This document provides an overview of statistical tests commonly used in neuroimaging such as t-tests, ANOVAs, and regression. It discusses the purposes of these tests and how they are applied. T-tests are used to compare means, for example to determine if the difference between two conditions is statistically significant. ANOVAs examine variances and can be used when comparing more than two groups. Regression allows describing and predicting the relationship between variables and is useful in the general linear model approach used in SPM. Key assumptions and calculations for each method are outlined.

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…