Recall: The Client

We have a client who is interested in determining what features (explanatory variables) in the data set are related to high energy use for residential buildings with at least 6000 sq feet of floor area. Recall that energy use is included in the EUI variable in the data set.

Since the client is particularly interested in large residential buildings, make a subset of your data called large_residential, which contains only residential buildings with at least 6000 sq feet of floor area.

The client asks you to create visualizations and/or use a model to explore this. The goal is to identify features related to high energy use so the client can explore ways to help buildings make changes and reduce energy consumption. They ask you specifically to look at the age of the building, but want you to examine other features as well.

While we will ultimately build a model with multiple predictors, it is useful to begin our exploration with one predictor. This allows us to explore each variable and assess model assumptions as we build the model.

Let’s start by fitting a simple linear regression model with your chosen feature. For example, if I start with age as an explanatory variable, I can fit linear regression with the lm function:

age_lm <- lm(site_eui ~ age, data = large_residential)

Let’s break that down.

• The lm part of the code tells R we want to build a linear regression model.
• The second piece of the code is our Y variable (site_eui).
• The third piece in the code is our X variable (age).
• We then provide the data set (data = large_residential) where the two variables can be found.
• We then store all of our results under the object called age_lm.

To get the coefficients (estimated intercept and slope) for the linear regression model, we use

age_lm$coefficients

Checking Assumptions

Linear regression works best when some assumptions about the relationship between our predictor(s) and response are satisfied. In particular, we often make the following assumptions:

• Shape: the true relationship is approximately linear
• Constant variance: the variance of the residuals is the same for different values of the predictor(s)
• Normality: the residuals are approximately normal

We can assess these assumptions with diagnostic plots, like residual plots (constant variance) and QQ plots (normality). For example, here is some code to make a residual plot and a QQ plot for the age_lm model above:

# residual plot
large_residential %>%
  ggplot(aes(x = predict(age_lm),
             y = resid(age_lm))) +
  geom_point() +
  geom_abline(slope = 0, intercept = 0, color="blue") +
  labs(x = "Predicted EUI",
       y = "Residual") +
  theme_bw()
# QQ plot
large_residential %>%
  ggplot(aes(sample = resid(age_lm))) +
  geom_qq() +
  geom_qq_line(color="blue") +
  labs(x = "Theoretical normal quantiles",
       y = "Observed residual quantiles") +
  theme_bw()

Assessing Model Fit

Once we have a fitted regression model for which the assumptions seem reasonable, we want to know whether it is actually useful at predicting EUI. For a simple least-squares linear regression (one predictor), we typically summarize model fit with the correlation ( \(r\) ) or coefficient of determination ( \(R^2\) ). For multiple least-squares regression, we use the adjusted coefficient of determination ( \(R^2_{adj}\) ), which accounts for the number of parameters in our model. \(R^2\) and \(R^2_{adj}\) can be found in the summary output in R.

summary(age_lm)

In general, we need more than one predictor to do a good job predicting the response. Let’s add more predictors to the model. When we add predictors, we also need to think about whether there are any interactions between explanatory variables. Remember that an interaction means that the relationship between an explanatory and the response changes depending on the value of another explanatory variable in the model.

Next steps

Exploring relationships by hand is important, but we probably can’t investigate every variable in the data this way. A linear relationship might also not be an appropriate choice for our data. In our next steps, we’ll talk about model selection techniques, and consider other types of regression.