appendix_b.Rmd

\label{appendix:b}

## Complete output of the Bayesian models  {-}
The complete results of the exploratory (Experiment 1) and confirmatory models (Experiment 2-3) are presented in Tables B1--B21.
<!-- \@ref(tab:exploratory-NAPbu)-\@ref(tab:confirmatory-NAPtd).  -->

Notice that the parameters are not entirely comparable accross models: (i) The intercept, $\alpha$, represents the mean log-RT of the first category for the ordinal models, but it is the grand mean for the  continuous model, $NAP_{bu}$. (ii) The size of the effect of well-formedness, $\beta$, represents the distance between two adjacent categories had they been equidistant for the ordinal models but it is the increase in log-scale for one unit in the well-formedness scale for the continuous model, $NAP_{bu}$. (iii) The parameter vector, $\zeta$, (present only in ordinal models) represents the normalized distances between consecutive predictor categories, so that the distance between the first and last category is $1$. (iv) For all models, the variance components are comparable: $\sigma$ represents the scale of the log-normal likelihood (or standard deviation of the distribution on the log scale), $\sigma_\alpha$ and $\sigma_\beta$ represent the by-participant adjustment to the intercept and slope respectively, and $\rho_{\alpha,\beta}$ represents the correlation between by-participant intercept and slope. 

For each parameter, `Bulk ESS`
and `Tail ESS` are effective sample size measures, and `Rhat` is the potential
scale reduction factor on split chains (at convergence, `Rhat`= 1, and `ESS` $> 10\%$ of post-warmup samples $=1200$).

\clearpage

```{r}

model_names <- c("NAP_bu" = "NAP_{bu}", "null"="Null", "SSP"="SSP_{exp}", "SSP_obs" = "SSP_{col}", "MSD" = "MSD_{exp}", "MSD_obs" = "MSD_{col}", "NAP_td" = "NAP_{td}")

model_tab <- function(models, type) {
   imap_chr(models, ~ {
  label = paste0("\\label{tab:",type,"-",str_remove(.y,"_"),"}")
  .x %>% {rbind(summary(.)$fixed,
                 summary(.)$mo,
                summary(.)$spec_pars,
             summary(.)$random$subj)} %>%
    round(2) %>%
  as_tibble(rownames= "Parameter") %>%
    mutate_at(vars(ends_with("ESS")), as.integer) %>%
  select(-Est.Error) %>%
    rename_all(~ str_replace_all(., "%","\\\\%")) %>% 
        rename_all(~ str_replace_all(., "_"," ")) %>% 
  mutate(Parameter = case_when(Parameter=="Intercept" ~ "$\\alpha$",
                          str_detect(Parameter,".*?\\[.\\]") ~ 
str_replace_all(Parameter,c(".*?\\[" = "$\\\\zeta_{","\\]" = "}$" )),
Parameter=="sigma" ~ "$\\sigma$",
Parameter =="sd(Intercept)" ~"$\\sigma_{\\alpha}$",
str_detect(Parameter,"sd.*?") ~"$\\sigma_{\\beta}$",
str_detect(Parameter, "cor.*?") ~ "$\\rho_{\\alpha,\\beta}$",
                          TRUE ~ "$\\beta$")) %>%
#   mutate(Parameter = case_when(Parameter=="Intercept" ~ "$\\hat\\alpha$",
#                           str_detect(Parameter,".*?\\[.\\]") ~ 
# str_replace_all(Parameter,c(".*?\\[" = "$\\\\hat\\\\zeta_{","\\]" = "}$" )),
# Parameter=="sigma" ~ "$\\hat\\sigma$",
# Parameter =="sd(Intercept)" ~"$\\hat\\sigma_{\\alpha}$",
# str_detect(Parameter,"sd.*?") ~"$\\hat\\sigma_{\\beta}$",
# str_detect(Parameter, "cor.*?") ~ "$\\hat\\rho_{\\alpha,\\beta}$",
#                           TRUE ~ "$\\hat\\beta$")) %>%
  papaja::apa_table(escape= FALSE,
                     caption = paste0("Results from ",type," model examining the results of the $",model_names[.y],"$ model. ",label= label, note= "See text for the interpretation of the parameters and column names."))}
) %>% 
  paste(collapse = "\n")
}

knitr::raw_latex(model_tab(m_pilot$models,"the exploratory"))
knitr::raw_latex(model_tab(m_german$models,"Experiment 2"))
knitr::raw_latex(model_tab(m_hebrew$models,"Experiment 3"))
```