**Sign Test & Wilcoxon Signed-Rank Test on Sleep Data:-**

```{r}
data(sleep)
```

```{r}
sleep
```

```{r}
drug1 <- sleep$extra[sleep$group == 1]
drug2 <- sleep$extra[sleep$group == 2]
```

```{r}
differences <- drug2 - drug1
```

```{r}
d <- differences[differences != 0]
```

```{r}
binom.test(
  sum(d > 0),
  length(d),
  p = 0.5
)
```

```{r}
wilcox.test(
  drug2,
  drug1,
  paired = TRUE,
  alternative = "two.sided",
  exact = FALSE
)
```

```{r}
wilcox.test(
  differences,
  mu = 0,
  alternative = "two.sided",
  exact = FALSE
)
```

**Wilcoxon Rank-Sum Test or Mann-Whitney U-Test on Air Quality Data:-**

```{r}
data(airquality)
```

```{r}
airquality
```

```{r}
may <- na.omit(airquality$Ozone[airquality$Month == 5])
august <- na.omit(airquality$Ozone[airquality$Month == 8])
```

```{r}
may
```

```{r}
wilcox.test(
  may,
  august,
  alternative = "two.sided",
  exact = FALSE
)
```

**Linear Regression & Multiple Linear Regression on Dental Data:-**

```{r}
data(ToothGrowth)
```

```{r}
ToothGrowth
```

```{r}
unique(ToothGrowth$supp)
```

```{r}
linear_model <- lm(len ~ dose, data = ToothGrowth)

summary(linear_model)
```

```{r}
plot(ToothGrowth$dose, ToothGrowth$len,
     xlab = "Supplement Dosage",
     ylab = "Tooth length",
     main = "Tooth length vs dose")

abline(linear_model)
```

```{r}
multiple_model <- lm(len ~ dose + supp, data = ToothGrowth)

summary(multiple_model)
```

```{r}
interaction_model <- lm(len ~ dose * supp, data = ToothGrowth)

summary(interaction_model)
```

```{r}
par(mfrow = c(2, 2))
plot(interaction_model)
par(mfrow = c(1, 1))
```

```{r}
# 2. Create a grid with a fine sequence of dose values for smooth regression lines
newdata_0 <- expand.grid(
  dose = seq(0.5, 2, by = 0.05),
  supp = levels(ToothGrowth$supp)
)

# 3. Predict values from the continuous model
newdata_0$predicted <- predict(interaction_model, newdata = newdata_0)

# 4. Plot the predicted linear trend lines
ggplot(newdata_0, aes(x = dose, y = predicted, color = supp, group = supp)) +
  geom_line(linewidth = 1) +
  labs(
    x = "Dose (mg/day)",
    y = "Predicted Tooth Length",
    color = "Supplement"
  ) +
  theme_pubr()
```

**Linear Regression with Interactions:-**

```{r}
factor_model <- lm(len ~ factor(dose) * supp, data = ToothGrowth)
summary(factor_model)
```

```{r}
par(mfrow = c(2, 2))
plot(factor_model)
par(mfrow = c(1, 1))
```

```{r}
library(ggplot2)
library(ggpubr)
```

```{r}
newdata <- expand.grid(
  dose = c(0.5, 1, 2),
  supp = levels(ToothGrowth$supp)
)

newdata$predicted <- predict(factor_model, newdata = newdata)

ggplot(newdata, aes(x = dose, y = predicted, color = supp, group = supp)) +
  geom_line(linewidth = 1) +
  geom_point(size = 3) +
  labs(
    x = "Dose",
    y = "Predicted tooth length",
    color = "Supplement"
  ) +
  theme_pubr()
```

**Variance Inflation Factor:-**

```{r}
library(car)
```

```{r}
data(mtcars)
```

```{r}
mtcars
```

```{r}
cars_model_1 <- lm(mpg ~ hp + wt, data = mtcars)
```

```{r}
vif(cars_model_1)
```

```{r}
cars_model_2 <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)

```

```{r}
vif(cars_model_2)
```

**When is Linear Regression Inadequate? Introduction to GLMs**

```{r}
set.seed(123)

n <- 200

x <- runif(n, 0, 10)
y <- 10 + 2*x + rnorm(n, mean = 0, sd = 3)

normal_data <- data.frame(x, y)
```

```{r}
normal_data
```

```{r}
normal_model <- lm(y ~ x, data = normal_data)

summary(normal_model)
```

```{r}
plot(
  normal_data$x,
  normal_data$y,
  pch = 19,
  xlab = "X",
  ylab = "Y",
  main = "Normally Distributed Continuous Data"
)

abline(normal_model, lwd = 2)
```

```{r}
par(mfrow = c(2, 2))
plot(normal_model)
par(mfrow = c(1, 1))
```

```{r}
hist(
  residuals(normal_model),
  main = "Distribution of Residuals",
  xlab = "Residuals"
)
```

```{r}
hist(
  normal_data$y,
  main = "Distribution of the Response Variable",
  xlab = "Response Variable"
)
```

```{r}
qqnorm(residuals(normal_model))
qqline(residuals(normal_model))
```

```{r}
set.seed(123)

n <- 200

x <- runif(n, 0, 10)

mu <- exp(1 + 0.3*x)

y_count <- rnbinom(
  n,
  size = 2,
  mu = mu
)

nb_data <- data.frame(x, y_count)
```

```{r}
nb_data
```

```{r}
hist(
  nb_data$y,
  main = "Distribution of the Response Variable",
  xlab = "Response Variable"
)
```

```{r}
mean(nb_data$y_count)
var(nb_data$y_count)
```

```{r}
nb_lm <- lm(y_count ~ x, data = nb_data)

summary(nb_lm)
```

```{r}
plot(
  nb_data$x,
  nb_data$y_count,
  pch = 19,
  xlab = "X",
  ylab = "Count",
  main = "Negative Binomial Counts with Linear Regression"
)

abline(nb_lm, lwd = 2)
```

```{r}
par(mfrow = c(2, 2))
plot(nb_lm)
par(mfrow = c(1, 1))
```

```{r}
library(MASS)

nb_model <- glm.nb(
  y_count ~ x,
  data = nb_data
)

summary(nb_model)
```

```{r}
par(mfrow = c(2, 2))
plot(nb_model)
par(mfrow = c(1, 1))
```