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Rmd	cb56a96	Paloma	2025-04-25	more figures
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Rmd	8108c43	Paloma	2025-04-25	improved figures
html	8108c43	Paloma	2025-04-25	improved figures
Rmd	0ed450f	Paloma	2025-04-25	updated input (myassays.com) including dilution
html	0ed450f	Paloma	2025-04-25	updated input (myassays.com) including dilution
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html	dd200fc	Paloma	2025-04-23	corrected figures
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Rmd	82ad928	Paloma	2025-04-17	upd
html	82ad928	Paloma	2025-04-17	upd
Rmd	16ce91c	Paloma	2025-04-10	recalc_evaluations
html	16ce91c	Paloma	2025-04-10	recalc_evaluations
html	bbb70a9	Paloma	2025-04-09	comparing methods
Rmd	ccad031	Paloma	2025-04-09	new_calc
html	ccad031	Paloma	2025-04-09	new_calc
Rmd	77c2ab5	Paloma	2025-04-08	cleaning test3
html	77c2ab5	Paloma	2025-04-08	cleaning test3

Summary

Final cortisol value calculations were conducted using three 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.
Sam’s Method (Method C): Adjusts for spiked samples using a different equation

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. This could be an artifact of an extremely high absorbance level caused by an excessive amount of spike. Non-spiked samples, on the other hand, result in values that are within the range found in similar studies of cortisol in human hair.

Summary	Nist et al.	(A) Standard	(B) Spike-Corrected	(C) Sam’s
Mean cort conc (pg/mg)	—	16.796	-0.1866	9.568
Median cort conc (pg/mg)	—	10.801	4.4386	10.058
Range cort conc (pg/mg)	—	2.7 to 60.4	-30.7005 to 12.0647	2.786 to 22.566

Weight (mg) of my samples
Range	11 to 37.1
Mean	23.54
Median	22.4

Conclusions: After accounting for differences in dilution and weight, our results suggest future Assays should use the optimal parameters listed below:

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

Explanation of each variable used in calculations

Ave_Conc_pg/ml: average ELISA reading per sample in pg/mL
Weight_mg: hair weight in mg
Buffer_nl: assay buffer volume in nL → we convert to mL
Spike: binary indicator (1 = spiked sample)
SpikeVol_uL: volume of spike added in µL
Dilution: dilution factor (already accounted for)
Vol_in_well.tube_uL: total volume in well/tube in µL (for spike correction)
std: standard reading value
extraction: methanol volume ratio = vol added / vol recovered (e.g. 1/0.75 ml)

Cortisol concentration calculations

Input is data with low quality samples flagged, but they get removed before continuing with calculations.

Parameters and unit transformations:

# Define volume of methanol used for cortisol extraction
# vol added / vol recovered (mL)
extraction <- 1 / 0.75

# Reading of spike standard and conversion to ug/dl
std <- 3139.5 # test 3 backfit
std_ul.dl <- std / 10000 # std in ul/dl

# Creating variables in indicated units

df$Buffer_ml <- c(df$Buffer_nl/1000) # dilution (buffer)
df$Ave_Conc_ug.dl <- c(df$Ave_Conc_pg.ml/10000) # Transform to μg/dl from assay output

Wells	Sample	Category	Binding.Perc	Ave_Conc_pg.ml	Ave_Conc_ug.dl	Weight_mg	Buffer_ml	Spike	SpikeVol_ul	Dilution	TotalVol_well_ul	Failed_samples	Sample_comparable
E5	11	NoSpike	71.6	513.2	0.05132	17.5	0.25	0	0	1	50	NA	14
F5	12	YesSpike	30.0	2728.0	0.27280	24.1	0.25	1	25	1	50	NA	17
G5	13	YesSpike	32.1	2477.0	0.24770	16.8	0.25	1	25	1	50	NA	14

(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 #####
##################################

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

Summary of all samples

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.786   8.021  10.801  16.796  15.729  60.483

	Wells	Sample	Category	Binding.Perc	Ave_Conc_pg.ml	Ave_Conc_ug.dl	Weight_mg	Buffer_ml	Spike	SpikeVol_ul	Dilution	TotalVol_well_ul	Failed_samples	Sample_comparable	Final_conc_pg.mg
27	H3	6	YesSpike	34.7	2204.0	0.22040	13.7	0.06	1	25	1	50	NA	2	12.870073
28	A5	7	YesSpike	30.5	2669.0	0.26690	16.4	0.06	1	25	1	50	NA	3	13.019512
29	B5	8	NoSpike	77.8	386.8	0.03868	15.3	0.25	0	0	1	50	NA	14	8.427015
30	C5	9	YesSpike	30.3	2693.0	0.26930	19.2	0.25	1	25	1	50	NA	15	46.753472

(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”

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

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

Summary of all samples

    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-30.7005  -9.6103   4.4386  -0.1866  10.1987  12.0647

Summary without spiked samples

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.786   5.217   9.101   8.111  10.754  12.065

	Wells	Sample	Category	Binding.Perc	Ave_Conc_pg.ml	Ave_Conc_ug.dl	Weight_mg	Buffer_ml	Spike	SpikeVol_ul	Dilution	TotalVol_well_ul	Failed_samples	Sample_comparable	Final_conc_pg.mg
28	A5	7	YesSpike	30.5	2669.0	0.26690	16.4	0.06	1	25	1	50	NA	3	-4.590244
29	B5	8	NoSpike	77.8	386.8	0.03868	15.3	0.25	0	0	1	50	NA	14	8.427015
30	C5	9	YesSpike	30.3	2693.0	0.26930	19.2	0.25	1	25	1	50	NA	15	-15.503472

(C) Sam’s calculation

Developed using Sam’s advice and logic. To facilitate the understanding of what is going on, here I do not transform the output values from pg/ml to ug/dL (as done in A and B).

Step 1: Calculate contribution of the spike
Step 2: Substract spike from plate reading values and calculate final values accounting for dilution of the sample, weight, and reconstitution

Step 1: Calculate contribution of spike

X * Y / Z / SPd = SP

SP = final value of spike contribution in pg/mL
X = volume of spike added (mL)
Y = concentration of the spike added (pg/mL)
SPd = if serially diluted, dilution factor for the spike (i.e: 1, 2, 4, 8, etc.)
Z = total volume (mL) in the well or tube, if spike is added before loading the plate (sample + spike)

# Transforming units
data$SpikeVol_ml <- data$SpikeVol_ul/1000           # X to mL
data$TotalVol_well_ml <- data$TotalVol_well_ul/1000 # Z to mL
      # SPd = dilution (in this case, is 1 for all)

 # Calculate spike contribution to each sample
      ##  ( Spike vol. x Spike Conc.)
      ##   ------------------------  / dilution = Spike contribution
      ##        Total vol. 

data$Spike.cont_pg.ml <- (((data$SpikeVol_ml * std ) / # X * Y /
                            data$TotalVol_well_ml) /   # Z /
                              data$Dilution)           # SPd 
summary(data$Spike.cont_pg.ml)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    0.0     0.0     0.0   627.9  1569.8  1569.8

The reading for standard 1 in this plate is 3139.5

The total contribution of the Spike to each sample is 1569.75 pg/mL

Step 2 : Substract spike and calculate final values

((A - SP)/B) * (C/D) * E * SLd = F

A = pg/ml from assay output;
SP = spike contribution (in pg/ml)
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 #####
##################################
dSpiked$Final_conc_pg.mg <- 
   ((dSpiked$Ave_Conc_pg.ml - dSpiked$Spike.cont_pg.ml) / # (A - spike) 
   dSpiked$Weight_mg) *                                   # / B *
   extraction *                                           # C / D
   dSpiked$Buffer_ml                 # E 
dSpiked[ , c("Sample", "Ave_Conc_pg.ml", "Buffer_ml","Spike.cont_pg.ml", "Final_conc_pg.mg")]

   Sample Ave_Conc_pg.ml Buffer_ml Spike.cont_pg.ml Final_conc_pg.mg
1      11          513.2      0.25             0.00         9.775238
2      12         2728.0      0.25          1569.75        16.020055
3      13         2477.0      0.25          1569.75        18.000992
4      14         2504.0      0.25          1569.75        22.566425
5      15          643.6      0.06             0.00         4.290667
6      16         3196.0      0.06          1569.75         5.559829
7      18          955.4      0.25             0.00        10.339827
8      19         3730.0      0.06          1569.75         6.194265
9      21         2540.0      0.25          1569.75        11.550595
10     22         2377.0      0.25          1569.75        12.515504
11     23          793.6      0.25             0.00        10.753388
12     24          680.2      0.25             0.00        11.114379
13     26         1991.0      0.06             0.00         7.885149
14     27         1393.0      0.06             0.00         5.159259
15     28          839.7      0.25             0.00        12.064655
16     29         2072.0      0.06             0.00         4.541370
17      3         2287.0      0.06          1569.75         5.216364
18    30A         2888.0      0.25          1569.75        14.845158
19     31         1149.0      0.06             0.00         4.335849
20     32         1197.0      0.25             0.00        10.754717
21     33         1100.0      0.25             0.00        10.848126
22     34         1124.0      0.25             0.00        10.553991
23     36         1062.0      0.25             0.00        11.346154
24     37         2076.0      0.06             0.00         5.392208
25     38         2444.0      0.06             0.00         5.634582
26      5          501.4      0.06             0.00         2.785556
27      6         2204.0      0.06          1569.75         3.703650
28      7         2669.0      0.06          1569.75         5.362195
29      8          386.8      0.25             0.00         8.427015
30      9         2693.0      0.25          1569.75        19.500868

Summary for all samples:

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.786   5.370  10.058   9.568  11.499  22.566

	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
21	33	10.848126	1100.0	0.00	52.3	33.8	0.25	0	1	50
22	34	10.553991	1124.0	0.00	51.7	35.5	0.25	0	1	50
23	36	11.346154	1062.0	0.00	53.2	31.2	0.25	0	1	50
24	37	5.392208	2076.0	0.00	36.1	30.8	0.06	0	1	50
25	38	5.634582	2444.0	0.00	32.5	34.7	0.06	0	1	50
26	5	2.785556	501.4	0.00	72.1	14.4	0.06	0	1	50
27	6	3.703650	2204.0	1569.75	34.7	13.7	0.06	25	1	50
28	7	5.362195	2669.0	1569.75	30.5	16.4	0.06	25	1	50
29	8	8.427015	386.8	0.00	77.8	15.3	0.25	0	1	50
30	9	19.500868	2693.0	1569.75	30.3	19.2	0.25	25	1	50

Plots

(A) Standard Calculation

Final cortisol concentrations not accounting for spike. Tags are sample numbers.

Expected results: a straight horizontal line showing that I obtained same cortisol concentration value in the Y axis, across different sample weights.

Version	Author	Date
8108c43	Paloma	2025-04-25
dd200fc	Paloma	2025-04-23
7240d2e	Paloma	2025-04-22
16ce91c	Paloma	2025-04-10

(B) Accounting for Spike

Final cortisol concentrations accounting for Spike as instructed in Nist et al. 2020.

Expected results: lower values than in the previous plot for the spiked samples, but not as low as negative samples (for all of them). Spiked and non-spiked samples should be aligned (same concentration across different weights)

Version	Author	Date
8108c43	Paloma	2025-04-25
dd200fc	Paloma	2025-04-23

(C) Sam’s calculation

Final cortisol concentration values using new method.

Expected results: one unique horizontal line, regardless of the spiking status and dilution. We see this line for the spiked samples that were reconstituted using 60 uL (i.e, the most concentrated samples). Perhaps the 250uL samples, by being more diluted and having a larger volume, present more variation if the cort distribution within the well/tube is not homogeneous.
Note that samples seem to be less aligned or more separated from each other than in previous plots. This is due to a difference in scale (A has values 0 to 60, while here all values fall between 2.5 and 22.5).

`geom_smooth()` using formula = 'y ~ x'

Version	Author	Date
cb56a96	Paloma	2025-04-25
8108c43	Paloma	2025-04-25
dd200fc	Paloma	2025-04-23

Evaluation using method C

`geom_smooth()` using formula = 'y ~ x'

Version	Author	Date
cb56a96	Paloma	2025-04-25

Version	Author	Date
cb56a96	Paloma	2025-04-25

The previous figure shows that:

results are relatively stable across weights, particularly for the samples where a dilution of 250 uL was used
For the 250 uL reconstitutions, about 18% of the variability in Final_conc_pg.mg is explained by Weight_mg
For the 60 uL reconstitutionsAbout 6.6% of the variability is explained by Weight_mg.
Reconstitution volume 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

Version	Author	Date
cb56a96	Paloma	2025-04-25

Optimal dilution (using method C results)

Error using samples w/0.06 mL buffer

Mean Absolute Error (MAE) 0.06 mL: 0.823

Standard Deviation of Residuals 0.06 mL: 1.182

Error using samples w/0.25 mL buffer

Mean Absolute Error (MAE) 0.25 mL: 2.56

Standard Deviation of Residuals 0.25 mL: 3.47

From this we conclude that using a 60 uL dilution produces more accurate/more consistent results

Error using spiked samples only

Mean Absolute Error (MAE) ALL: 5.47

Standard Deviation of Residuals ALL: 6.474

Error using non-spiked samples only

Mean Absolute Error (MAE) ALL: 2.64

Standard Deviation of Residuals ALL: 2.956

Error using all samples

Mean Absolute Error (MAE) ALL: 3.948

Standard Deviation of Residuals ALL: 4.983

sessionInfo()

R version 4.5.0 (2025-04-11)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.4.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1

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_1.0.8     ggplot2_3.5.2  
[5] knitr_1.50     

loaded via a namespace (and not attached):
 [1] sass_0.4.10       generics_0.1.3    tidyr_1.3.1       lattice_0.22-6   
 [5] stringi_1.8.7     digest_0.6.37     magrittr_2.0.3    evaluate_1.0.3   
 [9] grid_4.5.0        fastmap_1.2.0     Matrix_1.7-3      rprojroot_2.0.4  
[13] workflowr_1.7.1   jsonlite_2.0.0    whisker_0.4.1     backports_1.5.0  
[17] rematch2_2.1.2    promises_1.3.2    mgcv_1.9-1        purrr_1.0.4      
[21] scales_1.3.0      jquerylib_0.1.4   cli_3.6.4         rlang_1.1.6      
[25] munsell_0.5.1     splines_4.5.0     withr_3.0.2       cachem_1.1.0     
[29] yaml_2.3.10       tools_4.5.0       colorspace_2.1-1  httpuv_1.6.16    
[33] vctrs_0.6.5       R6_2.6.1          lifecycle_1.0.4   git2r_0.36.2     
[37] stringr_1.5.1     fs_1.6.6          pkgconfig_2.0.3   pillar_1.10.2    
[41] bslib_0.9.0       later_1.4.2       gtable_0.3.6      glue_1.8.0       
[45] Rcpp_1.0.14       xfun_0.52         tibble_3.2.1      tidyselect_1.2.1 
[49] rstudioapi_0.17.1 farver_2.1.2      nlme_3.1-168      htmltools_0.5.8.1
[53] rmarkdown_2.29    labeling_0.4.3    compiler_4.5.0

Calculation final values, Test3

Paloma Contreras

2025-04-28

Summary

Explanation of each variable used in calculations

Cortisol concentration calculations

(A) Standard Calculation

(B) Accounting for Spike

(C) Sam’s calculation

Plots

(A) Standard Calculation

(B) Accounting for Spike

(C) Sam’s calculation

Evaluation using method C

Optimal dilution (using method C results)