Labelling

We are not in favor of cut-off points (or ‘surrounding anchors’), but we need them when we want to label the height of support via the benchmarks, we propose the following:

We advise on using some kind of null model as the assumed population (possibly, using multiple ones). Then, you can see how extreme your finding is (or not) for this null population. The extremer your finding, the more support for your informative, theory-based, inequality-constrained hypothesis.

You can of course use other percentile levels (for which you think the finding is said to be extreme enough etc). But: Do make sure to define these before seeing the data, and preferably also pre-register them (together with your informative hypothesis/-es).

Use minimum effect

You may want to use a minimum effect. One option – the one we also advise, when it is doable – is to specify your hypothesis such that it inspects, for example, minimum differences between parameters (e.g., μ₁ − μ₂> 0.2 instead of μ₁ > μ₂, that is, μ₁ − μ₂> 0). Notably, one then possible does not need benchmarks anymore: Finding a ratio of GORIC(A) weights of 1 and higher would probably suffice.

A second option – although not something we advise on doing – is to investigate benchmarks using a minimum effect size (or looking at multiple ones). Then, you will compare your samples values with the distribution of (ratios of) GORIC(A) weights under a population with this minimum effect size.
In the function, you should specify the population parameter values (reflecting specific effect sizes). In case of an ANOVA model, you can also specify the effect size level(s) (for Cohen’s f). Note that the benchmarks differ when you assume different effect sizes or different population parameter estimates.

Sensitivity analysis

If of interest, as a sensitivity analysis, you can calculate the benchmarks for multiple sets of population parameter values (or population effect sizes). Note that this may also complicate drawing conclusions (especially when the assumed sets of population parameter values differ much, like when using multiple effect size heights). We advise on doing this for multiple null populations (setting some to all of the inequality restrictions to equalities).

If your preferred hypothesis does not have the highest fit and you want to inspect benchmarks, we also advise on inspecting multiple ratios of population parameter estimates, where some are in agreement with your hypothesis and others in agreement with the data.

Defaults

Once more, we advise on inspecting populations in which some of the inequalities of your hypothesis/-es of interest are set to equalities. This can also be very helpful if the log-likelihood values seem to be close: This way, you can also check for the support of a boundary hypothesis; as discussed in the third example below.

By default, two populations are used: i) a null population in which the population effect size is – or population parameter values are – set to 0 and ii) a population based on the observed effect size – or observed parameter values. One can overrule this by using the pop_es or pop_est argument, respectively.

Example 1 (ANOVA): H₁ vs its complement

# H1 vs complement - H1 is true
H1 <- "D1 > D2 > D3" # mu1 > mu2 > mu3

# Apply GORIC #
set.seed(123) 
results_1c <- goric(fit, hypotheses = list(H1), comparison = "complement")
results_1c

restriktor (0.6-10): generalized order-restricted information criterion: 

Results:
        model    loglik  penalty    goric  loglik.weights  penalty.weights  goric.weights
1          H1  -155.075    2.833  315.816           0.704            0.697          0.845
2  complement  -155.939    3.667  319.212           0.296            0.303          0.155

Conclusion:
The order-restricted hypothesis 'H1' has 5.46 times more support than its complement.

# Benchmarks based on null
benchmarks_1c <- benchmark(results_1c, model_type = "means", iter = 100)

Calculating means benchmark for effect-size = 0 (No-effect)
Calculating means benchmark for effect-size = 0.411 (Observed)

#benchmarks_1c # use in R file
print(benchmarks_1c, color = FALSE) # use in Rmd file, since Rmd cannot deal with colored text

Benchmark Results 
---------------------------------------------------------------------- 
Preferred Hypothesis: H1
Error probability Preferred Hypothesis vs. Complement: 0.155
Number of Groups: 3
Group Sizes: 40, 40, 40
Ratio of Population Means: 4.322, 2.000, 1.000
Population Effect-Sizes (Cohens f): 0.000, 0.411
Observed Effect-Size (Cohens f): 0.411


================================================================================== 
Benchmark: Percentiles of Ratio-of-GORIC-weights for the Preferred Hypothesis 'H1' 
---------------------------------------------------------------------------------- 
Population effect-size = 0
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 5.464  0.402 1.279 1.687 1.992 2.383

Population effect-size = 0.411
                  Sample 5%    35%   50%   65%   95%    hypothesis_rate
H1 vs. complement 5.464  2.297 3.643 4.939 8.121 35.607 0.990          

#plot(benchmarks_1c) 
plot(benchmarks_1c, x_lim = c(0, 57))

plot(benchmarks_1c, x_lim = c(0, 6))

From the goric output, you can conclude i) that there is support for H₁ : μ₁ > μ₂ > μ₃ and ii) that H₁ is 0.85 / 0.15 ≈ 5.46 times more supported than its complement. The probability that H₁ is not the best is 15.47% (namely, the goric.weight for the complement of H₁, which is also given in the benchmark output by $error_prob_pref_hypo, that is, the Error probability Preferred Hypothesis vs. complement). This already gives insight into the (un)certainty and, therefore, helps in qualifying the results. Additionally, the benchmarks can help:

Based on the benchmarks, you can check how plausible your finding is (given the assumed population parameters and given the sample size). If you want to compare your results with the situation in which one or more equalities hold true, use a null population (here, an effect size of 0, indicating that the three means are equal). Then, you obtain insight into how (un)likely / how extreme your finding (based on inequalities) is.

When assuming that there is no effect in the population (i.e., under the null), that is, all three group means are equal, the ratio of GORIC weights of H₁ versus its complement (i.e., 5.46) is larger than the 95th percentile (i.e., 2.38). Hence, our finding is very extreme if the null would be true. Using the table with cut-off values above, this indicates that there is very large (tremendous) support, when assuming no population effect size (given a group size of 40 for each of the 3 groups).

Additionally, when inspecting the distributions of the ratio of GORIC weights of H₁ versus its complement under the null population and under the population based on the observed effect size (see first plot), then we can see that they differ, that is, do not overlap much. This also indicates that our finding is very unlikely if the null would be true.

Notably, the gray line denotes a ratio of GORIC(A) weights of 1, which means that the support for both hypothesis is equal. In that case, you would be indecisive. The reason to plot this line is to remember that a ratio of GORIC(A) weights ranges from 0 to infinity, where 0 to 1 denotes the same support as 1 to infinity for the opposite comparison (e.g., a support of 0.2 for H₁ versus H_c is the same as a support of 1/0.2=5 for H_c versus H₁. One can also plot the log transformation of the ratio:

plot(benchmarks_1c, log_scale = TRUE)  

This then ranges from minus infinity to infinity, where minus infinity to 0 is comparable to 0 to infinity. Hence, equal support is now denoted by 0, that is, (again) the grey line in the plot.

Log-likelihood check:
Before inspecting the height of the support, one may want to establish whether there is support for the overlap or boundary of hypotheses. Since we evaluate an informative hypothesis H₁ versus its complement, we should check whether there is support for a boundary hypothesis (in which one or more inequalities in H₁ is replaced by an equality). For this, one should inspect the log-likelihood / fit values of the hypotheses. When these are close (i.e., the ratio of loglik weights is close to 1 or the difference in loglik values is close to 0), then there is support for (one of their) boundaries. In this case, the loglik values are -155.07 and -155.94, with corresponding loglik.weights of 0.7 and 0.3 (and thus a difference of approximately 0.86 and a ratio of approximately 2.37). Since it is hard to judge what is close, one can inspect the benchmarks for the ratio of log-likelihood (loglik) weights and/or for the differences in log-likelihood values. This should then be done for a null population in which such a boundary is true. We discuss this in the next section.
For now, we assume that the loglik values are not close.

Population information:
In the data generation, we used a ratio of population means of 3:2:1; implying that H₁ is correct. More specifically, we used population mean values of approximately 0.92, 0.61, and 0.31. This implies that Cohen’s f is .25; thus, there is a medium population effect size (which are in the same order as hypothesized). We then sampled 40 observations for each of the three groups, ran an ANOVA (with three groups), and applied the GORIC. Note that Cohen (1992) suggest that a minimum group size of 52 is needed to find a medium effect when doing null hypothesis testing.
Notably, the sample/observed effect size is .411 (with sample means of 1.12, 0.51, and 0.25), which can be seen as a high effect size. This also explains why we, despite the medium population effect size, find tremendous support for our hypothesis.
When we would sample more observations, the GORIC(A) weight for H₁ converges to 1 (denoting full support for H₁). Note that the benchmarks for the GORIC(A) weight for H₁ will remain the same for a null population and will go to 1 for a non-null population. Note that the error probability then goes to 0, and that the ratio of GORIC(A) weights of H₁ versus its complement then goes to infinity.

Example 2 (ANOVA): Overlapping hypotheses

# H1, H2, and unconstrained (default) - subset/overlap, that is, H1 is true
H1 <- "D1 > D2 > D3"               # mu1 > mu2 > mu3
H2 <- "D1 > D2" # H2: D1 > D2, D3  # mu1 > mu2,  mu3

# Apply GORIC #
set.seed(123) 
results_12u <- goric(fit, hypotheses = list(H1, H2))
results_12u

restriktor (0.6-10): generalized order-restricted information criterion: 

Results:
           model    loglik  penalty    goric  loglik.weights  penalty.weights  goric.weights  goric.weights_without_unc
1             H1  -155.075    2.833  315.816           0.333            0.548          0.548                      0.661
2             H2  -155.075    3.500  317.149           0.333            0.281          0.281                      0.339
3  unconstrained  -155.075    4.000  318.149           0.333            0.171          0.171                           

Conclusion:
- The order-restricted hypothesis 'H1' is the best in the set, as it has the highest GORIC(A) weight.

- Since 'H1' has a higher GORIC(A) weight than the unconstrained hypothesis, it is not considered weak. We can now inspect the relative support for 'H1' against the other order-restricted hypotheses:

  * 'H1' is 1.948 times more supported than 'H2' (This relative support reached its maximum, see Note.).

round(results_12u$ratio.gw, 3)

              vs. H1 vs. H2 vs. unconstrained
H1             1.000  1.948             3.211
H2             0.513  1.000             1.649
unconstrained  0.311  0.607             1.000

# Benchmarks
benchmarks_12u <- benchmark(results_12u, model_type = "means", iter = 100)

Calculating means benchmark for effect-size = 0 (No-effect)
Calculating means benchmark for effect-size = 0.411 (Observed)

#benchmarks_12u # R file
print(benchmarks_12u, color = FALSE) # Rmd file

Benchmark Results 
---------------------------------------------------------------------- 
Preferred Hypothesis: H1
Error probability Preferred Hypothesis vs. Complement: 0.155
Number of Groups: 3
Group Sizes: 40, 40, 40
Ratio of Population Means: 4.322, 2.000, 1.000
Population Effect-Sizes (Cohens f): 0.000, 0.411
Observed Effect-Size (Cohens f): 0.411


================================================================================== 
Benchmark: Percentiles of Ratio-of-GORIC-weights for the Preferred Hypothesis 'H1' 
---------------------------------------------------------------------------------- 
Population effect-size = 0
                     Sample 5%    35%   50%   65%   95%  
H1 vs. H2            1.948  0.451 1.576 1.831 1.948 1.948
H1 vs. unconstrained 3.211  0.689 2.097 2.464 2.796 3.211

Population effect-size = 0.411
                     Sample 5%    35%   50%   65%   95%   hypothesis_rate
H1 vs. H2            1.948  1.938 1.948 1.948 1.948 1.948 1.000          
H1 vs. unconstrained 3.211  3.195 3.211 3.211 3.211 3.211 1.000          

plot(benchmarks_12u, output_type = "rgw") 

plot_out <- plot(benchmarks_12u) # save all plots in object plot_out
plot(plot_out$plots$`H1 vs. H2`) # call separate plot

From the GORIC output, we can see that both hypotheses are not weak and that H₁ is the best. Additionally, we can see that the log-likelihood values are exactly the same, and thus there is support for the overlap of the hypotheses which is H₁ here (since H₁ is a subset of H₂).

If we would now check the GORIC(A) weights benchmarks (especially for non-nulls), we would find again that there is support for the overlap (here, H₁): Namely, the benchmarks for multiple percentiles (here, from the 35th percentile on) have the same value as our finding, that is, the sample value.

Notably, if there is support for the overlap or boundary, it is not meaningful to use the benchmarks to label the height of support for H₁, since the ratio of support reached its maximum. Hence, we do not proceed with labeling the height of support (by comparing the sample (ratio of) GORIC weights value(s) to the GORIC weights benchmarks under the null population). Instead, we conclude that there is support for the overlap.
If of interest, one can next inspect the support for the overlap, here H₁, versus its complement. When that does not result in support for a boundary, then the height of the support for H₁ versus its complement can be labelled using the benchmarks. This is then helpful information for future research: Generating a new theory and having another value to compare future results to.

Log-likelihood check:
Here, one clearly finds that the loglik / fit values are exactly the same. Hence, the ratio of loglik weights is exactly 1 and the difference in loglik values is exactly 0. Consequently, there is support for the overlap. In this case, we do not need to check the log-likelihood benchmarks.
Note that the log-likelihood benchmarks (under a null population) give insight into the distribution of loglik weights ratios and of the loglik differences in case some or all of the group means would be the same.

Population information:
In the data generation, we used a ratio of population means of 3:2:1; implying that H₁ is correct. More specifically, we used population mean values of approximately 0.92, 0.61, and 0.31. This implies that Cohen’s f is .25; thus, there is a medium population effect size (which are in the same order as hypothesized). We then sampled 40 observations for each of the three groups, ran an ANOVA (with three groups), and applied the GORIC. Note that Cohen (1992) suggest that a minimum group size of 52 is needed to find a medium effect when doing null hypothesis testing. Notably, the sample/observed effect size is .411 (with sample means of 1.12, 0.51, and 0.25), which can be seen as a high effect size.
When we would sample more observations, it does not (really) affect the GORIC(A) weights for H₁, H₂, and the unconstrained: It will converge to the bounds (i.e., the maximum support) it can take on. The benchmarks for the GORIC(A) weights will also attain the maximum value as will the ratio of weights; and it will for each positive population effect size. For a null population, the GORIC(A) weight benchmarks will remain the same.

Higher sample size

# Now, group size is 200 (instead of 40)

# H1 vs complement - border (nl., mu1 = mu2 > mu3) is true
H1 <- "D1 > D2 > D3" # mu1 > mu2 > mu3

# Apply GORIC #
set.seed(123) 
results_1c_border_200 <- goric(fit_border_200, hypotheses = list(H1), comparison = "complement")
results_1c_border_200

restriktor (0.6-10): generalized order-restricted information criterion: 

Results:
        model    loglik  penalty     goric  loglik.weights  penalty.weights  goric.weights
1          H1  -829.928    2.833  1665.522           0.509            0.697          0.704
2  complement  -829.962    3.667  1667.258           0.491            0.303          0.296

Conclusion:
The order-restricted hypothesis 'H1' has 2.38 times more support than its complement.

We will also now check whether there is support for the border of the two (non-overlapping) hypotheses:

# Specifying multiple null populations, that is,
# using all possibilities of setting inequalities to equalities.
# Here, we will use the default `model_type` (i.e., "asymp") which takes population parameter values (instead of population effect sizes) 
est <- coef(fit_border_200)
pop_est <- matrix(c(
                  mean(est[1:3]), mean(est[1:3]), mean(est[1:3]),
                  mean(est[1:2]), mean(est[1:2]), est[3],
                  mean(est[1:2]), est[2], mean(est[1:2]),
                  est[1], mean(est[2:3]), mean(est[2:3]),
                  est[1], est[2], est[3]
                  ),
                  byrow = TRUE, ncol = length(est))
rownames(pop_est) <- c("PE_123eq", "PE_12eq", "PE_13eq", "PE_23eq", "Observed")
benchmarks_1c_border_allpos_200 <- benchmark(results_1c_border_200, pop_est = pop_est, iter = 100)

Calculating asymptotic benchmark for population estimates = PE_123eq 
Calculating asymptotic benchmark for population estimates = PE_12eq 
Calculating asymptotic benchmark for population estimates = PE_13eq 
Calculating asymptotic benchmark for population estimates = PE_23eq 
Calculating asymptotic benchmark for population estimates = Observed 

# loglik difference
#print(benchmarks_1c_border_allpos_200, output_type = "ld") # R file
print(benchmarks_1c_border_allpos_200, output_type = "ld", color = FALSE) # Rmd file

Benchmark Results 
---------------------------------------------------------------------- 
Preferred Hypothesis: H1
Error probability Preferred Hypothesis vs. Complement: 0.296
Sample Size: 600
Number of Parameters: 3
Population Estimates (PE):
             D1     D2     D3
PE_123eq  0.729  0.729  0.729
PE_12eq   0.874  0.874  0.440
PE_13eq   0.874  0.861  0.874
PE_23eq   0.886  0.650  0.650
Observed  0.886  0.861  0.440


================================================================================================ 
Benchmark: Percentiles of Differences in Log-likelihood Values for the Preferred Hypothesis 'H1' 
------------------------------------------------------------------------------------------------ 
Population estimates = PE_123eq
                  Sample 5%     35%    50%    65%    95%  
H1 vs. complement 0.035  -2.034 -0.502 -0.240 -0.018 0.140

Population estimates = PE_12eq
                  Sample 5%     35%    50%    65%   95%  
H1 vs. complement 0.035  -1.437 -0.118 -0.015 0.043 0.684

Population estimates = PE_13eq
                  Sample 5%     35%    50%    65%    95%  
H1 vs. complement 0.035  -2.695 -0.506 -0.275 -0.139 0.104

Population estimates = PE_23eq
                  Sample 5%     35%    50%    65%   95%  
H1 vs. complement 0.035  -1.185 -0.207 -0.020 0.024 0.965

Population estimates = Observed
                  Sample 5%     35%        50%   65%   95%  
H1 vs. complement 0.035  -0.407 -2.344e-04 0.037 0.207 1.430

plot(benchmarks_1c_border_allpos_200, output_type = "ld", x_lim = c(-3, 3))

# ratio of loglik weights
#print(benchmarks_1c_border_allpos_200, output_type = "rlw") # R file
print(benchmarks_1c_border_allpos_200, output_type = "rlw", color = FALSE) # Rmd file

Benchmark Results 
---------------------------------------------------------------------- 
Preferred Hypothesis: H1
Error probability Preferred Hypothesis vs. Complement: 0.296
Sample Size: 600
Number of Parameters: 3
Population Estimates (PE):
             D1     D2     D3
PE_123eq  0.729  0.729  0.729
PE_12eq   0.874  0.874  0.440
PE_13eq   0.874  0.861  0.874
PE_23eq   0.886  0.650  0.650
Observed  0.886  0.861  0.440


=========================================================================================== 
Benchmark: Percentiles of Ratio-of-log-likelihood-weights for the Preferred Hypothesis 'H1' 
------------------------------------------------------------------------------------------- 
Population estimates = PE_123eq
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 1.035  0.131 0.605 0.787 0.983 1.150

Population estimates = PE_12eq
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 1.035  0.238 0.889 0.985 1.044 1.993

Population estimates = PE_13eq
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 1.035  0.068 0.603 0.759 0.871 1.109

Population estimates = PE_23eq
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 1.035  0.306 0.813 0.980 1.024 2.624

Population estimates = Observed
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 1.035  0.666 1.000 1.037 1.230 4.187

plot(benchmarks_1c_border_allpos_200, output_type = "rlw", x_lim = c(0, 3.6))

plot(benchmarks_1c_border_allpos_200, output_type = "rlw", log_scale = TRUE)

Also in this example, with a much higher sample size, the sample difference in log-likelihood values and the sample ratio of log-likelihood weights lies between the 5th and the 95th percentiles of the corresponding benchmarks (for all null populations). Based on this, we conclude that the log-likelihood values of H₁ and its complement are close (i.e., their loglik weights ratio is close to 1 and their loglik difference is close to 0). Additionally, one can argue that there is more support for a boundary hypothesis in which group means 1 and 2 are the same, since the sample value is the closest to the 50th percentile of that null distribution.

Higher sample size

# Now, group size is 200 (instead of 40)

# H1 vs complement
H1 <- "D1 > D2 > D3" # mu1 > mu2 > mu3

# Apply GORIC #
set.seed(123) 
results_1c_200 <- goric(fit_200, hypotheses = list(H1), comparison = "complement")
results_1c_200

restriktor (0.6-10): generalized order-restricted information criterion: 

Results:
        model    loglik  penalty     goric  loglik.weights  penalty.weights  goric.weights
1          H1  -829.928    2.833  1665.522           0.889            0.697          0.949
2  complement  -832.009    3.667  1671.351           0.111            0.303          0.051

Conclusion:
The order-restricted hypothesis 'H1' has 18.43 times more support than its complement.

Also here, we will check whether there is support for the border of the two (non-overlapping) hypotheses:

# Specifying multiple null populations, that is,
# using all possibilities of setting inequalities to equalities.
# Here, we will use the default `model_type` (i.e., "asymp") which takes population parameter values (instead of population effect sizes) 
est <- coef(fit_200)
pop_est <- matrix(c(
                  mean(est[1:3]), mean(est[1:3]), mean(est[1:3]),
                  mean(est[1:2]), mean(est[1:2]), est[3],
                  mean(est[1:2]), est[2], mean(est[1:2]),
                  est[1], mean(est[2:3]), mean(est[2:3]),
                  est[1], est[2], est[3]
                  ),
                  byrow = TRUE, ncol = length(est))
rownames(pop_est) <- c("PE_123eq", "PE_12eq", "PE_13eq", "PE_23eq", "Observed")
benchmarks_1c_allpos_200 <- benchmark(results_1c_200, pop_est = pop_est, iter = 100)

Calculating asymptotic benchmark for population estimates = PE_123eq 
Calculating asymptotic benchmark for population estimates = PE_12eq 
Calculating asymptotic benchmark for population estimates = PE_13eq 
Calculating asymptotic benchmark for population estimates = PE_23eq 
Calculating asymptotic benchmark for population estimates = Observed 

# loglik difference
#print(benchmarks_1c_allpos, output_type = "ld") # R file
print(benchmarks_1c_allpos_200, output_type = "ld", color = FALSE) # Rmd file

Benchmark Results 
---------------------------------------------------------------------- 
Preferred Hypothesis: H1
Error probability Preferred Hypothesis vs. Complement: 0.051
Sample Size: 600
Number of Parameters: 3
Population Estimates (PE):
             D1     D2     D3
PE_123eq  0.634  0.634  0.634
PE_12eq   0.755  0.755  0.392
PE_13eq   0.755  0.589  0.755
PE_23eq   0.921  0.491  0.491
Observed  0.921  0.589  0.392


================================================================================================ 
Benchmark: Percentiles of Differences in Log-likelihood Values for the Preferred Hypothesis 'H1' 
------------------------------------------------------------------------------------------------ 
Population estimates = PE_123eq
                  Sample 5%     35%    50%    65%    95%  
H1 vs. complement 2.081  -2.034 -0.502 -0.240 -0.018 0.140

Population estimates = PE_12eq
                  Sample 5%     35%    50%    65%   95%  
H1 vs. complement 2.081  -1.437 -0.118 -0.015 0.043 0.626

Population estimates = PE_13eq
                  Sample 5%     35%    50%    65%    95%   
H1 vs. complement 2.081  -5.577 -2.526 -1.756 -1.213 -0.071

Population estimates = PE_23eq
                  Sample 5%     35%    50%    65%   95%  
H1 vs. complement 2.081  -1.185 -0.207 -0.020 0.032 1.149

Population estimates = Observed
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 2.081  0.141 1.114 1.678 2.557 4.486

plot(benchmarks_1c_allpos_200, output_type = "ld", x_lim = c(-6, 4.6))

# ratio of loglik weights
#print(benchmarks_1c_allpos_200, output_type = "rlw") # R file
print(benchmarks_1c_allpos_200, output_type = "rlw", color = FALSE) # Rmd file

Benchmark Results 
---------------------------------------------------------------------- 
Preferred Hypothesis: H1
Error probability Preferred Hypothesis vs. Complement: 0.051
Sample Size: 600
Number of Parameters: 3
Population Estimates (PE):
             D1     D2     D3
PE_123eq  0.634  0.634  0.634
PE_12eq   0.755  0.755  0.392
PE_13eq   0.755  0.589  0.755
PE_23eq   0.921  0.491  0.491
Observed  0.921  0.589  0.392


=========================================================================================== 
Benchmark: Percentiles of Ratio-of-log-likelihood-weights for the Preferred Hypothesis 'H1' 
------------------------------------------------------------------------------------------- 
Population estimates = PE_123eq
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 8.012  0.131 0.605 0.787 0.983 1.150

Population estimates = PE_12eq
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 8.012  0.238 0.889 0.985 1.044 1.870

Population estimates = PE_13eq
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 8.012  0.004 0.080 0.173 0.297 0.932

Population estimates = PE_23eq
                  Sample 5%    35%   50%   65%   95%  
H1 vs. complement 8.012  0.306 0.813 0.980 1.032 3.156

Population estimates = Observed
                  Sample 5%    35%   50%   65%    95%   
H1 vs. complement 8.012  1.151 3.047 5.357 12.915 88.754

plot(benchmarks_1c_allpos_200, output_type = "rlw", x_lim = c(0, 10))

plot(benchmarks_1c_allpos_200, output_type = "rlw", log_scale = TRUE)

In this example with a higher sample size, the sample difference in log-likelihood values and the sample ratio of log-likelihood weights are higher than the 95th percentile of the corresponding benchmarks (for all null populations; see the benchmarks output table). Based on this, we conclude that the log-likelihood values of H₁ and its complement are not close. Thus, there is no support for a boundary hypothesis, only for H₁. In that case, we can inspect the height of the support via the GORIC(A) benchmarks (as was done in a previous section).