Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 154

STEP 3. Repeat Steps 1 and 2 for C times where C is suffi-
ciently large (i.e., from 200 to 1000).

Each FDM ( B, A ) calculated from a Bootstrap sample is written as a
vector with K ( K -1) /2 entries. The collection of all FDM ( B, A ) obtained
in this way is written as a matrix that has K ( K -1) /2 rows and C
columns. Each FDM is sorted according to the landmarks that define
the linear distance. Each column is a form difference matrix in vector
format obtained at the end of Step 2, and each row represents C form
difference ratios for a linear distance between a specified pair of land-
marks. To obtain a confidence interval for each linear distance, the
ratios in each row are sorted in increasing order. The 100(1-

)% confi-

dence interval is delimited by removing the first /2

% of the sorted entries. The minimum and maximum entries remain-
ing in that row constitute the lower and upper confidence limits for
that particular linear distance. If the interval of entries spanning the
lower and upper confidence limits contains the value 1, then it is like-
ly that the particular distance is not different in the two populations.
The interval also provides some idea as to the range of values that par-
ticular ratios might take. For example, if a confidence interval is
(1.3,1.7) we know that this particular linear distance is larger in sam-
ple B relative to sample A , but also that the linear distance in sample
B is likely to be 30% to 70% larger than the same distance in sample
A . Similar statements may be made separately for each row using the
values obtained to estimate the confidence interval for each linear dis-
tance.

/2 % and the last

4.8.3 The Monte Carlo method for obtaining confidence intervals.

As in the previous case, let A 1 , A 2 , A 3 ,… A n and B 1 , B 2 , B 3 …, B m denote
the two samples.

STEP 1. Estimate the mean form and the variance-covariance
matrix for samples A and B .
STEP 2. Use estimates of the mean form and variance-covari-
ance matrix based on sample A to generate a new sample of
observations (N n ) using the Gaussian perturbation model.

STEP 3 . Use the estimates of the mean form and variance-
covariance matrix based on sample B to generate a new
sample of observations (N

m ) using the Gaussian perturba-

tion model.