By default, the `RoBMA()`

function specifies models as a
combination of all supplied prior distributions (across null and
alternative specification), with their prior model weights being equal
to the product of prior distributions’ weights. This results in the 36
meta-analytic models using the default settings (Bartoš, Maier, et al., 2021)\(^1\). In another vignette, we illustrated that
RoBMA can be also utilized for reproducing Bayesian Model-Averaged
Meta-Analysis (BMA) (Bartoš, Gronau, et al.,
2021; Gronau et al., 2017, 2021). However, the package was built
as a framework for estimating highly customized meta-analytic model
ensembles. Here, we are going to illustrate how to do exactly that (see
Bartoš et al. (in press) for a tutorial
paper on customizing the model ensemble with JASP).

Please keep in mind that all models should be justified by theory. Furthermore, the models should be tested to make sure that the ensemble can perform as intended a priori to drawing inference from it. The following sections are only for illustrating the functionality of the package. We provide a completely discussion with the relevant sources in the Example section of Bartoš, Maier, et al. (2021).

To illustrate the custom model building procedure, we use data from the infamous Bem (2011) “Feeling the future” pre-cognition study. We use coding of the results as summarized by Bem in one of his later replies (Bem et al., 2011).

```
library(RoBMA)
#> Loading required namespace: runjags
#> Loading required namespace: mvtnorm
#> module RoBMA loaded
data("Bem2011", package = "RoBMA")
Bem2011#> d se study
#> 1 0.25 0.10155048 Detection of Erotic Stimuli
#> 2 0.20 0.08246211 Avoidance of Negative Stimuli
#> 3 0.26 0.10323629 Retroactive Priming I
#> 4 0.23 0.10182427 Retroactive Priming II
#> 5 0.22 0.10120277 Retroactive Habituation I - Negative trials
#> 6 0.15 0.08210765 Retroactive Habituation II - Negative trials
#> 7 0.09 0.07085372 Retroactive Induction of Boredom
#> 8 0.19 0.10089846 Facilitation of Recall I
#> 9 0.42 0.14752627 Facilitation of Recall II
```

We consider the following scenarios as plausible explanations for the data, and decide to include only those models into the meta-analytic ensemble:

- there is absolutely no precognition effect - a fixed effects model assuming the effect size to be zero (\(H_{0}^f\)),
- the experiments measured the same underlying precognition effect - a fixed effects model (\(H_{1}^f\)),
- each of the experiments measured a slightly different precognition effect - a random effects model (\(H_{1}^r\)),
- there is absolutely no precognition effect and the results can be
explained away with publication bias, modeled with one of the following
publication bias adjustments: - 4.1) one-sided selection operating on
significant
*p*-values (\(H_{1,\text{pb1}}^f\)), - 4.2) one-sided selection operating on significant and marginally significant*p*-values (\(H_{1,\text{pb2}}^f\)), - 4.3) PET correction for publication bias which adjusts for the relationship between effect sizes and standard errors (\(H_{1,\text{pb3}}^f\)), - 4.4) PEESE correction for publication bias which adjusts for the relationship between effect sizes and standard errors squared (\(H_{1,\text{pb4}}^f\)).

If we were to fit the ensemble using the `RoBMA()`

function and specifying all of the priors, we would have ended with 2
(effect or no effect) * 2 (heterogeneity or no heterogeneity) * 5 (no
publication bias or 4 ways of adjusting for publication bias) = 20
models. That is 13 models more than requested. Furthermore, we could not
specify different parameters for the prior distributions for each model,
which the following process allows (but we do not utilize it).

We start with fitting only the first model using the
`RoBMA()`

function and we will continuously update the fitted
object to include all of the models.

We initiate the model ensemble by specifying only the first model
with the `RoBMA()`

function. We explicitly specify prior
distributions for all components and set the prior distributions to
correspond to the null hypotheses and set seed to ensure reproducibility
of the results.

```
<- RoBMA(d = Bem2011$d, se = Bem2011$se, study_names = Bem2011$study,
fit priors_effect = NULL, priors_heterogeneity = NULL, priors_bias = NULL,
priors_effect_null = prior("spike", parameters = list(location = 0)),
priors_heterogeneity_null = prior("spike", parameters = list(location = 0)),
priors_bias_null = prior_none(),
seed = 1)
```

We verify that the ensemble contains only the single specified model
with the `summary()`

function by setting
`type = "models"`

.

```
summary(fit, type = "models")
#> Call:
#> RoBMA(d = Bem2011$d, se = Bem2011$se, study_names = Bem2011$study,
#> priors_effect = NULL, priors_heterogeneity = NULL, priors_bias = NULL,
#> priors_effect_null = prior("spike", parameters = list(location = 0)),
#> priors_heterogeneity_null = prior("spike", parameters = list(location = 0)),
#> priors_bias_null = prior_none(), seed = 1)
#>
#> Robust Bayesian meta-analysis
#> Models overview:
#> Model Prior Effect Prior Heterogeneity Prior Bias Prior prob. log(marglik)
#> 1 Spike(0) Spike(0) 1.000 -3.28
#> Post. prob. Inclusion BF
#> 1.000 Inf
```

Before we add the second model to the ensemble, we need to decide on
the prior distribution for the mean parameter. If pre-cognition were to
exist, the effect would be small since all casinos would be bankrupted
otherwise. The effect would also be positive, since any deviation from
randomness could be characterized as an effect. Therefore, we decide to
use a normal distribution with mean = 0.15, standard deviation 0.10, and
truncated to the positive range. This sets the prior density around
small effect sizes. To get a better grasp of the prior distribution, we
visualize it using the `plot())`

function (the figure can be
also created using the ggplot2 package by adding
`plot_type == "ggplot"`

argument).

`plot(prior("normal", parameters = list(mean = .15, sd = .10), truncation = list(lower = 0)))`

We add the second model to the ensemble using the
`update.RoBMA()`

function. The function can also be used to
many other purposes - updating settings, prior model weights, and
refitting failed models. Here, we supply the fitted ensemble object and
add an argument specifying the prior distributions of each components
for the additional model. Since we want to add Model 2 - we set the
prior for the \(\mu\) parameter to be
treated as a prior belonging to the alternative hypothesis of the effect
size component and the remaining priors treated as belonging to the
alternative hypotheses. If we wanted, we could also specify
`prior_weights`

argument, to change the prior probability of
the fitted model but we do not utilize this option here and keep the
default value, which sets the prior weights for the new model to
`1`

. (Note that the arguments for specifying prior
distributions in `update.RoBMA()`

function are
`prior_X`

- in singular, in comparison to
`RoBMA()`

function that uses `priors_X`

in
plural.)

```
<- update(fit,
fit prior_effect = prior("normal", parameters = list(mean = .15, sd = .10), truncation = list(lower = 0)),
prior_heterogeneity_null = prior("spike", parameters = list(location = 0)),
prior_bias_null = prior_none())
```

We can again inspect the updated ensemble to verify that it contains both models. We see that Model 2 notably outperformed the first model and attained all of the posterior model probability.

```
summary(fit, type = "models")
#> Call:
#> RoBMA(d = Bem2011$d, se = Bem2011$se, study_names = Bem2011$study,
#> priors_effect = NULL, priors_heterogeneity = NULL, priors_bias = NULL,
#> priors_effect_null = prior("spike", parameters = list(location = 0)),
#> priors_heterogeneity_null = prior("spike", parameters = list(location = 0)),
#> priors_bias_null = prior_none(), seed = 1)
#>
#> Robust Bayesian meta-analysis
#> Models overview:
#> Model Prior Effect Prior Heterogeneity Prior Bias Prior prob.
#> 1 Spike(0) Spike(0) 0.500
#> 2 Normal(0.15, 0.1)[0, Inf] Spike(0) 0.500
#> log(marglik) Post. prob. Inclusion BF
#> -3.28 0.000 0.000
#> 14.91 1.000 79422247.251
```

Before we add the remaining models to the ensemble using the
`update()`

function, we need to decide on the remaining prior
distributions. Specifically, on the prior distribution for the
heterogeneity parameter \(\tau\), and
the publication bias adjustment parameters \(\omega\) (for the selection models’
weightfunctions) and PET and PEESE for the PET and PEESE adjustment.

For Model 3, we use the usual inverse-gamma(1, .15) prior distribution based on empirical heterogeneity estimates (Erp et al., 2017) for the heterogeneity parameter \(\tau\). For Models 4.1-4.4 we use the default settings for the publication bias adjustments as outlined the Appendix B of (Bartoš, Maier, et al., 2021).

Now, we just need to add the remaining models to the ensemble using
the `update()`

function as already illustrated.

```
### adding Model 3
<- update(fit,
fit prior_effect = prior("normal", parameters = list(mean = .15, sd = .10), truncation = list(lower = 0)),
prior_heterogeneity = prior("invgamma", parameters = list(shape = 1, scale = .15)),
prior_bias_null = prior_none())
### adding Model 4.1
<- update(fit,
fit prior_effect_null = prior("spike", parameters = list(location = 0)),
prior_heterogeneity_null = prior("spike", parameters = list(location = 0)),
prior_bias = prior_weightfunction("one.sided", parameters = list(alpha = c(1, 1), steps = c(0.05))))
### adding Model 4.2
<- update(fit,
fit prior_effect_null = prior("spike", parameters = list(location = 0)),
prior_heterogeneity_null = prior("spike", parameters = list(location = 0)),
prior_bias = prior_weightfunction("one.sided", parameters = list(alpha = c(1, 1, 1), steps = c(0.05, 0.10))))
### adding Model 4.3
<- update(fit,
fit prior_effect_null = prior("spike", parameters = list(location = 0)),
prior_heterogeneity_null = prior("spike", parameters = list(location = 0)),
prior_bias = prior_PET("Cauchy", parameters = list(0, 1), truncation = list(lower = 0)))
### adding Model 4.4
<- update(fit,
fit prior_effect_null = prior("spike", parameters = list(location = 0)),
prior_heterogeneity_null = prior("spike", parameters = list(location = 0)),
prior_bias = prior_PEESE("Cauchy", parameters = list(0, 5), truncation = list(lower = 0)))
```

We again verify that all of the requested models are included in the
ensemble using the `summary())`

function with
`type = "models"`

argument.

```
summary(fit, type = "models")
#> Call:
#> RoBMA(d = Bem2011$d, se = Bem2011$se, study_names = Bem2011$study,
#> priors_effect = NULL, priors_heterogeneity = NULL, priors_bias = NULL,
#> priors_effect_null = prior("spike", parameters = list(location = 0)),
#> priors_heterogeneity_null = prior("spike", parameters = list(location = 0)),
#> priors_bias_null = prior_none(), seed = 1)
#>
#> Robust Bayesian meta-analysis
#> Models overview:
#> Model Prior Effect Prior Heterogeneity
#> 1 Spike(0) Spike(0)
#> 2 Normal(0.15, 0.1)[0, Inf] Spike(0)
#> 3 Normal(0.15, 0.1)[0, Inf] InvGamma(1, 0.15)
#> 4 Spike(0) Spike(0)
#> 5 Spike(0) Spike(0)
#> 6 Spike(0) Spike(0)
#> 7 Spike(0) Spike(0)
#> Prior Bias Prior prob. log(marglik)
#> 0.143 -3.28
#> 0.143 14.91
#> 0.143 12.85
#> omega[one-sided: .05] ~ CumDirichlet(1, 1) 0.143 13.70
#> omega[one-sided: .1, .05] ~ CumDirichlet(1, 1, 1) 0.143 12.58
#> PET ~ Cauchy(0, 1)[0, Inf] 0.143 15.75
#> PEESE ~ Cauchy(0, 5)[0, Inf] 0.143 15.65
#> Post. prob. Inclusion BF
#> 0.000 0.000
#> 0.168 1.210
#> 0.021 0.132
#> 0.050 0.318
#> 0.016 0.100
#> 0.391 3.845
#> 0.353 3.278
```

Finally, we use the `summary()`

function to inspect the
model results. The results from our custom ensemble indicate weak
evidence for the absence of the pre-cognition effect, \(\text{BF}_{10} = 0.584\) -> \(\text{BF}_{01} = 1.71\), moderate evidence
for the absence of heterogeneity, \(\text{BF}_{\text{rf}} = 0.132\) -> \(\text{BF}_{\text{fr}} = 7.58\), and
moderate evidence for the presence of the publication bias, \(\text{BF}_{\text{pb}} = 3.21\).

```
summary(fit)
#> Call:
#> RoBMA(d = Bem2011$d, se = Bem2011$se, study_names = Bem2011$study,
#> priors_effect = NULL, priors_heterogeneity = NULL, priors_bias = NULL,
#> priors_effect_null = prior("spike", parameters = list(location = 0)),
#> priors_heterogeneity_null = prior("spike", parameters = list(location = 0)),
#> priors_bias_null = prior_none(), seed = 1)
#>
#> Robust Bayesian meta-analysis
#> Components summary:
#> Models Prior prob. Post. prob. Inclusion BF
#> Effect 2/7 0.286 0.189 0.584
#> Heterogeneity 1/7 0.143 0.021 0.132
#> Bias 4/7 0.571 0.811 3.212
#>
#> Model-averaged estimates:
#> Mean Median 0.025 0.975
#> mu 0.036 0.000 0.000 0.226
#> tau 0.002 0.000 0.000 0.000
#> omega[0,0.05] 1.000 1.000 1.000 1.000
#> omega[0.05,0.1] 0.938 1.000 0.014 1.000
#> omega[0.1,1] 0.935 1.000 0.012 1.000
#> PET 0.820 0.000 0.000 2.601
#> PEESE 7.284 0.000 0.000 25.508
#> The estimates are summarized on the Cohen's d scale (priors were specified on the Cohen's d scale).
```

The finalized ensemble can be treated as any other `RoBMA`

ensemble using the `summary()`

, `plot()`

,
`plot_models()`

, `forest()`

, and
`diagnostics()`

functions. For example, we can use the
`plot.RoBMA()`

with the
`parameter = "mu", prior = TRUE`

arguments to plot the prior
(grey) and posterior distribution (black) for the effect size. The
function visualizes the model-averaged estimates across all models by
default. The arrows stand for the probability of a spike, here, at the
value 0. The secondary y-axis (right) shows the probability of the value
0, increasing from 0.71, to 0.81.

`plot(fit, parameter = "mu", prior = TRUE)`

We can also inspect the posterior distributions of the publication
bias adjustments. To visualize the model-averaged weightfunction, we
change set `parameter = weightfunction`

argument, with the
prior distribution in light gray and the posterior distribution in the
dark gray,

`plot(fit, parameter = "weightfunction", prior = TRUE)`

and the posterior estimate of the regression relationship between the
standard errors and effect sizes by setting
`parameter = "PET-PEESE"`

.

`plot(fit, parameter = "PET-PEESE", prior = TRUE)`

\(^1\) - The default setting used to produce 12 models in RoBMA versions < 2, which corresponded to earlier an article by Maier et al. (in press) in which we applied Bayesian model-averaging only across selection models.

Bartoš, F., Gronau, Q. F., Timmers, B., Otte, W. M., Ly, A., &
Wagenmakers, E.-J. (2021). Bayesian model-averaged meta-analysis in
medicine. *Statistics in Medicine*. https://doi.org/10.1002/sim.9170

Bartoš, F., Maier, M., Quintana, D. S., & Wagenmakers, E.-J. (in
press). Adjusting for publication bias in JASP &
R – selection models, PET-PEESE, and robust
Bayesian meta-analysis. *Advances in Methods and
Practices in Psychological Science*. https://doi.org/10.31234/osf.io/75bqn

Bartoš, F., Maier, M., Wagenmakers, E.-J., Doucouliagos, H., &
Stanley, T. D. (2021). *Robust Bayesian meta-analysis:
Model-averaging across complementary publication bias
adjustment methods*. PsyArXiv. https://doi.org/10.31234/osf.io/kvsp7

Bem, D. J. (2011). Feeling the future: Experimental evidence for
anomalous retroactive influences on cognition and affect. *Journal of
Personality and Social Psychology*, *100*(3), 407. https://doi.org/10.1037/a0021524

Bem, D. J., Utts, J., & Johnson, W. O. (2011). Must psychologists
change the way they analyze their data? *Journal of Personality and
Social Psychology*, *101*(4), 716. https://doi.org/10.1037/a0024777

Erp, S. van, Verhagen, J., Grasman, R. P., & Wagenmakers, E.-J.
(2017). Estimates of between-study heterogeneity for 705 meta-analyses
reported in Psychological Bulletin from
1990–2013. *Journal of Open Psychology Data*, *5*(1). https://doi.org/10.5334/jopd.33

Gronau, Q. F., Heck, D. W., Berkhout, S. W., Haaf, J. M., &
Wagenmakers, E.-J. (2021). A primer on Bayesian
model-averaged meta-analysis. *Advances in Methods and Practices in
Psychological Science*, *4*(3), 25152459211031256. https://doi.org/10.1177/25152459211031256

Gronau, Q. F., Van Erp, S., Heck, D. W., Cesario, J., Jonas, K. J.,
& Wagenmakers, E.-J. (2017). A Bayesian model-averaged
meta-analysis of the power pose effect with informed and default priors:
The case of felt power. *Comprehensive Results in Social
Psychology*, *2*(1), 123–138. https://doi.org/10.1080/23743603.2017.1326760

Maier, M., Bartoš, F., & Wagenmakers, E.-J. (in press). Robust
Bayesian meta-analysis: Addressing publication bias with
model-averaging. *Psychological Methods*. https://doi.org/10.31234/osf.io/u4cns