Standard error of parameter estimates

100 med1: the difference in effect between categories of medic 1 and 3 in s-th province med2: the difference in effect between categories of medic 2 and 3 in s-th province ULS: the effect of bad nutrition hotspot in s-th province : The poverty level k = 1, 2, or 3 of sub district in sth province, ith district, and jth observation as response variable outcome. : The characteristic of a sub district from sth province, ith district, and jth observation. Nested Generalized Linear Model for ordinal response is Nested Generalized Linear Mixed Models are also implemented for the same data as in Nested GLM with district as the random factor, . Nested GLMM in this application is as follows

4.4 Results and Discussion

4.4.1 Standard error of parameter estimates

Appendix 22 and Appendix 25 show the parameter estimates and their standard errors for Nested GLM and Nested GLMM, respectively. Nested Generalized Linear Model. Figure 25 shows the standard errors of model- based Nested GLM parameters with exchangeable, unstructured, and independent WCM. Standard errors of unstructured correlation models tend to be smaller than standard errors of independent and exchangeable models, while standard errors independent are the highest. The standard errors among those three models are not significantly different. The correlation structured does not give contribution to the standard errors. 101 Figure 25 Standard errors of model-based Nested GLM parameters Furthermore, Figure 26 shows the standard errors of robust Nested GLM parameters with exchangeable, unstructured, and independent WCM. It shows that there is no particular pattern among exchangeable, unstructured, and independent models for robust parameter estimates. For some parameters prov2, farm11, farm21, sch12, sch22, standard errors unstructured are extremely high. According to these two figures, parameter estimates of model-based are more stable than those of robust. The correlation structured gives contribution to the robust standard errors. Figure 26 Standard errors of robust Nested GLM parameters Figures 27, 28, and 29 show standard error comparisons between model- based and robust for exchangeable, unstructured and independent WCM, 0.00 1.00 2.00 3.00 4.00 5.00 6.00 th re s1 th re s2 p ro v 1 p ro v 2 fa rm1 1 fa rm2 1 fa rm1 2 fa rm1 3 fa rm2 3 sch 1 1 sch 2 1 sch 1 2 sch 2 2 sch 1 3 sch 23 me d 1 1 me d 2 1 me d 2 2 me d 1 3 me d 2 3 UL S 1 UL S 2 UL S 3 exchangeable unstructured independent 0.00 1.00 2.00 3.00 4.00 5.00 6.00 th re s1 th re s2 p ro v 1 p ro v 2 fa rm1 1 fa rm2 1 fa rm1 2 fa rm1 3 fa rm2 3 sch 1 1 sch 2 1 sch 1 2 sch 2 2 sch 1 3 sch 2 3 me d 1 1 me d 2 1 me d 2 2 me d 1 3 me d 2 3 UL S 1 UL S 2 UL S 3 exchangeable unstructered independent 102 respectively. Standard error of robust tends smaller than those of model based, except for unstructured WCM. Figure 27 Standard error of Nested GLM parameters of exchangeable WCM Figure 28 Standard error of Nested GLM parameters of unstructured WCM Figure 29 Standard error of Nested GLM parameters of independent WCM 0.00 1.00 2.00 3.00 4.00 5.00 6.00 th re s1 th re s2 p ro v 1 p ro v 2 fa rm1 1 fa rm2 1 fa rm12 fa rm1 3 fa rm2 3 sch 1 1 sch 2 1 sch 12 sch 2 2 sch 1 3 sch 2 3 me d 1 1 me d 2 1 me d 2 2 me d 1 3 me d 2 3 UL S 1 UL S 2 UL S 3 model based robust 0.00 1.00 2.00 3.00 4.00 5.00 6.00 th re s1 th re s2 p ro v 1 p ro v 2 fa rm1 1 fa rm2 1 fa rm1 2 fa rm1 3 fa rm2 3 sch 1 1 sch 2 1 sch 12 sch 2 2 sch 1 3 sch 2 3 me d 1 1 me d 2 1 me d 2 2 me d 1 3 me d 2 3 UL S 1 UL S 2 UL S 3 model based robust 0.00 1.00 2.00 3.00 4.00 5.00 6.00 th re s1 th re s2 p ro v 1 p ro v 2 fa rm1 1 fa rm2 1 fa rm1 2 fa rm1 3 fa rm2 3 sch 1 1 sch 21 sch 12 sch 2 2 sch 1 3 sch 2 3 m e d 1 1 m e d 2 1 m e d 2 2 m e d 1 3 m e d 2 3 UL S 1 UL S 2 UL S 3 model based robust 103 It is not easy to specify the correlation structure of the data, even though according to the nature of the data, unstructured WCM is the most appropriate due to some high correlations between sub districts in the same district Appendix 28. Until this point of analysis, it is not clear which structure is the most appropriate to the data. To continue the analysis, Table 17 is considered. This table shows the averages of standard errors for every WCM and estimation method. If exchangeable or independent WCM is used, robust estimation is better than model based estimation. Otherwise, if unstructured WCM is used, model based estimation is better than robust estimation. Table 17 Averages of standard errors of the Nested GLM exchangeable Unstructured Independent Model Based 0.89 0.82 1.05 Robust 0.53 1.24 fluctuating 0.56 Nested Generalized Linear Mixed Model. Standard errors of parameter estimates in Nested GLMM are equal for all WCMs, except there is a small difference for the threshold parameters of independent WCM. Figure 30 shows standard errors of robust tend smaller than those of model based. Table 18 shows the averages of standard errors of Nested GLMM, where robust estimation is better than model based estimation. Figure 30 Standard errors of Nested GLMM parameters 1 2 3 4 5 6 th res 1 th res 2 p ro v 1 far m1 1 far m2 1 far m1 3 far m2 3 sc h 1 1 sc h 2 1 sc h 1 2 sc h 2 2 sc h 1 3 sc h 2 3 med1 1 med2 1 med2 2 med 1 3 med2 3 ULS1 ULS2 ULS3 model based exchangeable, unstructured, independent robust exchangeable, unstructured, independent 104 Table 18 Averages of standard errors the Nested GLMM exchangeable Unstructured Independent Model Based 1.31 1.31 1.29 Robust 0.71 0.71 0.74 It is desirable to compare the fit of different working correlation structures within a GEE Nested GLM and Nested GLMM analysis using an informal comparison. An informal comparison is to compare the standard error of the robust SE R and model-based SE M . If SE R SE M is close to 1, this suggests the correlation structure is correctly modeled Bishop, Die Wang 2000. This study used the SE R SE M ratio to evaluate whether the exchangeable or unstructured WCM fit the data better than the independent correlation structure. There are no guidelines regarding the size of the ratio, but higher ratios reflect poorer model fit. This comparison is qualitative, but it is the best approach available at this time Koper and Manseau 2009. Figure 31 shows SE R SE M ratio for Nested GLM and GLMM, where some ratios of unstructured Nested GLM are close to 1, but it is more fluctuated, while exchangeable and independent are more stable. Figure 31 SE R SE M ratios for Nested GLM and GLMM Table 19 shows averages of SE R SE M ratios from all parameter estimates. According to this table, Nested GLM with independent working correlation structure gives the smallest ratio 0.56. Through the nature of the data, it would 1 2 3 4 5 6 th res 1 th res 2 p ro v 1 far m1 1 far m2 1 far m1 3 far m2 3 sc h 1 1 sc h 2 1 sc h 1 2 sc h 2 2 sc h 1 3 sc h 2 3 med1 1 med2 1 med2 2 med 1 3 med2 3 ULS1 ULS2 ULS3 GLM-exchangeable GLM-unstructured GLM-independent GLMM-exch. unstr. GLMM-independent 105 seem the data is appropriate to the unstructured correlation structure, but this value shows that the best model of Nested GLMs is the model with independent structure. Furthermore, random clustering effects include in modeling to control the correlation arise from recording multiple locations. The result is shown at Nested GLMM column, where ratios of exchangeable and unstructured are slightly smaller than independent. Exchangeable and unstructured WCM is not vehemently 2 rejected, because the ratios are smaller than independent and the difference is not significant. Table 19 Averages of SE R SE M Nested GLM Nested GLMM Exchange -able Unstruc -tured Indepen- dent Exchange -able Unstruc -tured Indepen- dent Average 0.63 1.5 0.56 0.57 0.57 0.59 Through this study, it is believed that the poverty data tends to have independent correlation structure. It can be seen through the value of SE R SE M in Nested GLM modeling, independent WCM has the smallest SE R SE M . After the random effect is included to the model Nested GLMM is used for modeling, where the correlation structure is controlled, Nested GLMM provides better fit to the data than Nested GLM with exchangeable and unstructured WCM, showed by the values of SE R SE M = 0,57 in GLMM, which is smaller than 0.63 and 1.5 in GLM.

4.4.2 Significance p-values