• Individual effect size functions
  • Inputting data
  • Direction of improvement
  • Standard error and confidence intervals
  • Other non-overlap indices
  • Further options for SMD()
  • Further options for log response ratios (LRRi() and LRRd())
  • Further options for LOR()
• calc_ES()
  • Further arguments
  • Long vs. wide format
• batch_calc_ES()
  • Data organization
  • Main arguments of batch_calc_ES()
  • Using grouping variables
  • Aggregating across replications
  • Dealing with outcome scales.
  • Further arguments
• References

The SingleCaseES package provides R functions for calculating basic, within-case effect size indices for single-case designs, including several non-overlap measures and parametric effect size measures, and for estimating the gradual effects model (Swan & Pustejovsky, 2018). Standard errors and confidence intervals are provided for the subset of effect sizes indices with known sampling distributions.

The package also includes two graphical user interfaces for interactive use (designed using Shiny), both of which are also available as web apps hosted through shinyapps.io:

• SCD_effect_sizes() opens an interactive calculator for the basic non-overlap indices and parametric effect sizes. It is also available at https://jepusto.shinyapps.io/SCD-effect-sizes/
• shine_gem_scd() opens an interactive calculator for the gradual effects model. It is also available at https://jepusto.shinyapps.io/gem-scd/

In this vignette, we introduce the package’s primary functions for carrying out effect size calculations. We demonstrate how to use the functions for calculating an effect size from a single data series, how to use the calc_ES() function for calculating multiple effect sizes from a single data series, and how to use batch_calc_ES() for calculating one or multiple effect sizes from multiple data series.

To start, be sure to load the package:

library(SingleCaseES)

Individual effect size functions

The SingleCaseES package includes functions for calculating the major non-overlap measures that have been proposed for use with single-case designs, as well as several parametric effect size measures. The following non-overlap measures are available (function names are listed in parentheses):

• Percentage of non-overlapping data (PND)
• Percentage of all non-overlapping data (PAND)
• Robust improvement rate difference (IRD)
• Percentage exceeding the median (PEM)
• Non-overlap of all pairs (NAP)
• Tau non-overlap (Tau)
• Baseline-corrected Tau (Tau_BC)
• Tau-U, which includes baseline trend adjustment (Tau_U)

The following parametric effect sizes are available:

• Within-case standardized mean difference (SMD)
• The increasing and decreasing versions of the log response ratio (LRRi and LRRd)
• Log odds ratio (LOR)
• Log ratio of medians (LRM)

All of the functions for calculating individual effect sizes follow the same syntax. For demonstration purposes, let’s take a look at the syntax for NAP(), which calculates the non-overlap of all pairs (Parker & Vannest, 2009):

args(NAP)
#> function (A_data, B_data, condition, outcome, baseline_phase = NULL, 
#>     intervention_phase = NULL, improvement = "increase", SE = "unbiased", 
#>     confidence = 0.95, trunc_const = FALSE) 
#> NULL

We will first demonstrate two methods for inputting data from a single SCD series, then explain the further arguments of the function.

Inputting data

There are two formats in which data can be provided to the functions: the A_data and B_data inputs, or the condition and outcome inputs. Both formats can be used for any of the non-overlap or parametric measures.

Using the A_data, B_data inputs

The first input format involves providing separate vectors for the data from each phase, where A corresponds to the baseline phase and B corresponds to the treatment phase.

Here are some hypothetical data from the A and B phases of a single-case data series:

A <- c(20, 20, 26, 25, 22, 23)
B <- c(28, 25, 24, 27, 30, 30, 29)

We can feed these data into the NAP function as follows:

NAP(A_data = A, B_data = B)
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.06900656 0.5973406 0.9860176

The result reports the NAP effect size estimate for these hypothetical data, along with a standard error and a 95% confidence interval.

Using the condition, outcome inputs

The second input format involves providing a single vector containing all of the outcome data from the series, along with a vector that describes the phase of each observation in the data. For example, the hypothetical data above contains 6 baseline phase observations and 7 treatment phase observations. Therefore, the condition input should consist of six entries of 'A' followed by seven entries of 'B':

phase <- c(rep("A", 6), rep("B", 7))
phase
#>  [1] "A" "A" "A" "A" "A" "A" "B" "B" "B" "B" "B" "B" "B"

This format also requires providing a single vector containing all of the outcome data from the series. Here is the hypothetical data from above, reformatted to follow this structure:

outcome_dat <- c(A, B)
outcome_dat
#>  [1] 20 20 26 25 22 23 28 25 24 27 30 30 29

We can feed these data into the NAP function as follows:

NAP(condition = phase, outcome = outcome_dat)
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.06900656 0.5973406 0.9860176

It’s important to note a few further distinctions that can be made when using the condition and outcome inputs. If the vector provided to condition has more than two values, the effect size function will assume that the first value of condition is the baseline phase and the second unique value of condition is the intervention phase.

phase2 <- c(rep("A", 5), rep("B", 5), rep("C",3))
NAP(condition = phase2, outcome = outcome_dat)
#> Warning in calc_ES(A_data = A_data, B_data = B_data, condition = condition, :
#> The 'condition' variable has more than two unique values. Treating 'B' as the
#> intervention phase.
#>    ES  Est    SE  CI_lower  CI_upper
#> 1 NAP 0.78 0.155 0.4115567 0.9423658

In some single-case data series, the initial observation might not be in the baseline phase. For example, an SCD with four cases might use a cross-over treatment reversal design, where two of the cases follow an ABAB design and the other two cases follow a BABA design. To handle this situation, we will need to specify the baseline phase using the baseline_phase argument:

phase_rev <- c(rep("B", 7), rep("A", 6))
outcome_rev <- c(B, A)
NAP(condition = phase_rev, outcome = outcome_rev, baseline_phase = "A")
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.06900656 0.5973406 0.9860176

In data series that include more than two unique phases, it is also possible to specify which one should be used as the intervention phase using the intervention_phase argument:

NAP(condition = phase2, outcome = outcome_dat, 
    baseline_phase = "A", intervention_phase = "C")
#>    ES Est         SE CI_lower CI_upper
#> 1 NAP   1 0.06346478        1        1

NAP(condition = phase2, outcome = outcome_dat, 
    baseline_phase = "B", intervention_phase = "C")
#>    ES Est         SE CI_lower CI_upper
#> 1 NAP   1 0.06346478        1        1

Direction of improvement

All of the effect size functions in SingleCaseES are defined based on some assumption about the direction of therapeutic improvement in the outcome (e.g., improvement would correspond to increases in on-task behavior but to decreases in aggressive behavior). For all of the effect size functions, it is important to specify the direction of therapeutic improvement for the data series by providing a value for the improvement argument, either “increase” or “decrease”:

NAP(A_data = A, B_data = B, improvement = "decrease")
#>    ES        Est         SE   CI_lower  CI_upper
#> 1 NAP 0.08333333 0.06900656 0.01398242 0.4026594

The NAP() function and most of the other effect size functions default to assuming that increases in the outcome correspond to improvements.

Standard error and confidence intervals

The NAP, Tau, and Tau_BC functions provide several possible methods for calculating the standard error. By default, the exactly unbiased standard errors are used. However, the function can also produce standard errors using the Hanley-McNeil estimator, the standard error under the null hypothesis of no effect, or no standard errors at all:

NAP(A_data = A, B_data = B, SE = "unbiased")
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.06900656 0.5973406 0.9860176

NAP(A_data = A, B_data = B, SE = "Hanley")
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.07739185 0.5973406 0.9860176

NAP(A_data = A, B_data = B, SE = "null")
#>    ES       Est        SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.1666667 0.5973406 0.9860176

NAP(A_data = A, B_data = B, SE = "none")
#>    ES       Est
#> 1 NAP 0.9166667

The functions also produce confidence intervals for NAP, Tau, and Tau_BC. By default, a 95% CI is calculated. This can be adjusted by setting the confidence argument to a value between 0 and 1. To omit the confidence interval all together, set the value to NULL:

NAP(A_data = A, B_data = B)
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.06900656 0.5973406 0.9860176

NAP(A_data = A, B_data = B, confidence = .99)
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.06900656 0.4875014 0.9907377

NAP(A_data = A, B_data = B, confidence = .90)
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.06900656 0.6591091 0.9822249

NAP(A_data = A, B_data = B, confidence = NULL)    
#>    ES       Est         SE
#> 1 NAP 0.9166667 0.06900656

Other non-overlap indices

The SingleCaseES package includes functions for calculating several other non-overlap indices in addition to NAP. All of the functions accept data in either the A_data, B_data format or the condition, outcome format with optional baseline specification, and all of the functions include an argument to specify the direction of improvement. Like the function for NAP, the functions for Tau (Tau) and baseline-corrected Tau (Tau_BC) can produce unbiased standard errors, Hanley-McNeil standard errors, standard errors under the null hypothesis of no effect, or no standard errors at all. Only NAP, Tau, and Tau_BC return standard errors and confidence intervals. The remaining non-overlap measures return only a point estimate:

Tau(A_data = A, B_data = B)
#>    ES       Est        SE  CI_lower  CI_upper
#> 1 Tau 0.8333333 0.1380131 0.1946812 0.9720352
Tau_BC(A_data = A, B_data = B)
#>       ES       Est        SE   CI_lower  CI_upper
#> 1 Tau-BC 0.2857143 0.3595159 -0.3260702 0.7180613
PND(A_data = A, B_data = B)
#>    ES       Est
#> 1 PND 0.7142857
PEM(A_data = A, B_data = B)
#>    ES Est
#> 1 PEM   1
PAND(A_data = A, B_data = B)
#>     ES       Est
#> 1 PAND 0.8461538
IRD(A_data = A, B_data = B)
#>    ES       Est
#> 1 IRD 0.6904762
Tau_U(A_data = A, B_data = B)
#>      ES       Est
#> 1 Tau-U 0.7380952

Further options for SMD()

The standardized mean difference parameter is defined as the difference between the mean level of the outcome in phase B and the mean level of the outcome in phase A, scaled by the within-case standard deviation of the outcome in phase A. As with all functions discussed so far, the SMD() function accepts data in either the A_data, B_data format or the condition, outcome format with optional baseline phase specification. In addition, direction of improvement can be specified as discussed above, with “increase” being the default. Changing the direction of the improvement does not change the magnitude of the effect size, but does change its sign:

SMD(A_data = A, B_data = B, improvement = "increase")
#>    ES      Est        SE  CI_lower CI_upper baseline_SD
#> 1 SMD 1.649932 0.6340935 0.4071314 2.892732    2.503331

SMD(A_data = A, B_data = B, improvement = "decrease")
#>    ES       Est        SE  CI_lower   CI_upper baseline_SD
#> 1 SMD -1.649932 0.6340935 -2.892732 -0.4071314    2.503331

The std_dev argument controls whether the effect size estimate is based on the standard deviation of the baseline phase alone (the default, std_dev = "baseline"), or based on the standard deviation after pooling across both phases (std_dev = "pool"):

SMD(A_data = A, B_data = B, std_dev = "baseline")
#>    ES      Est        SE  CI_lower CI_upper baseline_SD
#> 1 SMD 1.649932 0.6340935 0.4071314 2.892732    2.503331
SMD(A_data = A, B_data = B, std_dev = "pool")
#>    ES      Est        SE  CI_lower CI_upper pooled_SD
#> 1 SMD 1.876247 0.6374216 0.6269241 3.125571  2.431752

By default the SMD() function uses the Hedges’ g bias correction for small sample sizes. The bias correction can be turned off by specifying the argument bias_correct = FALSE.

The SMD() function also produces a 95% confidence interval by default. This can be adjusted by setting the confidence argument to a value between 0 and 1. To omit the confidence interval all together, set the argument to confidence = NULL.

Further options for log response ratios (LRRi() and LRRd())

The response ratio parameter is the ratio of the mean level of the outcome during phase B to the mean level of the outcome during phase A. The log response ratio is the natural logarithm of the response ratio. This effect size is appropriate for outcomes measured on a ratio scale, such that zero corresponds to the true absence of the outcome.

Direction of improvement

The package includes two versions of the LRR:

• LRR-increasing (LRRi()) is defined so that positive values correspond to therapeutic improvements

• LRR-decreasing (LRRd()) is defined so that negative values correspond to therapeutic improvements.

If you are estimating an effect size for a single series, pick the version of LRR that corresponds to the therapeutic improvement expected for your dependent variable. Similarly, if you are estimating effect sizes for a set of SCD series with the same therapeutic direction, pick the version that corresponds to your intervention’s expected change.

If you are estimating effect sizes for interventions where the direction of improvement depends upon the series or study, the choice between LRRi and LRRd is slightly more involved.

For example, imagine we have ten studies to meta-analyze. For eight studies, the outcome are initiations of peer interaction, so therapeutic improvements correspond to increases in behavior. For the other two studies, the outcomes were episodes of verbal aggression towards peers, so the therapeutic direction was a decrease. In this context it would be sensible to pick the LRRi() function, because most of the outcomes are positively valenced. For the final two studies, we would specify improvement = "decrease", which would ensure that the sign and magnitude of the outcomes were consistent with the direction of therapeutic improvement (i.e. a larger log-ratio represents a larger change in the desired direction). Conversely, if most of the outcomes had a negative valence and only a few had a positive valence, then we would use LRRd() and we would specify improvement = "increase" for the few series that had positive-valence outcomes.

Outcome scale

LRR differs from other effect size indices for single-case designs in that calculating it involves some further information about how the outcome variable was measured. One important piece of information to know is the scale of the outcome measurements. For outcomes that are measured by frequency counting, the scale might be expressed as a raw count (scale = "count") or as a standardized rate per minute (scale = "rate"). For outcomes that are measures of state behavior, where the main dimension of interest is the proportion of time that the behavior occurs, the scale might be expressed as a percentage (ranging from 0 to 100%; scale = "percentage") or as a proportion (ranging from 0 to 1; scale = "proportion"). For outcomes that don’t fit into any of these categories, set scale = "other".

The scale of the outcome variable has two important implications for how log response ratios are estimated. First, outcomes measured as percentages or proportions need to be coded so that the direction of therapeutic improvement is consistent with the direction of the effect size. Consequently, changing the improvement direction will alter the magnitude, in addition to the sign, of the effect size (see Pustejovsky, 2018, pp. 16–18 for further details). Here is an example:

A <- c(20, 20, 26, 25, 22, 23)
B <- c(28, 25, 24, 27, 30, 30, 29)

LRRi(A_data = A, B_data = B, scale = "percentage")
#>     ES       Est         SE   CI_lower  CI_upper
#> 1 LRRi 0.1953962 0.05557723 0.08646679 0.3043255

LRRi(A_data = A, B_data = B, improvement = "decrease", scale = "percentage")
#>     ES         Est         SE   CI_lower    CI_upper
#> 1 LRRi -0.06553504 0.01810144 -0.1010132 -0.03005687

Assuming that improvements correspond to increases, the LRRi value is positive and equal to 0.2. Assuming that improvements correspond to decreases, the LRRi value is negative and smaller in magnitude, equal to -0.07.

Note that if the outcome is a count (the default for both LRR functions) or rate, changing the improvement direction merely changes the sign of the effect size:

A <- c(20, 20, 26, 25, 22, 23)
B <- c(28, 25, 24, 27, 30, 30, 29)

LRRi(A_data = A, B_data = B, scale = "count")
#>     ES       Est         SE   CI_lower  CI_upper
#> 1 LRRi 0.1953962 0.05557723 0.08646679 0.3043255
LRRi(A_data = A, B_data = B, scale = "count", improvement = "decrease")
#>     ES        Est         SE   CI_lower    CI_upper
#> 1 LRRi -0.1953962 0.05557723 -0.3043255 -0.08646679

The scale of the outcome has one further important implication. To account for the possibility of a sample mean of zero, the LRRd() and LRRi() functions use a truncated sample mean, where the truncation level is determined by the scale of the outcome and some further details of how the outcomes were measured. For rates, the truncated mean requires specifying the length of the observation session in minutes:

A <- c(0, 0, 0, 0)
B <- c(28, 25, 24, 27, 30, 30, 29)
LRRd(A_data = A, B_data = B, scale = "rate")
#>     ES Est  SE CI_lower CI_upper
#> 1 LRRd NaN NaN      NaN      NaN
LRRd(A_data = A, B_data = B, scale = "rate", observation_length = 30)
#>     ES      Est        SE CI_lower CI_upper
#> 1 LRRd 8.672947 0.5010548 7.690897 9.654996

If no additional information is provided and there is a sample mean of 0, the function returns a value of NaN.

For outcomes specified as percentages or proportions, the argument intervals must be supplied. For interval recording methods such as partial interval recording or momentary time sampling, provide the number of intervals. For continuous recording, set intervals equal to 60 times the length of the observation session in minutes:

LRRd(A_data = A, B_data = B, scale = "percentage")
#>     ES Est  SE CI_lower CI_upper
#> 1 LRRd NaN NaN      NaN      NaN
LRRd(A_data = A, B_data = B, scale = "percentage", intervals = 180)
#>     ES      Est        SE CI_lower CI_upper
#> 1 LRRd 5.859536 0.5010548 4.877487 6.841586

You can also specify your own value for the constant used to truncate the sample mean using the D_const argument. If a vector, the mean will be used.

Additional arguments

Both LRR functions return a effect size that has been bias-corrected for small sample sizes by default. To omit the bias correction, set bias_correct = FALSE. Finally, as with the non-overlap measures, the confidence argument can be used to change the default 95% confidence interval, or set to NULL to omit confidence interval calculations.

Further options for LOR()

The odds ratio parameter is the ratio of the odds that the outcome occurs during phase B to the odds that the outcome occurs during phase A. The log-odds ratio (LOR) is the natural logarithm of the odds ratio. This effect size is appropriate for outcomes measured on a percentage or proportion scale. The LOR() function works almost identically to the LRRi() and LRRd() functions, but there are a few exceptions.

The LOR() function only works with outcomes that are on proportion or percentage scales:

A_pct <- c(20, 20, 25, 25, 20, 25)
B_pct <- c(30, 25, 25, 25, 35, 30, 25)

LOR(A_data = A_pct, B_data = B_pct, scale = "percentage")
#>    ES       Est         SE   CI_lower  CI_upper
#> 1 LOR 0.2852854 0.09790282 0.09339935 0.4771713

LOR(A_data = A_pct/100, B_data = B_pct/100, scale = "proportion")
#>    ES       Est         SE   CI_lower  CI_upper
#> 1 LOR 0.2852854 0.09790282 0.09339935 0.4771713

LOR(A_data = A_pct, B_data = B_pct, scale = "count")
#> Warning: LOR can only be calculated for proportions or percentages. It will
#> return NAs for other outcome scales.
#>    ES Est SE CI_lower CI_upper
#> 1 LOR  NA NA       NA       NA

LOR(A_data = A_pct, B_data = B_pct, scale = "proportion")
#> Error in `map()`:
#> ℹ In index: 1.
#> Caused by error in `calc_LOR()`:
#> ! Proportions must be between 0 and 1!

As with the LRR functions, LOR() includes an argument to specify the direction of therapeutic improvement, with the default assumption being that a therapeutic improvement is an increase in the behavior. In contrast to LRRi and LRRd, changing the direction of therapeutic improvement only reverses the sign of the LOR, but does not change its absolute magnitude:

LOR(A_data = A_pct, B_data = B_pct,
    scale = "percentage", improvement = "increase")
#>    ES       Est         SE   CI_lower  CI_upper
#> 1 LOR 0.2852854 0.09790282 0.09339935 0.4771713

LOR(A_data = A_pct, B_data = B_pct,
    scale = "percentage", improvement = "decrease")
#>    ES        Est         SE   CI_lower    CI_upper
#> 1 LOR -0.2852854 0.09790282 -0.4771713 -0.09339935

Similar to the LRR functions, LOR() will be calculated using truncated sample means for cases where phase means are close to the extremes of the scale. To use truncated means, the number of intervals per observation session must be specified using the intervals argument:

LOR(A_data = c(0,0,0), B_data = B_pct,
   scale = "percentage")
#>    ES Est  SE CI_lower CI_upper
#> 1 LOR NaN NaN      NaN      NaN
LOR(A_data = c(0,0,0), B_data = B_pct,
    scale = "percentage", intervals = 20)
#>    ES     Est       SE CI_lower CI_upper
#> 1 LOR 3.60657 0.676328 2.280992 4.932149

For data measured using continuous recording, set the number of intervals equal to 60 times the length of the observation session in minutes. Just like the LRR functions, it is possible to specify your own truncation constant using the D_const argument. By default the LOR() function uses a bias correction for small sample sizes, but this can be turned off by specifying the argument bias_correct = FALSE. The width of the confidence intervals is controlled via the confidence argument; set the argument to confidence = NULL to omit the confidence interval calculations.

calc_ES()

The calc_ES() function will calculate multiple effect sizes estimates for a single SCD series. Just as with the individual effect size functions, calc_ES() accepts data in either the A_data, B_data format or the condition, outcome format. Here we use the A_data, B_data format:

A <- c(20, 20, 26, 25, 22, 23)
B <- c(28, 25, 24, 27, 30, 30, 29)
calc_ES(A_data = A, B_data = B, ES = c("NAP","PND","Tau-U"))
#>      ES       Est         SE  CI_lower  CI_upper
#> 1   NAP 0.9166667 0.06900656 0.5973406 0.9860176
#> 2   PND 0.7142857         NA        NA        NA
#> 3 Tau-U 0.7380952         NA        NA        NA

Here is the same calculation in the condition, outcome format:

phase <- c(rep("A", length(A)), rep("B", length(B)))
outcome <- c(A, B)
calc_ES(condition = phase, outcome = outcome, baseline_phase = "A", 
        ES = c("NAP","PND","Tau-U"))
#>      ES       Est         SE  CI_lower  CI_upper
#> 1   NAP 0.9166667 0.06900656 0.5973406 0.9860176
#> 2   PND 0.7142857         NA        NA        NA
#> 3 Tau-U 0.7380952         NA        NA        NA

To specify which effect size to calculate, use the ES argument, which can include any of the following metrics: "LRRd", "LRRi", "LOR", "LRM", "SMD", "NAP", "PND", "PEM", "PAND", "IRD", "Tau", "Tau_BC" or "Tau-U".

calc_ES(A_data = A, B_data = B, ES = "SMD")
#>    ES      Est        SE  CI_lower CI_upper baseline_SD
#> 1 SMD 1.649932 0.6340935 0.4071314 2.892732    2.503331

To calculate multiple effect size estimates, provide a list of effect sizes to the ES argument.

calc_ES(A_data = A, B_data = B, ES = c("NAP", "PND", "Tau-U"))
#>      ES       Est         SE  CI_lower  CI_upper
#> 1   NAP 0.9166667 0.06900656 0.5973406 0.9860176
#> 2   PND 0.7142857         NA        NA        NA
#> 3 Tau-U 0.7380952         NA        NA        NA

Setting ES = "all" will return all available effect sizes:

calc_ES(A_data = A, B_data = B, ES = "all")
#> Error in `map()`:
#> ℹ In index: 11.
#> Caused by error in `calc_PoGO()`:
#> ! argument "goal" is missing, with no default

Setting ES = "NOM" will return all of the non-overlap measures.

calc_ES(A_data = A, B_data = B, ES = "NOM")
#>       ES       Est         SE   CI_lower  CI_upper
#> 1    NAP 0.9166667 0.06900656  0.5973406 0.9860176
#> 2    IRD 0.6904762         NA         NA        NA
#> 3   PAND 0.8461538         NA         NA        NA
#> 4    PND 0.7142857         NA         NA        NA
#> 5    PEM 1.0000000         NA         NA        NA
#> 6    Tau 0.8333333 0.13801311  0.1946812 0.9720352
#> 7  Tau-U 0.7380952         NA         NA        NA
#> 8 Tau-BC 0.2857143 0.35951593 -0.3260702 0.7180613

Setting ES = "parametric" will return all of the parametric effect sizes:

calc_ES(A_data = A, B_data = B, ES = "parametric")
#> Error in `map()`:
#> ℹ In index: 6.
#> Caused by error in `calc_PoGO()`:
#> ! argument "goal" is missing, with no default

If the ES argument is omitted, calc_ES() will return LRRd, LRRi, SMD, and Tau by default.

calc_ES(A_data = A, B_data = B)
#>     ES        Est         SE    CI_lower    CI_upper baseline_SD
#> 1 LRRd -0.1953962 0.05557723 -0.30432554 -0.08646679          NA
#> 2 LRRi  0.1953962 0.05557723  0.08646679  0.30432554          NA
#> 3  SMD  1.6499319 0.63409351  0.40713144  2.89273232    2.503331
#> 4  Tau  0.8333333 0.13801311  0.19468122  0.97203517          NA

Further arguments

All of the individual effect size functions have the further argument improvement, and several of them also have further optional arguments. Include these arguments in calc_ES() in order to pass them on to the individual effect size calculation functions. Any additional arguments included in calc_ES() will be used in the calculation of effect sizes for which they are relevant, but will be ignored if they are not relevant. For example, the direction of improvement can be changed from the default increase to decrease:

calc_ES(A_data = A, B_data = B, ES = "NOM", improvement = "decrease")
#>       ES         Est         SE    CI_lower   CI_upper
#> 1    NAP  0.08333333 0.06900656  0.01398242  0.4026594
#> 2    IRD  0.07142857         NA          NA         NA
#> 3   PAND  0.53846154         NA          NA         NA
#> 4    PND  0.00000000         NA          NA         NA
#> 5    PEM  0.00000000         NA          NA         NA
#> 6    Tau -0.83333333 0.13801311 -0.97203517 -0.1946812
#> 7  Tau-U -0.73809524         NA          NA         NA
#> 8 Tau-BC -0.28571429 0.35951593 -0.71806125  0.3260702

It is also possible to change the method for calculating the standard error for the NAP, Tau, and Tau_BC functions, as well as the coverage of the confidence interval. For example, to omit the confidence interval calculations for NAP and Tau, we can include the argument confidence = NULL:

calc_ES(A_data = A, B_data = B, ES = "NOM", improvement = "decrease", confidence = NULL)
#>       ES         Est         SE
#> 1    NAP  0.08333333 0.06900656
#> 2    IRD  0.07142857         NA
#> 3   PAND  0.53846154         NA
#> 4    PND  0.00000000         NA
#> 5    PEM  0.00000000         NA
#> 6    Tau -0.83333333 0.13801311
#> 7  Tau-U -0.73809524         NA
#> 8 Tau-BC -0.28571429 0.35951593

For SMD() there are several other inputs such as std_dev, bias_correct, and confidence which control how the effect size estimate is calculated, the usage of the Hedges’ g bias correction for small sample sizes, and the coverage of the confidence interval. The log response ratio and log odds ratio functions also include arguments for the outcome scale on which the input scores are measured and optional entries for session lengths and intervals. All of these additional options are discussed in more depth in the first section of this vignette.

Long vs. wide format

Finally, calc_ES() includes an option to change the format of the output. The function defaults to format = "long"; setting format = "wide" will return all of the results as a single line, rather than one line per effect size:

calc_ES(A_data = A, B_data = B, ES = c("NAP","PND","SMD"))
#>    ES       Est         SE  CI_lower  CI_upper baseline_SD
#> 1 NAP 0.9166667 0.06900656 0.5973406 0.9860176          NA
#> 2 PND 0.7142857         NA        NA        NA          NA
#> 3 SMD 1.6499319 0.63409351 0.4071314 2.8927323    2.503331

calc_ES(A_data = A, B_data = B, ES = c("NAP","PND","SMD"), format = "wide")
#>     NAP_Est     NAP_SE NAP_CI_lower NAP_CI_upper   PND_Est  SMD_Est    SMD_SE
#> 1 0.9166667 0.06900656    0.5973406    0.9860176 0.7142857 1.649932 0.6340935
#>   SMD_CI_lower SMD_CI_upper SMD_baseline_SD
#> 1    0.4071314     2.892732        2.503331

batch_calc_ES()

Most single-case studies include multiple cases, and many also include multiple dependent variables measured on each case. Thus, it will often be of interest to calculate effect size estimates for multiple data series from a study, or even from multiple studies. The batch_calc_ES() function does exactly this—calculating any of the previously detailed effect sizes for each of several data series. Its syntax is a bit more involved than the previous functions, and so we provide several examples here. In what follows, we will assume that you are already comfortable using the es_calc() function as well as the other individual effect size functions in the package.

Data organization

Unlike with the other functions in the package, the input data for batch_calc_ES() must be organized in a data frame, with one line corresponding to each observation within a series, and columns corresponding to different variables (e.g. outcome, phase, session number). One or more variables must be included that uniquely identify every data series. Let’s look at two examples.

McKissick

The McKissick dataset is data drawn from McKissick, Hawkins, Lentz, Hailley, & McGuire (2010), a single-case design study of a group contingency intervention. The study used a multiple baseline design across three classrooms. The outcome data are event counts of disruptive behaviors observed at the classroom level.

data(McKissick)

Here are the first few rows of the data:

Schmidt (2007)

The Schmidt2007 dataset are data drawn from Schmidt (2007). This data set is somewhat more complicated. It has two outcomes for each participant, and the outcomes differ in directions of therapeutic improvement and measurement scale. The study used an ABAB design, replicated across three participants. Each series therefore has four phases: a baseline phase, a treatment phase, a return to baseline phase, and a second treatment phase.

data(Schmidt2007)

Here are the first few rows of the data

The Schmidt (2007) dataset contains many variables, but for now let’s focus on the following:

• Case_Pseudonym uniquely identifies each of the three participants
• Behavior_type specifies whether the outcome is disruptive behavior or on-task behavior
• Session_number specifies the order of the sessions within each data series
• Outcome contains the dependent variable measurements
• Condition specifies whether the outcome is in a baseline (“A”) condition or a treatment (“B”) condition
• Phase_num specifies whether the session is in the first or second pair of phases in the design
• Metric specifies whether the dependent variable is percentage or count data
• Session_length specifies the length of the observation session
• direction specifies the direction of therapeutic improvement
• n_Intervals specifies the number of intervals per session for the dependent variable measured using partial interval recording.

Main arguments of batch_calc_ES()

Here are the arguments for the batch calculator function:

args(batch_calc_ES)
#> function (dat, grouping, condition, outcome, aggregate = NULL, 
#>     weighting = "equal", session_number = NULL, baseline_phase = NULL, 
#>     intervention_phase = NULL, ES = c("LRRd", "LRRi", "SMD", 
#>         "Tau"), improvement = "increase", scale = "other", intervals = NA, 
#>     observation_length = NA, goal = NULL, confidence = 0.95, 
#>     format = "long", warn = TRUE, ...) 
#> NULL

This function has a lot of arguments, but many of them are optional and only used for certain effect size metrics (these options are described in more detail in previous sections). For the moment, let’s focus on the first few arguments, which are all we need to get going.

• The argument dat should be a dataframe containing all of the observations for all of the data series of interest.

• The grouping argument should specify the set of variables that uniquely identify each series. For a single study consisting of several series, like the McKissick dataset, this might simply be a variable name that identifies the participant pseudonym. Specify using bare variable names (i.e., without quotes).

• The condition argument should be the variable that identifies the treatment condition for each observation in the series. Specify using a bare variable name. The values for the baseline and treatment phases should be uniform across all of the series within a dataset. That is, if some series are coded as “0” for baseline and “1” for treatment, whereas other series had “A” as baseline and “B” as treatment, you will first need to clean you data and standardize the coding.

• The outcome argument should be the variable that contains the outcomes of interest. Specify using a bare variable name.

• The ES argument allows you to specify which effect sizes to calculate. By default, the batch calculator generates estimates of LRRd, LRRi, SMD, and Tau. However, you’re probably going to want to specify your own effect sizes. Just as in calc_ES, specify your desired effect sizes as a character vector, with the individual options of "LRRd", "LRRi", "LOR", "LRM", "SMD", "NAP", "PND", "PEM", "PAND", "IRD", "Tau", "Tau_BC" or "Tau-U", in addition to "all" for all effect sizes, "NOM" for all non-overlap measures, and "parametric" for all parametric effect sizes.

All of the remaining arguments are truly optional, and we’ll introduce them as we go along.

Using grouping variables

Let’s try applying the function to the McKissick data. Remember that these data contains an identifier for each case (Case_pseudonym), a variable (Condition) identifying the baseline (“A”) and treatment (“B”) phases, and an outcome variable containing the values of the outcomes. The outcomes are disruptive behaviors, so a decrease in the behavior corresponds to therapeutic improvement. Just as with the calc_ES() function, we’ll need to specify the direction of therapeutic improvement using the improvement argument. In this example, we will calculate estimates of NAP and PND, to keep things simple:

mckissick_ES <- batch_calc_ES(dat = McKissick,
              grouping = Case_pseudonym, 
              condition = Condition,
              outcome = Outcome, 
              improvement = "decrease",
              ES = c("NAP", "PND"))

The output will always start with one or more columns corresponding to each unique combination of values from the grouping argument, followed by a column describing the effect size reported in each row. The column called Est contains the effect size estimates. If any of the requested effect sizes have standard errors and confidence intervals, there will also be columns corresponding to the standard error and the upper and lower limit. Here, PND has NA for each of those, because it does not have a known standard error or confidence interval.

Now let’s look at an example using the Schmidt data. Remember that these data contain a pseudonym that uniquely identifies each of the three participants (Case_Pseudonym) as well as a variable that specifies whether the outcome is disruptive behavior or on-task behavior (Behavior_type). Furthermore, these data come from a treatment reversal design with two pairs of AB phases for each combination of case and behavior type. Each pair of AB phases is labeled in the variable Phase_num. We’re going to want an effect size for each combination of pseudonym, behavior, and phase pair. The data also have an outcome variable (Outcome) and a variable identifying whether it was in the baseline (“A”) or treatment (“B”) phase (Condition). Finally, the the two different behavior types have different direction therapeutic improvement, so there is a variable called direction that specifies "increase" for on-task behavior or "decrease" for disruptive behavior.

Here’s an example of how to calculate NAP and LRRi for these data:

schmidt_ES <- batch_calc_ES(
  dat = Schmidt2007,
  grouping = c(Case_pseudonym, Behavior_type, Phase_num), 
  condition = Condition,
  outcome = Outcome, 
  improvement = direction,
  ES = c("NAP", "LRRi")
)

The syntax is similar to the example with the McKissick dataset, except for two things. Here, we’ve provided a vector of variable names for grouping that identify each series for which we want an effect size. Instead of providing a uniform direction of improvement to the improvement variable, we’ve provided a variable name, direction, which will account for the fact that the two behavior types have different directions of therapeutic improvement. Here is a table of the output:

The first three columns are the unique values from the variables supplied to grouping, followed by the effect size information.

Aggregating across replications

The Schmidt study used an ABAB design, and as a consequence we end up with not one but two effect size estimates for each case and each outcome. Under some circumstances, it may make sense to aggregate—or average together—the effect size estimates from the first and second AB pairs for each case. Doing so simplifies the structure of the resulting effect size dataset, so that there is just one effect size estimate per case per outcome. The batch_calc_ES function includes an optional argument called aggregate that allows you to aggregate effect size estimates across a grouping variable. To use it, specify the name of one or more variables across which to aggregate. These variables will then be treated as grouping variables for purposes of effect size calculation (just like those specified in the grouping argument), but the results will then be aggregated over the unique values of the variables.

Here’s an example of how to use aggregate with the Schmidt dataset (for simplicity, we will calculate only the NAP effect size). Rather than specifying Phase_num as a grouping variable, we specify it as an aggregate variable:

schmidt_ES_agg <- 
  batch_calc_ES(
    dat = Schmidt2007,
    grouping = c(Case_pseudonym, Behavior_type),
    aggregate = Phase_num,
    condition = Condition,
    outcome = Outcome, 
    improvement = direction,
    ES = "NAP"
  )

The package allows for several different weighting schemes:

• "equal" (the default) or "Equal": Equal weighting takes the simple arithmetic average of the effect size estimates.
• "1/V": Inverse variance weighting takes a weighted average of the effect size estimates with weights that are inversely proportional to the sampling variances of the estimates (i.e., the square of the standard error). This weighting scheme is the most efficient approach if the components being averaged together are all estimating the same underlying parameter. However, inverse variance weighting will not work for effect size estimates that do not have a known standard error, such as PND or PAND.
• "nA" or "n_A": uses the number of baseline phase observations as the weights for aggregating.
• "nB" or "n_B": uses the number of treatment phase observations as the weights for aggregating.
• "nAnB", "nA*nB", "nA * nB", "n_A*n_B", or "n_A * n_B": uses the product of the number of baseline and treatment phases as the weights for aggregating.
• "1/nA+1/nB", "1/nA + 1/nB", "1/n_A+1/n_B", or "1/n_A + 1/n_B": uses the sum of the inverse number of baseline phases and the inverse number of treatment phases as the weights for aggregating.

Here is an example of using equal weighting for calculating aggregated effect sizes across pairs of AB phases:

schmidt_ES_agg <- 
  batch_calc_ES(
    dat = Schmidt2007,
    grouping = c(Case_pseudonym, Behavior_type),
    aggregate = Phase_num,
    weighting = "equal",
    condition = Condition,
    outcome = Outcome, 
    improvement = direction,
    ES = "NAP"
  )

Dealing with outcome scales.

By default, the batch calculator assumes the outcome scale is "other". If using this default assumption, the log odd ratio and the log response ratio will not be calculated if a phase mean is equal to zero. Just as with calc_ES(), you may need to specify the outcome scales as well as things like the length of the observation session or the number of intervals in each observation session in order to calculate parametric effect sizes. If these values are the same for all observations in the dataset, you can specify them as further arguments to batch_calc_ES(). Here is an example using the McKissick dataset, where we specify that all of the outcomes are measured as counts during 20-minute observation periods:

mckissick_ES <- batch_calc_ES(dat = McKissick,
              grouping = Case_pseudonym, 
              condition = Condition,
              outcome = Outcome, 
              improvement = "decrease",
              scale = "count",
              observation_length = 20,
              ES = "parametric")
#> Error in batch_calc_ES(dat = McKissick, grouping = Case_pseudonym, condition = Condition, : You must provide the goal level of the behavior to calculate the PoGO effect size.

Once again, we have a column specifying the case to which the effect sizes correspond, as well as a column specifying the effect size metric. The log odds ratio returns all NAs, because the log odds ratio can’t be estimate for count outcomes.

Let’s suppose that we are interested in estimating effect sizes using data where the measurement scale—as well as perhaps measurement details like the observation length or the number of intervals—varies depending on the data series. The Schmidt data is one example of this. Remember that the Schmidt data has a variable specifying the measurement scale of the outcome (Metric) which is "percentage" for desirable behavior and "count" for disruptive behaviors. It also has a variable that specifies the length of the observation session (Session_length), and a variable that specifies the number of intervals per session for the dependent variable measured using partial interval recording (n_Intervals). The value of Session_length is NA for the percentage outcomes and the value of n_Intervals is NA for the count outcomes because those details are not relevant for those outcome measurement scales. Let’s try it out:

schmidt_ES <- batch_calc_ES(dat = Schmidt2007,
              grouping = c(Case_pseudonym, Behavior_type, Phase_num), 
              condition = Condition,
              outcome = Outcome, 
              improvement = direction,
              scale = Metric,
              observation_length = Session_length,
              intervals = n_Intervals,
              ES = c("parametric"))
#> Error in batch_calc_ES(dat = Schmidt2007, grouping = c(Case_pseudonym, : You must provide the goal level of the behavior to calculate the PoGO effect size.

Unlike the previous example, where we specified a uniform value for the scale and observation_length, we now have to specify variable names for scale, observation_length, and the number of intervals. Note that we get some warnings again about the LOR effect size. Let’s take a look at the output:

In this case, LOR is all NA for the outcomes that are disruptive behaviors because those are counts and therefore the LOR isn’t an appropriate effect size. However, for the percentage of on task behavior, the LOR was estimated.

Further arguments

Output format

We can also request the effect sizes in a wide format:

mckissick_wide_ES <- 
  batch_calc_ES(
    dat = McKissick,
    grouping = Case_pseudonym, 
    condition = Condition,
    outcome = Outcome, 
    improvement = "decrease",
    ES = c("NAP", "PND"),
    format = "wide"
  )

The default argument for the batch calculator is format = "long", but if you want each case to be on a single line, specifying format = "wide" will provide the output that way, just like calc_ES(). Here’s the output:

In this case there are columns for NAP, NAP’s standard error, and the upper and lower bounds of the confidence interval. PND only has a column for the estimate, but remember that the values for SE and upper and lower CI were all NA in the long format. Columns that would have all NA values are removed when specifying format = "wide".

Suppressing warnings

Remember how, when we asked for the LOR for counts, the calculator gave us a bunch of warning messages? If you’re asking for the LOR, and some of your outcomes are in a scale other than percentage or proportion, you can specify the argument warn = FALSE (by default it is set to TRUE) if you want to suppress the warning messages. You will still get NA for any series with an inappropriate outcome scale.

batch_calc_ES(dat = McKissick,
              grouping = Case_pseudonym, 
              condition = Condition,
              outcome = Outcome, 
              improvement = "decrease",
              scale = "count",
              observation_length = 20,
              ES = c("LRRi","LOR"),
              warn = FALSE)
#> # A tibble: 6 × 6
#>   Case_pseudonym ES       Est     SE CI_lower CI_upper
#>   <chr>          <chr>  <dbl>  <dbl>    <dbl>    <dbl>
#> 1 Period 1       LRRi   0.807  0.198   0.419      1.19
#> 2 Period 1       LOR   NA     NA      NA         NA   
#> 3 Period 2       LRRi   0.610  0.349  -0.0736     1.29
#> 4 Period 2       LOR   NA     NA      NA         NA   
#> 5 Period 3       LRRi   0.748  0.353   0.0550     1.44
#> 6 Period 3       LOR   NA     NA      NA         NA

Arguments to individual effect size functions

The ... argument allows you to specify arguments particular to an individual function such as std_dev for the SMD() function. For instance, compare the results of calculating a pooled SMD versus the default, baseline phase only SMD:

batch_calc_ES(dat = McKissick, 
              grouping = Case_pseudonym, 
              condition = Condition,
              outcome = Outcome, 
              ES = "SMD", 
              improvement = "decrease")
#> # A tibble: 3 × 7
#>   Case_pseudonym ES      Est    SE CI_lower CI_upper baseline_SD
#>   <chr>          <chr> <dbl> <dbl>    <dbl>    <dbl>       <dbl>
#> 1 Period 1       SMD    2.75 0.943   0.906      4.60        1.63
#> 2 Period 2       SMD    1.21 0.650  -0.0633     2.48        5.58
#> 3 Period 3       SMD    2.89 1.08    0.763      5.01        2.33

batch_calc_ES(dat = McKissick, 
              grouping = Case_pseudonym, 
              condition = Condition,
              outcome = Outcome, 
              ES = "SMD", 
              improvement = "decrease",
              std_dev = "pool")
#> # A tibble: 3 × 7
#>   Case_pseudonym ES      Est    SE CI_lower CI_upper pooled_SD
#>   <chr>          <chr> <dbl> <dbl>    <dbl>    <dbl>     <dbl>
#> 1 Period 1       SMD    2.58 0.853   0.909      4.25      2.74
#> 2 Period 2       SMD    1.12 0.588  -0.0345     2.27      6.97
#> 3 Period 3       SMD    2.34 0.727   0.920      3.77      2.95

Arguments common to several functions will be used when calculating any of the effect sizes for which they are relevant. For example, the bias_correct argument applies to all of the parametric effect sizes:

batch_calc_ES(dat = McKissick, 
              grouping = Case_pseudonym, 
              condition = Condition,
              outcome = Outcome, 
              ES = "parametric", 
              improvement = "decrease",
              scale = Procedure, 
              observation_length = Session_length,
              bias_correct = FALSE,
              warn = FALSE)
#> Error in batch_calc_ES(dat = McKissick, grouping = Case_pseudonym, condition = Condition, : You must provide the goal level of the behavior to calculate the PoGO effect size.

The bias_correct argument cannot be specified differently for different effect size functions. If you want to obtain bias-corrected values for the LRRd effect size but not for the SMD effect size, you would need to call batch_calc_ES() separately for the two different effect sizes.

Order of observations

The session_number argument orders the data within each series by the specified variable. This argument is only important if baseline-corrected Tau or Tau-U is being calculated. For these effect sizes, the ordering of the baseline phase is important because they involve adjustments for trend in the baseline phase. This argument is irrelevant for all of the other effect sizes.

Specifying a baseline phase

The baseline_phase argument works the same was as in the calc_ES() function. If nothing is specified, the first phase in each series will be treated as the baseline phase. However, if the baseline phase is not always the first phase in each series, such as an SCD with four cases that use a cross-over treatment reversal design, where two of the cases follow an ABAB design and the other two cases follow a BABA design, you will need to specify the baseline_phase in the same way as in the calc_ES() function.

Confidence levels

The confidence argument controls the confidence intervals in the same way as all the other functions. To skip calculating confidence intervals, specify confidence = NULL:

batch_calc_ES(dat = McKissick, 
              grouping = Case_pseudonym, 
              condition = Condition,
              outcome = Outcome, 
              ES = "parametric", 
              improvement = "decrease",
              scale = Procedure, 
              observation_length = Session_length,
              confidence = NULL,
              warn = FALSE)
#> Error in batch_calc_ES(dat = McKissick, grouping = Case_pseudonym, condition = Condition, : You must provide the goal level of the behavior to calculate the PoGO effect size.

References