310
18 Dating the Last Migration to New Zealand
latter to render them more comparable with the former. This leads to the following
standardized data, relating to an 1850 base:
1. Mataatua: 15, 20, 16, 12, 18, 13, 15, 16, 13, 17, 16, 16, 14, 15. These whakapapa
lengths give mean and standard deviation estimates x 1 = 15.429, ˆ
σ 1 = 2.102.
2. Te Arawa: 17, 18, 15, 16, 16, 18, 19, 16, 18, 17, 20, 14, 13, 13, 15, 16. These
give x 2 = 16.313, ˆ
σ 2 = 2.024.
3. Tainui: 20, 23, 17, 18, 15, 18, 16. These lead to x 3 = 18.143, ˆ
σ 3 = 2.673.
4. Aotea: 16, 17, 21, 17, 19, 16, 20. These lead to estimates x 4 = 18.000, ˆ
σ 4 = 2.000.
5. Tokomaru: 18, 14. These provide x 5 = 16.000, ˆ
σ 5 = 2.828.
6. Kurohaupo: 17, 13. These provide x 6 = 15.000, ˆ
σ 6 = 2.828.
18.5 Statistical Analysis
Standard statistical tests utilizing MINITAB treated the data as samples drawn from
six canoe populations. Probability plots of residuals were used to test for normality,
with 95% confidence intervals. These displayed no evidence of non-normality so it
was appropriate to use Snedecor’s F test for equal variances with the different data
sets. Again there was no evidence for any differences in the variances.
In view of the above, we are justified in using a standard one-way analysis of
variance to examine the null hypothesis H 0 that all canoe population means are
equal against the alternate hypothesis H 1 that at least two are different. This gives
rise to a p value of 0.058. That is, under the assumption that all six canoe population
means are the same, the probability of observing as much or more variation than
is displayed in our data is 0.058. Thus, using 95% confidence intervals, there is no
statistically significant evidence of difference between the mean lengths for the six
canoe genealogies.
Our data has a mean of 16.50 generations with corresponding standard deviation
of 2.345 generations. This provides the basis for a preliminary rough dating of the
Heke. First we note that standard calculation procedures based on generation counts
require some refinement. The head of a lineage and often the next generation will
have been born before the Heke. At the other end of the lineage, an informant in
the 19th or early 20th century will have been elderly, typically a grandparent. Thus
we have one and a fraction generations overcount at the beginning of a lineage and
two and a fraction generations undercount at the end of a lineage. The following
formulation allows us to refine our calculations to allow for such corrections.
For a given lineage from the Heke down to 1850 (a time t (years) later), the
successive birth epochs may be modelled as the events of an equilibrium renewal
process (see Cox [14]). The number N(t) of renewal events occurring then has mean
E(N(t) given exactly by E(N(t)) = t/μ where μ (years) represents the mean length of
a generation. That is, the time interval prior to 1850 that we wish to determine may
be obtained by multiplying the number of renewal events by the average length of
a generation. Taking the number of renewal points corresponds to adding one to the
16.50 generations to derive a mean of 17.50 generations. The mean has a standard
18 Dating the Last Migration to New Zealand
latter to render them more comparable with the former. This leads to the following
standardized data, relating to an 1850 base:
1. Mataatua: 15, 20, 16, 12, 18, 13, 15, 16, 13, 17, 16, 16, 14, 15. These whakapapa
lengths give mean and standard deviation estimates x 1 = 15.429, ˆ
σ 1 = 2.102.
2. Te Arawa: 17, 18, 15, 16, 16, 18, 19, 16, 18, 17, 20, 14, 13, 13, 15, 16. These
give x 2 = 16.313, ˆ
σ 2 = 2.024.
3. Tainui: 20, 23, 17, 18, 15, 18, 16. These lead to x 3 = 18.143, ˆ
σ 3 = 2.673.
4. Aotea: 16, 17, 21, 17, 19, 16, 20. These lead to estimates x 4 = 18.000, ˆ
σ 4 = 2.000.
5. Tokomaru: 18, 14. These provide x 5 = 16.000, ˆ
σ 5 = 2.828.
6. Kurohaupo: 17, 13. These provide x 6 = 15.000, ˆ
σ 6 = 2.828.
18.5 Statistical Analysis
Standard statistical tests utilizing MINITAB treated the data as samples drawn from
six canoe populations. Probability plots of residuals were used to test for normality,
with 95% confidence intervals. These displayed no evidence of non-normality so it
was appropriate to use Snedecor’s F test for equal variances with the different data
sets. Again there was no evidence for any differences in the variances.
In view of the above, we are justified in using a standard one-way analysis of
variance to examine the null hypothesis H 0 that all canoe population means are
equal against the alternate hypothesis H 1 that at least two are different. This gives
rise to a p value of 0.058. That is, under the assumption that all six canoe population
means are the same, the probability of observing as much or more variation than
is displayed in our data is 0.058. Thus, using 95% confidence intervals, there is no
statistically significant evidence of difference between the mean lengths for the six
canoe genealogies.
Our data has a mean of 16.50 generations with corresponding standard deviation
of 2.345 generations. This provides the basis for a preliminary rough dating of the
Heke. First we note that standard calculation procedures based on generation counts
require some refinement. The head of a lineage and often the next generation will
have been born before the Heke. At the other end of the lineage, an informant in
the 19th or early 20th century will have been elderly, typically a grandparent. Thus
we have one and a fraction generations overcount at the beginning of a lineage and
two and a fraction generations undercount at the end of a lineage. The following
formulation allows us to refine our calculations to allow for such corrections.
For a given lineage from the Heke down to 1850 (a time t (years) later), the
successive birth epochs may be modelled as the events of an equilibrium renewal
process (see Cox [14]). The number N(t) of renewal events occurring then has mean
E(N(t) given exactly by E(N(t)) = t/μ where μ (years) represents the mean length of
a generation. That is, the time interval prior to 1850 that we wish to determine may
be obtained by multiplying the number of renewal events by the average length of
a generation. Taking the number of renewal points corresponds to adding one to the
16.50 generations to derive a mean of 17.50 generations. The mean has a standard