Packet 5 Concept Map

Packet 5 Concept Map

1. Assumptions of Linear Regression Model:

   • 1. The average value of our residuals is equal to zero
     • 2. The variance of the residuals is a constant value of sigma squared for all possible values of x
   • 3. The distribution of values of the residuals is normal
     • 4. The errors associated with two different observations are independent

2. Detecting Lack of Fit: plot residuals versus fitted values

   • plot(regress_model$fitted.values,regress_model$residuals)
     • look to ensure that there is no trend in plotted values (like curvature / coning effect)

3. Detecting Unequal Variances

   • reference plot of residuals vs. fitted values: ensure that there is no heteroscedasticity (variance not constant)
     • If you detect unequal variance, transform the response variable
       • common transformations are sqrt(y) and ln(y)
         • example: lm(sqrt(species) ~ area + adjacent + ....)
           • Be careful when interpreting results after transformations!

4. Authors: Sam Jones, Claire DiOrio, Walker Utley, Abigail Stokes

5. Checking Normality Assumption

   • Method 1: make a histogram of residuals
     • Code: hist(regress_model$residuals)
       • Check if this is fairly normal (bell-shaped)
   • Method 2: Construct Normality-Probability Plot
     • Straight-line pattern indicates normality
       • qqnorm(regress_model$residuals)
         • qqline(regress_model$residuals)
   • Conduct formal hypothesis using Shapiro-Wilk test
     • Null is normality, alternative is non-normality
       • Shapiro.test(regress_model$residuals)

6. Detecting Residual Correlation: Durbin-Watson Test

   • Ho: Residuals are not correlated (independence assumption upheld!)
     • Ha: Residuals are correlated
       • DW test statistic will range from 0 to 4:

7. Detecting Outliers and Identifying Influential Outliers

   • Outliers: observation with z score >3 or < -3
   • Influential Data Points: points whose removal would cause large change in fit
     • Measure of influence of individual data points on regression is Cook's Distance

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