1. Lorelei Howard and Nick Wright MfD 2008
t-tests, ANOVA and regression
- and their application to the statistical analysis of fMRI data
Lorelei Howard and Nick Wright MfD 2008
Hello, my name is Lorelei. I’m in the first year of my PhD. Nick, who you met last week, and myself will be explaining various statistical tests and their application to the analysis of fMRI data.
In the first part of this session I will be talking to you about…(next slide)

2. Overview Why do we need statistics? P values T-tests ANOVA
…t-tests and ANOVAs.
Prior to this I will be discussing some basic principles in statistical testing.
This is mainly for the benefit of those of you who have not had much experience using statistics
I will start by explaining why we need statistics in science
For those of you who are familiar with this material, I apologise and ask you to bear with me
I will then discuss the use of p values and how these enable us to quantify our certainty in the generalisabilty of our results

3. Why do we need statistics?
To enable us to test experimental hypotheses
H0 = null hypothesis
H1 = experimental hypothesis
In terms of fMRI
Null = no difference in brain activation between these 2 conditions
Exp = there is a difference in brain activation between these 2 conditions
In each experiment there are two hypotheses…
The null hypothesis, denoted as H0
And the experimental hypothesis, denoted as H1
Alternatively the exp hypothesis can be referred to as alternative or research hypothesis.
This hypothesis makes a prediction that must be falsifiable
For example…
We may wish to investigate whether the presentation of famous names elicits more activation in the fusiform gyrus than the presentation of the names of unknown people.
In such a study the null hypothesis would be that…
And the experimental hypothesis would be that…
So…How do we establish which of theses two possible outcomes this is true?

4. 2 types of statistics Descriptive Stats Inferential statistics
e.g., mean and standard deviation (S.D)
Inferential statistics
t-tests, ANOVAs and regression
There are 2 types of statistics that can help us to determine whether our data support our exp hypothesis
- Descriptive Stats
Values such as the mean and the S.D allow us to summarise our data sets
Returning to our example…
for each voxel…
We could use the mean BOLD signal for the two conditions to describe any differences between the 2 conditions
Say that we were examining the activity in a voxel from the fusiform gyrus and we found the following…
Mean BOLD activity for condition 1 (famous names) = 500 units
Whereas the Mean BOLD activity for condition 2 (unknown names) = 498 units
We can see that these 2 numbers differ numerically
But we cannot comment on whether this difference is meaningful
- Inferential statistics
Include tests such as t-tests, ANOVAs and regression
Again, using the example from above…
A T-test could be used to determine whether the difference between these 2 conditions is statistically significant
Inferential stats also allow us to comment on the likelihood that the effects found in the current dataset are genuine
Linking sentence needed..

5. Issues when making inferences
It is important to acknowledge that the data collected represent only a single sample from a much larger population
It is therefore possible that if a different sample were used then different results could have been obtained

6. So how do we know whether the effect observed in our sample was genuine?
We don’t
Instead we use p values to indicate our level of certainty that our results represent a genuine effect present in the whole population Q
Explain that by genuine we mean that the observed effect was caused by a true effect present within the whole population A

7. P values
P values = the probability that the observed result was obtained by chance
i.e. when the null hypothesis is true
α level is set a priori (Usually 0.05)
If p < α level then we reject the null hypothesis and accept the experimental hypothesis
95% certain that our experimental effect is genuine
If however, p > α level then we reject the experimental hypothesis and accept the null hypothesis
P = Probability
this value tells us the probability that the observed result was obtained by chance
That there is no difference between the two groups
Each test result (e.g. t value) is associated with a particular p value
α level is set a priori
This is basically an acceptance level
Usually this is set to 0.05
But as I understand, α levels are usually much lower than this in fMRI
If p < α level then we reject the null hypothesis and accept the experimental hypothesis
- concluding that we are 95% certain that our experimental effect is genuine
If however, p > α level then we reject the experimental hypothesis and accept the null hypothesis
- that there was no sig diff in brain activation levels between the two conditions

8. Two types of errors Type I error = false positive
α level of 0.05 means that there is 5% risk that a type I error will be encountered
Type II error = false negative
Beware of errors
We must be aware that errors can occur during this process
Two types of errors
- type I error = false positive
Where we incorrectly reject the null hypothesis.
The pre-determined α level determines the risk of this type of error.
α level of 0.05 means that there is 5% risk that a type I error will be encountered.
The other type of error is…
- type II error = false negative
Where we incorrectly reject the exp hypothesis

9. t-tests Compare two group means
The type of statistical tests used depends on your hypotheses.
If you would like to compare differences between the means of two groups of data then a t-test would be appropriate as it compares two group means in context of their variability
I think the use of t-tests can be best explained with an example, here I will be using a hypothetical experiment to aid explanation…

10. Hypothetical experiment
Time
Explain diagram in full
Timeline of exp view faces of cartoon characters
Research Q….
Two conditions
H0 = no diff
H1 = diff
Q – does viewing pictures of the Simpson and the Griffin family activate the same brain regions?
Condition 1 = Simpson family faces
Condition 2 = Griffin family faces

11. Calculating T
Difference between the means divided by the pooled standard error of the mean
Group 1
Group 2
Predrag – To calculate the t score,
Take mean from group 1 (x bar)
Then the mean from group 2
Find the difference between these two
And divide this by their shared standard error
Calculate this by…
Taking the variance for group 1 and dividing it by the sample size for this group
Do the same for group 2 and add together.
Then finally take the square root of this and put the resulting value back into the original calculation to get your t value.

12. How do we apply this to fMRI data analysis?
If we just return to our hypothetical experiment…

13. Time NB: for each voxel…
Calculate the mean and variance for the activation levels associated with the Simpsons family members
And then for the Griffin family
Calculate the difference between the means
Calculate their shared S.E
Compute T
Once you have your T value you then need to work out your degrees of freedom…

14. Degrees of freedom = number of unconstrained data points
Which in this case = number of data points – 1.
Can use t value and df to find the associated p value
Then compare to the α level

15. Different types of t-test
2 sample t tests
Related = two samples related, i.e. same people in both conditions
Independent = two independent samples, i.e. diff people in 2 conditions
One sample t tests
compare the mean of one sample to a given value
2 sample t tests
Like e.g. above.
These may be related or independent
Related = two samples related, i.e. same people in both conditions
Independent = two independent samples, i.e. diff people in 2 conditions
Q – does fMRI always use related???
Also there are…
One sample t tests
- These compare the mean of one sample to a given value (e.g. 0)

16. Another approach to group differences
Analysis Of VAriance (ANOVA)
Variances not means
Multiple groups
e.g. Different facial expressions
H0 = no differences between groups
H1 = differences between groups
ANOVA uses variances instead of means to compare groups
Unlike t-tests where only 2 groups can be compared, ANOVA can compare across 3, 4, 5 groups or more
For example, do different facial expression elicit different neuronal activity?
Could compare the neural activity associated with happy, sad, and neutral faces

17. Calculating F
F = the between group variance divided by the within group variance
the model variance/error variance
for F to be significant the between group variance should be considerably larger than the within group variance
F = statistic computed in ANOVA
This can be defined as…
the between group variance divided by the within group variance
Or sometimes referred to as the model variance/error variance
If = 1 then means that the between and the within group variances are equal
So for F to be sig it must be noticeably larger than one, i.e. the between group variance should be considerably larger than the within group variance
Again this value and it’s associated degrees of freedom are used to find the p value

18. What can be concluded from a significant ANOVA?
There is a significant difference between the groups
NOT where this difference lies
Finding exactly where the differences lie requires further statistical analyses

19. Different types of ANOVA
One-way ANOVA
One factor with more than 2 levels
Factorial ANOVAs
More than 1 factor
Mixed design ANOVAs
Some factors independent, others related
The type that I have described is referred to as a one-way ANOVA because it has one factor which = cartoon characters (with more than 2 levels)
Can also have two-way, three-way ANOVAs
These = factorial ANOVAs
Allow for possible interactions between factors as well as main effects
For example you could have 2 factors with 2 levels each
This would = a 2 x 2 factorial design
Can also have related or independent designs or a mixture

20. Conclusions T-tests assess if two group means differ significantly
Can compare two samples or one sample to a given value
ANOVAs compare more than two groups or more complicated scenarios
They use variances instead of means

21. Further reading Acknowledgements
Howell. Statistical methods for psychologists
Howitt and Cramer. An introduction to statistics in psychology
Huettel. Functional magnetic resonance imaging (especially chapter 12)
Acknowledgements
MfD Slides 2005 – 2007

22. PART 2
Correlation
Regression
Relevance to GLM and SPM

23. Correlation
Strength and direction of the relationship between variables
Scattergrams Y X
Positive correlation
Negative correlation
No correlation

24. Describe correlation: covariance
A statistic representing the degree to which 2 variables vary together
Covariance formula
cf. variance formula
but…
the absolute value of cov(x,y) is also a function of the standard deviations of x and y.

25. Describe correlation: Pearson correlation coefficient (r)
Equation
r = -1 (max. negative correlation); r = 0 (no constant relationship); r = 1 (max. positive correlation)
Limitations:
Sensitive to extreme values, e.g.
r is an estimate from the sample, but does it represent the population parameter?
Relationship not a prediction.
s = st dev of sample

26. Summary
Correlation
Regression
Relevance to SPM

27. Regression
Regression: Prediction of one variable from knowledge of one or more other variables.
Regression v. correlation: Regression allows you to predict one variable from the other (not just say if there is an association).
Linear regression aims to fit a straight line to data that for any value of x gives the best prediction of y.

28. Best fit line, minimising sum of squared errors
Describing the line as in GCSE maths: y = m x + c
Here, ŷ = bx + a
ŷ : predicted value of y
b: slope of regression line
a: intercept
ŷ = bx + a ε
ε = residual
= y i , observed
= ŷ, predicted
Residual error (ε): Difference between obtained and predicted values of y (i.e. y- ŷ).
Best fit line (values of b and a) is the one that minimises the sum of squared errors (SSerror) (y- ŷ)2

29. How to minimise SSerror
Minimise (y- ŷ)2 , which is (y-bx+a)2
Plotting SSerror for each possible regression line gives a parabola.
Minimum SSerror is at the bottom of the curve where the gradient is zero – and this can found with calculus.
Take partial derivatives of (y-bx-a)2 and solve for 0 as simultaneous equations, giving:
Values of a and b
Sums of squared error (SSerror)
Gradient = 0
min SSerror

30. How good is the model? sy2 = sŷ2 + ser2 r2 = sŷ2 / sy2
We can calculate the regression line for any data, but how well does it fit the data?
Total variance = predicted variance + error variance
sy2 = sŷ2 + ser2
Also, it can be shown that r2 is the proportion of the variance in y that is explained by our regression model
r2 = sŷ2 / sy2
Insert r2 sy2 into sy2 = sŷ2 + ser2 and rearrange to get:
ser2 = sy2 (1 – r2)
From this we can see that the greater the correlation the smaller the error variance, so the better our prediction

31. 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 F
(dfŷ,dfer) =
sŷ2
ser2
r2 (n - 2)2
1 – r2
=......=
And it follows that:
r (n - 2)
So all we need to
know are r and n !
t(n-2) =
√1 – r2

32. Summary
Correlation
Regression
Relevance to SPM

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

34. = + Y = X × b + e data vector (Voxel) design matrix parameters
One voxel: The GLM
Our aim: Solve equation for β – tells us how much BOLD signal is explained by X
data vector (Voxel)
design matrix
parameters
error vector a m b3 b4 b5 b6 b7 b8 b9 = + Y = X × b + e

35. 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 = b0 + b1x1+ b2x2 +…..+ bnxn + 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

36. 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
This is what SPM does and will be explained soon…

37. Summary
Correlation
Regression
Relevance to SPM
Thanks!