T tests, ANOVAs and regression. Tom Jenkins Ellen Meierotto SPM Methods for Dummies 2007. Why do we need t tests?. Objectives. Types of error Probability distribution Z scores T tests ANOVAs. Error. Null hypothesis Type 1 error ( α ): false positive Type 2 error ( β ): false negative.
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Z=(xμ)/σfor one point compared to pop.
Pooled standard error of the mean
Variance
F test = Model variance /Error variance
Group 1 chance or is it stat. significant?
Group 1
Group 1
Group 2
Group 2
Group 2
Partitioning the variance
Total =
Model +
(Between groups)
Error
(Within groups)
Y chance or is it stat. significant?
X
ScattergramsY
Y
Y
Y
Y
X
X
Positive correlation
Negative correlation
No correlation
Covariance ~ chance or is it stat. significant?
DX * DY
Variance ~
DX * DX
Variance vs. Covariancex chance or is it stat. significant?
(
)(
)


y


x
x
y
y
x
x
y
y
i
i
i
i
0
3

3
0
0
2
2

1

1
1
3
4
0
1
0
4
0
1

3

3
6
6
3
3
9
å
=
=
7
y
3
=
x
3
Example CovarianceWhat does this number tell us?
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
ŷ chance or is it stat. significant? = ax + b
slope
ε
= y i , true value
ε =residual error
Bestfit Lineminimises distance between
data and fitted line, i.e.
the residuals
intercept
= ŷ, predicted value
Model line: ŷ = ax + b
a = slope, b = intercept
Residual (ε) = y  ŷ
Sum of squares of residuals = Σ (y – ŷ)2
Σ (y – ŷ)2
b chance or is it stat. significant?
Finding bb
ε
ε
b
b chance or is it stat. significant?
b
Finding ab
sums of squares (S) chance or is it stat. significant?
Gradient = 0
min S
Values of a and b
Minimising sums of squaresΣ(y  ax  b)2
r s chance or is it stat. significant?y
r = correlation coefficient of x and y
sy = standard deviation of y
sx = standard deviation of x
a =
sx
The solutiony = ax + b chance or is it stat. significant?
b = y – ax
b = y – ax
r sy
r = correlation coefficient of x and y
sy = standard deviation of y
sx = standard deviation of x
x
b = y 
sx
The solution cont.How well does this line fit the data, or how good is it at predicting y from x?
∑(ŷ – y) chance or is it stat. significant?2
∑(y – y)2
SSy
SSpred
sy2 =
sŷ2 =
=
=
n  1
n  1
dfŷ
dfy
∑(y – ŷ)2
SSer
serror2 =
=
n  2
dfer
How good is our model?This is the variance explained by our regression model
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
How good is our model cont. chance or is it stat. significant?
sy2 = sŷ2 + ser2
sŷ2 = r2 sy2
r2 = sŷ2 / sy2
ser2 = sy2 – r2sy2
= sy2 (1 – r2)
s chance or is it stat. significant?ŷ2
r2 (n  2)2
F
=
(dfŷ,dfer)
ser2
1 – r2
Is the model significant?complicated
rearranging
=......=
So all we need to
know are r and n !
r(n  2)
t(n2) =
(because F = t2)
√1 – r2
y = ax + b +ε
y = a1x1+ a2x2 +…..+ anxn + b + ε