boothomvart

Bootstrap Homogeneity of Variances Test T Analytical Approach.

The bootstrap is a way of estimating the variability of a statistic from a single data set by resampling it independently and with equal probabilities (Monte Carlo resampling). Allows the estimation of measures where the underlying distribution is unknown or where sample sizes are small. Their results are consistent with the statistical properties of those analytical methods (Efron and Tibshirani, 1993).

The name 'bootstrap' originates from the expression 'pulling yourself up by your own bootstraps' and refers to the basic idea of the bootstrap, sampling with replacement from the data. In this way a large number of 'bootstrap samples' is generated, each of the same size as the original data set. From each bootstrap sample the statistical parameter of interest is calculated (Wehrens and Van der Linden, 1997).

Here, we use the Non-parametric Bootstrap. Non-parametric bootstrap is simpler. It does not use the structure of the model to construct artificial data. The data is instead directly resampled with replecement.

The homogeneity of variances test is a useful tool in many scientific applications. Boos and Brownie (2004) and Conover et al. (1981) give a broad review.

Cahoy (2010) proposed a variance-based statistic that led to a bootstrap test for heterogeneity of variances, for any distribution and with a slight modification of the Alam and Cahoy's test (1999). This procedure, who used a generalized box-type acceptance region is shown to be more sensitive to slight deviations from the null specifications. Cahoy (2010) remarks that the properties of the test may change when there are more than four populations involved, and these populations are not from a location-scale family and may have different kurtosis. Meaning that experimenters should exercise caution when this method is used in practice. Within the boundaries of the study, he generally recommend the test T under most conditions.

As Cahoy (2010) did, here a m-file analytical procedure using bootstrap method is developed as an alternative to the homogeinity of variances test.

BOOTHOMVART treats NaN values as missing values, and removes them.

Syntax: function boothomvart(x,s,alpha)

Inputs:
x - data nx2 matrix (Col 1 = data; Col 2 = sample code)
s - boot times or number of Bootstrap simulations (resamplings)
alpha - significance level (default=0.05)

Outputs:
- Summary statistics from the samples
- Decision on the null-hypothesis tested

--We would appreciate any suggestions to improve this m-code in order to reduce the elapsed time. The execution of the example here presented, with 3000 resamplings takes near 38 seconds.--