91Ó°ÊÓ

Serial correlation, also known as autocorrelation, describes the extent to which the result in one period of a time series is related to the result in the next period. A time series with high serial correlation is said to be very predictable from one period to the next. If the serial correlation is low (or near zero), the time series is considered to be much less predictable. For more information about serial correlation, see the book Ibbotson \(S B B I\) published by Morningstar. A research veterinarian at a major university has developed a new vaccine to protect horses from West Nile virus. An important question is: How predictable is the buildup of antibodies in the horse's blood after the vaccination is given? A large random sample of horses from Wyoming were given the vaccination. The average antibody buildup factor (as determined from blood samples) was measured each week after the vaccination for 8 weeks. Results are shown in the following time series:$$\begin{array}{l|rrrrrrrr}\hline \text { Week } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\\hline \text { Buildup Factor } & 2.4 & 4.7 & 6.2 & 7.5 & 8.0 & 9.1 & 10.7 & 12.3 \\\\\hline\end{array}$$ To construct a serial correlation, we simply use data pairs \((x, y)\) where \(x=\) original buildup factor data and \(y=\) original data shifted ahead by 1 week. This gives us the following data set. since we are shifting 1 week ahead, we now have 7 data pairs (not 8 ). $$\begin{array}{c|ccccccc}\hline x & 2.4 & 4.7 & 6.2 & 7.5 & 8.0 & 9.1 & 10.7 \\\\\hline y & 4.7 & 6.2 & 7.5 & 8.0 & 9.1 & 10.7 & 12.3 \\\\\hline\end{array}$$ (a) Use the sums provided (or a calculator with least-squares regression) to compute the equation of the sample least-squares line, \(\hat{y}=a+b x .\) If the buildup factor was \(x=5.8\) one week, what would you predict the buildup factor to be the next week? (b) Compute the sample correlation coefficient \(r\) and the coefficient of determination \(r^{2}\). Test \(\rho>0\) at the \(1 \%\) level of significance. Would you say the time series of antibody buildup factor is relatively predictable from one week to the next? Explain.

Step by step solution

01

Organize the Data

The table provides paired data for weeks 1 to 8. Paired data allows us to use regression techniques to predict future values. Our pairs are (2.4, 4.7), (4.7, 6.2), (6.2, 7.5), (7.5, 8.0), (8.0, 9.1), (9.1, 10.7), and (10.7, 12.3).

02

Calculate Means and Sums

Compute the means \( \bar{x} \) and \( \bar{y} \). Then, calculate the sums \( \sum x \), \( \sum y \), \( \sum x^2 \), \( \sum y^2 \), and \( \sum xy \) needed for regression coefficients. The sums are: \( \sum x = 48.6 \), \( \sum y = 58.5 \), \( \sum x^2 = 353.4 \), \( \sum y^2 = 513.95 \), \( \sum xy = 407.55 \).

03

Calculate Regression Coefficients

Using the formulas for the regression coefficients: \[ b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ a = \bar{y} - b\bar{x} \] Compute \( b \) and \( a \) for \( \hat{y} = a + bx \). The calculated slope \( b \) and intercept \( a \) are used to form the regression equation.

04

Prediction for Week 9

Substitute \( x = 5.8 \) into the regression equation \( \hat{y} = a + bx \) to predict the buildup factor for the next week. Perform the calculation to get the predicted value.

05

Calculate the Correlation Coefficient

The correlation coefficient \( r \) is computed using \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]. Perform this calculation to find the strength and direction of the linear relationship.

06

Coefficient of Determination

Compute the coefficient of determination, \( r^2 \), which indicates the percentage of variation in \( y \) that can be explained by \( x \). Calculate \( r^2 \) and interpret the result.

07

Hypothesis Test for Serial Correlation

Test the hypothesis \( H_0: \rho = 0 \) vs. \( H_a: \rho > 0 \) using the correlation coefficient. Use a significance level of 1%. Calculate the t-statistic and compare it to critical t-value from the t-distribution with 5 degrees of freedom. Determine if the correlation is statistically significant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Time Series Analysis

Time series analysis is a method used to analyze data points collected or recorded at specific time intervals. This approach helps us understand the underlying patterns within the data, which is crucial for predicting future values. In the context of the exercise, time series analysis is employed to examine how the antibody buildup factor in horses' blood changes over several weeks. By doing so, researchers can determine if the vaccination has a consistent effect over time.

Using time series analysis, we can check for trends (long-term upward or downward movements), seasonality (repeated patterns at certain intervals), and serial correlation (how well current values predict future values).

• **Trend:** Reflects the general tendency of the dataset to move in a particular direction over time.
• **Seasonality:** Repeated fluctuations that occur regularly due to seasonal factors.
• **Serial correlation:** Measures the correlation of the series with its past values.

In the given exercise, serial correlation is of particular interest because it helps determine the predictability of the antibody levels from week to week.

Least-Squares Regression

Least-squares regression is a fundamental method used in statistics to find the best-fit line through a set of data points. This line minimizes the sum of the squares of the vertical distances (or 'residuals') between the observed values and the values predicted by the line. The primary purpose is to create a model that describes the relationship between variables.

In our exercise, the least-squares regression line is used to predict future antibody buildup factors. It takes the general form: \[ \hat{y} = a + bx \]where \(a\) is the intercept and \(b\) is the slope of the line.

• **Intercept (\(a\)):** The expected value of \(y\) when \(x = 0\).
• **Slope (\(b\)):** Indicates how much \(y\) changes for a one-unit increase in \(x\).

The regression equation provides insights into how predictable the antibody buildup is, given the initial data points. It allows us to make predictions for unseen instances, like determining the expected level of antibodies a week after an observed factor of 5.8.

Correlation Coefficient

The correlation coefficient, symbolized by \(r\), quantifies the degree to which two variables are linearly related. It ranges from -1 to 1. An \(r\) value of 1 indicates a perfect positive linear relationship, -1 signifies a perfect negative linear relationship, and 0 means no linear relationship.

The correlation coefficient helps assess the strength and direction of the association between the `x` and `y` variables in our exercise (weekly antibody levels).

• **Positive \(r\):** Indicates that as one variable increases, the other tends to increase as well.
• **Negative \(r\):** Suggests that as one variable increases, the other tends to decrease.
• **Zero \(r\):** Implies no linear correlation between the variables.

Understanding the correlation coefficient in this context informs us how well this week's buildup can predict next week's result. It plays a vital role in determining the reliability of our least-squares regression model.

Coefficient of Determination

The coefficient of determination, denoted as \(r^2\), is a measure that reflects the proportion of the variance in the dependent variable that is predictable from the independent variable. It is essentially the square of the correlation coefficient. This value ranges from 0 to 1, where a higher \(r^2\) percentage indicates a better fit of the regression line.

Following our analysis, the \(r^2\) value illustrates how well the data fits the regression model. For example, an \(r^2\) of 0.85 means that 85% of the variability in the antibody buildup can be explained by the changes in previous week's factor.

• A higher \(r^2\) value means greater predictability.
• A lower \(r^2\) value suggests the model might not explain much of the data's variability.

Thus, understanding and calculating the coefficient of determination is critical for the vaccine development process, as it quantifies the strength of the predictive model generated through our time series data analysis.