Lesson 8: Relationships Between Variables
Tyler Simko
05 July, 2021
1. What is a relationship anyway?
You will often hear that two variables of some kind are “related,” “linked,” or “associated” in some way. What does this mean?
Well, we generally mean that the relationship between them is predictable in some way. For example:
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p1 <- tibble(x = rnorm(1000, mean = 5, sd = 2),
y = x + rnorm(1000, mean = 5, sd = 2)) %>%
ggplot(aes(x, y)) +
geom_point() +
geom_smooth(method = "lm", col = "darkred") +
theme_bw() +
labs(subtitle = "As x increases, y tends to increase too.")
p2 <- tibble(x = rnorm(1000, mean = 5, sd = 2),
y = rnorm(1000, mean = 5, sd = 2) - x) %>%
ggplot(aes(x, y)) +
geom_point() +
geom_smooth(method = "lm", col = "darkred") +
theme_bw() +
labs(subtitle = "As x increases, y tends to decrease.")
p3 <- tibble(x = rnorm(1000, mean = 5, sd = 2),
y = sin(x) + rnorm(1000, mean = 1, sd = 0.5)) %>%
ggplot(aes(x, y)) +
geom_point() +
geom_smooth(col = "darkred") +
theme_bw() +
labs(subtitle = "As x increases, y changes in non-linear way.")
p4 <- tibble(x = rnorm(1000, mean = 2, sd = 2),
y = x^3 + rnorm(1000, mean = 50, sd = 25)) %>%
ggplot(aes(x, y)) +
geom_point() +
geom_smooth(col = "darkred") +
theme_bw() +
labs(subtitle = "As x increases, y increases non-linearly.")
p5 <- tibble(x = rnorm(1000, mean = 5, sd = 2),
y = rnorm(1000, mean = 5, sd = 2)) %>%
ggplot(aes(x, y)) +
geom_point() +
geom_smooth(method = "lm", col = "darkred") +
theme_bw() +
labs(subtitle = "No clear relationship between x and y.")
(p1 + p2) / (p3 + p4)## `geom_smooth()` using formula 'y ~ x'## `geom_smooth()` using formula 'y ~ x'## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'
## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'
## `geom_smooth()` using formula 'y ~ x'
2. Correlation vs. Causation
In scientific research (and for your final project), we are often interested in the relationship between several variables. For example, smoking and cancer or school funding and student outcomes.
However, just because we observe a clear relationship between two variables does not mean that they are causally related.
That is, correlation is not causation. This mistake can be very tempting when conducting your own analyses. However, some silly examples make it clear. Here are 3 from a website called Spurious Correlations1:
Often, seeming relationships will actually be caused by confounders. A confounding variable is one that is causally related to both your “explanatory” and “dependent” variables.2
This diagram is one way to visualize confounding. There may be no true relationship between X and Y, but a third variable Z can be causing both X and Y.
For example, imagine you notice that people who drink more bottles of Gatorade per week tend to live longer. You get hired by Gatorade to publish some beautiful plots made in R. The company CEO asks you - “There is a clear relationship between the number of pairs of running shoes a person owns and the number of years that they live. Is this causal?”
After taking this course, you should be prepared to say no! This is a case of confounding. In this situation, exercise is related to both drinking Gatorade and Life Expectancy.
Correlation is a statistical concept as well. Correlation measures the linear relationship between two variables, and is bounded between -1 and 1.
A correlation of 1 means that when one variable increases, the other always increases as well.
A correlation of -1 means that when one variable increases, the other always decreases. Values closer to 1 mean the relationship is stronger.
A correlation of 0 means there is no clear linear relationship. When one variable increases, the other may increase or decrease with no clear pattern.
This image from Wikipedia is helpful:3
The cor() function in R calculates correlations between two variables. The use = "complete.obs" argument means to only include rows which have both variables in it (and no NAs).
##
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## )## [1] 0.703137## [1] 0.2456559## [1] -0.0071616963. Controlling for Variables
You may have heard about “controlling” for a variable. Typically, controlling is done with a statistical model. These are beyond the scope of this course (although please ask me if you are interested!), but we can still understand conceptually what is happening when a variable is controlled for.
In the plot below, we see a clear upward trend between the number of Gatorades drunk per week and life expectancy. This is a potential relationship between two variables:
set.seed(80)
x <- tibble(exercise = rnorm(10000, mean = 40, sd = 8),
gatorade = rbinom(10000, size = 10,
prob = (exercise / max(exercise)) - 0.3 +
rnorm(10000, mean = 0.2, sd = 0.01)),
years = rnorm(10000, 70, sd = 6) + exercise / 4)
q <- quantile(x$exercise, probs = c(0.25, 0.5, 0.75, 1.0))
x %>%
ggplot(aes(x = gatorade, y = years, group = gatorade)) +
geom_boxplot() +
geom_smooth(data = x,
aes(x = gatorade, y = years, group = 1),
col = "darkred") +
labs(title = "Potential relationship between Gatorade and Life Expectancy",
x = "# of Gatorades / week",
y = "Years Lived") +
scale_x_continuous(breaks = 0:10,
labels = 0:10 %>% as.character()) +
theme_bw()## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'
However, what happens if we “control” for exercise? Well, in principle this means we want to ensure that we only compare people who exercise at similar rates. We can do that with plotting techniques that we already know:
x <- x %>%
mutate(income_label = case_when(exercise <= q[1] ~ "No Exercise",
exercise <= q[2] ~ "Some Exercise",
exercise <= q[3] ~ "Above Avg. Exercise",
exercise <= q[4] ~ "Fitness Fanatics"),
income_label = factor(income_label, levels = c("No Exercise",
"Some Exercise",
"Above Avg. Exercise",
"Fitness Fanatics")))
x %>% ggplot(aes(gatorade, years)) +
geom_jitter() +
geom_smooth(col = "darkred", method = "loess") +
theme_bw() +
facet_wrap(~income_label, nrow = 1) +
labs(subtitle = "Adjusting for a confounder can breakdown spurious correlations.",
x = "# of Gatorade / Week",
y = "Years Lived") +
theme_bw()## `geom_smooth()` using formula 'y ~ x'
Once we “control” for exercise by only comparing people who exercise at similar rates, the relationships between Gatorade and Life Expectancy within each group (the red lines) are essentially flat.
Exercises
Mistaking correlations for causation is one of the most common mistakes made in the social sciences. For your project, think of a plausible relationship between two variables. Use your dataset and create a plot with the relationship.
Now, try to think of a possible confounder in this relationship. Is there a variable that could be causing both your treatment and your outcome?
Examples from the following site.↩︎
For a reminder of what these terms mean, revisit the final project instructions.↩︎
