SLR Model Analysis: Prostate Cancer Data in R

Part a) To assess whether the model assumptions have been met or fail to be met, we need to consider the following assumptions:

1. Linearity: We assume that there is a linear relationship between the predictor variable (lcavol) and the response variable (lpsa). This assumption can be checked by plotting a scatter plot of lcavol against lpsa and visually inspecting if the points appear to follow a linear pattern.

2. Independence: We assume that the observations are independent of each other. This assumption can be difficult to verify, but it is often reasonable to assume independence if the data is collected randomly or through a well-designed study.

3. Homoscedasticity: We assume that the variability of the response variable is constant across all levels of the predictor variable. This assumption can be checked by plotting the residuals (the differences between the observed and predicted values of the response variable) against the predicted values. If the spread of the residuals is consistent across all levels of the predictor variable, then the assumption of homoscedasticity is likely valid.

4. Normality: We assume that the residuals are normally distributed. This assumption can be checked by plotting a histogram of the residuals and checking if it resembles a normal distribution. Additionally, a normal probability plot of the residuals can also be used to visually assess normality.

Part b) To fit a linear model in R, we can use the 'lm()' function. The model can be written as:

lpsa = β0 + β1 * lcavol + ε

Where lpsa is the response variable (log of the cancer volume), lcavol is the predictor variable (log of the prostate-specific hormone), β0 is the intercept, β1 is the slope coefficient, and ε is the error term.

Here is an example of how to fit the linear model in R:

library(Faraway)
data(prostate)

model <- lm(lpsa ~ lcavol, data = prostate)
summary(model)

Part c) To check the diagnostics and assess the goodness of fit of the linear model, we can perform the following:

1. Residual plot: Plot the residuals against the predicted values. If the spread of the residuals is consistent across all levels of the predictor variable, and the residuals appear randomly scattered around zero, then the assumption of homoscedasticity is likely valid.

2. Normality plot: Plot a histogram of the residuals and/or a normal probability plot of the residuals. If the residuals resemble a normal distribution, then the assumption of normality is likely valid.

Here is an example of how to create these plots in R:

par(mfrow = c(1, 2))  # Set up the plotting layout with 1 row and 2 columns

# Residual plot
plot(model, which = 1)

# Normality plot
plot(model, which = 2)

Part d) To implement a hypothesis test to determine if the slope coefficient is important to our model, we can conduct a t-test on the slope coefficient. The null hypothesis is that the slope coefficient is equal to zero (i.e., the predictor variable is not significantly associated with the response variable). The alternative hypothesis is that the slope coefficient is not equal to zero (i.e., the predictor variable is significantly associated with the response variable).

We can use the 'summary()' function in R to obtain the t-test results:

summary(model)

The t-test result will provide the t-value, degrees of freedom, and p-value associated with the slope coefficient. If the p-value is less than a chosen significance level (e.g., 0.05), we can reject the null hypothesis and conclude that the slope coefficient is significant in the model.