Model Summary
--------------------------------------------------------------------------
R 0.772 RMSE 45426.628
R-Squared 0.596 MSE 2063578544.951
Adj. R-Squared 0.594 Coef. Var 27.670
Pred R-Squared 0.579 AIC 4863.117
MAE 31692.288 SBC 4873.012
--------------------------------------------------------------------------
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 609852999259.857 1 609852999259.857 292.576 0.0000
Residual 412715708990.143 198 2084422772.677
Total 1022568708250.000 199
------------------------------------------------------------------------------------
Parameter Estimates
-------------------------------------------------------------------------------------------------
model Beta Std. Error Std. Beta t Sig lower upper
-------------------------------------------------------------------------------------------------
(Intercept) 16505.199 9262.237 1.782 0.076 -1760.095 34770.493
Living_Area 82.588 4.828 0.772 17.105 0.000 73.066 92.110
-------------------------------------------------------------------------------------------------BUA 345 - Lecture 10
Multiple Linear Regression Models in R
Housekeeping
HW 4 was due 2/12/2025 - 2 day grace period
Quiz 1 Will Take Place on Thursday 2/20 in class
- There is an asynchronous option.
Today’s plan
Review of SLR and MLR, Normal Distribution, and LN Transformation
SLR Model Output and Multiple Linear Regression (MLR)
Examining Regression Model Output
Understanding hypotheses being tested
Interpreting regression model output
Continue with Multiple Linear Regression
In-class Polling (Session ID: bua345s25)
Regression Model Assumptions
Review: For simple linear regression, there must be a linear relationship between X, the explanatory variable, and Y, the response variable.
The response variable, must be approximately normally distributed.
Recall that normally distributed means symmetric and bell-shaped.
What if it’s not.
One common solution is a linear transformation.
Financial data such as real estate data, prices, etc. are commonly right-skewed.
A good transformation for right-skewed data is the Natural Log (LN) Transformation.
In HW 5 (after Quiz 1) we will work through:
How the LN transformation ‘normalizes’ the distribution of the response.
How to ‘back-transform’ model results to return to original scale of the data, e.g. US dollars.
Review of Histograms of Different Distributions
Histograms are an effective tool for examining the distribution of the data.
LEFT SKEWED
Tail pulled out to LEFT
Low outliers
e.g. Human Lifespan
NORMAL/SYMMETRIC
Data appear in a symmetric bell-shaped curve
No graphic evidence of outliers
e.g. Test scores
RIGHT SKEWED
Tail pulled out to RIGHT
High outliers
e.g. Real Estate Data
Lecture 10 In-class Exercises - Q1
Below is the model output for a regression model relating the size of the living area of a house to it’s selling price.
What is the estimated selling price of a 2300 sq. ft. house, based on this model?
Round your answer to a whole dollar amount.
Lecture 10 In-class Exercises - Q1 con’t
Focus on the Parameter Estimates table to answer this question:
Regression Output Interpretation
- \(Est. Selling Price = 16505.199 + 82.588\times Living Area\)
Limitations of Simple Linear Regression
Simple Linear Regression - One X variable
In this case, X is the size of the living area.
This model says that regardless of other factors
a 2500 sq. ft house has a selling price of 222975.
The model ignores number of bathrooms, age of house, etc.
These factors may also be helpful in explaining selling price.
`geom_smooth()` using formula = 'y ~ x'
Correlation between Living Area and Selling price is 0.77
This a is strong correlation, but maybe we can explain more of the variability in the data.
Introduction to Multiple Linear Regression
This linear regression model format can also be used if there multiple explanatory (
X) variables.If a model has more than one
Xvariable, it is a MULTIPLE LINEAR REGRESSION model.We will examine one more dataset today to introduce this concept.
First let’s import and examine the data:
Code
# A tibble: 6 × 4
Price Living_Area Bathrooms House_Age
<dbl> <dbl> <dbl> <dbl>
1 217314 2498 2.5 14
2 238792 2250 2.5 10
3 222330 2712 3 1
4 206688 2284 2.5 17
5 88207 1480 1.5 14
6 236936 2300 2.5 16Simple Linear Regression vs. Multiple Linear Regression
Transitioning from SLR to MLR is Straightforward
In R and most software adding a variable to our model is as simple as addition.
The challenge is interpretation because we can no longer visualize the model.
There are 3-D visualization tools in R, BUT they are not always helpful.
Instead I recommend extending the SLR model output interpretation to the variables in the model.
One the next slide we’ll add number of bathrooms.
- Spoiler: Number of bathrooms is a huge deal when buying a house.
MLR Model with Two Variables
Code
Model Summary
--------------------------------------------------------------------------
R 0.815 RMSE 41412.317
R-Squared 0.665 MSE 1714980011.473
Adj. R-Squared 0.661 Coef. Var 25.289
Pred R-Squared 0.640 AIC 4828.109
MAE 30629.922 SBC 4841.302
--------------------------------------------------------------------------
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 679572705955.336 2 339786352977.668 195.157 0.0000
Residual 342996002294.664 197 1741096458.349
Total 1022568708250.000 199
------------------------------------------------------------------------------------
Parameter Estimates
---------------------------------------------------------------------------------------------------
model Beta Std. Error Std. Beta t Sig lower upper
---------------------------------------------------------------------------------------------------
(Intercept) -11553.295 9556.111 -1.209 0.228 -30398.701 7292.110
Living_Area 58.047 5.875 0.543 9.881 0.000 46.462 69.633
Bathrooms 38141.447 6027.411 0.348 6.328 0.000 26254.916 50027.977
---------------------------------------------------------------------------------------------------MLR Model with Two Variables
Code
Model Summary
--------------------------------------------------------------------------
R 0.815 RMSE 41412.317
R-Squared 0.665 MSE 1714980011.473
Adj. R-Squared 0.661 Coef. Var 25.289
Pred R-Squared 0.640 AIC 4828.109
MAE 30629.922 SBC 4841.302
--------------------------------------------------------------------------
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 679572705955.336 2 339786352977.668 195.157 0.0000
Residual 342996002294.664 197 1741096458.349
Total 1022568708250.000 199
------------------------------------------------------------------------------------
Parameter Estimates
---------------------------------------------------------------------------------------------------
model Beta Std. Error Std. Beta t Sig lower upper
---------------------------------------------------------------------------------------------------
(Intercept) -11553.295 9556.111 -1.209 0.228 -30398.701 7292.110
Living_Area 58.047 5.875 0.543 9.881 0.000 46.462 69.633
Bathrooms 38141.447 6027.411 0.348 6.328 0.000 26254.916 50027.977
---------------------------------------------------------------------------------------------------A closer look at the Parameter Estimates
Interpreting the New Model
Model: \[ Est. Selling Price = -11553.295 + 58.047\times Living Area + 38141.447 \times Bathrooms \]
Interpretation:
If number of bathrooms remains unchanged, each additional square foot is estimated to raise the selling price by about 58 dollars.
If living area remains unchanged, each additional bathroom will raise the estimated selling price by about 38 THOUSAND dollars.
Lecture 10 In-class Exercises - Q2
Based on this model, if a house is renovated to increase the square footage by 1000 square feet and two bathrooms are added, what would be estimated change in price?
Round your answer to a whole dollar amount.
Model: \[ Est. Selling Price = -11553.295 + 58.047\times Living Area + 38141.447 \times Bathrooms \]
Adding ANOTHER Term to our MLR
Next, we add age of the house to the model:
Code
Model Summary
--------------------------------------------------------------------------
R 0.821 RMSE 40864.224
R-Squared 0.673 MSE 1669884825.573
Adj. R-Squared 0.668 Coef. Var 25.018
Pred R-Squared 0.641 AIC 4824.780
MAE 30119.407 SBC 4841.271
--------------------------------------------------------------------------
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 688591743135.442 3 229530581045.147 134.704 0.0000
Residual 333976965114.558 196 1703964107.727
Total 1022568708250.000 199
------------------------------------------------------------------------------------
Parameter Estimates
--------------------------------------------------------------------------------------------------
model Beta Std. Error Std. Beta t Sig lower upper
--------------------------------------------------------------------------------------------------
(Intercept) 5775.299 12087.330 0.478 0.633 -18062.622 29613.220
Living_Area 60.614 5.918 0.567 10.243 0.000 48.943 72.285
Bathrooms 30089.928 6913.944 0.274 4.352 0.000 16454.654 43725.201
House_Age -235.721 102.458 -0.112 -2.301 0.022 -437.783 -33.658
--------------------------------------------------------------------------------------------------Examining the new model
Hopefully, the interpretation will seem redundant at this point…
The New Model
\[\begin{split} Est. Selling Price = \\ & 5775.299 + 60.614\times Living Area + \\ & 30089.928 \times Bathrooms - 235.721\times House Age \end{split}\]Interpretation:
If number of bathrooms and age of the house remain unchanged, each additional square foot is estimated to raise the selling price by about 61 dollars.
If living area and age of the house remain unchanged, each additional bathroom will raise the estimated selling price by about 30 THOUSAND dollars.
If living area and number of bathrooms remain unchanged, each additional year will LOWER the estimated selling price by about 236 dollars.
Lecture 10 In-class Exercises - Q3
What is the estimated price of a house that 2500 square feet with 4 bathrooms that is 20 years old?
Key Points from Today
Multiple Linear Regression (MLR) is an extension of SLR where we ADD more variables to the model.
- For MLR, the hypothesis test of each \(\beta\) is an indication of whether or not that variable is useful to the model.
A key assumption of SLR and MLR is that the response, Y, is normally distributed.
If the response is right-skewed which is common in data having to do with money, a good strategy is to use a natural log transformation.
This process will be illustrated and practiced in HW 5 after Quiz 1.
HW 4 was due 2/12/2025
To submit an Engagement Question or Comment about material from Lecture 10: Submit it by midnight today (day of lecture).