Estimating Hierarchical Linear Models for Single-Case Designs

scdhlm provides a set of tools for estimating hierarchical linear models and effect sizes based on data from single-case designs. The estimated effect sizes, as described in Pustejovsky, Hedges, and Shadish (2014), are comparable in principle to standardized mean differences (SMDs) estimated from between-subjects randomized experiments. The package includes functions for estimating design-comparable SMDs based on data from ABAB designs (or more general treatment reversal designs), multiple baseline designs, and multiple probe designs. Two estimation methods are available: moment estimation (Hedges, Pustejovsky, & Shadish, 2012; 2013) and restricted maximum likelihood estimation (Pustejovsky, Hedges, & Shadish, 2014). The package also includes an interactive web interface implemented using Shiny.


You can install the released version of scdhlm from CRAN with:

You can install the development version from GitHub with:

# install.packages("remotes")

You can access a local version of the interactive web-app by running the commands:


Here we demonstrate how to use scdhlm to calculate design-comparable SMDs based on data from different single-case designs. We will first demonstrate the recommended approach, which uses restricted maximum likelihood (REML) estimation. We will then demonstrate the older, moment estimation methods. The moment estimation methods were the originally proposed approach (described in Hedges, Pustejovsky, & Shadish, 2012, 2013). The package provides these methods for sake of completeness, but we no longer recommend them for general use.

Estimating SMDs using REML with g_mlm()

Laski, Charlop, and Schreibman (1988) used a multiple baseline across individuals to evaluate the effect of a training program for parents on the speech production of their autistic children, as measured using a partial interval recording procedure. The design included eight children. One child was measured separately with each parent; following Hedges, Pustejovsky, and Shadish (2013), only the measurements taken with the mother are included in our analysis.

For this study, we will estimate a design-comparable SMD using restricted maximum likelihood (REML) methods, as described by Pustejovsky and colleagues (2014). This is a two-step process. The first step is to estimate a hierarchical linear model for the data, treated the measurements as nested within cases. We fit the model using nlme::lme()


# Pustejovsky, Hedges, & Shadish (2014)
Laski_RML <- lme(fixed = outcome ~ treatment,
                 random = ~ 1 | case, 
                 correlation = corAR1(0, ~ time | case), 
                 data = Laski)
#> Linear mixed-effects model fit by REML
#>   Data: Laski 
#>   Log-restricted-likelihood: -519.1424
#>   Fixed: outcome ~ treatment 
#>        (Intercept) treatmenttreatment 
#>           39.07612           30.68366 
#> Random effects:
#>  Formula: ~1 | case
#>         (Intercept) Residual
#> StdDev:    15.68278  13.8842
#> Correlation Structure: AR(1)
#>  Formula: ~time | case 
#>  Parameter estimate(s):
#>      Phi 
#> 0.252769 
#> Number of Observations: 128
#> Number of Groups: 8

The summary of the fitted model displays estimates of the component parameters, including the within-case and between-case standard deviations, auto-correlation, and (unstandardized) treatment effect estimate. These estimated components will be used to calculate the effect size in next step.

The estimated variance components from the fitted model can be obtained using extract_varcomp():

varcomp_Laski_RML <- extract_varcomp(Laski_RML)
#> $Tau
#> $Tau$case
#> case.var((Intercept)) 
#>              245.9497 
#> $cor_params
#> [1] 0.252769
#> $var_params
#> numeric(0)
#> $sigma_sq
#> [1] 192.7711
#> attr(,"class")
#> [1] "varcomp"

The estimated between-case variance is 245.95, the estimated auto-correlation is 0.253, the estimated and the estimated within-case variance is 192.771. These estimated variance components will be used to calculate the effect size in next step.

The second step in the process is to estimate a design-comparable SMD using scdhlm::g_mlm(). The SMD parameter can be defined as the ratio of a linear combination of the fitted model’s fixed effect parameters over the square root of a linear combination of the model’s variance components. g_mlm() takes the fitted lme model object as input, followed by the vectors p_const and r_const, which specify the components of the fixed effects and variance estimates that are to be used in constructing the design-comparable SMD. Note that r_const is a vector of 0s and 1s which specify whether to use the variance component parameters for calculating the effect size: random effects variances, correlation structure parameters, variance structure parameters, and level-1 error variance. The function calculates an effect size estimate by first substituting maximum likelihood or restricted maximum likelihood estimates in place of the corresponding parameters, then applying a small-sample correction. The small-sample correction and the standard error are based on approximating the distribution of the estimator by a t distribution, with degrees of freedom given by a Satterthwaite approximation (Pustejovsky, Hedges, & Shadish, 2014). The g_mlm() function includes an option allowing use of the expected or average form of the Fisher information matrix in the calculations.

In this example, we use the treatment effect in the numerator of the effect size and the sum of the between-case and within-case variance components in the denominator of the effect size. The constants are therefore given by p_const = c(0, 1) and r_const = c(1, 0, 1). The effect size estimated is calculated as:

Laski_ES_RML <- g_mlm(Laski_RML, p_const = c(0, 1), r_const = c(1, 0, 1))

#>                           est    se
#> unadjusted effect size  1.465 0.299
#> adjusted effect size    1.405 0.286
#> degree of freedom      18.552

The adjusted SMD effect size estimate is 1.405 with standard error of 0.286 and degree of freedom 18.6.

A summary() method is included, which returns more detail about the model parameter estimates and effect size estimate when setting returnModel = TRUE (the default) in g_mlm():

#>                                           est      se
#>        245.950 142.179
#> cor_params                              0.253   0.100
#> sigma_sq                              192.771  28.265
#> total variance                        438.721 144.047
#> (Intercept)                            39.076   5.989
#> treatmenttreatment                     30.684   2.996
#> treatment effect at a specified time   30.684   2.996
#> unadjusted effect size                  1.465   0.299
#> adjusted effect size                    1.405   0.286
#> degree of freedom                      18.552        
#> constant kappa                          0.143        
#> logLik                               -519.142

The CI_g() calculates a symmetric confidence interval using a central t distribution (the default) or an asymmetric confidence interval using non-central t distribution (setting symmetric = FALSE).

#> [1] 0.8046224 2.0051521

CI_g(Laski_ES_RML, symmetric = FALSE)
#> [1] 0.9143684 2.0046719

The symmetric confidence interval is
0.805, 2.005
and the asymmetric confidence interval is
0.914, 2.005

Estimating SMDs using effect_size_ABk()

Lambert, Cartledge, Heward, and Lo (2006) tested the effect of using response cards (compared to single-student responding) during math lessons in two fourth-grade classrooms. The investigators collected data on rates of disruptive behavior for nine focal students, using an ABAB design. This example is discussed in Hedges, Pustejovsky, and Shadish (2012), who selected it because the design was close to balanced and used a relatively large number of cases. Their calculations can be replicated using the effect_size_ABk() function. To use this function, the user must provide the names of five variables:

  • the outcome variable,
  • a variable indicating the treatment condition,
  • a variable listing the case on which the outcome was measured,
  • a variable indicating the phase of treatment (i.e., each replication of a baseline and treatment condition), and
  • a variable listing the session number.

In the Lambert dataset, these variables are called respectively outcome, treatment, case, phase, and time. Given these inputs, the design-comparable SMD is calculated as follows:


Lambert_ES <- effect_size_ABk(outcome = outcome, treatment = treatment, id = case, 
                              phase = phase, time = time, data = Lambert)

#>                            est    se
#> unadjusted effect size  -2.525 0.202
#> adjusted effect size    -2.513 0.201
#> degree of freedom      164.492

The adjusted effect size estimate delta_hat is equal to -2.513; its variance V_delta_hat is equal to 0.041. A standard error for delta_hat can be calculated by taking the square root of V_delta_hat: sqrt(Lambert_ES$V_delta_hat) = 0.201. The effect size estimate is bias-corrected in a manner analogous to Hedges’ g correction for SMDs from a between-subjects design. The degrees of freedom nu are estimated based on a Satterthwaite-type approximation, which is equal to 164.5 in this example.

A summary() method is included to return more detail about the model parameter estimates and effect size estimates:

#>                                       est    se
#> within-case variance                4.534      
#> sample variance                     4.674      
#> intra-class correlation             0.030      
#> auto-correlation                    0.225      
#> numerator of effect size estimate  -5.458      
#> unadjusted effect size             -2.525 0.202
#> adjusted effect size               -2.513 0.201
#> degree of freedom                 164.492      
#> scalar constant                     0.145

Estimating SMDs using effect_size_MB()

Saddler, Behforooz, and Asaro (2008) used a multiple baseline design to investigate the effect of an instructional technique on the writing of fourth grade students. The investigators assessed the intervention’s effect on measures of writing quality, sentence complexity, and use of target constructions.

Design-comparable SMDs can be estimated based on these data using the effect_size_MB() function. The following code calculates a design-comparable SMD estimate for the measure of writing quality:


Saddler_quality <- subset(Saddler, measure=="writing quality")
quality_ES <- effect_size_MB(outcome, treatment, case, time, data = Saddler_quality)

#>                          est    se
#> unadjusted effect size 2.149 0.634
#> adjusted effect size   1.963 0.579
#> degree of freedom      8.918

The adjusted effect size estimate delta_hat is equal to 1.963, with sampling variance of V_delta_hat equal to 0.335 and a standard error of 0.579.

summary(quality_ES) returns more detail about the model parameter estimates and effect size estimates:

#>                                     est    se
#> within-case variance              0.349      
#> sample variance                   0.952      
#> intra-class correlation           0.633      
#> auto-correlation                  0.100      
#> numerator of effect size estimate 2.097      
#> unadjusted effect size            2.149 0.634
#> adjusted effect size              1.963 0.579
#> degree of freedom                 8.918      
#> scalar constant                   0.201


Gilmour, A. R., Thompson, R., & Cullis, B. R. (1995). Average information REML: An efficient algorithm for variance parameter estimation in linear mixed models. Biometrics, 51(4), 1440–1450.

Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2012). A standardized mean difference effect size for single case designs. Research Synthesis Methods, 3(3), 224-239. doi: 10.1002/jrsm.1052

Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2013). A standardized mean difference effect size for multiple baseline designs across individuals. Research Synthesis Methods, 4(4), 324-341. doi: 10.1002/jrsm.1086

Lambert, M. C., Cartledge, G., Heward, W. L., & Lo, Y. (2006). Effects of response cards on disruptive behavior and academic responding during math lessons by fourth-grade urban students. Journal of Positive Behavior Interventions, 8(2), 88-99. doi: 10.1177/10983007060080020701

Laski, K. E., Charlop, M. H., & Schreibman, L. (1988). Training parents to use the natural language paradigm to increase their autistic children’s speech. Journal of Applied Behavior Analysis, 21(4), 391–400. doi: 10.1901/jaba.1988.21-391

Pustejovsky, J. E., Hedges, L. V., & Shadish, W. R. (2014). Design-comparable effect sizes in multiple baseline designs: A general modeling framework. Journal of Educational and Behavioral Statistics, 39(4), 211-227. doi: 10.3102/1076998614547577

Saddler, B., Behforooz, B., & Asaro, K. (2008). The effects of sentence-combining instruction on the writing of fourth-grade students with writing difficulties. The Journal of Special Education, 42(2), 79–90. doi: 10.1177/0022466907310371