R Notebook

sample size

library(gtsummary)
library(dplyr)

#Determination of Formaldehyde

head(Formaldehyde)

require(stats); require(graphics)
plot(optden ~ carb, data = Formaldehyde,
     xlab = "Carbohydrate (ml)", ylab = "Optical Density",
     main = "Formaldehyde data", col = 4, las = 1)
abline(fm1 <- lm(optden ~ carb, data = Formaldehyde))

summary(fm1)

Call:
lm(formula = optden ~ carb, data = Formaldehyde)

Residuals:
        1         2         3         4         5         6 
-0.006714  0.001029  0.002771  0.007143  0.007514 -0.011743 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.005086   0.007834   0.649    0.552    
carb        0.876286   0.013535  64.744 3.41e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.008649 on 4 degrees of freedom
Multiple R-squared:  0.999, Adjusted R-squared:  0.9988 
F-statistic:  4192 on 1 and 4 DF,  p-value: 3.409e-07

opar <- par(mfrow = c(2, 2), oma = c(0, 0, 1.1, 0))
plot(fm1)

require(stats); require(graphics)
plot(optden ~ carb, data = Formaldehyde,
     xlab = "Carbohydrate (ml)", ylab = "Optical Density",
     main = "Formaldehyde data", col = 4, las = 1)
abline(fm1 <- lm(optden ~ carb, data = Formaldehyde))

summary(fm1)

Call:
lm(formula = optden ~ carb, data = Formaldehyde)

Residuals:
        1         2         3         4         5         6 
-0.006714  0.001029  0.002771  0.007143  0.007514 -0.011743 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.005086   0.007834   0.649    0.552    
carb        0.876286   0.013535  64.744 3.41e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.008649 on 4 degrees of freedom
Multiple R-squared:  0.999, Adjusted R-squared:  0.9988 
F-statistic:  4192 on 1 and 4 DF,  p-value: 3.409e-07

opar <- par(mfrow = c(2, 2), oma = c(0, 0, 1.1, 0))

plot(fm1)
par(opar)

Interpretation:

1. Scatterplot with regression line require(stats); require(graphics) plot(optden ~ carb, data = Formaldehyde, xlab = “Carbohydrate (ml)”, ylab = “Optical Density”, main = “Formaldehyde data”, col = 4, las = 1) abline(fm1 <- lm(optden ~ carb, data = Formaldehyde))

Loads stats and graphics (not strictly necessary since they’re base packages).

Plots carbohydrate concentration (carb) vs optical density (optden).

Fits a simple linear model fm1 (optden ~ carb).

Draws the regression line onto the scatterplot.

2. Model summary summary(fm1)

Displays regression coefficients, standard errors, 𝑡 t-values, 𝑅 2 R 2 , etc.

This lets you see how well carb predicts optden.

3. Diagnostic plots opar <- par(mfrow = c(2, 2), oma = c(0, 0, 1.1, 0)) plot(fm1)

Splits the plotting window into a 2×2 grid.

Produces the four standard diagnostic plots for linear models:

Residuals vs Fitted

Normal Q-Q

Scale-Location

Residuals vs Leverage

These help you check assumptions like linearity, normality, constant variance, and influential points.

4. Combined script require(stats); require(graphics) plot(optden ~ carb, data = Formaldehyde, xlab = “Carbohydrate (ml)”, ylab = “Optical Density”, main = “Formaldehyde data”, col = 4, las = 1) abline(fm1 <- lm(optden ~ carb, data = Formaldehyde)) summary(fm1) opar <- par(mfrow = c(2, 2), oma = c(0, 0, 1.1, 0)) plot(fm1) par(opar)

This is just a condensed version of everything above:

Fits and visualizes the regression line.

Summarizes the model.

Produces diagnostic plots.

Restores plotting parameters (par(opar)).