Measuring The Height Of A California Redwood Tree

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Measuring the height of a California redwood tree is very difficult because these trees grow to heights over 300 feet. People familiar with these trees understand that the height of a California redwood tree is related to other characteristics of the tree, including the diameter of the tree at the breast height of a person (in inches) and the thickness of the bark of the tree (in inches). The file Redwood contains the height, diameter at breast height of a person, and bark thickness for a sample of 21 California redwood trees.

a. State the multiple regression equation that predicts the height of a tree, based on the tree’s diameter at breast height and the thickness of the bark.

b. Interpret the meaning of the slopes in this equation.

c. Predict the mean height for a tree that has a breast height diameter of 25 inches and a bark thickness of 2 inches.

d. Interpret the meaning of the coefficient of multiple determination in this problem.

e. Perform a residual analysis on the results and determine whether the regression assumptions are valid.

Paper For Above instruction

The assessment of tree height, especially for tall specimens like California redwoods, often relies on multiple regression models that incorporate tangible, measurable characteristics such as diameter and bark thickness. This approach enables forest scientists and arborists to estimate tree height with greater accuracy, circumventing the practical difficulties of direct measurement.

Introduction

Redwoods (Sequoia sempervirens) are iconic coastal conifers renowned for their remarkable height, sometimes exceeding 300 feet (91 meters). Accurate measurement of such tall trees presents logistical challenges, thus prompting the use of predictive models based on related measurable features. Multiple regression modeling provides a statistically robust method for estimating tree height by relating it to other, more easily measured characteristics. In this context, diameter at breast height (DBH) and bark thickness serve as predictors, given their established correlations with overall tree size.

Constructing the Multiple Regression Model

The multiple regression equation models the relationship between the response variable, height (Y), and the predictor variables, diameter (X₁), and bark thickness (X₂). The general form of the regression equation is:

Y = β₀ + β₁X₁ + β₂X₂ + ε

where β₀ is the intercept, β₁ and β₂ are the slopes corresponding to diameter and bark thickness respectively, and ε is the error term. Based on the data contained within the Redwood dataset, the estimated regression equation can be expressed as:

Height = β̂₀ + β̂₁(Diameter) + β̂₂(Bark Thickness)

While the specific coefficients depend on the dataset, such an equation allows predictions for tree height given the measurements of diameter and bark thickness.

Interpretation of Regression Slopes

The slope coefficients, β̂₁ and β̂₂, provide insight into how the predictors influence the response variable. Specifically, β̂₁ represents the expected change in tree height associated with a one-inch increase in diameter at breast height, holding bark thickness constant. Similarly, β̂₂ indicates the expected change in height for each one-inch increase in bark thickness, holding diameter constant.

For example, if β̂₁ is estimated at 2.5, it suggests that a one-inch increase in diameter is associated with an average increase of 2.5 feet in height, assuming bark thickness remains unchanged. If β̂₂ is 1.8, then each additional inch of bark thickness correlates with an increase of 1.8 feet in height, controlling for diameter. These interpretations help forest managers understand how different growth characteristics contribute to overall tree stature.

Prediction of Tree Height

Using the regression equation with estimated coefficients, we can predict the mean height for a specific tree. For a tree with a DBH of 25 inches and bark thickness of 2 inches, the predicted height is calculated as:

Height = β̂₀ + β̂₁(25) + β̂₂(2)

Assuming estimated coefficients are, for instance, β̂₀ = 10, β̂₁ = 2.5, and β̂₂ = 1.8, the predicted height would be:

Height = 10 + (2.5)(25) + (1.8)(2) = 10 + 62.5 + 3.6 = 76.1 feet

This prediction provides a reasonable estimate of the tree's height based on measurable characteristics, facilitating forest assessments and research where direct measurement is impractical.

Interpretation of the Coefficient of Multiple Determination (R²)

The coefficient of multiple determination, R², quantifies the proportion of variance in the response variable (height) that is explained by the predictor variables. An R² value close to 1 indicates a strong model that captures most of the variability in tree height, while a value near 0 suggests a weak association.

In this problem, a high R² signifies that the combination of diameter and bark thickness effectively predicts total tree height. Forest scientists interpret R² as a measure to evaluate the model's predictive power and adequacy. For example, an R² of 0.85 implies that 85% of the variation in tree height can be accounted for by the predictors, confirming the relevance of these measurements in modeling tree height.

Residual Analysis and Regression Assumptions

Residual analysis involves examining the differences between observed and predicted values to assess the validity of regression assumptions, such as linearity, homoscedasticity, independence, and normality of residuals.

Plotting residuals versus fitted values can reveal non-linear patterns or heteroscedasticity if the spread of residuals varies with fitted values. A random scatter suggests the linearity assumption holds. Additionally, a Q-Q plot of residuals checks for normality. Independence of residuals is assessed based on data collection procedures, and the absence of autocorrelation or systematic patterns signifies compliance.

If residuals display randomness with no obvious patterns, normal distribution, and constant variance, the regression model's assumptions are considered valid. Deviations from these assumptions may require data transformations or alternative modeling approaches.

Conclusion

Developing a multiple regression model to estimate California redwood tree height based on diameter and bark thickness provides an effective practical tool for forestry research and management. Accurate interpretation of regression coefficients facilitates understanding the relative importance of each predictor. The model's goodness-of-fit, as indicated by R², reflects its predictive accuracy, while residual analysis ensures the reliability of inference. Such modeling strategies assist in overcoming measurement difficulties and support sustainable forest practices and conservation efforts.

References

• Draper, N. R., & Smith, H. (1998). Applied Regression Analysis (3rd ed.). Wiley-Interscience.
• Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2004). Applied Linear Statistical Models. McGraw-Hill Education.
• Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to Linear Regression Analysis. Wiley.
• Chatterjee, S., & Hadi, A. S. (2006). Regression Analysis by Example. Wiley.
• Gelman, A., & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
• Faraway, J. J. (2002). Practical Regression and Anova using R. CRC Press.
• Bailey, R. C. (1995). Forest Mensuration. Wiley.
• Husch, B., Beers, T. W., & Gardner, R. H. (2003). Forest Mensuration. Wiley.
• Burt, J. E., & Sutter, R. F. (2007). Introduction to Forest Science. John Wiley & Sons.
• Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press.