潜变量分析（进阶）

政务大数据应用与分析 (80700673)

胡悦

清华大学社会科学学院

概要

待解之题

观测变量与潜变量之间关系要非线性

方法

1. 项目反应理论(IRT)
2. 项目反应聚合估计

操作语言

• R¹
  • mirt
  • DCPO

1. 现存处理IRT的R packages已超过50个。

1 项目反应理论

1.1 超越因子分析

因子分析弊端：

1. 假定潜在变量是连续的；
2. 对于指标不区分变量类型；
3. 难以捕捉群组差异

4. EFA无法囊括指标间关系;
5. CFA“简略理论vs测量质量”矛盾

⇓

离散回应模型

CFA理论通常简略，只涉及一部分indices，但实际可能很复杂；当囊括更多indices测量质量会高，但不符合理论。

1.2 类型分析

基于单一潜在变量划分“三六九等” (Queen’s Gambit)

Frederic M. Lord

Item Response Theory (IRT)

1. 天生为二元指标设计（衍生适应定序变量和连续变量）；
2. 易与Bayesian inference结合，解决潜在变量scale不确定问题；
3. 易与跨群组估计结合，实现指标跨组可比

Frederic M. Lord, psychometrician, first worked for the Carnegie Foundation in 1944. By 1950, he began working for the Educational Testing Service (ETS). Lord’s research shaped the SAT, GRE, GMAT, LSAT and the TOEFL. Lord was called the “Father of Modern Testing” by the National Council on Measurement in Education.

1.3 Basic IRT

• 调查层次：个体
• Item: “一道题”
• Response: “对一道题（的选项）”
• 指标种类
  1. Yes/No
  2. 可以转化为二元的问题
  3. 定序问题（e.g., Liker scale questions）

Assumptions

1. Monotonicity
2. Unidimensionality
3. Local independence
4. Parameter invariance

1.4 Monotonicity

Item characteristic curve, 随潜在变量增加，获得1的可能性也随之增加

随着能力的提高，回答正确答案的机会也就越高

1.5 Unidimensionality

• 聚合的项目均指向同一个潜在变量¹
• 基于理论

1. 直到引入multidimensional IRT

1.6 Local Independence

\(P(Y_{ip}, Y_{iq}|\theta_i) = P(Y_{ip}|\theta_i)P(Y_{iq}|\theta_i),\)

• p, q: Items
• i: Respondent
• Y_ip: i对于项目p的的反应
• θ_i: 反应者i的潜在变量

对于不同项目的反应之间，关联性只来自共同的潜在变量。

换言之，控制潜在变量影响后，问题间响应相互独立

1.7 Parameter Invariance

• 项目特点（parameters of items，比如难度、梯度等）在项目间不变
• 项目特点在响应人群间不变¹

When parameters vary

当进行Multiple Group IRT时尤其容易被违反!

不想被违反的办法：足够多的参数；GPT 3.5 175B parameters； 4， 1.8T； deepseek R1 671 b

1. 通过基于Wald and likelihood-ratio approach来检测Differential item functioning (DIF).

1.8 IRT模型发展

Rasch Model (1PL)

Rasch /resh/

→ Two-Parameter Logistic Model (2PL)
→ Three-Parameter Logistic Model (3PL)
→ Four-Parameter Logistic Model (4PL)

↓

Multidimensional IRT
Ordinal IRT
Group IRT

1.9 Rasch Model

• y_iq∈{0,1}: 反应，i对问题q的回答
• θ_i∈{-∞, +∞}: 反应能力，(Unbounded latent trait)
• σ_q: Difficulty，问题难易程度，通常显示为z scores

🌰 不同难度的项目：

• 当面临重大公共卫生威胁时，政府是否应该及时响应，采取果断措施？
• 政府是否可以牺牲少数民众权利，来换取大多数社会成员的公共卫生安全？

\[ \text{1PL: }\color{red}{P(Y_{iq} = 1)} = \color{blue}{logist^{-1}(\theta_i - \sigma_q)}, \]

\[ \color{red}{项目反应} = \color{blue}{反应理论}. \]

1.10 演示案例 (BockLieberman1970?)

Law School Admissions Test
sec 7, 5个yes/no问题

     Item.1 Item.2 Item.3 Item.4 Item.5
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503       1      1      0      1      1
504       1      1      0      1      1
505       1      1      0      1      1
506       1      1      0      1      1
507       1      1      1      0      0
508       1      1      1      0      0
509       1      1      1      0      0
510       1      1      1      0      0
511       1      1      1      0      0
512       1      1      1      0      0
513       1      1      1      0      0
514       1      1      1      0      0
515       1      1      1      0      0
516       1      1      1      0      0
517       1      1      1      0      0
518       1      1      1      0      0
519       1      1      1      0      0
520       1      1      1      0      0
521       1      1      1      0      0
522       1      1      1      0      0
523       1      1      1      0      0
524       1      1      1      0      0
525       1      1      1      0      1
526       1      1      1      0      1
527       1      1      1      0      1
528       1      1      1      0      1
529       1      1      1      0      1
530       1      1      1      0      1
531       1      1      1      0      1
532       1      1      1      0      1
533       1      1      1      0      1
534       1      1      1      0      1
535       1      1      1      0      1
536       1      1      1      0      1
537       1      1      1      0      1
538       1      1      1      0      1
539       1      1      1      0      1
540       1      1      1      0      1
541       1      1      1      0      1
542       1      1      1      0      1
543       1      1      1      0      1
544       1      1      1      0      1
545       1      1      1      0      1
546       1      1      1      0      1
547       1      1      1      0      1
548       1      1      1      0      1
549       1      1      1      0      1
550       1      1      1      0      1
551       1      1      1      0      1
552       1      1      1      0      1
553       1      1      1      0      1
554       1      1      1      0      1
555       1      1      1      0      1
556       1      1      1      0      1
557       1      1      1      0      1
558       1      1      1      0      1
559       1      1      1      0      1
560       1      1      1      0      1
561       1      1      1      0      1
562       1      1      1      0      1
563       1      1      1      0      1
564       1      1      1      0      1
565       1      1      1      0      1
566       1      1      1      0      1
567       1      1      1      0      1
568       1      1      1      0      1
569       1      1      1      0      1
570       1      1      1      0      1
571       1      1      1      0      1
572       1      1      1      0      1
573       1      1      1      0      1
574       1      1      1      0      1
575       1      1      1      0      1
576       1      1      1      0      1
577       1      1      1      0      1
578       1      1      1      0      1
579       1      1      1      0      1
580       1      1      1      0      1
581       1      1      1      0      1
582       1      1      1      0      1
583       1      1      1      0      1
584       1      1      1      0      1
585       1      1      1      0      1
586       1      1      1      0      1
587       1      1      1      0      1
588       1      1      1      0      1
589       1      1      1      0      1
590       1      1      1      0      1
591       1      1      1      0      1
592       1      1      1      0      1
593       1      1      1      0      1
594       1      1      1      0      1
595       1      1      1      0      1
596       1      1      1      0      1
597       1      1      1      0      1
598       1      1      1      0      1
599       1      1      1      0      1
600       1      1      1      0      1
601       1      1      1      0      1
602       1      1      1      0      1
603       1      1      1      0      1
604       1      1      1      0      1
605       1      1      1      0      1
606       1      1      1      0      1
607       1      1      1      0      1
608       1      1      1      0      1
609       1      1      1      0      1
610       1      1      1      0      1
611       1      1      1      0      1
612       1      1      1      0      1
613       1      1      1      0      1
614       1      1      1      0      1
615       1      1      1      0      1
616       1      1      1      0      1
617       1      1      1      0      1
618       1      1      1      0      1
619       1      1      1      0      1
620       1      1      1      0      1
621       1      1      1      0      1
622       1      1      1      0      1
623       1      1      1      0      1
624       1      1      1      0      1
625       1      1      1      0      1
626       1      1      1      0      1
627       1      1      1      0      1
628       1      1      1      0      1
629       1      1      1      0      1
630       1      1      1      0      1
631       1      1      1      0      1
632       1      1      1      0      1
633       1      1      1      0      1
634       1      1      1      0      1
635       1      1      1      0      1
636       1      1      1      0      1
637       1      1      1      0      1
638       1      1      1      0      1
639       1      1      1      0      1
640       1      1      1      0      1
641       1      1      1      0      1
642       1      1      1      0      1
643       1      1      1      0      1
644       1      1      1      0      1
645       1      1      1      0      1
646       1      1      1      0      1
647       1      1      1      0      1
648       1      1      1      0      1
649       1      1      1      0      1
650       1      1      1      0      1
651       1      1      1      0      1
652       1      1      1      0      1
653       1      1      1      0      1
654       1      1      1      0      1
655       1      1      1      0      1
656       1      1      1      0      1
657       1      1      1      0      1
658       1      1      1      0      1
659       1      1      1      0      1
660       1      1      1      0      1
661       1      1      1      1      0
662       1      1      1      1      0
663       1      1      1      1      0
664       1      1      1      1      0
665       1      1      1      1      0
666       1      1      1      1      0
667       1      1      1      1      0
668       1      1      1      1      0
669       1      1      1      1      0
670       1      1      1      1      0
671       1      1      1      1      0
672       1      1      1      1      0
673       1      1      1      1      0
674       1      1      1      1      0
675       1      1      1      1      0
676       1      1      1      1      0
677       1      1      1      1      0
678       1      1      1      1      0
679       1      1      1      1      0
680       1      1      1      1      0
681       1      1      1      1      0
682       1      1      1      1      0
683       1      1      1      1      0
684       1      1      1      1      0
685       1      1      1      1      0
686       1      1      1      1      0
687       1      1      1      1      0
688       1      1      1      1      0
689       1      1      1      1      0
690       1      1      1      1      0
691       1      1      1      1      0
692       1      1      1      1      0
693       1      1      1      1      1
694       1      1      1      1      1
695       1      1      1      1      1
696       1      1      1      1      1
697       1      1      1      1      1
698       1      1      1      1      1
699       1      1      1      1      1
700       1      1      1      1      1
701       1      1      1      1      1
702       1      1      1      1      1
703       1      1      1      1      1
704       1      1      1      1      1
705       1      1      1      1      1
706       1      1      1      1      1
707       1      1      1      1      1
708       1      1      1      1      1
709       1      1      1      1      1
710       1      1      1      1      1
711       1      1      1      1      1
712       1      1      1      1      1
713       1      1      1      1      1
714       1      1      1      1      1
715       1      1      1      1      1
716       1      1      1      1      1
717       1      1      1      1      1
718       1      1      1      1      1
719       1      1      1      1      1
720       1      1      1      1      1
721       1      1      1      1      1
722       1      1      1      1      1
723       1      1      1      1      1
724       1      1      1      1      1
725       1      1      1      1      1
726       1      1      1      1      1
727       1      1      1      1      1
728       1      1      1      1      1
729       1      1      1      1      1
730       1      1      1      1      1
731       1      1      1      1      1
732       1      1      1      1      1
733       1      1      1      1      1
734       1      1      1      1      1
735       1      1      1      1      1
736       1      1      1      1      1
737       1      1      1      1      1
738       1      1      1      1      1
739       1      1      1      1      1
740       1      1      1      1      1
741       1      1      1      1      1
742       1      1      1      1      1
743       1      1      1      1      1
744       1      1      1      1      1
745       1      1      1      1      1
746       1      1      1      1      1
747       1      1      1      1      1
748       1      1      1      1      1
749       1      1      1      1      1
750       1      1      1      1      1
751       1      1      1      1      1
752       1      1      1      1      1
753       1      1      1      1      1
754       1      1      1      1      1
755       1      1      1      1      1
756       1      1      1      1      1
757       1      1      1      1      1
758       1      1      1      1      1
759       1      1      1      1      1
760       1      1      1      1      1
761       1      1      1      1      1
762       1      1      1      1      1
763       1      1      1      1      1
764       1      1      1      1      1
765       1      1      1      1      1
766       1      1      1      1      1
767       1      1      1      1      1
768       1      1      1      1      1
769       1      1      1      1      1
770       1      1      1      1      1
771       1      1      1      1      1
772       1      1      1      1      1
773       1      1      1      1      1
774       1      1      1      1      1
775       1      1      1      1      1
776       1      1      1      1      1
777       1      1      1      1      1
778       1      1      1      1      1
779       1      1      1      1      1
780       1      1      1      1      1
781       1      1      1      1      1
782       1      1      1      1      1
783       1      1      1      1      1
784       1      1      1      1      1
785       1      1      1      1      1
786       1      1      1      1      1
787       1      1      1      1      1
788       1      1      1      1      1
789       1      1      1      1      1
790       1      1      1      1      1
791       1      1      1      1      1
792       1      1      1      1      1
793       1      1      1      1      1
794       1      1      1      1      1
795       1      1      1      1      1
796       1      1      1      1      1
797       1      1      1      1      1
798       1      1      1      1      1
799       1      1      1      1      1
800       1      1      1      1      1
801       1      1      1      1      1
802       1      1      1      1      1
803       1      1      1      1      1
804       1      1      1      1      1
805       1      1      1      1      1
806       1      1      1      1      1
807       1      1      1      1      1
808       1      1      1      1      1
809       1      1      1      1      1
810       1      1      1      1      1
811       1      1      1      1      1
812       1      1      1      1      1
813       1      1      1      1      1
814       1      1      1      1      1
815       1      1      1      1      1
816       1      1      1      1      1
817       1      1      1      1      1
818       1      1      1      1      1
819       1      1      1      1      1
820       1      1      1      1      1
821       1      1      1      1      1
822       1      1      1      1      1
823       1      1      1      1      1
824       1      1      1      1      1
825       1      1      1      1      1
826       1      1      1      1      1
827       1      1      1      1      1
828       1      1      1      1      1
829       1      1      1      1      1
830       1      1      1      1      1
831       1      1      1      1      1
832       1      1      1      1      1
833       1      1      1      1      1
834       1      1      1      1      1
835       1      1      1      1      1
836       1      1      1      1      1
837       1      1      1      1      1
838       1      1      1      1      1
839       1      1      1      1      1
840       1      1      1      1      1
841       1      1      1      1      1
842       1      1      1      1      1
843       1      1      1      1      1
844       1      1      1      1      1
845       1      1      1      1      1
846       1      1      1      1      1
847       1      1      1      1      1
848       1      1      1      1      1
849       1      1      1      1      1
850       1      1      1      1      1
851       1      1      1      1      1
852       1      1      1      1      1
853       1      1      1      1      1
854       1      1      1      1      1
855       1      1      1      1      1
856       1      1      1      1      1
857       1      1      1      1      1
858       1      1      1      1      1
859       1      1      1      1      1
860       1      1      1      1      1
861       1      1      1      1      1
862       1      1      1      1      1
863       1      1      1      1      1
864       1      1      1      1      1
865       1      1      1      1      1
866       1      1      1      1      1
867       1      1      1      1      1
868       1      1      1      1      1
869       1      1      1      1      1
870       1      1      1      1      1
871       1      1      1      1      1
872       1      1      1      1      1
873       1      1      1      1      1
874       1      1      1      1      1
875       1      1      1      1      1
876       1      1      1      1      1
877       1      1      1      1      1
878       1      1      1      1      1
879       1      1      1      1      1
880       1      1      1      1      1
881       1      1      1      1      1
882       1      1      1      1      1
883       1      1      1      1      1
884       1      1      1      1      1
885       1      1      1      1      1
886       1      1      1      1      1
887       1      1      1      1      1
888       1      1      1      1      1
889       1      1      1      1      1
890       1      1      1      1      1
891       1      1      1      1      1
892       1      1      1      1      1
893       1      1      1      1      1
894       1      1      1      1      1
895       1      1      1      1      1
896       1      1      1      1      1
897       1      1      1      1      1
898       1      1      1      1      1
899       1      1      1      1      1
900       1      1      1      1      1
901       1      1      1      1      1
902       1      1      1      1      1
903       1      1      1      1      1
904       1      1      1      1      1
905       1      1      1      1      1
906       1      1      1      1      1
907       1      1      1      1      1
908       1      1      1      1      1
909       1      1      1      1      1
910       1      1      1      1      1
911       1      1      1      1      1
912       1      1      1      1      1
913       1      1      1      1      1
914       1      1      1      1      1
915       1      1      1      1      1
916       1      1      1      1      1
917       1      1      1      1      1
918       1      1      1      1      1
919       1      1      1      1      1
920       1      1      1      1      1
921       1      1      1      1      1
922       1      1      1      1      1
923       1      1      1      1      1
924       1      1      1      1      1
925       1      1      1      1      1
926       1      1      1      1      1
927       1      1      1      1      1
928       1      1      1      1      1
929       1      1      1      1      1
930       1      1      1      1      1
931       1      1      1      1      1
932       1      1      1      1      1
933       1      1      1      1      1
934       1      1      1      1      1
935       1      1      1      1      1
936       1      1      1      1      1
937       1      1      1      1      1
938       1      1      1      1      1
939       1      1      1      1      1
940       1      1      1      1      1
941       1      1      1      1      1
942       1      1      1      1      1
943       1      1      1      1      1
944       1      1      1      1      1
945       1      1      1      1      1
946       1      1      1      1      1
947       1      1      1      1      1
948       1      1      1      1      1
949       1      1      1      1      1
950       1      1      1      1      1
951       1      1      1      1      1
952       1      1      1      1      1
953       1      1      1      1      1
954       1      1      1      1      1
955       1      1      1      1      1
956       1      1      1      1      1
957       1      1      1      1      1
958       1      1      1      1      1
959       1      1      1      1      1
960       1      1      1      1      1
961       1      1      1      1      1
962       1      1      1      1      1
963       1      1      1      1      1
964       1      1      1      1      1
965       1      1      1      1      1
966       1      1      1      1      1
967       1      1      1      1      1
968       1      1      1      1      1
969       1      1      1      1      1
970       1      1      1      1      1
971       1      1      1      1      1
972       1      1      1      1      1
973       1      1      1      1      1
974       1      1      1      1      1
975       1      1      1      1      1
976       1      1      1      1      1
977       1      1      1      1      1
978       1      1      1      1      1
979       1      1      1      1      1
980       1      1      1      1      1
981       1      1      1      1      1
982       1      1      1      1      1
983       1      1      1      1      1
984       1      1      1      1      1
985       1      1      1      1      1
986       1      1      1      1      1
987       1      1      1      1      1
988       1      1      1      1      1
989       1      1      1      1      1
990       1      1      1      1      1
991       1      1      1      1      1
992       1      1      1      1      1
993       1      1      1      1      1
994       1      1      1      1      1
995       1      1      1      1      1
996       1      1      1      1      1
997       1      1      1      1      1
998       1      1      1      1      1
999       1      1      1      1      1
1000      1      1      1      1      1

mirt Workshop 1

1.11 Difficulty Parameter

       a1         d g u
Item.1  1 1.8680718 0 1
Item.2  1 0.7909134 0 1
Item.3  1 1.4608233 0 1
Item.4  1 0.5214399 0 1
Item.5  1 1.9927710 0 1

Item Characteristic Curves

检查各题affirmative的难易程度，看逐个是不是大体同一个趋势

1.12 Test Characteristic Curve

TCC： 所有ICC之和，体现how reliable, information 理想是形成一个钟形,顶尖处代表平均水平，据此对比个人θ可以判断这人是否是outlier SE(θ) = (test)^-1/2

1.13 1PL → 2PL

Rasch局限：Measurement error

\[ \begin{align} \text{Rasch: } P(Y_{iq} = 1) =& logist^{-1}(\theta_i - \sigma_q),\\ \text{2PL: } P(Y_{iq} = 1) =& logist^{-1}(\color{red}{\kappa_q}\theta_i - \sigma_q), \end{align} \]

κ_q: Discrimination (Parameter of dispersion)

另一种常见写法

\[Pr(y_{iq} = 1) = logist^{-1}[\frac{\theta_i - {\color{red}{\beta_q}}}{\color{red}{\alpha_q}}]\]

• α_q: Dispersion

Dispersion: magnitude of the measurement error

人们对同一个题理解不同，回答出affirmative答案可能性也不同。

Discrimination: how well a question can differentiate people from below to above;

Rule of thumb above 1 meaning a good question in terms of examination

1.14 Difficulty vs. Dispersion (Statistically)

\[Pr(y_{iq} = 1) = logist^{-1}[\frac{\theta_i - {\color{blue}{\beta_q}}}{\color{red}{\alpha_q}}]\]

• β_q: σ_q ⁄ κ_q, threshold(“difficulty”, 控制location)
• α_q: κ_q^-1, dispersion (控制斜率)

1.15 Difficulty vs. Dispersion (Substantively)

Difficulty

Dispersion

1.16 Estimation

Rasch

       a1         d g u
Item.1  1 1.8680718 0 1
Item.2  1 0.7909134 0 1
Item.3  1 1.4608233 0 1
Item.4  1 0.5214399 0 1
Item.5  1 1.9927710 0 1

2PL

              a1         d g u
Item.1 0.9879254 1.8560605 0 1
Item.2 1.0808847 0.8079786 0 1
Item.3 1.7058006 1.8042187 0 1
Item.4 0.7651853 0.4859966 0 1
Item.5 0.7357980 1.8545127 0 1

你真的需要2PL吗?

Likelihood-Ratio Test

	AIC	SABIC	HQ	logLik	df	p
m_lsat	5341.802	5352.192	5352.994	-2664.901	NA	NA
m_lsat2PL	5337.610	5354.927	5356.263	-2658.805	4	0.0159822

1.17 2PL → 3PL → 4PL

如果发给了初中生高中数学题？→ Three-Parameter Logistic Model (3PL)

全凭猜 → 大量低θ人群

\[Pr(y_{iq} = 1) = \color{red}{c_i + (1 - c_i)}logist^{-1}[\frac{(\theta_i - \beta_q)}{\alpha_q}],\] c_i：Item lower asymptote (“guessing”)

极大增加演算成本→通常需要1000以上观测点

如果有人不care咋办 → Four-Parameter Logistic Model (4PL)

\[Pr(y_{iq} = 1) = c_i + (\color{red}{d_i} - c_i)logist^{-1}[\frac{(\theta_i - \beta_q)}{\alpha_q}], \] d_i：Item upper asymptote (“carelessness”), d < 1

鉴于3PL已经需要1000-ish观测点……

1.18 IRT Diagnosis

• 测试层：Global fit
• 项目层：Item fit & residual
• 反应者层：Personal fit

1.19 Global Fit¹

\(G^2 = 2[\sum_l^s r_lln(\frac{r_l}{N\tilde{P}_l})],\)¹

N: 参与人数
l: 可能的反应
r: 做出特定反应的人数

当数据过于稀疏时(item > 10)，M2, M2*

            M2 df          p      RMSEA    RMSEA_5   RMSEA_95      SRMSR       TLI
stats 23.17287  9 0.00581954 0.03970314 0.02003961 0.05998303 0.04744033 0.9284234
            CFI
stats 0.9355811

REDO: Maydeu-Olivares (2013)

N is the number of subjects, L is number of possible response patterns, \(P_ l\) is the estimated probability of observing response pattern l, and \(r_ l\) is the number of subjects who have response pattern l

• χ sig
  
• Tucker-lewis index, 1 ideal, < .95 poor
  
• Comparative fit index, the same
  
• Root mean square error of approximation, 0 perfect, <.05 good, [.05, .08] ok, > .1 poor
  
• Standardized root mean square residual <.08 acceptable

1. RMSEA, SRMSR, CFI, TLI对于IRT同样使用.

1.20 Item Diagnostics

Covariation-based residuals

LD matrix (lower triangle) and standardized residual correlations (upper triangle)

Upper triangle summary:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 -0.064  -0.036   0.007   0.000   0.020   0.105 

       Item.1 Item.2 Item.3 Item.4 Item.5
Item.1        -0.017  0.020  0.022  0.019
Item.2  0.292         0.105 -0.042 -0.064
Item.3  0.389 10.976         0.007  0.007
Item.4  0.474  1.801  0.055        -0.052
Item.5  0.362  4.063  0.045  2.691       

• 多用于检验multidimensionality

看item residual的协变程度，unidimnesionality 聚合的项目均指向同一个潜在变量， 不应有关联

Single item/person fit

    item outfit z.outfit infit z.infit
1 Item.1  0.744   -3.597 0.939  -1.025
2 Item.2  0.758   -7.500 0.826  -6.303
3 Item.3  0.711   -5.420 0.860  -3.202
4 Item.4  0.770   -8.877 0.818  -7.962
5 Item.5  0.797   -2.572 0.993  -0.081

        outfit   z.outfit     infit     z.infit          Zh
1    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
2    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
3    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
4    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
5    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
6    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
7    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
8    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
9    0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
10   0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
11   0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
12   0.6420959 -0.9001783 0.6953215 -0.96202225  0.93429324
13   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
14   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
15   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
16   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
17   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
18   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
19   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
20   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
21   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
22   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
23   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
24   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
25   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
26   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
27   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
28   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
29   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
30   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
31   0.8178336 -0.6300162 0.8365515 -0.58965744  0.64983875
32   1.4702553  1.6126462 1.4447540  1.61101088 -1.73598830
33   1.3927122  1.1849894 1.3937871  1.27940982 -1.28637186
34   1.3927122  1.1849894 1.3937871  1.27940982 -1.28637186
35   1.3927122  1.1849894 1.3937871  1.27940982 -1.28637186
36   1.3927122  1.1849894 1.3937871  1.27940982 -1.28637186
37   1.3927122  1.1849894 1.3937871  1.27940982 -1.28637186
38   1.3927122  1.1849894 1.3937871  1.27940982 -1.28637186
39   1.3927122  1.1849894 1.3937871  1.27940982 -1.28637186
40   1.0428516  0.2464647 1.0613929  0.32004647 -0.21273742
41   1.0428516  0.2464647 1.0613929  0.32004647 -0.21273742
42   1.0428516  0.2464647 1.0613929  0.32004647 -0.21273742
43   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
44   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
45   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
46   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
47   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
48   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
49   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
50   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
51   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
52   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
53   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
54   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
55   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
56   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
57   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
58   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
59   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
60   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
61   1.0023472  0.1156527 0.9806210  0.03358782  0.03962718
62   1.6793416  1.8478659 1.5976464  1.80384093 -2.03724960
63   1.6793416  1.8478659 1.5976464  1.80384093 -2.03724960
64   1.6793416  1.8478659 1.5976464  1.80384093 -2.03724960
65   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
66   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
67   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
68   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
69   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
70   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
71   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
72   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
73   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
74   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
75   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
76   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
77   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
78   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
79   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
80   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
81   1.4319232  0.9040008 1.3542229  0.89387872 -0.83441775
82   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
83   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
84   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
85   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
86   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
87   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
88   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
89   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
90   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
91   1.3283603  1.1925623 1.3435562  1.29493299 -1.29902533
92   1.2804056  0.9003924 1.2754654  0.95116392 -0.90599298
93   1.2804056  0.9003924 1.2754654  0.95116392 -0.90599298
94   1.2804056  0.9003924 1.2754654  0.95116392 -0.90599298
95   1.2804056  0.9003924 1.2754654  0.95116392 -0.90599298
96   1.2804056  0.9003924 1.2754654  0.95116392 -0.90599298
97   1.9574000  2.4221949 1.8924908  2.48855647 -2.98286976
98   1.9574000  2.4221949 1.8924908  2.48855647 -2.98286976
99   1.9574000  2.4221949 1.8924908  2.48855647 -2.98286976
100  1.7753892  1.3847459 1.6548212  1.43482759 -1.53366231
101  1.7753892  1.3847459 1.6548212  1.43482759 -1.53366231
102  1.7753892  1.3847459 1.6548212  1.43482759 -1.53366231
103  1.7753892  1.3847459 1.6548212  1.43482759 -1.53366231
104  1.7753892  1.3847459 1.6548212  1.43482759 -1.53366231
105  1.7753892  1.3847459 1.6548212  1.43482759 -1.53366231
106  1.7753892  1.3847459 1.6548212  1.43482759 -1.53366231
107  1.5670351  1.5978712 1.4793247  1.50529970 -1.65687072
108  1.5670351  1.5978712 1.4793247  1.50529970 -1.65687072
109  1.5670351  1.5978712 1.4793247  1.50529970 -1.65687072
110  1.5670351  1.5978712 1.4793247  1.50529970 -1.65687072
111  1.5670351  1.5978712 1.4793247  1.50529970 -1.65687072
112  1.5670351  1.5978712 1.4793247  1.50529970 -1.65687072
113  1.5670351  1.5978712 1.4793247  1.50529970 -1.65687072
114  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
115  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
116  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
117  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
118  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
119  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
120  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
121  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
122  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
123  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
124  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
125  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
126  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
127  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
128  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
129  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
130  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
131  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
132  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
133  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
134  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
135  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
136  1.3208296  0.7321411 1.2189235  0.62398564 -0.55314427
137  2.2014754  1.9014751 1.8368418  1.73106099 -2.08890342
138  2.2014754  1.9014751 1.8368418  1.73106099 -2.08890342
139  2.2014754  1.9014751 1.8368418  1.73106099 -2.08890342
140  2.2014754  1.9014751 1.8368418  1.73106099 -2.08890342
141  2.2014754  1.9014751 1.8368418  1.73106099 -2.08890342
142  2.2014754  1.9014751 1.8368418  1.73106099 -2.08890342
143  2.2014754  1.9014751 1.8368418  1.73106099 -2.08890342
144  2.2014754  1.9014751 1.8368418  1.73106099 -2.08890342
145  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
146  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
147  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
148  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
149  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
150  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
151  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
152  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
153  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
154  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
155  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
156  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
157  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
158  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
159  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
160  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
161  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
162  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
163  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
164  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
165  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
166  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
167  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
168  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
169  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
170  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
171  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
172  1.7248030  1.0070459 1.3171687  0.66469738 -0.64120315
173  0.8741944 -0.3967143 0.8874432 -0.37066077  0.44763365
174  0.8741944 -0.3967143 0.8874432 -0.37066077  0.44763365
175  0.8741944 -0.3967143 0.8874432 -0.37066077  0.44763365
176  0.8741944 -0.3967143 0.8874432 -0.37066077  0.44763365
177  0.8741944 -0.3967143 0.8874432 -0.37066077  0.44763365
178  0.8741944 -0.3967143 0.8874432 -0.37066077  0.44763365
179  0.8741944 -0.3967143 0.8874432 -0.37066077  0.44763365
180  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
181  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
182  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
183  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
184  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
185  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
186  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
187  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
188  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
189  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
190  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
191  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
192  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
193  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
194  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
195  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
196  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
197  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
198  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
199  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
200  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
201  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
202  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
203  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
204  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
205  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
206  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
207  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
208  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
209  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
210  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
211  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
212  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
213  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
214  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
215  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
216  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
217  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
218  0.7925432 -0.5792464 0.8207813 -0.54337697  0.61448416
219  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
220  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
221  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
222  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
223  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
224  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
225  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
226  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
227  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
228  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
229  1.4695376  1.3709638 1.4378067  1.39677740 -1.46239261
230  1.1241484  0.4025567 1.2099241  0.60533914 -0.40933625
231  1.1241484  0.4025567 1.2099241  0.60533914 -0.40933625
232  1.1241484  0.4025567 1.2099241  0.60533914 -0.40933625
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970  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
971  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
972  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
973  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
974  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
975  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
976  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
977  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
978  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
979  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
980  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
981  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
982  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
983  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
984  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
985  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
986  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
987  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
988  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
989  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
990  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
991  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
992  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
993  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
994  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
995  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
996  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
997  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
998  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
999  0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589
1000 0.1482615 -0.7478428 0.1847431 -0.99648352  0.80314589

infit/outfit, close to 1 is good

Z_h > 0 better

1.21 如果出现问题

1. 通过S-χ²、local dependency等检查观测和估计数值差别
2. 增加模型复杂程度, 比如2PL → 3PL
3. 如果最初用binary，尝试polytomous或者nominal response models
4. 尝试non-parametric smoothing techniques

1.22 小结

• 项目反应理论（IRT）
  • 项目：题
  • 反应：做题
• IRT 假定
  • Monotonicity
  • Unidimensionality
  • Local independence
  • Parameter invariance

• IRT 模型演进
  • Rasch Model
  • 2PL, 3PL, 4PL
• IRT 诊断
  • Global fit
  • Item fit & residual
  • Personal fit

2 发展的项目反应理论

2.1 发展方向

• 个体层级
  • 一维到多维
  • 二分到多类
  • 单层到多层

• 群体层级
  • 更准确的、可比较的群体差异

2.2 一维到多维

Multidimentional IRT (MIRT, Chalmers 2015)

\[Pr(y_{iq} = 1) = logist^{-1}[\frac{\boldsymbol{\theta_i} - \beta_q}{\boldsymbol{\alpha_q}}]\]

Note

θ_i和α_q不再是单一值，而是一个矩阵。

2.3 二分到多类

Logit → Cumulative logit

\[P(Y_{iq} = 1) \rightarrow Pr(\frac{Y_{iq}\leq c}{Y_{iq}>c}).\]

三种主要类型

1. (Modified) Graded Response Model
   • 用于scoring rubrics，比如 Likert
2. (Generalized) Partial Credit Model，Rating Scale Model
   • 用于可转化为定序的分类变量
3. Nominal Response Model
   • 用于无序分类变量

https://stats.stackexchange.com/questions/402440/how-should-we-select-between-various-item-response-theory-models-e-g-rsm-grm

2.4 单层到多层

Multilevel Mixture IRT with Item Bias Effects (Stegmueller 2011)

在估测α_q时加入random effect.

Daniel Stegmueller, Duke U, poli sci

2.5 超越个体

Conquer the individual fallacy

Ecological fallacy (Simpson’s paradox)

再比如，民主、不平等、政治文化……

2.6 聚合层级上的尝试 (Caughey and Warshaw 2015)

Dynamic Group-level IRT（DGIRT)

聚合（Group-level）：

\(\eta_{ktq} = logit^{-1}(\frac{\color{red}{\bar{\theta}_{kt}}- \beta_q}{\sqrt{\alpha^2_q + \color{red}{(1.7\sigma_{kt})^2}}}),\)

\(\bar{\theta}_k\) 和 σ_kt 是潜在变量在组k时间t的均值和SD。

1. 在群组层面估测IRT；¹
2. 在估测IRT过程中加入群组级别变量；
3. 将时间变量融入IRT估测；
4. 用MrP给估测进行权重。

1.7: sd of probit is (π/3)^1/2 for logit, while Long 1997 found it is more close to 1.7 in actual estimations.

Mislevy, Robert J. 1983. “Item Response Models for Grouped Data.” Journal of Educational Statistics 8(4): 271–88.

η: eta

囊括时间与空间 (Dynamic)

\[ \begin{align} \bar{\theta}_k\sim& N(\xi_t + \boldsymbol{x'_k\gamma}, \sigma^2_{\bar{\theta}}), \\ \xi_t \sim& N(\xi_{t-1}, \sigma^2_{\gamma}), \\ \gamma_{pt} \sim& N(\gamma_{p,t-1}, \delta_t + \boldsymbol{z'_p.\eta_t}, \sigma^2_{\gamma}) \end{align} \]

ξ_t: xi

x 为群组级变量

t-1, dynamic linear model

z’_p.η_t: geography-level attributes, η是coefficients

n^*_kqt基于MrP

效果：

• 囊括诸多因素
• 可以部分平衡样本代表性问题
• 强大，但复杂

Caughey & Warshaw称会跑几个星期

1. 个体：\(p_{iq} = logist^{-1}[\frac{\theta_i - {\beta_q}}{\alpha_q}].\)

2.7 简化DGIRT (Claassen 2019)

\[ \begin{align} \eta_{ktq} = logit^{-1}&(\frac{\bar{\theta}'_{kt}- \beta_q}{\sqrt{\alpha^2_q + (1.7\sigma_{kt})^2}}).\\ \downarrow&\\ \eta_{ktq} = logit^{-1}&(\frac{\bar{\theta}'_{kt}- (\beta_q \color{red}{+ \delta_{kq}})}{\alpha_q}). \end{align} \]

δ_kq: 问题的difficulty随国家k变化。

1. 只作用于代表性样本和国家级别
   
2. 只对二分变量进行分析
3. 将国家作用从估测θ变为估测difficulty
   
4. 假定本地项目正态分布（忽略极化等现象😱）

2.8 聚合IRT修正版 (Solt 2020)

Dynamic Comparative Public Opinion

复杂程度：

Claasseen 2019 < DCPO < DGIRT

发表：

• APSR(Tai, Hu, and Solt 2024)
• BJPS(Woo, Goldberg, and Solt 2023)
• PSRM(Hu, Tai, and Solt 2024)
• SSQ(Woo et al. 2024)

……

2.9 操作

1. 收集survey数据，明确与感兴趣的变量相关的指标问题（手动）
2. 通过DCPOtools对数据进行预处理（半自动）
3. 通过DCPO进行数据分析（自动）
4. 通过shinystan诊断convergence（自动）

千万注意！

如同其他方法一样，聚合IRT也需要对结果与数据拟合程度进行诊断。 DCPO等聚合IRT方法使用贝叶斯统计框架，结果需要对reccurence, stationarity, aperodicity等进行检验。

2.10 效果与优势

Bounded: 使用logit归为0-1

2.11 最新进展 (“究极形态”？, Berwick and Caughey 2025)

Multidimensional, multilevel IRT

• Strengths
  • Flexible: Single-country or cross-national datasets
  • Extensible: Overdispersion, differential item functioning, and predictors.
• Considerations
  • Computationally intensive (Bayesian estimation).
  • Needs bridging items for comparability across groups/time.

2.12 总结

连续因子模型

• 解决问题：潜在变量描述与关系检验
• 探索性因子分析(EFA)
  • 潜在因子探索
• 验证性因子分析(CFA)
  • 统计推断
• 结构方程模型(SEM)
  • 潜在变量与外生变量关系

使用建议

• 没有高低之分
• 确有难易之别
• 关键是方法的恰当应用和诊断

离散回应模型

• 解决问题：Modeling非连续性指标
• IRT
  • Rasch，nPL
  • Diagnoses: 总体/item/respondent
  • 发展
    • 多维、多类、多层
• GIRT
  • 解决问题：Individual fallacy
  • DGIRT, Claassen model
  • DCPO, MODGIRT

参考文献

Berwick, Elissa, and Devin Caughey. 2025. “MODGIRT: Multidimensional Dynamic Scaling of Aggregate Survey Data.” Political Analysis, 1–16.

Caughey, Devin, and Christopher Warshaw. 2015. “Dynamic Estimation of Latent Opinion Using a Hierarchical Group-Level IRT Model.” Political Analysis 23 (2): 197–211. https://doi.org/10.1093/pan/mpu021.

Chalmers, R. Philip. 2015. “Extended Mixed-Effects Item Response Models with the MH-RM Algorithm.” Journal of Educational Measurement 52 (2): 200–222. https://doi.org/10.1111/jedm.12072.

Claassen, Christopher. 2019. “Estimating Smooth Country–Year Panels of Public Opinion.” Political Analysis 27 (1): 1–20.

Hu, Yue, Yuehong Cassandra Tai, and Frederick Solt. 2024. “Revisiting the Evidence on Thermostatic Response to Democratic Change: Degrees of Democratic Support or Researcher Degrees of Freedom?” Political Science Research and Methods, May, 1–7. https://doi.org/10.1017/psrm.2024.16.

Maydeu-Olivares, Alberto. 2013. “Goodness-of-Fit Assessment of Item Response Theory Models.” Measurement: Interdisciplinary Research & Perspective 11 (3): 71–101. https://doi.org/10.1080/15366367.2013.831680.

Solt, Frederick. 2020. “Modeling Dynamic Comparative Public Opinion.” SocArXiv.

Stegmueller, Daniel. 2011. “Apples and Oranges? The Problem of Equivalence in Comparative Research.” Political Analysis 19 (4): 471–87. https://doi.org/10.1093/pan/mpr028.

Tai, Yuehong ‘Cassandra’, Yue Hu, and Frederick Solt. 2024. “Democracy, Public Support, and Measurement Uncertainty.” American Political Science Review 118 (1): 512–18. https://doi.org/10.1017/S0003055422000429.

Woo, Byung-Deuk, Lindsey A. Goldberg, and Frederick Solt. 2023. “Public Gender Egalitarianism: A Dataset of Dynamic Comparative Public Opinion Toward Egalitarian Gender Roles in the Public Sphere.” British Journal of Political Science 53 (2): 766–75. https://doi.org/10.1017/S0007123422000436.

Woo, Byung-Deuk, Hyein Ko, Yuehong Cassandra Tai, Yue Hu, and Frederick Solt. 2024. “Public Support for Gay Rights Across Countries and over Time.” Social Science Quarterly 106 (1): e13478. https://doi.org/10.1111/ssqu.13478.

3 附录：群组加权平均计算

3.1 Individual → Aggregated

\[Y_{kq} = \frac{\sum Y_{ikq}}{n}.\]

不妥之处？

1. 如果群组过小，其平均值的代表意义不大
2. 不同的指标对于潜在变量贡献不一样

↓

经过群组信息（地理、人口）加权的平均值

Multilevel Regression and Post-stratification (MrP)

Gelman, Andrew, and Thomas C. Little. 1997. “Poststratification Into Many Categories Using Hierarchical Logistic Regression.” Survey Methods 23: 127–135.

3.2 Get the mean right

\[\theta_h = \frac{\sum_{j \in h} N_j \mu_j }{\sum_{j \in h} n_j},\]

N: 总体（来自普查）
n: 样本（来自sample）

1. 将总体（population）按群组（strata，如国家、地区）切分；
2. 估测对象为核心变量在每个群组中的平均值/比例， θ_h (h ∈ {1, H});
3. 已知各群组以人口变量j（如老年男性、青年女性等）划分，确定群组人口（N_j）或占总人口比；
4. 通过multilevel model进行估算各组总体平均值μ_j。

3.3 一个经济学🌰

数据：某年某市五区域2396家产业公司的财政信息
目标：估测每个区域的产业平均收入（记为θ_1~5）

公司规模和区域分布

	A	B	C	D	E
Big	30	13	1	16	23
Medium	180	121	111	187	138
Small	97	593	862	20	4

总体平均值（真值）

Zone	income
A	652.28
B	320.75
C	331.02
D	684.98
E	767.39

https://www.r-bloggers.com/gelmans-mrp-in-r-what-is-this-all-about/

3.4 样本

我们随机选取数据中1000个产业公司作为样本：

3.5 计算

Step I: Mr

\[\begin{align} 收入 = \beta_{0z}& + \beta_{1Z=z}公司规模_{iz} + \epsilon_{iz}, \\ \beta_{0z}& = \gamma_{00} + \gamma_{01}区域_z + u_{0z}. \end{align}\]

Output: Post-strata means (\(\mu_z\))

	A	B	C	D	E
Big	1274.74	1148.58	1189.59	1238.51	1251.95
Medium	706.19	580.03	621.03	669.96	683.40
Small	372.95	246.79	287.79	336.72	350.16

Step II: P \(\frac{N_z \times \mu_z}{n_z}\)

       A        B        C        D        E 
656.4551 318.3753 326.6951 680.8675 754.5761 

3.6 矫正效果

MrP 没有解决的问题

• 答题难度的地区差异
• 题目的scale
• Measurement error

4 附录：贝叶斯结果诊断

4.1 贝叶斯分析参数检验

• 最常见的Bayesian inference方法：Markov Chain Monte Carlo (MCMC)
• 数据拟合“底线”：Convergence
• 当Chain的posterior停留在一个相对稳定的区域内(ergodic chain)
  • \(\lim_{n\to \infty}p^n(\theta_i, \theta_j) = \pi(\theta_j), \forall \theta_i, \theta_j.\)
• 特征：
  • Reccurent
    • Homogeneous/Closed: At step m if the trasition probabilities at this step do not depend on m; for State A, B, p(A, B) = 0
    • Irreducible: If every reached point/point collection can be reached from every other reached point/point collection; p(θ_i, θ_j)≠ 0, ∀ θ_i, θ_j
  • Stationary: no autocorrelation
  • Aperodic: even with a long time there’s no identical cycle of chain values repeating

• Homogeneity: at step m the transition probability at this step do not depend on
• Ergodicity：遍历性

4.2 Converged时什么样：一个🌰

下雪啦天晴啦
下雪别忘穿棉袄
下雪啦天晴啦
天晴别忘戴草帽
带草帽~~~
—《心中的太阳》

今晴，明80%也晴；
今雪，明60%也雪。

		明天
		θ1	θ2
今天	θ1	0.8	0.2
	θ2	0.6	0.4

刘欢：《心中的太阳》, 《雪城》主题曲，1988年，倪萍主演

该曲采用再现三段体结构形式写成，外加一个高亢、悠扬的尾声。A段采用悠扬自由的散板节奏，具有思忖沉寂的意境。“大实话”式的歌词，饱含着对自然、社会、人生的不解、疑惑与诘问。B段开始转为充满活力的快板速度，歌词开始由诘问转变为对人间常态的肯定。A’段的旋律不变，只是节律变得富于急迫感、动力感，在这种动力的驱动下，继续开始诘问。高亢、悠扬的尾声，以反复咏唱的“啊太阳”结束，寓意作者对人间真理的无限向往与坚定追求

example of converged

起始点: [0.5 0.5]

\(S_1 = [0.5\; 0.5]\begin{bmatrix}0.8 & 0.2\\ 0.6 & 0.4 \end{bmatrix}=[0.7\; 0.3];\) \(S_2 = [0.7\; 0.3]\begin{bmatrix}0.8 & 0.2\\ 0.6 & 0.4 \end{bmatrix}=[0.74\; 0.26];\) \(S_3 = [0.74\; 0.26]\begin{bmatrix}0.8 & 0.2\\ 0.6 & 0.4 \end{bmatrix}=[0.748\; 0.252];\) \(S_4 = [0.748\; 0.252]\begin{bmatrix}0.8 & 0.2\\ 0.6 & 0.4 \end{bmatrix}=[0.749\; 0.250].\)

再往下算，会发现概率变化越来越小，趋于稳定（congverged）

4.3 检验Convergence

目前没有统计办法能够证明一个Markov Chain已经converged

1. 在给定时间内，无法保证Markov chain能够达到目标分布;
2. 无法预先确定一条Chain能够遍历目标分布的所有区域;
3. 只能诊断一条Chain是否未converged (诊断工具：Geweke’s G, Gelman-Rubin)

增加Convergence可能的方法

1. 足够的Burn-in rounds (诊断工具：Raftery-Lewis)
2. 尽可能排除autocorrelation（诊断工具：Heidelberger-Welch）

4.4 Geweke’s G

比较parameters在Chain早期和晚期两个不重叠的窗口内的均值;用语检验recurrence特征

\[G = \frac{\bar{\theta_1}- \bar{\theta_2}}{\sqrt{\frac{s_1}{n_1} + \frac{s_2}{n_2}}}.\]

A fancy difference of means

converge 则显示不显著差异，增加burn-in

4.5 Gelman & Rubin 1992

\[\hat{R}= \sqrt{\frac{\hat{var(\theta)}}{W}}\]

1. 跑多条chains（5~10），每条长2n
2. 对每一个感兴趣的parameter计算
   1. Within chain variance(W)
   2. Between chain variance(B)
3. 计算总体variance： var(θ) = (1 - 1/n)W + (1/n)B
4. 计算Scale reduction (亦称shrink factor)

Tip

• R趋近于1表示chains operating on same distribution
  • < 1.1或1.2是可以接受的

4.6 Burn-In

给与足够的burning in以到达目标分布；“炸毛的毛毛虫”（Fuzzy Caterpillar）

4.7 检验Burn-in是否足够：Raftery & Lewis (1991, 1996)

• 分别评价每一个Chain的每一个变量
  1. 根据Chain间的相关性，并据此提供一个迭代数（iteration number）
  2. 检验autocorrelation inflation
• 输出
  • Burn-in：一位数或两位数为佳
  • Total：建议的burn-in数，未考虑cross-chain，因此真正burnin要乘上chain数
  • Dependence Factor

4.8 Autocorrelation

特征：

1. Chain间高相关性
2. 单一parameter高相关性

1. Chain间高相关性：Slow convergence
2. 单一parameter的高相关：Individual nonconvergence

Autocorrelated

Not autocorrelated

4.9 Heidelberger and Welch

用于检验Stationarity

1. 确定一个迭代数N, 以及准确性（ε）和显著性数准(α)
2. 运行整个chain
3. 施用Cramér-von Mises Test，Null: Chain是stationary
4. 如检验未通过则依次略去10%、20%,乃至50%的迭代，再次检测；
5. 如结果表示部分数据不是stationary，则对该部分数据进行halfwidth检验
6. 如果halfwidth ratio < ε, 则通过检验

4.10 Thinning

Thinning 并不会提高Chain的运算速度、帮助convergence或增强估测质量

• 每个Chain记录多少samples
  • Chain将仅记录第k个值，越高丢失的信息越多
  • k通常取值：4，5，10
• 仅用于降低autocorrelation

何时调Thin

1. 迭代中autocorrelation太高
2. Chain的convergence太低
3. 并行运算
4. 模型维度过高