Number-line tests

1 Set up

Code

library(dplyr)
library(lubridate)
library(tidyr)
library(here)
library(ggplot2)
library(stringr)
library(purrr)
library(stringr)
library(readr)
library(moments)
library(patchwork)
library(scales)
library(kableExtra)
library(knitr)
library(glue)
library(openxlsx)

probe_dir <- here("02-Number_line")
data_dir   <- "Raw"
calib_file <- "Raw/00_number_line_coordinates.csv"
key_file   <- "Raw/00_number_line_answer_key.csv"
student_group_file <- "Raw/student_assessment_map.csv"

2 Question presentation

2.1 a) Bounded number line 0 - 100 (BNL0-100)

2.2 b) Bounded number line 0 - 20 (BNL0-20)

2.3 c) Unbounded number line chair (UBNL)

2.4 d) Unbounded number line no chair (UBNL)

3 Build conversion table from calibration points

Scale limit for each test type’s number line

n_left → Number on the left end
n_right → Number on the right end
Slope = units per pixel on number line (used to compute right edge for UBNL)

Code

coord_calibration_df <- read_csv(calib_file)

conv_tbl <- coord_calibration_df %>% 
  group_by(TestType) %>% # assumes two coordinate rows per test
  reframe(
    x_left = min(X_coord),  
    n_left = Number[which.min(X_coord)],
    x2 = max(X_coord), 
    n2 = Number[which.max(X_coord)],
    slope  = (n2 - n_left) / (x2 - x_left),    # units per pixel
    
    # Assume left end = x1, decide real right end for each test
    x_right = case_when(
      TestType == "UNL0-20" ~ 1000,             
      TRUE                   ~ x2
    ),
    n_right = n_left + slope * (x_right - x_left),   # infer right end number
  ) %>% 
  select(TestType, x_left, n_left, x_right, n_right, slope) %>% 
  distinct()

print(conv_tbl)

# A tibble: 3 × 6
  TestType x_left n_left x_right n_right  slope
  <chr>     <dbl>  <dbl>   <dbl>   <dbl>  <dbl>
1 BNL0-100  10         0    1089   100   0.0927
2 BNL0-20   11         0    1108    20   0.0182
3 UNL0-20    5.25      0    1000    34.4 0.0346

4 Read & tidy every student-response CSV

Code

nl_raw <- list.files(data_dir, pattern = "_responses\\.csv$", full.names = TRUE) %>%
  read_csv(show_col_types = FALSE, id = "source_file") %>%   # `id` keeps the file name
  select(-starts_with("..."))                  

# De-duplicate: keep only the latest per Identifier × Test Identifier × Question Identifier
tmp_nl <- nl_raw %>%
  mutate(.ts = dmy_hm(`Date Completed`))

nl_raw <- tmp_nl %>%
  group_by(Identifier, `Test Identifier`, `Question Identifier`) %>%
  slice_max(.ts, n = 1, with_ties = FALSE) %>%
  ungroup() %>%
  select(-.ts)

n_removed <- nrow(tmp_nl) - nrow(nl_raw)
message(n_removed, " duplicate rows removed; keeping only the most recent per Identifier/Test/Question")

# Pull out pixel coordinates
nl_raw <- nl_raw %>% 
  mutate(
    X_coord = str_match(`Answer Response`, '"X":\\s*"([0-9\\.]+)"')[, 2] %>% as.numeric(),
    Y_coord = str_match(`Answer Response`, '"Y":\\s*"([0-9\\.]+)"')[, 2] %>% as.numeric()
  )

# Number of students per file
# nl_raw %>%
#  count(source_file, name = "rows") 

# Normalise TestType
normalise_type <- function(id) {
  id %>%                                           
    sub("_\\d{4}.*$", "", .) %>%
    sub("-\\d{4}.*$", "", .) %>%
    sub("[Ff]$", "", .) %>%                        # drop prefix F and nc
    sub("-?nc", "", ., ignore.case = TRUE)
}

# Clean Question Identifier text string
clean_qid <- function(x) {
  x %>%
    str_replace(regex("^UNL[cn]*c?0-20", ignore_case = TRUE), "UBNL0-20") %>%
    str_remove("[-_]new[-_]?copy$") %>%     
    str_remove("[-_]copy$") %>% 
    str_remove("[-_]?new$")
}

# Replace exam group with the following:
student_df <- read_csv(student_group_file) %>% 
  select(-starts_with("..."))

student_clean <- student_df %>% 
  mutate(
    GradeGroup = case_when(
      str_detect(`Assessment Event Identifier`, "^year-1a-2025$")          ~ "Year 1A",
      str_detect(`Assessment Event Identifier`, "^year-1b-2025$")          ~ "Year 1B",
      str_detect(`Assessment Event Identifier`, "^foundation-a-2025$")     ~ "Foundation A",
      str_detect(`Assessment Event Identifier`, "^foundation-b-2025$")     ~ "Foundation B",
      TRUE                                                                 ~ NA_character_  
    )
  ) %>% 
  select(Identifier, GradeGroup) %>%          
  distinct()     

nl_tidy <- nl_raw %>%
  mutate(TestType = normalise_type(`Test Identifier`),
         `Question Identifier` = clean_qid(`Question Identifier`))


nl_tidy <- nl_tidy %>% 
  left_join(student_clean, by = "Identifier") %>% 
  select(-`Exam Group`)

# Fix grade groups
nl_tidy <- nl_tidy %>% 
  mutate(
    GradeGroup = case_when(
      `Test Identifier` == "BNL0-100_2025" & GradeGroup == "Foundation B" ~ "Year 1B",
      `Test Identifier` == "BNL0-20F_2025" & GradeGroup == "Year 1A" ~ "Foundation A",
      `Test Identifier` == "BNL0-20F_2025" & GradeGroup == "Year 1B" ~ "Foundation B",
      `Test Identifier` == "UNLC0-20F_2025" & GradeGroup == "Year 1B" ~ "Foundation B",
      `Test Identifier` == "UNLc0-20-2025" & GradeGroup == "Foundation B" ~ "Year 1B",
      `Test Identifier` == "UNLnc0-20F_2025" & GradeGroup == "Year 1A" ~ "Foundation A",
      TRUE ~ GradeGroup
    )
  )


# Group unbounded number line variants
nl_tidy <- nl_tidy %>% 
  mutate(
    TestType = case_when(
      TestType %in% c("UNLc0-20", "UNLC0-20") ~ "UNL0-20",   
      TRUE                                    ~ TestType     
    )
  ) %>% 
  select(-source_file)

5 Map number clicks and compute score

Code

# Map pixels to number
nl_clicks <- nl_tidy %>% 
  filter(!is.na(X_coord)) %>% 
  filter(!str_detect(`Question Identifier`, "TRANS_INTRO|_Prac|_Tch|-Prac|_prac")) %>% 
  left_join(conv_tbl, by = "TestType") %>% 
  # clamp clicks to both ends of line
  mutate(
    x_clamp = pmin(pmax(X_coord, x_left), x_right),
    Number_clicked = n_left + slope * (x_clamp - x_left)
  )

# Add answer key
answer_key_df <- read_csv(key_file, show_col_types = FALSE)

# Identify any Question Identifier missing from the key
missing_qids <- nl_clicks %>% 
  anti_join(answer_key_df, by = "Question Identifier") %>% 
  distinct(`Question Identifier`, TestType)  

# print(missing_qids)

# Compute PAE
nl_scored <- nl_clicks %>% 
  left_join(
    answer_key_df %>% select(`Question Identifier`, `Answer Key`),
    by = c(`Question Identifier` = "Question Identifier")
  ) %>% 
  mutate(
    abs_error = abs(Number_clicked - `Answer Key`),
    signed_error = Number_clicked - `Answer Key`,
    PAE = abs_error / (n_right - n_left),
    Score = 1 - PAE
  )

# Compute inverse efficiency
nl_scored <- nl_scored %>% 
  mutate(IE = `Answer Duration` / (1 - PAE))

6 Exploratory analysis

6.1 Student count per probe x grade group

See below for detailed probe summary statistics

Code

nl_scored %>% 
  group_by(GradeGroup, TestType) %>% 
  summarise(
    n_student = n_distinct(Identifier),
    mean_PAE = mean(PAE),
    mean_IE = mean(IE),
    .groups = "drop"
  ) %>% 
  arrange(GradeGroup)

# A tibble: 8 × 5
  GradeGroup   TestType n_student mean_PAE mean_IE
  <chr>        <chr>        <int>    <dbl>   <dbl>
1 Foundation A BNL0-20       1206    0.165   19.7 
2 Foundation A UNL0-20       1221    0.207   12.5 
3 Foundation B BNL0-20       1128    0.166   19.8 
4 Foundation B UNL0-20       1153    0.204   14.5 
5 Year 1A      BNL0-100      1260    0.140   25.4 
6 Year 1A      UNL0-20       1254    0.119    6.87
7 Year 1B      BNL0-100      1050    0.138   11.0 
8 Year 1B      UNL0-20       1063    0.126    7.41

Generate wide format

Code

# 1. Build one big wide table, keeping GradeGroup & TestType as keys
wide_by_group_test <- nl_scored %>%
  # extract the 3-digit item suffix
  mutate(item = str_extract(`Question Identifier`, "\\d{3}$")) %>%
  # keep only the columns we need
  select(Identifier, GradeGroup, `Test Identifier`, item,
         Number_clicked, `Answer Duration`, PAE) %>%
  # pivot to wide, with one block of columns per item
  pivot_wider(
    names_from  = item,
    values_from = c(Number_clicked, `Answer Duration`, PAE),
    names_glue  = "{.value}_item{item}"
  )

# 2. Split into a list by GradeGroup × Test Identifier
wide_list <- wide_by_group_test %>%
  group_by(GradeGroup, `Test Identifier`) %>%
  group_split(.keep = TRUE)

# capture matching sheet names
sheet_names <- wide_by_group_test %>%
  distinct(GradeGroup, `Test Identifier`) %>%
  mutate(sheet = glue("{GradeGroup}_{`Test Identifier`}")) %>%
  pull(sheet)

# 3. Create workbook and add each sub‐wide table as its own sheet
wb <- createWorkbook()
walk2(wide_list, sheet_names, function(df, sh) {
  addWorksheet(wb, sh)
  writeData(wb, sh, df)
})

# 4. Finally add the master answer‐key on its own sheet
addWorksheet(wb, "Answer Key")
writeData(wb, "Answer Key", answer_key_df)

# 5. Save the file
saveWorkbook(
  wb,
  file      = "numb_line_by_grade_test.xlsx",
  overwrite = TRUE
)

6.2 Distribution of response per item

Code

plot_response <- function(df, bins) {
  
  df <- df %>% 
    mutate(facet = paste0(`Question Identifier`, " (", `Answer Key`, ")"))
  
  ## -- build the plot *first* (we'll grab its stats in a moment)
  base_plot <- ggplot(
      df,
      aes(Number_clicked, group = facet)     # one histogram per item
    ) +
    geom_histogram(
      aes(y = after_stat(count / sum(count))),
      bins   = bins,
      boundary = 0,
      colour = "grey20", fill = "grey80"
    ) +
    geom_vline(aes(xintercept = `Answer Key`), colour = "red") +
    facet_wrap(~ facet, scales = "free_y", ncol = 4) +
    scale_y_continuous(labels = percent_format(accuracy = 0.1)) +
    labs(
      x = "Number clicked",
      y = "Percent of students",
      caption = "Red line = target number"
    ) +
    theme_bw()
  
  ## -- extract max proportion across all facets
  stats     <- ggplot_build(base_plot)$data[[1]]
  max_prop  <- max(stats$y, na.rm = TRUE)            # highest bar
  y_limit   <- max_prop * 1.05                       # +5 % head-room
  
  ## -- return the finished plot with a universal y-limit
  base_plot +
    coord_cartesian(ylim = c(0, y_limit), clip = "off")
}

bins_map <- c(
  "BNL0-100" = 25,
  "BNL0-20"  = 21,
  "UNL0-20"  = 34
)

unique(nl_scored$TestType) %>% 
  walk(function(tt) {
    
    df_probe <- nl_scored %>% filter(TestType == tt)
    bins     <- bins_map[tt]
    
    p <- plot_response(df_probe, bins)
    
    print(p)   
    
    ggsave(
      filename = file.path(probe_dir, glue("response_dist_{tt}.png")),
      plot     = p,
      width    = 12, height = 8, dpi = 300, units = "in"
    )
  })

6.3 Distribution of answer duration

Code

plot_duration <- function(df,
                               test_name,
                               bins      = 60,
                               trim_fast = 0,   
                               trim_slow = 60)  
{
  df <- df %>% 
    filter(`Answer Duration` >= trim_fast,
           `Answer Duration` <= trim_slow) %>%
    mutate(facet_label = paste0(`Question Identifier`, " (", `Answer Key`, ")"))

  ggplot(df, aes(x = `Answer Duration`)) +
    geom_histogram(
      aes(y = after_stat(count / sum(count))),
      bins   = bins,
      colour = "grey10",
      fill   = "steelblue"
    ) +
    facet_wrap(~ facet_label, scales = "free_y", ncol = 4) +
    scale_y_continuous(labels = scales::percent_format(accuracy = 0.01)) +
    labs(
      title   = paste("Answer duration distribution –", test_name),
      x       = "Seconds",
      y       = "Percent of students",
      caption = paste("Only", trim_fast, "–", trim_slow, "s shown")
    ) +
    theme_bw() +
    theme(panel.grid.y = element_line(colour = "grey90"),
          strip.text   = element_text(face = "bold"))
}
  
# Print and save plot
unique(nl_scored$TestType) %>%
  walk(function(tt) {
    df_test <- nl_scored %>% filter(TestType == tt)

    p <- plot_duration(df_test, tt)    

    print(p)

    ggsave(
      filename = file.path(probe_dir, glue("duration_dist_{tt}.png")),
      plot     = p,
      width    = 12, height = 8, dpi = 300, units = "in"
    )
  })

6.4 PAE and Inverse Efficiency per student by test type

Inverse Efficiency (IE) = Answer Duration / (1 - PAE). i.e. seconds per perfectly accurate response

Code

# Each student's mean score per probe
nl_student <- nl_scored %>%
  group_by(TestType, Identifier) %>%
  summarise(
    PAE = mean(PAE, na.rm = TRUE),
    IE  = mean(IE,  na.rm = TRUE),
    .groups = "drop"
  )

nl_long <- nl_student %>% 
  pivot_longer(c(PAE, IE), names_to = "Metric", values_to = "Value")

# Trim extreme outliers
k_trim <- 2

trim_iqr <- function(df, k_out = k_trim) {
  df %>% 
    group_by(TestType, Metric) %>% 
    mutate(
      Q1  = quantile(Value, 0.25, na.rm = TRUE),
      Q3  = quantile(Value, 0.75, na.rm = TRUE),
      hi  = Q3 + k_out * (Q3 - Q1),
      lo  = pmax(0, Q1 - k_out * (Q3 - Q1))
    ) %>% 
    ungroup() %>% 
    filter(Value >= lo, Value <= hi) %>% 
    select(-Q1, -Q3, -hi, -lo)
}

nl_trim <- trim_iqr(nl_long)

# Violin plot
vplot <- function(df, metric, y_lab, lab_fun) {
  ggplot(filter(df, Metric == metric),
         aes(TestType, Value, fill = TestType)) +
    geom_violin(trim = FALSE, alpha = .6, colour = NA) +
    geom_boxplot(width = .15, outlier.shape = NA, colour = "grey20") +
    scale_fill_brewer(palette = "Set2", guide = "none") +
    scale_y_continuous(labels = lab_fun,
                       expand = expansion(mult = c(0, .05))) +
    labs(title = metric, x = NULL, y = y_lab) +
    theme_bw() +
    theme(panel.grid.y = element_line(colour = "grey90"))
}

p_pae <- vplot(nl_trim, "PAE", "Percent of scale",
               lab_fun = percent_format(accuracy = 1))
p_ie  <- vplot(nl_trim, "IE",  "Inverse efficiency",
               lab_fun = comma_format(accuracy = 1))

p_probe_violin <- (p_pae | p_ie) +
  plot_annotation(
    title = paste0("PAE and IE by probe (student means, IQR-trim k = ", k_trim, ")"),
    theme = theme(plot.title = element_text(face = "bold"))
  )

p_probe_violin

Code

ggsave(
  filename = file.path(probe_dir, "probe_pae_ie_dist.png"),
  plot     = p_probe_violin,
  width    = 12, height = 6, dpi = 300, units = "in"
)

6.5 Plot clicks for questions

Code

plot_clicks_xy <- function(qid, data = nl_scored) {
  df <- data %>%
    filter(`Question Identifier` == qid) %>%
    arrange(Identifier) %>%
    mutate(student_idx = row_number())  

  answer <- unique(df$`Answer Key`)
  if (length(answer) != 1) {
    stop("Question ", qid, " has ", length(answer), " distinct answer keys!")
  }

  p <- ggplot(df, aes(x = Number_clicked, y = Y_coord)) +
    geom_point(alpha = 0.6, size = 1.8) +
    geom_vline(xintercept = answer,
               colour     = "red",
               linetype   = "dashed",
               size       = 1) +
    labs(
      title = paste("Click positions for", qid),
      x     = "Number clicked (converted from X coord)",
      y     = "Y coord clicked"
    ) +
    theme_minimal()

  print(p)
  invisible(p)
}

plot_clicks_xy("UBNL0-20_005")

7 Item summary statistics

7.1 Distribution of PAE and Inverse Efficiency per Item

Code

# Shorten item label
item_label <- function(qid, target) {
  item_no <- str_extract(qid, "(?<=_)(\\d{3})(?=$|[_-])")  # "001" part
  glue("Item {item_no} ({target})")
}

nl_item_long <- nl_scored %>% 
  mutate(
    ItemLab = item_label(`Question Identifier`, `Answer Key`)
  ) %>% 
  select(TestType, ItemLab, Target = `Answer Key`, PAE, IE)

# Boxplot
make_item_box <- function(df, metric, y_lab) {
  ggplot(df, aes(x = .data[[metric]],           
                 y = reorder(ItemLab, Target))) +
    geom_boxplot(fill = "steelblue", width = .6, outlier.size = .8) +
    labs(x = y_lab, y = NULL) +
    theme_bw() +
    theme(
      axis.text.y = element_text(size = 8),
      panel.grid.major.y = element_blank()
    )
}

# Trim outliers for inverse efficiency
k_trim_ie <- 2

trim_ie <- function(df, k = k_trim_ie) {
  df %>% 
    group_by(ItemLab) %>%                       
    mutate(
      Q1  = quantile(IE, .25, na.rm = TRUE),
      Q3  = quantile(IE, .75, na.rm = TRUE),
      hi  = Q3 + k * (Q3 - Q1),
      lo  = pmax(0, Q1 - k * (Q3 - Q1))
    ) %>% 
    ungroup() %>% 
    filter(IE >= lo, IE <= hi) %>% 
    select(-Q1, -Q3, -hi, -lo)
}

metrics <- c(PAE = "Percent absolute error", IE = "Inverse efficiency (s)")

nl_item_long %>% 
  split(.$TestType) %>% 
  iwalk(function(df_probe, probe_name) {

    walk2(names(metrics), metrics, function(metric, xlab) {
      
      df_use <- if (metric == "IE") trim_ie(df_probe) else df_probe
      
      p <- make_item_box(df_use, metric, xlab) +
           ggtitle(glue("{probe_name} – {metric}")) + 
           if (metric == "IE")
             labs(caption = paste0("Outliers beyond ± ", k_trim_ie, " × IQR per item removed"))

      print(p)    

      ggsave(
        file.path(probe_dir, glue("box_{probe_name}_{metric}.png")),
        plot   = p, width = 8, height = 6, units = "in"
      )
    })
  })

7.2 PAE

Code

nl_scored_item_pae_summary <- nl_scored %>% 
  group_by(`Question Identifier`) %>% 
  summarise(
    n      = n_distinct(Identifier),
    Target = first(`Answer Key`),
    pae_mean = mean(PAE),
    pae_sd   = sd(PAE),
    pae_min  = min(PAE),
    pae_max  = max(PAE),
    pae_skew = moments::skewness(PAE),
    pae_kurt = moments::kurtosis(PAE),
    .groups = "drop"
  )
  
nl_scored_item_pae_summary %>% 
  mutate(across(where(is.numeric), round, 3)) %>% 
  # plain kable
  kbl(
    caption   = "Item-level PAE summary",
    digits    = 3,
    align     = c("l", rep("r", 8)),  # left for ID, right for numbers
    col.names = c("Item", "n", "Target",
                  "Mean", "SD", "Min", "Max", "Skew", "Kurt")
  ) |>
  # kableExtra niceties
  kable_styling(full_width = FALSE, position = "left") %>% 
  scroll_box(width = "100%",  height = "500px")

Item-level PAE summary
Item	n	Target	Mean	SD	Min	Max	Skew	Kurt
BNL0-100_001	2232	59	0.085	0.086	0.000	0.590	1.931	7.733
BNL0-100_002	2201	20	0.171	0.137	0.000	0.788	1.436	5.402
BNL0-100_003	2220	46	0.103	0.095	0.000	0.521	1.518	5.249
BNL0-100_004	2232	13	0.168	0.146	0.000	0.866	1.454	5.497
BNL0-100_005	2219	72	0.113	0.114	0.000	0.718	2.013	7.769
BNL0-100_006	2223	31	0.130	0.121	0.000	0.690	1.691	6.107
BNL0-100_007	2241	97	0.200	0.173	0.001	0.968	1.699	5.872
BNL0-100_008	2225	85	0.168	0.132	0.000	0.850	1.522	6.505
BNL0-100_009	2216	68	0.128	0.112	0.000	0.680	1.590	6.392
BNL0-100_010	2254	4	0.090	0.121	0.000	0.951	3.909	21.462
BNL0-100_011	2209	13	0.179	0.147	0.000	0.870	1.486	5.806
BNL0-100_012	2207	68	0.126	0.112	0.000	0.678	1.521	5.649
BNL0-100_013	2194	85	0.170	0.149	0.000	0.811	1.324	4.589
BNL0-100_014	2197	31	0.134	0.122	0.000	0.684	1.577	5.562
BNL0-100_015	2191	72	0.127	0.120	0.000	0.720	1.720	6.262
BNL0-100_016	2187	97	0.198	0.178	0.000	0.970	1.559	5.218
BNL0-100_017	2220	20	0.166	0.152	0.000	0.800	1.595	5.693
BNL0-100_018	2216	59	0.107	0.096	0.000	0.558	1.427	5.022
BNL0-100_019	2248	4	0.098	0.126	0.000	0.959	3.330	16.146
BNL0-100_020	2214	46	0.126	0.102	0.000	0.540	1.203	4.207
BNL0-20_001	2194	2	0.107	0.168	0.000	0.900	2.457	9.018
BNL0-20_002	2038	5	0.124	0.121	0.000	0.750	2.602	11.865
BNL0-20_003	2113	16	0.209	0.184	0.000	0.800	1.215	3.903
BNL0-20_004	2157	9	0.148	0.126	0.000	0.550	1.012	3.259
BNL0-20_005	2164	13	0.171	0.137	0.000	0.650	1.252	4.475
BNL0-20_006	2180	7	0.142	0.123	0.000	0.650	1.701	6.405
BNL0-20_007	2195	4	0.121	0.138	0.000	0.800	2.531	10.154
BNL0-20_008	2177	19	0.251	0.240	0.000	0.950	0.941	2.986
BNL0-20_009	2140	18	0.223	0.224	0.000	0.900	1.101	3.348
BNL0-20_010	2162	11	0.161	0.133	0.000	0.550	0.821	2.734
UBNL0-20_001	4567	5	0.091	0.138	0.000	0.855	3.452	16.362
UBNL0-20_002	4426	16	0.198	0.141	0.000	0.535	0.652	2.447
UBNL0-20_003	4470	9	0.165	0.166	0.000	0.739	1.741	5.660
UBNL0-20_004	4448	13	0.209	0.167	0.000	0.623	0.840	2.758
UBNL0-20_005	4472	2	0.070	0.169	0.000	0.942	3.765	16.950
UBNL0-20_006	4444	7	0.136	0.157	0.000	0.797	2.272	8.478
UBNL0-20_007	4441	4	0.096	0.150	0.000	0.884	3.256	14.471
UBNL0-20_008	4414	19	0.234	0.138	0.000	0.552	0.077	1.882
UBNL0-20_009	4386	18	0.227	0.135	0.001	0.523	0.158	1.937
UBNL0-20_010	4432	11	0.191	0.161	0.000	0.681	1.197	3.980
UBNL0-20_011	2210	7	0.103	0.119	0.000	0.793	2.909	13.843
UBNL0-20_012	2203	19	0.194	0.124	0.000	0.552	0.444	2.304
UBNL0-20_013	2217	4	0.085	0.134	0.000	0.883	3.425	16.191
UBNL0-20_014	2199	9	0.120	0.128	0.000	0.738	2.283	9.299
UBNL0-20_015	2201	13	0.138	0.123	0.000	0.621	1.594	5.651
UBNL0-20_016	2202	5	0.096	0.124	0.000	0.853	3.038	14.480
UBNL0-20_017	2203	2	0.058	0.135	0.000	0.933	4.099	21.005
UBNL0-20_018	2183	16	0.163	0.116	0.000	0.535	0.876	3.335
UBNL0-20_019	2187	11	0.132	0.122	0.000	0.680	1.924	7.624
UBNL0-20_020	2191	18	0.176	0.121	0.000	0.523	0.677	2.683

7.3 Inverse Efficiency

Code

nl_scored_item_ie_summary <- nl_scored %>% 
  group_by(`Question Identifier`) %>% 
  summarise(
    n      = n_distinct(Identifier),
    Target = first(`Answer Key`),
    IE_mean = mean(IE),
    IE_sd   = sd(IE),
    IE_min  = min(IE),
    IE_max  = max(IE),
    IE_skew = moments::skewness(IE),
    IE_kurt = moments::kurtosis(IE),
    .groups = "drop"
  )

nl_scored_item_ie_summary %>% 
  mutate(across(where(is.numeric), round, 3)) %>% 
  kbl(
    caption   = "Item-level IE summary",
    digits    = 3,
    align     = c("l", rep("r", 8)),  # left for ID, right for numbers
    col.names = c("Item", "n", "Target",
                  "Mean", "SD", "Min", "Max", "Skew", "Kurt")
  ) |>
  # kableExtra niceties
  kable_styling(full_width = FALSE, position = "left") %>% 
  scroll_box(width = "100%",  height = "500px")

Item-level IE summary
Item	n	Target	Mean	SD	Min	Max	Skew	Kurt
BNL0-100_001	2232	59	48.326	414.360	3.041	18732.379	41.468	1853.533
BNL0-100_002	2201	20	22.888	24.941	2.189	326.970	4.607	39.765
BNL0-100_003	2220	46	15.837	16.878	1.083	352.943	6.000	85.190
BNL0-100_004	2232	13	13.561	13.424	1.084	174.244	3.336	22.820
BNL0-100_005	2219	72	14.368	111.880	1.161	5253.167	46.241	2165.514
BNL0-100_006	2223	31	10.880	10.275	1.016	120.056	3.414	21.618
BNL0-100_007	2241	97	14.098	20.274	1.272	394.544	9.468	135.477
BNL0-100_008	2225	85	12.254	84.984	1.037	3989.279	46.024	2153.591
BNL0-100_009	2216	68	50.856	1950.165	1.015	91873.442	47.073	2216.905
BNL0-100_010	2254	4	8.815	9.927	1.076	187.326	7.968	102.045
BNL0-100_011	2209	13	9.140	9.715	1.027	124.881	4.476	34.740
BNL0-100_012	2207	68	53.031	2102.634	1.006	98853.553	46.977	2207.918
BNL0-100_013	2194	85	8.355	7.021	1.004	113.288	3.974	36.167
BNL0-100_014	2197	31	48.912	1943.868	1.012	91203.042	46.882	2198.939
BNL0-100_015	2191	72	7.543	6.854	1.025	98.323	4.388	38.823
BNL0-100_016	2187	97	8.927	13.427	1.050	366.667	14.500	316.288
BNL0-100_017	2220	20	7.510	7.545	0.000	96.546	4.146	29.446
BNL0-100_018	2216	59	7.610	8.898	1.005	160.427	10.212	150.460
BNL0-100_019	2248	4	6.063	5.739	1.010	92.089	4.993	45.074
BNL0-100_020	2214	46	9.079	56.865	1.008	2664.628	45.979	2147.159
BNL0-20_001	2194	2	41.034	121.540	2.033	4725.990	27.959	1021.534
BNL0-20_002	2038	5	24.418	45.155	1.033	1712.420	26.620	961.107
BNL0-20_003	2113	16	27.786	31.000	1.059	642.862	6.504	91.113
BNL0-20_004	2157	9	17.680	16.280	1.007	204.633	4.085	33.965
BNL0-20_005	2164	13	17.898	19.112	1.003	289.794	4.695	42.001
BNL0-20_006	2180	7	13.020	12.584	1.028	148.525	3.848	26.346
BNL0-20_007	2195	4	10.565	9.912	1.011	131.992	4.118	32.665
BNL0-20_008	2177	19	19.489	31.290	1.019	545.170	8.575	110.453
BNL0-20_009	2140	18	14.198	18.171	1.017	281.831	5.614	53.869
BNL0-20_010	2162	11	11.493	12.581	1.001	352.253	11.027	259.080
UBNL0-20_001	4567	5	18.573	34.715	1.375	953.931	15.466	334.798
UBNL0-20_002	4426	16	16.160	47.399	1.007	2978.996	55.141	3438.742
UBNL0-20_003	4470	9	11.162	12.198	1.002	140.079	3.958	27.474
UBNL0-20_004	4448	13	11.668	13.349	0.000	228.133	4.786	47.500
UBNL0-20_005	4472	2	8.292	17.817	1.001	424.109	11.946	194.980
UBNL0-20_006	4444	7	8.809	11.543	1.002	192.405	6.896	79.157
UBNL0-20_007	4441	4	7.502	10.594	1.001	353.008	13.071	324.220
UBNL0-20_008	4414	19	10.791	14.004	1.007	454.201	11.299	277.666
UBNL0-20_009	4386	18	8.979	10.128	0.000	204.350	5.347	63.263
UBNL0-20_010	4432	11	8.518	11.227	0.000	317.308	11.446	263.430
UBNL0-20_011	2210	7	5.523	5.625	1.003	69.472	4.729	37.771
UBNL0-20_012	2203	19	6.623	7.531	0.000	179.450	7.409	134.453
UBNL0-20_013	2217	4	5.159	9.514	1.002	299.996	21.488	595.597
UBNL0-20_014	2199	9	5.361	5.259	1.002	71.540	4.612	40.179
UBNL0-20_015	2201	13	5.916	6.593	0.000	107.964	4.783	45.236
UBNL0-20_016	2202	5	4.958	5.673	1.000	96.417	7.983	101.623
UBNL0-20_017	2203	2	4.484	7.869	0.000	202.274	17.009	385.847
UBNL0-20_018	2183	16	6.345	7.159	0.000	87.223	4.268	30.188
UBNL0-20_019	2187	11	5.301	5.758	0.000	81.820	4.628	37.589
UBNL0-20_020	2191	18	6.088	6.436	1.001	68.049	3.635	21.744

7.4 Signed error

Code

nl_bias <- nl_scored %>%
  mutate(
    signed_error = Number_clicked - `Answer Key`
  )

ggplot(nl_bias, aes(x = `Answer Key`, y = signed_error)) +
  geom_point(alpha = 0.2, size = 0.8) +        # scatter of all clicks
  geom_smooth(method = "lm", colour = "red") + 
  facet_wrap(~ TestType, scales = "free") +
  labs(
    x = "Target number",
    y = "Signed error (clicked – target)",
    title = "Signed error vs target",
  ) +
  theme_bw()

7.5 Standard dev of abs error by target

Code

abs_error_sd <- nl_scored %>% 
  group_by(TestType, Target = `Answer Key`) %>% 
  summarise(
    sd_abs_error = sd(abs_error, na.rm = TRUE),
    .groups      = "drop"
  )

ggplot(abs_error_sd, aes(x = Target, y = sd_abs_error)) +
  geom_line(colour = "steelblue", size = 0.8) +
  geom_point(colour = "steelblue", size = 1.5) +
  facet_wrap(~ TestType, scales = "free_x", nrow = 1) +
  scale_x_continuous(breaks = pretty_breaks()) +
  labs(
    title =   "SD of Absolute Error by Target Number",
    x =       "Target number",
    y =       "SD of |clicked – target|",
    caption = "Each point = SD over all pupils for that target"
  ) +
  theme_bw() +
  theme(
    panel.grid.minor = element_blank(),
    strip.background = element_rect(fill = "grey90"),
    strip.text       = element_text(face = "bold")
  )

7.6 Click spread

Code

click_spread_df <- nl_scored %>%
  group_by(TestType, Identifier, Target = `Answer Key`) %>%
  summarise(
    n_attempts   = n(),                                
    click_spread = max(Number_clicked, na.rm = TRUE)
                 - min(Number_clicked, na.rm = TRUE),
    .groups = "drop"
  ) %>% 
  filter(TestType != "BNL0-20")

plot_spread <- function(df, probe_name) {
  df %>%
    mutate(Target = factor(Target, levels = sort(unique(Target)))) %>%
    ggplot(aes(x = click_spread,
               y = Target)) +
    geom_boxplot(fill = "steelblue", colour = "grey30", width = 0.6) +
    labs(
      title = paste0("Click spread by target number - ",probe_name),
      x     = "Click spread",
      y     = "Target number"
    ) +
    theme_bw() +
    theme(
      axis.text.y   = element_text(size = 8),
      panel.grid.minor = element_blank()
    )
}

click_spread_df %>%
  split(.$TestType) %>%
  iwalk(function(df_probe, probe_name) {
    p <- plot_spread(df_probe, probe_name)
    print(p)
    ggsave(
      filename = file.path(probe_dir, glue("click_spread_{probe_name}.png")),
      plot     = p,
      width    = 8, height = 6, units = "in", dpi = 300
    )
  })

7.7 Midpoint click position metrics

Code

nl_mid <- nl_scored %>%
  group_by(TestType, Identifier, `Answer Key`) %>%
  summarise(
    click_mean   = mean(Number_clicked, na.rm = TRUE),
    dur_mean     = mean(`Answer Duration`,   na.rm = TRUE),
    n_left       = first(n_left),
    n_right      = first(n_right),
    .groups      = "drop"
  ) %>%
  mutate(
    abs_error       = abs(click_mean - `Answer Key`),
    PAE_mid         = abs_error / (n_right - n_left),
    IE_mid          = dur_mean / (1 - PAE_mid)
  ) %>%
  select(TestType, Identifier, `Answer Key`,
         click_mean, dur_mean,
         abs_error, PAE_mid, IE_mid)

item_pae_mid_summary <- nl_mid %>%
  group_by(TestType, Target = `Answer Key`) %>%
  summarise(
    n        = n_distinct(Identifier),
    pae_mean = mean(PAE_mid, na.rm = TRUE),
    pae_sd   = sd(PAE_mid,   na.rm = TRUE),
    pae_min  = min(PAE_mid,  na.rm = TRUE),
    pae_max  = max(PAE_mid,  na.rm = TRUE),
    pae_skew = skewness(PAE_mid),
    pae_kurt = kurtosis(PAE_mid),
    .groups  = "drop"
  )

item_ie_mid_summary <- nl_mid %>%
  group_by(TestType, Target = `Answer Key`) %>%
  summarise(
    n       = n_distinct(Identifier),
    ie_mean = mean(IE_mid, na.rm = TRUE),
    ie_sd   = sd(IE_mid,   na.rm = TRUE),
    ie_min  = min(IE_mid,  na.rm = TRUE),
    ie_max  = max(IE_mid,  na.rm = TRUE),
    ie_skew = skewness(IE_mid),
    ie_kurt = kurtosis(IE_mid),
    .groups = "drop"
  )

item_pae_mid_summary %>% 
  mutate(across(where(is.numeric), ~ round(.x, 3))) %>% 
  kbl(
    caption   = "Item-level PAE (midpoint) summary",
    align     = c("l", rep("r", 7)),  
    col.names = c("Probe", "Target", "n",
                  "Mean_PAE", "SD_PAE", "Min_PAE", "Max_PAE", "Skew_PAE", "Kurt_PAE")
  ) %>% 
  kable_styling(full_width = FALSE, position = "left") %>% 
  scroll_box(width = "100%", height = "400px")

Item-level PAE (midpoint) summary
Probe	Target	n	Mean_PAE	SD_PAE	Min_PAE	Max_PAE	Skew_PAE	Kurt_PAE
BNL0-100	4	2295	0.094	0.111	0.000	0.955	3.231	16.564
BNL0-100	13	2288	0.172	0.137	0.000	0.870	1.266	4.957
BNL0-100	20	2301	0.163	0.134	0.000	0.794	1.328	5.030
BNL0-100	31	2293	0.122	0.110	0.000	0.684	1.569	5.818
BNL0-100	46	2299	0.101	0.084	0.000	0.540	1.537	5.826
BNL0-100	59	2308	0.079	0.073	0.000	0.590	1.855	7.912
BNL0-100	68	2284	0.115	0.099	0.000	0.673	1.514	6.044
BNL0-100	72	2293	0.109	0.103	0.000	0.719	1.865	7.567
BNL0-100	85	2286	0.166	0.127	0.000	0.850	1.229	4.953
BNL0-100	97	2292	0.200	0.162	0.001	0.965	1.484	5.117
BNL0-20	2	2194	0.108	0.168	0.000	0.900	2.448	8.971
BNL0-20	4	2195	0.121	0.138	0.000	0.800	2.524	10.105
BNL0-20	5	2038	0.124	0.121	0.000	0.750	2.597	11.820
BNL0-20	7	2180	0.142	0.123	0.000	0.650	1.700	6.392
BNL0-20	9	2157	0.148	0.126	0.000	0.550	1.015	3.266
BNL0-20	11	2162	0.161	0.133	0.000	0.550	0.820	2.734
BNL0-20	13	2164	0.171	0.136	0.000	0.650	1.258	4.509
BNL0-20	16	2113	0.208	0.183	0.000	0.800	1.218	3.923
BNL0-20	18	2140	0.223	0.224	0.000	0.900	1.102	3.350
BNL0-20	19	2177	0.250	0.240	0.000	0.950	0.943	2.992
UNL0-20	2	4529	0.074	0.168	0.000	0.942	3.588	15.902
UNL0-20	4	4507	0.100	0.148	0.000	0.884	3.056	13.293
UNL0-20	5	4604	0.096	0.137	0.000	0.855	3.231	15.068
UNL0-20	7	4507	0.136	0.152	0.000	0.797	2.195	8.248
UNL0-20	9	4542	0.162	0.162	0.000	0.739	1.697	5.580
UNL0-20	11	4491	0.187	0.159	0.000	0.681	1.214	4.028
UNL0-20	13	4515	0.202	0.164	0.000	0.623	0.883	2.866
UNL0-20	16	4501	0.192	0.137	0.000	0.535	0.708	2.597
UNL0-20	18	4462	0.216	0.136	0.000	0.523	0.270	1.981
UNL0-20	19	4467	0.226	0.138	0.000	0.552	0.157	1.911

Code

item_ie_mid_summary %>% 
  mutate(across(where(is.numeric), ~ round(.x, 3))) %>% 
  kbl(
    caption   = "Item-level IE (midpoint) summary",
    align     = c("l", rep("r", 7)),  
    col.names = c("Probe", "Target", "n",
                  "Mean_IE", "SD_IE", "Min_IE", "Max_IE", "Skew_IE", "Kurt_IE")
  ) %>% 
  kable_styling(full_width = FALSE, position = "left") %>% 
  scroll_box(width = "100%", height = "400px")

Item-level IE (midpoint) summary
Probe	Target	n	Mean_IE	SD_IE	Min_IE	Max_IE	Skew_IE	Kurt_IE
BNL0-100	4	2295	7.371	6.106	1.106	88.076	5.075	46.364
BNL0-100	13	2288	11.267	9.681	1.105	145.854	3.408	27.639
BNL0-100	20	2301	14.869	14.632	0.000	326.970	6.567	103.660
BNL0-100	31	2293	28.838	947.706	1.103	45388.901	47.850	2290.737
BNL0-100	46	2299	12.882	56.318	1.035	2664.628	45.469	2139.837
BNL0-100	59	2308	28.167	226.214	1.097	10456.514	42.812	1960.154
BNL0-100	68	2284	50.814	1421.811	1.011	49403.732	33.788	1143.579
BNL0-100	72	2293	10.650	46.432	1.284	2199.333	45.721	2154.944
BNL0-100	85	2286	10.106	35.844	1.242	1682.890	44.542	2076.931
BNL0-100	97	2292	11.087	11.453	1.467	304.066	10.228	209.937
BNL0-20	2	2194	41.104	121.838	2.033	4725.990	27.892	1016.591
BNL0-20	4	2195	10.567	9.934	1.011	131.992	4.112	32.541
BNL0-20	5	2038	24.409	45.263	1.033	1712.420	26.574	957.204
BNL0-20	7	2180	13.033	12.610	1.028	148.525	3.841	26.246
BNL0-20	9	2157	17.694	16.315	1.007	204.633	4.075	33.818
BNL0-20	11	2162	11.504	12.607	1.001	352.253	11.013	258.232
BNL0-20	13	2164	17.904	19.150	1.003	289.794	4.690	41.878
BNL0-20	16	2113	27.771	31.032	1.059	642.862	6.510	91.136
BNL0-20	18	2140	14.203	18.212	1.017	281.831	5.604	53.661
BNL0-20	19	2177	19.471	31.350	1.019	545.170	8.570	110.188
UNL0-20	2	4529	7.863	17.296	0.854	424.109	12.568	214.465
UNL0-20	4	4507	7.153	9.847	1.001	353.008	13.882	379.291
UNL0-20	5	4604	16.522	33.756	1.091	953.931	16.017	361.207
UNL0-20	7	4507	8.378	10.696	1.005	192.405	7.098	86.225
UNL0-20	9	4542	10.290	11.712	1.000	140.079	4.384	32.333
UNL0-20	11	4491	8.169	10.881	0.000	317.308	12.243	294.772
UNL0-20	13	4515	10.768	12.624	0.000	228.133	5.247	56.836
UNL0-20	16	4501	14.392	46.741	1.007	2978.996	56.841	3597.465
UNL0-20	18	4462	8.561	9.294	0.567	173.073	4.582	43.581
UNL0-20	19	4467	9.992	11.703	1.018	251.524	6.715	91.432

8 Probe summary statistics

Note: We first aggregate each student’s overall accuracy (PAE) and efficiency (IE) on each probe, then calculate the probe level statistics.

8.1 PAE

Code

nl_scored_probe_pae_summary <- nl_student %>% 
  group_by(TestType) %>% 
  summarise(
    n      = n_distinct(Identifier),
    pae_mean = mean(PAE),
    pae_sd   = sd(PAE),
    pae_min  = min(PAE),
    pae_max  = max(PAE),
    pae_skew = moments::skewness(PAE),
    pae_kurt = moments::kurtosis(PAE),
    .groups = "drop"
  )
  
nl_scored_probe_pae_summary %>% 
  mutate(across(where(is.numeric), round, 3)) %>% 
  kbl(
    caption   = "Probe-level PAE summary",
    digits    = 3,
    align     = c("l", rep("r", 7)),   # 1 left + 7 right = 8
    col.names = c("Probe", "n",
                  "Mean", "SD", "Min", "Max",
                  "Skew", "Kurt")
  ) %>% 
  kable_styling(full_width = FALSE, position = "left") %>% 
  scroll_box(width = "100%", height = "400px")

Probe-level PAE summary
Probe	n	Mean	SD	Min	Max	Skew	Kurt
BNL0-100	2310	0.141	0.073	0.003	0.514	1.031	4.140
BNL0-20	2334	0.168	0.101	0.006	0.773	1.199	4.805
UNL0-20	4691	0.166	0.095	0.006	0.688	1.466	6.386

8.2 Inverse efficiency

Code

nl_scored_probe_ie_summary <- nl_student %>% 
  group_by(TestType) %>% 
  summarise(
    n      = n_distinct(Identifier),
    IE_mean = mean(IE),
    IE_sd   = sd(IE),
    IE_min  = min(IE),
    IE_max  = max(IE),
    IE_skew = moments::skewness(IE),
    IE_kurt = moments::kurtosis(IE),
    .groups = "drop"
  )
  
nl_scored_probe_ie_summary%>% 
  mutate(across(where(is.numeric), round, 3)) %>% 
  kbl(
    caption   = "Probe-level IE summary",
    digits    = 3,
    align     = c("l", rep("r", 7)),   # 1 left + 7 right = 8
    col.names = c("Probe", "n",
                  "Mean", "SD", "Min", "Max",
                  "Skew", "Kurt")
  ) %>% 
  kable_styling(full_width = FALSE, position = "left") %>% 
  scroll_box(width = "100%", height = "400px")

Probe-level IE summary
Probe	n	Mean	SD	Min	Max	Skew	Kurt
BNL0-100	2310	19.430	177.640	2.333	5214.747	26.900	739.882
BNL0-20	2334	20.962	23.607	1.717	543.390	10.645	183.662
UNL0-20	4691	10.787	10.874	1.692	310.487	8.510	162.358

9 Export

# Prepare export tables
item_responses <- nl_scored %>% 
  select(Identifier, GradeGroup, `Test Identifier`, Test, 
         `Question Identifier`, `Answer Duration`, Number_clicked, 
         `Answer Key`, abs_error, PAE, IE, `Date Completed`)

item_summary <- nl_scored_item_pae_summary %>% 
  left_join(nl_scored_item_ie_summary %>% 
              select(-n, -Target),
            by = "Question Identifier")

probe_mean_summary <- nl_scored_probe_pae_summary %>% 
  left_join(nl_scored_probe_ie_summary %>% select(-n),
            by = "TestType")

mid_point_summary <- item_pae_mid_summary %>% 
  left_join(
    item_ie_mid_summary %>% select(-n), 
    by = c("TestType", "Target")
  )

# Bundle into list and export
csv_out <- list(
  Number_line_item_response      = item_responses,
  Number_line_item_summary       = item_summary,
  Number_line_probe_mean_summary = probe_mean_summary,
  Item_mid_point_summary = mid_point_summary,
  Click_spread = click_spread_df
)

imap(csv_out, ~ {
  out_path <- file.path(probe_dir, paste0(.y, ".csv"))
  write_csv(.x, out_path)
})

$Number_line_item_response
# A tibble: 132,736 × 12
   Identifier           GradeGroup `Test Identifier` Test  `Question Identifier`
   <chr>                <chr>      <chr>             <chr> <chr>                
 1 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_001          
 2 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_002          
 3 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_003          
 4 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_004          
 5 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_005          
 6 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_006          
 7 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_007          
 8 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_008          
 9 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_009          
10 00107783-010a-f011-… Foundatio… BNL0-20F_2025     Boun… BNL0-20_010          
# ℹ 132,726 more rows
# ℹ 7 more variables: `Answer Duration` <dbl>, Number_clicked <dbl>,
#   `Answer Key` <dbl>, abs_error <dbl>, PAE <dbl>, IE <dbl>,
#   `Date Completed` <chr>

$Number_line_item_summary
# A tibble: 50 × 15
   `Question Identifier`     n Target pae_mean pae_sd   pae_min pae_max pae_skew
   <chr>                 <int>  <dbl>    <dbl>  <dbl>     <dbl>   <dbl>    <dbl>
 1 BNL0-100_001           2232     59   0.0854 0.0864 0.000120    0.59      1.93
 2 BNL0-100_002           2201     20   0.171  0.137  0.000185    0.788     1.44
 3 BNL0-100_003           2220     46   0.103  0.0952 0.0000880   0.521     1.52
 4 BNL0-100_004           2232     13   0.168  0.146  0.000250    0.866     1.45
 5 BNL0-100_005           2219     72   0.113  0.114  0.000111    0.718     2.01
 6 BNL0-100_006           2223     31   0.130  0.121  0.000232    0.69      1.69
 7 BNL0-100_007           2241     97   0.200  0.173  0.000584    0.968     1.70
 8 BNL0-100_008           2225     85   0.168  0.132  0.0000463   0.85      1.52
 9 BNL0-100_009           2216     68   0.128  0.112  0.0000927   0.68      1.59
10 BNL0-100_010           2254      4   0.0904 0.121  0.000130    0.951     3.91
# ℹ 40 more rows
# ℹ 7 more variables: pae_kurt <dbl>, IE_mean <dbl>, IE_sd <dbl>, IE_min <dbl>,
#   IE_max <dbl>, IE_skew <dbl>, IE_kurt <dbl>

$Number_line_probe_mean_summary
# A tibble: 3 × 14
  TestType     n pae_mean pae_sd pae_min pae_max pae_skew pae_kurt IE_mean IE_sd
  <chr>    <int>    <dbl>  <dbl>   <dbl>   <dbl>    <dbl>    <dbl>   <dbl> <dbl>
1 BNL0-100  2310    0.141 0.0733 0.00335   0.514     1.03     4.14    19.4 178. 
2 BNL0-20   2334    0.168 0.101  0.00574   0.773     1.20     4.81    21.0  23.6
3 UNL0-20   4691    0.166 0.0952 0.00637   0.688     1.47     6.39    10.8  10.9
# ℹ 4 more variables: IE_min <dbl>, IE_max <dbl>, IE_skew <dbl>, IE_kurt <dbl>

$Item_mid_point_summary
# A tibble: 30 × 15
   TestType Target     n pae_mean pae_sd    pae_min pae_max pae_skew pae_kurt
   <chr>     <dbl> <int>    <dbl>  <dbl>      <dbl>   <dbl>    <dbl>    <dbl>
 1 BNL0-100      4  2295   0.0938 0.111  0.000148     0.955     3.23    16.6 
 2 BNL0-100     13  2288   0.172  0.137  0.000213     0.87      1.27     4.96
 3 BNL0-100     20  2301   0.163  0.134  0.000185     0.794     1.33     5.03
 4 BNL0-100     31  2293   0.122  0.110  0.00000927   0.684     1.57     5.82
 5 BNL0-100     46  2299   0.101  0.0839 0.0000572    0.54      1.54     5.83
 6 BNL0-100     59  2308   0.0787 0.0729 0.000102     0.59      1.85     7.91
 7 BNL0-100     68  2284   0.115  0.0995 0.000204     0.673     1.51     6.04
 8 BNL0-100     72  2293   0.109  0.103  0.0000185    0.719     1.87     7.57
 9 BNL0-100     85  2286   0.166  0.127  0.000139     0.85      1.23     4.95
10 BNL0-100     97  2292   0.200  0.162  0.000584     0.965     1.48     5.12
# ℹ 20 more rows
# ℹ 6 more variables: ie_mean <dbl>, ie_sd <dbl>, ie_min <dbl>, ie_max <dbl>,
#   ie_skew <dbl>, ie_kurt <dbl>

$Click_spread
# A tibble: 68,064 × 5
   TestType Identifier                           Target n_attempts click_spread
   <chr>    <chr>                                 <dbl>      <int>        <dbl>
 1 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e      4          2         2.04
 2 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     13          2         3.47
 3 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     20          2        11.1 
 4 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     31          2        45.4 
 5 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     46          2         8.79
 6 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     59          2        14.3 
 7 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     68          2        14.7 
 8 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     72          2         2.46
 9 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     85          2         6.47
10 BNL0-100 00137783-010a-f011-bcb9-8fb474f3e22e     97          2         1.01
# ℹ 68,054 more rows