Communicating the Updated Analysis

Biagio Palese

The last step

We are at the end of our journey.. we finally reached the top

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all of you feel better about R

We come a long way

Artwork by @allison_horst

Class Objective

Show the additional analysis performed.

Communicate the results of the updated data analysis.

Hotel Analysis Scenario

OptimaHotel Insights has been tasked by multiple hotel chains to develop a predictive model that can accurately forecast Average Daily Rates (ADR) based on various hotel features.

Updated ADR Analysis Deliverables

Update the presentation to address the OptimaHotel Insights representatives feedback.

Show the outcomes of the additional requested analyses from the OptimaHotel Insights representives.

Step 1: Exploring the Dataset

The hotel_data dataset used in this case study contains comprehensive information on various attributes and performance metrics of 3000 hotels across major U.S. cities.

It is meticulously curated¹ to support predictive analyses focused on predicting and optimizing the average daily rate (ADR) for these properties.

This dataset includes realistic and detailed data, encompassing a wide range of features crucial for developing predictive pricing models.

Dataset Dimensions
rows_number	columns_number
3000	22

Dataset Glimpse
Column Name	Data Type	Example Values
id	numeric	1, 2, 3
num_rooms	numeric	33, 71, 150
location_quality	character	Excellent, Excellent, Good
star_rating	numeric	1, 5, 3
review_score	numeric	1.84, 1.88, 2.93
occupancy_rate	numeric	71.65, 52.29, 97.43
amenities_score	numeric	3, 5, 1
business_facilities	numeric	0, 1, 1
leisure_facilities	numeric	1, 1, 1
distance_to_city_center	numeric	7.92, 4.85, 8.58
average_daily_rate	numeric	98.22, 98.73, 106.56
city	character	Chicago, New York, Los Angeles
hotel_chain	character	Hilton, Hyatt, Marriott
month	numeric	1, 5, 12
season	character	Winter, Spring, Winter
number_of_bookings	numeric	53, 61, 58
revPAR	numeric	70.37, 51.62, 103.82
average_length_of_stay	numeric	4.45, 3.09, 3.6
competitors_average_price	numeric	88.56, 84.46, 131.51
transport_accessibility_score	numeric	3.67, 6.3, 6.3
customer_satisfaction_index	numeric	81.57, 94.62, 91.12
cancellation_rate	numeric	0.01, 0.1, 0.14

ADR Descriptive Stats
min_average_daily_rate	avg_average_daily_rate	median_average_daily_rate	max_average_daily_rate	sd_average_daily_rate
49.87	95.87045	95.825	141.83	14.98696

ADR Descriptive Stats by City
city	min_average_daily_rate	avg_average_daily_rate	median_average_daily_rate	max_average_daily_rate	sd_average_daily_rate
Chicago	53.45	96.02526	96.045	141.83	15.48101
Los Angeles	49.87	95.56752	95.160	135.78	14.67615
Miami	54.23	96.19310	96.105	139.10	14.19395
New York	57.49	95.71646	95.930	140.88	15.10470
San Francisco	51.98	95.88067	96.085	140.77	15.45662

ADR Descriptive Stats by Hotel Chain
hotel_chain	min_average_daily_rate	avg_average_daily_rate	median_average_daily_rate	max_average_daily_rate	sd_average_daily_rate
Hilton	55.12	95.92	96.00	140.77	15.51
Hyatt	54.78	95.28	95.08	141.83	15.24
InterContinental	51.94	96.31	96.67	139.10	14.70
Marriott	57.47	95.87	96.32	140.88	14.52
Sheraton	49.87	96.01	95.93	139.65	14.98

Step 2: Manipulating the Data

Based on the above exploration the following minor manipulations¹ are required:

Transformed month from numeric to factor

hotel_clean <- hotel_data |> 
   filter(average_daily_rate    >60 &average_daily_rate <130) |> #removes outliers and gives normal distribution
   filter(revPAR    >5 &revPAR  <118) |> 
   mutate(miles_distance_to_city_center= distance_to_city_center/1.6,
          month= as_factor(month)) |> 
   select(-distance_to_city_center)

Step 3: Checking Correlations

Given that we are working on a new dataset and that we are not super familiar with hotel data, it is worth to take a look at the correlation of our numerical variables.

Correlation Matrix

Correlation Matrix Arranged By ADR
variables	number_of_bookings	num_rooms	star_rating	review_score	occupancy_rate	amenities_score	business_facilities	leisure_facilities	average_daily_rate	revPAR	average_length_of_stay	competitors_average_price	transport_accessibility_score	customer_satisfaction_index	cancellation_rate	miles_distance_to_city_center
average_daily_rate	0.0274	0.0043	0.1443	0.0301	0.2323	0.0145	0.2832	0.4994	1.0000	0.5704	-0.0028	0.5749	-0.0041	-0.0271	0.0105	0.0031
competitors_average_price	0.0047	-0.0096	0.0878	0.0044	0.1385	0.0080	0.1251	0.2872	0.5749	0.3305	0.0074	1.0000	-0.0087	-0.0033	0.0163	-0.0121
revPAR	-0.0080	0.0071	0.0790	0.0221	0.9228	0.0000	0.1011	0.1582	0.5704	1.0000	-0.0181	0.3305	-0.0175	-0.0023	0.0076	0.0003
leisure_facilities	0.0336	-0.0099	0.0177	0.0034	-0.0350	0.0122	-0.0179	1.0000	0.4994	0.1582	0.0042	0.2872	-0.0027	-0.0247	0.0020	-0.0020
business_facilities	-0.0125	-0.0061	0.0068	-0.0195	-0.0080	-0.0383	1.0000	-0.0179	0.2832	0.1011	-0.0120	0.1251	-0.0008	0.0062	-0.0243	0.0100
occupancy_rate	-0.0226	0.0065	0.0254	0.0148	1.0000	-0.0052	-0.0080	-0.0350	0.2323	0.9228	-0.0202	0.1385	-0.0220	0.0076	0.0013	-0.0005
star_rating	0.0138	0.0044	1.0000	-0.0049	0.0254	-0.0035	0.0068	0.0177	0.1443	0.0790	-0.0243	0.0878	0.0094	-0.0128	0.0085	0.0235
review_score	0.0129	0.0123	-0.0049	1.0000	0.0148	0.0063	-0.0195	0.0034	0.0301	0.0221	-0.0317	0.0044	-0.0135	0.0010	-0.0464	0.0005
number_of_bookings	1.0000	-0.0037	0.0138	0.0129	-0.0226	0.0175	-0.0125	0.0336	0.0274	-0.0080	0.0183	0.0047	-0.0121	0.0009	-0.0113	0.0100
amenities_score	0.0175	-0.0318	-0.0035	0.0063	-0.0052	1.0000	-0.0383	0.0122	0.0145	0.0000	-0.0304	0.0080	-0.0131	0.0304	-0.0188	0.0069
cancellation_rate	-0.0113	-0.0024	0.0085	-0.0464	0.0013	-0.0188	-0.0243	0.0020	0.0105	0.0076	0.0056	0.0163	-0.0037	0.0051	1.0000	0.0042
num_rooms	-0.0037	1.0000	0.0044	0.0123	0.0065	-0.0318	-0.0061	-0.0099	0.0043	0.0071	-0.0222	-0.0096	0.0135	0.0309	-0.0024	0.0068
miles_distance_to_city_center	0.0100	0.0068	0.0235	0.0005	-0.0005	0.0069	0.0100	-0.0020	0.0031	0.0003	-0.0263	-0.0121	0.0089	-0.0117	0.0042	1.0000
average_length_of_stay	0.0183	-0.0222	-0.0243	-0.0317	-0.0202	-0.0304	-0.0120	0.0042	-0.0028	-0.0181	1.0000	0.0074	0.0238	-0.0063	0.0056	-0.0263
transport_accessibility_score	-0.0121	0.0135	0.0094	-0.0135	-0.0220	-0.0131	-0.0008	-0.0027	-0.0041	-0.0175	0.0238	-0.0087	1.0000	-0.0210	-0.0037	0.0089
customer_satisfaction_index	0.0009	0.0309	-0.0128	0.0010	0.0076	0.0304	0.0062	-0.0247	-0.0271	-0.0023	-0.0063	-0.0033	-0.0210	1.0000	0.0051	-0.0117

Step 4: Data Splitting

When it comes to supervised data modeling one of the most adopted method to evaluate multiple models is to split the data into a training set and a test set from the beginning.

# Split the data
set.seed(123)# setting a seed will ensure reproducibility
hotel_split <- initial_split(hotel_clean, prop = 0.75) # define the split and its data proportion
hotel_train <- training(hotel_split) #create a train set
hotel_test <- testing(hotel_split)#create a test set

Hotel train

Training Dataset Dimensions
rows_number	columns_number
2192	22

Hotel test

Test Dataset Dimensions
rows_number	columns_number
731	22

Step 5: Setting up the Recipes

Creating recipes is a critical step because it allows to define the model formula and specify any preprocessing steps to the original dataset.

Recipe 1

recipe1 <- recipe(average_daily_rate ~ revPAR + competitors_average_price, data = hotel_train)

Recipe 2

recipe2 <- recipe(average_daily_rate ~ revPAR + competitors_average_price+ location_quality+ star_rating + season, data = hotel_train)|> 
  step_dummy(all_nominal_predictors())

Recipe 3

recipe3 <- recipe(average_daily_rate ~ ., data = hotel_train)|> 
  step_dummy(all_nominal_predictors())

NEW: Recipe 4

recipe4 <- recipe(average_daily_rate ~ ., data = hotel_train)|> 
  step_dummy(all_nominal_predictors()) |> 
  step_interact(terms = ~cancellation_rate:number_of_bookings) |> 
  step_interact(terms = ~occupancy_rate:number_of_bookings) |> 
  step_interact(terms= ~transport_accessibility_score:miles_distance_to_city_center)

Step 6: Specifying the Models

We specify five models using parsnip:

a lm engine linear regression;
a glmnet engine lasso regression¹;
a glment engine ridge regression (NEW);
a rpart engine decision tree (NEW);
a ranger engine random forest(NEW)².

Model 1

# Linear regression with lm
linear_mod_reg <- linear_reg() |>
  set_engine("lm") |>
  set_mode("regression")

Model 2

# Lasso regularized linear regression with glmnet
linear_mod_lasso <- linear_reg(penalty = 0.1, mixture = 1) |>
  set_engine("glmnet") |>
  set_mode("regression")

NEW: Model 3

# Ridge regularized linear regression with glmnet
linear_mod_ridge <- linear_reg(penalty = 0.1, mixture = 0) |>
  set_engine("glmnet") |>
  set_mode("regression")

NEW: Model 4

# Regression Decision Tree
dt <- decision_tree()|>
  set_engine("rpart") |>
  set_mode("regression")

NEW: Model 5

# Regression Random Forest
rf <- rand_forest()|>
  set_engine("ranger") |>
  set_mode("regression")

Step 7: Fitting Models using Workflows

Next, we fit the models to the hotel_train dataset, so that we can compare their predictions performance on hotel_test.

This is accomplished by embedding the model specifications within workflows that also incorporate our preprocessing recipes.

I will only use recipe 2, 3 & 4 for my workflow based on the results of the first presentation.

Step 8: Making Predictions

Then we use the models built on the training set to make predictions on our test set.

By doing so we will have both the actual values and the predicted values and we can assess the model performance.

Lasso recipe 4 model actual ADR values vs predicted

ADR actual vs predicted values
adr	pred_adr
84.520000	85.882901
79.160000	80.365714
107.740000	109.831862
104.800000	100.121869
97.880000	100.510787
126.980000	125.906419
76.970000	77.913506
101.360000	98.790749
93.440000	91.641639
119.410000	120.602377

Step 9: Assess Models Performance

While seeing the predictions next to the actual values can already provide some insights on the goodness of the model.

In regression analysis, model performance is evaluated using specific metrics that quantify the model’s accuracy and ability to generalize.

Three fundamental metrics are Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and R-squared (R²).

Full Metrics Table
.metric	.estimate	model
mae	2.9542	reg_model_recipe3
mae	2.9590	reg_model_recipe4
mae	2.9813	lasso_model_recipe4
mae	2.9816	lasso_model_recipe3
mae	4.6031	ridge_model_recipe4
mae	4.7870	ridge_model_recipe3
mae	6.3418	rf_model_recipe4
mae	6.4312	rf_model_recipe3
mae	7.8728	dt_model_recipe3
mae	7.8728	dt_model_recipe4
mae	7.9723	lasso_model_recipe2
mae	7.9748	reg_model_recipe2
mae	7.9845	ridge_model_recipe2
mae	8.1338	rf_model_recipe2
mae	8.9367	dt_model_recipe2
rmse	4.1066	reg_model_recipe3
rmse	4.1073	reg_model_recipe4
rmse	4.1121	lasso_model_recipe4
rmse	4.1123	lasso_model_recipe3
rmse	5.8296	ridge_model_recipe4
rmse	6.0467	ridge_model_recipe3
rmse	7.9047	rf_model_recipe4
rmse	7.9884	rf_model_recipe3
rmse	9.7966	reg_model_recipe2
rmse	9.8012	lasso_model_recipe2
rmse	9.8104	ridge_model_recipe2
rmse	9.8150	dt_model_recipe3
rmse	9.8150	dt_model_recipe4
rmse	10.0920	rf_model_recipe2
rmse	11.0944	dt_model_recipe2
rsq	0.3721	dt_model_recipe2
rsq	0.4772	rf_model_recipe2
rsq	0.5067	reg_model_recipe2
rsq	0.5068	dt_model_recipe3
rsq	0.5068	dt_model_recipe4
rsq	0.5068	lasso_model_recipe2
rsq	0.5071	ridge_model_recipe2
rsq	0.7105	rf_model_recipe3
rsq	0.7276	rf_model_recipe4
rsq	0.8234	ridge_model_recipe3
rsq	0.8363	ridge_model_recipe4
rsq	0.9133	reg_model_recipe4
rsq	0.9133	reg_model_recipe3
rsq	0.9145	lasso_model_recipe3
rsq	0.9145	lasso_model_recipe4

By looking at the metrics table recipe 3 lasso is still the best model (recipe4 is only slightly better but more complex).
We excluded the recipe 3 and 4 regression model because of a warning highlighting issues during its fit.
Adding complexity (interactions terms & more advanced models) doesn’t always pay off.

Step 10: Visualizing Prediction Results

Visualizing the prediction results can provide additional insights into model performance.

You should plot the predicted vs actual values and the residuals to visually assess how well the model is capturing the underlying relationship in the data.

Adding interaction terms (recipe4 lasso) didn’t result in the expected improvements in the residuals.

While the random forest unperformed in terms of prediction accuracy, our intuition to use it to address the slight non-linearity of our residuals was correct.

After adding interactions terms, we could go back and try some variable transformations on the most relevant variables. However, we will stop here for this demo.

Visual interpretations provide valuable insights into model performance but should be complemented with quantitative metrics (like RMSE, MAE and R²) for a comprehensive evaluation.

The plots help identifying patterns or biases not immediately apparent from numerical metrics alone.

Step 11: Model Interpretation

Given that I have already shown how to interpret recipe3 lasso regression, I want to use this updated presentation to interpret the linear regression and decision tree models (ridge is identical to lasso and random forest was beyond the scope of the class).

Zero Coefficients Variables do not have a statistically significant linear relationship with ADR after accounting for other variables and the regularization penalty.

Removed Predictors
term	estimate	penalty
num_rooms	0.000000	0.100000
number_of_bookings	0.000000	0.100000
average_length_of_stay	0.000000	0.100000
transport_accessibility_score	0.000000	0.100000
miles_distance_to_city_center	0.000000	0.100000
city_Los.Angeles	0.000000	0.100000
city_Miami	0.000000	0.100000
city_New.York	0.000000	0.100000
city_San.Francisco	0.000000	0.100000
hotel_chain_Hyatt	0.000000	0.100000
hotel_chain_Marriott	0.000000	0.100000
hotel_chain_Sheraton	0.000000	0.100000
month_X2	0.000000	0.100000
month_X3	0.000000	0.100000
month_X4	0.000000	0.100000
month_X5	0.000000	0.100000
month_X7	0.000000	0.100000
month_X12	0.000000	0.100000
season_Spring	0.000000	0.100000
season_Summer	0.000000	0.100000
season_Winter	0.000000	0.100000

Non-Zero Coefficients Variables are influencing factors for ADR and should be considered in pricing strategies.

Included Predictors
term	estimate	penalty
(Intercept)	83.397249	0.100000
star_rating	0.172368	0.100000
review_score	0.106223	0.100000
occupancy_rate	−1.143629	0.100000
amenities_score	0.015145	0.100000
business_facilities	1.712883	0.100000
leisure_facilities	2.758931	0.100000
revPAR	1.243878	0.100000
competitors_average_price	0.052473	0.100000
customer_satisfaction_index	−0.004735	0.100000
cancellation_rate	−0.576860	0.100000
location_quality_Excellent	1.881180	0.100000
location_quality_Good	0.948190	0.100000
location_quality_Poor	−0.762115	0.100000
hotel_chain_InterContinental	0.000765	0.100000
month_X6	0.026898	0.100000
month_X8	−0.005338	0.100000
month_X9	−0.010730	0.100000
month_X10	−0.220501	0.100000
month_X11	0.056927	0.100000

Intercept: $83.4 represent the expected ADR when the categorical predictors are in their baseline category (the first value in alphabetical order) and the numerical predictors are zero or at their lowest practical/observed levels.

Intercept Only Table
term	estimate
(Intercept)	83.397249

Star Rating: For one additional star, ADR increases by approximately $0.171, assuming all other variables remain constant.

This means that moving from 1-star to a 5-star hotel, which is an increase of 4 stars, would lead to an estimated increase of 0.171 x 4 = $0.684 in ADR¹.

Star Rating Only Table
term	estimate
star_rating	0.172368

Review Score: For one point increase in the review score (on a scale of 1 to 5), ADR increases by $0.106 given all other variables staying constant.

Review Score Only Table
term	estimate
review_score	0.106223

This shows the value customers place on higher-rated hotels, which can command higher prices but the impact is surprisingly small.

Occupancy Rate: For one percentage point increase in occupancy rate, ADR decreases by $1.14. Surprisingly, higher occupancy rates do not correspond with higher ADR.

Occupancy Rate Only Table
term	estimate
occupancy_rate	−1.143629

It’s possible that in an effort to maintain high occupancy levels, hotels might be underpricing their rooms, thus not maximizing potential revenue.

Business Facilities & Leisure Facilities: Hotels with business and leisure facilities can charge an additional $1.71 and $2.76, respectively, on ADR.

Business Facilities & Leisure Facilities Only Table
term	estimate
business_facilities	1.712883
leisure_facilities	2.758931

However, these amenities impact does not seem substantial at first glance, particularly when considering the potential costs associated with maintaining them (e.g., might test their interaction).

RevPAR (Revenue per Available Room): This variable is an important hospitality metrics. RevPAR combines the effects of room rates and occupancy levels. For one dollar increase in RevPAR, ADR increases by $1.24.

RevPAR Only Table
term	estimate
revPAR	1.243878

Competitors Average Price: For one dollar increase in competitors' average prices, ADR increases by only $0.0525.

Competitors Average Price Only Table
term	estimate
competitors_average_price	0.052473

The strategy of not matching the price increases fully could reflect a positioning approach where hotels aim to offer slightly lower prices to attract price-sensitive customers or to increase market share by becoming a more economical option within their competitive set.

Customer Satisfaction Index: For every one-unit increase in the customer satisfaction index, ADR decreases by approximately $0.005, assuming other factors remain constant.

Customer Satisfaction Index Only Table
term	estimate
customer_satisfaction_index	−0.004735

The negative coefficient suggests a potential under utilization of customer satisfaction. Thus, increases in customer satisfaction do not translate into higher ADR.

Cancellation Rate: For each percentage point increase in cancellation rates, ADR decreases by $0.570.

Cancellation Rate Only Table
term	estimate
cancellation_rate	−0.576860

While this drop in price can mitigate the risk of empty rooms, it also suggests potential revenue losses or instability in booking patterns.

Location Quality: Being in an Excellent location increases the ADR by $1.89, while being in a Good location increases it by $0.952, compared to our baseline, Average location.

Location Quality Only Table
term	estimate
location_quality_Excellent	1.881180
location_quality_Good	0.948190
location_quality_Poor	−0.762115

Being in a Poor location decreases it by $0.755 compared to Average. Hotels are clearly underselling their premium locations.

Linear Regression Model Interpretation

Before interpreting the individual variables, let’s break down the meaning of each column available in a linear regression results table:

Estimate:

Represents the average change in the dependent variable (ADR) for a one-unit increase in the independent variable, holding all other variables constant.
The value indicates the magnitude of the impact.
The sign indicates the direction of the impact.

Standard Error (std.error):

Reflects the variability or uncertainty of the estimated coefficient.
Smaller values indicate more precise estimates.

Statistic:

Also known as the t-value, this is the ratio of the estimate to its standard error.
Larger absolute values indicate that the variable is likely to be statistically significant.

p-value:
Indicates whether the variable is statistically significant.
Typically, a p-value below 0.05 means the variable has a significant relationship with the dependent variable.

Intercept:

When the categorical predictors are in their baseline category (the first value in alphabetical order) and the numerical are at zero or at their lowest practical/observed levels (e.g., revPAR = 0, competitors’ price = 0, baseline location quality = average, etc.), the predicted ADR is $53.44.

Intercept Only Table
term	estimate	std.error	statistic	p.value
(Intercept)	53.4413	1.0742	49.7477	0.0000

revPAR:

For every 1 dollar increase in revPAR, the ADR increases by $0.26, assuming all other variables remain constant.
Significance: With a p-value of < 0.05, this relationship is statistically significant.

RevPAR Only Table
term	estimate	std.error	statistic	p.value
revPAR	0.2580	0.0099	26.1377	0.0000

Competitors’ Average Price:

For every 1 dollar increase in competitors’ average price, ADR increases by $0.23, holding all else constant.
Significance: This is a strong, statistically significant predictor of ADR.

Competitors Average Price Only Table
term	estimate	std.error	statistic	p.value
competitors_average_price	0.2328	0.0088	26.3102	0.0000

Star Rating:

For every one-star increase in the hotel’s rating, ADR increases by $0.94, assuming all other variables remain constant.
Significance: This is a strong, statistically significant predictor of ADR.

Star Rating Only Table
term	estimate	std.error	statistic	p.value
star_rating	0.9389	0.1827	5.1376	0.0000

Location Quality: Hotels with an “Excellent” and “Good” location have a positive impact on ADR that is, on average, respectively 5.89 and 2.93 dollars higher than hotels with an “Average” location (the baseline category). While hotels with a “Poor” location have an ADR that is 1.50 dollars lower than those with an “Average” location, on average.

Location Quality Only Table
term	estimate	std.error	statistic	p.value
location_quality_Excellent	5.8876	0.6001	9.8104	0.0000
location_quality_Good	2.9307	0.4929	5.9462	0.0000
location_quality_Poor	−1.4989	0.7373	−2.0330	0.0422

All three are statistically significant predictors of ADR.

Seasons:

The ADR for Spring (15 cents) and Winter (10 cents) is, on average, higher than in the baseline season (Fall). While the ADR for Summer is, on average, $0.94 lower than in the Fall.

Location Quality Only Table
term	estimate	std.error	statistic	p.value
season_Spring	0.1534	0.5768	0.2660	0.7903
season_Summer	−0.9380	0.5762	−1.6278	0.1037
season_Winter	0.1009	0.5689	0.1774	0.8592

However, these results are not statistically significant (p > 0.05).

Decision Tree Interpretation

Decision tree are easier to interpret when you visualize them. See our decision tree for recipe2:

The following elements are key when interpreting tree based model results:

Nodes:
- Decision Nodes: Represent a split in the data based on a condition (e.g., competitors_average_price < 96).
- Terminal Nodes (Leaves): Represent the end of a branch, containing the predicted value for a subset of the data.

Splits:
- A split occurs when the tree partitions the data based on a predictor variable ( competitors_average_price, revPAR).
- Conditions at each split direct the data into smaller groups for improved homogeneity.

Branches:
- Left Branch: Indicates that the condition at the split is satisfied (e.g., competitors_average_price < 96).
- Right Branch: Indicates the condition is not satisfied (e.g., competitors_average_price >= 96).

Predictions:
- Each terminal node provides a predicted value for ADR.
- These predictions are based on the characteristics of the subset of data reaching the node.

Proportions:
- Each terminal node also shows the percentage of data points in that node relative to the total dataset.

One more time..

The sky is the limit if you become an