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Parameter Estimation, Dummies, & Model Fit

- We know mechanically how to “run a regression”…but how are the parameters actually estimated?
- How can we handle “categorical” explanatory (independent) variables?
- What is a measure of “goodness of fit” of a statistical model to data?

Example: Alien Species

- Exotic species cause economic and ecological damage
- Not all countries equally invaded
- Want to understand characteristics of country that make it more likely to be “invaded”.

Understanding Invasive Species

Steps to improving our understanding:

- Generate a set of hypotheses (so they can be “accepted” or “rejected”)
- Develop a statistical model. Interpret hypotheses in context of statistical model.
- Collect data. Estimate parameters of model.
- Test hypotheses.

2 Hypotheses (in words)

- We’ll measure “invasiveness” as proportion of Alien/Native species (article by Dalmazzone).
- Population density plays a role in a country’s invasiveness.
- Island nations are more invaded than mainland nations.

Variables

- Variables:
- Dependent: Proportion of number of alien species to native species in each country.
- Independent:
- Island?
- Population Density
- GDP per capita
- Agricultural activity

Computer Minimizes Sei2

- Remember, OLS finds coefficients that minimize sum squared residuals
- Graphical representation
- Why is this appropriate?
- Can show that this criterion leads to estimates that are most precise unbiased estimates.

Dummy Variable

- Generally:
- Male/Female; Pre-regulation/Post-regulation; etc..
- Use a “Dummy Variable”. Value = 1 if country is Island, 0 otherwise.
- More generally, if n categories, use n-1 dummies.
- E.g. if want to distinguish between 6 continents
- Problem: Lose “degrees of freedom”.

A Simple Model

- A simple linear model looks like this:
- Dummy changes intercept (explain).
- Interaction dummy variable?
- E.g. Invasions of island nations more strongly affected by agricultural activity.

Translating our Hypotheses

- 2 Hypotheses
- Hypothesis 1: Population: Focus on a3
- Hypothesis 2: Island: Focus on a2
- “Hypothesis Testing”… forthcoming in course.
- Parameter Estimates:

Value Std.Error t value Pr(>|t|)

(Intercept) -0.0184 0.0859 -0.2141 0.8326

Island 0.0623 0.0821 0.7588 0.4564

Pop.dens 0.0010 0.0002 6.2330 0.0000

GDP 0.0000 0.0000 3.3249 0.0032

Agr -0.0014 0.0015 -0.9039 0.3763

“Goodness of Fit”: R2

- “Coefficient of Determination”
- R2=Squared correlation between Y and OLS prediction of Y
- R2=% of total variation that is explained by regression, [0,1]
- OLS maximizes R2.
- Adding independent cannot R2
- Adjusted R2 penalizes for # vars.

Answers

- Island nations are more heavily invaded (.0623)
- Not significant (p=.46)
- Population density has impact on invasions (.001)
- Significant (p=.0000)
- R2=.80; about 80% of variation in dependent variable explained by model.
- Also, corr(A,Ahat)=.89

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