First, I fitted the data in R with a GLM for each of the 21 conditions (no pooling),

lm(y ~ x, …)

and got the following result for the effect of x:

```
Estimate Est.Error 2.5%ile 5%ile 95%ile 97.5%ile
```

condtion1 0.0008 0.0104 -0.0197 -0.0164 0.0180 0.0213

condtion2 0.0136 0.0130 -0.0122 -0.0080 0.0352 0.0394

condtion3 0.0004 0.0133 -0.0259 -0.0216 0.0224 0.0266

condtion4 -0.0038 0.0096 -0.0229 -0.0198 0.0122 0.0153

condtion5 -0.0131 0.0086 -0.0302 -0.0274 0.0013 0.0040

condtion6 -0.0085 0.0076 -0.0236 -0.0212 0.0042 0.0066

condtion7 0.0227 0.0114 0.0001 0.0038 0.0415 0.0452

condtion8 0.0050 0.0131 -0.0210 -0.0168 0.0268 0.0310

condtion9 0.0001 0.0116 -0.0228 -0.0191 0.0193 0.0231

condtion10 0.0094 0.0125 -0.0152 -0.0112 0.0301 0.0341

condtion11 -0.0181 0.0077 -0.0333 -0.0309 -0.0053 -0.0029

condtion12 0.0134 0.0139 -0.0140 -0.0096 0.0364 0.0409

condtion13 -0.0169 0.0081 -0.0329 -0.0303 -0.0035 -0.0009

condtion14 -0.0096 0.0111 -0.0315 -0.0280 0.0087 0.0122

condtion15 -0.0009 0.0071 -0.0151 -0.0128 0.0109 0.0132

condtion16 0.0074 0.0085 -0.0095 -0.0067 0.0216 0.0243

condtion17 0.0326 0.0148 0.0033 0.0080 0.0571 0.0618

**condtion18 0.0086 0.0141 -0.0193 -0.0148 0.0320 0.0365**

condtion19 -0.0083 0.0074 -0.0229 -0.0205 0.0039 0.0063

condtion20 0.0057 0.0094 -0.0129 -0.0099 0.0212 0.0243

condtion21 0.0180 0.0119 -0.0056 -0.0018 0.0377 0.0416

Then, I tried the same data with rstanarm (partial pooling),

stan_lmer(y~x+(1|subject)+(x|condition),…)

and had the following result for the effect of x:

```
Estimate Est.Error 2.5%ile 5%ile 95%ile 97.5%ile
```

condtion1 -0.0006 0.0075 -0.0155 -0.0127 0.0119 0.0141

condtion2 0.0072 0.0079 -0.0076 -0.0053 0.0210 0.0233

condtion3 0.0023 0.0077 -0.0134 -0.0107 0.0153 0.0169

condtion4 -0.0045 0.0080 -0.0195 -0.0173 0.0087 0.0116

condtion5 -0.0037 0.0080 -0.0203 -0.0176 0.0090 0.0114

condtion6 -0.0036 0.0079 -0.0194 -0.0169 0.0093 0.0122

condtion7 0.0105 0.0082 -0.0040 -0.0021 0.0244 0.0278

condtion8 0.0038 0.0077 -0.0111 -0.0088 0.0168 0.0190

condtion9 0.0013 0.0076 -0.0134 -0.0109 0.0136 0.0167

condtion10 0.0090 0.0081 -0.0064 -0.0041 0.0220 0.0250

condtion11 -0.0053 0.0081 -0.0218 -0.0188 0.0078 0.0102

condtion12 0.0069 0.0078 -0.0071 -0.0052 0.0199 0.0228

condtion13 -0.0040 0.0084 -0.0218 -0.0187 0.0092 0.0113

condtion14 -0.0029 0.0078 -0.0189 -0.0160 0.0102 0.0123

condtion15 0.0001 0.0078 -0.0155 -0.0134 0.0125 0.0150

condtion16 0.0032 0.0079 -0.0120 -0.0089 0.0163 0.0187

condtion17 0.0163 0.0091 -0.0004 0.0020 0.0325 0.0360

**condtion18 0.0180 0.0099 -0.0006 0.0020 0.0341 0.0373**

condtion19 -0.0018 0.0078 -0.0180 -0.0154 0.0109 0.0126

condtion20 0.0014 0.0080 -0.0142 -0.0113 0.0152 0.0181

condtion21 0.0068 0.0082 -0.0080 -0.0056 0.0209 0.0241

And here are the high-level effects:

Group-Level Effects:

~condition (Number of levels: 21)

Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat

sd(Intercept) 0.16 0.03 0.12 0.22 286 1.01

sd(x) 0.01 0.00 0.00 0.02 1227 1.00

cor(Intercept,x) 0.65 0.24 0.07 0.98 2000 1.00

~subject (Number of levels: 124)

Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat

sd(Intercept) 0.08 0.01 0.07 0.09 585 1.01

Population-Level Effects:

Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat

Intercept 0.17 0.04 0.09 0.25 154 1.01

x 0.00 0.01 -0.01 0.02 439 1.00

Most of effect estimates across the 21 conditions with the Bayesian model make sense to me: they were pooled toward the center. I plotted the posterior predictive check for both models, and the Bayesian model did a better fitting than GLM. However, I’m baffled by one particular case: **condition18** (in boldface in both results above) seems to have been pooled away from the center. Why is this? And how should I proceed to diagnose the situation?