Turn Taking Distributional Analyses
1 Experiment 1
In this study, children and adults answered questions while playing an iPad game. We varied
- Whether the questions’s final word was predicted or not (factor Pred/Unpred).
- Whether the context of the question meant that the length of the final word was predictable or not (factor Match/Mismatcb).
- The interaction of these two factors (i.e., a proper test of whether participants predict during conversation).
In this analysis, we fit hierarchical Bayesian models to characterize the distribution of reaction times
1.1 Distribution of Respose Times
The graphs below show the distribution of normaliszed RTs across conditions. When the answer was less predictable, participants were slower to respond. But there was no interaction in the non-distributional analyses.
## Warning: Removed 61 rows containing non-finite values (stat_bin).## Warning: Removed 6 rows containing missing values (geom_path).
1.2 Analysis
An ex-gaussian distribution convolves a normal distribution with an exponential – this gives it a long and heavy right tail. We aimed to fit an ex-Gaussian to the response time data, comparing adults with children, and hierarchically modeling the mu parameter, the sigma parameter, and the tau parameter. Mu is the mean of the normal and sigma its standard deviation, while tau is the rate of the exponential. We modeled how these varied across conditions, accounting for random subjec intercepts. Informally, our model had the form:
Mu ~ B0 + (B1 * Match) + (B2 * Pred) + (B3 * Age (5)) + (B4 * Age (3)) + … (the full set of interactions) + (1|Subject)
Sigma ~ B0 + (B1 * Match) + (B2 * Pred) + (B3 * Age (5))+ (B4 * Age (3)) + … (the full set of interactions) + (1|Subject)
Tau ~ B0 + (B1 * Match) + (B2 * Pred) + (B3 * Age (5)) + (B4 * Age (3)) + … (the full set of interactions) + (1|Subject)
This model initially did not converge that well. Following Jake Westfall’s ManyBabies Bayesian analysis (p.c.) we started using Beta distributions as priors for the standard deviation of the prior on By subject intercepts. Once we did that, things converged more nicely.
1.2.1 Ex-Gaussian analysis of All ages
z <- 0
convert_stan_to_dataframe <- function(stan_object){
sum.df <- data.frame(summary(stan_object, pars = c("beta0","beta","beta_s0","beta_s","beta_t0","beta_t"), probs = c(0.025,0.975))$summary)
colnames(sum.df) <- c("mean","se","sd","X2.5","X97.5","n_eff","R_hat")
sum.df <- sum.df[,c("mean","se","sd","n_eff","R_hat","X2.5","X97.5")]
sum.df$Diff_from_zero <- ifelse((sum.df$X2.5 * sum.df$X97.5) > 0, "***","-")
return(sum.df)
}
stan_col_names = c("Parameter","Mean","S.E.","S.D.","Eff. Samples","R_hat","2.5%","97.5%","Significance")
expt1.full.summary <- convert_stan_to_dataframe(eg_expt1_full)
kable(data.frame("Params" = age_factors,expt1.full.summary), digits = 2,
col.names = stan_col_names)| Parameter | Mean | S.E. | S.D. | Eff. Samples | R_hat | 2.5% | 97.5% | Significance | |
|---|---|---|---|---|---|---|---|---|---|
| beta0 | mu_intercept | -0.74 | 0.00 | 0.03 | 280.94 | 1.01 | -0.81 | -0.68 | *** |
| beta[1] | mu_Match | 0.00 | 0.00 | 0.01 | 2350.41 | 1.00 | -0.03 | 0.01 | - |
| beta[2] | mu_Pred | -0.05 | 0.00 | 0.02 | 1003.60 | 1.00 | -0.08 | -0.01 | *** |
| beta[3] | mu_Fives | 0.06 | 0.00 | 0.07 | 213.20 | 1.02 | -0.02 | 0.22 | - |
| beta[4] | mu_Threes | -0.01 | 0.00 | 0.03 | 594.80 | 1.00 | -0.11 | 0.04 | - |
| beta[5] | mu_MatchByPred | -0.01 | 0.00 | 0.01 | 2162.16 | 1.00 | -0.03 | 0.01 | - |
| beta[6] | mu_MatchByFives | 0.02 | 0.00 | 0.01 | 808.25 | 1.00 | 0.00 | 0.04 | - |
| beta[7] | mu_MatchByThrees | 0.00 | 0.00 | 0.01 | 4000.00 | 1.00 | -0.03 | 0.02 | - |
| beta[8] | mu_PredByFives | 0.02 | 0.00 | 0.02 | 1102.45 | 1.00 | 0.00 | 0.05 | - |
| beta[9] | mu_PredByThrees | 0.00 | 0.00 | 0.01 | 4000.00 | 1.00 | -0.02 | 0.02 | - |
| beta[10] | mu_MatchByPredByFives | -0.01 | 0.00 | 0.01 | 1208.87 | 1.00 | -0.03 | 0.01 | - |
| beta[11] | mu_MatchByPredByThrees | 0.01 | 0.00 | 0.01 | 1727.24 | 1.00 | -0.01 | 0.05 | - |
| beta_s0 | sig_intercept | -1.73 | 0.00 | 0.08 | 1279.66 | 1.00 | -1.89 | -1.58 | *** |
| beta_s[1] | sig_Match | 0.02 | 0.00 | 0.04 | 1448.54 | 1.00 | -0.04 | 0.13 | - |
| beta_s[2] | sig_Pred | 0.01 | 0.00 | 0.04 | 2217.81 | 1.00 | -0.06 | 0.10 | - |
| beta_s[3] | sig_Fives | -0.01 | 0.00 | 0.06 | 1623.11 | 1.00 | -0.17 | 0.09 | - |
| beta_s[4] | sig_Threes | 0.02 | 0.00 | 0.07 | 1000.95 | 1.00 | -0.08 | 0.20 | - |
| beta_s[5] | sig_MatchByPred | 0.00 | 0.00 | 0.03 | 2890.97 | 1.00 | -0.08 | 0.07 | - |
| beta_s[6] | sig_MatchByFives | 0.06 | 0.00 | 0.07 | 616.49 | 1.01 | -0.02 | 0.24 | - |
| beta_s[7] | sig_MatchByThrees | 0.00 | 0.00 | 0.04 | 2175.80 | 1.00 | -0.08 | 0.11 | - |
| beta_s[8] | sig_PredByFives | 0.03 | 0.00 | 0.05 | 992.19 | 1.00 | -0.03 | 0.17 | - |
| beta_s[9] | sig_PredByThrees | 0.01 | 0.00 | 0.04 | 2649.68 | 1.00 | -0.07 | 0.12 | - |
| beta_s[10] | sig_MatchByPredByFives | 0.00 | 0.00 | 0.04 | 2282.73 | 1.00 | -0.07 | 0.09 | - |
| beta_s[11] | sig_MatchByPredByThrees | 0.01 | 0.00 | 0.04 | 1675.77 | 1.00 | -0.07 | 0.12 | - |
| beta_t0 | tau_intercept | -0.16 | 0.00 | 0.10 | 458.64 | 1.01 | -0.36 | 0.03 | - |
| beta_t[1] | tau_Match | -0.02 | 0.00 | 0.04 | 4000.00 | 1.00 | -0.11 | 0.06 | - |
| beta_t[2] | tau_Pred | -0.31 | 0.00 | 0.06 | 1513.76 | 1.00 | -0.43 | -0.19 | *** |
| beta_t[3] | tau_Fives | -1.73 | 0.01 | 0.25 | 604.21 | 1.00 | -2.21 | -1.24 | *** |
| beta_t[4] | tau_Threes | 0.69 | 0.01 | 0.26 | 539.38 | 1.01 | 0.08 | 1.14 | *** |
| beta_t[5] | tau_MatchByPred | 0.07 | 0.00 | 0.05 | 2120.62 | 1.00 | -0.02 | 0.18 | - |
| beta_t[6] | tau_MatchByFives | 0.00 | 0.00 | 0.05 | 2346.86 | 1.00 | -0.10 | 0.10 | - |
| beta_t[7] | tau_MatchByThrees | -0.03 | 0.00 | 0.05 | 1478.16 | 1.00 | -0.15 | 0.06 | - |
| beta_t[8] | tau_PredByFives | -0.01 | 0.00 | 0.05 | 4000.00 | 1.00 | -0.12 | 0.09 | - |
| beta_t[9] | tau_PredByThrees | 0.03 | 0.00 | 0.05 | 2076.14 | 1.00 | -0.06 | 0.14 | - |
| beta_t[10] | tau_MatchByPredByFives | -0.01 | 0.00 | 0.04 | 4000.00 | 1.00 | -0.10 | 0.08 | - |
| beta_t[11] | tau_MatchByPredByThrees | -0.04 | 0.00 | 0.05 | 2186.64 | 1.00 | -0.16 | 0.05 | - |
1.2.2 Ex Gaussian analysis of Adults alone
no_age_factors <- c("mu_intercept","mu_Match","mu_Pred","mu_MatchByPred",
"sig_intercept","sig_Match","sig_Pred","sig_MatchByPred",
"tau_intercept","tau_Match","tau_Pred","tau_MatchByPred")
#expt1.adult.summary <- convert_stan_to_dataframe(eg_expt1_adults)
#kable(data.frame("Params" = no_age_factors,expt1.adult.summary), digits = 2,
# col.names = stan_col_names)1.2.3 Ex Gaussian analysis of Children alone (not broken up by age)
expt1.kids.summary <- convert_stan_to_dataframe(eg_expt1_kids_items)
z<-0
kable(data.frame("Params" = kid_factors,expt1.kids.summary), digits = 2,
col.names = stan_col_names)| Parameter | Mean | S.E. | S.D. | Eff. Samples | R_hat | 2.5% | 97.5% | Significance | |
|---|---|---|---|---|---|---|---|---|---|
| beta0 | mu_Intercept | -0.83 | 0 | 0.03 | 809.99 | 1.00 | -0.89 | -0.76 | *** |
| beta[1] | mu_Match | 0.01 | 0 | 0.01 | 2650.62 | 1.00 | -0.01 | 0.04 | - |
| beta[2] | mu_Pred | -0.06 | 0 | 0.02 | 2894.82 | 1.00 | -0.10 | -0.02 | *** |
| beta[3] | mu_Age | -0.01 | 0 | 0.02 | 1017.72 | 1.01 | -0.07 | 0.03 | - |
| beta[4] | mu_MatchByPred | 0.01 | 0 | 0.01 | 3587.75 | 1.00 | -0.02 | 0.04 | - |
| beta[5] | mu_MatchByAge | 0.00 | 0 | 0.01 | 4952.79 | 1.00 | -0.02 | 0.02 | - |
| beta[6] | mu_PredByAge | 0.01 | 0 | 0.01 | 4590.60 | 1.00 | -0.01 | 0.03 | - |
| beta[7] | mu_MatchByPredByAge | 0.01 | 0 | 0.01 | 2588.60 | 1.00 | -0.01 | 0.04 | - |
| beta_s0 | sigma_Intercept | -1.92 | 0 | 0.10 | 2524.41 | 1.00 | -2.12 | -1.75 | *** |
| beta_s[1] | sigma_Match | -0.02 | 0 | 0.06 | 1833.66 | 1.00 | -0.19 | 0.07 | - |
| beta_s[2] | sigma_Pred | -0.02 | 0 | 0.06 | 2377.78 | 1.00 | -0.19 | 0.08 | - |
| beta_s[3] | sigma_Age | 0.05 | 0 | 0.10 | 1312.83 | 1.00 | -0.06 | 0.33 | - |
| beta_s[4] | sigma_MatchByPred | 0.02 | 0 | 0.06 | 2564.98 | 1.00 | -0.07 | 0.19 | - |
| beta_s[5] | sigma_MatchByAge | 0.01 | 0 | 0.05 | 2743.29 | 1.00 | -0.08 | 0.15 | - |
| beta_s[6] | sigma_PredByAge | 0.02 | 0 | 0.06 | 2630.07 | 1.00 | -0.07 | 0.17 | - |
| beta_s[7] | sigma_MatchByPredByAge | 0.01 | 0 | 0.05 | 3212.20 | 1.00 | -0.10 | 0.15 | - |
| beta_t0 | tau_Intercept | 0.10 | 0 | 0.10 | 1170.33 | 1.00 | -0.08 | 0.31 | - |
| beta_t[1] | tau_Match | -0.02 | 0 | 0.05 | 2309.26 | 1.00 | -0.13 | 0.07 | - |
| beta_t[2] | tau_Pred | -0.26 | 0 | 0.08 | 3568.89 | 1.00 | -0.41 | -0.11 | *** |
| beta_t[3] | tau_Age | -0.02 | 0 | 0.09 | 1944.12 | 1.00 | -0.24 | 0.15 | - |
| beta_t[4] | tau_MatchByPred | 0.05 | 0 | 0.06 | 2947.50 | 1.00 | -0.03 | 0.18 | - |
| beta_t[5] | tau_MatchByAge | -0.01 | 0 | 0.04 | 4660.12 | 1.00 | -0.10 | 0.06 | - |
| beta_t[6] | tau_PredByAge | 0.02 | 0 | 0.04 | 4835.00 | 1.00 | -0.05 | 0.12 | - |
| beta_t[7] | tau_MatchByPredByAge | -0.04 | 0 | 0.05 | 2808.62 | 1.00 | -0.14 | 0.04 | - |
2 Experiment 2
In this study, children and adults answered questions while playing an iPad game. We varied
- Whether the context of the question made it possible to predict the answer (factor Pred/Unpred).
- Whether the form of the question made it possible to predict the answer (factor Early/Late).
- The interaction of these two factors (i.e., a proper test of whether participants predict during conversation).
- Whether the answer to the question was Yes/No (to ensure that participants’ attend to the questions).
In this analysis, we fit hierarchical Bayesian models to characterize the distribution of reaction times
2.1 Distribution of Respose Times
The graphs below show the distribution of normaliszed RTs across conditions (we exclude trials where participants were incorrect, and participants who produced fewer than 20 datapoints). When information comes early, adults respond reliably faster in predictable contexts (see non-distributional analyses). It is not clear if the same is true for children.
## Warning: Removed 104 rows containing non-finite values (stat_bin).## Warning: Removed 6 rows containing missing values (geom_path).
2.2 Analysis
We could not fit a whole group analysis, as the adult data were not well fit by an ex-gaussian. Instead, we fit a normal to the adults, and an ex-gaussian to the children. We were finally able to fit a whole group analysis, but separately exponentiating the intercept and the predictors of the scale (sigma and tau) terms. The final regression had roughly this structure:
Mu ~ B0 + (B1 * Early) + (B2 * Pred) + (B3 * Age) + … (the full set of interactions) + (1|Subject)
Sigma ~ B0 + (B1 * Early) + (B2 * Pred) + (B3 * Age) + … (the full set of interactions) + (1|Subject)
Tau ~ B0 + (B1 * Early) + (B2 * Pred) + (B3 * Age) + … (the full set of interactions) + (1|Subject)
2.2.1 Whole group analysis
load("./Expt2/eg_expt2_simpleage.RDATA")
z<-0levels= c("Intercept","Early","Pred","Age","EarlyByPred","EarlyByAge","PredByAge","EarlyByPredByAge")
whole_factors <- c(paste("mu_", levels, sep = ""),
paste("sigma_", levels, sep = ""),
paste("tau_", levels, sep = ""))
expt2.whole.summary <- convert_stan_to_dataframe(eg_expt2_full)
kable(data.frame("Params" = whole_factors,expt2.whole.summary), digits = 2,
col.names = stan_col_names)| Parameter | Mean | S.E. | S.D. | Eff. Samples | R_hat | 2.5% | 97.5% | Significance | |
|---|---|---|---|---|---|---|---|---|---|
| beta0 | mu_Intercept | -0.63 | 0.00 | 0.02 | 576.90 | 1.01 | -0.67 | -0.60 | *** |
| beta[1] | mu_Early | -0.11 | 0.00 | 0.02 | 867.04 | 1.00 | -0.15 | -0.08 | *** |
| beta[2] | mu_Pred | -0.05 | 0.00 | 0.03 | 630.46 | 1.00 | -0.11 | 0.01 | - |
| beta[3] | mu_Age | 0.13 | 0.00 | 0.04 | 576.50 | 1.00 | 0.06 | 0.20 | *** |
| beta[4] | mu_EarlyByPred | -0.04 | 0.00 | 0.01 | 4000.00 | 1.00 | -0.06 | -0.03 | *** |
| beta[5] | mu_EarlyByAge | 0.07 | 0.00 | 0.01 | 4000.00 | 1.00 | 0.05 | 0.08 | *** |
| beta[6] | mu_PredByAge | 0.01 | 0.00 | 0.03 | 779.98 | 1.01 | -0.04 | 0.07 | - |
| beta[7] | mu_EarlyByPredByAge | 0.01 | 0.00 | 0.01 | 4000.00 | 1.00 | 0.00 | 0.03 | - |
| beta_s0 | sigma_Intercept | -2.21 | 0.00 | 0.06 | 2205.38 | 1.00 | -2.32 | -2.10 | *** |
| beta_s[1] | sigma_Early | -0.18 | 0.00 | 0.07 | 3100.86 | 1.00 | -0.32 | -0.02 | *** |
| beta_s[2] | sigma_Pred | -0.02 | 0.00 | 0.07 | 3224.99 | 1.00 | -0.17 | 0.12 | - |
| beta_s[3] | sigma_Age | -0.27 | 0.00 | 0.12 | 1737.87 | 1.00 | -0.49 | -0.02 | *** |
| beta_s[4] | sigma_EarlyByPred | 0.04 | 0.00 | 0.05 | 4000.00 | 1.00 | -0.05 | 0.16 | - |
| beta_s[5] | sigma_EarlyByAge | 0.11 | 0.00 | 0.06 | 4000.00 | 1.00 | 0.00 | 0.24 | *** |
| beta_s[6] | sigma_PredByAge | 0.10 | 0.00 | 0.09 | 2448.33 | 1.00 | -0.05 | 0.30 | - |
| beta_s[7] | sigma_EarlyByPredByAge | 0.00 | 0.00 | 0.05 | 4000.00 | 1.00 | -0.10 | 0.09 | - |
| beta_t0 | tau_Intercept | -0.47 | 0.00 | 0.08 | 866.15 | 1.00 | -0.62 | -0.31 | *** |
| beta_t[1] | tau_Early | -0.25 | 0.00 | 0.06 | 2429.04 | 1.00 | -0.38 | -0.13 | *** |
| beta_t[2] | tau_Pred | -0.33 | 0.01 | 0.16 | 720.73 | 1.00 | -0.64 | -0.03 | *** |
| beta_t[3] | tau_Age | -1.81 | 0.01 | 0.15 | 753.98 | 1.00 | -2.10 | -1.52 | *** |
| beta_t[4] | tau_EarlyByPred | 0.05 | 0.00 | 0.04 | 4000.00 | 1.00 | -0.02 | 0.13 | - |
| beta_t[5] | tau_EarlyByAge | 0.01 | 0.00 | 0.04 | 4000.00 | 1.00 | -0.06 | 0.08 | - |
| beta_t[6] | tau_PredByAge | 0.01 | 0.00 | 0.11 | 1150.81 | 1.00 | -0.21 | 0.25 | - |
| beta_t[7] | tau_EarlyByPredByAge | -0.09 | 0.00 | 0.04 | 4000.00 | 1.00 | -0.17 | -0.02 | *** |
2.2.2 Distributional Analysis of Adults (normal, not ex-gaussian)
tt_adult <- subset(tt, Age == "Adult")
load("./Expt2/eg_expt2_simpleAdult.RDATA")
z<-0
convert_stan_to_dataframe_normal <- function(stan_object){
sum.df <- data.frame(summary(stan_object, pars = c("beta0","beta","beta_s0","beta_s"), probs = c(0.025,0.975))$summary)
colnames(sum.df) <- c("mean","se","sd","X2.5","X97.5","n_eff","R_hat")
sum.df <- sum.df[,c("mean","se","sd","n_eff","R_hat","X2.5","X97.5")]
sum.df$Diff_from_zero <- ifelse((sum.df$X2.5 * sum.df$X97.5) > 0, "***","-")
return(sum.df)
}z<-1
levels= c("Intercept","Early","Pred","EarlyByPred")
adult_factors <- c(paste("mu_", levels, sep = ""),
paste("sigma_", levels, sep = ""),
paste("tau_",levels, sep = ""))
expt2.adults.summary <- convert_stan_to_dataframe(eg_expt2_adult)
kable(data.frame("Params" = adult_factors,expt2.adults.summary), digits = 2,
col.names = stan_col_names)| Parameter | Mean | S.E. | S.D. | Eff. Samples | R_hat | 2.5% | 97.5% | Significance | |
|---|---|---|---|---|---|---|---|---|---|
| beta0 | mu_Intercept | -0.67 | 0.00 | 0.05 | 180.77 | 1.02 | -0.77 | -0.58 | *** |
| beta[1] | mu_Early | -0.10 | 0.00 | 0.02 | 714.46 | 1.00 | -0.14 | -0.05 | *** |
| beta[2] | mu_Pred | -0.06 | 0.00 | 0.08 | 284.54 | 1.00 | -0.24 | 0.06 | - |
| beta[3] | mu_EarlyByPred | -0.04 | 0.00 | 0.01 | 1683.57 | 1.00 | -0.07 | -0.01 | *** |
| beta_s0 | sigma_Intercept | -1.89 | 0.00 | 0.11 | 936.39 | 1.01 | -2.11 | -1.70 | *** |
| beta_s[1] | sigma_Early | -0.08 | 0.00 | 0.08 | 1463.15 | 1.00 | -0.26 | 0.04 | - |
| beta_s[2] | sigma_Pred | 0.00 | 0.00 | 0.11 | 1481.74 | 1.00 | -0.24 | 0.26 | - |
| beta_s[3] | sigma_EarlyByPred | 0.01 | 0.00 | 0.06 | 2623.38 | 1.00 | -0.12 | 0.15 | - |
| beta_t0 | tau_Intercept | -0.42 | 0.01 | 0.13 | 462.59 | 1.01 | -0.67 | -0.16 | *** |
| beta_t[1] | tau_Early | -0.26 | 0.00 | 0.11 | 980.31 | 1.00 | -0.46 | -0.04 | *** |
| beta_t[2] | tau_Pred | -0.27 | 0.01 | 0.24 | 261.99 | 1.01 | -0.79 | 0.12 | - |
| beta_t[3] | tau_EarlyByPred | -0.05 | 0.00 | 0.06 | 2308.83 | 1.00 | -0.18 | 0.06 | - |
2.2.3 Ex-Gaussian Analysis of Kids
tt_kids <- subset(tt, Age != "Adult")
z<-0
levels = c("Intercept","Early","Pred","AgeThree","EarlyByPred","EarlyByThree","PredByThree","EarlyByPredByThree")
kid_factors <- c(paste("mu_",levels, sep = ""),
paste("sigma_",levels, sep = ""),
paste("tau_",levels, sep = ""))
load("./Expt2/eg_expt2_simpleChild.RDATA")expt2.kids.summary <- convert_stan_to_dataframe(eg_expt2_child)
z<-0
kable(data.frame("Params" = adult_factors,expt2.kids.summary), digits = 2,
col.names = stan_col_names)| Parameter | Mean | S.E. | S.D. | Eff. Samples | R_hat | 2.5% | 97.5% | Significance | |
|---|---|---|---|---|---|---|---|---|---|
| beta0 | mu_Intercept | -0.75 | 0.00 | 0.02 | 579.79 | 1.01 | -0.78 | -0.71 | *** |
| beta[1] | mu_Early | -0.20 | 0.00 | 0.02 | 1094.51 | 1.00 | -0.25 | -0.16 | *** |
| beta[2] | mu_Pred | -0.05 | 0.00 | 0.03 | 662.64 | 1.01 | -0.11 | 0.01 | - |
| beta[3] | mu_EarlyByPred | -0.04 | 0.00 | 0.01 | 1839.89 | 1.00 | -0.07 | -0.02 | *** |
| beta_s0 | sigma_Intercept | -2.33 | 0.00 | 0.10 | 1289.27 | 1.00 | -2.54 | -2.16 | *** |
| beta_s[1] | sigma_Early | -0.26 | 0.00 | 0.14 | 1288.15 | 1.00 | -0.53 | 0.00 | - |
| beta_s[2] | sigma_Pred | -0.18 | 0.00 | 0.14 | 1682.20 | 1.00 | -0.48 | 0.06 | - |
| beta_s[3] | sigma_EarlyByPred | 0.07 | 0.00 | 0.10 | 1521.60 | 1.00 | -0.10 | 0.28 | - |
| beta_t0 | tau_Intercept | 0.11 | 0.00 | 0.09 | 515.36 | 1.01 | -0.04 | 0.31 | - |
| beta_t[1] | tau_Early | -0.23 | 0.00 | 0.07 | 1652.20 | 1.00 | -0.38 | -0.08 | *** |
| beta_t[2] | tau_Pred | -0.33 | 0.01 | 0.18 | 389.64 | 1.01 | -0.68 | 0.01 | - |
| beta_t[3] | tau_EarlyByPred | 0.18 | 0.00 | 0.05 | 1989.57 | 1.00 | 0.07 | 0.28 | *** |