## 2 Materials and Methods
### Statistical Analysis  
The first step in analysis was to estimate the structural model for \"\(c\)\", which was a confirmatory tri-factor model combining the bifactor structures for \"g\" and \"p\". Figure 1 shows the model configuration comprising \"\(c\)\", \"p\", \"g\", and all specific sub-factors, where all factors in the model are orthogonal to each other. Configuration for the \"g\" factor components was taken from [PERSON] et al. [29], and the \"p\" factor configuration was guided by [PERSON] et al. [13] with some adjustments. Both 2015 studies were conducted on the same sample (PNC), allowing much of the \"\(c\)\" factor structure to be known a priori. This is somewhat unusual insofar as factor analytic studies often begin with exploratory factor analysis, followed by confirmation of the structure (preferably in a different sample) using confirmatory factor analysis (CFA); however, because the structure of these cognitive tests and clinical diagnoses is known [2; 29], we were able to begin at the CFA stage.  
Figure 1: Model Configuration of “\(c\)” Factor and Associated Sub-Factors. Note. PTSD = posttraumatic stress disorder; Hyp = hyperactivity; WRAT = Wide Range Achievement Test.  
To explore the effect of age on the structure of the \"c\" factor model, we also conducted the above analyses on participants aged 8-12 and 13-21 separately. We compared coefficient estimates and model fit indices across these two models.  
The CFA for \"c\" was conducted in Mplus (v8.4) [51] using the mean- and variance-adjusted weighted least squares (wslmv) estimator. Model fit was assessed using the comparative fit index (CFI; >0.90 acceptable), root mean-square error of approximation (RSMEA; <0.08 acceptable), and standardized root mean-square residual (SRMR; <0.08 acceptable) [52]. Due to the \"Executive\" and \"Motor\" specific factors having only two indicators each (where three are required for identification in a bifactor model), factor loadings within these factors were constrained to equality. That is, the loadings of attention and working memory on the Executive factor were constrained to be equal to each other, and the same was true of the tapping and motor praxis tasks on the Motor factor.  
Finally, criterion validity of the c-factor scores was assessed by comparing its relationship with external criteria with the same relationships using \"p\" and \"g\". The specific validity criteria were average parent education, neighborhood socioeconomic status (SES), global assessment of functioning (GAF), and total intracranial volume (ICV). To assess the utility of the c factor in predicting functional outcome, we examined its association with longitudinal psychosis spectrum (PS) change at follow-up. Please see the Supplementary Materials for more discussion and rationale for using these specific validity criteria.  
Relationships with continuous criteria were assessed by Pearson correlations, all of which were expected to be positive (higher \"c\", higher SES/GAF/ICV/parent education); mean differences between follow-up groups (longitudinal PS change) were assessed using t-tests. All correlations and t-tests were performed using \"p\" and \"g\" scores both from a \"c\" factor model and in a model without a \"c\" factor. In the model including the \"c\" factor, \"p\" and \"g\" include only variance not already explained by \"c\", so they are akin to residuals of \"p\" and \"g\" after controlling for \"c\". We therefore also included \"p\" and \"g\" scores estimated without \"c\" to examine how strongly they associate with criteria when not controlled for \"c\".