Case study: Seattle house prices

Section 11.2 Case study: Seattle house prices

Kaggle.com is a machine learning and predictive modeling competition website that hosts datasets uploaded by companies, governmental organizations, and other individuals. One of their datasets is the “House Sales in King County, USA.” It consists of sale prices of homes sold between May 2014 and May 2015 in King County, Washington, USA, which includes the greater Seattle metropolitan area. This dataset is in the house_prices data frame included in the moderndive package.

The dataset consists of 21,613 houses and 21 variables describing these houses (for a full list and description of these variables, see the help file by running ?house_prices in the console). In this case study, we’ll create a multiple regression model where:

The outcome variable $y$ is the sale price of houses.
Two explanatory variables:
1. A numerical explanatory variable $x_1\text{:}$ house size sqft_living as measured in square feet of living space. Note that 1 square foot is about 0.09 square meters.
2. A categorical explanatory variable $x_2\text{:}$ house condition, a categorical variable with five levels where 1 indicates “poor” and 5 indicates “excellent.”
🔗

Subsection 11.2.1 Exploratory data analysis: part I

As we’ve said numerous times throughout this book, a crucial first step when presented with data is to perform an exploratory data analysis (EDA). Exploratory data analysis can give you a sense of your data, help identify issues with your data, bring to light any outliers, and help inform model construction. Recall the three common steps in an exploratory data analysis we introduced in Subsection 5.1.1:

Looking at the raw data values.
Computing summary statistics.
Creating data visualizations.

First, let’s look at the raw data using View() to bring up RStudio’s spreadsheet viewer and the glimpse() function from the dplyr package:

View(house_prices)
glimpse(house_prices)

Rows: 21,613
Columns: 21
$ id            <chr> "7129300520", "6414100192", "5631500400", "2487200875", "1954400510", "7237550310", "1321400060"…
$ date          <date> 2014-10-13, 2014-12-09, 2015-02-25, 2014-12-09, 2015-02-18, 2014-05-12, 2014-06-27, 2015-01-15,…
$ price         <dbl> 221900, 538000, 180000, 604000, 510000, 1225000, 257500, 291850, 229500, 323000, 662500, 468000,…
$ bedrooms      <int> 3, 3, 2, 4, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 5, 4, 3, 4, 2, 3, 4, 3, 5, 2, 3, 3, 3, 3, 3, 4, 3, 2, …
$ bathrooms     <dbl> 1.00, 2.25, 1.00, 3.00, 2.00, 4.50, 2.25, 1.50, 1.00, 2.50, 2.50, 1.00, 1.00, 1.75, 2.00, 3.00, …
$ sqft_living   <int> 1180, 2570, 770, 1960, 1680, 5420, 1715, 1060, 1780, 1890, 3560, 1160, 1430, 1370, 1810, 2950, 1…
$ sqft_lot      <int> 5650, 7242, 10000, 5000, 8080, 101930, 6819, 9711, 7470, 6560, 9796, 6000, 19901, 9680, 4850, 50…
$ floors        <dbl> 1.0, 2.0, 1.0, 1.0, 1.0, 1.0, 2.0, 1.0, 1.0, 2.0, 1.0, 1.0, 1.5, 1.0, 1.5, 2.0, 2.0, 1.5, 1.0, 1…
$ waterfront    <lgl> FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE…
$ view          <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ condition     <fct> 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 4, 5, 3, 5, 3, 3, 3, 3, …
$ grade         <fct> 7, 7, 6, 7, 8, 11, 7, 7, 7, 7, 8, 7, 7, 7, 7, 9, 7, 7, 7, 7, 7, 9, 8, 7, 8, 6, 8, 8, 7, 8, 8, 7,…
$ sqft_above    <int> 1180, 2170, 770, 1050, 1680, 3890, 1715, 1060, 1050, 1890, 1860, 860, 1430, 1370, 1810, 1980, 18…
$ sqft_basement <int> 0, 400, 0, 910, 0, 1530, 0, 0, 730, 0, 1700, 300, 0, 0, 0, 970, 0, 0, 0, 0, 760, 720, 0, 0, 0, 0…
$ yr_built      <int> 1955, 1951, 1933, 1965, 1987, 2001, 1995, 1963, 1960, 2003, 1965, 1942, 1927, 1977, 1900, 1979, …
$ yr_renovated  <int> 0, 1991, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ zipcode       <fct> 98178, 98125, 98028, 98136, 98074, 98053, 98003, 98198, 98146, 98038, 98007, 98115, 98028, 98074…
$ lat           <dbl> 47.5, 47.7, 47.7, 47.5, 47.6, 47.7, 47.3, 47.4, 47.5, 47.4, 47.6, 47.7, 47.8, 47.6, 47.7, 47.6, …
$ long          <dbl> -122, -122, -122, -122, -122, -122, -122, -122, -122, -122, -122, -122, -122, -122, -122, -122, …
$ sqft_living15 <int> 1340, 1690, 2720, 1360, 1800, 4760, 2238, 1650, 1780, 2390, 2210, 1330, 1780, 1370, 1360, 2140, …
$ sqft_lot15    <int> 5650, 7639, 8062, 5000, 7503, 101930, 6819, 9711, 8113, 7570, 8925, 6000, 12697, 10208, 4850, 40…

Here are some questions you can ask yourself at this stage of an EDA: Which variables are numerical? Which are categorical? For the categorical variables, what are their levels? Besides the variables we’ll be using in our regression model, what other variables do you think would be useful to use in a model for predicting house price?

Observe, for example, with the raw data that while the condition variable has values 1 through 5, these are saved in R as fct standing for “factors.” Recall this is one of R’s ways of saving categorical variables. So you should think of these as the “labels” 1 through 5 and not the numerical values 1 through 5.

Let’s now perform the second step in an EDA: computing summary statistics. Recall from Section 3.3 that summary statistics are single numerical values that summarize a large number of values. Examples of summary statistics include the mean, the median, the standard deviation, and various percentiles.

Let’s use the convenient tidy_summary() function from the moderndive package we first used in Subsection 6.1.1, being sure to only select() the variables of interest for our model:

house_prices |>
  select(price, sqft_living, condition) |>
  tidy_summary()

# A tibble: 7 × 11
  column        n group type    min     Q1    mean median     Q3     max      sd
  <chr>     <int> <chr> <chr> <dbl>  <dbl>   <dbl>  <dbl>  <dbl>   <dbl>   <dbl>
1 price     21613 <NA>  nume… 75000 321950 540088. 450000 645000 7700000 367127.
2 sqft_liv… 21613 <NA>  nume…   290   1427   2080.   1910   2550   13540    918.
3 condition    30 1     fact…    NA     NA     NA      NA     NA      NA     NA 
4 condition   172 2     fact…    NA     NA     NA      NA     NA      NA     NA 
5 condition 14031 3     fact…    NA     NA     NA      NA     NA      NA     NA 
6 condition  5679 4     fact…    NA     NA     NA      NA     NA      NA     NA 
7 condition  1701 5     fact…    NA     NA     NA      NA     NA      NA     NA

Observe that the mean price is larger than the median. This is because a small number of very expensive houses are inflating the average. In other words, there are “outlier” house prices in our dataset. (This fact will become even more apparent when we create our visualizations next.)

However, the median is not as sensitive to such outlier house prices. This is why news about the real estate market generally report median house prices and not mean/average house prices. We say here that the median is more robust to outliers than the mean. Similarly, while both the standard deviation and interquartile-range (IQR) are both measures of spread and variability, the IQR being based on quantiles as $Q3 - Q1$ is more robust to outliers.

Let’s now perform the last of the three common steps in an exploratory data analysis: creating data visualizations. Let’s first create univariate visualizations. These are plots focusing on a single variable at a time. Since price and sqft_living are numerical variables, we can visualize their distributions using a geom_histogram() as seen in Section 2.5 on histograms. On the other hand, since condition is categorical, we can visualize its distribution using a geom_bar(). Recall from Section 2.8 on barplots that since condition is not “pre-counted,” we use a geom_bar() and not a geom_col().

# Histogram of house price:
ggplot(house_prices, aes(x = price)) +
  geom_histogram(color = "white") +
  labs(x = "price (USD)", title = "House price")

# Histogram of sqft_living:
ggplot(house_prices, aes(x = sqft_living)) +
  geom_histogram(color = "white") +
  labs(x = "living space (square feet)", title = "House size")

# Barplot of condition:
ggplot(house_prices, aes(x = condition)) +
  geom_bar() +
  labs(x = "condition", title = "House condition")

In Figure 11.2.1, we display all three of these visualizations at once.

Three plots: histogram of house price (right-skewed), histogram of house size (right-skewed), and barplot of house condition (most houses are condition 3). — Figure 11.2.1. Exploratory visualizations of Seattle house prices data.

First, observe in the bottom plot that most houses are of condition 3, with a few more of conditions 4 and 5, and almost none that are 1 or 2.

Next, see in the histogram for price (the top-left plot) that a majority of houses are less than two million dollars. Observe also that the x-axis stretches out to 8 million dollars, even though there does not appear to be any houses close to that price. This is because there are a very small number of houses with prices closer to 8 million as noted in the tidy_summary(). These are the outlier house prices we mentioned earlier. We say that the variable price is right-skewed as exhibited by the long right tail.

Further, the histogram of sqft_living in the middle plot shows that most houses appear to have less than 5000 square feet of living space. For comparison, an American football field in the US is about 57,600 square feet, whereas a standard soccer/association football field is about 64,000 square feet. Observe also that this variable is also right-skewed, although not as drastically as the price variable.

For both the price and sqft_living variables, the right-skew makes distinguishing houses at the lower end of the x-axis hard. This is because the scale of the x-axis is compressed by the small number of quite expensive and immensely-sized houses.

So what can we do about this skew? Let’s apply a log10 transformation to these variables. In summary, log transformations allow us to alter the scale of a variable to focus on multiplicative changes instead of additive changes. In other words, they shift the view to be on relative changes instead of absolute changes. Such multiplicative/relative changes are also called changes in orders of magnitude.

Let’s create new log10 transformed versions of the right-skewed variable price and sqft_living using the mutate() function from Section 3.5, but we’ll give the latter the name log10_size, which is shorter and easier to understand than the name log10_sqft_living.

house_prices <- house_prices |>
  mutate(
    log10_price = log10(price),
    log10_size = log10(sqft_living)
  )

Let’s display the before and after effects of this transformation on these variables for only the first 10 rows of house_prices:

house_prices |>
  select(price, log10_price, sqft_living, log10_size)

# A tibble: 21,613 × 4
     price log10_price sqft_living log10_size
     <dbl>       <dbl>       <int>      <dbl>
 1  221900        5.35        1180       3.07
 2  538000        5.73        2570       3.41
 3  180000        5.26         770       2.89
 4  604000        5.78        1960       3.29
 5  510000        5.71        1680       3.23
 6 1225000        6.09        5420       3.73
 7  257500        5.41        1715       3.23
 8  291850        5.47        1060       3.03
 9  229500        5.36        1780       3.25
10  323000        5.51        1890       3.28
# ℹ 21,603 more rows

Observe in particular the houses in the sixth and third rows. The house in the sixth row has a price just above one million dollars. Since $10^6$ is one million, its log10_price is around 6. Contrast this with all other houses with log10_price less than six, since they all have price less than $1,000,000. The house in the third row is the only house with sqft_living less than 1000. Since $1000 = 10^3\text{,}$ it’s the lone house with log10_size less than 3.

Let’s now visualize the before and after effects of this transformation for price in Figure 11.2.2.

# Before log10 transformation:
ggplot(house_prices, aes(x = price)) +
  geom_histogram(color = "white") +
  labs(x = "price (USD)", title = "House price: Before")

# After log10 transformation:
ggplot(house_prices, aes(x = log10_price)) +
  geom_histogram(color = "white") +
  labs(x = "log10 price (USD)", title = "House price: After")

Two histograms side by side: before log10 transformation showing right-skewed house prices, and after showing a more symmetric bell-shaped distribution. — Figure 11.2.2. House price before and after log10 transformation.

Observe that after the transformation, the distribution is much less skewed, and in this case, more symmetric and more bell-shaped. Now you can more easily distinguish the lower priced houses.

Let’s do the same for house size, where the variable sqft_living was log10 transformed to log10_size.

# Before log10 transformation:
ggplot(house_prices, aes(x = sqft_living)) +
  geom_histogram(color = "white") +
  labs(x = "living space (square feet)", title = "House size: Before")

# After log10 transformation:
ggplot(house_prices, aes(x = log10_size)) +
  geom_histogram(color = "white") +
  labs(x = "log10 living space (square feet)", title = "House size: After")

Two histograms side by side: before log10 transformation showing right-skewed house sizes, and after showing a more symmetric distribution. — Figure 11.2.3. House size before and after log10 transformation.

Observe in Figure 11.2.3 that the log10 transformation has a similar effect of un-skewing the variable. We emphasize that while in these two cases the resulting distributions are more symmetric and bell-shaped, this is not always necessarily the case.

Given the now symmetric nature of log10_price and log10_size, we are going to revise our multiple regression model to use our new variables:

The outcome variable $y$ is the sale log10_price of houses.
Two explanatory variables:
1. A numerical explanatory variable $x_1\text{:}$ house size log10_size as measured in log base 10 square feet of living space.
2. A categorical explanatory variable $x_2\text{:}$ house condition, a categorical variable with five levels where 1 indicates “poor” and 5 indicates “excellent.”
🔗

Subsection 11.2.2 Exploratory data analysis: part II

Let’s now continue our EDA by creating multivariate visualizations. Unlike the univariate histograms and barplot in the earlier Figure 11.2.1, Figure 11.2.2, and Figure 11.2.3, multivariate visualizations show relationships between more than one variable. This is an important step of an EDA to perform since the goal of modeling is to explore relationships between variables.

Since our model involves a numerical outcome variable, a numerical explanatory variable, and a categorical explanatory variable, we are in a similar regression modeling situation as in Section 6.1 where we studied the UN member states dataset. Recall in that case the numerical outcome variable was fertility rate, the numerical explanatory variable was life expectancy, and the categorical explanatory variable was income group.

We thus have two choices of models we can fit: either (1) an interaction model where the regression line for each condition level will have both a different slope and a different intercept or (2) a parallel slopes model where the regression line for each condition level will have the same slope but different intercepts.

Recall from Subsection 6.1.3 that the geom_parallel_slopes() function is a special purpose function that Evgeni Chasnovski created and included in the moderndive package, since the geom_smooth() method in the ggplot2 package does not have a convenient way to plot parallel slopes models. We plot both resulting models in Figure 11.2.4, with the interaction model on the left.

# Plot interaction model
ggplot(house_prices,
       aes(x = log10_size, y = log10_price, col = condition)) +
  geom_point(alpha = 0.05) +
  geom_smooth(method = "lm", se = FALSE) +
  labs(y = "log10 price",
       x = "log10 size",
       title = "House prices in Seattle")

# Plot parallel slopes model
ggplot(house_prices,
       aes(x = log10_size, y = log10_price, col = condition)) +
  geom_point(alpha = 0.05) +
  geom_parallel_slopes(se = FALSE) +
  labs(y = "log10 price",
       x = "log10 size",
       title = "House prices in Seattle")

Two side-by-side scatterplots of log10 price vs log10 size colored by condition: the left plot shows the interaction model with five different-slope regression lines, and the right shows the parallel slopes model. — Figure 11.2.4. Interaction and parallel slopes models.

In both cases, we see there is a positive relationship between house price and size, meaning as houses are larger in size, they tend to be more expensive. Furthermore, in both plots it seems that houses of condition 5 tend to be the most expensive for most house sizes as evidenced by the fact that the line for condition 5 is highest, followed by conditions 4 and 3. As for conditions 1 and 2, this pattern isn’t as clear. Recall from the univariate barplot of condition in Figure 11.2.1, there are only a few houses of condition 1 or 2.

Let’s also show a faceted version of just the interaction model in Figure 11.2.5. It is now much more apparent just how few houses are of condition 1 or 2.

ggplot(house_prices,
       aes(x = log10_size, y = log10_price, col = condition)) +
  geom_point(alpha = 0.4) +
  geom_smooth(method = "lm", se = FALSE) +
  labs(y = "log10 price",
       x = "log10 size",
       title = "House prices in Seattle") +
  facet_wrap(~ condition)

This can be further checked using dplyr and its count() function:

house_prices |>
  count(condition)

# A tibble: 5 × 2
  condition     n
  <fct>     <int>
1 1            30
2 2           172
3 3         14031
4 4          5679
5 5          1701

Scatterplot of log10 price vs log10 size faceted by condition (1 through 5), showing very few houses of conditions 1 and 2. — Figure 11.2.5. Faceted plot of interaction model.

Which exploratory visualization of the interaction model is better, the one in the left-hand plot of Figure 11.2.4 or the faceted version in Figure 11.2.5? There is no universal right answer. You need to make a choice depending on what you want to convey, and own that choice, with including and discussing both also as an option as needed.

Subsection 11.2.3 Regression modeling

Which of the two models in Figure 11.2.4 is “better”? The interaction model in the left-hand plot or the parallel slopes model in the right-hand plot?

With model selection, we should only favor more complex models if the additional complexity is warranted. In this case, the more complex model is the interaction model since it considers five intercepts and five slopes total. This is in contrast to the parallel slopes model which considers five intercepts but only one common slope.

Is the additional complexity of the interaction model warranted? Looking at the left-hand plot in Figure 11.2.4, we’re of the opinion that it is, as evidenced by the slight x-like (crossing) pattern to some of the lines. Therefore, we’ll focus the rest of this analysis only on the interaction model. (This visual approach is somewhat subjective, however, so feel free to disagree!) What are the five different slopes and five different intercepts for the interaction model? We can get these values from the regression table. Recall our two-step process for getting the regression table:

price_interaction <- lm(log10_price ~ log10_size * condition, data = house_prices)
get_regression_table(price_interaction)

Table 11.2.6. Regression table for interaction model

term	estimate	std_error	statistic	p_value	lower_ci	upper_ci
intercept	3.330	0.451	7.380	0.000	2.446	4.215
log10_size	0.690	0.148	4.652	0.000	0.399	0.980
condition: 2	0.047	0.498	0.094	0.925	-0.930	1.024
condition: 3	-0.367	0.452	-0.812	0.417	-1.253	0.519
condition: 4	-0.398	0.453	-0.879	0.380	-1.286	0.490
condition: 5	-0.883	0.457	-1.931	0.053	-1.779	0.013
log10_size:condition2	-0.024	0.163	-0.148	0.882	-0.344	0.295
log10_size:condition3	0.133	0.148	0.893	0.372	-0.158	0.424
log10_size:condition4	0.146	0.149	0.979	0.328	-0.146	0.437
log10_size:condition5	0.310	0.150	2.067	0.039	0.016	0.604

Recall we saw in Subsection 6.1.2 how to interpret a regression table when there are both numerical and categorical explanatory variables. Let’s now do the same for all 10 values in the estimate column of Table 11.2.6.

In this case, the “baseline for comparison” group for the categorical variable condition are the condition 1 houses, since “1” comes first alphanumerically. Thus, the intercept and log10_size values are the intercept and slope for log10_size for this baseline group. Next, the condition2 through condition5 terms are the offsets in intercepts relative to the condition 1 intercept. Finally, the log10_size:condition2 through log10_size:condition5 are the offsets in slopes for log10_size relative to the condition 1 slope for log10_size.

Let’s simplify this by writing out the equation of each of the five regression lines using these 10 estimate values. We’ll write out each line in the following format:

\begin{equation*} \widehat{\log_{10}(\text{price})} = \hat{\beta}_0 + \hat{\beta}_{\text{size}} \cdot \log_{10}(\text{size}) \end{equation*}

Condition 1 (baseline):

\begin{equation*} \widehat{\log_{10}(\text{price})} = 3.33 + 0.69 \cdot \log_{10}(\text{size}) \end{equation*}
Condition 2:

\begin{equation*} \widehat{\log_{10}(\text{price})} = (3.33 + 0.047) + (0.69 - 0.024) \cdot \log_{10}(\text{size}) = 3.377 + 0.666 \cdot \log_{10}(\text{size}) \end{equation*}
Condition 3:

\begin{equation*} \widehat{\log_{10}(\text{price})} = (3.33 - 0.367) + (0.69+0.133)\cdot \log_{10}(\text{size}) = 2.963 + 0.823\cdot \log_{10}(\text{size}) \end{equation*}
Condition 4:

\begin{equation*} \widehat{\log_{10}(\text{price})} = (3.33 - 0.398) + (0.69+0.146)\cdot \log_{10}(\text{size}) = 2.932 + 0.836\cdot \log_{10}(\text{size}) \end{equation*}
Condition 5:

\begin{equation*} \widehat{\log_{10}(\text{price})} = (3.33 - 0.883) + (0.69+0.31)\cdot \log_{10}(\text{size}) = 2.447 + 1 \cdot \log_{10}(\text{size}) \end{equation*}

These correspond to the regression lines in the left-hand plot of Figure 11.2.4 and the faceted plot in Figure 11.2.5. For homes of all five condition types, as the size of the house increases, the price increases. This is what most would expect. However, the rate of increase of price with size is fastest for the homes with conditions 3, 4, and 5 of 0.823, 0.836, and 1, respectively. These are the three largest slopes out of the five.

Subsection 11.2.4 Making predictions

Say you’re a realtor and someone calls you asking you how much their home will sell for. They tell you that it’s in condition = 5 and is sized 1900 square feet. What do you tell them? Let’s use the interaction model we fit to make predictions!

We first make this prediction visually in Figure 11.2.7. The predicted log10_price of this house is marked with a black dot. This is where the following two lines intersect:

The regression line for the condition = 5 homes, and
The vertical dashed black line at log10_size equals 3.28, since our predictor variable is the log10 transformed square feet of living space of $\log_{10}(1900) \approx 3.28\text{.}$

Scatterplot of log10 price vs log10 size colored by condition, with five regression lines and a vertical dashed line at log10(1900) with a black dot indicating the predicted log10 price for a condition-5 house of 1900 square feet. — Figure 11.2.7. Interaction model with prediction.

Eyeballing it, it seems the predicted log10_price seems to be around 5.75. Let’s now find the exact numerical value for the prediction using the equation of the regression line for the condition = 5 houses, being sure to log10() the square footage first.

2.45 + 1 * log10(1900)

[1] 5.728754

This value is very close to our earlier visually made prediction of 5.75. But wait! Is our prediction for the price of this house $5.75? No, because we are using log10_price as our outcome variable! If we want a prediction in dollar units of price, we need to un-log this by taking a power of 10. This described in Section A.1 online.

10^(2.45 + 1 * log10(1900))

[1] 535492.8

Our predicted price for this home of condition 5 and of size 1900 square feet is approximately $536,000.

Subsection 11.2.5 Inference for multiple linear regression

Let’s next check the results of our multiple linear regression on house prices using both theory-based and simulation-based methods with hypothesis testing.

Subsubsection 11.2.5.1 Theory-based hypothesis testing for partial slopes

Recall the results of our theory-based inference that were shown when we interpreted the estimate column of get_regression_table(price_interaction). We can use these values to perform hypothesis tests on the partial slopes.

price_interaction <- lm(log10_price ~ log10_size * condition, data = house_prices)
get_regression_table(price_interaction)

Table 11.2.8. Regression table for interaction model (shown again)

term	estimate	std_error	statistic	p_value	lower_ci	upper_ci
intercept	3.330	0.451	7.380	0.000	2.446	4.215
log10_size	0.690	0.148	4.652	0.000	0.399	0.980
condition: 2	0.047	0.498	0.094	0.925	-0.930	1.024
condition: 3	-0.367	0.452	-0.812	0.417	-1.253	0.519
condition: 4	-0.398	0.453	-0.879	0.380	-1.286	0.490
condition: 5	-0.883	0.457	-1.931	0.053	-1.779	0.013
log10_size:condition2	-0.024	0.163	-0.148	0.882	-0.344	0.295
log10_size:condition3	0.133	0.148	0.893	0.372	-0.158	0.424
log10_size:condition4	0.146	0.149	0.979	0.328	-0.146	0.437
log10_size:condition5	0.310	0.150	2.067	0.039	0.016	0.604

For this model, we can perform hypothesis tests on the partial slopes with an $\alpha$ significance level set to 0.05. For example, we can test the null hypothesis that the partial slope for log10_size is zero. Looking at the $p$-value in the row corresponding to log10_size we see a value of 0 and a large statistic of 4.652. With $\alpha = 0.05\text{,}$ log10_size is the only regressor with a statistically significant relationship with log10_price in this theory-based model.

Subsubsection 11.2.5.2 Simulation-based hypothesis testing for partial slopes

We can also perform hypothesis tests on the partial slopes using simulation-based methods. We can use the fit() function and some of the other infer verbs to do so to check the results of our theory-based inference.

Let’s begin by retrieving the observed fit values from our price_interaction model:

observed_fit_coefficients <- house_prices |>
  specify(
    log10_price ~ log10_size * condition
    ) |>
  fit()
observed_fit_coefficients

# A tibble: 10 × 2
   term                    estimate
   <chr>                      <dbl>
 1 intercept              3.33046  
 2 log10_size             0.689534 
 3 condition2             0.0469644
 4 condition3            -0.367010 
 5 condition4            -0.398174 
 6 condition5            -0.883036 
 7 log10_size:condition2 -0.0241635
 8 log10_size:condition3  0.132593 
 9 log10_size:condition4  0.145632 
10 log10_size:condition5  0.309866

Next, we build a null distribution of the partial slopes using the generate() function and the fit() function, setting hypothesize() to null = "independence" which will shuffle the values of the response variable log10_price.

null_distribution_housing <- house_prices |>
  specify(
    log10_price ~ log10_size * condition
    ) |>
  hypothesize(null = "independence") |>
  generate(reps = 1000, type = "permute") |>
  fit()

We then visualize this null distribution and shade the observed fit values to see if they are statistically significant.

visualize(null_distribution_housing) +
  shade_p_value(obs_stat = observed_fit_coefficients,
                direction = "two-sided")

A grid of histograms, one per term in the regression model, each showing the null distribution of the partial slope with the observed value shaded to indicate statistical significance. — Figure 11.2.9. Null distributions for all partial slopes with observed fit values shaded.

Of the regressors, it appears that only log10_size has a statistically significant relationship with log10_price. This is evidenced by the fact that the observed fit value for log10_size is far outside the range of the null distribution.

Lastly, we can calculate the $p$-value for the partial slopes using the get_p_value() function.

null_distribution_housing |>
  get_p_value(obs_stat = observed_fit_coefficients, direction = "two-sided")

# A tibble: 10 × 2
   term                  p_value
   <chr>                    <dbl>
 1 condition2              0.948
 2 condition3              0.538
 3 condition4              0.508
 4 condition5              0.178
 5 intercept               0    
 6 log10_size              0.002
 7 log10_size:condition2   0.916
 8 log10_size:condition3   0.486
 9 log10_size:condition4   0.454
10 log10_size:condition5   0.148

The $p$-value matches up with this conclusion. Only log10_size appears to be a significant regressor for predicting log10_price in this model. This matches with the findings we had from our theory-based analysis for this same model. Remember that this won’t always be the case depending on the distributions of the variables and whether assumptions hold.

Exercises 11.2.5.3 Exercises

1.

Check that the LINE conditions are met for inference to be made in this Seattle house prices example with price_interaction <- lm(log10_price ~ log10_size * condition, data = house_prices).

Answer.

To check the LINE conditions, examine residual plots from get_regression_points(price_interaction). Look for: (L) a roughly linear pattern in the scatterplot; (I) independence of observations (generally assumed given random sampling); (N) near-normality in a histogram or QQ plot of residuals; and (E) roughly equal spread of residuals across all fitted values (homoscedasticity).

2.

Repeat the regression modeling in Subsection 11.2.3 and the prediction making in Subsection 11.2.4 on the house of condition 5 and size 1900 square feet, but using the parallel slopes model you visualized in Figure 11.2.4.

Answer.

Fit the parallel slopes model with price_parallel <- lm(log10_price ~ log10_size + condition, data = house_prices). Then use get_regression_table(price_parallel) and apply the condition 5 regression line equation to $\log_{10}(1900)$ to find the predicted log10_price, and convert with $10^{\hat{y}}$ to get the price in dollars.

3.

Interpret the results of the other rows in terms of inference in the get_regression_table(price_interaction) output that we did not interpret in Subsection 11.2.5.

Answer.

For each row other than log10_size, examine the $p$-value. A $p$-value greater than 0.05 suggests that the term is not statistically significantly different from zero at the $\alpha = 0.05$ level, meaning we cannot conclude that particular offset in intercept or slope differs from the baseline (condition 1).

4.

Create, visualize, and interpret confidence intervals using both theory-based and simulation-based approaches to mirror the hypothesis testing done in Subsection 11.2.5.

Answer.

For theory-based confidence intervals, use get_regression_table() which provides lower_ci and upper_ci columns. For simulation-based confidence intervals, use the infer workflow with specify(), generate(reps = 1000, type = "bootstrap"), and fit(), then apply get_confidence_interval() to the resulting bootstrap distribution.

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