Nested models

In some situations, a particular statistical model can be regarded as a special case of a more complex model (with more parameters). We will call the simpler model the small model, \(\mathcal{M}_S\), and say that it is nested in the more general big model, \(\mathcal{M}_B\). To compare these, we use the hypotheses,

Poisson parameter

The following table describes the number of defective items produced on a production line in 20 successive days.

1
2
3
4
2
3
2
5
5
2
4
3
5
1
2
4
0
2
2
6

It might be assumed that the data are a random sample from a \(\PoissonDistn(\lambda)\) distribution, and we might want to test whether the rate of defective items was \(\lambda = 2\) per day. Since the \(\PoissonDistn(2)\) distribution is a special case of the \(\PoissonDistn(\lambda)\) distribution,

Exponential means

Clinical records give the survival time, in months from diagnosis, of 30 sufferers from a certain disease as

9.73
5.56
4.28
4.87
1.55
6.20
1.08
7.17
28.65
6.10
16.16
9.92
2.40
6.19
7.67
1.11
4.66
4.35
7.31
3.28
13.38
3.08
0.41
4.33
2.16
4.49
0.75
 
4.45
10.29
0.90
 

In a clinical trial of a new drug treatment, 21 sufferers had survival times of:

22.07
12.47
6.42
8.15
0.64
20.04
17.49
2.22
3.00
28.09
3.94
8.59
4.26
32.82
8.32
2.12
18.53
 
9.95
4.25
 
3.70
5.82
 

Is there any difference in survival times for those using the new drug?

An exponential model might be considered a reasonable model for the data in each group. This would have a common death rate in both groups, \(\lambda\), if the drug had no effect on survival times, and different rates for the control group, \(\lambda_C\), and the group getting the new drug, \(\lambda_D\).

Exponential vs Weibull distribution

The \(\ExponDistn(\lambda)\) distribution is a special case of the \(\WeibullDistn(\alpha, \lambda)\) distribution corresponding to \(\alpha = 1\). Testing whether the failure rate is constant can therefore be done using the following small and big models.

Exponential vs Gamma

In a similar way, the \(\ExponDistn(\lambda)\) distribution is a special case of the \(\GammaDistn(\alpha, \lambda)\) distribution corresponding to \(\alpha = 1\).