Statistical Analysis of Species Diversity on Galapagos Islands using Poisson Regression - , Exams of Data Analysis & Statistical Methods

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Stat/F&W/Hort 572 Hwang/Zhao March 2, 2009

DISCUSSION 6

March 2, 2009

When the response is a count (a positive integer), we can use a count regression model to explain this response in terms of the given predictors. We will see how Poisson Regression works in this case.

Species diversity on the Galapagos Islands

M. P. Johnson and P. H. Raven (1973) “Species number and endemism: The Galapagos Archipelago revisited”, Science, 179, 893-895 For 30 Gal´apagos Islands, we have a count of the number of species of plants found on each island and the number that are endemic to that island. We also have five geographic varialbes for each island. The data was presented by Johnson and Raven(1973). There are 30 Gal´apagos islands and 6 variables in the dataset. The relationship between the number of plant species and several geographic variables is of interest. The original dataset contained several missing values which have been filled for convenience.

• Species : the number of plant species found on the island
• Area: the area of the island (km^2 )
• Elevation: the highest elevation of the island (m)
• Nearest: the distance from the nearest island (km)
• Scruz: the distance from Santa Cruz island (km)
• Adjacent: the area of the adjacent island (km^2 )

1. Make plots of each potential input variable versus the outcome number of species. Which variables appear to be most predictive? Try to fit a (standard) linear regression. Discuss the price we pay for fitting a standard linear regression.
2. Fit a Poisson regression model to predict number of species based on all other variables(do not use interac- tions). Report the coefficients of this model. Do the signs of the coefficients make sense?
3. To interpret coefficients, first find the standard deviation for each input variable in the previous model. What change in the predicted number of species is associated with a one standard deviation increase in each explanatory variable (holding other variables constant)?
4. Determine which island is at elevation 1000 or more, and find their values. Then use the model from (2) to predict the number of species in the island at elevation 1000. Show how to do these calculations by hand.
5. Check the residual plots. Do they raise any concern? Assess the fit from the model with a χ^2 distribution. What do you conclude?
6. Test each predictor’s coefficient using a Wald test, and using an analysis of deviance. Are the p-values from the two tests identical?
7. Refit the model above accounting for possible overdispersion, then test each predictor’s coefficient using both a Wald test and an F test.