Figure 1.
Schematic of the proposed framework. Each dot represents an individual. Here the samples were generated by the linear Gaussian model. First, the value of pathogenetic factors x 1 and x 2 are generated from a Gaussian distribution independently. Then, these factors are transformed into behavioral observations y 1 and y 2 with linear mapping Y = W X and adding some noise ϵ. The individuals are classified as patients if both behavioral observations y 1 and y 2 have larger values than h 1 and h 2, respectively (here we used the common cutoff point h 1 = h 2 = 0.5). The red dots represent the individuals of the patient group, and the gray dots represent the individuals of the control group. The conditional distributions of x 1 and x 2 given groups are plotted in the top left panel (gray lines for the control group and red lines for the patient group). Note that the conditional distributions no longer obey the Gaussian distribution. These pathogenetic factors are assumed to be observed with adding some estimation error δ, which also obeys a Gaussian distribution. The diagnostic category-based approach attempts to find a pathogenetic factor x j whose observed value differs between the patient group and control group (bottom left panel). The dimensional approach attempts to find a factor x j that correlates with a behavioral measure y i without using a category label (bottom right panel).
Figure 2.
Case 1-1 and Case 1-2: Two representative cases with two disorder categories. A) Schematic of the generative model of Case 1-1. This case assumes that there are two distinct pathogenetic factors corresponding to each disorder category. The common symptom y 2 has different pathogenetic factors x 1 and x 2. B) The statistical power of several approaches for detecting the target factor x 1 (Case 1-1). Method 1, Method 2, and Method 3 employ category-based approaches, which sample individuals based on some criteria and perform the statistical test of the difference in the means (t test). Method 1 uses the single behavioral observation y 2. This samples 20 individuals with y 2 < h for the control group and 20 individuals with y 2 ≥ h for the patient group. Method 2 uses the double criteria. The subjects in the patient group satisfy y 1 ≥ h and y 2 ≥ h. The subjects in the control group do not satisfy either of these criteria. Method 3 uses both disorder criteria, and individuals who fall into Disorder A but not Disorder B are sampled for the patient group. The control group has neither Disorder A nor Disorder B. Method 4 and Method 5 are dimensional approaches that sample the individuals randomly and perform a correlation analysis (Method 4) or a multiple regression (Method 5). The error bars indicating the 95% confidence intervals of the power estimate are plotted, although they are almost invisible because their length is very short. C) Schematic of the generative model of Case 1-2. This case assumes that there are three distinct pathogenetic factors, where each affects a single symptom. The common symptom y 2 has a single pathogenetic factor x 2. D) The statistical powers of several approaches for detecting the target factor x 2 (Case 1-2). The convention is identical with Case 1-1.
Figure 3.
Sample scatterplots that explain how category-based approaches perform differently in Case 1-1 (A–B) and Case 1-2 (C–D). The orange-filled circles represent the individuals classified as patient by both Method 1, which uses a single symptom, and Method 2, which uses double symptoms. The blue triangles represent the individuals classified as patient only by Method 1 but not by Method 2. Thus these individuals belong to the control group in Method 2. The individuals marked with gray dots belong to the control group in both methods. The marginal distributions of x 1 and x 2 for each group for each method (dashed line: Method 1; solid line: Method 2) are drawn in (A) and (C).
Figure 4.
Comparison of the statistical powers of the category-based and dimensional approaches in Case 2. A) The schematic of the generative model in Case 2. This case includes a single pathogenetic factor (N = 1) and a single behavioral observation (M = 1). B) Illustration of the category-based approach with a margin. C) The statistical power (with the significance level α = 0.01) of both methods as a function of the total number of subjects, with a variable margin m for the category-based approach. The lines represent the results obtained from analytical calculations (see Katahira & Yamashita, 2017, Appendix B). The symbols represent the results of the Monte Carlo simulations (see Appendix A). D) The effect of cutoff point h. The results of the standard dimensional approach, which does not use the cutoff point, are indicated by the horizontal chain line. The results of the dimensional (correlation) approach using the sample obtained from the category-based method (given h) are plotted by the dashed lines with inverted triangles. E) The distribution of the estimated pathogenetic factor for three h cases (with m = 0). The sample means of each group are indicated by the vertical dashed lines. d = Cohen’s d.
Figure 5.
The effect of the number of diagnostic criteria M in the category-based approach (Case 3). A) The schematic of the generative model in Case 3. Here the model includes two pathogenetic factors (N = 2; x 2 is irrelevant) and M behavioral observations. B) The distribution of the estimated pathogenetic factor for three M cases, with σ ϵ = 1.0. The sample means of each group are indicated by the vertical dashed lines. d = Cohen’s d. C) The statistical power (with significance level α = 0.01) of both methods as a function of M, with varying standard deviation of the noise σϵ. The horizontal lines at M = 1 represent the analytical results (see Katahira & Yamashita, 2017, Appendix B). The symbols and the lines connecting the symbols for M for the category-based approach represent the results of Monte Carlo simulations. Dashed lines with triangles show the power of the dimensional (correlation) approach that used the same sample as the corresponding category-based method.
Figure 6.
The effect of a mixture of pathogenetic factors (Case 4). A) The schematic of the generative model in Case 4. Here the model includes two pathogenetic factors (N = 2) and two behavioral observations (M = 2). The parameter c indicates the mixture degree. B) The scatterplot of Y for two c cases. C) The statistical power (with critical value α = 0.01) of both methods as a function of the mixture degree c. The dash-dotted lines for the dimensional approach and the dashed lines for the category-based approach with a single criterion (y 1 ≥ h) denote the results from the analytical calculations (see Katahira & Yamashita, 2017, Appendix B). Symbols and solid lines for the category-based approach using two criteria represent the results of the Monte Carlo simulations.
Figure 7.
The effect of the number of pathogenetic factors N. A) The schematic of the generative model in Case 5. The model includes N pathogenetic factors and one behavioral observation (M = 1). B) The statistical power for detecting the first pathogenetic factor x 1 (with critical value α = .01) as a function of N. The dash-dotted lines (for the category-based approach with a single criterion and the dimensional approach with a single regressor) represent the results obtained from analytical calculations. The symbols represent the results of the Monte Carlo simulations. The results of multiple regression using all N variables are marked with filled triangles. Only for the case with N = 10, the results of the intermediate situation between simple regression and full multiple regression, which uses a part of the observation of 10 factors (including the observed target factor, ), are shown in the left small panel.
