The scdhlm package (Pustejovsky, Chen, & Hamilton, 2020) provides several methods for estimating design-comparable standardized mean differences (SMDs) based on data from a single-case design. In principle, a design-comparable SMD is in the same metric as the SMD from a simple, between-groups randomized experiment performed on a comparable sample and with comparable outcome measures. Hedges, Pustejovsky, and Shadish (2012) proposed methods for estimating design-comparable SMDs based on data from an ABAB design (and, more generally, treatment reversal designs with an arbitrary number of phases). Hedges, Pustejovsky, and Shadish (2013) extended the methods to handle data from multiple baseline designs. In both cases, the proposed estimation methods are premised on a simple model for the data, which assumed that the outcome process is stable over time (lacking time trends) and that the treatment effect is constant across cases. Pustejovsky, Hedges, and Shadish (2014) proposed an approach to defining and estimating design-comparable SMDs under a more general model, which can allow for time trends and between-case variation in treatment effects and time trends.

In the scdhlm package, the original estimation methods proposed for
the ABAB design and multiple baseline design are implemented in the
`effect_size_ABk`

and `effect_size_MB`

functions,
respectively. Both of these functions take the raw data as input and
produce as output an effect size estimate, along with accompanying
standard error and some other auxiliary information. Thus, there is no
distinction between estimating the model and estimating the effect size.
In contrast, the more general methods proposed in Pustejovsky, Hedges,
and Shadish (2014) entail two steps: first, estimating a hierarchical
model for the data; and second, estimating a design-comparable effect
size based on the fitted model. The first step is accomplished using the
function `lme`

from the package `nlme`

by
Pinheiro, Bates, DebRoy, and Sarkar (2015). The second step is
accomplished using the function `g_mlm`

from the
`scdhlm`

package. This vignette demonstrates how to use all
of these functions to estimate design-comparable standardized mean
difference effect sizes. The R code presented below can be used to
replicate the examples found in the papers that proposed the methods. To
begin, the user must load the package:

`library(scdhlm)`

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 five pieces of data:

`outcome`

: the outcome variable,`treatment`

: a variable indicating the treatment condition,`id`

: a variable listing the case on which the outcome was measured,`phase`

: a variable indicating the phase of treatment (i.e., each replication of a baseline and treatment condition), and`time`

: 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:

```
data(Lambert)
<- effect_size_ABk(outcome = outcome, treatment = treatment,
Lambert_ES id = case, phase = phase, time = time,
data = Lambert)
print(Lambert_ES, 5)
```

```
## est se
## unadjusted effect size -1.16204 0.12345
## adjusted effect size -1.15660 0.12287
## degree of freedom 160.23777
```

The adjusted effect size estimate `delta_hat`

is equal to
-1.157; its variance `V_delta_hat`

is equal to 0.015. A
standard error for `delta_hat`

can be calculated by taking
the square root of `V_delta_hat`

:
`sqrt(Lambert_ES$V_delta_hat)`

= 0.123. 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 160.2 in this example.

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

`summary(Lambert_ES)`

```
## est se
## within-case variance 7.843
## sample variance 6.942
## intra-class correlation 0.000
## auto-correlation -0.316
## numerator of effect size estimate -3.062
## unadjusted effect size -1.162 0.123
## adjusted effect size -1.157 0.123
## degree of freedom 160.238
## scalar constant 0.104
```

By default, the `effect_size_ABk`

function calculates an
estimate of the first-order autocorrelation in the outcome series
(stored in the entry `phi`

) and an estimate of the
intra-class correlation, i.e., the ratio of the between-case variance in
the outcome to the total cross-sectional variance in the outcome (the
intra-class correlation estimate is stored in the entry
`rho`

). Optionally, the user can specify their own estimates
of these parameters as inputs to the function. In this example, the
auto-correlation estimate was -0.316. The following code examines the
sensitivity of the results to values of the auto-correlation that are
larger and smaller than the default estimate of -0.316.

```
<- effect_size_ABk(outcome = outcome, treatment = treatment,
Lambert_ES2 id = case, phase = phase, time = time,
data = Lambert, phi = 0.10)
print(Lambert_ES2, digits = 5)
```

```
## est se
## unadjusted effect size -1.16204 0.15280
## adjusted effect size -1.15725 0.15217
## degree of freedom 181.99265
```

```
<- effect_size_ABk(outcome = outcome, treatment = treatment,
Lambert_ES3 id = case, phase = phase, time = time,
data = Lambert, phi = 0.35)
print(Lambert_ES3, digits = 5)
```

```
## est se
## unadjusted effect size -1.16204 0.17968
## adjusted effect size -1.15626 0.17878
## degree of freedom 150.96192
```

The estimated auto-correlation has only a trivial effect on the effect size estimate and a minor effect on its estimated variance.

Anglesea, Hoch, and Taylor (2008) used an ABAB design to test the effect of using a pager prompt to reduce the rapid eating of three teenage boys with autism. The primary outcome was a measure of how quickly each participant consumed one serving of a familiar food. This example is discussed in Hedges, Pustejovsky, and Shadish (2012), who used it to illustrate the calculation of the design-comparable SMD when based on the minimum required number of cases. Their calculations can be replicated using the following code:

```
data(Anglesea)
<- effect_size_ABk(outcome, condition, case, phase, session, data = Anglesea)
Anglesea_ES Anglesea_ES
```

```
## est se
## unadjusted effect size 1.793 2.436
## adjusted effect size 1.150 1.562
## degree of freedom 2.340
```

Note that the standard error of the effect size estimate is quite large and the degrees of freedom corresponding to the denominator of the SMD estimate are very low. Both quantities are extreme due to the small number of cases used in this example.

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 syntax for this function is
similar to that of the `effect_size_ABk`

function, but does
not require the user to input information about the phase of the design
(because in the multiple baseline design, phase exactly corresponds to
treatment condition). The following code replicates the calculations
reported in Hedges, Pustejovsky, and Shadish (2013):

```
data(Saddler)
<- effect_size_MB(outcome, treatment, case, time,
quality_ES data = subset(Saddler, measure=="writing quality"))
<- effect_size_MB(outcome, treatment, case, time ,
complexity_ES data = subset(Saddler, measure=="T-unit length"))
<- effect_size_MB(outcome, treatment, case, time,
construction_ES data = subset(Saddler, measure=="number of constructions"))
data.frame(
quality = unlist(quality_ES),
complexity = unlist(complexity_ES),
construction = unlist(construction_ES)
c("delta_hat","V_delta_hat","nu","phi","rho"),] )[
```

```
## quality complexity construction
## delta_hat 1.96307272 0.78540043 0.74755356
## V_delta_hat 0.33491289 0.08023320 0.07847359
## nu 8.91814603 9.60204004 7.57981360
## phi 0.09965017 -0.07542229 -0.11159420
## rho 0.63321198 0.61453091 0.73123744
```

For multiple baseline designs, an alternative to using the
`effect_size_MB`

function is to estimate a hierarchical
linear model for the data and then use the `g_mlm`

function.
The two alternative approaches differ in how the model parameters and
effect size are estimated. Pustejovsky, Hedges, and Shadish (2014) found
that the latter approach (based on a hierarchical linear model) has
comparable mean-squared error to the former approach, while producing
better estimates of the variance of the effect size. The latter approach
is implemented in two steps, which we will demonstrate using the writing
quality measure. First, we estimate the hierarchical model with an AR(1)
within-case error structure using the `lme`

function:

```
<- lme(fixed = outcome ~ treatment,
quality_RML random = ~ 1 | case,
correlation = corAR1(0, ~ time | case),
data = subset(Saddler, measure=="writing quality"))
summary(quality_RML)
```

```
## Linear mixed-effects model fit by REML
## Data: subset(Saddler, measure == "writing quality")
## AIC BIC logLik
## 97.31383 105.6316 -43.65692
##
## Random effects:
## Formula: ~1 | case
## (Intercept) Residual
## StdDev: 0.6621631 0.6656806
##
## Correlation Structure: ARMA(1,0)
## Formula: ~time | case
## Parameter estimate(s):
## Phi1
## 0.3819756
## Fixed effects: outcome ~ treatment
## Value Std.Error DF t-value p-value
## (Intercept) 2.437206 0.3187517 34 7.646094 0
## treatmenttreatment 2.181029 0.2340101 34 9.320237 0
## Correlation:
## (Intr)
## treatmenttreatment -0.308
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.4700055 -0.4627404 -0.1907942 0.4286209 1.8063844
##
## Number of Observations: 41
## Number of Groups: 6
```

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. The next step is to combine these estimated components into an
effect size estimate using the `g_mlm`

function. This
function 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. For
details on how to choose these constants, see Pustejovsky, Hedges, and
Shadish (2014). In this example:

```
<- g_mlm(quality_RML, p_const = c(0,1), r_const = c(1,0,1))
quality_ES_RML summary(quality_ES_RML)
```

```
## est se
## Tau.case.case.var((Intercept)) 0.438 0.357
## cor_params 0.382 0.230
## sigma_sq 0.443 0.154
## total variance 0.882 0.360
## (Intercept) 2.437 0.319
## treatmenttreatment 2.181 0.234
## treatment effect at a specified time 2.181 0.234
## unadjusted effect size 2.323 0.595
## adjusted effect size 2.174 0.557
## degree of freedom 11.963
## constant kappa 0.249
## logLik -43.657
```

The adjusted SMD effect size is estimated as `g_AB`

=
2.174; its standard error is `SE_g_AB`

= 0.557; the estimated
auto-correlation is `phi`

= 0.382; and the estimated degrees
of freedom is `nu`

= 12. In this example, the RML effect size
estimate is about 10% larger than the estimate from
`effect_size_MB`

, with a slightly smaller variance estimate.
The RML estimate of the auto-correlation is substantially higher than
before, but `effect_size_MB`

uses a moment estimator that is
known to be biased towards zero and that does not perform well when
outcomes are intermittently missing for some sessions (as is the case
here).

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; for purposes of simplicity, and following Hedges, Pustejovsky, and Shadish (2013), only the measurements taken with the mother are included in the analysis.

The following code compares the estimates of the design-comparable
SMD effect size based on the Hedges, Pustejovsky, and Shadish (2013)
approach (using the `effect_size_MB`

function) to the
estimates based on the hierarchical linear modeling approach described
in Pustejovsky, Hedges, and Shadish (2014) (using the `g_mlm`

function).

```
data(Laski)
# Hedges, Pustejovsky, & Shadish (2013)
<- effect_size_MB(outcome, treatment, case, time, data= Laski)
Laski_ES_HPS
# Pustejovsky, Hedges, & Shadish (2014)
<- lme(fixed = outcome ~ treatment,
Laski_RML random = ~ 1 | case,
correlation = corAR1(0, ~ time | case),
data = Laski)
summary(Laski_RML)
```

```
## Linear mixed-effects model fit by REML
## Data: Laski
## AIC BIC logLik
## 1048.285 1062.466 -519.1424
##
## 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
## Fixed effects: outcome ~ treatment
## Value Std.Error DF t-value p-value
## (Intercept) 39.07612 5.989138 119 6.524498 0
## treatmenttreatment 30.68366 2.995972 119 10.241637 0
## Correlation:
## (Intr)
## treatmenttreatment -0.272
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.72642154 -0.69387388 0.01454473 0.69861200 2.14528141
##
## Number of Observations: 128
## Number of Groups: 8
```

```
<- g_mlm(Laski_RML, p_const = c(0,1), r_const = c(1,0,1))
Laski_ES_RML
# compare the estimates
data.frame(
HPS = with(Laski_ES_HPS, c(SMD = delta_hat, SE = sqrt(V_delta_hat),
phi = phi, rho = rho, nu = nu)),
RML = with(Laski_ES_RML, c(g_AB, SE_g_AB, theta$cor_params,
$Tau$case / (theta$Tau$case + theta$sigma_sq), nu))
theta )
```

```
## HPS RML
## SMD 1.38838504 1.4048872
## SE 0.31701943 0.2863249
## phi 0.01652579 0.2527690
## rho 0.69396648 0.5606064
## nu 13.10041496 18.5524061
```

As in the Saddler example, both methods produce very similar SMD estimates and variance estimates. The RML estimate of auto-correlation is substantially higher than the HPS estimate, while the intra-class correlation estimate is somewhat lower; in combination, these differences lead to larger degrees of freedom.

An advantage of the RML approach is that it is readily extended to more complex models. The above analysis was based on the assumption that the treatment effect is constant across cases. This assumption can be removed by fitting a model that includes a random treatment effect for each case:

```
<- lme(fixed = outcome ~ treatment,
Laski_RML2 random = ~ treatment | case,
correlation = corAR1(0, ~ time | case),
data = Laski)
summary(Laski_RML2)
```

```
## Linear mixed-effects model fit by REML
## Data: Laski
## AIC BIC logLik
## 1036.014 1055.868 -511.0069
##
## Random effects:
## Formula: ~treatment | case
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 21.79581 (Intr)
## treatmenttreatment 13.24108 -0.869
## Residual 11.94109
##
## Correlation Structure: AR(1)
## Formula: ~time | case
## Parameter estimate(s):
## Phi
## 0.02267299
## Fixed effects: outcome ~ treatment
## Value Std.Error DF t-value p-value
## (Intercept) 38.45243 7.881197 119 4.879008 0
## treatmenttreatment 31.59161 5.178366 119 6.100691 0
## Correlation:
## (Intr)
## treatmenttreatment -0.835
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -3.12357773 -0.50574784 -0.01605527 0.65004147 2.05388331
##
## Number of Observations: 128
## Number of Groups: 8
```

`anova(Laski_RML, Laski_RML2)`

```
## Model df AIC BIC logLik Test L.Ratio p-value
## Laski_RML 1 5 1048.285 1062.466 -519.1424
## Laski_RML2 2 7 1036.014 1055.868 -511.0069 1 vs 2 16.27091 3e-04
```

The fit of the two models can be compared using a likelihood ratio test, which rejects the model with a constant treatment effect. The second model, which allows the treatment effect to vary, is to be preferred. The following code estimates a design-comparable SMD based on the better-fitting model.

```
<- g_mlm(Laski_RML2, p_const = c(0,1), r_const = c(1,0,0,0,1))
Laski_ES_RML2 Laski_ES_RML2
```

```
## est se
## unadjusted effect size 1.271 0.386
## adjusted effect size 1.181 0.358
## degree of freedom 10.776
```

The adjusted effect size estimate that is 16% smaller than the estimate from the simpler model; with a standard error that is 25% larger. The difference between the two models is due to a difference in between-case variance across phases not captured by the assumptions of the simpler model. The between-case variation in the outcome appears to be substantially larger in the baseline phase than in the treatment phase. Maintaining the constant treatment effect assumption constrains the between-case variance to be constant across phases, and so the between-case variance is estimated by pooling across both phases. The constant treatment effect assumption therefore leads to a smaller variance estimate than the estimate based on allowing the between-case variance to differ by phase.

Schutte, Malouff, and Brown (2008) evaluated the effect of an emotion-focused therapy program for adults with prolonged fatigue using a multiple baseline across individuals. The design included 13 adults who met clinical criteria for prolonged fatigue. Fatigue severity was measured weekly using a self-reported scale that ranged from 1 to 63. Following Pustejovsky, Hedges, and Shadish (2014), the data for participant 4 are excluded from the analysis because nearly all of the measurements for this case are at the upper extreme of the scale. The data for the remaining participants are plotted below.

```
if (requireNamespace("ggplot2", quietly = TRUE)) {
library(ggplot2)
data(Schutte)
graph_SCD(case = case, phase = treatment,
session = week, outcome = fatigue,
design = "MB", data = Schutte) +
facet_wrap(~ case, ncol = 3) +
theme(legend.position = "bottom")
}
```

Time trends are apparent in the outcome series, as are changes in slope in the treatment condition. In order to operationally define a design-comparable SMD effect sizes in a model that includes time trends and treatment-by-time interactions, one will need to choose a time point A at which the treatment would begin and a time point B at which outcomes would be measured, both in a hypothetical between-subjects design based on the same population of participants. Here, we take A = 2 and B = 9; centering time at week 9 simplifies the effect size calculations.

```
# time-point constants
<- 2
A <- 9
B # center at follow-up time
# create time-by-trt interaction
<-
Schutte preprocess_SCD(case = case, phase = treatment,
session = week, outcome = fatigue,
design = "MB", center = B, data = Schutte)
```

Having completed the data-cleaning process, three different models will be considered, again following the example from Pustejovsky, Hedges, and Shadish (2014).

The initial model allows for a baseline time trend, treatment effect, and treatment-by-time interaction, all of which are assumed to be constant across the 12 cases; only the baseline intercept is assumed to vary across cases. This specification corresponds to Model MB3 from Pustejovsky, Hedges, and Shadish (2014)

```
<- lme(fixed = fatigue ~ week + treatment + week_trt,
hlm1 random = ~ 1 | case,
correlation = corAR1(0, ~ week | case),
data = Schutte,
method = "REML")
summary(hlm1)
```

```
## Linear mixed-effects model fit by REML
## Data: Schutte
## AIC BIC logLik
## 884.1744 904.354 -435.0872
##
## Random effects:
## Formula: ~1 | case
## (Intercept) Residual
## StdDev: 3.842612 9.950088
##
## Correlation Structure: AR(1)
## Formula: ~week | case
## Parameter estimate(s):
## Phi
## 0.8092339
## Fixed effects: fatigue ~ week + treatment + week_trt
## Value Std.Error DF t-value p-value
## (Intercept) 52.93290 4.422164 121 11.969907 0.0000
## week 0.48856 0.622153 121 0.785274 0.4338
## treatmenttreatment -1.37267 1.971510 121 -0.696253 0.4876
## week_trt -1.89613 0.938154 121 -2.021131 0.0455
## Correlation:
## (Intr) week trtmnt
## week 0.805
## treatmenttreatment -0.119 -0.069
## week_trt -0.761 -0.777 -0.263
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.69578897 -0.64000820 -0.06285371 0.65265931 2.01491696
##
## Number of Observations: 136
## Number of Groups: 12
```

The design-comparable standardized mean difference corresponds to the
treatment effect at week \(B = 9\),
after \(B - A = 7\) weeks of treatment.
The corresponding values of `p_const`

and
`r_const`

are specified below.

```
<- g_mlm(hlm1, p_const = c(0,0,1, B - A), r_const = c(1,0,1))
Schutte_g1 Schutte_g1
```

```
## est se
## unadjusted effect size -1.373 0.644
## adjusted effect size -1.337 0.627
## degree of freedom 29.152
```

It will be seen below that this initial model provides a poor fit to the data; thus, the effect size estimate based on it should not be trusted.

The next model (Model MB4) allows the baseline time trend to vary
across cases. This can be done by modifying the previous model with the
`update`

function.

```
<- update(hlm1, random = ~ week | case,
hlm2 control=lmeControl(msMaxIter = 50, apVar=FALSE, returnObject=TRUE))
summary(hlm2)
```

```
## Linear mixed-effects model fit by REML
## Data: Schutte
## AIC BIC logLik
## 875.9526 901.8978 -428.9763
##
## Random effects:
## Formula: ~week | case
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 9.783287 (Intr)
## week 1.412030 0.811
## Residual 5.421656
##
## Correlation Structure: AR(1)
## Formula: ~week | case
## Parameter estimate(s):
## Phi
## 0.3985953
## Fixed effects: fatigue ~ week + treatment + week_trt
## Value Std.Error DF t-value p-value
## (Intercept) 50.29150 4.073625 121 12.345637 0.0000
## week 0.20271 0.616194 121 0.328968 0.7427
## treatmenttreatment -0.54218 1.751668 121 -0.309520 0.7575
## week_trt -1.63198 0.655256 121 -2.490607 0.0141
## Correlation:
## (Intr) week trtmnt
## week 0.883
## treatmenttreatment -0.258 -0.200
## week_trt -0.567 -0.596 -0.178
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.33194245 -0.46272153 -0.05348694 0.32878028 4.08749914
##
## Number of Observations: 136
## Number of Groups: 12
```

`anova(hlm1, hlm2)`

```
## Model df AIC BIC logLik Test L.Ratio p-value
## hlm1 1 7 884.1744 904.3540 -435.0872
## hlm2 2 9 875.9526 901.8978 -428.9763 1 vs 2 12.22179 0.0022
```

A likelihood ratio test rejects the initial model in favor of this
more flexible model. An effect size estimate is calculated from the
fitted model as follows^{1}:

```
<- g_mlm(hlm2, p_const = c(0,0,1, B - A), r_const = c(1,0,0,0,1))
Schutte_g2 Schutte_g2
```

```
## est se
## unadjusted effect size -1.070 0.495
## adjusted effect size -1.013 0.469
## degree of freedom 14.302
```

The design-comparable SMD is estimated to be -1.013, with a standard error of 0.469.

The final model (Model MB5) is yet more flexible, in that it allows the treatment-by-time interactions to vary across cases. Given that the data contain only twelve cases, fitting a model with three random effects would be questionable if the goal were to make inferences about the full structure of the model. However, simulation evidence from Pustejovsky, Hedges, and Shadish (2014) suggests that using such a flexible parameterization may nonetheless be reasonable for the more limited purpose of effect size estimation.

```
<- update(hlm2, random = ~ week + week_trt | case,
hlm3 control=lmeControl(msMaxIter = 50, apVar=FALSE, returnObject=TRUE))
summary(hlm3)
```

```
## Linear mixed-effects model fit by REML
## Data: Schutte
## AIC BIC logLik
## 873.0245 907.6181 -424.5123
##
## Random effects:
## Formula: ~week + week_trt | case
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 6.1884596 (Intr) week
## week 0.3832993 0.162
## week_trt 1.7351247 0.162 0.999
## Residual 4.7477472
##
## Correlation Structure: AR(1)
## Formula: ~week | case
## Parameter estimate(s):
## Phi
## 0.2396137
## Fixed effects: fatigue ~ week + treatment + week_trt
## Value Std.Error DF t-value p-value
## (Intercept) 50.52566 2.8214130 121 17.907928 0.0000
## week 0.21901 0.3640857 121 0.601537 0.5486
## treatmenttreatment 0.02697 1.5999246 121 0.016855 0.9866
## week_trt -1.67113 0.7406249 121 -2.256378 0.0258
## Correlation:
## (Intr) week trtmnt
## week 0.722
## treatmenttreatment -0.357 -0.331
## week_trt -0.312 -0.283 -0.154
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.559123387 -0.446173508 -0.004457334 0.366743786 4.597278939
##
## Number of Observations: 136
## Number of Groups: 12
```

`anova(hlm2, hlm3)`

```
## Model df AIC BIC logLik Test L.Ratio p-value
## hlm2 1 9 875.9526 901.8978 -428.9763
## hlm3 2 12 873.0245 907.6181 -424.5123 1 vs 2 8.92808 0.0303
```

The REML estimation algorithm fails to converge because the estimate
of the correlation between the random slopes and the random
treatment-by-time interactions approaches 1. A likelihood ratio test
rejects the previous model (which constrained the treatment-by-time
interactions to be constant). However, this test cannot be trusted with
such small sample of cases, and given variance component estimates that
lie on an edge of the parameter space. An effect size estimate based on
this fitted model is obtained as follows.^{2}

```
<- g_mlm(hlm3, p_const = c(0,0,1,B - A), r_const = c(1,0,0,0,0,0,0,1))
Schutte_g3 Schutte_g3
```

```
## est se
## unadjusted effect size -1.496 0.933
## adjusted effect size -1.326 0.827
## degree of freedom 6.848
```

```
# compare effect size estimates
data.frame(
MB3 = round(unlist(Schutte_g1[c("g_AB","SE_g_AB","nu")]), 3),
MB4 = round(unlist(Schutte_g2[c("g_AB","SE_g_AB","nu")]), 3),
MB5 = round(unlist(Schutte_g3[c("g_AB","SE_g_AB","nu")]), 3)
)
```

```
## MB3 MB4 MB5
## g_AB -1.337 -1.013 -1.326
## SE_g_AB 0.627 0.469 0.827
## nu 29.152 14.302 6.848
```

The effect size estimate has magnitude comparable to the estimate from Model MB3. However, the variance of the estimate is extremely large, due to the fact that the variance components that go into the denominator of the SMD estimate are estimated with low precision. This final model may represent an outer limit on the complexity of models that can feasibility be estimated, given that most single-case designs have fewer cases than the study examined in this example.

Anglesea, M. M., Hoch, H., & Taylor, B. A. (2008). Reducing rapid
eating in teenagers with autism: Use of a pager prompt. *Journal of
Applied Behavior Analysis, 41*(1), 107–111. doi: 10.1901/jaba.2008.41-107

Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2012). A
standardized mean difference effect size for single case designs.
*Research Synthesis Methods, 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.

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.

Pinheiro J., Bates D., DebRoy S., Sarkar D. and R Core Team (2015).
*nlme: Linear and Nonlinear Mixed Effects Models*. R package
version 3.1-119. https://CRAN.R-project.org/package=nlme

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

Pustejovsky, J. E., Hamilton, B. J., & Chen, M. (2020).
*scdhlm: Estimating hierarchical linear models for single-case
designs*. R package version 0.3.2, https://jepusto.github.io/scdhlm/.

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

Note that the specification of the

`r_const`

argument here differs from the specification reported in Pustejovsky, Hedges, and Shadish (2014). The argument appears in a different order than reported in the article because of how the variance components are arranged in the`g_mlm()`

function. To see the order of the variance components used by`g_mlm()`

, use the`extract_varcomp()`

function:`unlist(extract_varcomp(hlm2))`

`## Tau.case.case.var((Intercept)) Tau.case.case.cov(week,(Intercept)) ## 95.7126977 11.2090726 ## Tau.case.case.var(week) cor_params ## 1.9938298 0.3985953 ## sigma_sq ## 29.3943508`

The order of parameter estimates appearing in the result is the same as assumed in

`g_mlm()`

. Thus, entries in`r_const`

should correspond to the order of the parameter estimates appearing here.↩︎Again, the specification of the

`r_const`

argument here differs from the specification reported in Pustejovsky, Hedges, and Shadish (2014). The entries in`r_const`

should correspond to the order of the parameter estimates appearing in`unlist(extract_varcomp(hlm3))`

.↩︎