Data 712 Week 5 HW
2025-03-15
Week 5 Homework: Marginal Effects and Predictions
# Load necessary packages
library(clarify)
# Filter dataset to remove NAs in Percentile_Rank_Combined_NYC
neighborhoods_clean <- subset(neighborhoods, !is.na(Percentile_Rank_Combined_NYC))
# Fit the linear regression model with the updated independent variables
nm0 <- lm(Percentile_Rank_Combined_NYC ~ Benzene_Concentration + Particulate_Matter_25 +
Traffic_Truck_Highways + Industrial_Land_Use + Low_Vegetative_Cover +
Housing_Vacancy_Rate + RMP_Sites,
data = neighborhoods_clean)
# Display summary of the regression model
summary(nm0)##
## Call:
## lm(formula = Percentile_Rank_Combined_NYC ~ Benzene_Concentration +
## Particulate_Matter_25 + Traffic_Truck_Highways + Industrial_Land_Use +
## Low_Vegetative_Cover + Housing_Vacancy_Rate + RMP_Sites,
## data = neighborhoods_clean)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54377 -0.15830 -0.02243 0.13258 0.76019
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.2494278 0.0103883 -24.010 < 2e-16 ***
## Benzene_Concentration 0.4647005 0.0289354 16.060 < 2e-16 ***
## Particulate_Matter_25 0.1252307 0.0213345 5.870 4.66e-09 ***
## Traffic_Truck_Highways 0.0816896 0.0125717 6.498 8.98e-11 ***
## Industrial_Land_Use 0.0783523 0.0102412 7.651 2.40e-14 ***
## Low_Vegetative_Cover 0.1534966 0.0220074 6.975 3.48e-12 ***
## Housing_Vacancy_Rate 0.0019183 0.0110989 0.173 0.863
## RMP_Sites 0.0005581 0.0001270 4.393 1.14e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2162 on 4771 degrees of freedom
## (139 observations deleted due to missingness)
## Multiple R-squared: 0.5246, Adjusted R-squared: 0.5239
## F-statistic: 752.2 on 7 and 4771 DF, p-value: < 2.2e-16The above code fits a linear regression model to the cleaned dataset,
predicting Percentile_Rank_Combined_NYC using various
independent variables. The summary of the model provides insights into
the coefficients and their significance. The table shows that the
independent variables have varying effects on the dependent variable,
with some being statistically significant. The most significant
predictors include Benzene_Concentration,
Particulate_Matter_25, and
Traffic_Truck_Highways. The coefficients indicate the
direction and magnitude of the relationships between the independent
variables and the dependent variable.
The intercept represents the expected value of
Percentile_Rank_Combined_NYC when all independent variables
are zero. A negative intercept suggests that, in the absence of any of
the predictors, the percentile rank would be below the average,
indicating a disadvantaged neighborhood which is impossible in this
context.
nm1 <- lm(Percentile_Rank_Combined_NYC ~ Benzene_Concentration + Particulate_Matter_25 +
Traffic_Truck_Highways * Low_Vegetative_Cover + Industrial_Land_Use +
Housing_Vacancy_Rate + RMP_Sites,
data = neighborhoods_clean)
summary(nm0)##
## Call:
## lm(formula = Percentile_Rank_Combined_NYC ~ Benzene_Concentration +
## Particulate_Matter_25 + Traffic_Truck_Highways + Industrial_Land_Use +
## Low_Vegetative_Cover + Housing_Vacancy_Rate + RMP_Sites,
## data = neighborhoods_clean)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54377 -0.15830 -0.02243 0.13258 0.76019
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.2494278 0.0103883 -24.010 < 2e-16 ***
## Benzene_Concentration 0.4647005 0.0289354 16.060 < 2e-16 ***
## Particulate_Matter_25 0.1252307 0.0213345 5.870 4.66e-09 ***
## Traffic_Truck_Highways 0.0816896 0.0125717 6.498 8.98e-11 ***
## Industrial_Land_Use 0.0783523 0.0102412 7.651 2.40e-14 ***
## Low_Vegetative_Cover 0.1534966 0.0220074 6.975 3.48e-12 ***
## Housing_Vacancy_Rate 0.0019183 0.0110989 0.173 0.863
## RMP_Sites 0.0005581 0.0001270 4.393 1.14e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2162 on 4771 degrees of freedom
## (139 observations deleted due to missingness)
## Multiple R-squared: 0.5246, Adjusted R-squared: 0.5239
## F-statistic: 752.2 on 7 and 4771 DF, p-value: < 2.2e-16In this model, we have added an interaction term between
Traffic_Truck_Highways and
Low_Vegetative_Cover. The interaction term allows us to
understand how the relationship between
Traffic_Truck_Highways and
Percentile_Rank_Combined_NYC changes depending on the level
of Low_Vegetative_Cover. The impact of this interaction can
be seen in the coefficients of the model. The coefficient for the
interaction term is positive, indicating that as
Low_Vegetative_Cover increases, the negative impact of
Traffic_Truck_Highways on
Percentile_Rank_Combined_NYC is mitigated. This suggests
that neighborhoods with higher levels of vegetative cover may be less
disadvantaged, even if they have high truck traffic.
nm2 <- lm(Percentile_Rank_Combined_NYC ~ Benzene_Concentration + Particulate_Matter_25 +
Traffic_Truck_Highways + Low_Vegetative_Cover * Industrial_Land_Use +
Housing_Vacancy_Rate + RMP_Sites,
data = neighborhoods_clean)
summary(nm1)##
## Call:
## lm(formula = Percentile_Rank_Combined_NYC ~ Benzene_Concentration +
## Particulate_Matter_25 + Traffic_Truck_Highways * Low_Vegetative_Cover +
## Industrial_Land_Use + Housing_Vacancy_Rate + RMP_Sites, data = neighborhoods_clean)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.63050 -0.13659 -0.01743 0.10905 0.78554
##
## Coefficients:
## Estimate Std. Error t value
## (Intercept) -0.0851800 0.0130156 -6.544
## Benzene_Concentration 0.4358662 0.0278680 15.640
## Particulate_Matter_25 0.0780742 0.0206584 3.779
## Traffic_Truck_Highways -0.2622183 0.0212425 -12.344
## Low_Vegetative_Cover -0.1692717 0.0267714 -6.323
## Industrial_Land_Use 0.0729819 0.0098536 7.407
## Housing_Vacancy_Rate -0.0359990 0.0108470 -3.319
## RMP_Sites 0.0008377 0.0001230 6.810
## Traffic_Truck_Highways:Low_Vegetative_Cover 0.7450897 0.0378399 19.691
## Pr(>|t|)
## (Intercept) 6.60e-11 ***
## Benzene_Concentration < 2e-16 ***
## Particulate_Matter_25 0.000159 ***
## Traffic_Truck_Highways < 2e-16 ***
## Low_Vegetative_Cover 2.80e-10 ***
## Industrial_Land_Use 1.52e-13 ***
## Housing_Vacancy_Rate 0.000911 ***
## RMP_Sites 1.10e-11 ***
## Traffic_Truck_Highways:Low_Vegetative_Cover < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2079 on 4770 degrees of freedom
## (139 observations deleted due to missingness)
## Multiple R-squared: 0.5604, Adjusted R-squared: 0.5596
## F-statistic: 759.9 on 8 and 4770 DF, p-value: < 2.2e-16In this model, we have added an interaction term between
Low_Vegetative_Cover and Industrial_Land_Use.
The interaction term allows us to understand how the relationship
between Low_Vegetative_Cover and
Percentile_Rank_Combined_NYC changes depending on the level
of Industrial_Land_Use. The impact of this interaction can
be seen in the coefficients of the model. The coefficient for the
interaction term is negative, indicating that as
Industrial_Land_Use increases, the positive impact of
Low_Vegetative_Cover on
Percentile_Rank_Combined_NYC is diminished. This suggests
that neighborhoods with higher levels of industrial land use may be more
disadvantaged, even if they have high vegetative cover.
library(texreg)## Version: 1.39.4
## Date: 2024-07-23
## Author: Philip Leifeld (University of Manchester)
##
## Consider submitting praise using the praise or praise_interactive functions.
## Please cite the JSS article in your publications -- see citation("texreg").screenreg(list(nm0, nm1, nm2), doctype = TRUE)##
## ==================================================================================
## Model 1 Model 2 Model 3
## ----------------------------------------------------------------------------------
## (Intercept) -0.25 *** -0.09 *** -0.19 ***
## (0.01) (0.01) (0.01)
## Benzene_Concentration 0.46 *** 0.44 *** 0.44 ***
## (0.03) (0.03) (0.03)
## Particulate_Matter_25 0.13 *** 0.08 *** 0.12 ***
## (0.02) (0.02) (0.02)
## Traffic_Truck_Highways 0.08 *** -0.26 *** 0.09 ***
## (0.01) (0.02) (0.01)
## Industrial_Land_Use 0.08 *** 0.07 *** -0.12 ***
## (0.01) (0.01) (0.02)
## Low_Vegetative_Cover 0.15 *** -0.17 *** 0.08 ***
## (0.02) (0.03) (0.02)
## Housing_Vacancy_Rate 0.00 -0.04 *** -0.00
## (0.01) (0.01) (0.01)
## RMP_Sites 0.00 *** 0.00 *** 0.00 ***
## (0.00) (0.00) (0.00)
## Traffic_Truck_Highways:Low_Vegetative_Cover 0.75 ***
## (0.04)
## Low_Vegetative_Cover:Industrial_Land_Use 0.34 ***
## (0.03)
## ----------------------------------------------------------------------------------
## R^2 0.52 0.56 0.53
## Adj. R^2 0.52 0.56 0.53
## Num. obs. 4779 4779 4779
## ==================================================================================
## *** p < 0.001; ** p < 0.01; * p < 0.05Using the sin_ame function from the clarify package
# Load necessary package
library(clarify)
# Set seed for reproducibility
set.seed(123)
# Step 1: Simulate coefficients from the model
sim_coefs <- sim(nm0, n = 1000) # Using nm0_fixed without County_binary
# Step 2: Compute Average Marginal Effects (AMEs) for Low_Vegetative_Cover
sim_est_lv <- sim_ame(sim_coefs, var = "Low_Vegetative_Cover", contrast = "rd", verbose = FALSE)
# Step 3: Display Results
summary(sim_est_lv)## Estimate 2.5 % 97.5 %
## E[dY/d(Low_Vegetative_Cover)] 0.153 0.111 0.194# Step 4: Plot Results
plot(sim_est_lv)
This plot shows the average marginal effects of
Low_Vegetative_Cover on
Percentile_Rank_Combined_NYC. The x-axis represents the
values of Low_Vegetative_Cover, and the y-axis represents
the average marginal effects. The shaded area around the line indicates
the confidence intervals for the average marginal effects. This plot
helps to visualize how changes in Low_Vegetative_Cover
impact the predicted values of
Percentile_Rank_Combined_NYC. What we can observe is that
as Low_Vegetative_Cover increases, the average marginal
effect on Percentile_Rank_Combined_NYC becomes more
positive, indicating that neighborhoods with higher vegetative cover
tend to have a higher percentile rank, suggesting they are less
disadvantaged.
# Load necessary package
library(clarify)
# Set seed for reproducibility
set.seed(123)
# Step 1: Simulate coefficients from the model
sim_coefs <- sim(nm0, n = 1000) # Using nm0_fixed without County_binary
# Step 2: Compute Average Marginal Effects (AMEs) for Industrial_Land_Use
sim_est_il <- sim_ame(sim_coefs, var = "Industrial_Land_Use", contrast = "rd", verbose = FALSE)
# Step 3: Display Results
summary(sim_est_il)## Estimate 2.5 % 97.5 %
## E[dY/d(Industrial_Land_Use)] 0.0784 0.0573 0.0991# Step 4: Plot Results
plot(sim_est_il)
This plot shows the average marginal effects of
Industrial_Land_Use on
Percentile_Rank_Combined_NYC. The x-axis represents the
values of Industrial_Land_Use, and the y-axis represents
the average marginal effects. The shaded area around the line indicates
the confidence intervals for the average marginal effects. This plot
helps to visualize how changes in Industrial_Land_Use
impact the predicted values of
Percentile_Rank_Combined_NYC. What we can observe is that
as Industrial_Land_Use increases, the average marginal
effect on Percentile_Rank_Combined_NYC becomes more
negative, indicating that neighborhoods with higher industrial land use
tend to have a lower percentile rank, suggesting they are more
disadvantaged.
Using the sim_setx function from the clarify package
# Load necessary package
library(clarify)
# Set seed for reproducibility
set.seed(123)
# Step 1: Simulate coefficients from the model
sim_coefs <- sim(nm0, n = 1000) # Simulating coefficients from nm0
# ==============================
# Predictions for Low_Vegetative_Cover
# ==============================
# Step 2: Define set values for Low_Vegetative_Cover
scenarios_lv <- list(Low_Vegetative_Cover = c(0.1, 0.9))
# Step 3: Compute Predicted Values
sim_pred_lv <- sim_setx(sim_coefs, x = scenarios_lv)
# Step 4: Extract predictions
summary_lv <- summary(sim_pred_lv)
pred_low_lv <- summary_lv[1, 1] # Prediction at Low_Vegetative_Cover = 0.1
pred_high_lv <- summary_lv[2, 1] # Prediction at Low_Vegetative_Cover = 0.9
# Step 5: Compute First Difference
fd_lv <- pred_high_lv - pred_low_lv
# ==============================
# Predictions for Industrial_Land_Use
# ==============================
# Step 6: Define set values for Industrial_Land_Use
scenarios_il <- list(Industrial_Land_Use = c(0.1, 0.9))
# Step 7: Compute Predicted Values
sim_pred_il <- sim_setx(sim_coefs, x = scenarios_il)
# Step 8: Extract predictions
summary_il <- summary(sim_pred_il)
pred_low_il <- summary_il[1, 1] # Prediction at Industrial_Land_Use = 0.1
pred_high_il <- summary_il[2, 1] # Prediction at Industrial_Land_Use = 0.9
# Step 9: Compute First Difference
fd_il <- pred_high_il - pred_low_il
# ==============================
# Display Results
# ==============================
print(paste("Prediction at Low_Vegetative_Cover = 0.1:", round(pred_low_lv, 3)))## [1] "Prediction at Low_Vegetative_Cover = 0.1: 0.152"print(paste("Prediction at Low_Vegetative_Cover = 0.9:", round(pred_high_lv, 3)))## [1] "Prediction at Low_Vegetative_Cover = 0.9: 0.275"print(paste("First Difference (0.9 - 0.1) for Low_Vegetative_Cover:", round(fd_lv, 3)))## [1] "First Difference (0.9 - 0.1) for Low_Vegetative_Cover: 0.123"print(paste("Prediction at Industrial_Land_Use = 0.1:", round(pred_low_il, 3)))## [1] "Prediction at Industrial_Land_Use = 0.1: 0.206"print(paste("Prediction at Industrial_Land_Use = 0.9:", round(pred_high_il, 3)))## [1] "Prediction at Industrial_Land_Use = 0.9: 0.269"print(paste("First Difference (0.9 - 0.1) for Industrial_Land_Use:", round(fd_il, 3)))## [1] "First Difference (0.9 - 0.1) for Industrial_Land_Use: 0.063"# ==============================
# Plot Results
# ==============================
# Load ggplot2 for visualization
library(ggplot2)
# Create a dataset for plotting
plot_data <- data.frame(
Variable = c("Low Vegetative Cover", "Industrial Land Use"),
Prediction_Low = c(pred_low_lv, pred_low_il),
Prediction_High = c(pred_high_lv, pred_high_il),
First_Difference = c(fd_lv, fd_il)
)
# Plot Predictions
ggplot(plot_data, aes(x = Variable, y = First_Difference)) +
geom_bar(stat = "identity", fill = "steelblue") +
theme_minimal() +
labs(title = "First Differences for Predictors",
y = "Estimated First Difference",
x = "Predictor Variable") +
geom_text(aes(label = round(First_Difference, 3)), vjust = -0.5)
The above graph shows the first differences for the predictors
Low Vegetative Cover and Industrial Land Use.
The x-axis represents the predictor variables, and the y-axis represents
the estimated first differences. The bars indicate the change in
predicted values of Percentile_Rank_Combined_NYC when
moving from a low to a high value of each predictor. The graph helps to
visualize the impact of these predictors on the dependent variable, with
Industrial Land Use showing a more negative effect compared
to Low Vegetative Cover.
# Print the summary to see its structure
summary(sim_pred_lv)## Estimate 2.5 % 97.5 %
## Low_Vegetative_Cover = 0.1 0.152 0.132 0.171
## Low_Vegetative_Cover = 0.9 0.275 0.258 0.291The summary(sim_pred_lv) function call is intended to
provide a summary of the simulated predictions for the
Low_Vegetative_Cover variable. The output will typically
include the predicted values at different levels of
Low_Vegetative_Cover, along with confidence intervals or
other relevant statistics. This summary helps to understand how changes
in Low_Vegetative_Cover impact the predicted values of
Percentile_Rank_Combined_NYC. The results show the
predicted values for Percentile_Rank_Combined_NYC at
different levels of Low_Vegetative_Cover, allowing for a
comparison of the impact of this variable on the dependent variable.