Tutorial: Delta-Delta

This vignette documents how dabestr is able to compute the calculation of delta-delta, an experimental function that allows the comparison between two bootstrapped effect sizes computed from two independent categorical variables.

Many experimental designs investigate the effects of two interacting independent variables on a dependent variable. The delta-delta effect size lets us distill the net effect of the two variables. To illustrate this, let’s delve into the following problem…

	Wild type	Mutant
Drug	\(X_D, W\)	\(X_D, M\)
Placebo	\(X_P, W\)	\(X_P, M\)

There are 2 Treatment conditions: the Placebo (control group) and the Drug (test group). There are 2 Genotypes: W (wild type population) and M (mutant population). Additionally, each experiment was conducted twice (Rep1 and Rep2). We will perform a few analyses to visualise these differences in a simulated dataset.

Create dataset for demo

set.seed(12345) # Fix the seed so the results are reproducible.
# pop_size = 10000 # Size of each population.
N <- 20 # The number of samples taken from each population

# Create samples
placebo <- rnorm(N / 2, mean = 4, sd = 0.4)
placebo <- c(placebo, rnorm(N / 2, mean = 2.8, sd = 0.4))
drug <- rnorm(N / 2, mean = 3, sd = 0.4)
drug <- c(drug, rnorm(N / 2, mean = 2.5, sd = 0.4))

# Add a `Genotype` column as the second variable
genotype <- c(rep("M", N / 2), rep("W", N / 2))

# Add an `id` column for paired data plotting.
id <- 1:N

# Add a `Rep` column as the first variable for the 2 replicates of experiments done
Rep <- rep(c("Rep1", "Rep2"), N / 2)

# Combine all columns into a DataFrame.
df <- tibble::tibble(
  Placebo = placebo,
  Drug = drug,
  Genotype = genotype,
  ID = id,
  Rep = Rep
)

df <- df %>%
  tidyr::gather(key = Treatment, value = Measurement, -ID, -Genotype, -Rep)

knitr::kable(head(df))

Genotype	ID	Rep	Treatment	Measurement
M	1	Rep1	Placebo	4.234211
M	2	Rep2	Placebo	4.283786
M	3	Rep1	Placebo	3.956279
M	4	Rep2	Placebo	3.818601
M	5	Rep1	Placebo	4.242355
M	6	Rep2	Placebo	3.272818

Loading Data

To make a delta-delta plot, you need to simply set delta2 = TRUE in the load() function. The colour parameter will be used to determine the colour of dots for the scattered plots or the colour of lines for the slopegraphs. The experiment parameter will be used to specify the grouping of the data. For delta-delta plots, the idx parameter is optional. Here’s an example:

Unpaired Data

unpaired_delta2 <- load(df,
  x = Genotype, y = Measurement,
  experiment = Treatment, colour = Genotype,
  delta2 = TRUE
)

The above function creates the following dabest object:

print(unpaired_delta2)
#> DABESTR v2025.3.14
#> ==================
#> 
#> Good morning!
#> The current time is 09:55 AM on Wednesday February 26, 2025.
#> 
#> ffect size(s) with 95% confidence intervals will be computed for:
#> 1. M Placebo minus W Placebo
#> 2. M Drug minus W Drug
#> 3. Drug minus Placebo (only for mean difference)
#> 
#> 5000 resamples will be used to generate the effect size bootstraps.

We can quickly check out the effect sizes:

unpaired_delta2.mean_diff <- mean_diff(unpaired_delta2)

print(unpaired_delta2.mean_diff)
#> DABESTR v2025.3.14
#> ==================
#> 
#> Good morning!
#> The current time is 09:55 AM on Wednesday February 26, 2025.
#> 
#> The character(0) mean difference between M Placebo and W Placebo is 1.032 [95%CI 0.731, 1.279].
#> The p-value of the two-sided permutation t-test is 0.0000, calculated for legacy purposes only.
#> 
#> The character(0) mean difference between M Drug and W Drug is 0.244 [95%CI -0.136, 0.666].
#> The p-value of the two-sided permutation t-test is 0.2695, calculated for legacy purposes only.
#> 
#> 5000 bootstrap samples were taken; the confidence interval is bias-corrected and accelerated.
#> Any p-value reported is the probability of observing the effect size (or greater),
#> assuming the null hypothesis of zero difference is true.
#> For each p-value, 5000 reshuffles of the control and test labels were performed.

dabest_plot(unpaired_delta2.mean_diff)

The horizontal axis in the above plot represents the Genotype condition, and the dot colour is also specified by Genotype. The left pair of scattered plots corresponds to the Placebo group, while the right pair is based on the Drug group. The bottom left axis contains the two primary deltas: the Placebo delta and the Drug delta.

It is evident that when only the placebo was administered, the mutant phenotype was around 1.23 [95% CI: 0.948, 1.52]. However, this difference was reduced to approximately 0.326 [95% CI: 0.0934, 0.584] when the drug was administered, indicating that the drug is effective in ameliorating the disease phenotype. Since the Drug did not completely eliminate the mutant phenotype, we need to calculate the net effect of the drug.

Delta-delta comes in handy in this situation. We use the Placebo delta as a reference for how much the mutant phenotype is supposed to be, and we subtract the Drug delta from it. The bootstrapped mean differences (delta-delta) between the Placebo and Drug group are plotted at the bottom right with a separate y-axis from other bootstrap plots. This effect size, at about -0.903 [95% CI: -1.28, -0.513], represents the net effect size of the drug treatment. In other words, treatment with drug A reduced the disease phenotype by 0.903.

The mean difference between mutants and wild types given the placebo treatment is:

\[\Delta_1 = \bar{X}_{P,M}-\bar{X}_{P,W}\]

The mean difference between mutants and wild types given the drug treatment is:

\[\Delta_2 = \bar{X}_{D,M}-\bar{X}_{D,W}\] The net effect of the drug on mutants is:

\[\Delta_\Delta = \Delta_1 - \Delta_2\] where \(\bar{X}\) is the sample mean, \(\Delta\) is the mean difference.

Specifying Grouping for Comparisons

In the example above, we used the convention of “test - control’ but you can manipulate the orders of experiment groups as well as the horizontal axis variable by setting experiment_label and x1_level.

unpaired_delta2_specified.mean_diff <- load(df,
  x = Genotype, y = Measurement,
  experiment = Treatment, colour = Genotype,
  delta2 = TRUE,
  experiment_label = c("Drug", "Placebo"),
  x1_level = c("M", "W")
) %>%
  mean_diff()

dabest_plot(unpaired_delta2_specified.mean_diff)

Paired Data

The delta-delta function also supports paired data, which can be useful for visualizing the data in an alternative way. If the placebo and drug treatment were administered to the same subjects, our data is paired between the treatment conditions. We can specify this by using Treatment as x and Genotype as experiment. Additionally, we can link data from the same subject with each other by specifying ID as id_col.

Since we have conducted two replicates of the experiments, we can colour the slope lines according to Rep to differentiate between the replicates.

Although idx is an optional parameter, it can still be included as an input to adjust the order of the data as opposed to using experiment_label and x1_level.

paired_delta2.mean_diff <- load(df,
  x = Treatment, y = Measurement,
  experiment = Genotype, colour = Rep,
  delta2 = TRUE,
  idx = list(
    c("Placebo W", "Drug W"),
    c("Placebo M", "Drug M")
  ),
  paired = "baseline", id_col = ID
) %>%
  mean_diff()

dabest_plot(paired_delta2.mean_diff,
  raw_marker_size = 0.5, raw_marker_alpha = 0.3
)

We see that the drug had a non-specific effect of -0.125 [95%CI -0.486 , 0.214] on the wild type subjects even when they were not sick, and it had a bigger effect of -0.913 [95%CI -1.24 , -0.577] in the mutant subjects. In this visualisation, we can see the delta-delta value of -0.789 [95%CI -1.3 , -0.317] as the net effect of the drug accounting for non-specific actions in healthy individuals.

The mean difference between drug and placebo treatments in wild type subjects is:

\[\Delta_1 = \bar{X}_{D,M}-\bar{X}_{P,W}\]

The mean difference between drug and placebo treatments in mutant subjects is:

\[\Delta_2 = \bar{X}_{D,M}-\bar{X}_{P,W}\] The net effect of the drug on mutants is:

\[\Delta_\Delta = \Delta_2 - \Delta_1\] where \(\bar{X}\) is the sample mean, \(\Delta\) is the mean difference.

Connection to ANOVA

The comparison we conducted earlier is reminiscent of a two-way ANOVA. In fact, the delta-delta is an effect size estimated for the interaction term between Treatment and Genotype. On the other hand, main effects of Treatment and Genotype can be determined through simpler, univariate contrast plots.

Omitting Delta-delta Plot

If for some reason you don’t want to display the delta-delta plot, you can easily do so by setting show_delta2 to FALSE:

dabest_plot(unpaired_delta2.mean_diff, show_delta2 = FALSE)

Other Effect Sizes

Since the delta-delta function is only applicable to mean differences, plots of other effect sizes will not include a delta-delta bootstrap plot.

# cohens_d(unpaired_delta2)

Statistics

You can find all the outputs of the delta - delta calculation by assessing the column named boot_result of the object dabest_effectsize_obj.

print(unpaired_delta2.mean_diff$boot_result)
#> # A tibble: 3 × 11
#>   control_group   test_group bootstraps nboots bca_ci_low bca_ci_high pct_ci_low
#>   <chr>           <chr>      <list>      <int>      <dbl>       <dbl>      <dbl>
#> 1 W Placebo       M Placebo  <dbl>        5000      0.731       1.28       0.748
#> 2 W Drug          M Drug     <dbl>        5000     -0.136       0.666     -0.151
#> 3 Delta2 Overall… Delta2 Ov… <dbl>        5000     -1.20       -0.337     -0.799
#> # ℹ 4 more variables: pct_ci_high <dbl>, ci <dbl>, difference <dbl>,
#> #   weight <dbl>

If you want to extract the permutations, permutation test’s p values, the statistical tests and the p value results, you can access them using the columns permutation_test_results, pval_permtest, pval_for_tests and pvalues respectively.

For instance, the P values for permutation tests pval_permtest:

print(unpaired_delta2.mean_diff$permtest_pvals$pval_permtest)
#> [1] 0.0000 0.2628

Or the permutation calculations and results could be accessed by:

print(unpaired_delta2.mean_diff$permtest_pvals$permutation_test_results)

A representative p value for statistical tests pval_for_tests:

print(unpaired_delta2.mean_diff$permtest_pvals$pval_for_tests)
#> $pvalue_welch
#> [1] 1.650881e-06
#> 
#> $pvalue_welch
#> [1] 0.2694873

Finally here the statistical test results and pvalues:

print(unpaired_delta2.mean_diff$permtest_pvals$pvalues)
#> [[1]]
#> [[1]]$pvalue_welch
#> [1] 1.650881e-06
#> 
#> [[1]]$statistic_welch
#>         t 
#> -6.971633 
#> 
#> [[1]]$students_t
#> 
#>  Welch Two Sample t-test
#> 
#> data:  control and test
#> t = -6.9716, df = 17.979, p-value = 1.651e-06
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -1.3435832 -0.7212793
#> sample estimates:
#> mean of x mean of y 
#>  2.914391  3.946822 
#> 
#> 
#> [[1]]$pvalue_students_t
#> [1] 1.650881e-06
#> 
#> [[1]]$statistic_students_t
#>         t 
#> -6.971633 
#> 
#> [[1]]$pvalue_mann_whitney
#> [1] 0.0002461281
#> 
#> [[1]]$statistic_mann_whitney
#> W 
#> 1 
#> 
#> 
#> [[2]]
#> [[2]]$pvalue_welch
#> [1] 0.2694873
#> 
#> [[2]]$statistic_welch
#>         t 
#> -1.139429 
#> 
#> [[2]]$students_t
#> 
#>  Welch Two Sample t-test
#> 
#> data:  control and test
#> t = -1.1394, df = 17.968, p-value = 0.2695
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -0.6928931  0.2056392
#> sample estimates:
#> mean of x mean of y 
#>  2.789728  3.033355 
#> 
#> 
#> [[2]]$pvalue_students_t
#> [1] 0.2694873
#> 
#> [[2]]$statistic_students_t
#>         t 
#> -1.139429 
#> 
#> [[2]]$pvalue_mann_whitney
#> [1] 0.3447042
#> 
#> [[2]]$statistic_mann_whitney
#>  W 
#> 37