BUA 345 - Lecture 14

Housekeeping

HW 5 was due 2/26/2025 - 2 day grace period

HW 6 was due 3/5/2025 - 2 day grace period

Quiz 2 will be on 4/1/2025 - Date has changed and syllabus has been updated.

Today’s plan 📋

• Review Parallel Lines Model

• Introduce Interaction term and Interaction Model

• Work through how to interpret model output

• Introduce HW 6

• Talk about next steps

Review Question - Import data

house_remodel <- read_csv("data/house_remodel.csv", show_col_types = F)
head(house_remodel, 3) |> kable()

house_remodel |> select(Remodeled) |> table()                     # number of obs by category

Remodeled
 No Yes 
 29  28 

house_remodel |> select(Price, Square_Feet) |> cor() |> round(2)  # correlation of price & sq. ft.

            Price Square_Feet
Price        1.00        0.75
Square_Feet  0.75        1.00

💥 Lecture 14 In-class Exercises - Q1 - Review 💥

Based on the Parameter Estimates table for the specified categorical regression model, which category is the baseline category?

                              Model Summary                                
--------------------------------------------------------------------------
R                           0.924       RMSE                    32079.481 
R-Squared                   0.854       MSE                1029093133.260 
Adj. R-Squared              0.848       Coef. Var                   8.223 
Pred R-Squared              0.836       AIC                      1352.620 
MAE                     28012.693       SBC                      1360.792 
--------------------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                                      ANOVA                                        
----------------------------------------------------------------------------------
                        Sum of                                                    
                       Squares        DF         Mean Square       F         Sig. 
----------------------------------------------------------------------------------
Regression    341892090457.686         2    170946045228.843     157.37    0.0000 
Residual       58658308595.823        54      1086264973.997                      
Total         400550399053.509        56                                          
----------------------------------------------------------------------------------

                                         Parameter Estimates                                          
-----------------------------------------------------------------------------------------------------
       model          Beta    Std. Error    Std. Beta      t        Sig          lower         upper 
-----------------------------------------------------------------------------------------------------
 (Intercept)    137549.093     17620.447                  7.806    0.000    102222.224    172875.963 
 Square_Feet       137.879        10.836        0.670    12.725    0.000       116.155       159.602 
RemodeledYes     90917.216      8834.268        0.542    10.291    0.000     73205.575    108628.858 
-----------------------------------------------------------------------------------------------------

A Comment About Formatted Output

• In HW 6 and below I use R coding to format the output to make it easier to read.

• The values are IDENTICAL to the unformatted output.

• Note: Formatted Output will differ in appearance depending on where it is viewed, i.e. slides, html file, or .Qmd file.

Quick Review of Categorical Regression

• On Tuesday we covered the Parallel Lines Model:

  • A Parallel Lines model has two X variables, one quantitative and one categorical variable.

  • Model estimates a separate SLR model for each category in the categorical variable.

  • Model assumes all categories have the same SLOPE.

  • Model estimates a separate INTERCEPT for each category.

  • Model output shows results of a hypothesis test to determine if each non-baseline category’s intercept is significantly different from baseline intercept.

Interactive Plot of House Remodel Data

Calculations from House Model

• By default R chooses baseline categories alphabetically

  • No is before Yes so un-Remodeled houses are the baseline

  • Un-Remodeled (No) SLR Model:

    • Est. Price = 137549.093 + 137.879 * Square_Feet

  • Remodeled (Yes) SLR Model:

    • Est. Price = 137549.093 + 137.879 * Square_Feet + 90917.216

    • Est. Price = 137549.093 + 90917.216 + 137.879 * Square_Feet

    • Est. Price = 228466.3 + 137.879 * Square_Feet

• Interpretation:

  • Prices of remodeled houses are about 91 thousand dollars more than similar houses without remodeling, after accounting for square footage.

  • This difference is statistically significant (P-value < 0.001)

HW 6 - Questions 1 - 6

• This part of HW 6 examines data similar to the House-Remodel data examined in Lecture 13 and the review question.

• The dataset is smaller and the numbers are different, but the questions are essentially the same.

Categorical Regression with Interactions

• The categorical models covered so far assume that the SLR models for all categories have the same slope.

• How do we examine that assumption?

• For example:

  • In the celebrity data in Lecture 13, the data showed a decrease in earnings as they got older.

  • Slope was assumed to be IDENTICAL for both males and females

  • That may not be true for all celebrities.

• In the following small dataset, we will look at male celebrities only and examine if actors and athletes salaries follow the same trend.

Import and Examine Celebrity Profession Data

# import and examine celeb profession dataset
celeb_prof <- read_csv("data/celeb_prof.csv", show_col_types=F) 
head(celeb_prof) |> kable()

# use table to summarize data by category
celeb_prof |> select(Profession) |> table()

Profession
  Actor Athlete 
      8       8 

Examine Correlations in Celebrity Professions Data

Note: If categories have different slopes, correlations for whole dataset will be misleading.

celeb_prof |> select(Earnings, Age) |> cor() |> round(2) # all data

         Earnings   Age
Earnings     1.00 -0.46
Age         -0.46  1.00

celeb_prof |> filter(Profession=="Actor") |>             # actors only
  select(Earnings, Age) |> cor() |> round(2)

         Earnings  Age
Earnings     1.00 0.99
Age          0.99 1.00

celeb_prof |> filter(Profession=="Athlete") |>           # athletes only
  select(Earnings, Age) |> cor() |> round(2)

         Earnings   Age
Earnings     1.00 -0.98
Age         -0.98  1.00

Explore and Plot Data

Interactive Model Plot - Celebrity Professions

Regression Model - Celebrity Professions

Now that we understand the data and linear trends, we can examine and interpret the regression model output.

                        Model Summary                          
--------------------------------------------------------------
R                       0.987       RMSE                2.640 
R-Squared               0.974       MSE                 6.968 
Adj. R-Squared          0.967       Coef. Var           6.058 
Pred R-Squared          0.951       AIC                86.467 
MAE                     2.265       SBC                90.330 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                                ANOVA                                 
---------------------------------------------------------------------
                Sum of                                               
               Squares        DF    Mean Square       F         Sig. 
---------------------------------------------------------------------
Regression    4131.949         3       1377.316    148.246    0.0000 
Residual       111.489        12          9.291                      
Total         4243.437        15                                     
---------------------------------------------------------------------

                                         Parameter Estimates                                           
------------------------------------------------------------------------------------------------------
                model       Beta    Std. Error    Std. Beta       t        Sig       lower      upper 
------------------------------------------------------------------------------------------------------
          (Intercept)    -50.297         9.054                  -5.555    0.000    -70.023    -30.571 
                  Age      1.824         0.179        0.983     10.170    0.000      1.433      2.215 
    ProfessionAthlete    227.218        12.389        0.263     18.340    0.000    200.224    254.212 
Age:ProfessionAthlete     -5.063         0.293       -1.487    -17.294    0.000     -5.701     -4.425 
------------------------------------------------------------------------------------------------------

Model Interpretation - Interpreting model coefficients (betas)

• Baseline category is first alphabetically

  • Actor comes before Athlete in the alphabet so Actor is the baseline category.

  • **Actor SLR Model:

    • Earnings = -50.297 + 1.824*Age**

• ProfessionAthlete term: Difference from Actor (baseline) model Intercept

• Age:ProfessionAthlete term: Difference from Actor model Slope

  • Athlete SLR Model requires some calculations:

    • Earnings = -50.297 + 1.824*Age + 227.218 - 5.063*Age

    • Earnings = (-50.297 + 227.218) + (1.824 - 5.063)*Age

    • Earnings = 176.921 - ____*Age

    • Athlete SLR Model: Earnings = 176.921 - ____*Age

💥 Lecture 14 In-class Exercises - Q2 💥

What is the slope term (estimated beta for Age) for the Athlete SLR model?

Specify answer to two decimal places.

Abridged Output

• ProfessionAthlete term: Difference from Actor (baseline) model Intercept

• Age:ProfessionAthlete term: Difference from Actor model Slope

  • Earnings = (-50.297 + 227.218) + (1.824 - 5.063)*Age

  • Athlete SLR Model: Earnings = 176.921 - ____*Age

Determining Statistical Significance

• In this case, we don’t need to examine the P-values because the model differences between groups are so clear (but we will).

• The two intercepts and two slopes are VERY different.

• Reminder of Hypothesis Testing concepts:

  • The SMALLER the P-value, the more evidence there is that the true value of Beta (model term) is not zero.

  • If this sentence is not clear to you, you are responsible for reviewing the Review Materials on Hypothesis Tests and Significance Tests (and other related topics):

    • MAS 261 Lecture Slides and Notes
    • Khan Academy - The Idea of Significance tests
    • Khan Academy - Comparing a P-value to a Significance level

Conclusions from Actor and Athlete Model

Abridged Output

• P-value for difference in intercepts (ProfessionAthlete): < 0.001

  • Actors SLR model and Athletes SLR model intercepts are significantly different.

• P-value for difference in slopes (Age:ProfessionAthlete): < 0.001

  • Actors SLR model and Athletes SLR model slopes are significantly different.

• These interpretations are in agreement with what we can easily see in the Interactive Model Plot.

Movie Genres and Costs

• Is length of a movie (Runtime) a good predictor of the movie budget?

• Does the relationship between movie length and budget differ by movie genre?

movies <- read_csv("data/movies.csv", show_col_types = F)  # Import and examine data
head(movies, 4) |> kable()

movies |> select(Genre) |> table()  # use table to examine categories in the data

Genre
           Action Suspense / Horror 
               10                10 

Examine Correlations in Movie Data

Note: If categories have different slopes, correlations for whole dataset will be misleading.

movies |> select(Budget, Runtime) |> cor() |> round(2) # all data

        Budget Runtime
Budget    1.00    0.76
Runtime   0.76    1.00

movies |> filter(Genre == "Action") |>                 # action movies
  select(Budget, Runtime) |> cor() |> round(2)

        Budget Runtime
Budget    1.00    0.95
Runtime   0.95    1.00

movies |> filter(Genre == "Suspense / Horror") |>      # suspense/horror movies
  select(Budget, Runtime) |> cor() |> round(2)

        Budget Runtime
Budget    1.00    0.99
Runtime   0.99    1.00

Explore and Plot Data

Interactive Model Plot - Movie Data

Movie Genres Regression Model

Again, we can examine and interpret the regression model output.

Abridged Output

💥 Lecture 14 In-class Exercises - Q3 💥

What is the intercept term (estimated beta) for the Suspense / Horror SLR model?

• Specify answer to two decimal places.

• GenreSuspense / Horror term: Difference from Action (baseline) model Intercept

• Runtime:GenreSuspense / Horror term: Difference from Action model Slope

  • Budget = (-286.67 + 218.75) + (3.25 - 2.35)*Runtime

  • Suspense / Horror SLR Model: Budget = ____ - 0.9*Runtime

Determining Statistical Significance

• In this case, we don’t need to examine the P-values because the model differences between groups are so clear (but we will).

• The two intercepts and two slopes are VERY different.

• Reminder of Hypothesis Testing concepts:

  • The SMALLER the P-value, the more evidence there is that the true value of Beta (model term) is not zero.

  • If this sentence is not clear to you, you are responsible for reviewing the Review Materials on Hypothesis Tests and Significance Tests (and other related topics):

    • MAS 261 Lecture Slides and Notes
    • Khan Academy - The Idea of Significance tests
    • Khan Academy - Comparing a P-value to a Significance level

Conclusions from Movie Genre Model

Abridged Output

• P-value for diff. in intercepts (GenreSuspense / Horror): < 0.001

  • Intercepts for these two distinct genre SLR models are _____ (Next Question).

• P-value for diff. in slopes (Runtime:GenreSuspense / Horror term): < 0.001

  • Slopes for for these two distinct genre SLR models are _____ (Next Question).

• These interpretations are in agreement with what we can easily see in the Interactive Model Plot.

💥 Lecture 14 In-class Exercises - Q4 💥

Abridged Output

The difference in intercepts and the difference in slopes between the model for Action movies and the model for Suspense / Horror movies are both _____.

💥 Lecture 14 In-class Exercises - Q5 💥

Recall that the cutoff for determining significance of a regression model term based on it’s P-value is 0.05.

Fill in the blank:

The smaller the P-value, the _____ evidence there is that the Beta coefficient is non-zero and the term is useful to the model.

HW 6 - Questions 7 - 16

• Dataset has three categories of Diamonds:

  • Colorless, Faint yellow, and Nearly colorless

• Colorless is first alphabetically so that is the baseline category by default.

  • Each color category has unique intercept AND a unique slope.

  • The interactive model plot and abridged regression output are provided.

  • All Blackboard questions can be answered by rendering .qmd file to examine .html output.

• Helpful TIP: In addition to other recommended options, change preview option (see next slide).

HW 6 - Change HTML Preview Option

• For HW 6 you do not have to write any R code.

• Instead you are expected to correctly interpret provided output.

• Quiz 2 will have similar output WITHOUT the interactive plots.

• Change the following option in the Basic tab of the R Markdown options:

Looking Ahead - What’s Next?

• This week, ALL of the categorical models could be simplified to multiple SLRs, with same or different slopes.

• ALL of the variables have had P-values less than 0.05 so the terms were all useful.

• There are many many model options where these two facts are not true.

• Time permitting, here’s a brief look at a dataset with many explanatory variables:

Insurance Data Model and Variable Selection

• There are 3 quantitative variables:

  • Age, BMI, and Children

• There are 3 categorical variables:

  • Sex, Smoker, Region

• There are literally hundreds of possible models including interaction terms.

• Note that an interaction can also be between two quantitative variables.

• You can also have interaction terms with three variables (but I try to avoid those).

• How do we sort through all of the possible options?

  • Software helps us pare down all the possible models to a few choices.
  • Analyst then uses critical thinking and examination of data to determine final model.

• Model and variable selection methods are the next set of topics.

Key Points from Today

• Categorical Interaction Model

  • Separate SLR for each group.
  • BOTH slopes and intercepts can differ by category
  • We can test if interaction term (slope difference) is significant.

• Next Topics

  • Comparing model goodness of fit
  • Introduction to variable selection

• HW 6 is now available and is due on Wed. 3/6.

• Date of Quiz 2 has been changed to Tuesday, 4/1.