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Demonstrating causation

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First, some reminders on

correlation

low correlation

high correlation

image adapted from Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. OTexts.com/fpp3. Accessed on 20 Jul 2024.

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One way to think about the strength of a correlation:

if you know the value of x, how accurately can you predict y?

low correlation

image adapted from Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. OTexts.com/fpp3. Accessed on 20 Jul 2024.

high correlation

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Correlations can also be

non-linear

image source: Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. OTexts.com/fpp3. Accessed on 20 Jul 2024.

Correlation doesn’t necessarily mean that the line on the graph is straight. All sorts of predictable relationships between variables can exist.

(Note to instructors: it can be worthwhile to spend some time asking students to analyze this chart and discuss what conclusions one can draw from it.)

This is a complex graph. The year 2014 was split into half-hour time periods -- 48 per day for the whole year -- and each of those 30-minute periods is represented by a dot on the chart. So some of these dots are from summer and some from winter, some from noon and some from midnight.

The most obvious pattern is that when the temperature is really high, electricity demand is also really high. This is apparent in the linearity of the upper right portion of the chart and the lack of any outliers at lower right. Electricity demand falls as the temperature falls, but then rises again when the temperature gets really low, likely because people begin using electricity to heat their homes.

Also interesting: the correlation between temperature and energy usage is very high when temperatures are very high, but much lower when temperatures are lower. In other words, if the temperature is high, you can predict energy demand pretty accurately, but if the temperature is low, it becomes much tougher to predict energy demand. Above 40 degrees celsius (which equals 104 degrees Fahrenheit), you can be almost certain that energy demand will be between 8.5 and 9.5 gigawatts, but at 10 degrees celsius (50 degrees Fahrenheit), it could be anywhere from 3 to 7 gigawatts. Much of the variability may stem from time of year and time of day – for example, it’s likely that 10-degree winter days use more electricity than 10-degree summer nights. In other words, we are only looking at two variables in a multi-variable system.

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To demonstrate

causation

you need to:

1

2

3

rule out randomness

show the direction of causality

rule out confounds

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1

rule out randomness

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1

rule out randomness

(always an argument from probability)

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Graph by Tyler Vigen (CC-BY-4.0)

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Graph by Tyler Vigen (CC-BY-4.0)

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Your correlation might be spurious if…

you searched through lots of variables looking for a correlation (Tyler Vigen’s database contains over 25,000 variables)

…

Graph by Tyler Vigen (CC-BY-4.0)

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Your correlation might be spurious if…

your sample size is small

(here, n=6)

…

Graph by Tyler Vigen (CC-BY-4.0)

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Your correlation might be spurious if…

you don’t account for outliers

(here, 2014 is an outlier

in both variables)

…

Graph by Tyler Vigen (CC-BY-4.0)

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Your correlation might be artificially amplified if…

you use line charts.

…

Graph by Tyler Vigen (CC-BY-4.0)

Ask students whether they can explain why the statement on the slide might be true. (Students respond and discuss)

Takeaway: Line charts are only appropriate for continuous variables – that is, variables where the thing being measured must move through intermediate values on its way from one data point to the next. Temperature is a continuous variable – if it was 60 degrees at 8 AM and it’s 90 degrees at noon, you know that at some point during the morning it must have been 70 degrees. But if Russell Crowe appeared in 2 movies in 2006 and 4 movies in 2007, it makes no sense to say that in between 2006 and 2007 he appeared in 3 movies. But that is what the line chart implies. This chart has only 16 data points for each variable. Drawing lines between the points makes the correlation seem much finer-grained than it actually is.

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1

rule out randomness

usually requires a randomized controlled study

and probability models typically only apply to random samples and randomized experiments

because making an argument from probability requires applying a probability model

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source: xkcd

CC-BY-NC 2.5

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source: xkcd

CC-BY-NC 2.5

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source: xkcd

CC-BY-NC 2.5

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source: xkcd

CC-BY-NC 2.5

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source: xkcd

CC-BY-NC 2.5

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2

rule out confounds

To demonstrate

causation

you need to:

(third variables that

affect both A and B)

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violent crimes reported

ice cream consumption

correlation: 0.74

Here’s a very famous spurious correlation – it’s been known for years that ice cream consumption and violent crime track each other closely. But unlike the spurious correlations we saw earlier, this one isn’t random. Ice cream and crime are likely to covary next year, and the year after that, because they are both being affected by a third variable. Can anyone think what it is? (Students respond and discuss)

(Monthly crime data for 2015 from the FBI Uniform Crime Reporting Section, accessed from https://www.crimeinamerica.net/top-months-for-violent-and-property-crime. Data for December are left out because some agencies report only once a year in December, inflating that month’s total. Monthly data for ice cream consumption for 1951-1953 from Hildreth, C. and J. Lu (1960) Demand relations with autocorrelated disturbances, Technical Bulletin No 2765, Michigan State University.)

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violent crimes reported

ice cream consumption

temperature

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ruling out confounds requires

deep knowledge of the subject

violent crimes reported

ice cream consumption

temperature

careful study design

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To demonstrate

causation

you need to:

3

show the direction of causality

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3

show the direction of causality

getting hungry

eating

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violent crimes

ice cream consumption

temperature

3

show the direction of causality

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global warming

greenhouse gas emissions

3

show the direction of causality

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global warming

greenhouse gas emissions

3

show the direction of causality

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amount of grass

3

show the direction of causality

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amount of grass

new grass seeding

increases

3

show the direction of causality

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amount of grass

new grass seeding

increases

3

show the direction of causality

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amount of grass

new grass seeding

soil erosion

increases

decreases

3

show the direction of causality

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amount of grass

number of grazers

decreases

increases

3

show the direction of causality

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amount of grass

new grass seeding

soil erosion

number of grazers

increases

decreases

increases

decreases

3

show the direction of causality

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show the direction of causality

amount of grass

new grass seeding

soil erosion

number of grazers

increases

decreases

increases

decreases

hint: it’s usually part of a complex system

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takeaway:

we can measure correlation

mathematically

but causation requires

logical / philosophical inference

(i.e. a deep understanding of complex systems,

data, experimental design, and statistics)