MOStest {vegan} | R Documentation |

Mitchell-Olds & Shaw test concerns the location of the highest (hump)
or lowest (pit) value of a quadratic curve at given points. Typically,
it is used to study whether the quadratic hump or pit is located
within a studied interval. The current test is generalized so that it
applies generalized linear models (`glm`

) with link
function instead of simple quadratic curve. The test was popularized
in ecology for the analysis of humped species richness patterns
(Mittelbach et al. 2001), but it is more general. With logarithmic
link function, the quadratic response defines the Gaussian response
model of ecological gradients (ter Braak & Looman 1986), and the test
can be used for inspecting the location of Gaussian optimum within a
given range of the gradient. It can also be used to replace Tokeshi's
test of “bimodal” species frequency distribution.

MOStest(x, y, interval, ...) ## S3 method for class 'MOStest': plot(x, which = c(1,2,3,6), ...) fieller.MOStest(object, level = 0.95) ## S3 method for class 'MOStest': profile(fitted, alpha = 0.01, maxsteps = 10, del = zmax/5, ...) ## S3 method for class 'MOStest': confint(object, parm = 1, level = 0.95, ...)

`x` |
The independent variable or plotting object in `plot` . |

`y` |
The dependent variable. |

`interval` |
The two points at which the test statistic is
evaluated. If missing, the extremes of `x` are used. |

`which` |
Subset of plots produced. Values `which=1` and
`2` define plots specific to `MOStest` (see Details), and
larger values select a graphs of `plot.lm` (minus2). |

`object, fitted` |
A result object from `MOStest` . |

`level` |
The confidence level required. |

`alpha` |
Maximum significance level allowed. |

`maxsteps` |
Maximum number of steps in the profile. |

`del` |
A step length parameter for the profile (see code). |

`parm` |
Ignored. |

`...` |
Other variables passed to functions. Function
`mitchell.olds.test` passes these to `glm` so that
these can include `family` . The other functions pass
these to underlying graphical functions. |

The function fits a quadratic curve *μ = b_0 + b_1 x + b_2 x^2*
with given `family`

and link function. If *b_2 < 0*,
this defines a unimodal curve with highest point at
*u = -b_2/(2 b_3)* (ter Braak & Looman 1986). If *b_2 > 0*,
the parabola has a minimum at *u* and the response is sometimes
called “bimodal”. The null hypothesis is that the extreme
point *u* is located within interval given by points *p_1* and
*p_2*. If the extreme point *u* is exactly at *p_1*, then
*b_1 = 0* on shifted axis *x - p_1*. In the test, origin of
`x`

is shifted to the values *p_1* and *p_2*, and the
test statistic is the value of the first degree coefficient with its
significance as estimated by the `summary.glm`

function(Mitchell-Olds & Shaw 1987).

The test is often presented as a general test for the location of the hump, but it really is dependent on the quadratic fitted curve. If the hump is of different form than quadratic, the test may be insignificant.

Because of strong assumptions in the test, you should use the support
functions to inspect the fit. Function `plot(..., which=1)`

displays the data points, fitted quadratic model, and its approximate
95% confidence intervals (2 times SE). Function `plot`

with
`which = 2`

(requires `ellipse.glm`

in
package ellipse) displays the approximate confidence interval of
the polynomial coefficients, together with two lines indicating the
combinations of the coefficients that produce the evaluated points of
`x`

. Moreover, the cross-hair shows the approximate confidence
intervals (2 times SE) for the polynomial coefficients ignoring their
correlations. Higher values of `which`

produce corresponding
graphs from `plot.lm`

. That is, you must add 2 to the
value of `which`

in `plot.lm`

.

Function `fieller.MOStest`

approximates the confidence limits
of the location of the extreme point (hump or pit) using Fieller's
theorem following ter Braak & Looman (1986). The test is based on
quasideviance except if the `family`

is `poisson`

or `binomial`

. Function `profile`

evaluates the profile
deviance of the fitted model, and `confint`

finds the profile
based confidence limits following Oksanen et al. (2001).

The test is typically used in assessing the significance of diversity hump against productivity gradient (Mittelbach et al. 2001). It also can be used for the location of the pit (deepest points) instead of the Tokeshi test. Further, it can be used to test the location of the the Gaussian optimum in ecological gradient analysis (ter Braak & Looman 1986, Oksanen et al. 2001).

The function is based on `glm`

, and it returns the result
of object of `glm`

amended with the result of the test. The new
items in the `MOStest`

are:

`isHump ` |
`TRUE` if the response is a
hump. |

`isBracketed` |
`TRUE` if the hump or the pit is bracketed by
the evaluated points. |

`hump` |
Sorted vector of location of the hump or the pit and the points where the test was evaluated. |

`coefficients` |
Table of test statistics and their significances. |

Function `fieller.MOStest`

is based on package optgrad in
the Ecological Archives
(http://www.esapubs.org/archive/ecol/E082/015/default.htm)
accompanying Oksanen et al. (2001). The Ecological Archive package
optgrad also contains profile deviance method for the location
of the hump or pit, but the current implementation of `profile`

and `confint`

rather follow the example of
`profile.glm`

and `confint.glm`

in
the MASS package.

Jari Oksanen

Mitchell-Olds, T. & Shaw, R.G. 1987. Regression analysis of natural
selection: statistical inference and biological
interpretation. *Evolution* 41, 1149–1161.

Mittelbach, G.C. Steiner, C.F., Scheiner, S.M., Gross, K.L., Reynolds,
H.L., Waide, R.B., Willig, R.M., Dodson, S.I. & Gough, L. 2001. What is
the observed richness between species richness and productivity?
*Ecology* 82, 2381–2396.

Oksanen, J., Läärä, E., Tolonen, K. & Warner, B.G. 2001. Confidence
intervals for the optimum in the Gaussian response
function. *Ecology* 82, 1191–1197.

ter Braak, C.J.F & Looman, C.W.N 1986. Weighted averaging, logistic
regression and the Gaussian response model. *Vegetatio* 65,
3–11.

The no-interaction model can be fitted with `humpfit`

.

## The Al-Mufti data analysed in humpfit(): mass <- c(140,230,310,310,400,510,610,670,860,900,1050,1160,1900,2480) spno <- c(1, 4, 3, 9, 18, 30, 20, 14, 3, 2, 3, 2, 5, 2) mod <- MOStest(mass, spno) ## Insignificant mod ## ... but inadequate shape of the curve op <- par(mfrow=c(2,2), mar=c(4,4,1,1)+.1) plot(mod) ## Looks rather like log-link with Poisson error and logarithmic biomass mod <- MOStest(log(mass), spno, family=quasipoisson) mod plot(mod) par(op) ## Infinite confidence limits (NA) fieller.MOStest(mod) ## Finite limits with the profile confint(mod) plot(profile(mod))

[Package *vegan* version 1.16-32 Index]