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By adding an interaction term for our binary predictor, we have made our model more complex and have therefore also added complexity to its interpretation.

Returning to our multiple regression equation with an interaction term from the last exercise, we have:

happy=12.11.0stress3.1exercise+0.4stressexercise\text{happy} = 12.1 - 1.0*\text{stress} - 3.1*\text{exercise} + 0.4*\text{stress}*\text{exercise}

We can rewrite this equation for the group that doesn’t exercise regularly (exercise = 0) and for the one that does (exercise = 1).

When exercise = 0, the last two terms become zero and go away:

happy=12.11.0stress3.10+0.4stress0happy=12.11.0stress0+0happy=12.11.0stress\begin{aligned} \text{happy} = 12.1 - 1.0*\text{stress} - 3.1*0 + 0.4*\text{stress}*0 \\ \text{happy} = 12.1 - 1.0*\text{stress} - 0 + 0 \\ \text{happy} = 12.1 - 1.0*\text{stress} \\ \end{aligned}

When exercise = 1, the intercept goes down by 3.1 and the coefficient on stress increases by 0.4:

happy=12.11.0stress3.11+0.4stress1happy=12.11.0stress3.1+0.4stresshappy=(12.13.1)+(1.0+0.4)stresshappy=9.00.6stress\begin{aligned} \text{happy} = 12.1 - 1.0*\text{stress} - 3.1*1 + 0.4*\text{stress}*1 \\ \text{happy} = 12.1 - 1.0*\text{stress} - 3.1 + 0.4*\text{stress} \\ \text{happy} = (12.1 - 3.1) + (- 1.0 + 0.4)*\text{stress} \\ \text{happy} = 9.0 - 0.6*\text{stress} \end{aligned}

We can see the coefficient on exercise tells us the difference in INTERCEPTS between the two exercise groups. In this case, the intercept of the regression line for the group that exercises (9.0) is 3.1 units lower than that of the group that doesn’t exercise (12.1).

On the other hand, the coefficient on the interaction term tells us the difference in SLOPES between the two regression lines. The slope on stress for the group that exercises (-0.6) is 0.4 units greater than that of the group that doesn’t exercise (-1.0). This would lead us to conclude that the happiness level of the group who exercises is less negatively impacted by stress.

Instructions

1.

The output of the regression predicting height from predictors weight and species with an interaction term from the last exercise is shown below.

# Output: # Intercept 8.168619 # species[T.B] -3.580515 # weight 1.658621 # weight:species[T.B] 1.115071

Write out and simplify the regression equation for species A. In script.py, save the value of the coefficient on weight as a variable named slopeA. What do we learn about the relationship between height and weight for species A?

2.

Write out and simplify the regression equation for species B. In script.py, save the value of the coefficient on weight as slopeB. Why might plants of different species show different relationships between their heights and weights?

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