smatr (version 2.1)

slope.com: Common slope test amongst several allometric lines

Description

Test if several major axis or standardised major axis lines share a common slope

Usage

slope.com(y, x, groups, method = 'SMA', alpha = 0.05, data = NULL, intercept = TRUE,
   V = array(0, c(2, 2, length(unique(groups)))), group.names = sort(unique(groups)),
   ci = TRUE, bs = TRUE)

Arguments

y

The Y-variable for all observations (as a vector)

x

The X-variable for all observations (as a vector)

groups

Coding variable identifying which group each observation belongs to (as a factor or vector)

method

The line fitting method: [object Object],[object Object],[object Object]

alpha

The desired confidence level for the 100(1-alpha)% confidence interval for the common slope. (Default value is 0.05, which returns a 95% confidence interval.)

data

(optional) data frame containing the data

intercept

(logical) Whether or not the line includes an intercept. [object Object],[object Object]

V

The estimated variance matrices of measurement error, for each group. This is a 3-dimensional array with measurement error in Y in the first row and column, error in X in the second row and column, and groups running along the third dimension. Default is

group.names

(optional: rarely required). A vector containing the labels for `groups'. (Only actually useful for reducing computation time in simulation work).

ci

(logical) Whether or not to return a confidence interval for the common slope.

bs

(logical) Whether or not to return the slopes for the separate groups, with confidence intervals.

Value

• lr
  The (Bartlett-corrected) likelihood ratio statistic testing for common slope
• p
  The P-value of the test. This is calculated assuming that lr has a chi-square distribution with (g-1) df, if there are g groups
• b
  The common slope estimate
• varb
  The sample variance of the common slope
• ci
  A 100(1-alpha)% confidence interval for the common slope
• lambda
  The error variance ratio - the ratio of error variance in y to error variance in x. For MA, this is assumed to be 1. for SMA, this is assumed to be $b^2$. For the `lamest' method, the error variance ratio is estimated from the data under the common slope assumption.
• bs
  The slopes and confidence intervals for data from each group.

Details

For several bivariate groups of observations, this function tests if the line-of-best-fit has a common slope for all samples, when the line-of-best-fit is estimated using the major axis, standardised major axis, or a more general version of these methods in which the error variance ratio is estimated from the data.

The test assumes the following:

1. each group of observations was independently sampled
2. y and x are linearly related within each group
3. residuals independently follow a normal distribution with equal variance at all points along the line, within each group

Note that we do not need to assume equal variance across groups, unlike in the standard test for common slope for linear regression.

The assumptions can be visually checked by plotting residuals against fitted axis scores, and by constructing a Q-Q plot* of residuals against a normal distribution. An appropriate residual variable is $y-\hat{\beta}_i\times x$, and for fitted axis scores $y+\hat{\beta}_i x$ (for SMA) or $\hat{\beta}_i y+lx$ (for MA or `lamest'), where $\hat{\beta}_i$ represents the estimated slope for group i, and l the error variance ratio.

If a plot of residuals against fitted axis scores is produced in which the estimated common slope (b) is used rather than the slopes estimated within each group ($\hat{\beta}_i$), this can be used as a visual test for common slope. If there is a distinct increasing or decreasing trend within any of the groups, this suggests that all groups do not share a common slope.

The common slope is estimated from a maximum of 100 iterations, convergence is reached when the change in b is $< 10^{-6}$.

References

Warton D. I. and Weber N. C. (2002) Common slope tests for bivariate structural relationships. Biometrical Journal 44, 161--174.

Warton D. I., Wright I. J., Falster D. S. and Westoby M. (2006) A review of bivariate line-fitting methods for allometry. Biological Reviews (in press)

See Also

line.cis, elev.com, shift.com

Examples

#load leaf longevity data
data(leaflife)

#plot the data, with different symbols for different groups.
plot(leaflife$lma, leaflife$longev, type='n', log='xy', xlab=
   'leaf mass per area [log scale]', ylab='leaf longevity [log scale]')
colours <- c('blue', 'red', 'green', 'yellow')
points(leaflife$lma, leaflife$longev,
   col=colours[as.numeric(leaflife$site)])
legend(55, 5, as.character(unique(leaflife$site)), col=colours,
   pch=rep(1,4))

#test for common SMA slope of log(leaf longevity) vs log(LMA),
#across species sampled at different sites:
fit <- slope.com(log10(longev), log10(lma), site, data = leaflife)
fit

#Residual vs fits plots for SMA fit of each site
y = log10(leaflife$longev)
x = log10(leaflife$lma)
site=leaflife$site
par( mfrow=c(2,2) )
plot(y[site==1] + fit$bs[1,1] * x[site==1], y[site==1] - fit$bs[1,1] 
   * x[site==1], xlab='fits (site 1)', ylab='residuals (site 1)')
plot(y[site==2] + fit$bs[1,2] * x[site==2], y[site==2] - fit$bs[1,2]
   * x[site==2], xlab='fits (site 2)', ylab='residuals (site 2)')
plot(y[site==3] + fit$bs[1,3] * x[site==3], y[site==3] - fit$bs[1,3]
   * x[site==3], xlab='fits (site 3)', ylab='residuals (site 3)')
plot(y[site==4] + fit$bs[1,4] * x[site==4], y[site==4] - fit$bs[1,4]
   * x[site==4], xlab='fits (site 4)', ylab='residuals (site 4)')

#Test for common SMA slope amongst species at low rainfall sites
#with different levels of soil nutrients
leaf.low.rain=leaflife[leaflife$rain=='low',]
slope.com(log10(longev), log10(lma), soilp, data=leaf.low.rain)

#test for common MA slope:
slope.com(log10(longev), log10(lma), site, data = leaflife,
   method='MA')

#test for common MA slope, and produce a 90% CI for the common slope:
slope.com(log10(longev), log10(lma), site, data = leaflife,
   method='MA', alpha=0.1)