Test for no mean shift along a common (standardised) major axis

Test if several groups of observations have no shift in location along major axis or standardised major axis lines with a common slope. This can now be done via sma(y~x+groups, type="shift"), see help on the sma function.

shift.com(
  y,
  x,
  groups,
  data = NULL,
  method = "SMA",
  intercept = TRUE,
  robust = FALSE,
  V = array(0, c(2, 2, length(unique(groups)))),
  group.names = sort(unique(groups))
)

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)

data

Deprecated. Use with() instead (see Examples).

method

The line fitting method:

'SMA' or 1: standardised major axis (this is the default)
'MA' or 2: major axis

intercept

(logical) Whether or not the line includes an intercept.

FALSE: no intercept, so the line is forced through the origin
TRUE: an intercept is fitted (this is the default)

robust

If TRUE, uses a robust method to fit the lines and construct the test statistic.

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 that there is no measurement error.

group.names

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

Value

stat: The Wald statistic testing for no shift along the common axis
p: The P-value of the test. This is calculated assuming that stat has a chi-square distribution with (g-1) df, if there are g groups
f.mean: The fitted axis means for each group

Details

Calculates a Wald statistic to test for no shift along several MA's or SMA's of common slope. This is done by testing for equal fitted axis means across groups.

Note that this test is only valid if it is reasonable to assume that the axes for the different groups all have the same slope.

The test assumes the following:

each group of observations was independently sampled
the axes fitted to all groups have a common slope
y and x are linearly related within each group
fitted axis scores 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 tests comparing several linear regression lines.

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, available using

plot.sma(sma.object,which="residual").

On a residual plot, 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.

A plot of residual scores against fitted axis scores can also be used as a visual test for no shift. If fitted axis scores systematically differ across groups then this is evidence of a shift along the common axis.

Setting robust=TRUE fits lines using Huber's M estimation, and modifies the test statistic as proposed in Taskinen & Warton (in review).

The common slope (\(\hat{\beta}\)) is estimated from a maximum of 100 iterations, convergence is reached when the change in \(\hat{\beta}\) is \(< 10^{-6}\).

References

Warton D. I., Wright I. J., Falster D. S. and Westoby M. (2006) A review of bivariate line-fitting methods for allometry. Biological Reviews 81, 259--291.

Taskinen, S. and D.I. Warton. in review. Robust tests for one or more allometric lines.

Author

Warton, D.I.David.Warton@unsw.edu.au, J. Ormerod, & S. Taskinen

Examples


#load leaf longevity data
data(leaflife)

#Test for common SMA slope amongst species at low rainfall sites
#with different levels of soil nutrients
leaf.low.rain=leaflife[leaflife$rain=='low',]
with(leaf.low.rain, slope.com(log10(longev), log10(lma), soilp))
#> $LR
#> [1] 3.942664
#> 
#> $p
#> [1] 0.04707617
#> 
#> $b
#> [1] 1.345644
#> 
#> $ci
#> [1] 1.064014 1.652252
#> 
#> $varb
#> [1] 0.01671713
#> 
#> $lambda
#> [1] 1.810757
#> 
#> $bs
#>                   high      low
#> slope        1.1825381 1.786551
#> lower.CI.lim 0.9398479 1.257257
#> upper.CI.lim 1.4878965 2.538672
#> 
#> $df
#> [1] 1
#> 

#Now test for no shift along the axes of common slope, for sites
#with different soil nutrient levels but low rainfall:
with(leaf.low.rain, shift.com(log10(longev), log10(lma), soilp))
#> $stat
#>           [,1]
#> [1,] 0.9125369
#> 
#> $p
#>           [,1]
#> [1,] 0.3394422
#> 
#> $f.mean
#> [1] 3.097899 3.259749
#> 
#> $df
#> [1] 1
#> 

#Now test for no shift along the axes of common slope, for sites
#with different soil nutrient levels but low rainfall:
with(leaf.low.rain,shift.com(log10(longev), log10(lma), soilp, method='MA'))
#> $stat
#>          [,1]
#> [1,] 1.030127
#> 
#> $p
#>           [,1]
#> [1,] 0.3101288
#> 
#> $f.mean
#> [1] 2.376975 2.571328
#> 
#> $df
#> [1] 1
#>