Dominance Analysis

The focus of the domir package is on dominance analysis (DA). DA is a method that resolves the indeterminacy of trying to ascribe a the results from a predictive model’s fit metric, referred to as a ‘value’ in the package, to individual predictive factors (i.e., independent variables/IVs, predictors, features), referred to as ‘names’ in the package.

A challenge for many predictive models and fit metrics are that there is no way to analytically decompose the fit statistic/metric given correlations between the predictive factors that are naturally present in the data or are introduced by the model. When there is no way to analytically separate the fit statistic to ascribe it to predictive factors, a methodological approach could be applied where values are ascribed by including the factors in the model sequentially. As each predictive factor is included, the change in the fit metric is ascribed to that predictive factor.

One issue with the sequential approach is that the sequence chosen to ascribe the fit statistic to predictive factors determines how much of the fit statistic is ascribed to the factor. When the analyst has good reason to choose a specific inclusion order, this approach produces a useful result.

Using a single inclusion order can be problematic when there is not a good reason to choose one specific inclusion order over another. When the inclusion order is effectively arbitrary, changing the order changes the values ascribed to the predictive factors in ways that have implications on inferences from the model.

A solution to this problem is to consider all possible ways of including the predictive factor. This method for resolving this issue is the approach used by Shapley value decomposition from Cooperative Game Theory (see Grömping (2007) for a discussion) which seeks to find a solution to the problem of how to subdivide payoffs to players in a cooperative game based on their relative contribution when it is not possible to separate relative contributions analytically.

DA uses the idea of comparing across inclusion orders as a methodological, and almost experimental design-like, approach to determining importance. DA also extends on the classic Shapley value decomposition methodology by adding two, more difficult to achieve, importance criteria. All three different importance criteria, known as dominance designations, are discussed in greater detail in the context of an example discussed below after a brief introduction to the domir function.

DA Implementation with domir

The domir function is an API for applying DA to predictive modeling functions in R. The sections below will use this function to illustrate how DA is implemented and discuss conceptual details of each computation.

Fit Statistic Results

Overall Value:      0.8113023

The first result domin prints is related to the overall fit statistic value for the model. In game theory terms, this value is the total payoff all players/IVs produced in the cooperative game/model.

The value produced serves as the fit statistic “to be decomposed” by the DA and is limiting value for how much each IV will be able to explain. The DA will ascribe the three IVs in this model separate components of this ~\(.8113\) value related to their contributions to predicting mpg.

Other fit statistic value adjustments are reported in this section as well in particular those associated with the all subsets and constant model adjustments (to be discussed further below; sections under development) when used in the DA.

General Dominance Statistics

General Dominance Values:
     General Dominance Standardized Ranks
am           0.2156848    0.2658501     2
cyl          0.4173094    0.5143698     1
carb         0.1783081    0.2197801     3

The second result printed reports the general dominance statistics related to how the overall fit statistic’s value is divided up among the IVs. These also represent the Shapley value decompositions of the fit statistic showing how each player/IV is ascribed a component of the payoff/fit statistic from the game/model.

The General Dominance column of statistics can be interpreted in terms of the fit metric it decomposed. For example, am has a value of ~\(0.2157\) which means am is associated with an \(R^2\) of about twenty-two percentage points of mpg’s variance given the predictive model and other IVs.

The Standardized column of statistics expresses the general dominance statistic value as a percentage of the overall fit statistic value and thus sums to 100%. am’s contributions to the \(R^2\)’s total value is ~27% (i.e., \(\frac{.2157}{.8113} = .2659\)).

The final Ranks column is most relevant to the focal purpose of determining the relative importance of the IVs in this model as it provides a rank ordering of the IVs based on their general dominance statistics. It is here that DA moves beyond Shapley value decomposition in that DA, and this rank ordering based on general dominance statistics, allows for applying labels to the relationships between IVs in terms of their importance.

am is ranked second because it has a smaller general dominance statistic than the first ranked cyl. As such, am “is generally dominated by” cyl. By contrast, am is ranked higher than third ranked carb and thus am “generally dominates” carb. These labels are known as general dominance designations.

Conditional Dominance Statistics

Conditional Dominance Values:
     Include At: 1 Include At: 2 Include At: 3
am       0.3597989     0.2164938    0.07076149
cyl      0.7261800     0.4181185    0.10762967
carb     0.3035184     0.1791172    0.05228872

The third section reported on by domin prints the conditional dominance statistics associated with each IV. Each IV has a separate conditional dominance statistic related to position at which it is included in the sequence of IVs in the model.

The conditional dominance matrix can be used to designate importance in a way that is more stringent/harder to achieve than the general dominance statistics. To determine importance with the conditional dominance matrix each IV is compared to each other in a ‘row-wise’ fashion. If the value of each entry for a row/IV is greater than the value of another row/IV at the same position (i.e., comparing IVs at the same inclusion position) than an IV is said to “conditionally dominate” the other IV. The matrix above shows that am “is conditionally dominated by” cyl as its conditional dominance statistics are smaller than cyl’s at positions 1, 2, and 3. Conversely, am “conditionally dominates” carb as its conditional dominance statistics are greater than carb’s at positions 1, 2, and 3.

Complete Dominance Proportions

Complete Dominance Proportions:
       > am > cyl > carb
am >     NA     0      1
cyl >     1    NA      1
carb >    0     0     NA

The fourth section reported on by domir prints the complete dominance proportions associated with each IV pair. Each IV is compared to each other IV and has two entries in this matrix. The IV noted in the row labels is the ‘dominating’ IV as is implied by the greater than symbol (i.e., \(>\)) preferring it. The IV noted in the column labels is the ‘dominated’ IV as is implied by the greater than symbol not preferring it. The values reported are the proportion of sub-models in which the IV in the row obtains a larger value than the IV in the column.

The complete dominance designations are useful beyond the general and conditional dominance results as they are the most stringent sets of comparisons. Complete dominance reflects the relative performance of each IV to another IV in all the sub-models where their relative predictive performance can be compared. When a value of 1 is obtained, the IV in the row is said to “completely dominate” the IV in the column. Conversely, when a value of 0 is obtained, the IV in the row is said to be “completely dominated by” the IV in the column.

Computational Implementation: All Possible Orders

DA requires evaluating the contribution IVs make to prediction given all possible orders in which they are included in the prediction model. As was noted above, this is an experimental design-like approach where all possible combinations of the IVs included or excluded are estimated as sub-models. When there are \(p\) IVs in the model there will be \(2^p\) sub-models estimated. The experimental design-like approach of the method makes it widely applicable across predictive models and fit statistic values but is computationally expensive as each additional IV added to the model results in a geometric increase in the number of required sub-models.

DA Results: The Full-Factorial Design

The DA results related to the lm model with three IVs discussed above is composed of 8 sub-models and their \(R^2\) values. The domir function, if supplied a predictive modeling function that can record each sub-model’s results, can be adapted to capture each sub-model’s \(R^2\) value along with the IVs that comprise it.

The code below constructs a wrapper function to export results from each sub-model to an external data frame. The code to produce these results is complex and each line is commented to note its purpose. This wrapper function then replaces lm in the call to domir so that, as the DA is executed, all the sub-models’ data are captured for the illustration to come.

lm_capture <- 
  function(formula, data, ...) { # wrapper program that accepts formula, data, and ellipsis arguments
    count <<- count + 1 # increment counter in enclosing environment
    lm_obj <- lm(formula, data = data, ...) # estimate 'lm' model and save object
    DA_results[count, "formula"] <<- 
      deparse(formula) # record string version of formula passed in 'DA_results' in enclosing environment
    DA_results[count, "R^2"] <<- 
      summary(lm_obj)[["r.squared"]] # record R^2 in 'DA_results' in enclosing environment
    summary(lm_obj)[["r.squared"]] # return R^2
  }

count <- 0 # initialize the count indicating the row in which the results will fill-in

DA_results <- # container data frame in which to record results
  data.frame(formula = rep("", times = 2^3-1), 
             `R^2` = rep(NA, times = 2^3-1), 
             check.names = FALSE)

lm_da <- domir(mpg ~ am + cyl + carb, # implement the DA with the wrapper
               lm_capture, 
               data = mtcars)

DA_results
#>                 formula       R^2
#> 1 mpg ~ am + cyl + carb 0.8113023
#> 2              mpg ~ am 0.3597989
#> 3             mpg ~ cyl 0.7261800
#> 4        mpg ~ am + cyl 0.7590135
#> 5            mpg ~ carb 0.3035184
#> 6       mpg ~ am + carb 0.7036726
#> 7      mpg ~ cyl + carb 0.7405408

The printed result from DA_results shows that domir runs 7 sub-models; each a different combination of the IVs. Note that, by default, the sub-model where all IVs are excluded is assumed to result in a fit statistic value of 0 and is not estimated directly (which can be changed with the .adj argument).

The \(R^2\) values recorded in DA_results are used to compute the dominance statistics and designations reported on above.

lm_da$Complete_Dominance
#>        >_am >_cyl >_carb
#> am_>     NA     0      1
#> cyl_>     1    NA      1
#> carb_>    0     0     NA

lm_da$Conditional_Dominance
#>      include_at_1 include_at_2 include_at_3
#> am      0.3597989    0.2164938   0.07076149
#> cyl     0.7261800    0.4181185   0.10762967
#> carb    0.3035184    0.1791172   0.05228872

lm_da$General_Dominance
#>        am       cyl      carb 
#> 0.2156848 0.4173094 0.1783081

Strongest Dominance Designations

Whereas all dominance designations have been made in the example above, the strongest designation between two IVs is likely of primary interest as the strongest designation, as is noted above, implies all weaker designations.

To access the strongest dominance designations, the DA object can be submitted to the summary function.

summary(lm_da)$Strongest_Dominance
#>                                                                            
#>  "am"                         "am"                   "cyl"                 
#>  "is completely dominated by" "completely dominates" "completely dominates"
#>  "cyl"                        "carb"                 "carb"

The result the summary function produces in the Strongest_Dominance element is consistent with expectation in that all three IV interrelationships have complete dominance designations between them.

Criticality: Model Selection

In many cases, methods that focus on relative importance are probably best used for model selection. When applied to model selection, a method would identify when an IV should be included in the model or not. The process of determining whether or not an IV should be included in the model is desribed by Azen et al. as reflecting IV Criticality.

In the view of the author, methods such as posterior inclusion probability, Akaike weights, and permutation importance are actually IV criticality, as opposed to importance, measures. These methods are criticality methods as they tend to be informative for identifying whether an IV has trivial or non-trivial conribution to prediction but is less informative for identifying the magnitude of their contribution.

Importance: Model Evaluation

Model evaluation differs from model selection in that it seeks not to determine whether IVs should be in the model, but rather compares them in terms of their impact in the model conditional on their being included. Thus, importance methods assume that a model has passed through a model selection phase and that all the predictors in the model have non-trivial effects.

The DA method implemented by the domir function is an importance method in this sense in that it assumes that the predictive model used is “pre-selected” or has passed through model selection procedures and the user is confident that the IVs model and are, in fact, reasonable to include. The results from DA and similar methods then provide more information about relative contribution to prediction which assists in model evaluation.