Question

Now, calculate the model equation for that line of best fit, using your dataframe avg and the lm() function. What rate of emissions was an average Japanese city projected to produce in the year 0 CE? How much does the average carbon footprint increase with every passing year, according to your model’s beta coeficient?

Answer

m <- avg %>%
  lm(formula = footprint ~ year)

m

## 
## Call:
## lm(formula = footprint ~ year, data = .)
## 
## Coefficients:
## (Intercept)         year  
##   -307.2731       0.1583

The average Japanese city was projected to produce -307.27 kilotons of carbon emissions per 1000 residents in the year CE. With every passing year, the average carbon footprint increases by 0.16 kilotons per 1000 residents.

Question

Examine the model table, using summary() or get_regression_table(). How likely is it that these two statistics were just that extreme by chance?

Answer

m %>%
  get_regression_table()

The alpha coefficient, -307.27, was so extreme that there is just a 0.036 probability (3.6% chance) that this statistic occurred due to chance. The effect of each year (beta = 0.16) was so extreme that there is just a 0.031 probability (3.1% chance) that this statistic occurred due to chance. Both effects are therefore statistical significant at the p < 0.05 level.

Question

Evaluate this model, using summary() or glance(). What percentage of variation in average carbon footprints does it explain? Does it fit better than an intercept model? Is this model’s F statistic statistically significant? How do you know?

Answer

glance(m)

This model explained 39% of the variation in average carbon footprints (\(R^{2}\) = 0.39). It fits much better than an intercept model, according to its F-statistic value of ``6.26. In fact, there’s just a 0.031 probability (3.1% chance) that this statistic occurred due to chance, so we can be at least 96.9% confident that this statistic did not occur due to chance. This model has a statistically significant fit.

Question

Using filter() and arrange(), find the names of the 3 municipalities with the largest population in Japan in 2017.

Then, for those 3 largest municipalities in Japan, visualize the relationship between year and carbon footprints, adding a separate line of best fit for each city. You can do this using facet_wrap(), aes(color = ...), or both! How do these trends compare with the average city in Japan?

Answer

jp %>%
  filter(year == 2017) %>%
  arrange(desc(pop)) %>%
  head(3)
# write down names of top 3 cities

jp %>%
  # Filter to top 3 cities
  filter(muni %in% c("Kanagawa-ken Yokohama-shi", 
                     "Osaka-fu Osaka-shi",
                     "Aichi-ken Nagoya-shi") ) %>%
  # plot them
  ggplot(mapping = aes(x = year, y = footprint,
                       color = muni)) +
  geom_point() +
  facet_wrap(~muni) +
  # Add line of best fit!
  geom_smooth(method = "lm", se = FALSE) +
  # if you every want to get rid of a legend, 
  # just say 'attribute' = "none"
  guides(color = "none") +
  theme_classic() +
  labs(x = "Year", y = "Carbon Emissions\nper 1000 residents")

Looks like Nagoya has sharply reduced their emissions over time, while Yokohama and Osaka have seen much variation. Though Yokohama and Osaka reduced their emissions recently in 2015, it’s just a recent trend; the line of best fit still predicts increases in emissions based on their past.

Question

Now, filtering to each of these 3 municipalities separately, please calculate the model equation for your lines of best fit. (You’ll need to use filter() and lm() 3 times. You can’t use group_by() for this!) Report the city name, alpha coefficient, beta coefficients, and R-squared for each model in the table below. For Alpha, Beta, and the F-statistic, please report the p-value too in parentheses.

What story does this table tell? In which city did each passing year decrease its carbon footprint the most? (See beta). Which city’s emissions are best predicted by these models? Which city’s emissions are poorly captured by these models?

Answer

m1 <- jp %>%
  filter(muni == "Kanagawa-ken Yokohama-shi") %>%
  lm(formula = footprint ~ year)

m1 %>% summary()

m2 <- jp %>%
  filter(muni == "Osaka-fu Osaka-shi") %>%
  lm(formula = footprint ~ year)

m2 %>% summary()

m3 <- jp %>%
  filter(muni == "Aichi-ken Nagoya-shi") %>%
  lm(formula = footprint ~ year)
m3 %>% summary()

Nagoya decreased its emissions the most each year, at -0.07 fewer kilotons of emissions per 1000 residents each year. That effect was statistically significant (p < 0.01). In contrast, Osaka increased its carbon footprint at a rate of 0.09 kilotons of emissions per 1000 residents each year. That effect was somewhat statistically significant (p < 0.10). Finally, Yokohama increased its emissions slightly over time, at 0.03 kilotons per 1000 residents a year. However, that effect was not statistically significant (p > 0.240). The best fitting model was for Nagoya, as it explained 51% of the variation in emissions rates.