Table 1. Heat intensity effects on pain across all participants (n=40)*.
| Predictors | Estimates | Confidence intervals | P-Value / probability of direction | Bayesian estimates | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| LMER† | NLME‡ | BRMS§ | LMER† | NLME‡ | BRMS§ | LMER† | NLME‡ | BRMS§ | % in ROPE | Rhat | ESS | |
| (Intercept) | 3.64 | 3.646 | 3.641 | 3.40–3.88 | [3.408, 3.885] | [ 3.438, 3.849] | <0.001 | 0.000 | 100.00% | 0 | 1.005 | 1597.656 |
| Group | 0.11 | 0.101 | 0.112 | –0.13–0.35 | [–0.144, 0.347] | [–0.092, 0.318] | 0.356 | 0.409 | 80.83% | 82.783 | 1.001 | 1632.917 |
| Heat Level | 2.09 | 2.078 | 2.083 | 1.90–2.27 | [1.897, 2.258] | [ 1.929, 2.238] | <0.001 | 0.000 | 100.00% | 0 | 1 | 4205.817 |
| Cue | 0.32 | 0.293 | 0.323 | 0.17–0.47 | [0.148, 0.437] | [ 0.203, 0.447] | <0.001 | 0.000 | 100.00% | 12.692 | 1 | 9174.052 |
| Phase | 0.1 | 0.107 | 0.099 | 0.01–0.19 | [0.022, 0.193] | [ 0.026, 0.170] | 0.026 | 0.014 | 98.40% | 99.842 | 1.001 | 7792.54 |
| Group x Heat Level | –0.04 | –0.064 | –0.042 | –0.23–0.14 | [–0.245, 0.116] | [–0.195, 0.113] | 0.636 | 0.484 | 67.15% | 97.225 | 1.001 | 4410.973 |
| Group x Cue | 0.1 | 0.115 | 0.098 | –0.05–0.25 | [–0.029, 0.259] | [–0.025, 0.220] | 0.197 | 0.119 | 90.05% | 96.508 | 1 | 8745.241 |
| Heat Level x Cue | 0.16 | 0.128 | 0.158 | –0.03–0.35 | [–0.055, 0.311] | [ 0.002, 0.307] | 0.097 | 0.169 | 95.27% | 79.3 | 1 | 17260.576 |
| Group x Phase | 0 | –0.001 | –4.95E-04 | –0.09–0.09 | [–0.086, 0.085] | [–0.074, 0.071] | 0.999 | 0.985 | 50.43% | 100 | 1 | 8609.937 |
| Heat Level * Phase | –0.06 | –0.055 | –0.059 | –0.15–0.04 | [–0.139, 0.03] | [–0.133, 0.017] | 0.23 | 0.205 | 89.04% | 100 | 1 | 15179.351 |
| Cue * Phase | 0.58 | 0.615 | 0.575 | 0.39–0.77 | [0.424, 0.806] | [ 0.410, 0.731] | <0.001 | 0.000 | 100.00% | 0.025 | 1 | 9874.616 |
| (Group * Heat Level) * Cue | –0.04 | –0.033 | –0.036 | –0.22–0.15 | [–0.216, 0.15] | [–0.185, 0.112] | 0.709 | 0.724 | 64.35% | 98.242 | 1 | 17007.967 |
| (Group * Heat Level) * Phase | –0.03 | –0.037 | –0.029 | –0.12–0.07 | [–0.121, 0.047] | [–0.105, 0.049] | 0.546 | 0.390 | 73.22% | 100 | 1 | 14459.934 |
| (Group *Cue) * Phase | 0.23 | 0.249 | 0.231 | 0.04–0.42 | [0.058, 0.44] | [ 0.073, 0.392] | 0.017 | 0.011 | 98.79% | 52.8 | 1 | 10768.063 |
| (Heat Level *Cue) *Phase | 1.79 | 1.774 | 1.782 | 1.60–1.97 | [1.59, 1.958] | [ 1.634, 1.939] | <0.001 | 0.000 | 100.00% | 0 | 1 | 21294.817 |
| (Group *Heat Level *Cue) *Phase | –0.2 | –0.167 | –0.203 | –0.39 to –0.01 | [–0.351, 0.018] | [-0.359,–0.054] | 0.038 | 0.076 | 98.48% | 63.242 | 1 | 23048.272 |
This table presents results of linear mixed models predicting subjective pain as a function of Heat Level (High vs Medium vs Low), Group (Instructed vs Uninstructed), Cue (Original High vs Original Low), and Phase (Original vs Reversed). All predictors were dummy-coded and mean centered to facilitate interpretation of coefficients and interactions. Model specification was based on Bayesian model comparison. We compared three types of linear mixed models: frequentist analysis using the “lmer” function of lme4 (Bates et al., 2015), frequentist analysis using the “lme” function of nlme (Pinheiro et al., 2021) accounting for autoregression, and Bayesian estimation using mildly informative conservative priors (i.e. centered on 0 for all effects). Effects that are both statistically and practically significant are bolded, whereas effects that are statistically significant but not practically significant (i.e. >2.5% in the region of partial equivalence (ROPE)) are italicized.
Estimates based on a linear mixed effects model implemented in the “lmer” function of lme4 (Bates et al., 2015) using the following code: lmer(Pain~Group*Templevels*Cue*Phase+(1+Templevels + Cue*Phase||Subject)). Confidence intervals were obtained using the “tab_model” function from sjPlot (Lüdecke, 2021) and corresponds to the 95% confidence interval.
Estimates based on a linear mixed effects model implemented in the “lme” function of nlme (Pinheiro et al., 2021) including autoregression using the following code: lme(Pain~Group*Templevels*Cue*Phase, random = ~1 + Templevels +Cue*Phase|Subject, correlation = corAR1(), na.action=na.exclude). Confidence intervals were obtained using the ‘intervals’ function from nlme (Pinheiro et al., 2021) and corresponds to the 95% confidence interval.
Estimates based on Bayesian model linear mixed models using the “brms” function (Bürkner, 2017) using the following code: brm Pain~Group*Templevels*Cue*Phase+(1+Templevels + Cue*Phase|Subject,prior = set_prior("normal(0,2.5)", class="b"), save_all_pars = TRUE, silent = TRUE, refresh = 0, iter = 4000, warmup = 1000). Posterior estimates, including the probable direction (which is roughly equivalent to 1- frequentist p-value), 89% confidence intervals, and the ROPE were obtained using the “describe_posterior” function from the package BayesTestR (Makowski et al., 2019a) and interpreted as in Makowski et al., 2019b. The Region of Partial Equivalence (ROPE) was defined as [–0.237, 0.237]. We report the median estimate for each parameter.