Section B.1 Motivating Regression
Now, imagine that we play a game. Imagine I have all the respondents waiting in a room, and I randomly call one of them to the stage. You’re sitting in the audience, and you have to guess the level of homicide (HR90) for that respondent. Imagine that I pay \(£150\) to the student that gets the closest to the right value. What would you guess if you only have one guess and you knew (as we do) how homicide in the 90s is distributed?
ggplot(ncovr, aes(x = HR90)) +
geom_density() +
geom_vline(xintercept = median(ncovr$HR90), linetype = "dashed", size = 1,
color="red") + # median = 4.377
geom_vline(xintercept = mean(ncovr$HR90), linetype = "dashed", size = 1,
color="blue") + # mean = 6.183
ggtitle("Density estimate, mean and median of homicide rate 1990")
summary(ncovr$HR90)
## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 0.00 1.33 4.38 6.18 8.94 71.38
If I only had one shot, you could go for the median (given the skew) but the mean perhaps would be your second best. Most of the areas here have values clustered around those values, which is another way of saying they are bound to be not too far from them.
Imagine, however, that now when someone is called to the stage, you are told the level of resource deprivation in the county — so the value of the RD90 variable for the individual that has been selected (for example 4). Imagine as well that you have the scatterplot that we produced earlier in front of you. Would you still go for the value of “4.377” as your best guess for the value of the selected county?
I certainly would not go with the overall mean or median as my prediction anymore. If somebody said to me: the value RD90 for the selected respondent is 4, I would be more inclined to guess the mean value for the level of homicide with that level of resource deprivation (the conditional mean), rather than the overall mean across all the counties. Wouldn’t you?
If we plot the conditional means, we can see that the mean of homicide rate for counties that report a value of 4 in RD90 is around 22. So you may be better off guessing that.
library(grid)
ggplot() +
geom_point(data=ncovr, aes(x = RD90, y = HR90), alpha=.2) +
geom_line(data=ncovr, aes(x = round(RD90/0.12)*0.12, y = HR90),
stat='summary', fun.y=mean, color="red", size=1) +
annotate("segment", x=3, xend = 4, y = 25, yend= 22, color = "blue",
size = 2, arrow = arrow()) +
annotate("text", x = 3, y = 29, label = "Pick this one!",
size =7, colour = "blue")
Linear regression tackles this problem using a slightly different approach. Rather than focusing on the conditional mean (smoothed or not), it draws a straight line that tries to capture the trend in the data. If we focus on the regions of the scatterplot that are less sparse, we see that this is an upward trend, suggesting that as resource deprivation increases, so does the homicide rate.
Simple linear regression draws a single straight line of predicted values as the model for the data. This line would be a model, a simplification of the real world like any other model (e.g., toy pistol, architectural drawing, subway map), that assumes that there is approximately a linear relationship between X and Y. Let’s draw the regression line using the function
geom_smooth(). This asks for a method= parameter where we specify lm for the linear regression line. We also set se= to FALSE, so we ask for just the line to be printed but not the standard error around it. The other arguments specify the colour and thickness of the line.
ggplot(data = ncovr, aes(x = RD90, y = HR90)) +
geom_point(alpha = .2) +
geom_smooth(method = "lm", se = FALSE, color = "red", size = 1)
What that line is doing is giving you guesses (predictions) for the values of homicide based on the information that we have about the level of resource deprivation. It gives you one possible guess for the value of homicide for every possible value of resource deprivation and links them all together in a straight line.
The linear model then is a model that takes the form of the equation of a straight line through the data. The line does not go through all the points.
Our regression line underpredicts at low levels of resource deprivation and does not seem to capture well the variability at higher levels of resource deprivation. But imperfect as a model as it might be it simplifies well the overall growing trend for homicide as resource deprivation increases.
As [246] highlight: “like all models of the real world, the line will be wrong, wrong in the sense that it can’t match reality exactly. But it can help us understand how the variables are associated” (p. 179). A map is never a perfect representation of the world, the same happens with statistical models. Yet, as with maps, models can be helpful.
