R Practice: Using Linear Models Determine the Relationship Between Biodiversity and Socioeconomic Status (Part I: Exploring Linear Model Outputs)

Loading Packages and Data

First, let's load any libraries we will be using (here just tidyverse) and our dataset. 

library(tidyverse) #Loads the tidyverse package into R's brain. This pacakge contains other packages that allow us to do data wrangling, transformation, and visualization.

bird <- read.csv(url("https://bio.libretexts.org/@api/deki/files/65130/GINI_BIRDS.csv?origin=mt-web"), sep = ',', head = TRUE)
#This loads in the dataset and stores it in R's "brain," that way we can use the data within it for analyses and figure making.

tibble(bird) # Visualizes the dataset in a table.

Creating a Regression Plot

Now, let's visualize the relationship using a linear regression. Afterwards, we will have to test whether this relationship is statistically significant. 

bird %>%
  ggplot(aes(x = GiniRatio_1969, y = DecResBird_1966)) + #plotting two continuous variables, Gini ratio and Declining resident bird species onto a plot.
  geom_point() + #adding points to my plot.
  geom_smooth( method = "lm" ) # Creates a regression line for a linear model.

Before we move on, we should take a look at our plot and see whether it is likely to fit the assumptions of a linear model. Is the relationship linear? Are the data homoscedastic (e.g., the variance of points stays similar across values of the y-axis). Learn more about the assumptions of linear models on Wikipedia. 

Running a Linear Model

The code below generates some key linear regression statistics, including the p-value, the slope of the relationship, and the r-squared value. The p-value tells us how likely it is that our data occurred due to chance. A higher p-value indicates a stronger probability that the results we see are due to chance. Most fields used a cut-off off p-value of 0.05 to consider a relationship statistically significant. Anything less than 0.05 means that the probability of our results occurring by chance is less than 5%. The R-squared value is tells us how much of the dependent variable can be explained by the independent variable. For example, if we were to calculate an r² value of .75, that would mean that 75% of the variance in the declining resident bird population can be explained by the Gini Ratio. Lastly, the slope tells us how fast the dependent variable increases or decreases with every one unit change in the independent variable. 

To find the linear regression statistics, we use the formula:

new_data_frame = lm( y-axis_Variable ~ x-axis_Variable, data = old_data_frame)

We then have to display the summary statistics, which we do by using the code:

summary(new_data_frame)

Now let's try it our with our data! 

bird_mod = lm(DecResBird_1966 ~ GiniRatio_1969, data = bird) # creates a new data frame called bird mod, which calculates  linear model statistics using the data in DecResBird_1966 and GiniRatio_1969.

summary(bird_mod) # shows us a summary of the statistics present in bird mod.

Interpreting a Linear Model

Now we have our linear regression statistics, what can we interpret from them?

First, let's take a look at our p-value. Our p-value is 0.05266, which presents an interesting dilemma. As mentioned previously, many scientists infer that there is statistically significant correlation between two variables if the p-value is less than 0.05. This cut-off is somewhat arbitrary, however, and has received a lot of pushback from some scientists. Our p-value means that there was a 5.266 percent likelihood of our results occurring by chance. So while we cannot claim our results are statistically significant, it does seem like there might be a pattern here.

Next, let's look at our r-squared value. We can determine that there is a weak positive correlation between the Gini Ratio and declining resident bird species given our low r² value of 0.08454. This value suggests that only 8.5% of the difference in %declining resident bird species across states can be explained by the Gini Ratio.

The last value we want to look at is our Estimate, or our slope. Here, our Estimate is 22.578. This represents the slope between the Gini Ratio and the % of declining resident bird species. In other words, for every increase of 1 in the Gini Ratio, the percent of resident bird species showing population decline increases by 22.58%. Because the Gini Ratio ranges from 0 to 1, this suggests that a state with perfect inequality (Ratio = 1) has 22.58% more of its resident bird species declining than a state with perfect equality (Ratio = 0). 

We can also use this table to write out the linear model that describes the relationship we are observing, using the form y = mx + b where:

• y is our dependent variable (declining resident bird species)
• x is our independent variable (Gini Ratio)
• m is our slope 
• b is out y-intercept

In the R outline, the column Estimate gives us our slope and y-intercept. The y-intercept is the estimate for "intercept", while the slope is the Estimate for "GiniRatio_1969". 

A male Kirtland's Warbler in a jack pine forest in Michigan, USA by snowmanradio is licensed under CC BY 2.0. Kirkland's Warblers are near threatened species that are dependent on jack pine habitat.

References:

Holland, T. G., Peterson, G. D., & Gonzalez, A. (2009). A cross-national analysis of how  economic inequality predicts Biodiversity loss. Conservation Biology, 23(5), 1304–1313. https://doi.org/10.1111/j.1523-1739.2009.01207.x

Hope, D., Gries, C., Zhu, W., Fagan, W. F., Redman, C. L., Grimm, N. B., Nelson, A. L., Martin, C., & Kinzig, A. (2003). Socioeconomics Drive Urban Plant Diversity. Proceedings of the National Academy of Sciences, 100(15), 8788–8792. https://doi.org/10.1073/pnas.1537557100

Kuras, E. R., Warren, P. S., Zinda, J. A., Aronson, M. F. J., Cilliers, S., Goddard, M. A., Nilon, C. H., & Winkler, R. (2020). Urban socioeconomic inequality and biodiversity often converge, but not always: A global meta-analysis. Landscape and Urban Planning, 198. https://doi.org/10.1016/j.landurbplan.2020.103799

Mikkelson, G. M., Gonzalez, A., & Peterson, G. D. (2007). Economic inequality predicts biodiversity loss. PloS one, 2(5), e444.