Over a decade of research on the topic of informative hypotheses has resulted in a bunch of methodological, tutorial, and applied papers, books (e.g., Hoijtink, 2012), and several software packages (see informative-hypotheses.sites.uu.nl). Informative hypotheses with simple order constraints, such as $H_i\text{: }\mu_1 > \mu_2 > \mu_3$, $H_i: \mu_1 > (\mu_2, \mu_3)$ have proven their use: not only do these hypotheses represent researchers' expectations better, they also have more statistical power than classical hypotheses (Vanbrabant et al., 2015). The Bayes Factor (BF) is a Bayesian measure of support used for model selection (Kass and raftery, 1995). The BF can also be used for the evaluation of informative hypotheses. Using the BF, the support in the data for an informative hypothesis ($H_i$) versus the unconstrained alternative ($H_u$) can be calculated. As was shown by Klugkist et al. (2005):
$$ BF_{{H_i,H_u}} = \frac{f_i}{c_i}, $$
where $f_i$ denotes fit for the hypothesis $H_i$, and $c_i$ denotes complexity for the hypothesis $H_i$. $f_i$ is a measure of agreement between the data and $H_i$, it is the posterior probability of the hypothesis given the data. $f_i$ can only be calculated after the data is observed. $c_i$ refers to the proportion of the parameter space in agreement with the hypothesis by chance: the a priori probability of the hypothesis. Complexity can be calculated before conducting the analysis.
In case of multivariate analyses, BIEMS (Mulder et al., 2012) can be used to estimate the BF. For structural equation models, BF-SEM has been developed (Gu et al., submitted). The computation of the BF for structural equation models calculated in Mplus (Muthén and Muthén, 1998-2012) is described by Van de Schoot et al. (2012). As these authors show, fit can be calculated by means of MplusAutomation (Hallquist, 2013), and complexity can be calculated manually. With few parameters and few constraints, this manual computation of $c_i$ is easy. When the number of parameters and constraints increases, however, it is preferable to automate the process. Therefore, we published the CRAN package complexity (Zondervan-Zwijnenburg, 2017). The code for the complexity function is provided in Appendix A. We will consider two examples in which the goal is to obtain the complexity for the hypothesis of interest to demonstrate the complexity package.
The first example includes four regression coefficients: $\beta_1, \beta_2, \beta_3$, and $\beta_4$. The expectation is that $\beta_1$ is smaller than $\beta_2$ and that $\beta_3$ is smaller than $\beta_4$, which can be expressed as, $H_i: (\beta_1 < \beta_2$) \& ($\beta_3 < \beta_4)$.
Obtain all possible ways in which the parameters can be ordered; for the first example there are $4! = 4\times 3 \times 2 \times 1= 24$ different ways of ordering the parameters.
Count the number of possible orderings that are in line with each of the informative hypotheses: in this example it is six.
Divide the value obtained in step 2 by the value obtained in step 1. In our example: $c_i = 6/24 = 0.25$.
complexity(4,1,2,3,4)
. Here, the first 4
stands for the four parameters involved in the analysis. Every following set of two numbers, represents a constraint in which the first parameter is constrained to be lower than the second parameter. Hence, 1,2
refers to ($\beta_1 < \beta_2$) and 3,4
refers to ($\beta_3 < \beta_4$). If the hypothesis would have been ($\beta_1 < \beta_2$) \& ($\beta_2 > \beta_4$), the specification would be complexity(4,1,2,4,2)
. Parameter 3 is then unrestricted. For $\beta_1<\beta_2<\beta_3<\beta_4$ the correct specification is: complexity(4,1,2,2,3,3,4)
.
complexity(4,1,2,3,4)
generates the following output: library(complexity) complexity(4,1,2,3,4)
The output shows:
the list of permutations in accordance with the request, where the column represents parameter and the numbers represent it's position.
the total number of permutations.
the number of permutations in agreement with the hypothesis.
the percentage of permutations in agreement with the hypothesis ($= c_i$).
The second example comes from Johnson et al. (2015) and extends on the first example with two extra parameters that we hypothesize to be ordered as follows: $(\beta_1 < \beta_2)$ \& $(\beta_2, \beta_3 > \beta_4)$ \& $(\beta_4, \beta_5 < \beta_6)$. Step two of the manual method will show that there are 720 ways in which these parameters can be ordered. Writing down all possibilities and counting the number of possibilities in agreement with the hypothesis is time consuming and prone to errors. Hence, we switch to the automated procedure. To determine the correct input, we need to specify all the separate constraints in terms of parameters smaller than other parameters. In this case: $(\beta_1 < \beta_2)$ \& $(\beta_4 < \beta_2)$ \& $(\beta_4 < \beta_3)$ \& $(\beta_4 < \beta_6)$ \& $(\beta_5 < \beta_6)$. The number of parameters involved is six. Hence, the required input is: complexity(6, 1,2,4,2,4,3,4,6,5,6)
. Almost immediately, the complexity package generates the following output:
complexity(6, 1,2,4,2,4,3,4,6,5,6) #`total number of permutations' #[1] 720 #$`number true' #[1] 66 #$`complexity (proportion)' #[1] 0.09166667
The output shows that 66 out of 720 possibilities are in agreement with our hypothesis, which results in a complexity of .09. In addition, the output provides all permutations in agreement with the hypothesis for visual inspection of the results.
The complexity package also includes a function that launches a Shiny (Chang et al., 2016) application with runShiny()
. The output is the same as the output obtained with the complexity
function.
Although it is possible to calculate the proportion of the parameter space in line with the hypothesis by chance (i.e., $c_i$) manually, the second example showed that especially when the number of parameters involved becomes larger or when the constraints are more complex, it is easier and more reliable to use the complexity function in R to obtain the complexity for the Bayes Factor.
To compute the Bayes Factor for Example 2, the remaining element to compute is $f_i$. To obtain $f_i$ we need to calculate the proportion of iterations in the Gibbs sampler that is in agreement with $(\beta_1 < \beta_2)$ \& $(\beta_2, \beta_3 > \beta_4)$ \& $(\beta_4, \beta_5 < \beta_6)$. Using MplusAutomation we find that for \cite{Johnson_2015} $f_i$ is .353.
$$ BF_{{H_i,H_u}} = \frac{.353}{.092} = 3.837. $$
Thus, taking the model complexity into account, $H_i\text{: }(\beta_1 < \beta_2)$ \& $(\beta_2, \beta_3 > \beta_4)$ \& $(\beta_4, \beta_5 < \beta_6)$ fits better than $H_u\text{: }\beta_1, \beta_2, \beta_3, \beta_4, \beta_5, \beta_6$: the informative hypothesis receives more than three times the support.
The complexity of an informative hypothesis is required to calculate $BF_{{H_i,H_u}}$. The complexity package enables the user to calculate $c_i$ quickly, even for more complicated hypotheses.
MZ is supported by the Consortium Individual Development (CID), which is funded through the Gravitation program of the Dutch Ministry of Education, Culture, and Science and the Netherlands Organization for Scientific Research (NWO grant number 024.001.003). RS is supported by a VIDI grant from the Netherlands Organization for Scientific Research (NWO grant number 452.14.006).
Mariëlle Zondervan-Zwijnenburg
Department of Methodology & Statistics
Utrecht University
The Netherlands
Alan R. Johnson
Nord University Business School
Bodø, Norway
and
RATIO Research Institute
Stockholm, Sweden
Rens van de Schoot
Department of Methodology & Statistics
Utrecht University
The Netherlands
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M. Zondervan-Zwijnenburg. complexity: Calculate the Proportion of Permutations in Line with an Informative Hypothesis, 2017. URL https://CRAN.R-project.org/package=complexity. R package version 1.1.2.
complexity <- function(npar,...){ require(combinat) values <- c(1:npar) #the parameters to permute perm <- permn(values) #all permutations in a list length <- length(perm) #number of permutations perm.matrix <- matrix(unlist(perm),length, byrow=TRUE) #permutations in rows of matrix dim <- dim(perm.matrix) #length and width of permutation matrix z <- matrix(unlist(list(...)),ncol=2,byrow=TRUE) #one set of restriction values in each row dimz <- dim(z) n <- dimz[1] #number of restrictions logical <- logical.last <- rep(0, nrow=dimz[1], ncol=1) #empty vectors with length perm.matrix for (i in 1:n){ logical <- logical.last #fill logical with logical.last z1<- z[i,1] #get first restriction value z2<- z[i,2] #get second restriction value logical <- #restrictions, TRUEs and FALSES saved in logical perm.matrix[,z1]<perm.matrix[,z2] #see if 1st restr. value column < 2nd restr. perm.matrix <- matrix(perm.matrix[logical],ncol=dim[2]) #save TRUE selection in perm.matrix dimp <- dim(perm.matrix) #dimensions of new matrix logical.last <- rep(0,nrow=dimp[1],ncol=1) #new empty vector with length new matrix } y <- sum(logical) #number of TRUEs in set true <- perm.matrix #matrix with TRUE permutations prop <- y/length #TRUE n / total n = complexity list("true permutations" = true, "total number of permutations"=length, "number true"=y, "complexity (proportion)"=prop)}
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