Question

The relationship between yield of maize, date of planting, and planting density was investigated in the article "Development of a Model for Use in Maize Replant Decisions" (Agronomy Journal [1980]: 459-464). Let \(\begin{aligned} y &=\text { percent maize yield } \\ x_{1} &=\text { planting date }(\text { days after April 20 }) \\ x_{2} &=\text { planting density (plants/ha) } \end{aligned}\) The regression model with both quadratic terms \((y=\alpha+\) \(\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+e\) where \(x_{3}=x_{1}^{2}\) and \(x_{4}=x_{2}^{2}\) ) provides a good description of the relationship between \(y\) and the independent variables. a. If \(\alpha=21.09, \beta_{1}=.653, \beta_{2}=.0022, \beta_{3}=-.0206\), and \(\beta_{4}=.00004\), what is the population regression function? b. Use the regression function in Part (a) to determine the mean yield for a plot planted on May 6 with a density of 41,180 plants/ha. c. Would the mean yield be higher for a planting date of May 6 or May 22 (for the same density)? d. Is it legitimate to interpret \(\beta_{1}=.653\) as the true average change in yield when planting date increases by one day and the values of the other three predictors are held fixed? Why or why not?

Short answer

a) The population regression function is \(y = 21.09 + 0.653x_{1} + 0.0022x_{2} - 0.0206x_{3} + 0.00004x_{4} + e\). b) The mean yield for a plot planted on May 6 with a density of 41,180 plants/ha can be calculated by substituting these values into the regression function. c) Comparing the mean yields for planting dates of May 6 and May 22 (at the same density) involves substituting these dates into the regression function and seeing which yields more. d) The value of \(\beta_{1}=0.653\) can be interpreted as the average change in yield when the planting date increases by one day and the values of the other three predictors are held constant.

Step-by-step solution

Write the Regression Model Function

The regression model given is \(y=\alpha+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{4}x_{4}+e\) where \(x_{3}={x_{1}}^{2}\) and \(x_{4}={x_{2}}^{2}\). The question gives the values for \(\alpha, \beta_{1},\beta_{2}, \beta_{3}, \beta_{4}\) which are 21.09, 0.653, 0.0022, -0.0206 and 0.00004, respectively. Substituting these values into the regression model, we get: \(y = 21.09 + 0.653x_{1} + 0.0022x_{2} - 0.0206x_{3} + 0.00004x_{4} + e\).

Calculate Yield for Given Date and Density

To find the mean yield for a plot planted on May 6, we need to substitute the planting date \(x_{1}\) as the number of days after April 20 until May 6, which is 16 days, and the planting density \(x_{2}\) as 41180 plants/ha into the regression function. Now, solve for \(y\). Remember that \(x_{3}\) and \(x_{4}\) are the squares of \(x_{1}\) and \(x_{2}\) respectively.

Compare Yields for Different Planting Dates

For the same planting density, use the regression function to get yields for May 6 and May 22 (32 days after April 20) to compare which yields more.

Interpret the Coefficient \(\beta_{1}\)

Interpret \(\beta_{1}=0.653\) in terms of changing the planting date while the other variables are held constant. The coefficient \(\beta_{1}\) shows the relationship between maize yield and planting date when all other predictors are held constant. It indicates how much the output (yield) changes for each unit increase in planting date, keeping other predictors fixed.

Key concepts

Statistical Modeling

Statistical modeling is a mathematical representation of a process or relationship between variables. In the context of agriculture, models are created to predict outcomes like crop yields based on various factors such as planting date and planting density. These models are crucial for making informed decisions that can lead to improved efficiency and productivity.

In the exercise provided, a statistical model is used to describe the relationship between the percent yield of maize and two main independent variables: planting date (\(x_1\)) and planting density (\(x_2\)). The model also includes the quadratic terms of these variables to account for the non-linear effects as the planting date and density change, enhancing the accuracy of the predictions.

The key to a useful statistical model in agriculture is the ability to quantify and predict how changes in the inputs (\(x_1\text{ and }x_2\text{ in this case}\)) will affect the output (\(y\text{, the yield of maize}\)). By understanding these relationships, farmers and agricultural scientists can optimize conditions to maximize crop yields.

Agronomic Data Analysis

Agronomic data analysis involves interpreting data collected from agricultural experiments or field observations to make sound decisions. Variables can include soil properties, weather conditions, crop growth measurements, and agricultural practices. In our exercise example, the analysis is focused on how the date of planting and the planting density impact the maize yield.

The understanding of agronomic data is critical for farmers and researchers. By analyzing this data, they can optimize planting schedules and density to achieve higher yields. The regression model constructed using agronomic data assists in revealing significant patterns and relationships.

For students working with agronomic data, it is essential to recognize the real-world variability that can affect results. Things like differences in soil type, local weather patterns, and farming techniques can all influence the data and should be considered when building models and making projections based on the data collected.

Multiple Regression

Multiple regression is a statistical technique that models the relationship between a dependent variable and two or more independent variables. This method adjusts for the influence of various factors simultaneously, providing a more comprehensive understanding of their individual and combined effects.

In the case of the maize yield example, multiple regression allows us to see how both planting date and planting density, along with their squared terms, contribute to the variability in yield. Whether you're looking to predict the output measure (\(y\text{, percent maize yield}\)) or understand the intensity of each factor’s influence, multiple regression can aid in quantifying those relationships.

Furthermore, interpreting the coefficients in a multiple regression (\( \beta_1, \beta_2, ... \beta_n \text{ in the exercise}\)) gives insights into how significant each independent variable is when other variables are accounted for. This is crucial for agronomists and farmers when deciding which levers to pull to improve crop outcomes. In the educational context, it's imperative to not just calculate these coefficients but to also understand their implications in real-world agricultural practices.