bootgmregress

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.

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 vector [yi, xi] is instead directly resampled with replecement. The parameters are constructed from these pairs.

Model II regression should be used when the two variables in the regression equation are random and subject to error, i.e. not controlled by the researcher. Model I regression using ordinary least squares underestimates the slope of the linear relationship between the variables when they both contain error. According to Sokal and Rohlf (1995), the subject of Model II regression is one on which research and controversy are continuing and definitive recommendations are difficult to make.

BOOTGMREGRESS is a bootstrap Model II procedure. It takes s bootstrap samples and standardize variables before the slope is computed. Each of the two variables is transformed to have a mean of zero and a standard deviation of one. The resulting slope is the geometric mean of the linear regression coefficient of Y on X. Ricker (1973) coined this term and gives an extensive review of Model II regression. It is also known as Bootstrap Standard Major Axis.

[B,BINTR,BINTJM] = BOOTGMREGRESS(S,X,Y,ALPHA) returns the vector B of bootstrap regression coefficients in the linear Model II and a matrix BINT of the given confidence intervals for B by the Ricker (1973) [uses the Student t-distribution with n-2 df] and Jolicoeur and Mosimann (1968)-McArdle (1988) [uses the asymmetrical F distribution] procedure. The confidence intervals are based on the bias-correction method.

Confidence intervals is a better way of describing variability when when dealing with non-normal distributions. A 95confidence interval contains the middle 95of the numbers in a list.

Efron and Tibshirani (1993) suggest to do 100 iterations for determination of standard error and 1000 for confidence intervals are usually adequate.

The herein Matlab bootstrap procedure is by first time developed using the statistical fundamentals given by Efron and Tibshirani (1993). Taking advantage of the bootstrp (Matlab Statistics Toolbox) and gmregress
(http://www.mathworks.com/matlabcentral/fileexchange/27918-gmregress)
m-functions. Plotnick (1989) gives a broad review of this procedure and it is applied to the simple allometry equation transformed into a linear regression by log-transformation of the data.

As well as Plotnick (1989) did, this Matlab procedure can be applied to other models relating to linear variables X-Y.

BOOTGMREGRESS treats NaNs in X or Y as missing values, and removes them.

Syntax: function [b,bintr,bintjm] = bootgmregress(s,x,y,c,alpha)

Inputs:
s - boot times or number of Bootstrap simulations (resamplings)
x - independent variable
y - dependent (response) variable
g - graphical display option:histograms (1=not/2=yes) [default=1]
l - bins [default=50]
alpha - significance level (default=0.05)

Outputs:
b - bootstrap geometric mean regression parameters (intercept and slope)
bintr - CI by the Ricker's procedure
bintjm - CI by the Jolicoeur-Mosimann's procedure
Histograms at your choice. You can modify it size by adjusting the margins or simply maximazing it.

Alternatively, at your choice, you can call as an output:
bootstat - bootstrap statistics of the covariance matrix
bootsamp - index matrix of bootstrap samples
R - correlations by the bootstrap covariance matrices
bs - pairs intercept-slope of each re-samling
bintrin - Ricker's intercepts confidence interval
bintrsl - Ricker's slopes confidence interval
bintjmin - Jolicoeur-Mosimann's intercepts confidence interval
bintjmsl - Jolicoeur-Mosimann's slopes confidence interval