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
Group 1
Group 2
Group 2
Group 2
Partitioning the variance
Total =
Model +
(Between groups)
Error
(Within groups)
(
)(
)


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
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
Gradient = 0
min S
Values of a and b
Minimising sums of squaresΣ(y  ax  b)2
r = correlation coefficient of x and y
sy = standard deviation of y
sx = standard deviation of x
a =
sx
The solutionb = 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?
∑(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
sy2 = sŷ2 + ser2
sŷ2 = r2 sy2
r2 = sŷ2 / sy2
ser2 = sy2 – r2sy2
= sy2 (1 – r2)
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