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, \(\mathcal{M}_B\).
In this section, we will describe a general method of performing a hypothesis test to compare these two models.
We now give a few examples of small and big models.
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 \(\alpha = 1\).