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Table 2 Criterion validity assessment: Lorenz-1 status and inpatient hospital use

From: Use of power-law analysis to predict abuse or diversion of prescribed medications: proof-of-concept mathematical exploration

Power law zone Alprazolam (n = 463,203) Gabapentin (n = 267,693) Opioid (n = 2,077,393)  
PLG1 PLG2 PLG3 PLG4 PLG1 PLG2 PLG3   PLG1 PLG2 PLG3  
N of cases 241,992 161,677 49,603 9931 174,989 90,590 2114   1,833,724 186,258 57,411  
Lorenz-1 in follow-up 0.0 0.0 1.6 45.3 0.0 1.4 58.5   0.0 0.5 34.5  
Lorenz-1 sensitivity, specificity, PPV, NPVa (%) 83.5, 98.8, 45.3, 99.8 48.9, 99.7, 58.5, 99.5 95.4, 98.2, 34.5, 100.0
 Incident Lorenz-1 in follow-up 0.0 0.0 1.6 20.7 0.0 1.1 0.0   0.0 0.5 9.9  
Incident Lorenz-1 sensitivity, specificity, PPV, NPVa (%) 69.9, 98.3, 20.7, 99.8 0.0, 99.2, 0.0, 99.6 85.5, 97.5, 9.9, 100.0
 IPH observation year 3.5 5.0 6.6 6.3 7.0 8.7 9.5   5.7 9.3 8.4  
 IPH follow-up 3.6 5.0 6.6 7.3 6.7 8.4 9.4   4.5 8.1 8.2  
 IPH % change 2.9 0.0 0.0 15.9 − 4.3 − 3.4 − 1.1   − 21.1 − 12.9 − 2.4  
IPH sensitivity, specificity, PPV, NPV (%) 3.5, 97.9, 7.3, 95.6 1.0, 99.2, 9.4, 92.7 4.6, 97.3, 8.2, 95.2
 SUDb IPH observation year 0.6 1.2 2.0 2.0 1.4 1.7 2.6   0.7 2.3 2.6  
 SUDb IPH follow-up 0.6 1.2 2.2 2.8 1.3 1.6 2.5   0.7 2.2 2.9  
 SUDb IPH % change 0.0 0.0 10.0 40.0 − 7.1 − 5.9 − 3.8   0.0 − 4.3 11.5  
SUD IPH sensitivity, specificity, PPV, NPVa (%) 5.7, 97.9, 2.8, 99.0 1.4, 99.2, 2.5, 98.6 9.4, 97.3, 2.9, 99.2
Utilization quartile Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4
N of cases 133,446 108,543 103,532 117,680 71,889 60,802 65,223 69,779 486,760 537,304 528,704 524,625
Lorenz-1 in follow-up 0.0 0.0 0.0 4.6 0.0 0.0 0.0 3.6 0.0 0.0 0.0 4.0
Lorenz-1 sensitivity, specificity, PPV, NPVa (%) 99.5, 75.5, 4.6, 100.0 99.6, 74.6, 3.6, 100.0 99.9, 75.5, 4.0, 100.0
 Incident Lorenz-1 in follow-up 0.0 0.0 0.0 2.5 0.0 0.0 0.0 1.4 0.0 0.0 0.0 1.3
Incident Lorenz-1 sensitivity, specificity, PPV, NPVa (%) 99.2, 75.1, 2.5, 100.0 99.1, 74.2, 1.4, 100.0 99.7, 75.0, 1.3, 100.0
 IPH observation year 3.2 4.0 4.7 6.0 6.4 7.1 8.2 8.9 2.8 4.7 7.3 9.3
 IPH follow-up 3.4 3.9 4.7 6.1 6.1 6.9 7.7 8.5 3.2 3.8 5.2 7.4
 IPH % change 6.2 − 2.5 0.0 1.7 − 4.7 − 2.8 − 6.1 − 4.5 14.3 − 19.1 − 28.8 − 20.4
IPH sensitivity, specificity, PPV, NPVa (%) 34.6, 75.0, 6.1, 96.1 30.5, 74.3, 8.5, 93.1 38.0, 75.4, 7.4, 95.9
 SUDb IPH observation year 0.5 0.8 1.1 1.7 1.2 1.4 1.5 1.7 0.3 0.5 0.8 1.8
 SUDb IPH follow-up 0.5 0.8 1.1 1.9 1.2 1.3 1.4 1.7 0.4 0.5 0.7 1.8
 SUDb IPH % change 0.0 0.0 0.0 11.8 0.0 − 7.1 − 6.7 0.0 33.3 0.0 − 12.5 0.0
SUD IPH sensitivity, specificity, PPV, NPVa (%) 45.9, 74.8, 1.9, 99.2 31.6, 74.0, 1.7, 98.7 52.4, 75.0, 1.8, 99.4
  1. Earliest baseline year from January 1, 2013, through December 31, 2013, with follow-up from January 1, 2014, through June 30, 2014. Latest baseline year from July 1, 2014, through June 30, 2015, with follow-up from July 1, 2015, through December 31, 2015
  2. IPH inpatient hospital, NPV negative predictive value, PLG power-law group, PPV positive predictive value, Q quartile, SUD substance use disorder
  3. aAssuming that top PLG category and fourth quartile are predicted as at risk
  4. bDiagnosis codes in Additional file 1: Appendix S4