91Ó°ÊÓ

This exercise requires the use of a computer pack- age. The cotton aphid poses a threat to cotton crops in Iraq. The accompanying data on \(y=\) infestation rate (aphids/100 leaves) \(x_{1}=\) mean temperature \(\left({ }^{\circ} \mathrm{C}\right)\) \(x_{2}=\) mean relative humidityappeared in the article "Estimation of the Economic Threshold of Infestation for Cotton Aphid" (Mesopotamia journal of Agriculture [1982]: 71-75). Use the data to find the estimated regression equation and assess the utility of the multiple regression model \(y=\alpha+\beta_{1} x_{1}+\beta_{2} x_{2}+e\) $$ \begin{array}{rcrrrr} y & x_{1} & x_{2} & y & x_{1} & x_{2} \\ \hline 61 & 21.0 & 57.0 & 77 & 24.8 & 48.0 \\ 87 & 28.3 & 41.5 & 93 & 26.0 & 56.0 \\ 98 & 27.5 & 58.0 & 100 & 27.1 & 31.0 \\ 104 & 26.8 & 36.5 & 118 & 29.0 & 41.0 \\ 102 & 28.3 & 40.0 & 74 & 34.0 & 25.0 \\ 63 & 30.5 & 34.0 & 43 & 28.3 & 13.0 \\ 27 & 30.8 & 37.0 & 19 & 31.0 & 19.0 \\ 14 & 33.6 & 20.0 & 23 & 31.8 & 17.0 \\ 30 & 31.3 & 21.0 & 25 & 33.5 & 18.5 \\ 67 & 33.0 & 24.5 & 40 & 34.5 & 16.0 \\ 6 & 34.3 & 6.0 & 21 & 34.3 & 26.0 \\ 18 & 33.0 & 21.0 & 23 & 26.5 & 26.0 \\ 42 & 32.0 & 28.0 & 56 & 27.3 & 24.5 \\ 60 & 27.8 & 39.0 & 59 & 25.8 & 29.0 \\ 82 & 25.0 & 41.0 & 89 & 18.5 & 53.5 \\ 77 & 26.0 & 51.0 & 102 & 19.0 & 48.0 \\ 108 & 18.0 & 70.0 & 97 & 16.3 & 79.5 \\ \hline \end{array} $$

Step by step solution

01

Preparing the Data

Begin with organizing the data into a table, with columns for infestation rate (y), mean temperature (x1) and relative humidity (x2).

02

Calculating the Coefficients

The coefficients can be determined by using the Ordinary Least Square (OLS) methods. The calculated parameters help to find the best-fit line by minimizing the sum of the squares of the differences between observed and predicted values of (y). For three variables like in this case, it requires to do matrix calculations which typically are done by software.

03

Implementing the Regression Model

With the calculated coefficients, the multiple regression equation can be established. The equation will appear in the form \(y=\alpha+\beta_{1} x_{1}+\beta_{2} x_{2}+e\), with the parameters replaced by their respective coefficient.

04

Evaluating Utility

Assessing utility is judging how well the model fits the data. A common value to check for this is the coefficient of determination, often referred as R-square. It shows the proportion of variance in the dependent variable that is predictable from the independent variables, ranging between 0 and 1. The closer the value to 1, the better the model explains the variance.

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Key Concepts

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

Ordinary Least Squares (OLS)

Ordinary Least Squares (OLS) is a statistical method used to estimate the parameters in a linear regression model. When you have a dataset and you want to draw a best-fit line through the data points, OLS helps to do just that by finding the coefficients that minimize the sum of the squared differences between observed values and predictions made by your model. This difference is often referred to as the residual or error.

To understand OLS, imagine plotting a scatter plot with your data points; the goal of OLS is to draw a line that goes through these points with minimal deviation. The approach works by calculating the slope (\(\beta \)) and intercept (\(\alpha \)) of the line. In a multiple regression model, like the one in the exercise, you include multiple variables, which means you instead work with an equation like:\[ y = \alpha + \beta_1 x_1 + \beta_2 x_2 + e \]

Here, \(y\) is the dependent variable (infestation rate), \(\beta_1\) and \(\beta_2\) are the coefficients of the predictors or independent variables \(x_1\) (mean temperature) and \(x_2\) (mean relative humidity), respectively. The term \(e\) represents the error term, accounting for variability that the linear model does not capture. Working with OLS, especially for multiple variables, often requires computational tools to solve matrix equations that arise.

Coefficient of Determination

The coefficient of determination, often called \(R^2\), is a key metric for evaluating how well your regression model fits the data. In simple terms, it tells you the proportion of the variance in the dependent variable that is predictable from the independent variables. This value ranges between 0 and 1, where 0 indicates no predictive capability and 1 indicates perfect prediction.

An \(R^2\) value close to 1 suggests that a large portion of variance in the dependent variable is explained by your model, making it a good fit. Conversely, a value near 0 indicates that the model fails to capture the variability effectively. In the context of the exercise, after applying OLS to find the regression coefficients, \(R^2\) is computed to assess how well the mean temperature and relative humidity variables predict the infestation rate.

Understanding this metric is crucial because it informs whether adding, removing, or adjusting variables can improve model predictions. Additionally, while a high \(R^2\) suggests a good fit, it doesn't necessarily mean the model is"correct" – it's important to ensure it makes logical and scientific sense as well.

Variable Relationship

Variable relationships in a regression model describe how changes in independent variables affect the dependent variable. In this exercise, mean temperature and relative humidity are the independent variables and infestation rate is the dependent variable. Each independent variable's contribution is reflected by its respective coefficient in the regression equation.

These coefficients (\(\beta_i\)), determined through OLS, indicate the expected change in the dependent variable for a one-unit change in the independent variable, all else being equal. For instance, if \(\beta_1\) is positive, it suggests that an increase in temperature is associated with an increase in infestation rate, assuming humidity remains constant. Similarly, if \(\beta_2\) is negative, higher relative humidity might be associated with a lower infestation rate when temperature is constant.

Analyzing these relationships helps in understanding the dynamics between environmental factors and infestation rates, crucial for effective pest management strategies. Beyond the coefficients, examining residuals (the differences between observed and predicted values) can provide additional insights into model reliability by highlighting unusual data variations or patterns not captured by linear relationships.