In this video we would like to get an idea of how
to interpret the multiple linear regression model output,
as well as getting a geometric understanding of what the model is doing.
The geometry of this model can be visualized in the following way, with two variables,
x_1 and x_2 over here, being used to predict the output value, y. Values of x_1, x_2,
and y are collected ahead of time and then the least squares models now forms a plane
in the three-dimensional space given by x_1, x_2, and y. That plane will have some points
that lie above it and some points that lie below it,
so that that sum of squares of errors is minimized.
Anytime we make a prediction using this model that prediction will be from a point
on that blue plane, shown here in the diagram.
The interpretation of our model coefficient b_1 is the slope as seen from the perspective
of the x_1 axis, and the b_2 coefficient is the slope seen from the perspective of the x_2 axis.
When we want to interpret what these b_1 and b_2 values, these slope coefficients,
we have to be a little bit more specific than in the case where we were dealing
with just a single input variable.
Let me give you an example.
Imagine we are trying to predict the yield, in micrograms, from our bioreactor.
We have two variables that we are using, volume of the reactor, and temperature.
When we run some experiments, which we will see in a later part of this course,
we might obtain the linear model that Y = b_V times the volume V plus b_T,
times temperature T. The b_V coefficient is the equivalent of b_1,
and the b_T coefficient is the equivalent of b_2.
When using the data, we might find that that slope coefficient, b_V is negative,
indicating that for higher tank volumes, we obtain a lower yield.
The b_T coefficient, the temperature effect, might be positive,
indicating that at higher temperatures we achieve a higher yield.
So we have learned something about our process from that model.
Now that's the general interpretation, which is correct.
Let's go take a look at some numbers and be a bit more specific.
Imagine that we measure the volume, V in litres, and we find that that coefficient is -0.5.
That negative coefficient of -0.5 indicates that the yield is expected to decrease, on average,
by 0.5 micrograms for every one litre increase in the tank volume, holding temperature fixed.
It is very important that we specify that last phrase, "holding temperature fixed."
When we were looking at regression models in the single variable case,
we did not need that extra piece, because there was only one variable in use in the model.
For multiple linear regression, we have to be cognizant of the fact
that multiple variables can affect the output, Y. We must specify that we have
to hold the other variables fixed.
What is your interpretation of the 4.2 coefficient,
as in "4.2 times the temperature, T"?
You should have hopefully said that that coefficient, b_T equal to 4.2,
indicates that the yield will increase by 4.2 micrograms, on average,
for every 1 Kelvin increase in temperature, when holding the tank volume constant.
That's quite a mouthful, but always make sure you have two parts in your interpretation:
that you recognize that it is the increase or decrease in the Y value, on average,
and then always remember to specify that you are holding all the other variables constant.
In this particular course we are not going to look
into the theoretical derivation of the confidence intervals.
But I would like to show how we can calculate them using computer software,
as well as how to interpret them.
Let's say for a moment that when we calculate the confidence interval for the b_V coefficient
of -0.5, we find that it spans zero.
Then we should say, if we were writing this in an message to our colleagues,
that the effect of volume, controlling for temperature, is not significant.
Here again, it is important to add the "controlling for" part.
It indicates that controlled variables were considered in the model.
In other words, that we controlled for temperature, and when controlling for it,
we found that the volume effect was not significant.
So let's wrap up and look at that confidence interval for temperature: it does not span zero.
So our interpretation should be that the effect of temperature, controlling for volume,
is to increase the yield from the reactor by 4.2 micrograms, on average,
for every 1 Kelvin increase in temperature.
And that that is a significant effect.
So this video then shows how to interpret the coefficients and confidence intervals
from the multiple linear regression models.
In the next video, we are going to take our knowledge a step further
and look at integer variables.
The interpretation of these variables is a little bit different,
but the confidence intervals are interpreted in exactly the same way.