Session 6 - Assignment 3

Calling Packages

library(tidyverse)

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library(ggplot2)
library(plotly)

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Week 5 Lab’s Simulated Data Parameters

# Define function simulate plant growth based on
#soil moisture content
simulate_plant_growth <- function(n = 500, 
                                  alpha = 2, 
                                  beta1 = 20, 
                                  rse = 8, 
                                  moisture_range = c(0.1, 5.0)) {
  # Generates randomized soil moisture values within range
  soil_moisture <- runif(n, min = moisture_range[1], max = moisture_range[2])
  # Generates random error values using normal distribution
  error <- rnorm(n, mean = 8, sd = rse)
  # Growth rate linear regression equation
  growth_rate <- alpha + beta1 * soil_moisture + error
  
  # Add soil moisture and growth rate to same data frame
  data.frame(
    soil_moisture = soil_moisture,
    growth_rate = growth_rate
  )
}

Run simulation w/Seed Number as Student Number

Simulate Week 5 Lab’s Data

# Simulate plant growth data w/baseline parameters
data_single <- simulate_plant_growth()

head(data_single)

##   soil_moisture growth_rate
## 1     2.2074530    46.70344
## 2     0.1299629    14.43521
## 3     1.4455090    45.87660
## 4     3.7324001    84.94240
## 5     1.0463439    33.10439
## 6     1.9395232    50.88101

Week 5 Lab’s Fitting Linear Regression

fit_single <- lm(growth_rate ~ soil_moisture, data = data_single)

# Extract fitted values and residuals
data_single <- data_single %>%
  mutate(
    fitted = predict(fit_single),
    resid_model = growth_rate - fitted, # Residuals from line (SSE)
    resid_mean  = growth_rate - mean(growth_rate) 
  ) # Deviations from Y_bar (SST)

Question 1. a) & 1. b)

# Summary table
summary(fit_single)

## 
## Call:
## lm(formula = growth_rate ~ soil_moisture, data = data_single)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -28.8926  -5.2028  -0.2195   5.4681  22.9886 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     9.4114     0.7187   13.09   <2e-16 ***
## soil_moisture  20.3529     0.2466   82.53   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.809 on 498 degrees of freedom
## Multiple R-squared:  0.9319, Adjusted R-squared:  0.9317 
## F-statistic:  6811 on 1 and 498 DF,  p-value: < 2.2e-16

#ANOVA table
anova(fit_single)

## Analysis of Variance Table
## 
## Response: growth_rate
##                Df Sum Sq Mean Sq F value    Pr(>F)    
## soil_moisture   1 415261  415261  6810.5 < 2.2e-16 ***
## Residuals     498  30365      61                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Questions 2-7: Equations

Question 2. Equation for yi

\[ Y_i = \beta_0 + \beta_1X_i + \varepsilon_i \] ## Question 3. ### Display Regression Coefficients

coef(fit_single)

##   (Intercept) soil_moisture 
##      9.411372     20.352868

Equation for yi w/hat

\[ \hat{Y}_i = b_0 + b_1X_i = 9.411372 + 20.352868X_i \]

Questions 4-7

#Total Variance
SST <- sum(data_single$resid_mean^2)

#Residual Variance
SSE <- sum(data_single$resid_model^2)

#Sum Of Squares For Regression
SSR <- SST - SSE

#Mean Square Error
MSE <- SSE / df.residual(fit_single)

#F Statistic
F_Stat <- (SSR / 1) / MSE

#Coefficient Of Determination
R2 <- SSR / SST 
##Or
R2 <- 1 - (SSE / SST)

Question 8. Plot Regression (From Week 5)

ggplot(data_single, aes(x = soil_moisture, y = growth_rate)) +
  geom_point(alpha = 0.4) +
  geom_smooth( #Linear Regression line, and Confidence Band
    method = "lm", 
    formula = y ~ x, 
    color = "red", 
    se = TRUE, 
    fill = "blue", alpha = 0.7, linewidth = 0.5
  ) +
    geom_segment( #Residual diff between observed/actual growth rate
    aes(xend = soil_moisture, yend = predict(fit_single)),
    alpha = 0.4,
    color = "green"
  ) +
  theme_classic() +
  labs(
    title = "Model Residuals (Vertical Distance to Regression Line)",
    x = "Soil Moisture Content",
    y = "Growth Rate"
  )