Fig 1.
The panels show functions fl(g[ϕ]) and fu(1 − g[ϕ]). In the left panel fl(g[ϕ]) = fu(g[ϕ]) = log g[ϕ] while in the right panel fl(g[ϕ]) = fu(g[ϕ]) = log(g[ϕ]/(1 − g[ϕ])).
Table 1.
Maximum entropy distributions derived from constraints posed by structures with essential minimum and/or essential maximum.
Fig 2.
Each panel shows a qualitatively different outcome of a 2 × 2 factorial design. The first panel (from left) shows two main effects without an interaction. The second panel shows an (additive) interaction but no main effect. The remaining three panels show different constellations of two main effects accompanied by an interaction. Crucially the relative order of the conditions differs across these three constellations. The five displayed outcomes are used in the simulations.
Table 2.
Parameter values used in simulation.
Fig 3.
Results obtained with Generalized Gamma distribution.
The panels are referred to in order from left to right, from top to bottom. Horizontal axis in each panel shows cl ( to be precise). The colored polygons in the first panel show the mean group difference (thick line) and its 95% CI (surface). The blue color shows results with log-transformed data while the red color shows results with raw data. The second panel shows Cohen’s d along with CIs. The layout is similar to the layout of the first panel. The third panel shows results of the TOST equivalence testing procedure. The vertical axis shows the proportion of cases which supported group equivalence. In the fourth panel, the vertical axis shows the probability of the hypothesis of group mean equivalence relative to the hypothesis of a group mean difference. The blue and the red color in the third and fourth panel designate whether the methods are applied to transformed or raw data. The vertical axis in the fifth and sixth panel shows the proportion of rejections of null hypothesis (group equivalence). Fifth panel shows results from t-test (red), log-t-test (blue), MW test (green) and trimmed t-test (yellow). The sixth panel show results from three-level-F-test with transformed data (blue) and raw data (red). The green line shows results of KW test. The values in panels one, two and four were obtained as median across replications.
Fig 4.
Results obtained with Wald distribution.
The figure layout follows the layout of Fig 3. Refer to Fig 3 for details.
Fig 5.
Results obtained with Beta Prime distribution.
The figure layout follows the layout of Fig 3. Refer to Fig 3 for details.
Fig 6.
Results obtained with Beta distribution.
The figure layout follows the layout of Fig 3. Refer to Fig 3 for details.
Fig 7.
Results obtained with Beta-binomial distribution.
The figure layout follows the layout of Fig 3. Refer to Fig 3 for details.
Fig 8.
The figure layout follows the layout of Fig 3. Refer to Fig 3 for details.
Fig 9.
Performance with 2 × 2 factor design.
Figure shows the comparison of detection rate of rank-based SRHT (green) with ANOVA (red) and with ANOVA on transformed data (blue). Further details are provided in the text.
Fig 10.
Parameter recovery with Ordered logistic regression and Beta-binomial distribution.
Figure shows the estimate of group difference on vertical axis that was obtained by fitting OLRM (left panel) and Beta-binomial distribution (right panel) to the data generated with OLRM by manipulating the magnitude of ceiling effect cu. cu is shown on the horizontal axis and the magnitude of the ceiling effect increases from left to right. In reference to the precise notation in section 2.3, the black line shows the median estimate of the group difference
, while the gray surface shows the estimate’s 95% interval. The blue line shows the true value used by OLRM that generated the data and the true value corresponds to an OLRM estimate of group difference in [1] (i.e. median estimate of d0). The crosses show the difficulty of the six items (i.e.
) in the original study (green [1]) and in the replication study (red [2]), while ci and grid lines of the horizontal axis show the OLRM thresholds obtained by pooling the original and replication data.
Fig 11.
Effect of CFE on mean, variance and skewness.
Figure shows how the magnitude of floor effect affects the expectation, variance and skewness of the distributions shown in columns. In each panel, the magnitude of the floor effect increases from right to the left. Further details are provided in the text.