Multiple regression - Interpreting Coefficients
Hi. I'm Dr Nic, and in this video I'm going to explain how to interpret coefficients in
regression models. This video follows on from Multiple Regression, Overview.
And from Multiple regression - dummy or indicator variables
Check in the description below for more videos about multiple regression.
A regression model is like a formula. For example the Prius price prediction
model introduced in the Overview video was: Price = 23340 - 1028 Age - 25 Distance
The Income model in the dummy variable video was: Income = -58200 + 2700 Age + 34300 Degree
These models were developed from sample data and help us to understand the relationship between
the predictor or independent variables and the response or dependent variable.
It is important to be able to interpret the coefficients of the predictor variables in the model.
In the Prius prediction model, the value , negative 1028, which is the coefficient of Age
tells us that for each year older that a car is, the selling price is $1028 less…
all other things being equal.
The coefficient of distance, negative 25 means that for every extra unit of distance,
in this case one thousand kilometres, that the car has travelled, the price is decreased by $25, all other things being equal.
There are several things to consider when interpreting regression coefficients
They are: Observation, units, right way around, other things equal and causation.
First, observation. The term "observation" refers to the thing that the model applies to,
such as one car or one person. Here are some examples of different
observations or data points - you can think of them as lines of data in a spreadsheet.
For the Prius model, an observation is a car for sale.
For the Income model, an observation is a person whose income we are interested in.
An observation could be a branch of a business - to find out what
factors are associated with profitability.
An observation could be a single orange, when we are examining juice yield.
If we were looking at electricity consumption, an observation might be a single household.
An observation could be a period of time such as a month if we were looking at influences on sales,
or it could be a district in a city or town if we were studying health or crime.
Units Next we need to think
about the units in which things are measured. In the Prius model, the unit for age was one year,
the unit for distance was one thousand km and the unit for price was New Zealand dollars.
In the Income model, the unit for age is one year, income was in dollars,
and Degree is a dummy or indicator variable that expresses the presence or absence of a degree qualification.
When the units are percentages and rates, it can make interpretation more challenging.
For example, attempts to explain health outcomes may use data on regions or towns.
For each observation there might be number of heart attacks per 10,000 population in a year
as the dependent variable. A potential predictor variable could
be the percentage of green space in the area. Say the coefficient of greenspace was negative 7,
that would be interpreted that for every increase of one percent of green space in a region,
the number of heart attacks per 10,000 people decreases by 7 heart attacks per year.
Note that the green space is not increasing by 1 percent
but rather the percentage of green space is one percent higher - for example going from 4% to 5%.
Similarly, the number of heart attacks is not going down by seven, but rather by 7 for each
10,000 people in the area.
Right way around.
It is very easy to get this confused and I have seen many many students confuse this.
It is the predictor variable - the one inside the formula that increases by one,
to give an increase of its coefficient value in the dependent or result variable.
So when we interpret the Prius model we say: For every increase of one unit of age,
the price increases by negative 1028 dollars. Here is another example
Oranges are weighed in grams and then squeezed and the weight of the juice in grams is also recorded.
The regression model is: Juice = 0.44*whole -25
This means that for each extra gram of orange, we would expect about 0.44 grams more juice.
Other things equal, or its fancier Latin version ceteris paribus
means that we are assuming that nothing else changes when the value of the predictor variable changes.
For example with the income model we say that if the two people are identical but one has a degree but the other does not,
then the difference in income would be predicted to be around $34,300.
However, if one were older, or one was a man and the other a woman, or one had a disability
or a myriad other possibilities, then the $34,300 would not apply.
You always need to think hard about the sensible interpretation of a regression model.
Which brings us to causation. Causation is a big idea and
one we always need to be aware of with interpreting regression coefficients.
You can see more about this in my videos listed in the description below.
For example it is not unusual to have a model of crime rate,
where an increased amount spent on policing is associated with a higher level of crime.
People can jump to the wrong conclusion that increasing funding on policing is
going to cause an increase in the crime rate. This makes no sense.
A possible explanation is that areas that have a high crime rate for other reasons
have a higher police budget in response to the crime rate, not the other way around.
A higher police budget could be an indicator of a higher crime rate without being a cause.
So when we interpret regression coefficients we need to be careful to choose our words so as not to imply causation.
For example in the income model, we found that the age coefficient was 2700.
It would be incorrect to say, each year a person gets older causes their income to increase by $2700.
A better way to put it is:
An additional year of age is associated with an extra income
of $2700, all other things being equal.
This video has been about interpreting regression coefficients and is one of several videos about multiple regression.
If you have other aspects you would like covered, please comment below and I will see what I can do.
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