Calibration

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[1] "/Users/mb312/dev_trees/excel-but-simpler"

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The first thing we need to do, it so load the various tidyverse packages. We do this with the R library command. library loads code packages so that we can use commands from those packages in our R session:

library('tidyverse')

package ‘tidyverse’ was built under R version 3.2.5Note: the specification for S3 class “difftime” in package ‘lubridate’ seems equivalent to one from package ‘hms’: not turning on duplicate class definitions for this class.
Loading tidyverse: ggplot2
Loading tidyverse: tibble
Loading tidyverse: tidyr
Loading tidyverse: readr
Loading tidyverse: purrr
Loading tidyverse: dplyr
package ‘ggplot2’ was built under R version 3.2.5package ‘tibble’ was built under R version 3.2.5package ‘tidyr’ was built under R version 3.2.5package ‘readr’ was built under R version 3.2.5package ‘purrr’ was built under R version 3.2.5Conflicts with tidy packages ------------------------------
filter(): dplyr, stats
lag():    dplyr, stats

Let’s load the data first:

calibration = read_csv('calibration.csv')

Parsed with column specification:
cols(
  BSA = col_integer(),
  Absorbance = col_double()
)

calibration

Next we would like to do a simple scatterplot of the data. Do do this we will use the ggplot function from the ggplot2 package. ggplot2 is part of the tidyverse set of packages, so we have already loaded it with our library('tidyverse') command above. Have a look at the command output above to confirm.

ggplot(calibration, aes(x=BSA, y=Absorbance)) +
    geom_point(shape=1)      # Use hollow circles

We would also like to see a regression line through these points. We do this by adding a lm (Linear Model) component to the plot, like this:

ggplot(calibration, aes(x=BSA, y=Absorbance)) +
    geom_point(shape=1) +    # Use hollow circles
    geom_smooth(method=lm)   # linear model regression line

What is the formula for this line? We can show this (the least squares regression line) by estimating it again using R’s lm command (Linear Model again):

fit = lm(Absorbance ~ BSA, data=calibration)
summary(fit)

essentially perfect fit: summary may be unreliable

Call:
lm(formula = Absorbance ~ BSA, data = calibration)

Residuals:
       Min         1Q     Median         3Q        Max 
-3.348e-17 -1.118e-17 -7.213e-18  5.671e-18  5.574e-17 

Coefficients:
             Estimate Std. Error   t value Pr(>|t|)    
(Intercept) 1.850e-17  1.227e-17 1.508e+00    0.175    
BSA         6.000e-03  4.952e-19 1.212e+16   <2e-16 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.6e-17 on 7 degrees of freedom
Multiple R-squared:      1, Adjusted R-squared:      1 
F-statistic: 1.468e+32 on 1 and 7 DF,  p-value: < 2.2e-16

Notice the Intercept and BSA slope in the Estimate column above.

So far we have the line for predicting Absorbance from BSA - but it’s a simple switch to predict BSA from Absorbance:

reverse_fit = lm(BSA ~ Absorbance, data=calibration)
summary(reverse_fit)

essentially perfect fit: summary may be unreliable

Call:
lm(formula = BSA ~ Absorbance, data = calibration)

Residuals:
       Min         1Q     Median         3Q        Max 
-6.646e-15 -2.927e-15 -2.687e-15  1.480e-15  1.645e-14 

Coefficients:
             Estimate Std. Error   t value Pr(>|t|)    
(Intercept) 7.105e-15  3.382e-15 2.101e+00   0.0738 .  
Absorbance  1.667e+02  2.274e-14 7.330e+15   <2e-16 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.163e-15 on 7 degrees of freedom
Multiple R-squared:      1, Adjusted R-squared:      1 
F-statistic: 5.372e+31 on 1 and 7 DF,  p-value: < 2.2e-16

What it the relationship between the estimate of slope for this line, and the estimate for the Absorbance ~ BSA line?

We can use this fit to predict BSA from other values of Absorbance. To do this, we make a new data table with the Absorbance values to predict for. The tibble function makes a new data table. We pass it the name of the new column - in our case, Absorbance:

new_absorbance = tibble(Absorbance=c(0.25, 0.04, 0.4))
new_absorbance

Now we can predict BSA values from the new Absorbance values:

predict(reverse_fit, new_absorbance)

        1         2         3 
41.666667  6.666667 66.666667