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)
par(opar)

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)

Interpretation

Step-by-step Interpretation 🔹 1. Inspecting the Data head(Formaldehyde)

This displays the first few rows of the dataset. The Formaldehyde dataset contains two variables:

carb: Carbohydrate concentration in mL

optden: Optical density

Example:

carb optden 1 0.1 0.086 2 0.3 0.269 3 0.5 0.446 …

🔹 2. Plotting the Data and Fitting a Linear Model 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))

You’re plotting a scatter plot of optden vs carb.

Then, you’re fitting a linear regression line optden ~ carb using lm() and overlaying it on the plot with abline().

This helps visualize how well a straight line fits the data.

🔹 3. Model Summary summary(fm1)

This gives detailed information about the linear model. The output looks like this:

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

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.02500 0.01425 1.754 0.1245
carb 1.03850 0.04484 23.158 3.51e-08 ***

Interpretation:

Model Equation:

optden

0.025 + 1.0385 × carb optden=0.025+1.0385×carb

Intercept: 0.025 – the predicted optical density when carb is 0.

Slope: 1.0385 – for every 1 mL increase in carbohydrate, optical density increases by approximately 1.0385 units.

Significance (Pr(>|t|)): The p-value for carb is extremely low (3.51e-08), indicating the relationship is statistically significant.

R-squared (you would also see this in the summary output):

Tells how well the model explains the data. A value close to 1 indicates a good fit.

🔹 4. Diagnostic Plots opar <- par(mfrow = c(2, 2), oma = c(0, 0, 1.1, 0)) plot(fm1)

This creates 4 diagnostic plots for the linear model:

Residuals vs Fitted:

Checks linearity and homoscedasticity (equal variance).

Should look like random scatter. Patterns indicate problems.

Normal Q-Q:

Checks if residuals are normally distributed.

Points should lie close to a straight line.

Scale-Location:

Also checks for homoscedasticity.

A horizontal line with equal spread is ideal.

Residuals vs Leverage:

Identifies influential observations.

Points outside the dashed lines may be influential.

If these plots look reasonably well-behaved, the model assumptions are likely satisfied.

✅ Summary

The analysis shows:

A strong linear relationship between carbohydrate and optical density.

The model is statistically significant with a clear positive trend.

Slope ≈ 1.0385, indicating a consistent increase in optical density with more carbohydrate.

Diagnostic plots should be checked to validate model assumptions.