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
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
Pooled standard error of the mean

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
F test = Model variance /Error variance

17. Partitioning the variance
Group 1
Group 2
Group 1
Group 2
Group 1
Group 2
Total =
Model
(Between groups)
Error
(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
We often want to know whether various variables are ‘linked’, i.e., correlated.
This can be interesting in itself, but is also important if we want to predict one variable’s value given a value of the other.

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
means that correlation cannot be validly used to infer a causal relationship between the variables
not be taken to mean that correlations cannot indicate causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown
A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health? Or does good health lead to good mood? Or does some other factor underlie both? Or is it pure coincidence? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.

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

26. Variance vs. Covariance
Do two variables change together?
Variance ~
DX * DX
Variance is just a definition. The reason we’re squaring it is to have it get a positive value whether dx is negative or positive, so that we can sum them and positives and negatives will not cancel out.
Variance is spread around a mean, covariance is the measure of how much x and y change together; very similar: multiply 2 variables rather than square 1
Covariance ~
DX * DY

27. 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

28. Example Covariance What does this number tell us? x ( )( ) 3 2 1 4 6 9y i - ( )( ) 3 2 1 4 6 9 = å 7
What does this number tell us?

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

30. Pearson’s R Covariance does not really tell us anything
Solution: standardise this measure
Pearson’s R: standardise by adding std to equation:
Can only compare covariances between different variables to see which is greater.

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
The distance of r from 0 indicates strength of correlation
r = 1 or r = (-1) means that we can predict y from x and vice versa with certainty; all data points are on a straight line. i.e., y = ax + b
The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.
the correlation coefficient detects only linear dependencies between two variables

33. Limitations of r r is actually r = true r of whole population
= estimate of r based on data
r is very sensitive to extreme values:
X av = 3, y av = 2
If extreme value such as y = 5 is possible (graph)
X – 1,2,3,4,5,
Y – 1,2,3,4,0

34. In the real world…
r is never 1 or –1
interpretations for correlations in psychological research (Cohen)
Correlation Negative Positive
Small to to 0.29
Medium to to 0.49
Large to to 1.00
the interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.9 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences where there may be a greater contribution from complicating factors.

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 ŷ = ax + b ε = ŷ, predicted value = y i , true 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 ε
ŷ = ax + b
ε = residual error
= y i , true value
slope
intercept
So to understand the relationship between two variables, we want to draw the ‘best’ line through the cloud – find the best fit.
This is done using the principle of least squares
= ŷ, predicted value

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 b ε 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 squaresb 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
Values of a and b
sums of squares (S)
Gradient = 0
min S
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.

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

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? Variance of predicted y values (ŷ):
sy2 =
∑(y – y)2
n - 1
SSy
dfy =
Total variance of y:
Variance of predicted y values (ŷ):
This is the variance explained by our regression model
sŷ2 =
∑(ŷ – y)2
n - 1
SSpred
dfŷ =
Error variance:
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
serror2 =
∑(y – ŷ)2
n - 2
SSer
dfer =

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 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) =
(because F = t2)
√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…