Getting data

Data used here were from a subset of Experiment 1b in Jin et al., (2022). You may get the data by installing the custom package remotes::install_github("haiyangjin/psychr") and getting the data psychr::jin2022noncon. (Alternatively, you may download the data directly from GitHub.)

# to install psychr package used here:
# remotes::install_github("haiyangjin/psychr")
df_clean <- psychr::jin2022noncon 

head(df_clean, 10)

There are three independent variables:

• Congruency: con vs inc;
• Alignment: ali vs mis;
• SD: same vs diff (we will treat same as signal and diff as noise.)

The dependent variables are:

• isSame: whether the response was “same” (1) or “different” (0);
• Correct: whether the response was “correct” (1) or “incorrect” (0);
• RT: response time (this DV will not be used in this post).

Apply signal detection theory in R

In this application, we treat same trials as “signal” and diff trials as “noise”. The general steps for calculating the sensitivity d’ are:

1. Calculate the hit (hit) and false alarm rates (fa);
2. Apply corrections if needed;
3. Convert rates (in percentage) to standard (Z) values (z_hit & z_fa);
4. Get the sensitivity d’ with z_hit - z_fa.

It is noteworthy that we may use either isSame or Correct to calculate hit and false alarm rates. So we try both here and can compare both approaches.

# load packages
library(tidyverse)

## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ✔ forcats   1.0.0     ✔ stringr   1.5.1
## ✔ ggplot2   3.5.1     ✔ tibble    3.2.1
## ✔ lubridate 1.9.3     ✔ tidyr     1.3.1
## ✔ purrr     1.0.2     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

df_sdt_correct <- df_clean %>% 
  group_by(SubjID, Congruency, Alignment, SD) %>% 
  summarize(acc = mean(Correct), # hit and correct rejection
            count = n(), # number of trials in each condition
            .groups = "drop") %>% 
  mutate(acc = if_else(SD=="diff", 1-acc, acc), # hit and false alarm (Step 1)
         acc = if_else(SD=="same"&acc==1, (2*count-1)/(2*count), acc), # correction for hit
         acc = if_else(SD=="diff"&acc==0, 1/(2*count), acc), # correction for false alarm
         z = qnorm(acc)) %>% # convert to standard Z value
  pivot_wider(id_cols = c(SubjID, Congruency, Alignment), # make hit and false alarm into two columns
              names_from = SD,   
              values_from = z) %>% 
  mutate(d = same - diff, # calculate d from hit and false alarm
         d = round(d,2)) %>% 
  select(-c(same, diff))
  
# display results as wide format
df_sdt_correct %>% 
  pivot_wider(names_from = c(Congruency, Alignment),
              values_from = d)

df_sdt_issame <- df_clean %>% 
  group_by(SubjID, Congruency, Alignment, SD) %>% 
  summarize(acc = mean(isSame), # hit and false alarm
            count = n(), # number of trials in each condition
            .groups = "drop") %>% 
  mutate(acc = if_else(SD=="same"&acc==1, (2*count-1)/(2*count), acc),
         acc = if_else(SD=="diff"&acc==0, 1/(2*count), acc), # correction for false alarm
         z = qnorm(acc)) %>% # convert to standard Z value
  pivot_wider(id_cols = c(SubjID, Congruency, Alignment), # make hit and false alarm into two columns
              names_from = SD,   
              values_from = z) %>% 
  mutate(d = same - diff, # calculate d from hit and false alarm
         d = round(d,2)) %>% 
  select(-c(same, diff))
  
df_sdt_issame %>% 
  pivot_wider(names_from = c(Congruency, Alignment),
              values_from = d) 

all.equal(df_sdt_correct, df_sdt_issame)

## [1] TRUE

Use library(psychr)

In addition to calculating d’ manually in R, I also made an R function (psychr::sdt()) to calculate d’ in R.

# to install psychr package used here:
# remotes::install_github("haiyangjin/psychr")
library(psychr)

An example with the same data set is:

sdt_clean <- psychr::sdt(df_clean, 
                         SN = "SD", # the column containing the signal/noise information
                         isSignal = "isSame", # the column contains "isSignal"
                         SubjID = "SubjID",
                         group_other = c("Congruency", "Alignment"),
                         signal = "same") # which level in SN is signal

sdt_clean$df %>% 
  mutate(d = round(d,2)) %>% 
  pivot_wider(names_from = c(Congruency, Alignment),
              values_from = d) 

all.equal(df_sdt_correct, sdt_clean$df)

## [1] "Component \"d\": Mean relative difference: 0.001039788"

all.equal(df_sdt_issame, sdt_clean$df)

## [1] "Component \"d\": Mean relative difference: 0.001039788"

The data frame obtained from psychr::sdt() is the same as those from the first two approaches. So you may either use custom codes or psychr::sdt() to apply SDT in R!