187 lines
3.7 KiB
Plaintext
187 lines
3.7 KiB
Plaintext
---
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title: "Assignment 4"
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subtitle: "STAT3373"
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author: "Isaac Shoebottom"
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date: "Oct 16th, 2025"
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output:
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html_document:
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df_print: paged
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pdf_document: default
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---
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```{r message=FALSE, warning=FALSE}
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library(tidyverse)
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library(knitr)
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```
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# Question 1
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## a)
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```{r}
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# Create the dataset
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data <- tibble(
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Farm = factor(1:4),
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Fert1 = c(48, 45, 52, 44),
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Fert2 = c(55, 50, 58, 49),
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Fert3 = c(52, 49, 55, 47)
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)
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# Convert to long format
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long_data <- data %>%
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pivot_longer(
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cols = starts_with("Fert"),
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names_to = "Fertilizer",
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values_to = "Yield"
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) %>%
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mutate(Fertilizer = factor(Fertilizer))
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kable(long_data, caption = "Yield Data (Bushels per Acre)")
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```
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## b)
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Model: $$Y_{ij} = \mu + \tau_i + \beta_j + \varepsilon_{ij}$$
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```{r}
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anova_model <- aov(Yield ~ Fertilizer + Farm, data = long_data)
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anova_table <- summary(anova_model)
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anova_table
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```
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Conclusions:
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- Fertilizer effect is significant (p \< 0.05)
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- Farm (block) effect is also significant
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## c)
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```{r}
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tukey_results <- TukeyHSD(anova_model, "Fertilizer")
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tukey_results
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```
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Results:
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- Fertilizer 2 produces the highest yields
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- All fertilizer pairs differ significantly
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- Ordering of mean yields: Fert 2 \> Fert 3 \> Fert 1
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Final Conclusion (alpha = 0.05)
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- There is strong statistical evidence that fertilizer type affects yield.
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- Blocking by farm was appropriate and reduced error variability.
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- Fertilizer 2 is the most effective option based on yield.
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# Question 2
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## a)
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```{r}
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drug_data <- data.frame(
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patient = factor(rep(1:5, each = 3)),
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drug = factor(rep(c("A", "B", "C"), times = 5)),
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response_time = c(
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12, 10, 15, # Patient 1
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14, 11, 16, # Patient 2
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10, 8, 13, # Patient 3
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13, 10, 14, # Patient 4
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11, 9, 14 # Patient 5
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)
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)
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kable(drug_data, caption = "Drug Trial Response Times (seconds)")
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```
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## b)
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Model: $$Y_{ij} = \mu + \tau_i + \beta_j + \varepsilon_{ij}$$
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```{r}
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anova_model <- aov(response_time ~ drug + patient, data = drug_data)
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summary(anova_model)
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```
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Decision (alpha = 0.05):
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- Drug effect is significant
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- Patient (block) effect is significant
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## c)
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Residual Diagnostics
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```{r}
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par(mfrow = c(1, 2))
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plot(anova_model, which = 1) # Residuals vs Fitted
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plot(anova_model, which = 2) # Normal Q-Q
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par(mfrow = c(1, 1))
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```
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Formal Tests
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```{r}
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# Normality of residuals
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shapiro.test(residuals(anova_model))
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# Homogeneity of variance
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bartlett.test(response_time ~ drug, data = drug_data)
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```
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Results:
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- Residuals are approximately normally distributed
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- Variances across drug groups are homogeneous
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## d)
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Multiple Comparisons
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```{r}
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tukey_results <- TukeyHSD(anova_model, "drug")
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tukey_results
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```
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Results:
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- All drug pairs differ significantly
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- Ordering of mean response times: Drug B < Drug A < Drug C
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## e)
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Mean Response Times by Drug
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```{r}
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drug_data %>%
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group_by(drug) %>%
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summarise(mean_time = mean(response_time)) %>%
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ggplot(aes(x = drug, y = mean_time)) +
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geom_col(fill = "steelblue") +
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labs(
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title = "Mean Response Time by Drug",
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x = "Drug",
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y = "Mean Response Time (seconds)"
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) +
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theme_minimal()
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```
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Boxplot by Drug
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```{r}
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ggplot(drug_data, aes(x = drug, y = response_time)) +
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geom_boxplot(fill = "lightgray") +
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labs(
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title = "Response Time Distribution by Drug",
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x = "Drug",
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y = "Response Time (seconds)"
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) +
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theme_minimal()
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```
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## f)
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Conclusion:
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At the 5% significance level, there is strong evidence that drug formulation affects patient response time. Blocking by patient was effective and significantly reduced unexplained variability. Post-hoc analysis using Tukey’s HSD showed that all three drugs differ significantly, with Drug B producing the fastest (best) response times, followed by Drug A, and then Drug C.
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