# Data simulation descriptions

## multi-level path analysis (equation representation)
### Level-1: multiple *t*arget paintings within a given *i*ndividual participant

$$VAL_{ti} = \overset{VAL}{\beta_{0i}} + \overset{VAL}{\beta_{1i}} DIS_{ti} + \overset{VAL}{\epsilon_{ti}} $$

$$INT_{ti} = \overset{INT}{\beta_{0i}} + \overset{INT}{\beta_{1i}} DIS_{ti}+ \overset{INT}{\beta_{2i}} VAL_{ti} + \overset{INT}{\epsilon_{ti}} $$

### Level-2: different *i*ndividual particpants
$$\overset{VAL}{\beta_{0i}} = \overset{VAL}{\gamma_{00}} + \overset{VAL}{\upsilon_{0i}}$$

$$\overset{VAL}{\beta_{1i}} = \overset{VAL}{\gamma_{10}} + \overset{VAL}{\upsilon_{1i}}$$

$$\overset{INT}{\beta_{0i}} = \overset{INT}{\gamma_{00}} + \overset{INT}{\upsilon_{0i}}$$

$$\overset{INT}{\beta_{1i}} = \overset{INT}{\gamma_{10}} + \overset{INT}{\upsilon_{1i}}$$

$$\overset{INT}{\beta_{2i}} = \overset{INT}{\gamma_{20}} + \overset{INT}{\upsilon_{2i}}$$

## data generating process at the individual level under flawed design

Each **i**ndividual participant is represented by the subscript $i$ and the **t**arget stimuli he/she judged are indexed by the subscript $t$.

1. $\text{Disturbing}_t$: inherent disturbingness of Painting $t$
+ = the average disturbing rating of Painting $t$ calculated from the dataset generated by any of the survey
3. $\text{Disturbing}_{ti}$ (participant $i$'s personal disturbing experience inudced by Painting $t$) $=r_{i}^{DD}*\text{Disturbing}_t+\epsilon^{Dis}_{ti}$
+ $\epsilon^{Dis}_{ti} \sim N(0, (1-r^{DD}_i\times{r^{DD}_i})\times{}\sigma^2_{\text{Disturbing}_t})$
4. $\text{Aroused}_{ti} = r^{DA}_i*\text{Disturbing}_{ti} + \epsilon^{Aro}_{ti}$
5. $\text{Pleasant}_{ti} = r^{DP}_i*\text{Disturbing}_{ti} + \epsilon^{Ple}_{ti}$
+ assuming $|r^{da}_i|= |r^{dp}_i|$

4. $\text{Attentive}_{ti} = r^{AA}_i*\text{Aroused}_{ti} + \epsilon^{Att}_{ti}$

5. $\text{Interest}_{ti}$ = $a_i$ + $b_i*\text{Pleasant}_{ti}$ + $c_i*\text{Attentive}_{ti}$ + $\epsilon^{int}_{ti}$

## Goal:
We would need 1000 simulated studies. Each simulated study comprises the data from $i=1~to~120$ simuated participants, each of which "reacts" to $t=1~to~13$ target paintings. Thus, the dataset for a simulated study would be a tidy dataframe with 120 $\times$ 13 rows and three columns (i.e., $\text{Disturbing}_{ti}$, $\text{Pleasant}_{ti}$ and $\text{Interest}_{ti}$).

Then we run the single predictor and dual-predictor regressions on these stimualted dataset to see if we can recover $b_i$ in equation (6)