Transcript t test (one sample)

LECTURE 3
LINEAR REGRESSION &
CORRELATION
Parametric hypothesis tests(cont); Linear
regression and correlation
Supplementary Readings: Wilks, chapters 5,6
Bevington, P.R., Robinson, D.K., Data Reduction and
Error Analysis for the Physical Sciences, McGraw-Hill,
1992.
t test (one sample)
Consider a
Gaussian sample
of data of size n
We’re interested in whether or not the data are consistent
with a population with some specified mean m
Under the assumption of Gaussian statistics, the
statistic
follows a “t” distribution
xm
t
var(x )
1/ 2
var( x )  s / n
2
t test (one sample)
(f= n)
Note that the test could
be either one or twosided depending on the
situation at hand
xm
t
var(x )
1/ 2
Note: t approaches Z
for large n…
follows a “t” distribution
t test (two sample)
Consider two
independent
Gaussian samples of
data of size n1 and n2
We’re interested in whether or not they appear to have
been drawn from parent distributions with different
means
The standard deviation of the difference between the
means under the assumption of Gaussian statistics is:
t test (two sample)
Consider two
independent
Gaussian samples of
data of size n1 and n2
This motivates a
modified form of
the t statistic:
t test (two sample)
Consider two
independent
Gaussian samples of
data of size n1 and n2
This motivates a
modified form of
the t statistic:
(f= n1+n2 for var1=var2 )
F test
What if we’re interested in whether or not two subsets of
the data appear to have been drawn from parent
distributions with different variances?
 /v
F
 /v
2
1
1
2
2
2
F test
 /v
F
 /v
2
1
1
2
2
2
F distribution
F distribution
Note that Confidence Levels Associated with
the F-test and t-test Depend on the
Assumption of some nominal (N) degrees of
freedom
Frequently the true number of degrees of
freedom N’ is less than N owing to serial
correlation
We will discuss how to correct for this
later…
Recall Chi-Squared
Suppose that we’re not
interested in the model of a
constant mean process (y=m),
but rather, a process that has a
linear trend (y=a+bx)
2

y

m

N
2
i

 
i1  





What is the appropriately modified
chi-squared?





N
1

P
P 

i  2 
1,..., N








2

y m
1
 
exp    i
 
2    

Recall Chi-Squared
y: “dependent variable”
x: “independent variable
y a bx
N
i
i
 2 

i1


















What is the appropriately modified
chi-squared?





N
1

P
P 

i  2 
1,..., N








2

y m
1
 
exp    i
 
2    

2
Recall Method of Maximum Likelihood
Most probable value for the statistic of
interest is given by the peak value of the
joint probability distribution.
Easiest to work with the Log-Likelihood function:


2
1



L(m, )   N ln  N ln 2 
y

m


2  
2   i 





For model of a constant mean, we want to maximize L
relative to m:
L(m, ) 0
m
Method of Maximum Likelihood
Most probable value for the statistic of
interest is given by the peak value of the
joint probability distribution.
Easiest to work with the Log-Likelihood function:
















L(a,b)  N ln  N ln 2  1 2  y a bx
i
i
2






For model of a linear trend, we want to maximize L
relative to a and b:
L(a,b)  0
a
L(a,b)  0
b






2
Linear Regression
This amounts to the minimization of chi-squared,
 2(a,b)  0
a
 2(a,b)  0
b
















1
L(a,b)  N ln  N ln 2  
2











y a bx
i
i

For model of a linear trend, we want to maximize L
relative to a and b:
L(a,b)  0
a
L(a,b)  0
b











2
Linear Regression
This amounts to the minimization of chi-squared,
n  y a bx 
i
 2(a,b)    i



i 1

2
 
2





 yi nab xi
2
 2(a,b)  0
a




y  a  bx  0
i
i
y  a bx
Linear Regression
This amounts to the minimization of chi-squared,
n  y a bx 
i
 2(a,b)    i



i 1

2
 
2





2
 2(a,b)  0
b




( y  a  bx )( x )  0
i
i
i
2
y
x

a
x

b
x
 ii  i  i
Linear Regression
y  a bx
2
y
x

a
x

b
x
 ii  i  i
We can write this as a matrix equation,






n
x
i



x
y
 i  a    
i
   

 



2
x
y
x
  b 

i 
i i

Linear Regression






n
x
i
 xi  a   y 
i
   

2    y x 
x
  b 

i 
i i

The solution is:
n y x  y  x
i i
i i
b
2
n x 2  x
i
i






If
x  y 0






a  y bx
y x
b i i
we have:
x 2
i
a 0
Linear Regression
Linear Correlation
















yx

1
i i
r
n s s
x y
If
















x  y 0
s
b x
s
y
y x
b i i
we have:
x 2
i
a 0
Linear Regression
Linear Correlation
What if the independent variable (“x”)
is time?
Determination of Trend
y x
b i i
If x  y  0 we have:
a

0
x
2

i
Linear Regression
n  y a bx 
i
 2(a,b)    i



i 1

2
Define:
  y  a  bx
i
i
i
We call these residuals
What should we require of them?
Linear Regression
GAUSSIAN
What should we require of them?
Chi-Squared
( N  2) / 2
y  a  bx
x
exp(

x
/
2
)
N
i
i
P ( x) 
2 

N
N
/
2
i

1
( N / 2)2







2(n=5)
Gaussian
data
m 2 v








2
This is an important
feature in the
Analysis of Variance
(“ANOVA”)
2(n=5)
y  a  bx
N
i
i
2 

i1







Gaussian
data
m 2 v








2