1. T tests, ANOVAs and
regression
Tom Jenkins
Ellen Meierotto
SPM Methods for Dummies 2007

2. Why do we need t tests?

3. Objectives
 Types of error
 Probability distribution
 Z scores
 T tests
 ANOVAs

5. Error
 Null hypothesis
 Type 1 error (α): false positive
 Type 2 error (β): false negative

6. Normal distribution

7. Z scores
 Standardised normal distribution
 µ = 0, σ = 1
 Z scores: 0, 1, 1.65, 1.96
 Need to know population standard
deviation
Z=(x-μ)/σ for
one point
compared to pop.

8. T tests
 Comparing means
 1 sample t
 2 sample t
 Paired t

9. Different sample variances

10. 2 sample t tests
x1 − x 2
t =
s x1 −x2
2 2
Pooled standard
s1 s 2
= +
error of the mean
s x1 − x2
n1 n2

11. 1 sample t test

12. The effect of degrees of
freedom on t distribution

13. Paired t tests

14. T tests in SPM: Did the observed signal
change occur by chance or is it stat.
significant?
 Recall GLM. Y= X β + ε
 β1 is an estimate of signal change over time
attributable to the condition of interest
 Set up contrast (cT) 1 0 for β1: 1xβ1+0xβ2+0xβn/s.d
 Null hypothesis: cTβ=0 No significant effect at each
voxel for condition β1
 Contrast 1 -1 : Is the difference between 2 conditions
significantly non-zero?
 t = cTβ/sd[cTβ] – 1 sided

16. ANOVA
 Variances not means
 Total variance= model variance + error variance
 Results in F score- corresponding to a p value
Variance
n
∑ ( xi − x ) 2 F test = Model variance /Error
variance
s2 = i =1
n −1

17. Partitioning the variance
Group Group Group Group Group Group
1 2 1 2 1 2
Total = Model + Error
(Between groups) (Within groups)

18. T vs F tests
 F tests- any differences between
multiple groups, interactions
 Have to determine where differences
are post-hoc
 SPM- T- one tailed (con)
 SPM- F- two tailed (ess)

20. Conclusions
 T tests describe how unlikely it is that experimental
differences are due to chance
 Higher the t score, smaller the p value, more unlikely
to be due to chance
 Can compare sample with population or 2 samples,
paired or unpaired
 ANOVA/F tests are similar but use variances instead
of means and can be applied to more than 2 groups
and other more complex scenarios

21. Acknowledgements
 MfD slides 2004-2006
 Van Belle, Biostatistics
 Human Brain Function
 Wikipedia

22. Correlation and Regression

23. 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 it to predict
y from x?
 Regression
 Is the relationship we have described statistically
significant?
 F- and t-tests
 Relevance to SPM
 GLM

24. Relationship between x and y
 Correlation describes the strength and
direction of a linear relationship between two
variables
 Regression tells you how well a certain
independent variable predicts a dependent
variable

CORRELATION ≠ CAUSATION
 In order to infer causality: manipulate independent
variable and observe effect on dependent variable

25. Scattergrams
Y Y Y
Y Y Y
X X X
Positive correlation Negative correlation No correlation

26. Variance vs. Covariance
 Do two variables change together?
n
Variance ~
∑(x i − x) 2
S =
2
x
i =1
DX * DX n
n
Covariance ~ ∑(x i − x)( yi − y )
DX * DY
cov( x, y ) = i =1
n

27. Covariance
n
∑(x i − x)( yi − y )
cov( x, y ) = i =1
n
 When X and Y : cov (x,y) = pos.
 When X and Y : cov (x,y) = neg.
 When no constant relationship: cov (x,y)
=0

28. Example Covariance
7
6 x y xi − x yi − y ( xi − x )( yi − y )
5
0 3 -3 0 0
4
2 2 -1 -1 1
3
2
3 4 0 1 0
1
4 0 1 -3 -3
0
6 6 3 3 9
0 1 2 3 4 5 6 7
x=3 y=3 ∑= 7
n
∑(x i − x)( yi − y ))
7 What does this
cov( x, y ) = i =1
= = 1.4 number tell us?
n 5

29. Example of how covariance value
relies on variance
High variance data Low variance data
Subject x y x error * y x y X error * y
error 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

30. Pearson’s R
− ∞ ≤ cov( x, y ) ≤ ∞
 Covariance does not really tell us
anything
 Solution: standardise this measure
 Pearson’s R: standardise by adding std
to equation: cov( x, y )
rxy =
sx s y

31. Basic assumptions
 Normal distributions
 Variances are constant and not zero
 Independent sampling – no autocorrelations
 No errors in the values of the independent
variable
 All causation in the model is one-way (not
necessary mathematically, but essential for
prediction)

32. Pearson’s R: degree of linear
dependence
n n
∑ (x i − x)( yi − y ) ∑ ( x − x)( y − y)
i i
cov( x, y ) = i =1
rxy = i =1
n nsx s y
−1 ≤ r ≤ 1
n
∑Z xi * Z yi
rxy = i =1
n

33. Limitations of r
 ˆ
r is actually r
 r = true r of whole population

ˆ
r = estimate of r based on data

r is very sensitive to extreme values:
5
4
3
2
1
0
0 1 2 3 4 5 6

34. In the real world…
 r is never 1 or –1
 interpretations for correlations in
psychological research (Cohen)
Correlation Negative Positive
Small -0.29 to -0.10 00.10 to 0.29
Medium -0.49 to -0.30 0.30 to 0.49
Large -1.00 to -0.50 0.50 to 1.00

35. 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!

36. Best-fit Line
 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
ŷ = ax + b
minimises distance between slope intercept
data and fitted line, i.e.
the residuals ε
= ŷ, predicted value
= y i , true value
ε = residual error

37. 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)
Model line: ŷ = ax + b a = slope, b = intercept
Residual (ε) = y - ŷ
Sum of squares of residuals = Σ (y – ŷ)2
 we must find values of a and b that minimise
Σ (y – ŷ)2

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

39. 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

40. Minimising sums of squares
 Need to minimise Σ(y–ŷ)2
 ŷ = ax + b
 so need to minimise:
sums of squares (S)
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 Gradient = 0
min S
Values of a and b
 So the min sum of squares is
at the bottom of the curve,
where the gradient is zero.

41. The maths bit
 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

42. The solution
 Doing this gives the following equations for a and b:
r sy r = correlation coefficient of x and y
a= s sy = standard deviation of y
x sx = standard deviation of x
 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)

43. 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 into this giving:
r sy r = correlation coefficient of x and y
b=y- x sy = standard deviation of y
sx sx = standard deviation of x
 The smaller the correlation, the closer the intercept is to the mean
of y

44. Back to the model
 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?

45. How good is our model?
∑(y – y)2 SSy
 Total variance of y: sy2 = =
n-1 dfy
 Variance of predicted y values (ŷ):
∑(ŷ – y)2 SSpred This is the variance
sŷ2 = = explained by our
n-1 dfŷ regression model
 Error variance: This is the variance of the error
between our predicted y values
∑(y – ŷ)2 SSer and the actual y values, and
serror =
2
= thus is the variance in y that is
n-2 dfer NOT explained by the
regression model

46. How good is our model cont.
 Total variance = predicted variance + error variance
sy2 = sŷ2 + ser2
 Conveniently, via some complicated rearranging
sŷ2 = r2 sy2
r2 = sŷ2 / sy2
 so r2 is the proportion of the variance in y that is explained
by our regression model

47. How good is our model cont.
 Insert r2 sy2 into sy2 = sŷ2 + ser2 and rearrange to get:
ser2 = sy2 – r2sy2
= sy2 (1 – r2)
 From this we can see that the greater the
correlation the smaller the error variance, so the
better our prediction

48. 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: complicated
rearranging
sŷ2 r2 (n - 2)2
F(df ,df ) = =......=
ŷ er
ser2 1 – r2
 And it follows that:
r (n - 2) So all we need to
(because F = t 2) t(n-2) = know are r and n !
√1 – r2

49. 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

50. 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

51. 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 will be explained soon…