1. Linear RegressionSlide #1 Example - Rabbit Metabolic Rate Katzner et al. (1997; J. Wildl. Man. 78:1053-1062) examined the metabolic rate of pygmy rabbits (Brachylagus idahoensis) in the laboratory. In particular, they wanted to determine if the variability in resting metabolic rate (ml O 2 g -1 h -1 ) at 20 o C could be adequately explained by body mass (g). What is the response variable? –Resting metabolic rate What is the explanatory variable? –Body mass 1 2

2. Linear RegressionSlide #2 Example - Rabbit Metabolic Rate Y = 1.41 - 0.00124X R-Sq = 55.4 % 400450500 0.8 0.9 1.0 Mass Metabolic Rate In terms of the variables of the problem, what is the equation of the best-fit line? MetRate = 1.41-0.00124Mass 3

3. Linear RegressionSlide #3 Example - Rabbit Metabolic Rate Y = 1.41 - 0.00124X R-Sq = 55.4 % 400450500 0.8 0.9 1.0 Mass Metabolic Rate In terms of the variables of the problem, interpret the value of the slope? For each additional gram of mass, the metabolic rate decreases 0.00124 ml O 2 g -1 h -1 on average 4

4. Linear RegressionSlide #4 Example - Rabbit Metabolic Rate Y = 1.41 - 0.00124X R-Sq = 55.4 % 400450500 0.8 0.9 1.0 Mass Metabolic Rate In terms of the variables of the problem, interpret the value of the y- intercept? Rabbits with no mass have a metabolic rate of 1.41 ml O 2 g -1 h -1 on average 5

5. Linear RegressionSlide #5 Example - Rabbit Metabolic Rate Y = 1.41 - 0.00124X R-Sq = 55.4 % 400450500 0.8 0.9 1.0 Mass Metabolic Rate What is the predicted metabolic rate for a mass of 450 g? 6 (450,0.85) MetRate = 1.41-0.00124Mass MetRate = 1.41-0.00124*450 MetRate = 0.85

6. Linear RegressionSlide #6 Example - Rabbit Metabolic Rate Y = 1.41 - 0.00124X R-Sq = 55.4 % 400450500 0.8 0.9 1.0 Mass Metabolic Rate What is the predicted metabolic rate for a mass of 600 g? 7

7. Linear RegressionSlide #7 Example - Rabbit Metabolic Rate Y = 1.41 - 0.00124X R-Sq = 55.4 % 400450500 0.8 0.9 1.0 Mass Metabolic Rate What is the residual for a mass of 425 g and a metabolic rate of 0.82 ml O 2 g -1 h -1 ? 8 (425,0.82) (425,0.88) Resid = Obs Y – Pred Y Resid = 0.82-(1.41-0.00124*425) Resid = 0.82-0.88 Resid = -0.06

8. Linear RegressionSlide #8 Example - Rabbit Metabolic Rate Y = 1.41 - 0.00124X R-Sq = 55.4 % 400450500 0.8 0.9 1.0 Mass Metabolic Rate What proportion of the variability in metabolic rate is explained by knowing mass? r 2 = 0.554 9 What is the correlation between metabolic rate and mass? r = 0.554 0.5 10 r = 0.744 -

9. Linear RegressionSlide #9 Example - Rabbit Metabolic Rate Y = 1.41 - 0.00124X R-Sq = 55.4 % 400450500 0.8 0.9 1.0 Mass Metabolic Rate Does anything about this regression concern you? 11