Question

Use the data given in Exercises 6-7 (Exercises 17-18, Section 12.1). Construct the ANOVA table for a simple linear regression analysis, showing the sources, degrees of freedom, sums of squares, and mean sauares. $$\begin{array}{l|rrrrrrr}x & -2 & -1 & 0 & 1 & 2 \\\\\hline y & 1 & 1 & 3 & 5 & 5\end{array}$$

Short answer

Answer: The equation of the regression line is y = 3 + 1.4x. The mean square of regression is 19.6, and the mean square of error is 0.67.

Step-by-step solution

Calculate necessary sums

First, we need to compute the necessary sums and other quantities required for calculating slope, intercept, and sum of squares. | x | y | x^2 | xy | |---|----|-----|----| |-2 | 1 | 4 | -2 | |-1 | 1 | 1 | 1 | | 0 | 3 | 0 | 0 | | 1 | 5 | 1 | 5 | | 2 | 5 | 4 | 10 | Now we will compute the sums: $$ \sum x = 0, \, \sum y = 15,\, \sum x^2 = 10,\, \sum xy = 14 $$ And we have n = 5.

Calculate slope and intercept

We will now use the formulas mentioned earlier to calculate the slope and the intercept of the regression line. $$ a = \frac{\sum y \cdot \sum x^2 - \sum x \cdot \sum xy}{n \cdot \sum x^2 - (\sum x)^2} = \frac{15 \cdot 10 - 0 \cdot 14}{5 \cdot 10 - 0^2} = \frac{150}{50} = 3 $$ $$ b = \frac{n \cdot \sum xy - \sum x \cdot \sum y}{n \cdot \sum x^2 - (\sum x)^2} = \frac{5 \cdot 14 - 0 \cdot 15}{5 \cdot 10 - 0^2} = \frac{70}{50} = 1.4 $$ Therefore, the regression line equation is: $$ y = 3 + 1.4x $$

Calculate SST, SSR, and SSE

Now we will calculate the Total Sum of Squares (SST), Regression Sum of Squares (SSR) and Error Sum of Squares (SSE). $$ \begin{aligned} SST &= \sum(y - \bar{y})^2\\ SSR &= \sum(\hat{y} - \bar{y})^2\\ SSE &= \sum(y - \hat{y})^2\\ \end{aligned} $$ Where: \(\bar{y}\) = mean of y, \(\hat{y}\) = predicted values using the regression line equation. First, calculate the mean of y: $$ \bar{y} = \frac{\sum y}{n} = \frac{15}{5} = 3 $$ Now find the predicted values using the regression line equation: | x | y | \(\hat{y}\) | \((y - \bar{y})^2\) | \((\hat{y} - \bar{y})^2\) | \((y - \hat{y})^2\) | |----|----|-----------|-------------------|--------------------------|-------------------| | -2 | 1 | 0.2 | 4 | 7.84 | 0.64 | | -1 | 1 | 1.6 | 4 | 1.96 | 0.36 | | 0 | 3 | 3 | 0 | 0 | 0 | | 1 | 5 | 4.4 | 4 | 1.96 | 0.36 | | 2 | 5 | 5.8 | 4 | 7.84 | 0.64 | $$ \begin{aligned} SST &= \sum(y - \bar{y})^2 = 16\\ SSR &= \sum(\hat{y} - \bar{y})^2 = 19.6\\ SSE &= \sum(y - \hat{y})^2 = 2\\ \end{aligned} $$

Create the ANOVA table

Now that we have SST, SSR, and SSE, we can create the ANOVA table: | Source | DF| SS | MS | |--------------|---|------|------| | Regression | 1 | 19.6 | 19.6 | | Error | 3 | 2 | 0.67 | | Total | 4 | 21.6 | | Where: DF (Degrees of freedom) = n-1 for Total, 1 for Regression, and (n-1)-1 for Error. MS (Mean square) = SS / DF for Regression and Error. In conclusion, the ANOVA table for the given data have been constructed.

Key concepts

Analysis of Variance

Analysis of Variance, commonly known as ANOVA, is a statistical method used to compare means across different groups and determine if any of those means are statistically significantly different from each other. This technique is particularly useful when dealing with more than two groups. In the context of simple linear regression, ANOVA allows us to test if there is a statistically significant linear relationship between an independent variable (X) and a dependent variable (Y).

The basic principle behind ANOVA for regression is that the total variation in the dependent variable, known as the Total Sum of Squares (SST), can be partitioned into two parts: the variation explained by the regression model, known as the Regression Sum of Squares (SSR), and the unexplained variation, or Error Sum of Squares (SSE). The formula for SST is \( SST = SSR + SSE \).

By constructing an ANOVA table, we can view a summary that includes the sources of variation, the associated degrees of freedom (DF), and the mean squares (MS), which are the sums of squares divided by their respective degrees of freedom. In a simple linear regression, there is only one independent variable, so the DF for regression is 1. The Error DF is calculated by subtracting 2 from the total number of observations, reflecting the reduction for estimating two parameters, the slope and the intercept of the regression line.

Simple Linear Regression

Simple linear regression is a way to model the linear relationship between two quantitative variables. The model aims to explain the variation in a dependent variable (Y) as a function of an independent variable (X). In this type of regression, we assume that there is a straight-line relationship between X and Y, which can be described by an equation of the form \( y = a + bx \), where \( a \) is the y-intercept and \( b \) is the slope of the line. This equation is commonly referred to as the regression line.

The slope (\( b \)) indicates the average change in the dependent variable for each one-unit change in the independent variable. The y-intercept (\( a \)) represents the expected value of Y when X is zero. One of the main objectives of simple linear regression is to find the values of the intercept and slope that best fit the data. This is typically done using the least squares method, which minimizes the sum of the squared differences between the observed values and those predicted by the regression model. In the textbook exercise, we calculated the regression coefficients using summation formulas derived from the least squares criterion.

Sum of Squares

The sum of squares is a key concept in statistics that quantifies the amount of variation in a set of data. When we conduct simple linear regression analysis, we are particularly interested in three types of sum of squares:

• Total Sum of Squares (SST): This measures the total variation in the dependent variable and is calculated as the sum of the squared differences between each observed value and the overall mean.
• Regression Sum of Squares (SSR): This reflects the amount of variation in the dependent variable that is explained by the independent variable (or the regression line). SSR is the sum of the squared differences between the predicted values and the overall mean.
• Error Sum of Squares (SSE): This represents the unexplained variation after fitting the regression line and is calculated as the sum of the squared differences between each observed value and its corresponding predicted value.

Understanding the distinction between these types of sums of squares is crucial for interpreting the ANOVA table in regression analysis. By comparing the SSR and SSE, we can evaluate the strength and significance of the relationship between the variables.

Regression Coefficients

Regression coefficients are the numerical values that multiply the predictor variables in the equation of a regression line. They constitute the slope (\( b \)) and y-intercept (\( a \)) in a simple linear regression model. These coefficients are crucial because they tell us both the direction and the strength of the relationship between the independent variable and the dependent variable.

The y-intercept (\( a \)) indicates where the regression line crosses the Y-axis, while the slope (\( b \)) gives the rate at which Y changes for each unit change in X. In the textbook exercise, the coefficients were calculated using the sums of products and sums of squares of the observed values. Once we know these coefficients, we can predict the value of the dependent variable for any given value of the independent variable. Moreover, in statistical inference, we commonly perform hypothesis tests on these coefficients to determine if they are significantly different from zero, which would indicate a meaningful relationship between the variables.