Question

The ability of ecologists to identify regions of greatest species richness could have an impact on the preservation of genetic diversity, a major objective of the World Conservation Strategy. The article “Prediction of Rarities from Habitat Variables: Coastal Plain Plants on Nova Scotian Lakeshores" (Ecology [1992]: \(1852-\) 1859) used a sample of \(n=37\) lakes to obtain the estimated regression equation $$ \begin{aligned} \hat{y}=& 3.89+.033 x_{1}+.024 x_{2}+.023 x_{3} \\ &+.008 x_{4}-.13 x_{5}-.72 x_{6} \end{aligned} $$ where \(y=\) species richness, \(x_{1}=\) watershed area, \(x_{2}=\) shore width, \(x_{3}=\) drainage \((\%), x_{4}=\) water color \((\) total color units), \(x_{5}=\) sand \((\%),\) and \(x_{6}=\) alkalinity. The coefficient of multiple determination was reported as \(R^{2}=.83 .\) Use a test with significance level .01 to decide whether the chosen model is useful.

Short answer

Based on the high \(R^{2}\) value, the regression model is likely useful for predicting species richness. However, a specific statistical test is required to conclusively reject the null hypothesis at the 0.01 significance level. Assuming the p-value is less than 0.01, the null hypothesis can be rejected, supporting that the model is useful.

Step-by-step solution

Understand the model

The provided regression equation shows the relationship of species richness (\(y\)) with several environmental factors (\(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}\)). The test of significance of this regression model essentially tests whether all the regression coefficients are zero. If they are zero, it means the predictors do not influence the dependent variable.

Set up null and alternative hypotheses

The null hypothesis (\(H_{0}\)) assumes that all the predictor variables are not associated with the response variable, meaning all the coefficients (except the constant coefficient) are zero. In contrast, the alternative hypothesis (\(H_{a}\)) suggests that at least one predictor is significantly associated with the response variable, meaning at least one coefficient is not zero. Mathematically, \(H_{0}\) : \(\beta_{1} = \beta_{2} = \beta_{3} = \beta_{4} = \beta_{5} = \beta_{6} = 0\) versus \(H_{a}\) : at least one \(\beta_{i} \neq 0\), \(i = 1, 2, 3, 4, 5, 6\).

Choose significance level

As given in the question, the significance level (\(\alpha\)) is chosen as 0.01 (1%). This will be used as the threshold to decide whether the observed result is statistically significant.

Decide on the basis of \(R^{2}\)

The given coefficient of determination (\(R^{2}\)) value is 0.83 or 83%. This means that 83% of the variability in species richness could be explained by variations in the predictor variables. An \(R^{2}\) value closer to 1 demonstrates stronger prediction power of the model.

Conclusion

Since the \(R^{2}\) value is fairly high, it suggests that the regression model is useful at predicting the outcome. The final decision will depend on the test statistic for the hypothesis test (like F-test), which is not provided. However, assuming that the p-value is less than 0.01 (based on the given \(R^{2}\) and the sample size), we can reject the null hypothesis. This would mean that there is at least one predictor that has a significant relationship with the species richness.

Key concepts

Species Richness and Its Importance in Ecology

When it comes to understanding biodiversity, one concept that comes into play is species richness. This term refers to the number of different species represented in an ecological community, landscape, or region. Why is this important? Because species richness is a crucial component for conservation strategies. More diverse ecosystems are often considered healthier, more resilient to disturbances like climate change, and better providers of ecosystem services.

Ecologists use various methods, including statistical models, to predict species richness across different habitats. This is vital, as predicting areas with high species richness can guide conservation efforts to protect the greatest number of species, especially in situations where resources for conservation are limited. The exercise refers to a model that estimates species richness (y) by relating it to various habitat variables like watershed area, shore width, and others.

By studying the relationships between environmental factors and species richness, ecologists can identify, preserve, and manage habitats that are critical for maintaining biodiversity. The maintenance of these rich areas is not just about saving plants and animals; it's also about preserving the genetic diversity that is needed for species to adapt and survive long term.

Coefficient of Determination in Multiple Regression

The coefficient of determination, denoted as R^2, is a statistic that provides us with the percentage of variance in the dependent variable that's explained by all the independent variables in the model. In the context of ecology and species richness, an R^2 value can reveal how well environmental factors can predict the biodiversity of an area.

An R^2 value of 0.83, like in the given exercise, indicates a strong relationship - 83% of variance in species richness is explained by the model's variables. This high value is indicative of a powerful model that can be very useful in making predictions. It's important to note that while a high R^2 suggests a good fit, it doesn't imply causation, and it doesn't mean the model is perfect. Factors such as overfitting and the need for a balance between model complexity and explanatory power also need to be considered.

Hypothesis Testing in Regression Analysis

In the realm of statistics, hypothesis testing is a fundamental concept used to infer whether there is enough evidence to support a particular belief or hypothesis about a population, based on sample data. In the context of our multiple regression model, hypothesis testing is used to determine if at least one of the predictors has a statistically significant relationship with the dependent variable, species richness, in this case.

During the test, we set up two hypotheses: the null hypothesis (H_0), which states that there is no effect or relationship (all coefficients are zero), and the alternative hypothesis (H_a), suggesting that at least one predictor is truly associated with the response variable. We then use a significance level to decide how strong our evidence must be to reject the null hypothesis. A common choice, as seen in the exercise, is the 0.01 significance level. This means that for the null hypothesis to be rejected, the probability of the observed data, given that the null hypothesis is true (the p-value), must be less than 1%.

Assuming that the unseen test statistic (like the F-test) provides a p-value less than 0.01 with the high R^2 value, it suggests that the regression model is not only statistically significant but also practically useful. The multiple regression analysis has therefore provided us with good indicators that the predictors used are meaningful in explaining species richness, and this approach allows ecologists to make informed decisions on conservation strategies.