Sign test for two-sample paired data interpretation and median differences

Question

I have paired data with measurement of time in seconds for unstructured type (variable clean_final_Unstructured_time_ID_only_aggregated_ordered$Time) and structured type (variable clean_final_Structured_time_ID_only_aggregated_ordered$Time).

First I calculate median differences:

(median_Unstructured_agreg <- median(clean_final_Unstructured_time_ID_only_aggregated_ordered$Time))
(median_Structured_agreg <-median(clean_final_Structured_time_ID_only_aggregated_ordered$Time))

median_Unstructured_agreg - median_Structured_agreg

output:

> (median_Unstructured_agreg <- median(clean_final_Unstructured_time_ID_only_aggregated_ordered$Time))
[1] 219
> (median_Structured_agreg <-median(clean_final_Structured_time_ID_only_aggregated_ordered$Time))
[1] 319.5
> 
> median_Unstructured_agreg - median_Structured_agreg
[1] -100.5
> 

After I am conducting "Sign Test for Two-sample Paired Data" (as described here: https://rcompanion.org/handbook/F_07.html)

library(BSDA)

SIGN.test(x = clean_final_Unstructured_time_ID_only_aggregated_ordered$Time,
          y = clean_final_Structured_time_ID_only_aggregated_ordered$Time,
          alternative = "two.sided",
          conf.level = 0.95)

output:

    Dependent-samples Sign-Test

data:  clean_final_Unstructured_time_ID_only_aggregated_ordered$Time and clean_final_Structured_time_ID_only_aggregated_ordered$Time
S = 4, p-value = 0.1185
alternative hypothesis: true median difference is not equal to 0
95 percent confidence interval:
 -224.23839   38.26659
sample estimates:
median of x-y 
          -99 

Achieved and Interpolated Confidence Intervals: 

                  Conf.Level    L.E.pt   U.E.pt
Lower Achieved CI     0.8815 -211.5000 -32.0000
Interpolated CI       0.9500 -224.2384  38.2666
Upper Achieved CI     0.9648 -227.0000  53.5000

As I understand median of x-y from my output is the same as "paired differences" from here: https://statistics.laerd.com/spss-tutorials/sign-test-using-spss-statistics.php and there they interpret that value as I understand difference of medians "The carbohydrate-protein drink elicited a statistically significant median increase in distance run (0.113 km) compared to the carbohydrate-only drink, p = .004."

Question:

1. Why the difference of two medians that I calculated -100.5 is different from BSDA R package median of x-y value which is -99? I understand that that is not much difference but anyway it is and I want to know why.

2. Can I present that median differences as effect size?

Answer 1

1. There's an important distinction between what you calculated ($\text{median}(x) - \text{median}(y)$) and what the sign test looks at ($\text{median}(d)$, where $d_i=x_i-y_i$).

   The first thing is a difference of medians. The second thing is the median difference.

   Because taking the median is not a linear operation, it doesn't commute with subtraction (taking the difference) - that is a difference of medians is distinct from a median difference.

2. Can I present that median differences as effect size?

   You could indeed present the median of the pair-differences as a raw effect size (since that's what the test looks at), but that was not what you calculated; you calculated the difference of the medians. Be careful not to conflate the two.