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Transcript
t-tests, ANOVA and regression
- and their application to the statistical analysis of fMRI data
Lorelei Howard and Nick Wright
MfD 2008
Overview
•
•
•
•
Why do we need statistics?
P values
T-tests
ANOVA
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
2 types of statistics
• Descriptive Stats
– e.g., mean and standard deviation (S.D)
• Inferential statistics
– t-tests, ANOVAs and regression
Issues when making inferences
• 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
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
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
t-tests
• Compare two group means
Hypothetical experiment
Time
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
Calculating T
Difference between the means divided by the pooled
standard error of the mean
x1  x 2
t
s x1  x2
2
Group 1
Group 2
s x1  x2 
2
s1
s2

n1
n2
How do we apply this to fMRI
data analysis?
Time
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
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
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
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
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
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
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
Further reading
•
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
PART 2
• Correlation
• Regression
• Relevance to GLM and SPM
Correlation
• Strength and direction of the relationship
between variables
• Scattergrams
Y
Y
Y
X
Positive correlation
Y
Y
Y
X
Negative correlation
No correlation
•
Describe correlation:
covariance
A statistic representing the degree to which 2
variables vary together
– Covariance formula
n
cov( x, y ) 
 ( x  x)( y
i 1
i
i
 y)
n
n
– cf. variance formula
S x2 
2
(
x

x
)
 i
i 1
n
but…
• the absolute value of cov(x,y) is also a function of the
standard deviations of x and y.
Describe correlation: Pearson
correlation coefficient (r)
• Equation
cov( x, y)
s = st dev of sample
rxy 
sx s y
– r = -1 (max. negative correlation); r = 0 (no constant
relationship); r = 1 (max. positive correlation)
• Limitations:
5
4
3
– Sensitive to extreme values, e.g.
2
1
0
0
1
2
3
4
– r is an estimate from the sample, but does it
represent the population parameter?
– Relationship not a prediction.
5
6
Summary
• Correlation
• Regression
• Relevance to SPM
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.
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
= ŷ, predicted
= y i , observed
ε = residual
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
• Minimise (y- ŷ)2 , which is (ybx+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 (ybx-a)2 and solve for 0 as
simultaneous equations, giving:
rs y
a  y  bx
b
sx
Sums of squared error (SSerror)
How to minimise SSerror
Gradient = 0
min SSerror
Values of a and b
How good is the model?
• 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
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(df ,df ) =
ŷ
er
sŷ2
ser2
• And it follows that:
r (n - 2)
t(n-2) =
√1 – r2
complicated
rearranging
r2 (n - 2)2
=......=
1 – r2
So all we need to
know are r and n !
Summary
• Correlation
• Regression
• Relevance to SPM
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
One voxel: The GLM
Our aim: Solve equation for β – tells us how much BOLD signal is explained by X
a
m
b3
b4
b5
=
b6
+
b7
b8
b9
Y
=
X
×
b
+
e
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
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…
Summary
• Correlation
• Regression
• Relevance to SPM
Thanks!