Hello, it's Margriet here.

In today's video,

I would like to talk you through

how to build a linear model in R.

To do that, we will use some data

from Miller and Haden, which you

might remember from a previous video.

It contains information from 25 participants,

25 children on their reading ability,

their IQ, how often they read books at

home and how often they watch TV at home.

And here you just see the scores

of the first six participants.

Now, the usual thing about,

um, building a linear model,

is that it allows you to figure out

whether you can say something sensible

sensible about the dependent variable,

which in our example is

going to be reading ability.

So can you say something sensible about

reading ability based on IQ as a predictor?

So we're going to fit a linear model

to see whether if we know a child's IQ,

whether we can reliably say

something or predict something

about their reading ability.

Now to start off with,

it is always useful to visualise your data,

and for that we can use a scatterplot (which

you have encountered in previous lectures).

So here we go, a scatter plot with reading

ability on the Y axis and IQ on the X axis.

Here and here we have one

data point for each child.

So this child here has an IQ of 100 and a

reading score of something like 46 or 47,

something like that.

So one data point for each child.

And then what the process of fitting

the model does is that it estimates

this line of best fit and,

if it then test whether the predictor

(IQ) allows you to say something

sensible about the dependent

variable (reading ability).

Now you might remember this

formula from the previous

lecture and it defines a line.

So here we have the dependent variable Y.

And so here we have Y which

is the dependent variable.

So reading ability in our case.

We have the predictor

X (IQ in our case).

And what linear regression does is that

it estimates these parameters so that it

allows you to define this line of best fit.

We have beta zero. The intercept,

that is where the line crosses the Y

axis at X is 0 and we have beta one

which is the weight for the predictor

which defines the slope of the line.

And of course there is also

some error in the data,

so that's the the E here that we

don't need to do anything about that.

Now the way you build the

model in R is somewhat similar to this.

Um, so here you can see the function.

So we used the function lm().

Uh, and then you need to tell lm() a

few things so that it can do its job.

First of all,

you need to tell it which data to use.

So here we say data equals 'mh'

because that's what I've called

the Miller Haden data frame.

And then here, before the comma,

you tell it what you want to model.

And um, you do that as follows.

So here we have a little squiggle in the

middle and the dependent variable sits

to the left of the squiggle.

So here reading ability and the

predictor or predictor in case of

multiple regression sits on the

right of that squiggle and you

can read that as follows.

You can say model reading

ability as a function of IQ.

Now the next thing you

would do is that um ohh.

So here we have that function and we

assign the output of that function to

an object called 'mod' for model.

It could have called that something else now.

Then we used a second function called

summary() to pull out the relevant

information from the output.

So we say run summary(),

give us a summary of our model and

assign it to a new object which I

have just called mod underscore

summary and then I just call that

and and then uh we get the output.

So this is what the output looks like.

This is what you will see in the

console if you've typed that and

pressed enter or if you've run that

line of code of your in your script.

First of all, in the output it helpfully

reminds you what you've actually modelled.

So here you can see what data you've

used and what model you've asked for.

Always helpful to check.

And then let's first look at the bottom.

So here you have the statistics that give

you information about the overall model.

So overall, the overall model

statistics,

is the F statistic and and that has a

certain number of degrees of freedom.

And here we have a P value

for the overall model.

Actually you can see this P value is

smaller than .05, which means that

the model as a whole is significant.

So, um, all the predictors that you've put

in your model together say something, um,

sensible about your dependent variable.

In that case, we only have one predictor,

but you could have more predictors.

Um, the other useful thing that it gives

you here at the bottom is multiple

R-squared or adjusted R-squared.

And it is in most cases,

it is better to look at the adjusted squared,

which tells you how much variance

the model as a whole explains.

So how much of the variance in

your dependent variable.

So in our case,

how much of the variance in reading

ability is explained by whatever we've

put in our model. So in our case, IQ.

So you can see that it is .1689,

similar to what we saw for correlation,

you can multiply it by 100 and then you

have the percent variance accounted for.

So here that would be about 17%

of the variance.

The final bit that I want to talk

about today is this coefficients table.

So here you have the

estimates of the coefficients

to specify the line.

So here you have to intercept,

so this is where the line of best

fit crosses the Y axis at X is 0,

so this is beta zero.

And here you have the beta one.

So you can see that's

for IQ that is

.3036. Sorry about 3.

You can also see it's positive,

which suggests that there is a positive

relationship between IQ and our

dependent variable reading ability.

Um, it gives you more information.

It also gives you the standard error

of that estimate and the T value

associated with it and the probability

that the T value is significant.

So this is the significance of our predictor.

And as you can see, it has a little star,

which means that it is significant

at the .05 level.

We can see that because it is .0236,

so clearly smaller than

.5 so this significant.

So from that you can conclude that IQ is a

significant predictor of reading ability.

Now, if you would have more than

one predictor in this model,

some of the predictors might be

significant and others might not.

OK, so that's what the output looks like,

and that gives you all the information

that you need to report as well.

Um, now it's also important

to check a few assumptions

when you've fitted your model. So

previously with with correlation and stuff,

we checked the assumptions beforehand.

With fitting model,

you first fit the model and then check

whether it meets the assumptions.

So the first assumption is that

there is a linear relationship

between IQ and reading ability,

and this CR plot tells you as much.

So as long as this pink line sits

close enough to the dashed blue line,

it suggests that that the relationship

between the two is linear.

The second assumption asks you whether

the residuals are normally distributed.

Now this is an important one to get right.

When it comes to linear regression,

we don't care so much about the

distribution of the variable we

or variables, we care about the

distribution of the residuals.

So you can do that by creating a QQ

plot and and the residuals are actually

also stored in this object that we

created when we ran the 'lm' function.

So we say, OK, take that object,

go to the residuals and use that to

create a quantile-quantile, or QQ plot.

So QQ plot is the function to do that.

And mind the capital P here.

This is what you get.

So as long as these data points fall

quite closely to this blue line and

kind of within this shaded blue area,

you can conclude that the residuals

are normally distributed.

Now finally we want to check whether

the residuals, again the residuals,

show constant variance, or also

referred to as homoscedasticity.

That basically means that the variance.

So if we look kind of in the middle here,

the variance here is between this

and bottom value and this top value.

So here we we we have this amount

of variance and we basically

want to know whether across all

the values that we have fitted.

So between just under 50 and just above 60,

the variance is roughly similar.

So you know the data points fall similarly

far from this midline um and that's

also indicated kind of by this,

this, this blue line.

And you can see that the variance

at the extremes is not quite

the same as in the middle.

But overall from this plot you would

still conclude that this is close enough,

so that the residuals show

homoscedasticity or constant variance.

So from this you can conclude that

it meets all the assumptions of

a linear regression model so that

your model is valid. Here you

have the code to do that.

So again we take our object that

we've called 'mod', which is the

output of the linear regression and

we say residualPlot with a capital

P and then you get this, this plot.

Now finally, um, how would you report that,

um, this is how you would report it.

So you would say something along the

lines of "A simple linear regression

was performed with reading ability

as the outcome variable and IQ

as the predictor variable."

And here you can see that our report

the means and standard deviations

of each of these variables.

"The results of the regression indicated

that the model significantly predicted

the reading ability ..." and then you

give the overall model statistics,

so the F value with the 1 and 23 degrees of

freedom, and then the exact P value of .024.

And here I've reported the R-squared

rather than the adjusted R square because

there's only one predictor that was .2,

accounting for 20% of the variance.

"IQ was a significant positive predictor ...",

so than a report the estimate

of beta one and its P value.

So this is the way that the APA would

like you to report a regression.

OK. Thank you very much.