In this example we apply the t Student-$t$-test on page 162) to a subset of the iris samples, to see if the different species can be reasonably separated using their petal lengths. Shown in Table 9.2 is a very small subset of the full iris petal-length data. Are the types Virginica and Versicolor longer than the type Setosa ? Is the Virginica longer than Versicolor ? For each of these, we need to specify the model, determine the best estimate for the parameters of the model, and then compare the distributions.
The model we will use is the simple Normal model,
d
a
t
a
=
Normal(mean=
true value
,unknown
σ
)
which is the same as the previous example, except that the deviation, σ
The resulting distribution for the “true value”, μ Student-t distribution (Section 7.3),
P
(
μ
|
d
a
t
a
)
=
S
t
u
d
e
n
t
d
o
f
=
N
−
1
(
x
¯
,
S
/
N
)
The best estimates for the true length-values of each type is given by their sample means,
μ
^
s
e
t
o
s
a
=
1.4
+
1.4
+
1.3
+
1.5
+
1.4
5
=
1.40
μ
^
v
i
r
g
i
n
i
c
a
=
6.0
+
5.1
+
5.9
+
5.6
+
5.8
5
=
5.68
μ
^
v
e
r
s
i
c
o
l
o
r
=
4.7
+
4.5
+
4.9
+
4.0
+
4.6
5
=
4.54
and the sample deviations for each is given by
S
s
e
t
o
s
a
=
1
5
−
1
·
(
(
1.4
−
1.40
)
2
+
(
1.4
−
1.40
)
2
+
(
1.3
−
1.40
)
2
+
(
1.5
−
1.40
)
2
+
(
1.4
−
1.40
)
2
)
=
=
0.07
S
v
i
r
g
i
n
i
c
a
=
1
5
−
1
·
(
(
6.0
−
5.68
)
2
+
(
5.1
−
5.68
)
2
+
(
5.9
−
5.68
)
2
+
(
5.6
−
5.68
)
2
+
(
5.8
−
5.68
)
2
)
=
=
0.36
S
v
e
r
s
i
c
o
l
o
r
=
1
5
−
1
·
(
(
4.7
−
4.54
)
2
+
(
4.5
−
4.54
)
2
+
(
4.9
−
4.54
)
2
+
(
4.0
−
4.54
)
2
+
(
4.6
−
4.54
)
2
)
=
0.34
The posterior probability distributions, shown in Figure 9.1, have the following form:
P
(
μ
s
e
t
o
s
a
|
d
a
t
a
)
=
S
t
u
d
e
n
t
d
o
f
=
4
(
1.40
,
0.07
/
5
)
P
(
μ
v
i
r
g
i
n
i
c
a
|
d
a
t
a
)
=
S
t
u
d
e
n
t
d
o
f
=
4
(
5.68
,
0.36
/
5
)
P
(
μ
v
e
r
s
i
c
o
l
o
r
|
d
a
t
a
)
=
S
t
u
d
e
n
t
d
o
f
=
4
(
4.64
,
0.34
/
5
)
It is clear from the picture that they are very well separated, but we can quantify this by looking at the probability that the difference between their means is greater than zero.
Figure 9.1. Probability distributions for the subset of iris petal lengths. Each distribution follows a Student-t form.The probability of their difference approximately takes the form of a Student's t distribution, with the same center and deviation shown for the Normal in Section 7.2. Here we do the calculation between the closest two iris types, Virginica and Versicolor :
Note
This approximation is called Welch's method. The exact analysis is beyond this book, but numerically one can calculate it and it doesn't differ from this approximate analysis in any significant way. Essentially you calculate P ( μ v e r s i c o l o r > μ v i r g i n i c a | d a t a ) P ( μ v e r s i c o l o r | d a t a ) × P ( μ v i r g i n i c a | d a t a )
μ
d
i
f
f
=
5.68
−
4.64
=
1.04
σ
d
i
f
f
=
0.36
2
5
+
0.34
2
5
=
0.22
The degrees of freedom used for this Student's t distribution is approximately the smallest one from the two samples, or in this case (since both samples have the same number of data points), dof=4. The resulting posterior probability distribution for the difference of means is shown in Figure 9.2.
We observe that the difference of the means is over 4 times the deviation away from zero, so even with 4 degrees of freedom, this is significant at the 99% level. We can be highly certain that these two species have different petal lengths, and that the difference observed is not just a product of the random sample.
Figure 9.2. Probability distributions for the difference between iris petal lengths for the closest two iris types, Virginica and Versicolor . The distribution follows a Student-t form, and clearly shows significant probability (greater than 99%) for being greater than zero. 