Question

Use this information to fill in all values in an analysis of variance for regression table as shown. $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS } & \text { F-statistic } & \text { p-value } \\ \hline \text { Model } & & & & & \\ \hline \text { Error } & & & & & \\ \hline \text { Total } & & & & & \\ \hline \end{array} $$ SSModel \(=800\) with SSTotal \(=5820\) and a sample size of \(n=40\)

Short answer

The completed Analysis of Variance (ANOVA) for regression table is:\[\begin{array}{|l|c|c|c|c|c|}\hline \text{Source} & \text{df} & \text{SS} & \text{MS} & \text{F-statistic} & \text{p-value} \ \hline \text{Model} & 1 & 800 & 800 & 6.055 & ? \ \hline \text{Error} & 38 & 5020 & 132.105 & & \ \hline \text{Total} & 39 & 5820 & & & \ \hline \end{array} \]

Step-by-step solution

Calculate Degrees of Freedom (df)

Degrees of Freedom (df) refers to the number of independent pieces of information that went into the calculation of an estimate. Generally, in a regression table, df for Model is \(k-1\), df for Error is \(n-k\), and df for Total is \(n-1\). Here, where the sample size \(n = 40\), and the regression is a simple linear regression (only one predictor), so \(k = 2\).Therefore df for the model would be \(k-1 = 2-1 = 1\), df for the error would be \(n-k = 40-2 = 38\), df for the Total would be \(n-1 = 40-1 = 39\).

Calculate Remaining Sum of Squares (SSError)

The Sum of Squares Error (SSError) is the sum of the squared differences between the predicted and actual observation. It's represented as the SSTotal - SSModel. In this case, SSError would be 5820 (SSTotal) - 800 (SSModel) = 5020.

Calculate Mean Sum of Squares (MS)

The Mean Sum of Squares (MS) is the average sum of squared errors. It can be calculated as the SS divided by its degree of freedom (df).Here, MS for Model = SSModel/dfModel = 800/1 = 800, MS for Error = SSError/dfError = 5020/38 = approximately 132.105.

Calculate F-statistic

The F-statistic is the ratio of the Model Mean Square to the Error Mean Square. So, F-statistic would be MSModel/MSError = 800/132.105 = approximately 6.055.

Calculate p-value

The p-value is determined from the F-statistic value and corresponding degrees of freedom using an F-distribution table or statistical software. Since the p-value calculation usually involves complex calculations or the use of a statistics software, it is not feasible to compute in this context. We can denote it with '?' for the purposes of this exercise.

Key concepts

Degrees of Freedom (df)

When diving into statistical analysis, the concept of Degrees of Freedom (df) is essential to understand. It refers to the number of independent values or quantities that can vary in an analysis without breaking any constraints. In the context of regression analysis, the df for the model, error, and total helps set the stage for further calculations and interpreting results.

For the model df, we subtract one from the number of predictors, including the intercept, since we're estimating parameters. For error df, we consider the total sample size minus the number of estimated parameters. Lastly, the total df is simply one less than our sample size because we're using one piece of data to estimate the mean. Understanding these intricacies allows for correct set up in an analysis of variance for regression.

Sum of Squares (SS)

The Sum of Squares (SS) holds a key role in understanding variability within your data. Essentially, it's a measure of the total variation in the dataset. When we calculate SS, we're squaring the difference between each observed value and the mean (for total SS) or the predicted values (for model SS), then summing all those squared values.

There are different types of SS in regression analysis – the SS of the model indicates how well the model explains the data, while the SS of the error shows how much variation is unexplained by the model. In the given exercise, we've seen how the SS for the model and SS for error add up to the total SS, which represents all variation in the data, both explained and unexplained.

Mean Sum of Squares (MS)

Once we grasp the concept of SS, we can move on to the Mean Sum of Squares (MS), which is derived by dividing the SS by their corresponding df.

It's the average square of these deviations and is crucial for comparing models. This calculation allows us to assess whether the model significantly reduces the error when predicting our dependent variable. In other words, we can see how well our model performs by looking at how much the MS for the model deviates from the MS of the error. The given problem shows the direct application by dividing SS of Model and Error by their respective df's to obtain their MS values.

F-statistic

The F-statistic is a powerful test statistic that comes into play when comparing statistical models. It is calculated by dividing the MS of the model by the MS of the error.

This ratio tells us if the additional complexity of our model is justified by a significant reduction in residuals or unexplained variance. A high F-statistic signifies that the model explains a significant portion of the variance in the data, which in turn suggests that the model is a good fit. It's how we quantitatively decide if our model is significantly better than the baseline or not, making it a cornerstone of regression analysis.

p-value

Last but not least, we have the p-value, which might just be the most recognized term in statistical hypothesis testing. The p-value tells us about the likelihood or probability of observing our results (or more extreme) given that the null hypothesis is true.

In the context of ANOVA and regression, it helps us determine whether the patterns found in the sample data are strong enough to be considered statistically significant in the population. A small p-value (typically ≤ 0.05) indicates that there is strong evidence against the null hypothesis, leading us to reject it. In other words, if our F-statistic is significantly high, the corresponding p-value helps confirm whether our findings are likely to be valid.