82
7 Regression Models and Hypothesis Testing
The integral is given by the gamma function, defined in (7.20), as + 1)/2), such
that we finally arrive at
ν (t) =
1
√
νπ
(
ν+1
2
)
(
ν
2
)
1
1 + t 2 /ν
ν+1
2
,
(7.33)
which is the commonly found form of Student’s t-distribution. Note, that it is
expressed in terms of ν, the number of degrees of freedom, rather than the number of samples n = ν + 1.
It is instructive to consider the limiting cases of the distribution. In particular, for
n = ν + 1 = 2 we recover the Lorentz, Breit-Wigner, or Cauchy-distribution and
for infinitely many degrees of freedom it can be shown that we recover the Gaussian distribution. The distributions for other values of ν lie between these extremes.
Figure 7.5 shows ν (t) for ν = 1, 2, and 100.
In order to assess the reliability of testing only a few samples we need to calculate the probability that our test statistic t = ( ¯
X n − μ)/(S n /
√
n) lies within the
central range of the distribution. If we specify this central range to lie within ±ˆ t this
probability is given by the integral
A(ˆ t, ν) =
ˆ
t
−ˆ t
ν (t)dt = 1 − I x
ν
2
,
1
2
(7.34)
where I x (a, b) is the (regularized) incomplete beta function [4] and x = ν/(ν + ˆ
t
2
).
The second equality follows from the substitution s = t
2 and the definition of the
incomplete beta function as an integral with the same integrand [4]. On the left-hand
side in Fig. 7.6 we show A(ˆ t, ν) for ν = 1, 2, 5, and 100 as a function of ˆ
t. The solid
black line corresponds to the case ν = 100, where ν (t) approaches a Gaussian,
which contains 95% of the distribution within two standard deviations and corresponds to ˆ
t ≈ 2. The range containing a percentage, say 95%, of the distribution,
defines the 95% confidence level. When taking fewer samples n = ν + 1 the probability to find a value of t within ±ˆ t is reduced compared to the Gaussian. For example,
if we only test three samples (ν = 2) the 95% confidence level is only reached at
|ˆ t| ≈ 4.3. For convenience, we show the values |ˆ t| where A(ˆ t, ν) as a function of ν
reaches 95, 90, and 80% on the right-hand plot in Fig. 7.6. The solid black curve
corresponds to the 95% confidence level and indeed we find the point mentioned
above at ν = 2 and |ˆ t| ≈ 4.3 on it. Furthermore, for large values of ν it approaches
ˆ
t ≈ 2 as expected for Gaussian distributions. Below the curve for the 95% level, we
find the curves for 90 and 80%. They correspond to a smaller area around the center
of the distribution function and consequently a smaller probability of finding a value
of t closer to the center. We point out that for ν > 10 the curves rapidly approach
their asymptotic values, which agree with those of a Gaussian. This is the reason for
the common method to use two standard deviations to specify the 95% confidence
level.
7 Regression Models and Hypothesis Testing
The integral is given by the gamma function, defined in (7.20), as + 1)/2), such
that we finally arrive at
ν (t) =
1
√
νπ
(
ν+1
2
)
(
ν
2
)
1
1 + t 2 /ν
ν+1
2
,
(7.33)
which is the commonly found form of Student’s t-distribution. Note, that it is
expressed in terms of ν, the number of degrees of freedom, rather than the number of samples n = ν + 1.
It is instructive to consider the limiting cases of the distribution. In particular, for
n = ν + 1 = 2 we recover the Lorentz, Breit-Wigner, or Cauchy-distribution and
for infinitely many degrees of freedom it can be shown that we recover the Gaussian distribution. The distributions for other values of ν lie between these extremes.
Figure 7.5 shows ν (t) for ν = 1, 2, and 100.
In order to assess the reliability of testing only a few samples we need to calculate the probability that our test statistic t = ( ¯
X n − μ)/(S n /
√
n) lies within the
central range of the distribution. If we specify this central range to lie within ±ˆ t this
probability is given by the integral
A(ˆ t, ν) =
ˆ
t
−ˆ t
ν (t)dt = 1 − I x
ν
2
,
1
2
(7.34)
where I x (a, b) is the (regularized) incomplete beta function [4] and x = ν/(ν + ˆ
t
2
).
The second equality follows from the substitution s = t
2 and the definition of the
incomplete beta function as an integral with the same integrand [4]. On the left-hand
side in Fig. 7.6 we show A(ˆ t, ν) for ν = 1, 2, 5, and 100 as a function of ˆ
t. The solid
black line corresponds to the case ν = 100, where ν (t) approaches a Gaussian,
which contains 95% of the distribution within two standard deviations and corresponds to ˆ
t ≈ 2. The range containing a percentage, say 95%, of the distribution,
defines the 95% confidence level. When taking fewer samples n = ν + 1 the probability to find a value of t within ±ˆ t is reduced compared to the Gaussian. For example,
if we only test three samples (ν = 2) the 95% confidence level is only reached at
|ˆ t| ≈ 4.3. For convenience, we show the values |ˆ t| where A(ˆ t, ν) as a function of ν
reaches 95, 90, and 80% on the right-hand plot in Fig. 7.6. The solid black curve
corresponds to the 95% confidence level and indeed we find the point mentioned
above at ν = 2 and |ˆ t| ≈ 4.3 on it. Furthermore, for large values of ν it approaches
ˆ
t ≈ 2 as expected for Gaussian distributions. Below the curve for the 95% level, we
find the curves for 90 and 80%. They correspond to a smaller area around the center
of the distribution function and consequently a smaller probability of finding a value
of t closer to the center. We point out that for ν > 10 the curves rapidly approach
their asymptotic values, which agree with those of a Gaussian. This is the reason for
the common method to use two standard deviations to specify the 95% confidence
level.