Q1

## 1
# Load necessary libraries
library(dplyr) # For data manipulation
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(ggplot2) # For visualization

# Load the dataset
housing_data <- read.csv("pacdev_data.csv")

# Fit a simple linear regression model
model <- lm(price ~ sqft, data = housing_data)

# Display the coefficients
summary(model)
## 
## Call:
## lm(formula = price ~ sqft, data = housing_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -622948 -151283   -1650  138951  804553 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 40623.019  15862.454   2.561   0.0105 *  
## sqft          269.345      7.742  34.791   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 210300 on 4989 degrees of freedom
## Multiple R-squared:  0.1953, Adjusted R-squared:  0.1951 
## F-statistic:  1210 on 1 and 4989 DF,  p-value: < 2.2e-16
## 2
# Load necessary libraries
library(dplyr) # For data manipulation
library(ggplot2) # For visualization

# Load the dataset
housing_data <- read.csv("pacdev_data.csv")

# Create a new centered variable in the dataset
housing_data$centered_sqft <- housing_data$sqft - mean(housing_data$sqft)

# Fit the model using the newly created centered variable
model_centered <- lm(price ~ centered_sqft, data = housing_data)

# Display coefficients for the centered model
summary(model_centered)
## 
## Call:
## lm(formula = price ~ centered_sqft, data = housing_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -622948 -151283   -1650  138951  804553 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   5.827e+05  2.976e+03  195.77   <2e-16 ***
## centered_sqft 2.693e+02  7.742e+00   34.79   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 210300 on 4989 degrees of freedom
## Multiple R-squared:  0.1953, Adjusted R-squared:  0.1951 
## F-statistic:  1210 on 1 and 4989 DF,  p-value: < 2.2e-16

The intercept in the linear regression is a value that represents the dependent variable when independent variables are zero. In this scenario we are centering the predictor ‘sqft’ and this means that the center point of the variable is moved because the reference point, for example, is now moving from what it previously was.

Q2

# Load necessary libraries
library(dplyr) # For data manipulation

# Load the dataset
housing_data <- read.csv("pacdev_data.csv")

# Centering numeric predictors (assuming 'bed' and 'bath' are numeric)
housing_data$centered_bed <- scale(housing_data$bed, center = TRUE, scale = FALSE)
housing_data$centered_bath <- scale(housing_data$bath, center = TRUE, scale = FALSE)
housing_data$centered_sqft<- scale(housing_data$sqft, center = TRUE, scale = FALSE)

# Assuming 'garage' is a factor/categorical variable - create dummy variables
housing_data <- cbind(housing_data, model.matrix(~ garage - 1, data = housing_data))

# Renaming the dummy columns for better readability
colnames(housing_data)[(ncol(housing_data) - 1):ncol(housing_data)] <- c("no_garage", "yes_garage")

# Fit the multiple regression model with all predictors (including centered and dummy variables)
model_multiple <- lm(price ~ centered_sqft + centered_bed + centered_bath + no_garage + yes_garage + pool + city, data = as.data.frame(housing_data))

# Display coefficients
summary(model_multiple)
## 
## Call:
## lm(formula = price ~ centered_sqft + centered_bed + centered_bath + 
##     no_garage + yes_garage + pool + city, data = as.data.frame(housing_data))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -539286 -137407   -3532  124838  852187 
## 
## Coefficients: (1 not defined because of singularities)
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      507404.206   4502.633 112.691   <2e-16 ***
## centered_sqft       271.561      7.515  36.138   <2e-16 ***
## centered_bed         41.553   8420.185   0.005   0.9961    
## centered_bath     -3092.909   9439.900  -0.328   0.7432    
## no_garage         14195.911   6120.799   2.319   0.0204 *  
## yes_garage               NA         NA      NA       NA    
## poolyes           10124.630   6760.090   1.498   0.1343    
## citySanta Monica 190239.704   6757.751  28.151   <2e-16 ***
## cityWestwood      88020.719   6794.984  12.954   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 195000 on 4983 degrees of freedom
## Multiple R-squared:  0.3085, Adjusted R-squared:  0.3075 
## F-statistic: 317.6 on 7 and 4983 DF,  p-value: < 2.2e-16

Q3

# Adding the interaction term between centered_sqft and city to the existing model
model_interaction <- lm(price ~ centered_sqft * city + centered_bed + centered_bath + no_garage + yes_garage + pool, data = as.data.frame(housing_data))

# Display coefficients for the interaction model
summary(model_interaction)
## 
## Call:
## lm(formula = price ~ centered_sqft * city + centered_bed + centered_bath + 
##     no_garage + yes_garage + pool, data = as.data.frame(housing_data))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -536726 -138828   -2505  124312  868332 
## 
## Coefficients: (1 not defined because of singularities)
##                                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                    507303.45    4491.69 112.943  < 2e-16 ***
## centered_sqft                     240.63      10.14  23.726  < 2e-16 ***
## citySanta Monica               189904.29    6741.32  28.170  < 2e-16 ***
## cityWestwood                    88037.11    6778.98  12.987  < 2e-16 ***
## centered_bed                      235.26    8403.39   0.028   0.9777    
## centered_bath                   -3099.26    9419.64  -0.329   0.7422    
## no_garage                       14205.98    6108.49   2.326   0.0201 *  
## yes_garage                            NA         NA      NA       NA    
## poolyes                         10817.17    6744.71   1.604   0.1088    
## centered_sqft:citySanta Monica     90.09      17.49   5.152 2.68e-07 ***
## centered_sqft:cityWestwood         37.95      18.05   2.103   0.0356 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 194500 on 4981 degrees of freedom
## Multiple R-squared:  0.3122, Adjusted R-squared:  0.311 
## F-statistic: 251.2 on 9 and 4981 DF,  p-value: < 2.2e-16
# Create a visualization of the interaction between sqft and city
library(ggplot2)

ggplot(housing_data, aes(x = centered_sqft, y = price, color = city)) +
  geom_point() +
  geom_smooth(method = "lm", formula = y ~ x) +
  labs(title = "Interaction between sqft and city", x = "Centered Sqft", y = "Price")

Q4

# Generate the residual plot for the interaction model
plot(model_interaction, which = 1)  # Plot of model residuals

## Based on the produced graph above of fitted values versus residuals, this is a good model. There is seemingly no pattern between these two values which implies no correlation or relationship. This also suggests that the model is capturing the variability in the dataset.

Q5

## Based on the regression model it appears that the primary factor affecting housing prices is 'sqft'. It is not the only impactful variable as the other predictors have positive effects on housing prices as well (or in other words are positively correlated, so increasing the predictor increases price). The model does show that there is a varied effect of the impact of sqft on housing prices in various cities. This means that there is a varying relationship across residential areas and that the model is capturing this relationship. 

## From the city relationship plot, it appears that Santa Monica has the strongest positive relationship between sqft and housing prices. This is explained by the steeper slope in that graph, along with the line of best fit being significantly higher at its intercept than the other best fit lines. 

## The model itself, per Q4, also shows that there is a significant, and non-influenced, impact of these predictors on housing prices. The plot showing residuals randomly scattered against fitted values confirms this as well. 

## I would recommend that Andrew increase the price per sqft in Santa Monica because of the stronger positive relationship between sqft and housing remodel prices. It would be prudent that it be the city where prices are raised because it will maximize profits for the company as this clientele base is likely willing to pay more as they are exposed to these prices more often. If Andrew were to recommend the higher prices per sqft in Westwood it could have a negative effect on the number of customers who go through with a remodel, or push them to competitors. In order to stay competitive I would recommend he increase prices in Santa Monica per sqft.

Challenge