Assignment 5 – Multivariable linear models with interactions - 2019

Put all your answers in a word document

Note: Generally, while we should always report the betas and C.I. from our analyses on main effects, I believe that if interactions are non-significant, it is enough to say so (with the p-value) and take the interaction OUT of the model before extracting betas from main effects. The reason is because interactions make everything so much more complicated.

Truth for all data

Case 1 - Data

In this example, you have a continuous x-variable (e.g., road density [roads/sq. km]) and one categorical x-variable (Hardwood forest vs. Pine forest) and you are looking at their effect on squirrel density (squirrels / hectare). In this system, there is an interaction between road density and forest type.

1.      Write a linear (statistical) model that describes the system; be sure you specify what each ‘x’ represents.

2.      What do the various β’s mean in this model?

3.      Run the full model in R. Report whether or not the interaction is significant of not (including p-value)

4.   Analyze data from the two forest types separately and describe the results using the standard sentences.

Bonus: Note that the sq. dens is higher in pine at low road density and lower in pine at high road density. At what road density is squirrel density the same between both forest types?

Case 2 - Data

In this example you have two categorical factors (e.g., Sex and Treatment) and each factor has two levels (male + female, control (placebo) + hormone injection). The dependent variable might be something like adult size. In this system, there is an interaction between sex and treatment

1.      Write a linear (statistical) model that describes the system; be sure you specify what each ‘x’ represents.

2.      What do the various β’s mean in this model?

3.      Run the full model in R. Report whether or not the interaction is significant of not (including p-value)  

4. Analyze data from the two sexes separately and describe the results using the standard sentences.

Case 3 - Data

In this example you have two continuous variables (e.g., Elevation - meters; and Latitude - km). The dependent variable is lotus flower size. In this system, there is an interaction between elevation and latitude.

1.      Write a linear (statistical) model that describes the system; be sure you specify what each ‘x’ represents.

2.      What do the various β’s mean in this model?

3.      Write example sentences that you might include in a manuscript for publication that describes the observed results. First describe the effect of elevation (at zero latitude). Then describe the effect of latitude (at zero elevation). Finally, describe how the effect of elevation changes as latitude increases (or vice versa). The following sentence might help:

We found a significant interaction (p = [p-value for interaction term]), such that for each 1 [x1 units] increase in [x1], we observed that the slope of the [x2]-[y] relationship [increased/decreased] by [beta for interaction][y units]/[x2 units] ([95% C.I. of beta for interaction]; +/-95% C.I.).