Last updated: 2025-04-08

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Summary

Cortisol value calculations were conducted using two methods:

  • Standard Method (Method A): Calculates cortisol concentration without correction for spiked samples.
  • Spike-Corrected Method (Method B): Adjusts for spiked samples to account for addition of a known amount of cortisol, following Nist et al. 2020.

Results: As we see below, the formula used by Nist et al. results in negative values, which would mean that there is no cortisol in original samples.

Summary Nist et al. My samples Non-spiked only
Mean cort conc (pg/mg) 23.74 -0.18 7.9
Range cort conc (pg/mg) 2.1 to 124.9* -29.3 to 11.76 2.71 to 11.76
Weight range (mg) 0.4 to 10.9 11 to 37.1 12 to 37
Sample size X 30 18

This could be an artifact of an extremely high absorbance level. Non-spiked samples, however, result in values that are within the range found in similar studies of cortisol in human hair. After accounting for differences in dilution and weight, our results suggest some optimal parameters.

Conclusions:

  • Dilution of 250uL is preferable over 60uL
  • Non-spiked samples seem to generate expected results

Concerns

  • Spike results in unrealistic values
  • Could be explained by the higher weight of our samples
  • Dilution of 250uL results in values that are twice as big as with 60uL, but they should be very similar or at least overlap
# DATA SET 
current_test <- "Test3"
data_path <- file.path("./data", current_test)
# Define volume of methanol used for cortisol extraction
# vol added / vol recovered (mL)
extraction <- 1.3 / 1

# Reading of spike standard and conversion to ug/dL
std <- (3133 + 3146) / 2. # test 3 backfit
std.r <- std / 10000

Loading files and transforming units, including low quality data

df <- read.csv(file.path(data_path,"Data_QC_flagged.csv"))
kable(tail(df))
Sample Wells Raw.OD Binding.Perc Conc_pg.ml Ave_Conc_pg.ml CV.Perc SD SEM Category Weight_mg Buffer_nl Spike TotalVol_well_uL SpikeVol_uL Dilution CV_categ Binding.Perc_categ Failed_samples
34 4 F3 1.070 80.6 295.8 334.5 16.30 54.6 38.6 NoSpike 11.7 250 0 50 0 1 HIGH CV ABOVE 80% binding HIGH CV;ABOVE 80% binding
35 5 G3 0.944 72.1 488.0 501.4 3.79 19.0 13.4 NoSpike 14.4 60 0 50 0 1 NA NA NA
36 6 H3 0.490 34.7 2099.0 2204.0 6.75 149.0 105.0 YesSpike 13.7 60 1 50 25 1 NA NA NA
37 7 A5 0.436 30.5 2551.0 2669.0 6.25 167.0 118.0 YesSpike 16.4 60 1 50 25 1 NA NA NA
38 8 B5 1.030 77.8 354.9 386.8 11.70 45.1 31.9 NoSpike 15.3 250 0 50 0 1 NA NA NA
39 9 C5 0.432 30.3 2590.0 2693.0 5.46 147.0 104.0 YesSpike 19.2 250 1 50 25 1 NA NA NA 
# Creating variables in indicated units
# dilution (buffer)
df$Buffer_ml <- c(df$Buffer_nl/1000)

# remove unnecessary information 
data <- df %>%
    filter(CV.Perc < 15) %>% 
  filter(Binding.Perc < 80 & Binding.Perc > 20) %>%
    dplyr::select(Wells, Sample, Category, Binding.Perc, Ave_Conc_pg.ml, Weight_mg, Buffer_ml, Spike, SpikeVol_uL, Dilution, TotalVol_well_uL, Failed_samples) 

kable(tail(data, 10))
Wells Sample Category Binding.Perc Ave_Conc_pg.ml Weight_mg Buffer_ml Spike SpikeVol_uL Dilution TotalVol_well_uL Failed_samples
21 B11 33 NoSpike 52.3 1100.0 33.8 0.25 0 0 1 50 NA
22 C11 34 NoSpike 51.7 1124.0 35.5 0.25 0 0 1 50 NA
23 E11 36 NoSpike 53.2 1062.0 31.2 0.25 0 0 1 50 NA
24 F11 37 NoSpike 36.1 2076.0 30.8 0.06 0 0 1 50 NA
25 G11 38 NoSpike 32.5 2444.0 34.7 0.06 0 0 1 50 NA
26 G3 5 NoSpike 72.1 501.4 14.4 0.06 0 0 1 50 NA
27 H3 6 YesSpike 34.7 2204.0 13.7 0.06 1 25 1 50 NA
28 A5 7 YesSpike 30.5 2669.0 16.4 0.06 1 25 1 50 NA
29 B5 8 NoSpike 77.8 386.8 15.3 0.25 0 0 1 50 NA
30 C5 9 YesSpike 30.3 2693.0 19.2 0.25 1 25 1 50 NA 
dim(data)
[1] 30 12
# remove duplicates
data.og <- data[!is.na(data$Binding.Perc), ]

(A) Standard Calculation

Formula:

((A/B) * (C/D) * E * 10,000) = F

  • A = μg/dl from assay output;
  • B = weight (in mg) of hair subjected to extraction;
  • C = vol. (in ml) of methanol added to the powdered hair;
  • D = vol. (in ml) of methanol recovered from the extract and subsequently dried down;
  • E = vol. (in ml) of assay buffer used to reconstitute the dried extract;
  • F = final value of hair CORT Concentration in pg/mg.
##################################
##### Calculate final values #####
##################################

# Transform to μg/dl from assay output
data$Ave_Conc_ug.dL <- c(data$Ave_Conc_pg.ml/10000)

data$Final_conc_pg.mg <- c(
    ((data$Ave_Conc_ug.dL) / data$Weight_mg) * # A/B *
      extraction *                                  # C/D  *     
      data$Buffer_ml * 10000 )                 # E * 10000
data <- data[order(data$Sample),]
write.csv(data, file.path(data_path, "Data_cort_values_methodA.csv"), row.names = F)

# summary for all samples
summary(data$Final_conc_pg.mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.716   7.820  10.531  16.376  15.336  58.971 
kable(tail(data, 7))
Wells Sample Category Binding.Perc Ave_Conc_pg.ml Weight_mg Buffer_ml Spike SpikeVol_uL Dilution TotalVol_well_uL Failed_samples Ave_Conc_ug.dL Final_conc_pg.mg
24 F11 37 NoSpike 36.1 2076.0 30.8 0.06 0 0 1 50 NA 0.20760 5.257403
25 G11 38 NoSpike 32.5 2444.0 34.7 0.06 0 0 1 50 NA 0.24440 5.493718
26 G3 5 NoSpike 72.1 501.4 14.4 0.06 0 0 1 50 NA 0.05014 2.715917
27 H3 6 YesSpike 34.7 2204.0 13.7 0.06 1 25 1 50 NA 0.22040 12.548321
28 A5 7 YesSpike 30.5 2669.0 16.4 0.06 1 25 1 50 NA 0.26690 12.694024
29 B5 8 NoSpike 77.8 386.8 15.3 0.25 0 0 1 50 NA 0.03868 8.216340
30 C5 9 YesSpike 30.3 2693.0 19.2 0.25 1 25 1 50 NA 0.26930 45.584635 
dim(data)
[1] 30 14

(B) Accounting for Spike

We followed the procedure described in Nist et al. 2020:

“Thus, after pipetting 25μL of standards and samples into the appropriate wells of the 96-well assay plate, we added 25μL of the 0.333ug/dL standard to all samples, resulting in a 1:2 dilution of samples. The remainder of the manufacturer’s protocol was unchanged. We analyzed the assay plate in a Powerwave plate reader (BioTek, Winooski, VT) at 450nm and subtracted background values from all assay wells. In the calculations, we subtracted the 0.333ug/dL standard reading from the sample readings. Samples that resulted in a negative number were considered nondetectable. We converted cortisol levels from ug/dL, as measured by the assay, to pg/mg—based on the mass of hair collected and analyzed using the following formula:

A/B * C/D * E * 10,000 * 2 = F

where - A = μg/dl from assay output; - B = weight (in mg) of collected hair; - C = vol. (in ml) of methanol added to the powdered hair; - D = vol. (in ml) of methanol recovered from the extract and subsequently dried down; - E = vol. (in ml) of assay buffer used to reconstitute the dried extract; 10,000 accounts for changes in metrics; 2 accounts for the dilution factor after addition of the spike; and - F = final value of hair cortisol concentration in pg/mg”

dSpike <- data

##################################
##### Calculate final values #####
##################################

dSpike$Final_conc_pg.mg <- 
  ifelse(
    dSpike$Spike == 1,    ## Only spiked samples
      ((dSpike$Ave_Conc_ug.dL - (std.r)) / # (A-spike) / B
        dSpike$Weight_mg) 
        * extraction *                  # C / D
        dSpike$Buffer_ml * 10000 * 2,    # E * 10000 * 2
        dSpike$Final_conc_pg.mg  
    )


write.csv(dSpike, file.path(data_path, "Data_cort_values_methodB.csv"), row.names = F)

# summary for all samples
summary(dSpike$Final_conc_pg.mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
-29.933  -9.370   4.328  -0.182   9.944  11.763 
dSpikeSub <- data[c(data$Spike == 0), ]
summary(dSpikeSub$Final_conc_pg.mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.716   5.087   8.874   7.908  10.486  11.763 
kable(tail(dSpike, 10))
Wells Sample Category Binding.Perc Ave_Conc_pg.ml Weight_mg Buffer_ml Spike SpikeVol_uL Dilution TotalVol_well_uL Failed_samples Ave_Conc_ug.dL Final_conc_pg.mg
21 B11 33 NoSpike 52.3 1100.0 33.8 0.25 0 0 1 50 NA 0.11000 10.576923
22 C11 34 NoSpike 51.7 1124.0 35.5 0.25 0 0 1 50 NA 0.11240 10.290141
23 E11 36 NoSpike 53.2 1062.0 31.2 0.25 0 0 1 50 NA 0.10620 11.062500
24 F11 37 NoSpike 36.1 2076.0 30.8 0.06 0 0 1 50 NA 0.20760 5.257403
25 G11 38 NoSpike 32.5 2444.0 34.7 0.06 0 0 1 50 NA 0.24440 5.493718
26 G3 5 NoSpike 72.1 501.4 14.4 0.06 0 0 1 50 NA 0.05014 2.715917
27 H3 6 YesSpike 34.7 2204.0 13.7 0.06 1 25 1 50 NA 0.22040 -10.652409
28 A5 7 YesSpike 30.5 2669.0 16.4 0.06 1 25 1 50 NA 0.26690 -4.475488
29 B5 8 NoSpike 77.8 386.8 15.3 0.25 0 0 1 50 NA 0.03868 8.216340
30 C5 9 YesSpike 30.3 2693.0 19.2 0.25 1 25 1 50 NA 0.26930 -15.115885

(C) Skip unit transformation

##################################
##### Calculate final values #####
################################## 
datac <- data
datac$Final_conc_pg.mg <- c(
    (datac$Ave_Conc_pg.ml / datac$Weight_mg) * # A/B *
      extraction *                                  # C/D  *     
      datac$Buffer_ml)                 # E 
datac <- datac[order(datac$Sample),]
write.csv(datac, file.path(data_path, "Data_cort_values_methodC.csv"), row.names = F)

# summary for all samples
summary(datac$Final_conc_pg.mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.716   7.820  10.531  16.376  15.336  58.971

(D) Account for Spike contribution (uses vol. of spike, conc. of spike, and total reconstit. vol.)

Spike contribution (pg/mL) = (Vol. spike (mL) x Conc. spike (pg/mL) ) / Vol. reconstitution (mL) or total vol. in well (50uL) (depending on where the spike was added)

# Calculate contribution of spike according to the different volumes in which it was added
# Consider that contribution of spike in serial dilutions gets smaller 

# Vol. of spike transformed to mL
data$SpikeVol_ml <- data$SpikeVol_uL/1000
# Concentration of the spike:
std
[1] 3139.5
# Vol. reconstitution (mL) is the total volume in tube or well (sample + spike), after adding spike.
# transform to mL
data$TotalVol_well_mL <- data$TotalVol_well_uL/1000


  ##( Spike vol. x Spike Conc.)
  ## ------------------------  / dilution = Spike contribution
  ##      Total vol. 
  
# Cortisol added by spike in wells: 0.0025 mL x 3200 pg/mL = 80 pg
# Calculate cort contribution of spike to each sample
data$Spike.cont_pg.mL <- (((data$SpikeVol_ml * std ) / # Volume of spike * Spike concentration
                            data$TotalVol_well_mL) / # divided by the total volume (spike + sample)
                              data$Dilution) # resulting number changes depending on the dilution

dSpiked <- data
                          
##################################
##### Calculate final values #####
##################################


dSpiked$Final_conc_pg.mg <- 
  ifelse(
    dSpiked$Spike == 1,           ## Only spiked samples
      ((dSpiked$Ave_Conc_pg.ml - dSpiked$Spike.cont_pg.mL) / # (A - spike) / B
      dSpiked$Weight_mg) 
    * extraction *      # C / D
      dSpiked$Buffer_ml,    # E * 
    dSpiked$Final_conc_pg.mg  
)

write.csv(dSpiked, file.path(data_path, "Data_cort_values_methodD.csv"), row.names = F)

# summary for all samples
summary(dSpiked$Final_conc_pg.mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.716   5.235   9.806   9.329  11.212  22.002 
kable(tail(dSpiked[!is.na(dSpiked$Final_conc_pg.mg) , c("Sample", "Final_conc_pg.mg", "Ave_Conc_pg.ml", "Spike.cont_pg.mL", "Binding.Perc", "Weight_mg", "Buffer_ml", "SpikeVol_uL", "Dilution", "TotalVol_well_uL")],20))
Sample Final_conc_pg.mg Ave_Conc_pg.ml Spike.cont_pg.mL Binding.Perc Weight_mg Buffer_ml SpikeVol_uL Dilution TotalVol_well_uL
11 23 10.484553 793.6 0.00 60.9 24.6 0.25 0 1 50
12 24 10.836520 680.2 0.00 64.8 20.4 0.25 0 1 50
13 26 7.688020 1991.0 0.00 37.3 20.2 0.06 0 1 50
14 27 5.030278 1393.0 0.00 46.0 21.6 0.06 0 1 50
15 28 11.763039 839.7 0.00 59.5 23.2 0.25 0 1 50
16 29 4.427836 2072.0 0.00 36.2 36.5 0.06 0 1 50
17 3 5.085954 2287.0 1569.75 33.9 11.0 0.06 25 1 50
18 30A 14.474029 2888.0 1569.75 28.8 29.6 0.25 25 1 50
19 31 4.227453 1149.0 0.00 51.2 21.2 0.06 0 1 50
20 32 10.485849 1197.0 0.00 50.0 37.1 0.25 0 1 50
21 33 10.576923 1100.0 0.00 52.3 33.8 0.25 0 1 50
22 34 10.290141 1124.0 0.00 51.7 35.5 0.25 0 1 50
23 36 11.062500 1062.0 0.00 53.2 31.2 0.25 0 1 50
24 37 5.257403 2076.0 0.00 36.1 30.8 0.06 0 1 50
25 38 5.493718 2444.0 0.00 32.5 34.7 0.06 0 1 50
26 5 2.715917 501.4 0.00 72.1 14.4 0.06 0 1 50
27 6 3.611058 2204.0 1569.75 34.7 13.7 0.06 25 1 50
28 7 5.228140 2669.0 1569.75 30.5 16.4 0.06 25 1 50
29 8 8.216340 386.8 0.00 77.8 15.3 0.25 0 1 50
30 9 19.013346 2693.0 1569.75 30.3 19.2 0.25 25 1 50

Plots

(A) Standard Calculation

(B) Accounting for Spike

(C) Simplified Standard Calculation

(D) New calculation (substract half of the spike from A)

Evaluation Non-spiked Samples Only

Since all spiked samples produce negative values, we will continue our analyses using only non-spiked samples.

# non-spiked samples only
data2 <-data[data$Spike == "No", ]

#two datasets, separated by dilution
data2.06 <- data2[data2$Buffer == "60 uL", ]
data2.25 <- data2[data2$Buffer == "250 uL", ]

#### fit models ####

# model Buffer = 0.06
model06 <- lm(Final_conc_pg.mg ~ Weight_mg, 
              data = data2.06)
r_squared06 <- summary(model06)$r.squared

# model Buffer = 0.25
model25 <- lm(Final_conc_pg.mg ~ Weight_mg, 
              data = data2.25)
r_squared25 <- summary(model25)$r.squared

# Calculate residuals
residuals06 <- residuals(model06)
residuals25 <- residuals(model25)

# Quantify residuals
# Mean Absolute Error
mae06 <- mean(abs(residuals06))          
# Standard Deviation of Residuals
std_dev06 <- sd(residuals06)   

# Mean Absolute Error
mae25 <- mean(abs(residuals25))      
# Standard Deviation of Residuals
std_dev25 <- sd(residuals25)                      



# scatterplot

ggplot(data2, aes(y = Final_conc_pg.mg, 
                  x = Weight_mg, 
                  color = Buffer, 
                  fill = Buffer)) +
  geom_point(size = 2.5) +  
  geom_text(label = c(data2$Sample), nudge_y = 0.75, nudge_x = -0.5) +
  geom_smooth(method = "lm", 
              color = "gold3", 
              se = TRUE,
              alpha = 0.1) + 
  geom_hline(yintercept = mean(data2$Final_conc_pg.mg), 
             color = "gray80",
             linetype = "dashed") +
  geom_hline(yintercept = mean(data2.06$Final_conc_pg.mg), 
             color = "lightblue3",
             linetype = "dashed") +
  geom_hline(yintercept = mean(data2.25$Final_conc_pg.mg), 
             color = "lightpink3",
             linetype = "dashed") +
  theme_minimal() +  
  xlim(5, max(data2$Weight_mg) + 4) +
  ylim(0, max(data2$Final_conc_pg.mg)+4) + 
  labs(
    title = "Final Cort Concentration and Weight
    (Non-spiked only)",
    y = "Final Concentration (pg/mg)",
    x = "Weight (mg)"  
  ) +
  theme(
    plot.title = element_text(hjust = 0.5, size = 16, face = "bold"),  
    axis.title = element_text(size = 14),  
    axis.text = element_text(size = 12) 
  ) +
  # Add R^2 annotation
  annotate("text", x = max(data2$Weight_mg) * 0.7, 
           y = min(data2$Final_conc_pg.mg) * 1.5,
           label = paste("R² =", round(r_squared06, 3)), 
           size = 5, color = "black") +
  annotate("text", x = max(data2$Weight_mg) * 0.7, 
           y = max(data2$Final_conc_pg.mg) * 0.84,
           label = paste("R² =", round(r_squared25, 3)), 
           size = 5, color = "black")

The previous figure shows that:

  • results are very stable across weights, particularly for the samples where a dilution of 250 uL was used
  • there is more error when using a dilution of 60 uL
  • dilution affects estimation of cortisol concentration in a significant way: even though final concentration numbers account for differences in the dilutions, the results we observe for each group do not overlap
  • the average value when using 250 uL of buffer is twice as big as when using 60 uL

Optimal dilution

Error using 0.06 mL buffer

Mean Absolute Error (MAE) 0.06 mL: 0.873 
Standard Deviation of Residuals 0.06 mL: 1.377

Error using 0.25 mL buffer

Mean Absolute Error (MAE) 0.25 mL: 0.65 
Standard Deviation of Residuals 0.25 mL: 0.86

From this we conclude that using a 250 uL dilution provides more consistent results


sessionInfo()
R version 4.4.3 (2025-02-28)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.4

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Detroit
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] dplyr_1.1.4     paletteer_1.6.0 broom_1.0.7     ggplot2_3.5.1  
[5] knitr_1.49     

loaded via a namespace (and not attached):
 [1] sass_0.4.9        utf8_1.2.4        generics_0.1.3    tidyr_1.3.1      
 [5] prismatic_1.1.2   lattice_0.22-6    stringi_1.8.4     digest_0.6.37    
 [9] magrittr_2.0.3    evaluate_1.0.1    grid_4.4.3        fastmap_1.2.0    
[13] Matrix_1.7-2      rprojroot_2.0.4   workflowr_1.7.1   jsonlite_1.8.9   
[17] whisker_0.4.1     backports_1.5.0   rematch2_2.1.2    promises_1.3.0   
[21] mgcv_1.9-1        purrr_1.0.2       fansi_1.0.6       scales_1.3.0     
[25] jquerylib_0.1.4   cli_3.6.3         rlang_1.1.4       splines_4.4.3    
[29] munsell_0.5.1     withr_3.0.2       cachem_1.1.0      yaml_2.3.10      
[33] tools_4.4.3       colorspace_2.1-1  httpuv_1.6.15     vctrs_0.6.5      
[37] R6_2.5.1          lifecycle_1.0.4   git2r_0.35.0      stringr_1.5.1    
[41] fs_1.6.5          pkgconfig_2.0.3   pillar_1.9.0      bslib_0.8.0      
[45] later_1.3.2       gtable_0.3.6      glue_1.8.0        Rcpp_1.0.13-1    
[49] xfun_0.49         tibble_3.2.1      tidyselect_1.2.1  rstudioapi_0.17.1
[53] farver_2.1.2      nlme_3.1-167      htmltools_0.5.8.1 rmarkdown_2.29   
[57] labeling_0.4.3    compiler_4.4.3