# Tutorial: Mini-Meta Delta

When scientists conduct replicates of the same experiment, the effect size of each replicate often varies, which can make interpreting the results more challenging. This vignette documents how `dabestr` is able to compute the meta-analyzed weighted effect size given multiple replicates of the same experiment. This feature can help resolve differences between replicates and simplify interpretation.

This function uses the generic inverse-variance method to calculate the effect size, as follows:

\[\theta_{weighted} = \frac{\sum{\hat{\theta_i}}w_i}{\sum{w_i}}\] where:

\[\hat{\theta_i}=\text{Mean difference for replicate } i\] \[w_i=\text{Weight for replicate } i = \frac{1}{s_i^2}\] \[s_i^2=\text{Pooled variance for replicate } i = \frac{(n_{test}-1)s_{test}^2 + (n_{control}-1)s_{control}^2} {n_{test}+n_{control}-2}\] \[n = \text{sample size and } s^2 = \text{variance for control/test}\] Note that `dabestr` uses the fixed-effects model of meta-analysis, as opposed to the random-effects model. This means that all variation between the results of each replicate is assumed to be due solely to sampling error. We recommend using this function only for replications of the same experiment, where it can be safely assumed that each replicate estimates the same population mean, \(\mu\).

The `dabestr` package can only compute weighted effect size for mean difference only, and not standardized measures such as Cohen’s d.

For more information on meta-analysis, please refer to Chapter 10 of the Cochrane handbook:

    library(dabestr)

## 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
    c1 <- rnorm(N, mean = 3, sd = 0.4)
    c2 <- rnorm(N, mean = 3.5, sd = 0.75)
    c3 <- rnorm(N, mean = 3.25, sd = 0.4)

    t1 <- rnorm(N, mean = 3.5, sd = 0.5)
    t2 <- rnorm(N, mean = 2.5, sd = 0.6)
    t3 <- rnorm(N, mean = 3, sd = 0.75)

    # Add a `gender` column for coloring the data.
    gender <- c(rep("Male", N / 2), rep("Female", N / 2))

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

    # Combine samples and gender into a DataFrame.
    df <- tibble::tibble(
      `Control 1` = c1, `Control 2` = c2, `Control 3` = c3,
      `Test 1` = t1, `Test 2` = t2, `Test 3` = t3,
      Gender = gender, ID = id
    )

    df <- df %>%
      tidyr::gather(key = Group, value = Measurement, -ID, -Gender)

We now have 3 Control groups and 3 Test groups, simulating 3 replicates of the same experiment. Our dataset also includes a non-numerical column indicating gender and another column indicating the identity of each observation.

This is known as a ‘long’ dataset. See this writeup for more details.

    knitr::kable(head(df))

Gender | ID | Group | Measurement
--- | --- | --- | ---
Male | 1 | Control 1 | 3.234211
Male | 2 | Control 1 | 3.283786
Male | 3 | Control 1 | 2.956279
Male | 4 | Control 1 | 2.818601
Male | 5 | Control 1 | 3.242355
Male | 6 | Control 1 | 2.272818

## Loading Data

Next, we load data as we would typically do using `load()`. This time, however, we also specify the argument `minimeta = TRUE`. As we are loading three experiments’ worth of data, `idx` is passed as a list of vectors, as shown below:

    unpaired <- load(df,
      x = Group, y = Measurement,
      idx = list(
        c("Control 1", "Test 1"),
        c("Control 2", "Test 2"),
        c("Control 3", "Test 3")
      ),
      minimeta = TRUE
    )

When this `dabest` object is printed, it should show that effect sizes will be calculated for each group, as well as the weighted delta. Note once again that the weighted delta will only be calculated for the mean difference.

    print(unpaired)
    #> DABESTR v2025.3.15
    #> ==================
    #>
    #> Good morning!
    #> The current time is 11:47 AM on Tuesday October 28, 2025.
    #>
    #> ffect size(s) with 95% confidence intervals will be computed for:
    #> 1. Test 1 minus Control 1
    #> 2. Test 2 minus Control 2
    #> 3. Test 3 minus Control 3
    #> 4. weighted delta (only for mean difference)
    #>
    #> 5000 resamples will be used to generate the effect size bootstraps.

After applying the `mean_diff()` function to the `dabest` object, you can view the mean differences for each group as well as the weighted delta by printing the `dabest_effectsize_obj`.

    unpaired.mean_diff <- mean_diff(unpaired)

    print(unpaired.mean_diff)
    #> DABESTR v2025.3.15
    #> ==================
    #>
    #> Good morning!
    #> The current time is 11:47 AM on Tuesday October 28, 2025.
    #>
    #> The character(0) mean difference between Test 1 and Control 1 is 0.585 [95%CI 0.307, 0.869].
    #> The p-value of the two-sided permutation t-test is 0.0004, calculated for legacy purposes only.
    #>
    #> The character(0) mean difference between Test 2 and Control 2 is -1.058 [95%CI -1.52, -0.577].
    #> The p-value of the two-sided permutation t-test is 0.0002, calculated for legacy purposes only.
    #>
    #> The character(0) mean difference between Test 3 and Control 3 is -0.254 [95%CI -0.626, 0.169].
    #> The p-value of the two-sided permutation t-test is 0.2167, 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.

You can view the details of each experiment by accessing `dabest_effectsize_obj$boot_results`, as shown below. This also contains details of the weighted delta.

    unpaired.mean_diff$boot_result
    #> # A tibble: 4 × 11
    #>   control_group   test_group bootstraps nboots bca_ci_low bca_ci_high pct_ci_low
    #>   <chr>           <chr>      <list>      <int>      <dbl>       <dbl>      <dbl>
    #> 1 Control 1       Test 1     <dbl>        5000      0.307       0.869     0.302
    #> 2 Control 2       Test 2     <dbl>        5000     -1.52       -0.577    -1.52
    #> 3 Control 3       Test 3     <dbl>        5000     -0.626       0.169    -0.645
    #> 4 Minimeta Overa… Minimeta … <dbl>        5000     -0.186       0.238     0.0355
    #> # ℹ 4 more variables: pct_ci_high <dbl>, ci <dbl>, difference <dbl>,
    #> #   weight <dbl>

## Unpaired Data

Simply calling the `dabest_plot()` function it will generate a Cumming estimation plot showing the data for each experimental replicate as well as the calculated weighted delta.

    dabest_plot(unpaired.mean_diff)

You can also hide the weighted delta by passing the argument `show_mini_meta = FALSE`. In this case, the resulting graph would be identical to a multiple two-groups plot:

    dabest_plot(unpaired.mean_diff, show_mini_meta = FALSE)

## Paired Data

The tutorial up to this point has dealt with unpaired data. If your data is paired data, the process for loading, plotting and accessing the data is the same as for unpaired data, except that you need to pass the argument `paired = "sequential"` or `paired = "baseline"` and an appropriate `id_col` during the `load()` step, as follows:

    paired.mean_diff <- load(df,
      x = Group, y = Measurement,
      idx = list(
        c("Control 1", "Test 1"),
        c("Control 2", "Test 2"),
        c("Control 3", "Test 3")
      ),
      paired = "baseline", id_col = ID,
      minimeta = TRUE
    ) %>%
      mean_diff()

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