Fitting Bayesian Multilevel Single Case models using bmscstan

The package bmscstan provides useful functions to fit Bayesian Multilevel Single Case models (BMSC) using as backend Stan (Carpenter et al. 2017).

This approach is based on the seminal approach of the Crawford’s tests (Crawford and Howell 1998; Crawford and Garthwaite 2005; Crawford et al. 2010), using a small control sample of individuals, to see whether the performance of the single case deviates from them. Unfortunately, Crawford’s tests are limited to a number of specific experimental designs that do not allow researchers to use complex experimental designs.

The BMSC approach is born mainly to deal with this problem: its purpose is, in fact, to allow the fitting of models with the same flexibility of a Multilevel Model, with single case and controls data.

The core function of the bmscstan package is BMSC, whose theoretical assumptions, and its validation are reported in (Scandola and Romano 2021).

The syntax used by the BMSC function is extremely similar to the syntax used in the lme4 package. However, the specification of random effects is limited, but it can cover the greatest part of cases (for further details, please see ?bmscstan::randomeffects).

Example on real data

In order to show an example on the use of the bmscstan package, the datasets in this package will be used.

In these datasets we have data coming from a Body Sidedness Effect paradigm (Ottoboni et al. 2005; Tessari et al. 2012), that is a Simon-like paradigm useful to measure body representation.

In this experimental paradigm, participants have to answer to a circle showed in the centre of the computer screen, superimposed to an irrelevant image of a left or right hand, or to a left or right foot.

The circle can be of two colors (e.g. red or blue), and participants have to press one button with the left when the circle is of a specific colour, and with the right hand when the circle is of the another colour.

When the irrelevant background image (foot or hand) is incongruent with the hand used to answer, the reaction times and frequency of errors are higher.

The two irrelevant backgrounds are administered in different experimental blocks.

This is considered an effect of the body representation.

In the package there are two datasets, one composed by 16 healthy control participants, and the other one by an individual affected by right unilateral brachial plexus lesion (however, s/he could independently press the keyboard buttons).

Explore the data

The datasets are called data.pt for the single case, and data.ctrl for the control group, and they can be loaded using data(BSE).

In these datasets there are the Reaction Times RT, a Body.District factor with levels FOOT and HAND, a Congruency factor (levels: Congruent, Incongruent), and a Side factor (levels: Left, Right). In the data.ctrl dataset there also is an ID factor, representing the different 16 control participants.

library(ggplot2)
library(bmscstan)

data(BSE)

str(data.pt)

str(data.ctrl)

ggplot(data.pt, aes(y = RT, x = Body.District:Side , fill = Congruency))+
  geom_boxplot()

ggplot(data.ctrl, aes(y = RT, x = Body.District:Side , fill = Congruency))+
  geom_boxplot()+
  facet_wrap( ~ ID , ncol = 4)

These data seem to have some outliers. Let see if they are normally distributed.

qqnorm(data.ctrl$RT, main = "Controls")
qqline(data.ctrl$RT)

qqnorm(data.pt$RT, main = "Single Case")
qqline(data.pt$RT)

They are not normally distributed. Outliers will be removed by using the boxplot.stats function.

out <- boxplot.stats( data.ctrl$RT )$out
data.ctrl <- droplevels( data.ctrl[ !data.ctrl$RT %in% out , ] )

out <- boxplot.stats( data.pt$RT )$out
data.pt <- droplevels( data.pt[ !data.pt$RT %in% out , ] )

qqnorm(data.ctrl$RT, main = "Controls")
qqline(data.ctrl$RT)

qqnorm(data.pt$RT, main = "Single Case")
qqline(data.pt$RT)

They are not perfect, but definitively better.

Deciding the contrasts and the random effects part

First of all, there is the necessity to think to our hypotheses, and setting the contrast matrices consequently.

In all cases, our factors have only two levels. Therefore, we set the factors with a Treatment Contrasts matrix, with baseline level for Side the Left level, for Congruency the Congruent level, and for Body.District the FOOT level.

In this way, each coefficient will represent the difference between the two levels.

contrasts( data.ctrl$Side )          <- contr.treatment( n = 2 )
contrasts( data.ctrl$Congruency )    <- contr.treatment( n = 2 )
contrasts( data.ctrl$Body.District ) <- contr.treatment( n = 2 )

contrasts( data.pt$Side )            <- contr.treatment( n = 2 )
contrasts( data.pt$Congruency )      <- contr.treatment( n = 2 )
contrasts( data.pt$Body.District )   <- contr.treatment( n = 2 )

The use of the BMSC function, for those who are used to lme4 or brms syntax should be straightforward.

In this case, we want to fit the following model:

RT ~ Body.District * Congruency * Side + (Congruency * Side | ID : Body.District)

Unfortunately, BMSC does not directly allow the syntax ID : Body.District in the specification of the random effects.

Therefore, it is necessary to create a new variable for ID : Body.District

data.ctrl$BD_ID <- interaction( data.ctrl$Body.District , data.ctrl$ID )

and the model would be:

RT ~ Body.District * Congruency * Side + (Congruency * Side | BD_ID)

For further details concerning the random effects available in bmscstan, please type ?bmscstan::randomeffect.

Fitting the BMSC model

At this point, fitting the model is easy, and it can be done with the use of a single function.

mdl <- BMSC(formula = RT ~ Body.District * Congruency * Side +
             (Congruency * Side | BD_ID),
             data_ctrl = data.ctrl,
             data_sc = data.pt,
             chains = 2,
             cores = 1,
             seed = 2020)

After fitting the model, we should check its quality by means of Posterior Predictive P-Values (Gelman 2013) with the bmscstan::pp_check function.

Thanks to this graphical function, we will see if the observed data and the data sampled from the posterior distributions of our BMSC model are similar.

If we observe strong deviations, it means that your BMSC model is not adequately describing your data. In this case, you might want to change the priors distribution (see the help page), change the random effects structure, or transform your dependent variable (using the logarithm or other math functions).

pp_check( mdl )

In both the controls and the single case data, the Posterior Predictive P-Values check seems to adequately resemble the observed data.

A further control on our model is given by checking the Effective Sample Size (ESS) for each coefficient and the R̂ diagnostic index (Gelman and Rubin 1992).

The ESS is the “effective number of simulation draws” for any coefficient, namely the approximate number of independent draws, taking into account that the various simulations in a Monte Carlo Markov Chain (MCMC) are not independent each other. For further details, see an introductory book in Bayesian Statistics. A good ESS estimates should be ESS > 100 or ESS > 10% of the total draws (remembering that you should remove the burn-in simulations from the total iterations counting).

The R̂ is an index of the convergence of the MCMCs. In BMSC the default is 4. Usually, MCMCs are considered convergent when R̂ < 1.1 (Stan default).

In order to check these values, the summary.BMSC function is needed (see next section).

The summary.BMSC output

The output of the brmscstan::summary.BMSC function is divided in four main parts:

  1. In the first part, the model and the selected priors are recalled.
  2. In the second part, the coefficients of the fixed effects for the control group are shown.
  3. In the third part, the coefficients of the fixed effects for the single case are shown.
  4. In the fourth and last part, the fixed effects coefficients for the difference between the single case and the control group are shown.
print( sum_mdl <- summary( mdl ) , digits = 3 )

In the second and fourth part of the output, we can observe a descriptive summary reporting the mean, the standard error, the standard deviation, the 2.5%, 25, 50%, 75% and 97.5% of the posterior distributions of each coefficient. If we want the 95% Credible Interval, we can consider only the 2.5% and 97.5% extremes. Then, two diagnostic indexes are reported: the n_eff parameter, that is the ESS, and the Rhat (R̂). Finally, the Savage-Dickey Bayes Factor is reported (BF10).

In the third part the diagnostic indexes are not reported because these coefficients are computed as marginal probabilities from the probabilities summarized in the second and fourth part.

Understanding the summary.BMSC output

Checking the diagnostic indexes

The first step should be controlling the diagnostic indexes.

In this model, all n_eff are greater than the 10% of the total iterations (default iterations: 4000, default warmup iterations: 2000, default chains: 4 = 800). Also, all R̂s < 1.1. Finally, we already saw that the Posterior Predictive P-values are showing that the model is representative of the data.

The Control Group results

Then, observing what the fixed effects of the Control group are showing is important before of seeing the differences with the single case.

In this analysis, there are 5 fixed effects which BF10 is greater than 3 (Raftery 1995).

tmp <- sum_mdl[[1]][sum_mdl[[1]]$BF10 > 3,c("BF10","mean","2.5%","97.5%")]

colnames(tmp) <- c("$BF_{10}$", "$\\mu$", "low $95\\%~CI$", "up $95\\%~CI$")

knitr::kable(
  tmp,
  digits = 3
)

We can have a general overview of the coefficients of the model with the plot.BMSC function.

plot( mdl , who = "control" )

The interaction between Body District and Congruency needs a further analysis to better understand the phenomenon. It comes useful the function pairwise.BMSC.

pp <- pairwise.BMSC(mdl = mdl , contrast = "Body.District2:Congruency2" ,
                    who = "control")

print( pp , digits = 3 )

The output of this function is divided in two parts:

  • a first part, called “Marginal distributions”, where the marginal distributions of each level of the coefficients are reported with a BF10 against zero.
  • a second part, called “Table of contrasts”, with all possible pairwise comparisons and their BF10.

It is also possible to plot the results of this function with the use of plot.pairwise.BMSC.

plot( pp )

Finally, it is possible to plot marginal posterior distributions for each effects with BF10 > 3.

p1 <- pairwise.BMSC(mdl , contrast = "Body.District2" ,  who = "control" )

plot( p1 )[[1]] +
  ggtitle("Body District" , subtitle = "Marginal effects") 

plot( p1 )[[2]] +
  ggtitle("Body District" , subtitle = "Contrasts") 

p2 <- pairwise.BMSC(mdl , contrast = "Congruency2" ,  who = "control" )

plot( p2 )[[1]] +
  ggtitle("Congruency" , subtitle = "Marginal effects")

plot( p2 )[[2]] +
  ggtitle("Congruency" , subtitle = "Contrasts")

p3 <- pairwise.BMSC(mdl , contrast = "Side2" ,  who = "control" )

plot( p3 )[[1]] +
  ggtitle("Side" , subtitle = "Marginal effects")

plot( p3 )[[2]] +
  ggtitle("Side" , subtitle = "Contrasts")

The differences between the Control Group and the Single Case

Finally, the difference between the Control Group and the Single Case is of interest.

A general plot can be obtained in the following way, plotting both the Control Group and the Single Case:

plot( mdl ) +
  theme_bw( base_size = 18 )+
  theme( legend.position = "bottom",
         legend.direction = "horizontal")

or plotting only the difference

plot( mdl ,who = "delta" ) +
  theme_bw( base_size = 18 )

The relevant coefficients are:

tmp <- sum_mdl[[3]][sum_mdl[[3]]$BF10 > 3,c("BF10","mean","2.5%","97.5%")]

colnames(tmp) <- c("$BF_{10}$", "$\\mu$", "low $95\\%~CI$", "up $95\\%~CI$")

knitr::kable(
  tmp,
  digits = 3
)

The Intercept coefficient is showing us that the single case is generally slower than the Control Sample (generally speaking, when you analyse healthy controls against a single case with a specific disease, the single case is slower).

All the main effects can be further analysed by simply looking at their estimates (knowing the contrast matrix and the direction of the estimate you can understand which level is greater than the other), or by means of the pairwise.BMSC function, if you also want marginal effects and automatic plots.

The interactions require the use of the pairwise.BMSC function.

The Body District : Congruency interaction:

p4 <- pairwise.BMSC(mdl , contrast = "Body.District2:Congruency2" ,
                    who = "delta")

print( p4 , digits = 3 )

The pairwise.BMSC function shows that in all cases the marginal effects of the RTs where greater than zero, but the differences where present only in the comparison between FOOT Congruent and the other cases.

plot( p4 , type = "interval")

plot( p4 , type = "area")

plot( p4 , type = "hist")

In this case we can observe that the single case was more facilitated by the FOOT Congruent condition than the Control Group.

If the interpretation of the results is difficult, it can be useful look what happens in the Single Case marginal effects.

p5 <- pairwise.BMSC(mdl , contrast = "Body.District2:Congruency2" ,
                    who = "singlecase")

plot( p5 , type = "hist")[[1]]

The Body District : Congruency interaction:

p6 <- pairwise.BMSC(mdl , contrast = "Body.District2:Side2" , who = "delta")

print( p6 , digits = 3 )

plot( p6 , type = "hist")[[1]] +
  theme_bw( base_size = 18)+
  theme( strip.text.y = element_text( angle = 0 ) )

In this case, we can see that the left - right difference in the single case is always present, with faster RTs in the left foot than in the other cases.

The Body District : Congruency : Side interaction:

p7 <- pairwise.BMSC(mdl ,
                    contrast = "Body.District2:Congruency2:Side2" ,
                    who = "delta")

print( p7 , digits = 3 )

plot( p7 , type = "hist")[[1]] +
  theme_bw( base_size = 18)+
  theme( strip.text.y = element_text( angle = 0 ) )

Here we can see that the effect was pushed by the facilitation that the single case had in the Left Congruent Foot condition compared to the Control Group.

Efficient approximate leave-one-out cross-validation for fitted BMSC models

The bmscstan package has wrapper functions to interface with the loo package, to diagnostic and compare BMSC models.

Leaving-One-Out scores, diagnostics and comparisons are separately computed for the Control group and the Single Case data.

In order to see the Leaving-One-Out and the Pareto smoothed importance sampling (PSIS), it is possible to use the function loo.BMSC:

print( loo1 <- BMSC_loo( mdl ) )

plot( loo1 )

Model comparison can be done by means of the BMSC_loo_compare function:

mdl.null <- BMSC(formula = RT ~ 1 +
             (Congruency * Side | BD_ID),
             data_ctrl = data.ctrl,
             data_sc = data.pt,
             cores = 1,
             chains = 2,
             seed = 2021)

print( loo2 <- BMSC_loo( mdl.null ) )

plot( loo2 )

BMSC_loo_compare( list( loo1, loo2 ) )

Further details on LOO, PSIS and their use can be found in the loo package and in Vehtari, Gelman, and Gabry (2017) and Vehtari et al. (2015).

A binomial case

In this section, a brief example on how to use the package for binomial data.

We start simulating the data.

######################################
# simulation of controls' group data
######################################

# Number of levels for each condition and trials
NCond  <- 2
Ntrials <- 20
NSubjs  <- 40

betas <- c( 0.5 , 0 )

data.sim <- expand.grid(
  trial      = 1:Ntrials,
  ID         = factor(1:NSubjs),
  Cond      = factor(1:NCond)
)

### d.v. generation
y <- rep( times = nrow(data.sim) , NA )

# cheap simulation of individual random intercepts
set.seed(1)
rsubj <- rnorm(NSubjs , sd = 0.1)

for( i in 1:length( levels( data.sim$ID ) ) ){
  
  sel <- which( data.sim$ID == as.character(i) )
  
  mm  <- model.matrix(~ 1 + Cond , data = data.sim[ sel , ] )
  
  set.seed(1 + i)
  y[sel] <- mm %*% as.matrix(betas + rsubj[i]) +
    rnorm( n = Ntrials * NCond )
  
}

data.sim$y <- y
data.sim$bin <- sapply(
  LaplacesDemon::invlogit(data.sim$y),
  function(x) rbinom( 1, 1, x)
  )

data.sim.bin <- aggregate( bin ~ Cond * ID, data = data.sim, FUN = sum)
data.sim.bin$n <- aggregate( bin ~ Cond * ID,
                             data = data.sim, FUN = length)$bin

######################################
# simulation of patient data
######################################

betas.pt <- c( 0 , 2  )

data.pt <- expand.grid(
  trial      = 1:Ntrials,
  Cond      = factor(1:NCond)
)

### d.v. generation
mm  <- model.matrix(~ 1 + Cond , data = data.pt )

set.seed(5)
data.pt$y <- (mm %*% as.matrix(betas.pt + betas) +
  rnorm( n = Ntrials * NCond ))[,1]
data.pt$bin <- sapply(
  LaplacesDemon::invlogit(data.pt$y),
  function(x) rbinom( 1, 1, x)
  )

data.pt.bin <- aggregate( bin ~ Cond, data = data.pt, FUN = sum)
data.pt.bin$n <- aggregate( bin ~ Cond,
                             data = data.pt, FUN = length)$bin


plot(x = data.sim.bin$Cond, y = data.sim.bin$bin, ylim = c(0,20))
points(x = data.pt.bin$Cond, y = data.pt.bin$bin, col = "red")

The boxplot represents the control participants, the red dot the single case.

Now, we can specify the model:

cbind(bin, n) ~ Cond

The right-hand side of the formula follows the usual lmer- and brms-like syntax. In the left-hand side of the formula, brms and lme4 have divergent notations.

In future, the bmscstan package will be able to use both notations, for the moment it is necessary the lme4 notation cbind(bin, n) where:

  • bin is the number of observations
  • n is the total number of trials
mdlBin <- BMSC(formula = cbind(bin, n) ~ 1 + Cond,
            data_ctrl = data.sim.bin, data_sc = data.pt.bin, seed = 2022,
            chains = 2,
            family = "binomial", cores = 1)

print( summary( mdlBin ) , digits = 3 )

Conclusions

In this vignette we have seen how to use the package bmscstan and its functions to analyse and make sense of Single Case data.

The output of the main functions is rich of information, and the Bayesian Inference can be done by taking into account the Savage-Dickey BF10, or the 95% CI (see Kruschke 2014 for further details).

In this vignette there is almost no discussion concerning how to test the Single Case fixed effects (third part of the main output), but it was used to better understand what happens in the differences between the single case and the control group.

However, if your hypotheses focus on the behaviour of the patient, and not only on the differences between single case and the control group, it will be important to analyse in detail also that part.

References

Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. 2017. “<i>Stan</i> : A Probabilistic Programming Language.” Journal of Statistical Software 76 (1). https://doi.org/10.18637/jss.v076.i01.
Crawford, John R., and Paul H. Garthwaite. 2005. “Evaluation of criteria for classical dissociations in single-case studies by Monte Carlo simulation.” Neuropsychology 19 (5): 664–78. https://doi.org/10.1037/0894-4105.19.5.664.
Crawford, John R., Paul H Garthwaite, Sara Porter, and Sara Porter Point. 2010. “Point and interval estimates of effect sizes for the case-controls design in neuropsychology : Rationale , methods , implementations , and proposed reporting standards Point and interval estimates of effect sizes for the case-controls design in neuropsych” 3294. https://doi.org/10.1080/02643294.2010.513967.
Crawford, John R., and David C Howell. 1998. “Comparing an Individual ’ s Test Score Against Norms Derived from Small Samples.” The Clinical Neuropsychologist 12 (4): 482–86. https://doi.org/10.1076/clin.12.4.482.7241.
Gelman, Andrew. 2013. “Understanding posterior p-values.” Electronic Journal of Statistics, 1–8. https://doi.org/10.1214/13-EJS854.
Gelman, Andrew, and Donald B Rubin. 1992. “Inference from Iterative Simulation Using Multiple Sequences.” Statistical Science 7 (4): 457–72.
Kruschke, John K. 2014. Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan, second edition. 2nd ed. Elsevier Inc. https://doi.org/10.1016/B978-0-12-405888-0.09999-2.
Ottoboni, Giovanni, Alessia Tessari, Roberto Cubelli, and Carlo Umiltà. 2005. “Is handedness recognition automatic? A study using a Simon-like paradigm.” Journal of Experimental Psychology: Human Perception and Performance 31 (4): 778–89. https://doi.org/10.1037/0096-1523.31.4.778.
Raftery, Adrian E. 1995. “Bayesian Model Selection in Social Research.” In Sociological Methodology, edited by P V Marsden, 25:111–63. 1995. Cambridge: Blackwells.
Scandola, Michele, and Daniele Romano. 2021. “Bayesian multilevel single case models using ‘Stan’. A new tool to study single cases in neuropsychology.” Neuropsychologia 156 (June): 107834. https://doi.org/10.1016/j.neuropsychologia.2021.107834.
Tessari, Alessia, Giovanni Ottoboni, Giulia Baroni, Ed Symes, and Roberto Nicoletti. 2012. “Is access to the body structural description sensitive to a body part’s significance for action and cognition? A study of the sidedness effect using feet.” Experimental Brain Research 218 (4): 515–25. https://doi.org/10.1007/s00221-012-3045-4.
Vehtari, Aki, Andrew Gelman, and Jonah Gabry. 2017. “Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC.” Statistics and Computing 27 (5): 1413–32. https://doi.org/10.1007/s11222-016-9696-4.
Vehtari, Aki, Daniel Simpson, Andrew Gelman, Yuling Yao, and Jonah Gabry. 2015. “Pareto Smoothed Importance Sampling,” no. July (July). http://arxiv.org/abs/1507.02646.