Analyzing proportions with the Arrington et al. 2002 example
Arrington, Winemiller, Loftus, & Akin (2002) published a data set available from the web. It presents species of fish and what proportion of them were empty stomached when catched. The dataset contained 36000+ catches, which where identified by their Location (Africa, North America, rest of America), by their Trophism (their diet, Detritivore, Invertivore, Omnivore, Piscivore) and by the moment of feeding (Diel: Diurnal or Nocturnal).
The compiled scores can be consulted with
## Location Trophism Diel s n
## 1 Africa Detritivore Diurnal 16 217
## 2 Africa Invertivore Diurnal 76 498
## 3 Africa Invertivore Nocturnal 55 430
## 4 Africa Omnivore Diurnal 2 87
## 5 Africa Piscivore Diurnal 673 989
## 6 Africa Piscivore Nocturnal 221 525
## 7 Central/South America Detritivore Diurnal 68 1589
## 8 Central/South America Detritivore Nocturnal 9 318
## 9 Central/South America Invertivore Diurnal 706 7452
## 10 Central/South America Invertivore Nocturnal 486 2101
## 11 Central/South America Omnivore Diurnal 293 6496
## 12 Central/South America Omnivore Nocturnal 82 203
## 13 Central/South America Piscivore Diurnal 1275 5226
## 14 Central/South America Piscivore Nocturnal 109 824
## 15 North America Detritivore Diurnal 142 1741
## 16 North America Invertivore Diurnal 525 3368
## 17 North America Invertivore Nocturnal 231 1539
## 18 North America Omnivore Diurnal 210 1843
## 19 North America Omnivore Nocturnal 7 38
## 20 North America Piscivore Diurnal 536 1289
## 21 North America Piscivore Nocturnal 19 102One first difficulty with this dataset is that some of the cells are missing (e.g., African fish that are Detrivore during the night). As is the case for other sorts of analyses (e.g., ANOVAs), data with missing cells cannot be analyzed because the error terms cannot be computed.
One solution adopted by Warton & Hui (2011) was to impute the missing value. We are not aware if this is an adequate solution, and if so, what imputation would be acceptable. Consider the following with adequate care.
Warton imputed the missing cells with a very small proportion. In ANOPA, both the proportions and the group sizes are required. We implemented a procedure that impute a count of 0.05 (fractional counts are not possible from observations, but are not forbidden in ANOPA) obtained from a single observation.
Consult the default option with
## [1] 0.05 1.00The analysis is obtained with
The fyi message lets you know that cells are missing;
the Warning message lets you know that these cells were
imputed (you can suppress messages with
options("ANOPA.feedback"="none").
To see the result, use summary(w) (which shows the
corrected and uncorrected statistics) or uncorrected(w) (as
the sample is quite large, the correction will be immaterial…),
## MS df F pvalue
## Location 0.027449 2 0.961802 0.382203
## Diel 0.029715 1 1.041227 0.307536
## Trophism 0.095656 3 3.351781 0.018102
## Location:Diel 0.005277 2 0.184900 0.831187
## Location:Trophism 0.029485 6 1.033146 0.401285
## Diel:Trophism 0.073769 3 2.584868 0.051365
## Location:Diel:Trophism 0.011297 6 0.395837 0.882184
## Error(between) 0.028539 InfThese suggests an interaction Diel : Trophism close to significant.
You can easily make a plot with all the factors using
Figure 1. The proportions in the Arrington et al. 2002 data. Error bars show difference-adjusted 95% confidence intervals.
The missing cells are absent from the plot. To highlight the interaction, restrict the plot to
Figure 1. The proportions as a function of class and Difficulty. Error bars show difference-adjusted 95% confidence intervals.
which shows clearly massive difference between Trophism, and small differences between Omnivorous and Piscivorous fishes with regards to Location.
This can be confirmed by examining simple effects (a.k.a. expected marginal analyzes):
(but it will have to wait for the next version of ANOPA ;-)