91Ó°ÊÓ

Some plant viruses are spread by insects and tend to spread from the edges of a field inward. The data on \(x=\) distance from the edge of the field (in meters) and \(y=\) proportion of plants with virus symptoms that appeared in the paper "Patterns of Spread of Two NonPersistently Aphid-Borne Viruses in Lupin Stands" \((A n-\) nals of Applied Biology [2005]: \(337-350\) ) was used to fit a least-squares regression line to describe the relationship between \(x\) and \(y^{\prime}=\ln \left(\frac{p}{1-p}\right) .\) Minitab output resulting from fitting the least-squares line is given below. The regression equation is \(\ln (p /(1-p))=-0.917-0.107\) Distance to Crop Edge \(\begin{array}{lrrrr}\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P. } \\ \text { Constant } & -0.9171 & 0.1249 & -7.34 & 0.000\end{array}\) \begin{tabular}{lrrrr} Constant & -0.9171 & 0.1249 & -7.34 & 0.000 \\ Distance to Crop Edge & -0.10716 & 0.01062 & -10.09 & 0.000 \\ \hline \end{tabular} \(\mathrm{S}=0.387646 \quad \mathrm{R}-\mathrm{Sq}=72.8 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{adj})=72.1 \%\) a. What is the logistic regression function relating \(x\) and the proportion of plants with virus symptoms? b. What would you predict for the proportion of plants with virus symptoms at a distance of 15 meters from the edge of the field? (Note: the \(x\) values in the data set ranged from 0 to \(20 .)\)

Step by step solution

01

Formulate Logistic Regression Function

Using the provided regression equation \(\ln(p /(1-p))=-0.917-0.107x\), we can isolate \(p\) to get the logistic regression function as follows: \n- First, rewrite the equation to solve for \(p\): \n \(\ln(p /(1-p)) = -0.917-0.107x\) \n \(\Rightarrow p /(1-p) = \exp(-0.917-0.107x)\) \n \(\Rightarrow p = 1/(1+\exp(0.917+0.107x))\) \nSo, the logistic regression function relating \(x\) and the proportion of plants with virus symptoms is \(p = 1/(1+\exp(0.917+0.107x))\)

02

Predict the Proportion

Now that we have the logistic regression function, we can use it to predict the proportion of plants with virus symptoms at a distance of 15 meters from the edge of the field. We are given \(x=15\) in the question and this is possible, because all the \(x\) values in the dataset ran from 0 to 20. So we can plug this value into the function as follows: \n \(p = 1/(1+\exp(0.917+0.107*15))\)

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Key Concepts

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

Least-Squares Regression

Least-squares regression is a statistical method used to determine the best-fitting line through a set of points on a graph, ordinarily used when the relationship between the variables is linear. In logistic regression, the outcome variable is binary, and the least-squares technique is adjusted to accommodate this by using the natural logarithm of the odds of the dependent variable, rather than the dependent variable itself. This method is crucial because it minimizes the sum of the squares of the differences between the observed values and those predicted by the model.

This minimization ensures that the resulting parameters produce the most accurate predictions possible for new, unseen data, within the range of the observed data. When applied to logistic regression, we usually transform the binary outcome into log-odds (or logit) and then fit the least-squares line, which can later be converted back to a probability.

Proportion of Plants with Virus Symptoms

The proportion of plants with virus symptoms is a statistical measure that represents the ratio of the number of plants infected by a virus to the total number of plants observed. In the landscape of logistic regression, this proportion is the dependent variable that we try to predict, using one or more independent variables, such as the distance from the edge of the field.

The key to understanding this concept is to recognize that these proportions, when transformed into log-odds, can be used within a logistic regression framework to quantify the effect of predictor variables on the likelihood of plants displaying virus symptoms. By doing so, researchers and agriculturists can gain insights into the spatial dynamics of plant viral infections, which is crucial for developing effective disease management strategies.

Predictor Variables

Predictor variables, sometimes referred to as independent variables, are the inputs of our statistical model that we believe are influencing our dependent variable—in this case, the proportion of plants with virus symptoms. The distance from the edge of the field is an example of a predictor variable. It is selected based on the hypothesis that this distance may have an effect on the likelihood of a plant being infected with a virus.

When identifying predictor variables, it is important to choose those that are relevant to the context of the study and are most likely to have a causal or correlational relationship with the dependent variable. Once identified, these variables are included in the logistic regression model to estimate their association with the probability of the outcome and help in making predictions or understanding the data patterns.

Statistical Analysis

Statistical analysis encompasses a wide range of techniques to explore, describe, and infer conclusions from data. In the context of logistic regression, statistical analysis is used to estimate the parameters of the model, assess its fit, and evaluate the significance of the predictor variables.

The outputs of a logistic regression, including the coefficients, standard errors, and p-values, tell us not only the strength and direction of the relationship between each predictor and the outcome but also how confident we can be about these estimates. In the given exercise, the significance of the predictors is denoted by p-values, where values close to zero suggest that the corresponding variable plays a significant role in predicting the outcome.
Goodness-of-fit statistics, such as the R-squared, illustrate how well the model explains the variability in the data. In complex analyses, advanced techniques such as cross-validation may also be employed to ensure the model's predictions are robust and will generalize to new data.