Question to ponder
• As our ability to collect and store huge
amounts of data increases, should we add
more environmental covariates to our
ecological models to make them more
realistic?
Data exploration and simple
linear regression
Todays goals
• Describe steps to take in exploring data prior
to analysis
• Review linear regression
– Most of the remainder of this course is based on
the concepts underlying linear regression
Steps in sta6s6cal analysis
1.
2.
3.
4.
5.
Develop hypothesis
Data exploration
Model fitting
Model selection
Model diagnostics
Data exploration
• Reasons to explore data
– Quality control
– Check for outliers
– Detect patterns in data
– Ensure assumptions of analysis are met
Histograms
Dotplots
Boxplots
Colinearity
Linear regression
• Used to determine the extent to which there
is a linear relationship between a dependent
variable (response) and one, or more,
‘independent’ variables (predictor)
Linear Models
• Relationship between variables is a Linear
Function
Y-Intercept
Random
Error
Slope
𝑌! = 𝛽" + 𝛽# 𝑋! + 𝜀!
Dependent
(Response)
Variable
Independent or Explanatory
Variable (or Covariate)
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Regression
𝑌! = 𝛽" + 𝛽# 𝑋! + 𝜀!
𝛽! = Slope
(rise-over-run, ΔY/ ΔX)
Y
𝛽" = y-intercept
(where line crosses Y-axis)
Quinn and Keough 2002
X
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Regression
𝑌! = 𝛽" + 𝛽# 𝑋! + 𝜀!
Y
𝜀! = unexpected error
Residual = observed – expected
𝜀#̂ = 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = 𝑌# − 𝑌/#
Quinn and Keough 2002
X
How do we fit this model?
• Ordinary least squares
– Method of fitting a model in which you
minimize the residual sums of squares (RSS)
𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = 𝜀# = 𝑌# − 𝑌/#
%
𝑅𝑆𝑆 = 2(𝑌# − 𝑌/# )&
#$!
14
• Khan academy video
Maximum likelihood
• Gaussian (normal distribution)
• µ = mean
• s2 = variance
Simple example
• Consider 3 data points
– y1=1, y2=0.5, y3=1.5
• Likelihood is the
product of the
individual probabilities
of each data point
given a parameter
estimate
Maximum likelihood
• Normal distribution likelihood equation
• In practice, for numerical purposes, we
minimize the negative log-likelihood
Example data
R code
• lm(fishSize~fishDens, data=sim, weights=frequency)
Call:
lm(formula = fishSize ~ fishDens, data = sim)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 17.5375 0.3263 53.755 <2e-16 ***
fishDens -1.3171 0.6462 -2.038 0.0418 *
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 5.147 on 1259 degrees of freedom
Multiple R-squared: 0.003288, Adjusted R-squared: 0.002497
F-statistic: 4.154 on 1 and 1259 DF, p-value: 0.04176
Assumptions
1. Normality
2. Homogeneity of
variance (HOV)
Y
3. Independence
4. Correct model
specification
5. Fixed X
X
Zuur et al. 2009
22
1. Normality
• The errors are assumed to be normally
distributed with a mean of zero
– Needed for parametric statistical tests
23
1. Normality
• Diagnostics
– Graphical evaluation (best)
• histogram of residuals
24
2. Homogeneity of variance
• Diagnostics
– Graphical evaluation is best
• Residual plots or conditional boxplots of residuals
25
2. Homogeneity of variance
• Diagnostics
– Graphical evaluation is best
• Residual plots or conditional boxplots of residuals
26
3. Independence of errors
• Collec\on of one sample is assumed to have
NO effect, influence, or correla\on with
another sample
• Sta\s\cal methods rely on samples being
random and independent
27
3. Independence of errors
• Examples of non-independence:
– Nested/hierarchical design;
– repeated measures;
– temporal correlation;
– spatial correlation
– “pseudoreplication”
Data set
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3. Independence of errors
Diagnostics
– Understanding of sampling design!
– Graphical evaluation
• Plot Y (or residuals) versus time/space (e.g., bubble plots)
• Auto-correlation plots
Spatial correlation of residuals
Autocorrelation function acf()
Residuals
Y-coordinates
•
X-coordinates
Time lag (years)
29