Lake Eutrophication and a Golf Course

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Lake Eutrophication and a Golf Course

• Chlorophyll-a (C) widely used indicator measure
  of eutrophication
• Nitrogen (N) associated with eutrophication
• Q Golf Course Development. Nitrogen expected to
  ?. By how much will C increase/decrease in the
  local lake?

• Lets look at data from other lakes and fit a
  linear relation between C and N
• Slope of relationship will give us the expected
  effect on C of a unit increase in N

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Ordinary Least Squares (OLS) Regression

• Estimators have many properties.
• 6 is an estimator, but not a very good one.
• Two main properties we care about
• Unbiased The expected distance of estimator from
  thing it is estimating is 0.
• Efficient Small variance (uncertainty)
• 6 is biased, but has a very small variance
  (zero).

• Also called Classical Linear Regression Model
  (CLRM)
• Find the intercept and slope parameters such that
  the sum of squared residuals is as small as
  possible
• OLS is an estimator for the parameters of the
  model

Given certain assumptions are satisfied, OLS
estimator is unbiased and has minimum variance of
all unbiased estimators.
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Fraction of variance in Chlorophyll-a explained
by model
Standard deviation of residuals
P-value for model as a whole
1-sample t-tests
P-value for intercept
P-value for slope parameter
Parameter estimates
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Is something wrong here?
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Multiple regression

• Eutrophication may be affected by both nitrogen
  and phosphorus
• We are interested in the effects of N on C, while
  holding P constant
• We cant get that independence directly from the
  data N and P are correlated
• Multiple regression is the key
• Equation on board
• Slopes are partial coefficients
• In simple regression, slope is marginal
  coefficient

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Back to the golf course

• Use data to estimate parameter values that give
  best fit b0-9.4, b10.3, b21.2
• Answer A one unit increase in N, results in
  about a 1.2 unit increase in C.
• Importance Omitting phosphorus from model
  introduced significant bias!!!

• But theres a lot of uncertainty in the estimate
  of the effect of N
• 95 CI ranges from about 1.2 to about 3.6
• Question does nitrogen have any effect on
  chlorophyll A in these lakes?

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Does nitrogen have an effect?

• In multiple regression, cant tell just from
  looking at the P values of the individual
  coefficients
• If two independent variables are collinear
  (correlated), then the P values will be inflated
  or deflated
• Removing one may decrease P value of other
• Instead, look at effect of removing each
  variable, one at a time, from the model
• Uses F-test to test null hypothesis that
  increased goodness of fit from that variable is
  just due to chance

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Enhance understanding
Make predictions
Estimate parameters
Describe patterns and relationships in data
Select models
Test statistical hypotheses
Test theories
Make decisions
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Assumptions of OLS regression

• Model is linear in parameters
• The data are a random sample of the population
• The residuals are statistically independent from
  one another
• The expected value of the residuals is always
  zero
• The independent variables are not too strongly
  collinear
• The residuals have constant variance

• If assumptions 1-4 are satisfied, then OLS
  estimator is unbiased
• If assumption 5 is also satisfied, then
• OLS estimator has minimum variance of all
  unbiased estimators.
• How can we test these assumptions?
• If assumptions are violated,
• what does this do to our conclusions?
• how do we fix the problem?

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What makes it linear regression?

• Model is linear in parameters
• Parameters cant occur inside a nonlinear
  function
• Residuals are additive

• Nonlinearity in variables is OK

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Dummy variables

• How can we handle categorical explanatory
  (independent) variables in a regression?
• Dichotomous
• Male/Female
• Pre-regulation/Post-regulation
• Island/Mainland
• Polytomous
• Continent
• Political party
• Soil type

• Answer make a dummy!

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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.

• Well measure invasiveness as fraction of
  species that are Alien
• Two hypotheses
• Human population density plays a role in a
  countrys invasiveness.
• Island nations are more invaded than mainland
  nations.

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Island
Mainland
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A Simple Model

• ISL is a Dummy variable, coded 0 if mainland, 1
  if island
• Dummy changes intercept (explain).
• Interaction dummy variable?
• E.g. Invasions of island nations more strongly
  affected by population density.

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A model with interactions
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What about a polytomous variable (e.g.,
Continent)?
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