This presentation uses simple linear regression to analyze an example in biology.

The goal is to understand whether leaf length can help predict plant height.

This type of analysis is commonly used in biology to study relationships between measurable traits.

leaf_length <- c(4.1, 4.5, 5.0, 5.2, 5.8, 6.0, 6.3, 6.8, 7.1, 7.5)
plant_height <- c(10.2, 11.0, 12.5, 12.9, 13.8, 14.4, 15.1, 16.0, 16.8, 17.5)
plants <- data.frame(leaf_length, plant_height)
plants

##    leaf_length plant_height
## 1          4.1         10.2
## 2          4.5         11.0
## 3          5.0         12.5
## 4          5.2         12.9
## 5          5.8         13.8
## 6          6.0         14.4
## 7          6.3         15.1
## 8          6.8         16.0
## 9          7.1         16.8
## 10         7.5         17.5

ggplot(plants, aes(x = leaf_length, y = plant_height)) +
  geom_point() + 
  labs(
    title = "Leaf Length vs. Plant Height",
    x = "Leaf Length (cm)",
    y = "Plant Height (cm)"
)

ggplot(plants, aes(x = leaf_length, y = plant_height)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  labs(
    title = "Regression Line for Plant Height",
    x = "Leaf Length (cm)",
    y = "Plant Height (cm)"
  )

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

\[ y = \beta_0 + \beta_1 x + \epsilon \]

In this example, \(y\) is plant height and \(x\) is leaf length.

\[ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]

This formula calculates the slope of the regression line.

plot_ly(
  plants,
  x = ~leaf_length,
  y = ~plant_height,
  type = "scatter",
  mode = "markers"
)

ggplot(plants, aes(x = leaf_length, y = plant_height)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE)

Simple linear regression shows the relationship between two quantitative variables.

In this biology example, longer leaf length appears to be associated with greater plant height. This indicates that there is a positive relationship between the two variables.