The function 'emProportions()' performs a simple effect analyses of proportions after an omnibus analysis has been obtained with 'anopa()' according to the ANOPA framework. Alternatively, it is also called an expected marginal analysis of proportions. See Laurencelle and Cousineau (2023) for more.

emProportions(w, formula)

Arguments

w

An ANOPA object obtained from anopa();

formula

A formula which indicates what simple effect to analyze. Only one simple effect formula at a time can be analyzed. The formula is given using a vertical bar, e.g., " ~ factorA | factorB " to obtain the effect of Factor A within every level of the Factor B.

Value

An ANOPA table of the various simple main effets and if relevant, of the simple interaction effets.

Details

emProportions() computes expected marginal proportions and analyzes the hypothesis of equal proportion. The sum of the _F_s of the simple effects are equal to the interaction and main effect _F_s, as this is an additive decomposition of the effects.

References

Laurencelle L, Cousineau D (2023). “Analysis of frequency tables: The ANOFA framework.” The Quantitative Methods for Psychology, 19, 173--193. doi:10.20982/tqmp.19.2.p173 .

Examples


# -- FIRST EXAMPLE --
# This is a basic example using a two-factors design with the factors between 
# subjects. Ficticious data present the number of success according
# to Class (three levels) and Difficulty (two levels) for 6 possible cells
# and 72 observations in total (equal cell sizes of 12 participants in each group).
twoWayExample
#>   Class Difficulty success total
#> 1 First       Easy      11    12
#> 2 First   Moderate       9    12
#> 3 First  Difficult       6    12
#> 4  Last       Easy      10    12
#> 5  Last   Moderate       8    12
#> 6  Last  Difficult       3    12

# As seen the data are provided in a compiled format (one line per group).
# Performs the omnibus analysis first (mandatory):
w <- anopa( {success;total} ~ Difficulty * Class, twoWayExample) 
summary(w)
#>                        MS  df        F   pvalue correction    Fcorr pvalcorr
#> Difficulty       0.136787   2 6.839333 0.001071   1.027778 6.654486 0.001288
#> Class            0.032569   1 1.628455 0.201917   1.013889 1.606147 0.205034
#> Difficulty:Class 0.003660   2 0.183006 0.832763   1.243056 0.147223 0.863102
#> Error(between)   0.020000 Inf                                               

# The results shows an important interaction. You can visualize the data
# using anopaPlot:
anopaPlot(w)

# The interaction is overadditive, with a small differences between Difficulty
# levels in the first class, but important differences between Difficulty for 
# the last class.

# Let's execute the simple effect of Difficulty for every levels of Class
e <- emProportions(w, ~ Difficulty | Class )
#> Not yet programmed...
summary(e)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>     -99     -99     -99     -99     -99     -99 


# -- SECOND EXAMPLE --
# Example using the Arrington et al. (2002) data, a 3 x 4 x 2 design involving 
# Location (3 levels), Trophism (4 levels) and Diel (2 levels), all between subject.
ArringtonEtAl2002
#>                 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  102

# first, we perform the omnibus analysis (mandatory):
w <- anopa( {s;n} ~ Location * Trophism * Diel, ArringtonEtAl2002) 
#> ANOPA::fyi(1): Combination of cells missing. Adding: 
#>       Location    Trophism      Diel s n
#>         Africa Detritivore Nocturnal 0 0
#>         Africa    Omnivore Nocturnal 0 0
#>  North America Detritivore Nocturnal 0 0
#> Warning: ANOPA::warning(1): Some cells have zero over zero data. Imputing...
summary(w)
#>                              MS  df        F   pvalue correction    Fcorr
#> Location               0.027449   2 0.961802 0.382203   1.000112 0.961694
#> Trophism               0.095656   3 3.351781 0.018102   1.000115 3.351396
#> Diel                   0.029715   1 1.041227 0.307536   1.000049 1.041176
#> Location:Trophism      0.029485   6 1.033146 0.401285   1.013842 1.019041
#> Location:Diel          0.005277   2 0.184900 0.831187   1.010164 0.183040
#> Trophism:Diel          0.073769   3 2.584868 0.051365   1.012197 2.553721
#> Location:Trophism:Diel 0.011297   6 0.395837 0.882184   1.055660 0.374967
#> Error(between)         0.028539 Inf                                      
#>                        pvalcorr
#> Location               0.382245
#> Trophism               0.018111
#> Diel                   0.307548
#> Location:Trophism      0.410515
#> Location:Diel          0.832735
#> Trophism:Diel          0.053559
#> Location:Trophism:Diel 0.895351
#> Error(between)                 

# There is a near-significant interaction of Trophism * Diel (if we consider
# the unadjusted p value, but you really should consider the adjusted p value...).
# If you generate the plot of the four factors, we don't see much:
anopaPlot(w)


#... but a plot specifically of the interaction helps:
anopaPlot(w, ~ Trophism * Diel )

# it seems that the most important difference is for omnivorous fishes
# (keep in mind that there were missing cells that were imputed but there does not
# exist to our knowledge agreed-upon common practices on how to impute proportions...
# Are you looking for a thesis topic?).

# Let's analyse the simple effect of Trophism for every levels of Diel and Location
e <- emProportions(w, ~ Trophism | Diel )
#> Not yet programmed...
summary(e)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>     -99     -99     -99     -99     -99     -99 


# You can ask easier outputs with
corrected(w) # or summary(w) for the ANOPA table only
#>                              MS  df        F correction    Fcorr pvalcorr
#> Location               0.027449   2 0.961802   1.000112 0.961694 0.382245
#> Trophism               0.095656   3 3.351781   1.000115 3.351396 0.018111
#> Diel                   0.029715   1 1.041227   1.000049 1.041176 0.307548
#> Location:Trophism      0.029485   6 1.033146   1.013842 1.019041 0.410515
#> Location:Diel          0.005277   2 0.184900   1.010164 0.183040 0.832735
#> Trophism:Diel          0.073769   3 2.584868   1.012197 2.553721 0.053559
#> Location:Trophism:Diel 0.011297   6 0.395837   1.055660 0.374967 0.895351
#> Error(between)         0.028539 Inf                                      
explain(w)   # human-readable ouptut ((pending))
#> [1] "method explain not yet done..."