Chi Squared Test
\(H_0\): Observed number of deaths at each age are consistent with expected number predicted by graduated rates
Test statistic:
\[ X = \sum_x z_x^2\sim\chi^2_{n-p} \]
- General goodness of fitness test
- Does not tell us about direction of any bias or grouping of directional deviations.
- Squared statistic, so if it was consistently biased in one direction but the deviations were small it would show as good in this test.
- A few large deviations could be offset by a lot of very small deviations
Cumulative Deviations Test
\(H_0\): Observed number of deaths at each age are consistent with expected number predicted by graduated rates
Test statistic:
\[ \frac{\sum_x A_x-E_x}{\sqrt{\left(\sum_xVar(D_x)\right)}}\] * General goodness of fit test * Detects overall bias or long runs of deviations of the same sign + However positive deviations over one age may cancel out negatives * A high value of the test statistic indicates that the rates are too biased or that the actual variance is higher than that preducted by the assumed model (possibly due to duplicate policies)
Standardised Deviations Test
\(H_0\): The observed pattern of the individual standardised deviations is consistent with the standard normal distribution
Test Statistic:
\[ X = \sum_n\frac{(\text{actual - expected})^2}{\text{expected}}\sim\chi^2_{n-1}\] * Compare the observed amount of deviations in each interval to see the distribution of deviations compared to standard normal distribution.
Signs Test
- \(H_0\):
- \(m\) is total number of deviations
\[ P\sim Bin(m,0.5)\]
- Test statistic:
\[ P = \text{Number of }z_x\text{ that are positive}\]
- For large \(m\) we can use the normal approximation:
- Note we use a continuity approximation of \(P-0.5\) when calculating critical value:
\[ P\sim N(0.5m, 0.25m)\]
- Tests the balance between positive and negative deviations
- If number of positive deviations \(P\) is very high or very low, this indicates that the rates are on average too low or too high, giving information about direction of bias.
- Pattern of signs can indicate range of ages where bias is worst.
- Qualitative rather than quantitative
- Does not give indication of size of signs, but rather how many exist
Grouping of Signs Test (Steven’s Test)
\(H_0\): The given \(n_1\) positive and \(n_2\) negative deviations are in random order
Test Statistic:
\[ G = \text{Number of Groups of Positive } z_x's\]
- Normal Approximation (Noting we need to use continuity adjustment):
\[ G\sim N\left(\frac{n_1(n_2+1)}{(n_1+n_2)},\frac{(n_1n_2)^2}{(n_1+n_2)^3}\right)\]
- Test for overgraduation/groups of deviations with the same signs
- One sided test
- Can have different results if you consider positive or negative deviations as reference
Serial Correlation Test
\(H_0\): The individual standardised deviations at consecutive ages are independent
Test Statistic:
\[ r_j\sim N\left(0,\frac{1}{m}\right)\]
Test for overgraduation/dependence between lagged correlations
- In overgraduation the graduated curve will tend to stay on the same side of the crude rates for relatively long periods of time
One sided test
Positive correlations in one age range can be cancelled out by opposite correlations in another part, which often happens resulting in this being a rather weak test. Therefore the signs test and grouping of signs test is typucally more powerful in detecting overgraduation.
Comparing Experience with Standard Table
- We can do these tests with reference to standard table rates rather than graduated rates if we are interested to see if the standard table rates are a good fit for the data themselves.
*\(H_0\): The true mortality rates are the same as those in the standard tables