Corr-and-Regress (1).ppt

Correlation and
Regression
Davina Bristow &
Angela Quayle

Topics Covered:
 Is there a relationship between x and y?
 What is the strength of this relationship
 Pearson’s r
 Can we describe this relationship and use this to predict y from
x?
 Regression
 Is the relationship we have described statistically significant?
 t test
 Relevance to SPM
 GLM

The relationship between x and y
 Correlation: is there a relationship between 2
variables?
 Regression: how well a certain independent
variable predict dependent variable?
 CORRELATION  CAUSATION
In order to infer causality: manipulate independent
variable and observe effect on dependent variable

Scattergrams
Y
X
Y
X
Y
X
Y
Y Y
Positive correlation Negative correlation No correlation

Variance vs Covariance
 First, a note on your sample:
 If you’re wishing to assume that your sample is
representative of the general population (RANDOM
EFFECTS MODEL), use the degrees of freedom (n – 1)
in your calculations of variance or covariance.
 But if you’re simply wanting to assess your current
sample (FIXED EFFECTS MODEL), substitute n for
the degrees of freedom.

Variance vs Covariance
 Do two variables change together?
1
)
)(
(
)
,
cov( 1






n
y
y
x
x
y
x
i
n
i
i
Covariance:
• Gives information on the degree to
which two variables vary together.
• Note how similar the covariance is to
variance: the equation simply
multiplies x’s error scores by y’s error
scores as opposed to squaring x’s error
scores.
1
)
( 2
1
2





n
x
x
S
n
i
i
x
Variance:
• Gives information on variability of a
single variable.

Covariance
 When X and Y : cov (x,y) = pos.
 When X and Y : cov (x,y) = neg.
 When no constant relationship: cov (x,y) = 0
1
)
)(
(
)
,
cov( 1






n
y
y
x
x
y
x
i
n
i
i

Example Covariance
x y x
xi
 y
yi
 ( x
i
x  )( y
i
y  )
0 3 -3 0 0
2 2 -1 -1 1
3 4 0 1 0
4 0 1 -3 -3
6 6 3 3 9
3

x 3

y  7
75
.
1
4
7
1
))
)(
(
)
,
cov( 1








n
y
y
x
x
y
x
i
n
i
i What does this
number tell us?
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7

Problem with Covariance:
 The value obtained by covariance is dependent on the size of
the data’s standard deviations: if large, the value will be
greater than if small… even if the relationship between x and y
is exactly the same in the large versus small standard
deviation datasets.

Example of how covariance value
relies on variance
High variance data Low variance data
Subject x y x error * y
error
x y X error * y
error
1 101 100 2500 54 53 9
2 81 80 900 53 52 4
3 61 60 100 52 51 1
4 51 50 0 51 50 0
5 41 40 100 50 49 1
6 21 20 900 49 48 4
7 1 0 2500 48 47 9
Mean 51 50 51 50
Sum of x error * y error : 7000 Sum of x error * y error : 28
Covariance: 1166.67 Covariance: 4.67

Solution: Pearson’s r
 Covariance does not really tell us anything
 Solution: standardise this measure
 Pearson’s R: standardises the covariance value.
 Divides the covariance by the multiplied standard deviations of
X and Y:
y
x
xy
s
s
y
x
r
)
,
cov(


Pearson’s R continued
1
)
)(
(
)
,
cov( 1






n
y
y
x
x
y
x
i
n
i
i
y
x
i
n
i
i
xy
s
s
n
y
y
x
x
r
)
1
(
)
)(
(
1






1
*
1




n
Z
Z
r
n
i
y
x
xy
i
i

Limitations of r
 When r = 1 or r = -1:
 We can predict y from x with certainty
 all data points are on a straight line: y = ax + b
 r is actually
 r = true r of whole population
 = estimate of r based on data
 r is very sensitive to extreme values:
0
1
2
3
4
5
0 1 2 3 4 5 6
r̂
r̂

Regression
 Correlation tells you if there is an association
between x and y but it doesn’t describe the
relationship or allow you to predict one
variable from the other.
 To do this we need REGRESSION!

Best-fit Line
= ŷ, predicted value
 Aim of linear regression is to fit a straight line, ŷ = ax + b, to data that
gives best prediction of y for any value of x
 This will be the line that
minimises distance between
data and fitted line, i.e.
the residuals
intercept
ε
ŷ = ax + b
ε = residual error
= y i , true value
slope

Least Squares Regression
 To find the best line we must minimise the sum of
the squares of the residuals (the vertical distances
from the data points to our line)
Residual (ε) = y - ŷ
Sum of squares of residuals = Σ (y – ŷ)2
Model line: ŷ = ax + b
 we must find values of a and b that minimise
Σ (y – ŷ)2
a = slope, b = intercept

Finding b
 First we find the value of b that gives the min
sum of squares
ε ε
b
b
b
 Trying different values of b is equivalent to
shifting the line up and down the scatter plot

Finding a
 Now we find the value of a that gives the min
sum of squares
b b b
 Trying out different values of a is equivalent to
changing the slope of the line, while b stays
constant

Minimising sums of squares
 Need to minimise Σ(y–ŷ)2
 ŷ = ax + b
 so need to minimise:
Σ(y - ax - b)2
 If we plot the sums of squares
for all different values of a and b
we get a parabola, because it is a
squared term
 So the min sum of squares is at
the bottom of the curve, where
the gradient is zero.
Values of a and b
sums
of
squares
(S)
Gradient = 0
min S

The maths bit
 The min sum of squares is at the bottom of the curve
where the gradient = 0
 So we can find a and b that give min sum of squares
by taking partial derivatives of Σ(y - ax - b)2 with
respect to a and b separately
 Then we solve these for 0 to give us the values of a
and b that give the min sum of squares

The solution
 Doing this gives the following equations for a and b:
a =
r sy
sx
r = correlation coefficient of x and y
sy = standard deviation of y
sx = standard deviation of x
 From you can see that:
 A low correlation coefficient gives a flatter slope (small value of
a)
 Large spread of y, i.e. high standard deviation, results in a
steeper slope (high value of a)
 Large spread of x, i.e. high standard deviation, results in a flatter
slope (high value of a)

The solution cont.
 Our model equation is ŷ = ax + b
 This line must pass through the mean so:
y = ax + b b = y – ax
 We can put our equation for a into this giving:
b = y – ax
b = y -
r sy
sx
r = correlation coefficient of x and y
sy = standard deviation of y
sx = standard deviation of x
x
 The smaller the correlation, the closer the
intercept is to the mean of y

Back to the model
 If the correlation is zero, we will simply predict the mean of y for every
value of x, and our regression line is just a flat straight line crossing the
x-axis at y
 But this isn’t very useful.
 We can calculate the regression line for any data, but the important
question is how well does this line fit the data, or how good is it at
predicting y from x
ŷ = ax + b =
r sy
sx
r sy
sx
x + y - x
r sy
sx
ŷ = (x – x) + y
Rearranges to:
a b
a a

How good is our model?
 Total variance of y: sy
2 =
∑(y – y)2
n - 1
SSy
dfy
=
 Variance of predicted y values (ŷ):
 Error variance:
sŷ
2 =
∑(ŷ – y)2
n - 1
SSpred
dfŷ
=
This is the variance
explained by our
regression model
serror
2 =
∑(y – ŷ)2
n - 2
SSer
dfer
=
This is the variance of the error
between our predicted y values and
the actual y values, and thus is the
variance in y that is NOT explained
by the regression model

 Total variance = predicted variance + error variance
sy
2 = sŷ
2 + ser
2
 Conveniently, via some complicated rearranging
sŷ
2 = r2 sy
2
r2 = sŷ
2 / sy
2
 so r2 is the proportion of the variance in y that is explained by
our regression model
How good is our model cont.

How good is our model cont.
 Insert r2 sy
2 into sy
2 = sŷ
2 + ser
2 and rearrange to get:
ser
2 = sy
2 – r2sy
2
= sy
2 (1 – r2)
 From this we can see that the greater the correlation
the smaller the error variance, so the better our
prediction

Is the model significant?
 i.e. do we get a significantly better prediction of y
from our regression equation than by just predicting
the mean?
 F-statistic:
F(dfŷ,dfer) =
sŷ
2
ser
2
=......=
r2 (n - 2)2
1 – r2
complicated
rearranging
 And it follows that:
t(n-2) =
r (n - 2)
√1 – r2
(because F = t2)
So all we need to
know are r and n

General Linear Model
 Linear regression is actually a form of the
General Linear Model where the parameters
are a, the slope of the line, and b, the intercept.
y = ax + b +ε
 A General Linear Model is just any model that
describes the data in terms of a straight line

Multiple regression
 Multiple regression is used to determine the effect of a number
of independent variables, x1, x2, x3 etc, on a single dependent
variable, y
 The different x variables are combined in a linear way and
each has its own regression coefficient:
y = a1x1+ a2x2 +…..+ anxn + b + ε
 The a parameters reflect the independent contribution of each
independent variable, x, to the value of the dependent variable,
y.
 i.e. the amount of variance in y that is accounted for by each x
variable after all the other x variables have been accounted for

SPM
 Linear regression is a GLM that models the effect of one
independent variable, x, on ONE dependent variable, y
 Multiple Regression models the effect of several independent
variables, x1, x2 etc, on ONE dependent variable, y
 Both are types of General Linear Model
 GLM can also allow you to analyse the effects of several
independent x variables on several dependent variables, y1, y2,
y3 etc, in a linear combination
 This is what SPM does and all will be explained next week!

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