We now use a maximum likelihood test to compare two random samples from exponential distributions.
Example
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 |
Assuming that survival times are exponentially distributed, perform a likelihood ratio test for whether the death rate is different for those getting the new drug.
The assumption of exponential distributions means that
\[ \{X_{C,1}, \dots, X_{C,30}\} \;\;\sim\;\; \ExponDistn(\lambda_C) \\ \{X_{D,1}, \dots, X_{D,21}\} \;\;\sim\;\; \ExponDistn(\lambda_D) \]The hypotheses of that we want to test are:
The log-likelihood for the big model is
\[ \begin{align} \ell(\lambda_C, \lambda_D) \;\;&=\;\; \left(30 \log \lambda_C - \left(\sum_{i}{x_{C,i}}\right) \lambda_C \right) + \left(21 \log \lambda_D - \left(\sum_{i}{x_{D,i}}\right) \lambda_D \right) \\ &=\;\; 30 \log \lambda_C - 182.48 \lambda_C + 21 \log \lambda_D - 222.87 \lambda_D \end{align} \]which simplifies for the small model to be
\[ \ell(\lambda) \;\;=\;\; 51 \log \lambda - 405.35 \lambda \]Big model, \(\mathcal{M}_B\)
For the big model, the maximum likelihood estimates of the two unknown parameters are
\[ \hat{\lambda}_C \;\;=\;\; \frac{n_C}{\sum{x_{C,i}}} \;\;=\;\; \frac{30}{182.48} \;\;=\;\; 0.1644 \\ \hat{\lambda}_D \;\;=\;\; \frac{n_D}{\sum{x_{D,i}}} \;\;=\;\; \frac{21}{222.87} \;\;=\;\; 0.0942 \]The maximum possible value for the log-likelihood is found by replacing \(\lambda_C\) and \(\lambda_D\) in the log-likelihood with these values,
\[ \ell(\mathcal{M}_B) \;\;=\;\; \ell(\hat{\lambda}_C,\hat{\lambda}_D) \;\;=\;\; -154.77 \]Small model, \(\mathcal{M}_S\)
When the two exponential parameters are equal, their maximum likelihood estimate is
\[ \hat{\lambda} \;\;=\;\; \frac{n}{\sum{x_i}} \;\;=\;\; \frac{51}{405.35} \;\;=\;\; 0.1258 \]giving log-likelihood
\[ \ell(\mathcal{M}_S) \;\;=\;\; \ell(\hat{\lambda}) \;\;=\;\; -156.72 \]Likelihood ratio test
The test statistic is
\[ \chi^2 \;\;=\;\; 2\left(\ell(\mathcal{M}_B) - \ell(\mathcal{M}_S)\right) \;\;=\;\; 3.906 \]Since there is one more unknown parameter in the big model, this should be compared to the \(\ChiSqrDistn(1 \text{ df})\) distribution. Its upper tail probability above 3.906 is 0.048 and this is the p-value for the test.
Since the p-value is just under 5%, we should conclude that there is only moderately strong evidence that the survival rate is different for those getting the new drug treatment.