91Ó°ÊÓ

As part of a study of the effects of timber management strategies (Ecological Applications [2003]: IIIOII123) investigators used satellite imagery to study abundance of the lichen Lobaria oregano at different elevations. Abundance of a species was classified as "common" if there were more than 10 individuals in a plot of land. In the table below, approximate proportions of plots in which Lobaria oregano were common are given. Proportions of Plots Where Lobaria oregano Are Common \begin{tabular}{lrrrrrrr} \hline Elevation (m) & 400 & 600 & 800 & 1000 & 1200 & 1400 & 1600 \\ Prop. of plots & \(0.99\) & \(0.96\) & \(0.75\) & \(0.29\) & \(0.077\) & \(0.035\) & \(0.01\) \\ with lichen & & & & \end{tabular} with lichen \begin{tabular}{l} with lichen \\ common \\ \hline \end{tabular} a. As elevation increases, does the proportion of plots for which lichen is common become larger or smaller? What aspect(s) of the table support your answer? b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the elevations and fit the line \(y^{\prime}=a+b(\) Elevation). What is the equation of the best-fit line? c. Using the best-fit line from Part (b), estimate the proportion of plots of land on which Lobaria oregano are classified as "common" at an elevation of \(900 \mathrm{~m} .\)

Step by step solution

01

Identify the Trend

By looking at the table, a decline in the proportion of plots with Lobaria oregano being classified as 'common' can be observed as the elevation increases. Specifically, at 400m elevation, the proportion is \(0.99\), but at 1600m, the proportion drastically decreases to \(0.01\). Thus, as the elevation increases, the proportion of plots for which the lichen is common becomes smaller.

02

Calculation of \(y'\)

In the second part of the question, we are required to calculate \(y' = ln(\frac{p}{1-p})\) for each of the elevations. Here, \(p\) is the proportion of plots. Carry out these calculations for each of the given elevation points.

03

Line Fitting

The values of \(y'\) computed in the previous step have to be used to fit the line \(y'= a + b(\) Elevation). This can be achieved using a linear regression model. The equation of the line that fits the data best can be found using software such as R or Python, or a statistical calculator that can perform linear regressions.

04

Proportion Estimation

For the final part, we need to estimate the proportion of plots where Lobaria oregano is common at an elevation of \(900m\). This can be done by substituting \(900m\) into the equation of the fitted line obtained in Step 3. Then, the inverse of the logistic function can be applied to the result to retrieve the estimated proportion. That is, if the computed \(y'\) for \(900m\) is \(y'_{900}\), then the estimated proportion \(p_{900}\) can be computed as \(p_{900} = \frac{e^{y'_{900}}}{1+e^{y'_{900}}}\).

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Key Concepts

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

Elevation Effect

The elevation effect on the abundance of Lobaria oregano is intriguing because it shows how environmental factors influence species distribution. As elevation increases, the plot proportions with common Lobaria oregano decrease. This trend emerges clearly from the provided data, where at 400 meters the proportion is high at 0.99, and it diminishes sharply to 0.01 at 1600 meters. Understanding this phenomenon is crucial for ecological studies and timber management strategies.

• Lower elevations provide conditions more favorable for lichen growth.
• Higher elevations may pose challenging conditions, such as colder temperatures and less optimal growth habitats.

This exercise underscores the importance of examining multiple variables, like elevation, to grasp the complex interplay affecting ecosystem dynamics.

Proportion Calculation

Proportion calculations in logistic regression are essential to understanding relationships between variables. In this exercise, we compute the logarithm of the odds for the occurrence of an event, specifically, the presence of common Lobaria oregano at various elevations. This is mathematically represented as:
\[ y' = \ln\left(\frac{p}{1-p}\right) \]where \( p \) is the proportion of plots. This calculation transforms the proportions into a format that linear regression models can handle.

• Calculate \( y' \) for each elevation point using the formula.
• Transforming data into a linear relationship aids in creating predictive models.

Proportion calculation lays the groundwork for assessing how variable changes, like elevation, impact species abundance in ecological studies.

Linear Regression Model

Linear regression models are powerful tools for interpreting data and making predictions. Once we have transformed and calculated our logarithm of the odds \( y' \) for different elevations, our next task is to fit these values to a linear regression model:
\[ y' = a + b \, (\text{Elevation}) \]This equation represents the best-fit line, where \( a \) is the intercept and \( b \) is the slope coefficient calculated through statistical software or calculators.

• The slope \( b \) indicates how much \( y' \) changes with a unit change in elevation.
• Using the fitted model, we can predict \( y' \) for unknown elevations.
• This model helps estimate unknown proportions, illustrating how calculations inform ecological predictions.

By analyzing data with a linear regression model, researchers can develop insights and make informed management decisions based on predicted ecological patterns.