In this homework assignment, you will work with the TALIS 2018 (PISA linked) to answer the following questions. Please show your work and provide the R code, model output, and relevant interpretations for your analyses.

Level-1 (teacher) variables
- math (outcome variable): math achievement score
- gender: teacher gender
- motivation: teacher motivation, a composite score based on teacher’s responses on multiple items of Question 7 in the teacher questionnaire. The original scale of the survey item includes Not important at all, Of low importance, Of moderate importance, and Of high importance.
- satisfaction: teacher job satisfaction, a composite score based on their responses on multiple items of Question 53 in the teacher questionnaire. The original scale of the survey item includes Strongly disagree, Disagree, Agree, and Strongly agree.
Level-2 (school) variables
- public: whether the school is publicly- or privately-managed from Question 12 of the principal questionnaire.
- climate: principal’s conception of school climate, a composite score based on principals’ responses on multiple items of Question 26 in the principal questionnaire. The original scale of the survey item includes Strongly disagree, Disagree, Agree, and Strongly agree.
Level-3 (country) variables

1 Two-level random coefficient model (9 pts)

1.1 The model (2pts)

We first ignore the level-3 units (country level) and just focus on teachers nested within schools. Fit a random intercept and random slope model with satisfaction as the level-1 dependent variable and motivation as a level-1 predictor (without centering). Show the summary output of the model. Provide the population model and the estimation model in separate levels or in a composite model form.

# R code here for intercept and slopes coefficents
head(TALIS, 8) %>%
  kbl(booktabs = T, digits = 1, linesep = " ") %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                latex_options = "hold_position")

countryID	schID	teacherID	gender	motivation	satisfaction	public	climate
Colombia	17000125	700101	Female	2.3	2.4	Public	2.4
Colombia	17000125	700103	Male	1.9	2.1	Public	2.4
Colombia	17000125	700104	Male	1.9	2.5	Public	2.4
Colombia	17000125	700105	Male	2.4	2.8	Public	2.4
Colombia	17000125	700108	Male	2.1	3.0	Public	2.4
Colombia	17000125	700109	Female	1.6	2.6	Public	2.4
Colombia	17000125	700110	Female	2.3	3.0	Public	2.4
Colombia	17000125	700111	Male	1.9	2.2	Public	2.4

summary(TALIS)

##           countryID         schID         teacherID        gender         
##  Turkey        :2872   79200092:   29   700201 :    9   Length:12584      
##  Czech Republic:2258   79200021:   27   700202 :    9   Class :character  
##  Colombia      :1799   79200033:   27   700308 :    9   Mode  :character  
##  Australia     :1615   79200078:   27   700904 :    9                     
##  Viet Nam      :1152   79200008:   26   701901 :    9                     
##  Georgia       : 871   79200034:   26   704702 :    9                     
##  (Other)       :2017   (Other) :12422   (Other):12530                     
##    motivation     satisfaction      public             climate    
##  Min.   :0.000   Min.   :0.000   Length:12584       Min.   :0.00  
##  1st Qu.:1.857   1st Qu.:1.750   Class :character   1st Qu.:2.00  
##  Median :2.286   Median :2.125   Mode  :character   Median :2.20  
##  Mean   :2.185   Mean   :2.134                      Mean   :2.22  
##  3rd Qu.:2.571   3rd Qu.:2.500                      3rd Qu.:2.50  
##  Max.   :3.000   Max.   :3.000                      Max.   :3.00  
##

psych::describe(TALIS[, c("gender", "motivation", "satisfaction")]) %>%
  dplyr::select(vars, n, mean, sd, min, max, se) %>%
  kbl(booktabs = T, digits = 3)%>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                latex_options = "hold_position")

	vars	n	mean	sd	min	max	se
gender*	1	12584	1.385	0.487	1	2	0.004
motivation	2	12584	2.185	0.553	0	3	0.005
satisfaction	3	12584	2.134	0.519	0	3	0.005

mod_1 <- lmer(satisfaction ~ motivation + ( 1 | schID), data = TALIS)
summary(mod_1)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: satisfaction ~ motivation + (1 | schID)
##    Data: TALIS
## 
## REML criterion at convergence: 18184.5
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.3949 -0.6237 -0.0091  0.7177  2.5534 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  schID    (Intercept) 0.03867  0.1966  
##  Residual             0.22762  0.4771  
## Number of obs: 12584, groups:  schID, 978
## 
## Fixed effects:
##              Estimate Std. Error        df t value Pr(>|t|)    
## (Intercept) 1.905e+00  2.026e-02 8.939e+03   94.02   <2e-16 ***
## motivation  1.106e-01  8.566e-03 1.233e+04   12.91   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##            (Intr)
## motivation -0.923

For the level 1 predictor of motivation, the intercept for satisfaction is 1.905-e and for every increase of unit of motivation there is a predicted increase of 1.106-e increase in satisfaction.

The population model is Yij = Gamma00 + Gamma10Motivationij + u0j + u1jMotivationij +Rijk

The estimation model is Satisfactionij Hat = 1.905-e(00) + 1.106-e(10) + + .144tau0^2 + .014 Tau1^2 + .228(sigma^2)

\[ \hat{Y}_{ij} = \gamma_{00} + \gamma_{10} \cdot \text{Motivation}_{ij} \]

Response here:

1.2 Interpretation (2 pts)

Please report and interpret the intercept and slope coefficients.

#code ofr random slopes and correlation
mod_rsm_combo <- lmer(satisfaction ~ motivation + (motivation | schID), data = TALIS)
summary(mod_rsm_combo)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: satisfaction ~ motivation + (motivation | schID)
##    Data: TALIS
## 
## REML criterion at convergence: 18141.3
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.4417 -0.6229 -0.0128  0.7199  2.7429 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev. Corr 
##  schID    (Intercept) 0.14409  0.3796        
##           motivation  0.01385  0.1177   -0.88
##  Residual             0.22418  0.4735        
## Number of obs: 12584, groups:  schID, 978
## 
## Fixed effects:
##              Estimate Std. Error        df t value Pr(>|t|)    
## (Intercept)   1.90064    0.02351 764.34909   80.84   <2e-16 ***
## motivation    0.11107    0.00961 777.45447   11.56   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##            (Intr)
## motivation -0.945

1.3 Random slopes (2 pts)

Do the slopes vary from school to school? Please provide statistical evidence to support your conclusion.

Slopes vary from school to school based on the predicted variance of .01385, probably meaning that the effect of motivation on satisfaction for teachers varies across schools. This is because the slope is not zero.

Response here:

1.4 Correlation between random intercept and random slope (2 pts)

What is the correlation between the intercepts and slopes? Do schools with large intercepts (e.g., high job satisfaction) also have large slopes (e.g., strong relationship between motivation and job satisfaction)? Please provide statistical evidence to support your conclusion. HINT: How to test for the correlation between random intercept and random slope?

The correlation between the random intercept and the random slope is -0.88, meaning that there is a high negative correlation between school teacher satisfaction and school teacher motivation. This means that schools with higher teacher satisfaction rates have predicted weaker slopes for motivation.

# R code here for empirical bayes
coef(mod_rsm_combo)$schID %>%
  kbl(booktabs = T) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                latex_options = "hold_position")

	(Intercept)	motivation
3200009	2.364110	0.0161977
3200011	2.093119	0.0959608
3200013	2.161271	0.0747837
3200016	2.171795	0.0529590
3200020	2.437207	0.0197293
3200029	2.136726	0.1046395
3200032	2.232924	0.0355927
3200034	2.081806	0.0779581
3200035	2.002598	0.0801574
3200060	1.828996	0.1253593
3200075	2.156916	0.0787794
3200086	2.085641	0.0814092
3200087	1.836029	0.1437401
3200095	2.079866	0.0855243
3200105	2.150896	0.0596999
3200111	2.180511	0.0614838
3200114	2.173239	0.0372956
3200118	2.075604	0.0865300
3200124	2.340485	-0.0157801
3200129	2.265138	0.0639673
3200131	2.078609	0.0495133
3200133	2.087210	0.0632373
3200134	2.127784	0.0896985
3200138	2.443820	-0.0004488
3200149	2.391776	0.0077585
3200152	2.172032	0.0356197
3200168	2.107062	0.0947727
3200170	2.155374	0.0893860
3200177	2.176806	0.0620930
3200184	2.619943	-0.0389525
3200207	2.401928	-0.0211110
3200214	2.249871	0.0379512
3200224	2.292438	0.0105698
3200236	2.036040	0.0884819
3200240	2.275566	0.0295031
3200241	2.202216	0.0693705
3200257	2.691642	-0.0652107
3200263	2.161603	0.0708719
3200268	2.221630	0.0089581
3200281	2.419061	-0.0091513
3200296	2.215920	0.0460894
3200297	2.256394	0.0125586
3200300	2.412996	0.0011245
3200301	2.262222	0.0703357
3200319	2.065250	0.0881763
3200338	2.102735	0.0364149
3200340	2.163556	-0.0217485
3200342	2.287591	0.0433953
3200355	2.184924	0.0470605
3200361	2.079302	0.0761503
3200363	2.415307	-0.0394060
3200374	2.012002	0.0601211
3200376	2.531168	-0.0311777
3200380	2.473515	0.0075492
3200382	2.301681	0.0243654
3200403	2.424476	-0.0130071
3200405	2.218313	0.0455693
3200407	2.516457	-0.0487721
3200415	2.204530	0.0484410
3200423	2.053450	0.0959698
3200428	1.984534	0.1012595
3200440	2.302497	0.0409412
3200452	2.261438	0.0236987
3200454	2.465144	-0.0250138
3200455	2.090345	0.0489787
3600001	1.898143	0.1183184
3600005	2.339067	-0.0328725
3600006	1.870946	0.1056740
3600013	2.041552	0.0395875
3600015	2.270627	0.0355675
3600019	2.184407	0.0411498
3600022	1.849201	0.1089593
3600026	2.301837	0.0034114
3600029	1.991925	0.0971101
3600044	1.781452	0.1953435
3600047	1.556943	0.1701777
3600051	2.329820	-0.0087239
3600057	1.729277	0.1262951
3600060	1.777015	0.1226759
3600062	1.763462	0.1215374
3600063	1.994378	0.1016476
3600064	1.776897	0.1151922
3600065	1.977226	0.1120375
3600073	1.549337	0.1457821
3600076	1.788740	0.1259559
3600082	2.241829	0.0742958
3600092	1.826054	0.1331127
3600093	1.949561	0.0869367
3600108	1.958173	0.0994734
3600109	1.731029	0.1330767
3600111	2.005473	0.0894547
3600116	2.220959	0.0455968
3600137	1.896525	0.1217982
3600142	1.889302	0.0942961
3600144	1.754499	0.1365522
3600149	1.953504	0.0947818
3600156	1.666982	0.1400214
3600166	1.739888	0.1351625
3600170	1.582737	0.1429152
3600171	1.880823	0.0924881
3600175	2.111631	0.1188364
3600191	1.579464	0.1640183
3600195	1.903065	0.0817323
3600209	1.675998	0.1587511
3600210	1.757452	0.0956487
3600211	1.688908	0.1405154
3600218	2.288172	0.0096948
3600223	2.306232	0.0442670
3600228	1.884888	0.0741787
3600229	1.927380	0.0777410
3600242	1.714129	0.1399887
3600245	1.987564	0.0922393
3600247	2.099468	0.0603652
3600251	1.537497	0.1831413
3600256	1.809142	0.1274508
3600262	1.879133	0.1288737
3600265	1.728980	0.1402660
3600266	2.151193	0.0914314
3600270	2.274452	0.0316135
3600278	1.633595	0.1528055
3600280	1.698971	0.1608769
3600283	1.728884	0.1034588
3600304	1.745956	0.1448324
3600308	2.027179	0.0805665
3600311	2.286716	0.0386574
3600312	2.026011	0.0776941
3600321	1.634438	0.1706639
3600323	1.721574	0.1654274
3600330	2.119772	0.0426160
3600344	2.314235	0.0048384
3600352	1.762247	0.1463329
3600357	2.039741	0.0662870
3600364	1.921203	0.0918450
3600367	1.909518	0.1053046
3600370	1.553687	0.1889732
3600375	1.912544	0.1143124
3600383	2.034573	0.0860791
3600391	2.090300	0.0686546
3600393	1.778961	0.1292142
3600400	1.847212	0.1396228
3600401	2.008592	0.0765216
3600408	2.155042	0.0444879
3600412	2.215785	0.0607554
3600426	1.916264	0.0827922
3600434	1.948517	0.0918372
3600437	1.720814	0.1842600
3600438	1.597354	0.1767749
3600450	1.882039	0.1358371
3600460	2.145348	0.1228853
3600473	2.124009	0.1078749
3600476	2.368166	0.0467296
3600488	1.710235	0.1698743
3600493	1.886342	0.0940046
3600499	1.797638	0.0911863
3600515	1.946960	0.0776819
3600518	2.114457	0.0227256
3600522	1.871260	0.1218040
3600528	1.913175	0.1297550
3600529	1.880059	0.1104848
3600532	1.829648	0.0896124
3600533	1.754973	0.1730004
3600540	2.166752	0.0729562
3600545	1.715524	0.1208083
3600549	1.671372	0.1521412
3600556	2.143350	-0.0073418
3600561	1.817082	0.1229783
3600562	1.967028	0.1010729
3600568	1.668978	0.1623430
3600569	1.860789	0.1261990
3600589	1.926233	0.1075793
3600593	1.821356	0.1464166
3600602	1.511919	0.1850528
3600604	1.847851	0.1064949
3600609	1.852436	0.1191940
3600612	2.060859	0.0741496
3600614	1.886527	0.1408520
3600620	1.812257	0.1292702
3600640	1.994608	0.0863751
3600642	1.607690	0.1751861
3600649	1.818108	0.1253437
3600654	2.208197	0.0691086
3600670	1.608702	0.1508938
3600671	2.270729	0.0283898
3600679	2.206085	0.0954553
3600684	1.712317	0.1409467
3600700	1.628859	0.1313072
3600731	2.191902	0.0636918
3600732	2.024337	0.1109579
3600737	1.918165	0.0955177
3600743	1.616426	0.1556244
3600744	1.799111	0.1266924
3600748	2.091324	0.0570386
17000007	1.864080	0.1018885
17000009	2.363697	-0.0053981
17000011	2.255624	0.0003512
17000012	1.849399	0.1394313
17000013	2.125230	0.0833886
17000016	1.745711	0.1464350
17000017	1.913273	0.1178639
17000018	2.099455	0.1376820
17000023	2.257347	0.0820309
17000025	1.825330	0.1169302
17000028	2.147507	0.0638859
17000029	2.286426	0.0525203
17000032	1.578476	0.2442451
17000034	2.042143	0.1019540
17000035	1.929648	0.0797929
17000037	1.997016	0.1001581
17000040	2.283354	0.0630685
17000041	1.938705	0.1468650
17000043	1.414997	0.2263755
17000045	2.233662	0.0482074
17000046	2.303401	0.0483063
17000048	2.102679	0.0891633
17000050	2.234188	0.0287862
17000051	2.549095	-0.0222058
17000053	1.918346	0.1061062
17000054	2.227256	0.0554550
17000055	2.286300	-0.0039430
17000057	2.335448	0.0134005
17000058	2.331435	0.0199172
17000060	1.975950	0.1071760
17000061	2.215914	0.0477805
17000064	1.947553	0.1374480
17000065	2.185867	0.0618137
17000066	1.798266	0.1559446
17000067	1.959612	0.0979000
17000070	2.233034	0.0447234
17000072	2.047230	0.1413534
17000073	1.796216	0.1843839
17000074	1.828743	0.1425918
17000075	1.996773	0.0509096
17000076	2.302334	0.0619781
17000077	1.897497	0.1383221
17000079	2.042317	0.0523407
17000080	2.436351	-0.0014829
17000081	2.095249	0.0580186
17000085	1.735192	0.1398497
17000086	2.109920	0.0887552
17000087	1.731031	0.1372684
17000088	2.088154	0.0875697
17000089	2.446333	-0.0011922
17000090	2.016400	0.1028609
17000091	2.134293	0.0876678
17000095	2.189661	0.0368019
17000096	1.845715	0.1509786
17000097	2.269614	0.0983172
17000098	1.984849	0.1045447
17000099	2.404655	0.0770152
17000102	2.221678	0.0592962
17000103	2.128768	0.0809195
17000104	2.144691	0.0702415
17000112	2.321874	0.0304434
17000113	2.154546	0.0867861
17000115	1.978762	0.0834277
17000116	2.234133	0.0616098
17000118	2.401662	0.0403048
17000120	2.299835	0.0502851
17000121	2.197454	0.0921965
17000124	1.939662	0.1250016
17000125	2.204800	0.0647511
17000126	1.919055	0.1380837
17000127	2.215577	0.0533428
17000128	2.345242	0.0550615
17000129	2.312346	0.0384464
17000131	1.710114	0.1359669
17000133	2.009480	0.0522949
17000134	1.984941	0.1082877
17000135	2.217113	0.0303151
17000136	2.339623	0.0128509
17000137	1.966161	0.1114241
17000139	2.049712	0.0894426
17000141	1.904756	0.1262244
17000142	2.302123	0.0358343
17000144	2.404104	0.0295858
17000147	2.227831	0.0858771
17000148	1.984857	0.0920604
17000151	1.962884	0.1180379
17000152	2.053104	0.0917731
17000153	2.093836	0.0748145
17000158	2.193169	0.1060068
17000159	2.228501	0.0477960
17000161	2.134612	0.0786891
17000162	2.039388	0.1010011
17000164	1.991920	0.1151333
17000165	2.057944	0.1007083
17000167	1.984091	0.1541446
17000168	2.212265	0.0734138
17000169	2.098089	0.0922654
17000171	1.984318	0.1060303
17000172	2.266638	0.0658022
17000177	2.353636	-0.0216412
17000178	2.011066	0.0967732
17000181	2.003250	0.1359886
17000184	2.165928	0.0949682
17000186	1.992939	0.0849424
17000188	2.168415	0.0778002
17000190	2.341983	0.0736509
17000191	2.144667	0.0984026
17000194	2.459422	0.0000234
17000195	2.043793	0.1430610
17000196	2.279250	0.0203550
17000198	2.177945	0.0866806
17000200	2.264973	0.0345243
17000201	1.988503	0.1198606
17000203	1.973712	0.1632327
17000204	2.093039	0.0491186
17000206	2.143513	0.0516639
17000207	2.348740	0.0459009
17000210	1.835883	0.1540999
17000213	2.309662	0.0126415
17000215	2.122716	0.0851435
17000216	2.076626	0.0559671
17000217	2.169802	0.0761464
17000219	2.156620	0.0767155
17000220	2.436278	0.0374594
17000221	1.947352	0.0948077
17000225	2.137140	0.0625480
17000226	2.428912	0.0280121
17000227	1.978529	0.1402423
17000228	2.220792	0.0983612
17000230	2.174893	0.0664266
17000231	2.227694	0.0128124
17000232	2.110810	0.0945513
17000234	2.031231	0.0987963
17000237	2.330886	0.0576838
17000238	2.047924	0.0931852
17000240	2.093629	0.0907763
17000242	1.966270	0.1144191
17000244	1.751791	0.1409022
17000245	1.928492	0.1323078
17000248	2.333102	0.0139264
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70400137	1.510412	0.1952049
70400138	1.719119	0.1365839
70400139	1.884586	0.1300015
70400140	1.918425	0.1085579
70400141	1.570379	0.1491165
70400142	1.987856	0.0960276
70400143	1.676743	0.1369819
70400144	1.806793	0.1418415
70400145	1.673726	0.1424704
70400146	2.035923	0.1344003
70400147	1.934943	0.1214711
70400148	1.721305	0.1591199
70400149	2.032561	0.1126565
70400151	1.549895	0.1615543
70400152	1.615300	0.1567078
79200001	1.961270	0.0124924
79200002	1.990183	0.1155663
79200003	1.713697	0.1856032
79200004	1.796058	0.1162965
79200005	1.423713	0.2572355
79200006	1.539154	0.2498412
79200007	1.380320	0.1819930
79200008	1.513471	0.1906366
79200009	1.964834	0.1100190
79200010	1.874317	0.0932474
79200012	1.436060	0.2075876
79200013	2.153581	0.0050716
79200014	1.748335	0.1143294
79200015	1.361278	0.2110462
79200016	1.628404	0.1298818
79200017	1.911464	0.1414468
79200018	1.433945	0.1933245
79200021	1.533868	0.1671411
79200022	1.539860	0.1478451
79200023	2.134691	0.1052276
79200024	1.510717	0.1858292
79200026	1.724372	0.1251691
79200027	1.991155	0.0593796
79200028	1.937602	0.0680053
79200030	1.868175	0.1218496
79200033	1.750092	0.1548247
79200034	2.194246	0.0288494
79200035	1.468829	0.1413083
79200036	1.574531	0.1741631
79200037	1.962479	0.0002095
79200038	1.505105	0.2179381
79200040	1.439251	0.2013055
79200041	1.472606	0.1058898
79200043	1.271781	0.2457959
79200044	1.421747	0.2230666
79200046	1.533181	0.1261561
79200048	1.271107	0.2690510
79200049	1.982716	0.1196213
79200051	1.232381	0.2426476
79200052	1.598394	0.1830097
79200053	1.826355	0.1357563
79200056	1.244401	0.1500572
79200057	1.193475	0.2176815
79200058	1.761680	0.1147797
79200059	1.664888	0.1237269
79200060	2.149056	0.1210909
79200061	1.991916	0.0439991
79200064	1.513177	0.2174788
79200065	1.669208	0.1845714
79200067	1.582287	0.1619708
79200069	1.976666	0.0634032
79200071	1.781789	0.1380098
79200072	1.647553	0.1644116
79200077	2.055569	0.1109993
79200078	1.965897	0.0872996
79200080	1.747330	0.1097701
79200081	1.429463	0.2546015
79200082	1.761857	0.1324316
79200083	1.615394	0.1480896
79200084	1.633912	0.1560592
79200085	1.609993	0.1355659
79200086	1.791584	0.0889259
79200087	1.843893	0.1208998
79200089	1.612633	0.1479050
79200090	1.752323	0.0704878
79200091	1.328391	0.2038342
79200092	1.441451	0.2124034
79200094	1.625696	0.1085496
79200095	1.226408	0.2544926
79200097	1.799502	0.1644347
79200098	1.908148	0.1120751
79200099	1.529089	0.1635277
79200102	1.649201	0.1597787
79200103	2.152052	0.0575613
79200104	1.563614	0.1899085
79200105	1.319456	0.2926959
79200106	1.447412	0.2301923
79200107	1.641683	0.1620439
79200108	1.667164	0.1276343
79200109	1.638706	0.1846715
79200111	1.349577	0.2409677
79200112	2.016323	0.0735799
79200113	1.662209	0.1229101
79200114	1.427240	0.2033803
79200116	1.610807	0.2256889
79200117	1.474252	0.2586217
79200118	1.429569	0.1898561
79200119	1.565931	0.1714164
79200121	1.995902	0.1189376
79200122	1.744383	0.1729743
79200123	1.955266	0.0861780
79200124	1.628290	0.1366600
79200125	1.512643	0.2152392
79200126	1.288312	0.2713939
79200127	2.398902	-0.0243934
79200129	1.685511	0.1301672
79200130	1.955985	0.1037338
79200132	1.600153	0.1837437
79200133	1.817659	0.1517466
79200135	1.778000	0.1291792
79200136	1.811934	0.1189688
79200139	1.656699	0.1411255
79200140	1.691350	0.1346657
79200141	1.867189	0.1778892
79200142	1.309501	0.1909318
79200143	1.455956	0.1559460
79200144	1.243521	0.2486893
79200145	2.061470	0.0337069
79200146	1.655995	0.1387458
79200147	1.945562	0.0532753
79200148	1.868294	0.1172422
79200149	1.855775	0.1630573
79200150	1.862693	0.1151168
79200151	1.651670	0.1441694
79200153	1.778194	0.1450429
79200154	1.689980	0.1517552
79200155	1.536053	0.1574281
79200157	1.069851	0.2553948
79200158	1.832144	0.1627441
79200159	1.178757	0.2847159
79200161	1.549083	0.1696378
79200162	1.544871	0.1451498
79200163	1.417697	0.2178172
79200164	1.611437	0.1498001
79200165	1.543933	0.1371138
79200169	2.087134	0.0901811
79200170	1.456454	0.1755094
79200171	1.981341	0.0583243
79200172	1.720183	0.1518963
79200173	2.128694	0.0452401
79200174	1.240160	0.2967374
79200175	1.955370	0.1321964
79200177	1.833360	0.1155398
79200179	1.529016	0.2384238
79200180	1.812145	0.1792011
79200181	1.535567	0.1751603
79200182	1.370352	0.2003055
79200183	1.440776	0.1393419
79200184	1.391412	0.2210140
79200186	1.549858	0.1570000
79200188	1.668386	0.1710634

school_coef <- coef(mod_rsm_combo)$schID

summary_RSM_combo <- data.frame(
  variable = colnames(school_coef),
  mean = apply(school_coef, 2, mean),
  median = apply(school_coef, 2, median),
  sd = apply(school_coef, 2, sd),
  min = apply(school_coef, 2, min),
  max = apply(school_coef, 2, max)
)
summary(summary_RSM_combo)

##    variable              mean            median             sd         
##  Length:2           Min.   :0.1111   Min.   :0.1118   Min.   :0.05532  
##  Class :character   1st Qu.:0.5585   1st Qu.:0.5595   1st Qu.:0.10553  
##  Mode  :character   Median :1.0059   Median :1.0072   Median :0.15574  
##                     Mean   :1.0059   Mean   :1.0072   Mean   :0.15574  
##                     3rd Qu.:1.4532   3rd Qu.:1.4549   3rd Qu.:0.20595  
##                     Max.   :1.9006   Max.   :1.9026   Max.   :0.25616  
##       min                max        
##  Min.   :-0.06521   Min.   :0.2967  
##  1st Qu.: 0.21855   1st Qu.:0.8955  
##  Median : 0.50232   Median :1.4942  
##  Mean   : 0.50232   Mean   :1.4942  
##  3rd Qu.: 0.78609   3rd Qu.:2.0929  
##  Max.   : 1.06985   Max.   :2.6916

Response here:

1.5 Empirical Bayes (1 pt)

Obtain the Empirical Bayes’ estimates of the random school effects, including the EB for the random intercepts and the slopes. Please report the summary statistics of these estimates, such as mean, median, sd, min, and max.

# R code here.

Response here:

1.6 Extra credit (1 pt)

In this random coefficient model, calculate and report the correlation between two randomly selected teachers from a randomly selected school, one teacher with motivation of 0 and another teacher with motivation of 1. HINT: Follow the definition, find the covariance in the satisfaction between an individual with motivation of 0 and an individual with motivation of 1, find the variance of satisfaction for an individual with motivation of 0 and an individual with motivation of 1, then calculate the correlation.

# R code here.

Response here:

2 Three-level models (3 points)

Fit a three-level random intercept model with satisfaction as outcome variable, motivation as level-1 predictor variable, climate and public as level-2 predictors. Please report and interpret the slope coefficients. Compute the expected correlation between teachers from the same school and the expected correlation between teachers from the same country.

# R code here.
mod_empty <- lmer(satisfaction ~ 1 + ( 1 | countryID / schID), data = TALIS)
mod_empty <- lmer(satisfaction ~ 1 + ( 1 | countryID) + (1 | schID), data = TALIS)
summary(mod_empty)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: satisfaction ~ 1 + (1 | countryID) + (1 | schID)
##    Data: TALIS
## 
## REML criterion at convergence: 17991
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.2876 -0.6299 -0.0151  0.7141  2.4628 
## 
## Random effects:
##  Groups    Name        Variance Std.Dev.
##  schID     (Intercept) 0.01826  0.1351  
##  countryID (Intercept) 0.02060  0.1435  
##  Residual              0.23136  0.4810  
## Number of obs: 12584, groups:  schID, 978; countryID, 9
## 
## Fixed effects:
##             Estimate Std. Error      df t value Pr(>|t|)    
## (Intercept)  2.15346    0.04834 7.98784   44.55 7.32e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#checking that climate and public are school level variables
TALIS %>%
  group_by(schID) %>%
  summarize(n = n_distinct(climate), m = n_distinct(public)) %>%
  summarize(max_n = max(n), max_m = max(m))

## # A tibble: 1 × 2
##   max_n max_m
##   <int> <int>
## 1     1     1

#confirmed

mod_lvl1 <- lmer(satisfaction ~ motivation + ( 1 | countryID / schID), data = TALIS)
summary(mod_lvl1)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: satisfaction ~ motivation + (1 | countryID/schID)
##    Data: TALIS
## 
## REML criterion at convergence: 17791.9
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.3989 -0.6312 -0.0114  0.7220  2.5314 
## 
## Random effects:
##  Groups          Name        Variance Std.Dev.
##  schID:countryID (Intercept) 0.0179   0.1338  
##  countryID       (Intercept) 0.0254   0.1594  
##  Residual                    0.2276   0.4771  
## Number of obs: 12584, groups:  schID:countryID, 978; countryID, 9
## 
## Fixed effects:
##              Estimate Std. Error        df t value Pr(>|t|)    
## (Intercept) 1.878e+00  5.687e-02 1.011e+01   33.01 1.24e-11 ***
## motivation  1.279e-01  8.853e-03 1.245e+04   14.44  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##            (Intr)
## motivation -0.336

mod_lvl1 <- lmer(satisfaction ~ motivation + (1 | countryID) + ( 1 | schID), data = TALIS)
summary(mod_lvl1)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: satisfaction ~ motivation + (1 | countryID) + (1 | schID)
##    Data: TALIS
## 
## REML criterion at convergence: 17791.9
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.3989 -0.6312 -0.0114  0.7220  2.5314 
## 
## Random effects:
##  Groups    Name        Variance Std.Dev.
##  schID     (Intercept) 0.0179   0.1338  
##  countryID (Intercept) 0.0254   0.1594  
##  Residual              0.2276   0.4771  
## Number of obs: 12584, groups:  schID, 978; countryID, 9
## 
## Fixed effects:
##              Estimate Std. Error        df t value Pr(>|t|)    
## (Intercept) 1.878e+00  5.687e-02 1.011e+01   33.01 1.24e-11 ***
## motivation  1.279e-01  8.853e-03 1.245e+04   14.44  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##            (Intr)
## motivation -0.336

mod_lvl2 <- lmer(satisfaction ~ motivation + climate + public + (1 |countryID) + (1 | schID), data = TALIS)
summary(mod_lvl2)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: satisfaction ~ motivation + climate + public + (1 | countryID) +  
##     (1 | schID)
##    Data: TALIS
## 
## REML criterion at convergence: 17772.9
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.4178 -0.6314 -0.0082  0.7199  2.5429 
## 
## Random effects:
##  Groups    Name        Variance Std.Dev.
##  schID     (Intercept) 0.01664  0.1290  
##  countryID (Intercept) 0.02502  0.1582  
##  Residual              0.22768  0.4772  
## Number of obs: 12584, groups:  schID, 978; countryID, 9
## 
## Fixed effects:
##                Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)   1.732e+00  6.874e-02  2.209e+01  25.190  < 2e-16 ***
## motivation    1.285e-01  8.852e-03  1.245e+04  14.512  < 2e-16 ***
## climate       8.131e-02  1.688e-02  8.926e+02   4.818 1.71e-06 ***
## publicPublic -4.748e-02  1.651e-02  1.005e+03  -2.876  0.00412 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) motvtn climat
## motivation  -0.267              
## climate     -0.545 -0.009       
## publicPublc -0.183 -0.035  0.030

VarCorr(mod_lvl2)

##  Groups    Name        Std.Dev.
##  schID     (Intercept) 0.12898 
##  countryID (Intercept) 0.15818 
##  Residual              0.47715

var_comps <- VarCorr(mod_lvl2)
tau_sq <- as.numeric(var_comps)[2]
omega_sq <- as.numeric(var_comps)[1]
sigma_sq <- attr(var_comps, "sc")^2

tau_sq / (tau_sq + omega_sq + sigma_sq) #between-country variance

## [1] 0.09290113

#countries have a 9.29% variance of teacher satisfaction, depending on the country.

omega_sq / (tau_sq + omega_sq + sigma_sq) #between school variance

## [1] 0.06177129

#schools have a 6.18% variance from school to school for teacher satisfaction depending on the school

sigma_sq / (tau_sq + omega_sq + sigma_sq) #residual variance (within school)

## [1] 0.8453276

#the withhin school variance is 84.53% for individual teacher satisfaction

Response here:

The intercept for the 3 level model with predictors for teacher satisfaction is 1.73166 and for every unit increase of motivation teacher satisfaction is predicted to increase by .12846 when holding climate constant and school type constant. The slope for climate for the 3 level model means that when climate increases by one unit then the teacher satisfaction increases by .08131 when holding school type constant. When schools have 0 motivation, 0 climate, and the school is private the predicted intercept for teacher satisfaction is 1.73166

Countries have a 9.29% variance of teacher satisfaction, depending on the country. Schools have a 6.18% variance from school to school for teacher satisfaction depending on the school The withhin school variance is 84.53% for individual teacher satisfaction

3 Introduction of the project (3 pts)

Describe the context and motivation for your class project.
Explain the research question(s) that your project investigates.
Provide a few key references.

Fall 2025 HLM - Assignment 2

1 Two-level random coefficient model (9 pts)

1.1 The model (2pts)

1.2 Interpretation (2 pts)

1.3 Random slopes (2 pts)

1.4 Correlation between random intercept and random slope (2 pts)

1.5 Empirical Bayes (1 pt)

1.6 Extra credit (1 pt)

2 Three-level models (3 points)

3 Introduction of the project (3 pts)