02     (0.06)

Accelerated     0.11 *     (0.05)     0.1     (0.09)     0.05     (0.1)

5th Grd State Test     0.37 *     (0.05)     0.5 *     (0.09)     0.53 *     (0.09)

Pretest     0.15 *     (0.05)     0.2 *     (0.09)     0.18     (0.09)

EIP     –0.06 *     (0.03)     –0.08     (0.06)     –0.12 *     (0.06)

ESOL     0.01     (0.04)     0.02     (0.06)     0.04     (0.06)

Gifted     0.05     (0.03)     0.07     (0.06)     0.07     (0.06)

IEP     0.02     (0.03)     -0     (0.05)     0.01     (0.05)

$\sqrt{6^{th}\mbox{ Grd Days Absent}}$     –0.02     (0.03)     –0.03     (0.06)     –0.04     (0.06)

Log(Pretest Avg. Time)     0.07 *     (0.03)     0.08     (0.06)     0.1     (0.06)

Math Anxiety     –0.06     (0.04)     –0.08     (0.07)     –0.06     (0.07)

Math Self-Efficacy     0.04     (0.03)     0.04     (0.07)     0.07     (0.07)

Perceptual Sensitivity     0.14 *     (0.04)     0.12     (0.08)     0.12     (0.08)

$\sigma_U$     0.49          1          1

$R^2$     0.8          0.6          0.61

* Note. * Central 95% credible interval excludes 0. Teacher effects are omitted.

Since the covariates were Z-scored prior to model fitting, the coefficient estimates in Table <ref> are all on the same scale within each model; however, the variance of $\eta_T$
differed between the Rasch model and the 2PL and GRM models, so the sizes of coefficients in the Rasch model are not comparable to those in the other two. The most important predictor appears to be students’ scores on the state 5th-grade standardized math test; students who scored higher in 5th grade were more likely to answer ASSISTments problems correctly and less likely to receive feedback. Student pretest scores showed a similar pattern, although with a smaller coefficient. Male students and students in early intervention programs (EIP) may have been less likely than their peers to answer problems correctly on the first try, while students who spent more time on task during the pretest and scored higher on a perceptual sensitivity (coefficients for these predictors were significant with p<0.05 for some models but not others, but point estimates were similar across models).

Modeling Outcomes and Estimating Effects

Table <ref> shows estimates of causal parameters $\tau_0$  and $\tau_1$, structural parameter $\omega$, and covariate coefficients $\boldsymbol{\gamma}$ (excluding teacher fixed effects) from the three FLPS models. The outcome was standardized prior to fitting the models, so the treatment effects are in standard deviation units. The parameter $\tau_0$ can be thought of as the average treatment effect when $\eta_T$
takes its mean value–in other words, the average effect of assignment to the “Immediate” condition for the average student. The 2PL and GRM estimate small positive effects while the Rasch FLPS model estimates a small negative effect; however, the standard errors on all four estimates are sufficiently large that both small positive and negative effects are consistent with the data–in fact, all four models rule out effects for average students that are greater than 0.1 standard deviations in either direction.

Counterintuitively, all three models estimate positive $\tau_1$: that the effect of assignment to the “Immediate” condition is higher, on average, for students who had a greater propensity to answer problems correctly on the first try, and hence to receive less feedback. However, the data are also consistent with values of $\tau_1$ that are slightly negative or null–the posterior probability that $\tau_1>0$ ranges from 0.81 (Rasch) to 0.96 (2PL).

A positive value of $\tau_1$ would not necessarily imply that receiving feedback decreases the effect of the treatment assignment; in the principal stratification framework, $\eta_T$ is taken as student baseline characteristic rather than a manipulable behavior. Students with high values of $\eta_T$ may differ from their low-$\eta_T$ peers in other ways that may lead to positive effects of immediate feedback. For instance, students who answer problems correctly with greater frequency may be more conscientious and may benefit more from the feedback they get. Along similar lines, prior mathematics test scores are correlated with $\eta_T$, suggesting that students who receive less feedback began the study with greater math ability, on average, which may allow them to better understand hints and error messages.

The moderator-like role of $\eta_T$ in principal stratification is illustrated in Figure <ref>. For each of the four models, Figure <ref> plots each student’s $\eta_T$  against their posttest score.
The plotted $\eta_{T}$ are all taken from a random posterior draw for students in both the Immediate and Delayed feedback groups: values for students assigned to Immediate feedback were estimated based on the amount of feedback received, using either the Rasch, 2PL, or GRM model, and values for students assigned to delayed feedback were imputed as a function of their baseline covariates, using the models described in Table <ref>. In all three models, for both treatment arms, $\eta_{T}$ is positively correlated with posttest scores (consistent with positive slope estimates $\omega$ shown in Table <ref>). However, the slopes are slightly different between the two treatment arms–i.e. a treatment interaction–the distance between the two regression lines shown for any given value for $\eta_T$ or $\hat{p}$ represents the estimated principal effect.

Previous work has shown little to no difference in state test scores between the Immediate and Delayed feedback conditions in this RCT; however, it was unclear whether the reason for this null result is that when it comes to the state test, there really is no difference between the two conditions, or that students in Immediate arm the RCT just didn't receive enough feedback—make enough errors or request enough hints—for the effect to be detectable.
If the reason for the null effect was a low dosage, then we would hope to see negative estimates of $\tau_1$ in FLPS models. Instead, $\tau_1$ was estimated to be close to zero; if anything, $\tau_1$ was more likely to be positive than negative. These results suggest that, at least in the context of this particular RCT, there really was little causal difference between the two conditions on the state test.
If there was an effect, it was most likely largest for students who received less feedback but were better equipped to understand it and make use of it.

In fitting FLPS models, it made little difference what measurement model to use—estimates of $\eta_T$, $\tau_0$, and $\tau_1$ were highly similar between the three models.
This comforting regularity suggests a robustness to misspecification of the measurement model; future research may determine whether this reflects an underlying property of FLPS models or was merely fortuitous.
In any event, the results of this illustration underscore the conclusions in Section <ref>, that FLPS models are feasible in real-life data analyses.

Structural Parameters from outcome submodels.

Rasch          2PL          GRM

(Intercept)     9.05 *     (0.03)     9.04 *     (0.03)     9.03 *     (0.03)

$\omega$     0.46 *     (0.06)     0.35 *     (0.04)     0.32 *     (0.05)

$\tau_0$     –0.01     (0.04)     0.02     (0.04)     0.03     (0.04)

$\tau_1$     0.03     (0.04)     0.06     (0.04)     0.04     (0.04)

Male     0.04 *     (0.02)     0.03     (0.02)     0.03     (0.02)

Hispanic/Latino     -0     (0.02)     –0.01     (0.02)     –0.01     (0.02)

Asian     –0.04     (0.03)     –0.02     (0.03)     –0.01     (0.03)

Other     –0.02     (0.02)     –0.02     (0.02)     –0.03     (0.02)

Accelerated     0.09 *     (0.04)     0.11 *     (0.03)     0.13 *     (0.03)

5th Grd State Test     0.24 *     (0.04)     0.27 *     (0.04)     0.28 *     (0.04)

Pretest     0.05     (0.04)     0.06     (0.03)     0.07 *     (0.03)

EIP     –0.01     (0.02)     –0.01     (0.02)     –0.01     (0.02)

ESOL     –0.04     (0.02)     –0.04     (0.02)     –0.04     (0.02)

Gifted     0.05 *     (0.02)     0.05 *     (0.02)     0.05 *     (0.02)

IEP     –0.05 *     (0.02)     –0.04 *     (0.02)     –0.05 *     (0.02)

$\sqrt{6^{th}\mbox{ Grd Days Absent}}$     –0.02     (0.02)     –0.02     (0.02)     –0.02     (0.02)

Log(Pretest Avg. Time)     –0.05 *     (0.02)     –0.04 *     (0.02)     –0.04 *     (0.02)

Math Anxiety     -0     (0.02)     –0.01     (0.02)     –0.02     (0.02)

Math Self-Efficacy     0.03     (0.02)     0.04     (0.02)     0.03     (0.02)

Perceptual Sensitivity     –0.01     (0.03)     0.02     (0.03)     0.03     (0.03)

$\sigma_Y$     0.45          0.44          0.45

$R^2$     0.53          0.62          0.63

* Note. * Central 95% credible interval excludes 0. Covariate coefficients and teacher fixed-effects omitted.

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