Module 4.6: More Regression Practice
All materials can be found at alexcardazzi.github.io.
We will keep working with the Ames data. First, load the packages, read in the data, and rename the columns as before:
library("scales")
library("modelsummary")
options("modelsummary_factory_default" = "kableExtra")
ames <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/openintro/ames.csv")
keep_columns <- c("price", "area", "Bedroom.AbvGr", "Year.Built", "Overall.Cond")
ames <- ames[,keep_columns]
colnames(ames) <- c("price", "sqft", "bedrooms", "yr_built", "condition")Next, let’s estimate models with different explanatory variables. First, we’ll estimate the relationship between sale price and number of bedrooms.
r_price_bedrooms <- lm(price ~ bedrooms, ames)
r_lprice_bedrooms <- lm(log(price) ~ bedrooms, ames)
regz <- list(`Price` = r_price_bedrooms,
`log(Price)` = r_lprice_bedrooms)
coefz <- c("bedrooms" = "# of Bedrooms",
"(Intercept)" = "Constant")
gofz <- c("nobs", "r.squared")
modelsummary(regz,
title = "Effect of Bedrooms on Sale Price",
estimate = "{estimate}{stars}",
coef_map = coefz,
gof_map = gofz)| Price | log(Price) | |
|---|---|---|
| # of Bedrooms | 13889.495*** | 0.089*** |
| (1765.042) | (0.009) | |
| Constant | 141151.743*** | 11.767*** |
| (5245.395) | (0.027) | |
| Num.Obs. | 2930 | 2930 |
| R2 | 0.021 | 0.033 |
Breaking down the output:
***).Next, we can estimate models where a home’s condition is the explanatory variable.
r_price_condition <- lm(price ~ condition, ames)
r_lprice_condition <- lm(log(price) ~ condition, ames)
regz <- list(`Price` = r_price_condition,
`log(Price)` = r_lprice_condition)
coefz <- c("condition" = "# of Condition",
"(Intercept)" = "Constant")
gofz <- c("nobs", "r.squared")
modelsummary(regz,
title = "Effect of Condition on Sale Price",
estimate = "{estimate}{stars}",
coef_map = coefz,
gof_map = gofz)| Price | log(Price) | |
|---|---|---|
| # of Condition | −7309.010*** | −0.018** |
| (1321.319) | (0.007) | |
| Constant | 221457.104*** | 12.119*** |
| (7495.923) | (0.038) | |
| Num.Obs. | 2930 | 2930 |
| R2 | 0.010 | 0.002 |
First of all, using the condition variable in the regression does not lead to interpretable coefficients. Condition is an ordinal (and subjective) variable, which makes a one unit change meaningless.
That said, the results of these models are suspicious. Since higher numbers mean better condition, the coefficient estimates should still be positive even if it’s an ordinal variable. Remember the four reasons you might find a correlation between two variables. In this case, there is likely a third variable that is driving both condition and sale price.
Let’s try to investigate and figure out why this might be.
First, let’s check out the distribution of the condition variable.
plot(table(ames$condition), las = 1,
main = "Distribution of Condition",
xlab = "Condition", ylab = "Frequency")
Clearly, there are many homes with a condition of 5. In fact, 56% of homes are rated to be a 5.
Next, let’s examine the average sale price for each possible value for condition.
agg <- aggregate(list(price = ames$price), list(condition = ames$condition), mean)
plot(agg$condition, agg$price/1000, las = 1, pch = 19,
xlab = "Condition", ylab = "Price (in Thousands)")
There appears to be an upward trend in price as condition increases. However, there is an outlier value when condition equals 5. There must be a bunch of high price homes that have been given a condition of 5.
Theoretically, the condition of a home should be related to its age (or the year it was built). Let’s examine the relationship between year built and condition by finding the average condition for each possible year built.
agg <- aggregate(list(condition = ames$condition), list(yr = ames$yr_built), mean)
plot(agg$yr, agg$condition, las = 1, pch = 19,
col = alpha("mediumseagreen", 0.6),
xlab = "Year Built", ylab = "Average Condition")
There seems to be a negative relationship between year built and condition. However, the correlation between year built and price is positive. It’s likely that year built is driving both price and condition simultaneously. The negative coefficient between condition and price is due to the negative relationship between year built and condition.
What do models look like where year built is the main explanatory variable? Using the year a home was built does not make as much intuitive sense as thinking about the age of a home. Since all of these homes were sold between 2006 and 2010, according to the data description, we can calculate age of a property by subtracting the year built from 2011. Let’s run some models with age as the explanatory variable.
ames$age <- 2011 - ames$yr_built
r_price_age <- lm(price ~ age, ames)
r_lprice_age <- lm(log(price) ~ age, ames)
r_price_lage <- lm(price ~ log(age), ames)
r_lprice_lage <- lm(log(price) ~ log(age), ames)
regz <- list(`Level - Level` = r_price_age,
`Log - Level` = r_lprice_age,
`Level - Log` = r_price_lage,
`Log - Log` = r_lprice_lage)
coefz <- c("age" = "Age",
"log(age)" = "log(Age)",
"(Intercept)" = "Constant")
gofz <- c("nobs", "r.squared")
modelsummary(regz,
title = "Effect of Age on Sale Price",
estimate = "{estimate}{stars}",
coef_map = coefz,
gof_map = gofz)| Level - Level | Log - Level | Level - Log | Log - Log | |
|---|---|---|---|---|
| Age | −1474.964*** | −0.008*** | ||
| (40.493) | (0.000) | |||
| log(Age) | −47030.296*** | −0.251*** | ||
| (1098.021) | (0.005) | |||
| Constant | 239269.065*** | 12.350*** | 333731.719*** | 12.838*** |
| (2018.987) | (0.010) | (3753.501) | (0.019) | |
| Num.Obs. | 2930 | 2930 | 2930 | 2930 |
| R2 | 0.312 | 0.379 | 0.385 | 0.423 |
Interpreting these models:
A final point for this module is what happens to coefficient estimates when apply different transformations before estimation. So far, we’ve shown the effect of using a log() transformation. Now we’ll look at the effect of addition/subtraction and multiplication/division.
r_price_sqft <- lm(price ~ sqft, ames)
r_price_sqftdiv <- lm(price ~ I(sqft/1000), ames)
r_pricediv_sqft <- lm((price/1000) ~ sqft, ames)
r_price_sqftsub <- lm(price ~ I(sqft - 1500), ames)
r_pricesub_sqft <- lm(I(price - 180000) ~ sqft, ames)
regz <- list(`Original` = r_price_sqft,
`Sq Ft. / 1000` = r_price_sqftdiv,
`Price / 1000` = r_pricediv_sqft,
`Sq Ft. - 1500` = r_price_sqftsub,
`Price - 180K` = r_pricesub_sqft)
coefz <- c("sqft" = "Sq Ft.",
"I(sqft/1000)" = "Sq Ft.",
"I(sqft - 1500)" = "Sq Ft.")
gofz <- c("nobs", "r.squared")
modelsummary(regz,
title = "Variable Manipulation",
estimate = "{estimate}{stars}",
coef_map = coefz,
gof_map = gofz)| Original | Sq Ft. / 1000 | Price / 1000 | Sq Ft. - 1500 | Price - 180K | |
|---|---|---|---|---|---|
| Sq Ft. | 111.694*** | 111694.001*** | 0.112*** | 111.694*** | 111.694*** |
| (2.066) | (2066.073) | (0.002) | (2.066) | (2.066) | |
| Num.Obs. | 2930 | 2930 | 2930 | 2930 | 2930 |
| R2 | 0.500 | 0.500 | 0.500 | 0.500 | 0.500 |