Impact of Intragroup (AS4) Correlation on the Significance of Relevant Unemployment Factors in Australia
A Generalised Linear Mixed Model Approach (GLMM)
Ana María Jaime Rivera
2020-06-29
1 Overview
In response to the Australian Social Impact Investors’ (ASII) request, the Data Scientists for Social Good (DSSG) conducted a modelling exercise that examined the factors affecting unemployment in Australia. Logistic regression was applied to that aim given scope, time, and budget restrictions.
On reflection of the limitations brought by the above-mentioned approach, this article implements a Generalised Linear Mixed Model (GLMM) to investigate the impact of having assumed independence of observations within an AS4 (Statistical Area Level 4).
2 From GLM to GLMM
2.1 Initial Considerations
Data sources and structure, and pre-processing techniques implemented in this article are based on those described by the DSSG.
2.2 Justification of Methodology
Generalised Linear Models’ (GLM) assume a zero correlation between the observations. As individuals within AS4 territories share a milieu conditioned to common key elements that might generate patterns on unemployment -e.g. local governments, economic activities, customs, population density-, independence of observations is not reasonable.
Recognising the need of incorporating intuition to guide data analysis, the variance on the effects of \(age\), \(sex\), \(education\) and \(ingp\) across AS4 territories are all deemed plausible. As an example, an unobservable sexist bias in a region might cause an increased effect of \(sex\) in unemployment for that particular territory. Consequently, relaxing that assumption as much as possible would permit correcting initial estimations.
2.3 Objectives & Research Questions
This piece aims to model heterogeneity in effects across AS4 territories to answer the following questions:
How significance and direction1 of the odds ratios’ estimates (fixed effects) change when inducing an intragroup correlation structure?
What patterns, and their potentially related variables, could be recognised by analysing random intercepts?
3 Initial GLM Approach
The GLM defined by DSSG predicts the log-odds of \(unemployed_i=1\) for observation \(i\), as indicated in equation (3.1):
\[\begin{equation} \left.\begin{aligned} g(\hat{p}_{unemployed_i})&=log\left(\frac{\hat{p}_{unemployed_i}}{1-\hat{p}_{unemployed_i}}\right) \\ logit(\hat{p}_{unemployed_i}) &= \hat{\beta}_0 + \hat{\beta}_{age}age_i + \hat{\beta}_{sex}sex_i + \hat{\beta}_{education}education_i + \hat{\beta}_{ingp}ingp_i\\ &\quad+\hat{\beta}_{IEODecile}IEODecile_i+\hat{\beta}_{remotenessindex}remotenessindex_i + \hat{\beta}_{capital}capital_i\\ &\quad+ \hat{\beta}_{age*sex}age_i*sex_i+\hat{\beta}_{age*education}age_i*education_i + \hat{\beta}_{age*ingp}age_i*ingp_i \\ &\quad+ \hat{\beta}_{sex*education}sex_i*education_i+\hat{\beta}_{sex*ingp}sex_i*ingp_i\\ &\quad+ \hat{\beta}_{education*ingp}education_i*ingp_i + \hat{\beta}_{age*capital}age_i*capital_i\\ &\quad+\hat{\beta}_{sex*capital}sex_i*capital_i + \hat{\beta}_{ingp*capital}ingp_i*capital_i \\ \end{aligned}\right. \tag{3.1} \end{equation}\]
Please note:
\(\hat{\beta}_0\) denotes the intercept estimate
\(\hat{\beta}_{variable}\) denotes the slope estimate of \(variable\)
Formulation in R equals to:
[1] "Formula"
unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capitalPlease refer to GLM Summary for details on results.
4 Building a GLMM Approach
4.1 Modelling
To select the best GLMM formulation, models are constructed starting from the simplest approach (only random intercept) and continuing with the iterative addition of random slopes -i.e. Forward Selection-. Please note that fixed effects from Initial GLM Approach are maintained in line to the Objectives & Research Questions. To evaluate the significance of the difference between models, a likelihood ratio test (“Chisq”) is implemented (Sonderegger, Wagner, & Torreira, 2018). Similarly, the Akaike Information Criterion (AIC) is considered as it adjusts by the number of parameters, favouring simpler models (Manning,2007).
Despite simplicity given by only random intercept modelling, random effects were not only deemed necessary but also kept to the maximal2. Justification can be summarised as follows:
The intragroup correlation could be not resolved with mixed models if important fixed or random effects are not taken into consideration. Accordingly, authors such as Barr et al. (2013) and Harrison et al. (2018) recommend including all random slopes that are plausible or relevant to the interpretation of the study unless convergence is not reached.
Researchers have proven through simulations that multilevel models that only consider random intercepts present higher Type I error rates (Winter, n.d.).
Additionally, correlations of random effects are evaluated in each case to ensure no values of +-1 are encountered, as they would indicate either a) null variation within the grouping, b) non-convergence (Matuscheck et al., 2017). Considering that a) applies for \(IEODecile\), \(remotenessindex\), and \(capital\), these are not considered.
This section summarises final GLMM selection. Please refer to GLMMs Summary for details on results.
Model A:
[1] "Formula"
unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 | sa4)
Model B:
[1] "Formula"
unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age |
sa4)
[1] "Random Effects Correlation Matrix"| (Intercept) | age20-29 years | age30-39 years | age40-49 years | age50-59 years | |
|---|---|---|---|---|---|
| (Intercept) | 1.0000000 | -0.3810525 | -0.5164655 | -0.5336610 | -0.5390193 |
| age20-29 years | -0.3810525 | 1.0000000 | 0.8201441 | 0.6635950 | 0.5878674 |
| age30-39 years | -0.5164655 | 0.8201441 | 1.0000000 | 0.8849971 | 0.7660772 |
| age40-49 years | -0.5336610 | 0.6635950 | 0.8849971 | 1.0000000 | 0.9395329 |
| age50-59 years | -0.5390193 | 0.5878674 | 0.7660772 | 0.9395329 | 1.0000000 |
[1] "Hypothesis Testing"
AIC Pr(>Chisq)
at3_model_a 43780
at3_model_b 40036 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1A likelihood ratio test shows that AS4s differ significantly in the effect of \(age\) \((p< 2.2e-16)\). An AIC reduction for model B \((40036)\), compared to model A \((43780)\), indicates an improvement of fit.
Model C:
[1] "Formula"
unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age +
sex | sa4)
[1] "Random Effects Correlation Matrix"| (Intercept) | age20-29 years | age30-39 years | age40-49 years | age50-59 years | sexMale | |
|---|---|---|---|---|---|---|
| (Intercept) | 1.0000000 | -0.4615148 | -0.5711161 | -0.5719806 | -0.5738917 | -0.0637823 |
| age20-29 years | -0.4615148 | 1.0000000 | 0.8237266 | 0.6687505 | 0.5872168 | 0.2674464 |
| age30-39 years | -0.5711161 | 0.8237266 | 1.0000000 | 0.8858468 | 0.7648781 | 0.2063752 |
| age40-49 years | -0.5719806 | 0.6687505 | 0.8858468 | 1.0000000 | 0.9387253 | 0.1828554 |
| age50-59 years | -0.5738917 | 0.5872168 | 0.7648781 | 0.9387253 | 1.0000000 | 0.1208082 |
| sexMale | -0.0637823 | 0.2674464 | 0.2063752 | 0.1828554 | 0.1208082 | 1.0000000 |
[1] "Hypothesis Testing"
AIC Pr(>Chisq)
at3_model_b 40036
at3_model_c 38317 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1A likelihood ratio test shows that AS4s differ significantly in the effect of \(sex\) \((p< 2.2e-16)\). An AIC reduction for model C \((38317)\), compared to model B \((40036)\), indicates an improvement of fit.
Model D:
[1] "Formula"
unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age +
sex + education | sa4)
[1] "Random Effects Correlation Matrix"| (Intercept) | age20-29 years | age30-39 years | age40-49 years | age50-59 years | sexMale | educationhs-not-finished | |
|---|---|---|---|---|---|---|---|
| (Intercept) | 1.0000000 | -0.5669196 | -0.6746014 | -0.6503010 | -0.5931649 | -0.0378451 | -0.4413982 |
| age20-29 years | -0.5669196 | 1.0000000 | 0.8608334 | 0.7231557 | 0.6204294 | 0.2489127 | 0.3758299 |
| age30-39 years | -0.6746014 | 0.8608334 | 1.0000000 | 0.8958171 | 0.7594933 | 0.1717879 | 0.4516639 |
| age40-49 years | -0.6503010 | 0.7231557 | 0.8958171 | 1.0000000 | 0.9326056 | 0.1568105 | 0.3832347 |
| age50-59 years | -0.5931649 | 0.6204294 | 0.7594933 | 0.9326056 | 1.0000000 | 0.1087562 | 0.2206278 |
| sexMale | -0.0378451 | 0.2489127 | 0.1717879 | 0.1568105 | 0.1087562 | 1.0000000 | 0.0254617 |
| educationhs_not_finished | -0.4413982 | 0.3758299 | 0.4516639 | 0.3832347 | 0.2206278 | 0.0254617 | 1.0000000 |
[1] "Hypothesis Testing"
AIC Pr(>Chisq)
at3_model_c 38317
at3_model_d 36251 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1A likelihood ratio test shows that AS4s differ significantly in the effect of \(education\) \((p< 2.2e-16)\). An AIC reduction for model D \((36251)\), compared to model C \((38317)\), indicates an improvement of fit.
Model E:
[1] "Formula"
unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age +
sex + education + ingp | sa4)
[1] "Random Effects Correlation Matrix"| (Intercept) | age20-29 years | age30-39 years | age40-49 years | age50-59 years | sexMale | educationhs-not-finished | ingpNon-Indigenous | |
|---|---|---|---|---|---|---|---|---|
| (Intercept) | 1.0000000 | -0.0868327 | 0.1140377 | 0.1948135 | 0.2021416 | 0.1474821 | 0.3062938 | -0.8717418 |
| age20-29 years | -0.0868327 | 1.0000000 | 0.8448542 | 0.6875529 | 0.5788329 | 0.2344428 | 0.4074246 | -0.1413111 |
| age30-39 years | 0.1140377 | 0.8448542 | 1.0000000 | 0.8923160 | 0.7668547 | 0.1646408 | 0.4844798 | -0.4019561 |
| age40-49 years | 0.1948135 | 0.6875529 | 0.8923160 | 1.0000000 | 0.9455944 | 0.1527898 | 0.3936088 | -0.4606818 |
| age50-59 years | 0.2021416 | 0.5788329 | 0.7668547 | 0.9455944 | 1.0000000 | 0.0965349 | 0.3100408 | -0.4484222 |
| sexMale | 0.1474821 | 0.2344428 | 0.1646408 | 0.1527898 | 0.0965349 | 1.0000000 | 0.0707023 | -0.1270621 |
| educationhs_not_finished | 0.3062938 | 0.4074246 | 0.4844798 | 0.3936088 | 0.3100408 | 0.0707023 | 1.0000000 | -0.5430379 |
| ingpNon-Indigenous | -0.8717418 | -0.1413111 | -0.4019561 | -0.4606818 | -0.4484222 | -0.1270621 | -0.5430379 | 1.0000000 |
[1] "Hypothesis Testing"
AIC Pr(>Chisq)
at3_model_d 36251
at3_model_e 31188 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1A likelihood ratio test shows that AS4s differ significantly in the effect of \(ingp\) \((p< 2.2e-16)\). An AIC reduction for model E \((31188)\), compared to model D \((36251)\), indicates an improvement of fit.
Overall, no problems were encountered regarding convergence. The following table summarises significance and fit testing. Thus, model E is selected as the best GLMM approach.
[1] "Hypothesis Testing"
AIC Pr(>Chisq)
at3_model_a 43780
at3_model_b 40036 < 2.2e-16 ***
at3_model_c 38317 < 2.2e-16 ***
at3_model_d 36251 < 2.2e-16 ***
at3_model_e 31188 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Figure 4.1 allows to compare estimates of fixed effects (Ludecke, 2020) for models A-E. As anticipated, induced adjustments on fixed effects correct coefficients of both, individual variables, and interactions terms. From the plot, it is possible to conclude that the biggest alterations are induced when incorporating \(education\) (GLMM.D) and \(ingp\) (GLMM.E), which is supported by models D and E reducing AIC more significantly. Another element of interest is the indirect adjustment of the fixed coefficients related to \(capital\), which appears once variations on \(age\) effect are allowed between AS4s.
Figure 4.1: Coefficients (Fixed), GLMMs
4.2 Final Model
The final GLMM predicts the log-odds of \(unemployed_i=1\) for observation \(i\), within AS4 \(k\), as indicated in equation (4.1):\[\begin{equation} \left.\begin{aligned} g(\hat{p}_{unemployed_i\mid k[i]})&=log\left(\frac{\hat{p}_{unemployed_i\mid k[i]}}{1-\hat{p}_{unemployed_i\mid k[i]}}\right) \\ logit(\hat{p}_{unemployed_i\mid k[i]}) &= \hat{\beta}_0 +\hat{\delta}_{0,k[i]}\\ &\quad+(\hat{\beta}_{age}+\hat{\delta}_{age,k[i]})age_i\\ &\quad+(\hat{\beta}_{sex}+\hat{\delta}_{sex,k[i]})sex_i\\ &\quad+(\hat{\beta}_{education}+\hat{\delta}_{education,k[i]})education_i\\ &\quad+(\hat{\beta}_{ingp}+\hat{\delta}_{ingp,k[i]})ingp_i\\ &\quad+\hat{\beta}_{IEODecile}IEODecile_{k[i]} + \hat{\beta}_{remotenessindex}remotenessindex_{k[i]} + \hat{\beta}_{capital}capital_{k[i]}\\ &\quad+\hat{\beta}_{age*sex}age_i*sex_i + \hat{\beta}_{age*education}age_i*education_i + \hat{\beta}_{age*ingp}age_i*ingp_i\\ &\quad+\hat{\beta}_{sex*education}sex_i*education_i + \hat{\beta}_{sex*ingp}sex_i*ingp_i\\ &\quad+\hat{\beta}_{education*ingp}education_i*ingp_i+\hat{\beta}_{age*capital}age_i*capital_i\\ &\quad+ \hat{\beta}_{sex*capital}sex_i*capital_i + \hat{\beta}_{ingp*capital}ingp_i*capital_i\\ \end{aligned}\right. \tag{4.1} \end{equation}\]
Please note:
\(\hat{\beta}_0\) denotes the fixed intercept estimate
\(\hat{\beta}_{variable}\) denotes the fixed slope estimate of \(variable\)
\(\hat{\delta}_{0,k}\) denotes the random intercept estimate for AS4 \(k\)
\(\hat{\delta}_{variable,k}\) denotes the random slope estimate of \(variable\), for AS4 \(k\)
5 GLM vs. GLMM - Goodness of Fit
Considering that the likelihood ratio test (“Chisq”) is also supported to compare GLM and GLMM models (Bolker, 2020), it is implemented to evaluate significance of the differences between the models. The same applies for AIC, which is adequate as it allows to compare models, including non-mixed (Turner, 2008).
[1] "Hypothesis Testing"
AIC Pr(>Chisq)
at2_model 63264
at3_model 31188 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1A likelihood ratio test shows that intercepts and effects differ significantly across AS4s \((p< 2.2e-16)\). Similarly, an AIC reduction for the GLMM approach \((31188)\), compared to GLM approach \((63264)\), indicates an important improvement of fit.
In addition to the evaluation measures implemented in Modelling, classification performance is evaluated using precision, recall and F1, with F1 being predominant as it is a weighted measure. These metrics are selected given class imbalance. Improvement in F1 score from \((0.9148)\) to \((0.9670)\)indicate the significance of random effects to improve the fit of the model, nonetheless, increase is not substantially enough to ensure this would help an improvement in model generalisability3.
| GLM | Predicted Employed | Predicted Unemployed |
| Actual Employed | 9,520,199 | 61,811 |
| Actual Unemployed | 61,809 | 663,266 |
| GLMM | Predicted Employed | Predicted Unemployed |
| Actual Employed | 9,558,052 | 23,958 |
| Actual Unemployed | 23,962 | 701,113 |
| Model | Recall | Precision | F1 |
|---|---|---|---|
| GLM | 0.9148 | 0.9148 | 0.9148 |
| GLMM | 0.9670 | 0.9670 | 0.9670 |
Even though generalisability might not be drastically enhanced, the GLMM approach is proved to be adequate as variance across AS4 territories is significant for all the effects tested, and overall model fit is improved. More importantly, this approach is preferred considering that the assumption of null correlation of observations is not appropriate.
Adjusting for the potential bias is recommended when possible unless interpretability is comprised, or model complexity prevents convergence. For this article in particular, none of these are a matter of concern. Main complexity on interpretability is related to interactions included for the initial GLM approach. Also, as stated previously, no inconveniences on computational power nor time cost are encountered for convergence in any for any of the GLMM models tested.
6 From Numbers to Meaning
6.1 Impact of Intragroup Correlation on Unemployment Factors
The fixed effects estimates in mixed models with nonlinear link functions have an interpretation conditional on the random effects (Rizopoulos, n.d.). This is, overall effect is subject to the random effect calculated for a particular AS4 \(( \hat{\beta}_{variable}+ \hat{\delta}_{variable,k})\) in the equation. As per the first research question, comparison of interest concerns fixed effects ( \(\hat{\beta}_{variable}\) in equations (3.1) and (4.1))4.
Adjustments of fixed effects significance, when correcting by correlation within AS4, can be detected by comparing fixed effects estimates -i.e. model coefficients for GLM and fixed effects for GLMM-.
[1] "GLM Approach - Summary"
Call:
glm(formula = unemployment/labour_force ~ age + sex + education +
ingp + IEO_Decile + remoteness_index + capital + age * sex +
age * education + age * ingp + sex * education + sex * ingp +
education * ingp + age * capital + sex * capital + ingp *
capital, family = "binomial", data = use_dataset, weights = labour_force)
Deviance Residuals:
Min 1Q Median 3Q Max
-16.5473 -2.4222 -0.5863 1.4704 21.0410
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.665391 0.021323 -78.104 < 2e-16 ***
age20-29 years -0.578106 0.019240 -30.047 < 2e-16 ***
age30-39 years -1.030477 0.021515 -47.897 < 2e-16 ***
age40-49 years -1.168602 0.022263 -52.490 < 2e-16 ***
age50-59 years -1.531104 0.025964 -58.969 < 2e-16 ***
sexMale 0.350908 0.014416 24.342 < 2e-16 ***
educationhs_not_finished 0.621432 0.014434 43.054 < 2e-16 ***
ingpNon-Indigenous 0.085527 0.020651 4.141 3.45e-05 ***
IEO_Decile -0.141819 0.001576 -89.968 < 2e-16 ***
remoteness_index -0.104425 0.002094 -49.858 < 2e-16 ***
capitalNonCapitalCity 0.331909 0.015858 20.930 < 2e-16 ***
age20-29 years:sexMale -0.145641 0.007814 -18.637 < 2e-16 ***
age30-39 years:sexMale -0.472949 0.008445 -56.004 < 2e-16 ***
age40-49 years:sexMale -0.351550 0.008512 -41.299 < 2e-16 ***
age50-59 years:sexMale -0.072699 0.008818 -8.244 < 2e-16 ***
age20-29 years:educationhs_not_finished 0.259070 0.008098 31.990 < 2e-16 ***
age30-39 years:educationhs_not_finished 0.490332 0.008751 56.031 < 2e-16 ***
age40-49 years:educationhs_not_finished 0.192822 0.008676 22.226 < 2e-16 ***
age50-59 years:educationhs_not_finished -0.059820 0.008998 -6.648 2.97e-11 ***
age20-29 years:ingpNon-Indigenous -0.283516 0.018055 -15.703 < 2e-16 ***
age30-39 years:ingpNon-Indigenous -0.348060 0.020241 -17.196 < 2e-16 ***
age40-49 years:ingpNon-Indigenous -0.309445 0.020918 -14.793 < 2e-16 ***
age50-59 years:ingpNon-Indigenous -0.032813 0.024538 -1.337 0.181
sexMale:educationhs_not_finished -0.156564 0.005337 -29.334 < 2e-16 ***
sexMale:ingpNon-Indigenous -0.104705 0.012577 -8.325 < 2e-16 ***
educationhs_not_finished:ingpNon-Indigenous -0.313381 0.013341 -23.490 < 2e-16 ***
age20-29 years:capitalNonCapitalCity 0.125653 0.007948 15.810 < 2e-16 ***
age30-39 years:capitalNonCapitalCity 0.158292 0.008736 18.119 < 2e-16 ***
age40-49 years:capitalNonCapitalCity 0.151364 0.008788 17.224 < 2e-16 ***
age50-59 years:capitalNonCapitalCity 0.205146 0.009062 22.638 < 2e-16 ***
sexMale:capitalNonCapitalCity 0.073985 0.005238 14.125 < 2e-16 ***
ingpNon-Indigenous:capitalNonCapitalCity -0.628600 0.014573 -43.134 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 294706 on 3516 degrees of freedom
Residual deviance: 44292 on 3485 degrees of freedom
(3 observations deleted due to missingness)
AIC: 63264
Number of Fisher Scoring iterations: 4
[1] "GLMM Approach - Summary"Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: binomial ( logit )
Formula: unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age + sex + education + ingp | sa4)
Data: use_dataset
Weights: labour_force
Control: glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e+05))
AIC BIC logLik deviance df.resid
31187.7 31607.0 -15525.9 31051.7 3449
Scaled residuals:
Min 1Q Median 3Q Max
-5.2163 -1.1270 -0.1702 0.9912 9.5438
Random effects:
Groups Name Variance Std.Dev. Corr
sa4 (Intercept) 0.20943 0.4576
age20-29 years 0.02058 0.1434 -0.09
age30-39 years 0.03809 0.1952 0.11 0.84
age40-49 years 0.03134 0.1770 0.19 0.69 0.89
age50-59 years 0.03921 0.1980 0.20 0.58 0.77 0.95
sexMale 0.01195 0.1093 0.15 0.23 0.16 0.15 0.10
educationhs_not_finished 0.02020 0.1421 0.31 0.41 0.48 0.39 0.31 0.07
ingpNon-Indigenous 0.24842 0.4984 -0.87 -0.14 -0.40 -0.46 -0.45 -0.13 -0.54
Number of obs: 3517, groups: sa4, 88
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.654390 0.087954 -18.810 < 2e-16 ***
age20-29 years -0.659990 0.031130 -21.201 < 2e-16 ***
age30-39 years -1.168217 0.038900 -30.032 < 2e-16 ***
age40-49 years -1.315311 0.037607 -34.975 < 2e-16 ***
age50-59 years -1.670063 0.043276 -38.591 < 2e-16 ***
sexMale 0.359199 0.024296 14.784 < 2e-16 ***
educationhs_not_finished 0.575686 0.021542 26.724 < 2e-16 ***
ingpNon-Indigenous -0.105131 0.082995 -1.267 0.20526
IEO_Decile -0.166084 0.031297 -5.307 1.12e-07 ***
remoteness_index -0.236947 0.037339 -6.346 2.21e-10 ***
capitalNonCapitalCity 0.144356 0.124551 1.159 0.24645
age20-29 years:sexMale -0.154010 0.007873 -19.562 < 2e-16 ***
age30-39 years:sexMale -0.483027 0.008504 -56.798 < 2e-16 ***
age40-49 years:sexMale -0.362949 0.008561 -42.395 < 2e-16 ***
age50-59 years:sexMale -0.087697 0.008871 -9.886 < 2e-16 ***
age20-29 years:educationhs_not_finished 0.239815 0.008312 28.850 < 2e-16 ***
age30-39 years:educationhs_not_finished 0.436987 0.008994 48.588 < 2e-16 ***
age40-49 years:educationhs_not_finished 0.149333 0.008892 16.793 < 2e-16 ***
age50-59 years:educationhs_not_finished -0.101220 0.009249 -10.944 < 2e-16 ***
age20-29 years:ingpNon-Indigenous -0.204022 0.019090 -10.687 < 2e-16 ***
age30-39 years:ingpNon-Indigenous -0.173759 0.021770 -7.982 1.44e-15 ***
age40-49 years:ingpNon-Indigenous -0.123377 0.022371 -5.515 3.49e-08 ***
age50-59 years:ingpNon-Indigenous 0.154496 0.026215 5.893 3.78e-09 ***
sexMale:educationhs_not_finished -0.151554 0.005471 -27.703 < 2e-16 ***
sexMale:ingpNon-Indigenous -0.092894 0.013545 -6.858 6.98e-12 ***
educationhs_not_finished:ingpNon-Indigenous -0.194102 0.014531 -13.358 < 2e-16 ***
age20-29 years:capitalNonCapitalCity 0.184684 0.034096 5.417 6.08e-08 ***
age30-39 years:capitalNonCapitalCity 0.198511 0.043881 4.524 6.07e-06 ***
age40-49 years:capitalNonCapitalCity 0.187351 0.041831 4.479 7.51e-06 ***
age50-59 years:capitalNonCapitalCity 0.223410 0.047952 4.659 3.18e-06 ***
sexMale:capitalNonCapitalCity 0.064439 0.027744 2.323 0.02020 *
ingpNon-Indigenous:capitalNonCapitalCity -0.349468 0.106553 -3.280 0.00104 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Crucial results include:
ingpNon-IndigenousandcapitalNonCapitalCitystop being statistically significantsexMale: capitalNonCapitalCitycontinues to be statistically significant but only at a level of \(\alpha = 0.05\)ingpNon-Indigenous : capitalNonCapitalCitycontinues to be statistically significant but only at a level of \(\alpha = 0.01\)age50-59 years: ingpNon-Indigenousbecomes statistically significant at a level of \(\alpha = 0.001\)
Please refer to Final Model Additional Results for values and standard deviation of random effects.
On the other hand, figure 6.1 facilitates the identification of adjustments in direction, which are of key interest as they would indicate corrected effect is opposite to the initially interpreted effect. In this case, odds ratios are plotted for easier interpretation (Ludecke, 2020). Subsequently, the direction of effects is defined as being positive if odds ratios are greater than 1 and negative otherwise (null if equal to 1). As per the visualisation, the direction only shifts foringpNon-Indigenous and age50-59 years: ingpNon-Indigenous, which is related to the above-stated results. Precise quantification of overall regression coefficients follows in equations (3.1) and (4.1) and is conditioned to a unique combination of variables, given the complexity brought by multiple interactions.
Figure 6.1: Coefficients (Fixed), GLM vs. GLMM
All in, what these results imply is that interpretation of some variables under the initial approach is biased as a consequence of intragroup correlation. This generates effects of variables such as
capitalNonCapitalCity to be overestimated. Moreover, conclusions on their relevance5 to understand unemployment in Australia have proven to be unreliable, which indicates they are confounding variable. In other words, they capture erroneously the effect of other underlying variables unless AS4 variance is stated explicitly under the GLMM approach. This last statement is supported by the fact that deviations for interactions considering capitalNonCapitalCity are not ignorable.
7 Conclusion
All things considered, the GLMM approach presented in this article enhances the analysis conducted by the DSSG, by explicitly incorporating variance of intercepts and effects across AS4. This is relevant to identify potential significant biases on the initial interpretation of fixed effects estimates generated by intragroup correlation (Guo, 2005). Furthermore, relaxing the assumption of independence within observations is deemed reasonable and necessary as unemployment is likely to be conditioned to elements shared within the milieu such as industry mix. Indeed, an analysis on random intercepts facilitate the identification of unveiled variables of interest, as the aforementioned.
8 References
Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory and Language, 68(3), 255–278. https://www-sciencedirect-com.ezproxy.lib.uts.edu.au/science/article/pii/S0749596X12001180
Bolker, B., & others. (2020). GLMM. GitHub Pages. https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#testing-significance-of-random-effects
Guo, S. (2005). Analyzing grouped data with hierarchical linear modelling. Children and Youth Services Review, 27(6), 637-652. https://www.sciencedirect.com/science/article/pii/S0190740904002506?casa_token=DtkN1lSjgiAAAAAA:2ZbW99DYn0o5I6NxauLLFvtCT9hXW0TYuKZMgwE9FAhgJ6-nHa4QV4FILdQl4M-Elpgeame7yw
Harrison, X. A., Donaldson, L., Correa-Cano, M. E., Evans, J., Fisher, D. N., Goodwin, C. E., … & Inger, R. (2018). A brief introduction to mixed effects modelling and multi-model inference in ecology. PeerJ, 6, e4794. https://search.lib.uts.edu.au/discovery/search?tab=CentralIndex&search_scope=CentralIndex&vid=61UTS_INST:61UTS&lang=en
Manning, C. (2007). Generalized Linear Mixed Models. https://nlp.stanford.edu/manning/courses/ling289/GLMM.pdf
Matuscheck, H., Kliegl, R., Vasishth, S., Baayen, H. & Bates, D. (2017). Balancing type I error and power in linear mixed models. Journal of Memory and Language, 94, 305-315. https://www.sciencedirect.com/science/article/pii/S0749596X17300013
Ludecke, D. (2020). Plotting Estimates (Fixed Effects) of Regression Models. https://cran.r-project.org/web/packages/sjPlot/vignettes/plot_model_estimates.html
Rizopoulos, D. (n.d.). Statistical Analysis of Repeated Measurements Data. Erasmus University Medical Center. http://www.drizopoulos.com/courses/EMC/CE08.pdf
Sonderegger, M., Wagner, M., & Torreira, F. (2018). Quantitative methods for linguistic data. prosody.lab | prosody at mcgill and beyond. https://people.linguistics.mcgill.ca/~morgan/book/mixed-effects-logistic-regression.html#evaluation-measure-1-likelihood-ratio-test
Turner, H. (2008). Introduction to Generalized Linear Models. University of Warwick. https://statmath.wu.ac.at/courses/heather_turner/glmCourse_001.pdf
Winter, B. (n.d.). Linear models and linear mixed effects models in R with linguistic applications. University of California. https://arxiv.org/ftp/arxiv/papers/1308/1308.5499.pdf
9 Appendix
9.1 Accessible Unemployment Insurance
A project by Data Scientists for Social Good (DSSG), for the Australian Social Impact Investors (ASII).Download Accessible Unemployment Insurance - DSSG.pdf
9.2 GLM Summary
9.2.1 Initial Model
Call:
glm(formula = unemployment/labour_force ~ age + sex + education +
ingp + IEO_Decile + remoteness_index + capital + age * sex +
age * education + age * ingp + sex * education + sex * ingp +
education * ingp + age * capital + sex * capital + ingp *
capital, family = "binomial", data = use_dataset, weights = labour_force)
Deviance Residuals:
Min 1Q Median 3Q Max
-16.5473 -2.4222 -0.5863 1.4704 21.0410
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.665391 0.021323 -78.104 < 2e-16 ***
age20-29 years -0.578106 0.019240 -30.047 < 2e-16 ***
age30-39 years -1.030477 0.021515 -47.897 < 2e-16 ***
age40-49 years -1.168602 0.022263 -52.490 < 2e-16 ***
age50-59 years -1.531104 0.025964 -58.969 < 2e-16 ***
sexMale 0.350908 0.014416 24.342 < 2e-16 ***
educationhs_not_finished 0.621432 0.014434 43.054 < 2e-16 ***
ingpNon-Indigenous 0.085527 0.020651 4.141 3.45e-05 ***
IEO_Decile -0.141819 0.001576 -89.968 < 2e-16 ***
remoteness_index -0.104425 0.002094 -49.858 < 2e-16 ***
capitalNonCapitalCity 0.331909 0.015858 20.930 < 2e-16 ***
age20-29 years:sexMale -0.145641 0.007814 -18.637 < 2e-16 ***
age30-39 years:sexMale -0.472949 0.008445 -56.004 < 2e-16 ***
age40-49 years:sexMale -0.351550 0.008512 -41.299 < 2e-16 ***
age50-59 years:sexMale -0.072699 0.008818 -8.244 < 2e-16 ***
age20-29 years:educationhs_not_finished 0.259070 0.008098 31.990 < 2e-16 ***
age30-39 years:educationhs_not_finished 0.490332 0.008751 56.031 < 2e-16 ***
age40-49 years:educationhs_not_finished 0.192822 0.008676 22.226 < 2e-16 ***
age50-59 years:educationhs_not_finished -0.059820 0.008998 -6.648 2.97e-11 ***
age20-29 years:ingpNon-Indigenous -0.283516 0.018055 -15.703 < 2e-16 ***
age30-39 years:ingpNon-Indigenous -0.348060 0.020241 -17.196 < 2e-16 ***
age40-49 years:ingpNon-Indigenous -0.309445 0.020918 -14.793 < 2e-16 ***
age50-59 years:ingpNon-Indigenous -0.032813 0.024538 -1.337 0.181
sexMale:educationhs_not_finished -0.156564 0.005337 -29.334 < 2e-16 ***
sexMale:ingpNon-Indigenous -0.104705 0.012577 -8.325 < 2e-16 ***
educationhs_not_finished:ingpNon-Indigenous -0.313381 0.013341 -23.490 < 2e-16 ***
age20-29 years:capitalNonCapitalCity 0.125653 0.007948 15.810 < 2e-16 ***
age30-39 years:capitalNonCapitalCity 0.158292 0.008736 18.119 < 2e-16 ***
age40-49 years:capitalNonCapitalCity 0.151364 0.008788 17.224 < 2e-16 ***
age50-59 years:capitalNonCapitalCity 0.205146 0.009062 22.638 < 2e-16 ***
sexMale:capitalNonCapitalCity 0.073985 0.005238 14.125 < 2e-16 ***
ingpNon-Indigenous:capitalNonCapitalCity -0.628600 0.014573 -43.134 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 294706 on 3516 degrees of freedom
Residual deviance: 44292 on 3485 degrees of freedom
(3 observations deleted due to missingness)
AIC: 63264
Number of Fisher Scoring iterations: 4
9.3 GLMMs Summary
9.3.1 Model A - Random Intercept
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: binomial ( logit )
Formula: unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 | sa4)
Data: use_dataset
Weights: labour_force
Control: glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e+05))
AIC BIC logLik deviance df.resid
43780.4 43983.9 -21857.2 43714.4 3484
Scaled residuals:
Min 1Q Median 3Q Max
-12.5858 -1.7286 -0.4705 1.2004 16.5840
Random effects:
Groups Name Variance Std.Dev.
sa4 (Intercept) 0.03776 0.1943
Number of obs: 3517, groups: sa4, 88
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.626240 0.045327 -35.878 < 2e-16 ***
age20-29 years -0.608709 0.019298 -31.542 < 2e-16 ***
age30-39 years -1.057996 0.021580 -49.026 < 2e-16 ***
age40-49 years -1.181390 0.022312 -52.948 < 2e-16 ***
age50-59 years -1.536353 0.026002 -59.086 < 2e-16 ***
sexMale 0.352457 0.014443 24.403 < 2e-16 ***
educationhs_not_finished 0.643433 0.014489 44.408 < 2e-16 ***
ingpNon-Indigenous -0.001272 0.020745 -0.061 0.951102
IEO_Decile -0.135067 0.028829 -4.685 2.8e-06 ***
remoteness_index -0.105208 0.027359 -3.846 0.000120 ***
capitalNonCapitalCity 0.236185 0.063420 3.724 0.000196 ***
age20-29 years:sexMale -0.148646 0.007831 -18.982 < 2e-16 ***
age30-39 years:sexMale -0.478944 0.008460 -56.610 < 2e-16 ***
age40-49 years:sexMale -0.356752 0.008528 -41.835 < 2e-16 ***
age50-59 years:sexMale -0.077092 0.008833 -8.728 < 2e-16 ***
age20-29 years:educationhs_not_finished 0.255798 0.008122 31.495 < 2e-16 ***
age30-39 years:educationhs_not_finished 0.479997 0.008776 54.695 < 2e-16 ***
age40-49 years:educationhs_not_finished 0.177821 0.008698 20.444 < 2e-16 ***
age50-59 years:educationhs_not_finished -0.078006 0.009024 -8.645 < 2e-16 ***
age20-29 years:ingpNon-Indigenous -0.274448 0.018109 -15.156 < 2e-16 ***
age30-39 years:ingpNon-Indigenous -0.334635 0.020303 -16.482 < 2e-16 ***
age40-49 years:ingpNon-Indigenous -0.296257 0.020968 -14.129 < 2e-16 ***
age50-59 years:ingpNon-Indigenous -0.024190 0.024580 -0.984 0.325058
sexMale:educationhs_not_finished -0.150979 0.005348 -28.229 < 2e-16 ***
sexMale:ingpNon-Indigenous -0.106610 0.012602 -8.460 < 2e-16 ***
educationhs_not_finished:ingpNon-Indigenous -0.303820 0.013390 -22.690 < 2e-16 ***
age20-29 years:capitalNonCapitalCity 0.156687 0.007986 19.619 < 2e-16 ***
age30-39 years:capitalNonCapitalCity 0.183667 0.008772 20.938 < 2e-16 ***
age40-49 years:capitalNonCapitalCity 0.163724 0.008812 18.580 < 2e-16 ***
age50-59 years:capitalNonCapitalCity 0.216800 0.009087 23.859 < 2e-16 ***
sexMale:capitalNonCapitalCity 0.074923 0.005248 14.275 < 2e-16 ***
ingpNon-Indigenous:capitalNonCapitalCity -0.522173 0.014776 -35.339 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
9.3.2 Model B - Random Intercept, Random Effect (age)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: binomial ( logit )
Formula: unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age | sa4)
Data: use_dataset
Weights: labour_force
Control: glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e+05))
AIC BIC logLik deviance df.resid
40035.6 40325.4 -19970.8 39941.6 3470
Scaled residuals:
Min 1Q Median 3Q Max
-11.3692 -1.6502 -0.4015 1.1456 10.0161
Random effects:
Groups Name Variance Std.Dev. Corr
sa4 (Intercept) 0.05299 0.2302
age20-29 years 0.01761 0.1327 -0.38
age30-39 years 0.03214 0.1793 -0.52 0.82
age40-49 years 0.02754 0.1660 -0.53 0.66 0.88
age50-59 years 0.03512 0.1874 -0.54 0.59 0.77 0.94
Number of obs: 3517, groups: sa4, 88
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.599108 0.052282 -30.586 < 2e-16 ***
age20-29 years -0.618737 0.029034 -21.311 < 2e-16 ***
age30-39 years -1.083642 0.036309 -29.845 < 2e-16 ***
age40-49 years -1.219279 0.035040 -34.796 < 2e-16 ***
age50-59 years -1.565886 0.040054 -39.094 < 2e-16 ***
sexMale 0.356639 0.014453 24.676 < 2e-16 ***
educationhs_not_finished 0.656049 0.014536 45.132 < 2e-16 ***
ingpNon-Indigenous -0.023793 0.020936 -1.136 0.255750
IEO_Decile -0.145217 0.031953 -4.545 5.50e-06 ***
remoteness_index -0.100023 0.030212 -3.311 0.000931 ***
capitalNonCapitalCity 0.169148 0.075292 2.247 0.024668 *
age20-29 years:sexMale -0.152248 0.007838 -19.425 < 2e-16 ***
age30-39 years:sexMale -0.483320 0.008468 -57.075 < 2e-16 ***
age40-49 years:sexMale -0.361000 0.008533 -42.304 < 2e-16 ***
age50-59 years:sexMale -0.081904 0.008840 -9.266 < 2e-16 ***
age20-29 years:educationhs_not_finished 0.250229 0.008240 30.367 < 2e-16 ***
age30-39 years:educationhs_not_finished 0.451940 0.008904 50.756 < 2e-16 ***
age40-49 years:educationhs_not_finished 0.159914 0.008819 18.134 < 2e-16 ***
age50-59 years:educationhs_not_finished -0.091904 0.009164 -10.029 < 2e-16 ***
age20-29 years:ingpNon-Indigenous -0.263595 0.018531 -14.225 < 2e-16 ***
age30-39 years:ingpNon-Indigenous -0.298235 0.020884 -14.281 < 2e-16 ***
age40-49 years:ingpNon-Indigenous -0.251518 0.021432 -11.735 < 2e-16 ***
age50-59 years:ingpNon-Indigenous 0.018246 0.025061 0.728 0.466582
sexMale:educationhs_not_finished -0.151040 0.005350 -28.233 < 2e-16 ***
sexMale:ingpNon-Indigenous -0.107615 0.012610 -8.534 < 2e-16 ***
educationhs_not_finished:ingpNon-Indigenous -0.303840 0.013404 -22.668 < 2e-16 ***
age20-29 years:capitalNonCapitalCity 0.203204 0.029958 6.783 1.18e-11 ***
age30-39 years:capitalNonCapitalCity 0.251941 0.039810 6.329 2.47e-10 ***
age40-49 years:capitalNonCapitalCity 0.229087 0.037016 6.189 6.06e-10 ***
age50-59 years:capitalNonCapitalCity 0.263575 0.041551 6.343 2.25e-10 ***
sexMale:capitalNonCapitalCity 0.075730 0.005250 14.424 < 2e-16 ***
ingpNon-Indigenous:capitalNonCapitalCity -0.524575 0.014797 -35.452 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
9.3.3 Model C - Random Intercept, Random Effects (age, sex)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: binomial ( logit )
Formula: unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age + sex | sa4)
Data: use_dataset
Weights: labour_force
Control: glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e+05))
AIC BIC logLik deviance df.resid
38317.4 38644.2 -19105.7 38211.4 3464
Scaled residuals:
Min 1Q Median 3Q Max
-10.2601 -1.5449 -0.3408 1.0678 9.7598
Random effects:
Groups Name Variance Std.Dev. Corr
sa4 (Intercept) 0.05119 0.2263
age20-29 years 0.01728 0.1315 -0.46
age30-39 years 0.03239 0.1800 -0.57 0.82
age40-49 years 0.02788 0.1670 -0.57 0.67 0.89
age50-59 years 0.03527 0.1878 -0.57 0.59 0.76 0.94
sexMale 0.01219 0.1104 -0.06 0.27 0.21 0.18 0.12
Number of obs: 3517, groups: sa4, 88
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.601238 0.050962 -31.420 < 2e-16 ***
age20-29 years -0.617488 0.028892 -21.372 < 2e-16 ***
age30-39 years -1.084585 0.036400 -29.797 < 2e-16 ***
age40-49 years -1.219328 0.035170 -34.669 < 2e-16 ***
age50-59 years -1.563807 0.040110 -38.988 < 2e-16 ***
sexMale 0.353488 0.023045 15.339 < 2e-16 ***
educationhs_not_finished 0.658691 0.014548 45.276 < 2e-16 ***
ingpNon-Indigenous -0.030065 0.020989 -1.432 0.15204
IEO_Decile -0.156967 0.030441 -5.156 2.52e-07 ***
remoteness_index -0.116048 0.029121 -3.985 6.75e-05 ***
capitalNonCapitalCity 0.171469 0.072781 2.356 0.01848 *
age20-29 years:sexMale -0.153595 0.007866 -19.527 < 2e-16 ***
age30-39 years:sexMale -0.482949 0.008496 -56.843 < 2e-16 ***
age40-49 years:sexMale -0.362670 0.008554 -42.398 < 2e-16 ***
age50-59 years:sexMale -0.086939 0.008864 -9.808 < 2e-16 ***
age20-29 years:educationhs_not_finished 0.250410 0.008242 30.381 < 2e-16 ***
age30-39 years:educationhs_not_finished 0.451965 0.008907 50.741 < 2e-16 ***
age40-49 years:educationhs_not_finished 0.159388 0.008821 18.069 < 2e-16 ***
age50-59 years:educationhs_not_finished -0.093050 0.009166 -10.152 < 2e-16 ***
age20-29 years:ingpNon-Indigenous -0.264494 0.018531 -14.273 < 2e-16 ***
age30-39 years:ingpNon-Indigenous -0.297626 0.020889 -14.248 < 2e-16 ***
age40-49 years:ingpNon-Indigenous -0.250880 0.021438 -11.703 < 2e-16 ***
age50-59 years:ingpNon-Indigenous 0.018541 0.025068 0.740 0.45952
sexMale:educationhs_not_finished -0.152764 0.005432 -28.122 < 2e-16 ***
sexMale:ingpNon-Indigenous -0.093266 0.012982 -7.184 6.75e-13 ***
educationhs_not_finished:ingpNon-Indigenous -0.304849 0.013408 -22.736 < 2e-16 ***
age20-29 years:capitalNonCapitalCity 0.203459 0.029705 6.849 7.42e-12 ***
age30-39 years:capitalNonCapitalCity 0.253091 0.039952 6.335 2.37e-10 ***
age40-49 years:capitalNonCapitalCity 0.230748 0.037231 6.198 5.73e-10 ***
age50-59 years:capitalNonCapitalCity 0.264943 0.041638 6.363 1.98e-10 ***
sexMale:capitalNonCapitalCity 0.078571 0.024448 3.214 0.00131 **
ingpNon-Indigenous:capitalNonCapitalCity -0.525524 0.014800 -35.507 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
9.3.4 Model D - Random Intercept, Random Effects (age, sex, education)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: binomial ( logit )
Formula: unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age + sex + education | sa4)
Data: use_dataset
Weights: labour_force
Control: glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e+05))
AIC BIC logLik deviance df.resid
36250.8 36620.7 -18065.4 36130.8 3457
Scaled residuals:
Min 1Q Median 3Q Max
-9.7125 -1.4216 -0.2994 1.0045 10.4152
Random effects:
Groups Name Variance Std.Dev. Corr
sa4 (Intercept) 0.06189 0.2488
age20-29 years 0.02020 0.1421 -0.57
age30-39 years 0.03746 0.1936 -0.67 0.86
age40-49 years 0.03041 0.1744 -0.65 0.72 0.90
age50-59 years 0.03588 0.1894 -0.59 0.62 0.76 0.93
sexMale 0.01214 0.1102 -0.04 0.25 0.17 0.16 0.11
educationhs_not_finished 0.02800 0.1673 -0.44 0.38 0.45 0.38 0.22 0.03
Number of obs: 3517, groups: sa4, 88
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.573011 0.051285 -30.672 < 2e-16 ***
age20-29 years -0.626108 0.030543 -20.499 < 2e-16 ***
age30-39 years -1.072776 0.038160 -28.113 < 2e-16 ***
age40-49 years -1.203544 0.036499 -32.974 < 2e-16 ***
age50-59 years -1.544130 0.041491 -37.216 < 2e-16 ***
sexMale 0.368636 0.023928 15.406 < 2e-16 ***
educationhs_not_finished 0.604204 0.023211 26.031 < 2e-16 ***
ingpNon-Indigenous -0.099776 0.021056 -4.739 2.15e-06 ***
IEO_Decile -0.166399 0.029629 -5.616 1.95e-08 ***
remoteness_index -0.136249 0.029685 -4.590 4.44e-06 ***
capitalNonCapitalCity 0.175449 0.070823 2.477 0.0132 *
age20-29 years:sexMale -0.154851 0.007869 -19.679 < 2e-16 ***
age30-39 years:sexMale -0.483806 0.008499 -56.923 < 2e-16 ***
age40-49 years:sexMale -0.363753 0.008556 -42.514 < 2e-16 ***
age50-59 years:sexMale -0.088313 0.008866 -9.961 < 2e-16 ***
age20-29 years:educationhs_not_finished 0.238791 0.008311 28.731 < 2e-16 ***
age30-39 years:educationhs_not_finished 0.437033 0.008992 48.602 < 2e-16 ***
age40-49 years:educationhs_not_finished 0.148611 0.008892 16.712 < 2e-16 ***
age50-59 years:educationhs_not_finished -0.100991 0.009247 -10.922 < 2e-16 ***
age20-29 years:ingpNon-Indigenous -0.242473 0.018567 -13.059 < 2e-16 ***
age30-39 years:ingpNon-Indigenous -0.277234 0.020944 -13.237 < 2e-16 ***
age40-49 years:ingpNon-Indigenous -0.240798 0.021490 -11.205 < 2e-16 ***
age50-59 years:ingpNon-Indigenous 0.014684 0.025101 0.585 0.5585
sexMale:educationhs_not_finished -0.153360 0.005470 -28.035 < 2e-16 ***
sexMale:ingpNon-Indigenous -0.103927 0.013012 -7.987 1.38e-15 ***
educationhs_not_finished:ingpNon-Indigenous -0.214814 0.013867 -15.491 < 2e-16 ***
age20-29 years:capitalNonCapitalCity 0.196787 0.033521 5.871 4.34e-09 ***
age30-39 years:capitalNonCapitalCity 0.208419 0.043342 4.809 1.52e-06 ***
age40-49 years:capitalNonCapitalCity 0.191005 0.040531 4.713 2.45e-06 ***
age50-59 years:capitalNonCapitalCity 0.233079 0.046064 5.060 4.19e-07 ***
sexMale:capitalNonCapitalCity 0.065269 0.027410 2.381 0.0173 *
ingpNon-Indigenous:capitalNonCapitalCity -0.501182 0.014860 -33.727 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
9.3.5 Model E - Random Intercept, Random Effects (age, sex, education, ingp)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: binomial ( logit )
Formula: unemployment/labour_force ~ age + sex + education + ingp + IEO_Decile +
remoteness_index + capital + age * sex + age * education +
age * ingp + sex * education + sex * ingp + education * ingp +
age * capital + sex * capital + ingp * capital + (1 + age + sex + education + ingp | sa4)
Data: use_dataset
Weights: labour_force
Control: glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e+05))
AIC BIC logLik deviance df.resid
31187.7 31607.0 -15525.9 31051.7 3449
Scaled residuals:
Min 1Q Median 3Q Max
-5.2163 -1.1270 -0.1702 0.9912 9.5438
Random effects:
Groups Name Variance Std.Dev. Corr
sa4 (Intercept) 0.20943 0.4576
age20-29 years 0.02058 0.1434 -0.09
age30-39 years 0.03809 0.1952 0.11 0.84
age40-49 years 0.03134 0.1770 0.19 0.69 0.89
age50-59 years 0.03921 0.1980 0.20 0.58 0.77 0.95
sexMale 0.01195 0.1093 0.15 0.23 0.16 0.15 0.10
educationhs_not_finished 0.02020 0.1421 0.31 0.41 0.48 0.39 0.31 0.07
ingpNon-Indigenous 0.24842 0.4984 -0.87 -0.14 -0.40 -0.46 -0.45 -0.13 -0.54
Number of obs: 3517, groups: sa4, 88
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.654390 0.087954 -18.810 < 2e-16 ***
age20-29 years -0.659990 0.031130 -21.201 < 2e-16 ***
age30-39 years -1.168217 0.038900 -30.032 < 2e-16 ***
age40-49 years -1.315311 0.037607 -34.975 < 2e-16 ***
age50-59 years -1.670063 0.043276 -38.591 < 2e-16 ***
sexMale 0.359199 0.024296 14.784 < 2e-16 ***
educationhs_not_finished 0.575686 0.021542 26.724 < 2e-16 ***
ingpNon-Indigenous -0.105131 0.082995 -1.267 0.20526
IEO_Decile -0.166084 0.031297 -5.307 1.12e-07 ***
remoteness_index -0.236947 0.037339 -6.346 2.21e-10 ***
capitalNonCapitalCity 0.144356 0.124551 1.159 0.24645
age20-29 years:sexMale -0.154010 0.007873 -19.562 < 2e-16 ***
age30-39 years:sexMale -0.483027 0.008504 -56.798 < 2e-16 ***
age40-49 years:sexMale -0.362949 0.008561 -42.395 < 2e-16 ***
age50-59 years:sexMale -0.087697 0.008871 -9.886 < 2e-16 ***
age20-29 years:educationhs_not_finished 0.239815 0.008312 28.850 < 2e-16 ***
age30-39 years:educationhs_not_finished 0.436987 0.008994 48.588 < 2e-16 ***
age40-49 years:educationhs_not_finished 0.149333 0.008892 16.793 < 2e-16 ***
age50-59 years:educationhs_not_finished -0.101220 0.009249 -10.944 < 2e-16 ***
age20-29 years:ingpNon-Indigenous -0.204022 0.019090 -10.687 < 2e-16 ***
age30-39 years:ingpNon-Indigenous -0.173759 0.021770 -7.982 1.44e-15 ***
age40-49 years:ingpNon-Indigenous -0.123377 0.022371 -5.515 3.49e-08 ***
age50-59 years:ingpNon-Indigenous 0.154496 0.026215 5.893 3.78e-09 ***
sexMale:educationhs_not_finished -0.151554 0.005471 -27.703 < 2e-16 ***
sexMale:ingpNon-Indigenous -0.092894 0.013545 -6.858 6.98e-12 ***
educationhs_not_finished:ingpNon-Indigenous -0.194102 0.014531 -13.358 < 2e-16 ***
age20-29 years:capitalNonCapitalCity 0.184684 0.034096 5.417 6.08e-08 ***
age30-39 years:capitalNonCapitalCity 0.198511 0.043881 4.524 6.07e-06 ***
age40-49 years:capitalNonCapitalCity 0.187351 0.041831 4.479 7.51e-06 ***
age50-59 years:capitalNonCapitalCity 0.223410 0.047952 4.659 3.18e-06 ***
sexMale:capitalNonCapitalCity 0.064439 0.027744 2.323 0.02020 *
ingpNon-Indigenous:capitalNonCapitalCity -0.349468 0.106553 -3.280 0.00104 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
9.4 Final Model Additional Results
9.4.1 Random Effects Values
| AS4 | (Intercept) | age20-29 years | age30-39 years | age40-49 years | age50-59 years | sexMale | educationhs-not-finished | ingpNon-Indigenous |
|---|---|---|---|---|---|---|---|---|
| Sydney - Northern Beaches | -1.3600178 | -0.1944590 | -0.1423744 | 0.0001002 | 0.0201977 | -0.0877179 | -0.1480407 | 1.0800456 |
| Melbourne - Inner South | -0.7868524 | 0.0635811 | -0.1711872 | -0.1563354 | -0.1286678 | -0.0019024 | -0.0703106 | 0.9567988 |
| Melbourne - Inner East | -0.7392526 | 0.0463022 | -0.0931416 | -0.1707941 | -0.2214768 | 0.0383447 | -0.1712890 | 1.0976614 |
| Sydney - Sutherland | -0.7052567 | -0.1391774 | -0.1711416 | -0.0755043 | -0.0142096 | -0.1079666 | 0.0283826 | 0.2904409 |
| Geelong | -0.6751825 | -0.0263004 | -0.1250327 | -0.1114924 | -0.1626974 | -0.0178550 | 0.0363829 | 0.6703276 |
| Hume | -0.6327012 | 0.1349869 | 0.1512263 | 0.1353106 | 0.1220942 | -0.1772732 | 0.1445244 | 0.3525700 |
| Sydney - Inner West | -0.6255845 | 0.1354196 | -0.0286202 | 0.0209724 | 0.1217146 | -0.0284468 | -0.1135498 | 0.7009712 |
| Ballarat | -0.6018200 | 0.1969238 | 0.0750506 | 0.0077460 | -0.0430063 | 0.0699516 | 0.0396380 | 0.6505785 |
| Mornington Peninsula | -0.5241393 | -0.1479350 | -0.1536839 | -0.0913826 | -0.0606440 | -0.1624588 | 0.0230023 | 0.3840902 |
| Sydney - Baulkham Hills and Hawkesbury | -0.5162790 | -0.1697069 | -0.1201590 | -0.1305577 | -0.1244284 | -0.0765164 | -0.4693607 | 0.4836564 |
| South East | -0.5002942 | -0.1065337 | -0.1051841 | 0.0339804 | 0.1004941 | 0.0853487 | -0.3825284 | 0.7481213 |
| Sydney - North Sydney and Hornsby | -0.4617809 | -0.2014918 | -0.0750760 | 0.0594749 | 0.0804684 | -0.0738909 | -0.1894052 | 0.5340866 |
| Melbourne - Outer East | -0.4568530 | -0.1426467 | -0.1467225 | -0.1292705 | -0.1614721 | -0.0412965 | -0.1061903 | 0.4612457 |
| Sydney - Ryde | -0.4187982 | 0.1029729 | 0.0260989 | -0.0591519 | -0.0251819 | -0.0013769 | -0.2650963 | 0.6785165 |
| West and North West | -0.3733382 | 0.0760796 | 0.0595314 | 0.1427976 | 0.1707309 | 0.2233196 | -0.2512935 | 0.4406607 |
| Central Coast | -0.3563686 | -0.1252555 | -0.1791573 | -0.1476546 | -0.1148859 | -0.0827178 | 0.0508898 | 0.2983085 |
| Warrnambool and South West | -0.3169516 | 0.1617987 | 0.2124886 | 0.1295025 | 0.1055748 | -0.0756329 | 0.0801544 | -0.0972506 |
| Melbourne - North West | -0.3116465 | -0.0549785 | -0.0780367 | -0.1136141 | -0.1983324 | -0.0394344 | -0.0751513 | 0.4764697 |
| Bunbury | -0.2990137 | 0.1882435 | 0.2896245 | 0.2748765 | 0.3125570 | 0.0529596 | 0.0845887 | -0.0038169 |
| Southern Highlands and Shoalhaven | -0.2816488 | 0.1098882 | 0.2026671 | 0.1713578 | 0.1866369 | 0.0000876 | 0.0577873 | -0.0924434 |
| Gold Coast | -0.2786703 | -0.1940122 | -0.2402180 | -0.1684697 | -0.1062953 | -0.0758940 | -0.0847490 | 0.4672726 |
| Hunter Valley exc Newcastle | -0.2601732 | -0.0963315 | -0.1003616 | -0.0685747 | -0.0781709 | -0.1091986 | 0.0190353 | 0.1977810 |
| Shepparton | -0.2490580 | 0.1216818 | 0.1528819 | -0.0089279 | -0.1532630 | -0.0574544 | 0.1803291 | -0.0836643 |
| Sydney - Inner South West | -0.2481510 | 0.0520795 | 0.0492869 | 0.0952271 | 0.1538047 | -0.0594887 | -0.0904107 | 0.3596070 |
| Sydney - Outer West and Blue Mountains | -0.2413170 | -0.0538345 | -0.0473756 | -0.0481931 | -0.0161416 | -0.1282986 | -0.1068767 | -0.0837959 |
| Sydney - South West | -0.2173141 | -0.1020883 | 0.0160451 | 0.0768114 | 0.1372332 | -0.0783602 | -0.0742050 | 0.2933656 |
| Riverina | -0.2032251 | 0.1028692 | 0.1645470 | 0.0527820 | 0.0124169 | -0.0700414 | 0.0912099 | -0.0674051 |
| Richmond - Tweed | -0.1997892 | 0.1365570 | 0.1143957 | 0.1691713 | 0.1937746 | 0.1127489 | -0.0453046 | 0.0754958 |
| Newcastle and Lake Macquarie | -0.1987085 | -0.1856538 | -0.2431830 | -0.2525147 | -0.3019800 | 0.1490559 | 0.0541936 | 0.1487360 |
| South Australia - South East | -0.1939293 | 0.1833448 | 0.2216073 | 0.2013097 | 0.2153837 | 0.0403067 | 0.1428776 | -0.1567654 |
| Hobart | -0.1775058 | -0.0758393 | -0.2277309 | -0.3699948 | -0.4828243 | 0.2626712 | -0.0892555 | 0.3757048 |
| Capital Region | -0.1645150 | 0.3492210 | 0.4146935 | 0.3029353 | 0.3399013 | 0.0769020 | 0.1849917 | -0.5013153 |
| Brisbane - West | -0.1578081 | 0.0439103 | -0.1670533 | -0.2446003 | -0.2934055 | 0.0627455 | -0.1507207 | 0.5443319 |
| Latrobe - Gippsland | -0.0943652 | 0.1821787 | 0.2103873 | 0.1949021 | 0.1919446 | 0.0631109 | 0.0936063 | -0.0943012 |
| Sydney - Parramatta | -0.0892663 | 0.1197948 | 0.2294411 | 0.1544475 | 0.1531386 | -0.0968671 | -0.0586344 | 0.2687233 |
| Sydney - Outer South West | -0.0808078 | -0.0390485 | -0.0128866 | -0.0386295 | -0.0263843 | -0.2221264 | -0.1960274 | 0.0382211 |
| Illawarra | -0.0773471 | -0.2027914 | -0.4168171 | -0.4020017 | -0.4306703 | -0.0119836 | -0.1575447 | 0.3017666 |
| Mid North Coast | -0.0749366 | 0.2271236 | 0.2534744 | 0.2426135 | 0.3107314 | 0.1388683 | 0.0518491 | -0.1377904 |
| Brisbane - East | -0.0638361 | -0.0054196 | -0.0964216 | -0.1214340 | -0.0336487 | 0.0695583 | 0.0418173 | -0.1012224 |
| Bendigo | -0.0609582 | 0.0273151 | -0.0159906 | -0.0796278 | -0.1272876 | 0.0088399 | 0.0521780 | 0.1161961 |
| Central West | -0.0589801 | 0.0662517 | 0.1080431 | 0.0914524 | 0.0882144 | -0.0203726 | 0.1805270 | -0.1549772 |
| Sunshine Coast | -0.0468893 | -0.0710899 | -0.1246274 | -0.0686248 | 0.0308850 | -0.0549942 | -0.0496832 | 0.1578608 |
| Australian Capital Territory | -0.0336385 | -0.1402627 | -0.2070098 | -0.2707794 | -0.2346601 | 0.0388021 | 0.0606054 | -0.0165283 |
| Adelaide - South | -0.0261125 | -0.0065157 | -0.0444339 | -0.1168054 | -0.1483833 | 0.1915729 | 0.0876126 | -0.0313903 |
| Moreton Bay - North | -0.0083706 | 0.0344433 | -0.0263910 | -0.1085615 | -0.1407134 | -0.0974572 | 0.0546962 | 0.0435094 |
| Logan - Beaudesert | -0.0070522 | -0.1581670 | -0.2332150 | -0.3300913 | -0.4008268 | -0.1819834 | 0.0359073 | 0.2551297 |
| Moreton Bay - South | 0.0189218 | -0.3123523 | -0.4507463 | -0.4540830 | -0.4582261 | -0.1182146 | 0.0596619 | 0.2517878 |
| Mandurah | 0.0245819 | -0.0090396 | 0.1009111 | 0.0980630 | 0.0724950 | -0.0401484 | 0.0464739 | 0.0041192 |
| Melbourne - West | 0.0291658 | 0.0501359 | 0.0000521 | 0.0255012 | 0.0393750 | -0.1654650 | -0.0046378 | 0.1670459 |
| Murray | 0.0521736 | 0.1659516 | 0.2225174 | 0.1529055 | 0.1027471 | -0.0561037 | 0.1150757 | -0.3143181 |
| Adelaide - West | 0.0610071 | 0.1156947 | 0.1433109 | 0.0973350 | 0.0632708 | 0.1863763 | 0.0786192 | -0.1686846 |
| Melbourne - South East | 0.0693007 | -0.0593079 | -0.1296192 | -0.1757146 | -0.1552100 | -0.1603155 | -0.2067787 | 0.2265140 |
| Brisbane Inner City | 0.0725460 | -0.0560164 | -0.0141924 | 0.0207439 | 0.0076765 | 0.0976694 | -0.0702210 | 0.1072779 |
| Darling Downs - Maranoa | 0.0833210 | 0.2383924 | 0.2388806 | 0.1713081 | 0.1544931 | -0.1567960 | 0.2241831 | -0.3314104 |
| Adelaide - North | 0.0842095 | 0.0543792 | 0.0443258 | -0.0051881 | -0.1008824 | 0.1563139 | 0.0981745 | -0.1744672 |
| Melbourne - North East | 0.1093993 | -0.0212606 | -0.0898507 | -0.1520447 | -0.2058881 | -0.0207431 | -0.1975457 | 0.0914138 |
| Coffs Harbour - Grafton | 0.1222782 | 0.1051295 | 0.1225410 | 0.0689576 | 0.1124084 | 0.0818951 | 0.0566889 | -0.0920297 |
| Far West and Orana | 0.1362365 | 0.1288822 | 0.3064535 | 0.2747315 | 0.2492753 | 0.0736911 | 0.2538665 | -0.5651718 |
| Wide Bay | 0.1388287 | 0.2117239 | 0.2247934 | 0.1311156 | 0.1830806 | 0.1113868 | 0.1682748 | -0.1199439 |
| Launceston and North East | 0.1410398 | -0.0775011 | -0.1976579 | -0.1272287 | -0.1058787 | 0.1568761 | -0.2465647 | 0.3006108 |
| Central Queensland | 0.1649181 | 0.1527057 | 0.2636173 | 0.2375103 | 0.2210377 | 0.0247878 | 0.1036642 | -0.2334495 |
| Sydney - Blacktown | 0.1723783 | -0.0227536 | -0.0095176 | -0.0249557 | 0.0104743 | -0.1990617 | -0.0690032 | -0.1621308 |
| Adelaide - Central and Hills | 0.1803248 | 0.0992611 | 0.0626532 | -0.0805843 | -0.2285324 | 0.0997319 | -0.1275975 | 0.0257467 |
| North West | 0.2023868 | 0.1472526 | 0.2751321 | 0.1373022 | 0.0371632 | 0.0569113 | 0.1931242 | -0.4040034 |
| Barossa - Yorke - Mid North | 0.2105210 | 0.1125799 | 0.0830572 | 0.0617567 | 0.0428358 | 0.0319875 | 0.1922620 | -0.3537957 |
| Brisbane - North | 0.2339809 | -0.0723615 | -0.0430509 | -0.0053041 | 0.0142466 | 0.1161644 | 0.0864764 | -0.3007150 |
| Ipswich | 0.2471397 | -0.2088020 | -0.2412550 | -0.2986585 | -0.3808323 | -0.1161336 | 0.1147795 | 0.0670222 |
| Brisbane - South | 0.2505241 | 0.0917738 | -0.0405335 | -0.0800536 | -0.1337967 | 0.0583896 | 0.0223248 | -0.0137205 |
| Toowoomba | 0.2554054 | -0.0194840 | 0.0146952 | -0.1471277 | -0.2789266 | 0.0030061 | 0.0196428 | -0.0764704 |
| Perth - South West | 0.2590236 | 0.0176310 | 0.1611172 | 0.1936857 | 0.1830849 | 0.0710297 | 0.0303307 | -0.2695565 |
| Melbourne - Inner | 0.2661977 | -0.3357054 | -0.5948466 | -0.3897029 | -0.3944398 | 0.0206133 | -0.1162458 | 0.4246775 |
| Mackay - Isaac - Whitsunday | 0.3216783 | 0.0379802 | 0.0932064 | 0.1443196 | 0.2046647 | -0.1622533 | 0.0892374 | -0.2057717 |
| Sydney - Eastern Suburbs | 0.3326114 | -0.2593740 | -0.5141560 | -0.3028668 | -0.2366593 | 0.0140157 | -0.2060630 | -0.0341787 |
| New England and North West | 0.3329626 | 0.1366664 | 0.1344509 | 0.1057903 | 0.0949215 | -0.0989056 | 0.0906297 | -0.4531587 |
| Perth - Inner | 0.3883132 | -0.0262698 | 0.0683888 | 0.1045972 | 0.0339882 | 0.0653207 | -0.0379369 | -0.1252128 |
| Perth - North East | 0.4201859 | -0.0339494 | 0.0578539 | 0.0723829 | 0.0881474 | 0.0570727 | -0.0415985 | -0.4210112 |
| Western Australia - Wheat Belt | 0.4543680 | 0.0547873 | 0.1896824 | 0.2094023 | 0.2084607 | 0.0532615 | 0.1902048 | -0.7316049 |
| Perth - North West | 0.4642840 | -0.0825643 | 0.0662419 | 0.1010610 | 0.0774315 | 0.0729424 | -0.0168265 | -0.3817261 |
| Darwin | 0.5254543 | -0.1427314 | -0.0158008 | 0.0611453 | 0.1061262 | -0.0596672 | -0.0255519 | -0.5064669 |
| Perth - South East | 0.5462605 | 0.0468800 | 0.1826525 | 0.2057684 | 0.1690583 | 0.0134394 | -0.1385359 | -0.4398957 |
| Sydney - City and Inner South | 0.5516886 | -0.2194195 | -0.3920704 | -0.1164411 | 0.0375688 | -0.0151765 | -0.0672987 | -0.2176187 |
| South Australia - Outback | 0.6201709 | 0.0656942 | 0.1428943 | 0.1990999 | 0.2182408 | 0.1745461 | 0.1461804 | -0.5798892 |
| Cairns | 0.7775748 | 0.1168538 | 0.1601262 | 0.2402312 | 0.3121241 | 0.1447983 | 0.0959393 | -0.6980668 |
| Townsville | 0.9034626 | -0.0786256 | 0.0518741 | 0.0173679 | -0.0056468 | 0.0277713 | 0.1602889 | -0.4768637 |
| Western Australia - Outback (South) | 0.9594223 | 0.0601852 | 0.1175323 | 0.2396607 | 0.3119757 | 0.0780614 | 0.0843699 | -0.8669404 |
| Queensland - Outback | 0.9687057 | -0.0625477 | 0.0795368 | 0.1531526 | 0.2237109 | -0.0372335 | 0.2219915 | -1.2280866 |
| Western Australia - Outback (North) | 1.2031366 | -0.2140361 | -0.0946072 | 0.0471548 | 0.1473626 | -0.2040884 | -0.0567727 | -1.3063389 |
| Northern Territory - Outback | 1.3612164 | -0.1245471 | 0.0668770 | 0.1201455 | 0.1510600 | 0.1160209 | 0.1543394 | -1.9563254 |
9.4.2 Random Effects Standard Deviation
| AS4 | (Intercept) | age20-29 years | age30-39 years | age40-49 years | age50-59 years | sexMale | educationhs-not-finished | ingpNon-Indigenous |
|---|---|---|---|---|---|---|---|---|
| Sydney - Northern Beaches | 0.2256716 | 0.0469049 | 0.0515394 | 0.0441323 | 0.0492355 | 0.0303800 | 0.0354323 | 0.2241622 |
| Melbourne - Inner South | 0.2086273 | 0.0322783 | 0.0364716 | 0.0326243 | 0.0354417 | 0.0201608 | 0.0251969 | 0.2073206 |
| Melbourne - Inner East | 0.2366047 | 0.0313960 | 0.0361750 | 0.0328707 | 0.0358754 | 0.0198853 | 0.0276202 | 0.2357979 |
| Sydney - Sutherland | 0.1468027 | 0.0460578 | 0.0524888 | 0.0445016 | 0.0498986 | 0.0320711 | 0.0343094 | 0.1426348 |
| Geelong | 0.1180421 | 0.0354329 | 0.0402264 | 0.0356701 | 0.0396599 | 0.0239519 | 0.0251807 | 0.1147124 |
| Hume | 0.1214034 | 0.0489007 | 0.0554699 | 0.0477401 | 0.0524334 | 0.0331280 | 0.0350697 | 0.1150174 |
| Sydney - Inner West | 0.1500097 | 0.0402086 | 0.0444866 | 0.0414301 | 0.0452369 | 0.0224749 | 0.0299141 | 0.1460445 |
| Ballarat | 0.1329848 | 0.0447440 | 0.0515092 | 0.0451183 | 0.0503583 | 0.0302136 | 0.0315948 | 0.1273223 |
| Mornington Peninsula | 0.1095226 | 0.0358233 | 0.0402195 | 0.0350971 | 0.0384215 | 0.0241886 | 0.0250611 | 0.1057143 |
| Sydney - Baulkham Hills and Hawkesbury | 0.1549492 | 0.0415002 | 0.0470211 | 0.0404061 | 0.0455242 | 0.0285002 | 0.0330476 | 0.1524507 |
| South East | 0.1516107 | 0.0866833 | 0.1038203 | 0.0882399 | 0.0982070 | 0.0589255 | 0.0631601 | 0.1337010 |
| Sydney - North Sydney and Hornsby | 0.1812429 | 0.0367325 | 0.0389326 | 0.0356316 | 0.0385792 | 0.0215213 | 0.0304998 | 0.1792220 |
| Melbourne - Outer East | 0.1203103 | 0.0271389 | 0.0303279 | 0.0273917 | 0.0299788 | 0.0183809 | 0.0196406 | 0.1184534 |
| Sydney - Ryde | 0.2168265 | 0.0449940 | 0.0502338 | 0.0460359 | 0.0510576 | 0.0274529 | 0.0384394 | 0.2144930 |
| West and North West | 0.0775453 | 0.0509705 | 0.0582330 | 0.0504065 | 0.0561480 | 0.0345512 | 0.0375918 | 0.0617218 |
| Central Coast | 0.0536075 | 0.0312678 | 0.0351203 | 0.0315098 | 0.0343043 | 0.0214121 | 0.0222190 | 0.0465725 |
| Warrnambool and South West | 0.1304348 | 0.0558340 | 0.0643019 | 0.0550741 | 0.0614694 | 0.0390608 | 0.0415567 | 0.1218603 |
| Melbourne - North West | 0.1209681 | 0.0292235 | 0.0322206 | 0.0302966 | 0.0343408 | 0.0190955 | 0.0205686 | 0.1185083 |
| Bunbury | 0.0857717 | 0.0437626 | 0.0470870 | 0.0422613 | 0.0464914 | 0.0280709 | 0.0290555 | 0.0766801 |
| Southern Highlands and Shoalhaven | 0.0816208 | 0.0522755 | 0.0594206 | 0.0512743 | 0.0567475 | 0.0356579 | 0.0375237 | 0.0670886 |
| Gold Coast | 0.0544969 | 0.0228038 | 0.0253366 | 0.0236720 | 0.0256064 | 0.0153882 | 0.0160259 | 0.0515623 |
| Hunter Valley exc Newcastle | 0.0520845 | 0.0334582 | 0.0373426 | 0.0339670 | 0.0380229 | 0.0230214 | 0.0244785 | 0.0426506 |
| Shepparton | 0.1028391 | 0.0508302 | 0.0576801 | 0.0502339 | 0.0575517 | 0.0360049 | 0.0389508 | 0.0913590 |
| Sydney - Inner South West | 0.0899849 | 0.0256572 | 0.0279774 | 0.0269437 | 0.0288311 | 0.0153394 | 0.0175310 | 0.0874507 |
| Sydney - Outer West and Blue Mountains | 0.0586680 | 0.0326623 | 0.0368130 | 0.0338136 | 0.0380975 | 0.0230544 | 0.0236937 | 0.0521905 |
| Sydney - South West | 0.0811712 | 0.0278788 | 0.0305322 | 0.0286986 | 0.0312385 | 0.0177105 | 0.0186678 | 0.0780824 |
| Riverina | 0.0745728 | 0.0477682 | 0.0545255 | 0.0487829 | 0.0553660 | 0.0334992 | 0.0354465 | 0.0606911 |
| Richmond - Tweed | 0.0622873 | 0.0393403 | 0.0433513 | 0.0380540 | 0.0406469 | 0.0253390 | 0.0261564 | 0.0522591 |
| Newcastle and Lake Macquarie | 0.0481482 | 0.0280646 | 0.0318410 | 0.0294373 | 0.0326666 | 0.0195187 | 0.0201901 | 0.0413117 |
| South Australia - South East | 0.0875727 | 0.0458217 | 0.0510285 | 0.0449795 | 0.0489685 | 0.0301223 | 0.0319630 | 0.0771533 |
| Hobart | 0.0689729 | 0.0371838 | 0.0423243 | 0.0387282 | 0.0440729 | 0.0258354 | 0.0267132 | 0.0598134 |
| Capital Region | 0.0727929 | 0.0447500 | 0.0499605 | 0.0439557 | 0.0483442 | 0.0299863 | 0.0312007 | 0.0611960 |
| Brisbane - West | 0.1352761 | 0.0362386 | 0.0431223 | 0.0380379 | 0.0442947 | 0.0262741 | 0.0342337 | 0.1329307 |
| Latrobe - Gippsland | 0.0832136 | 0.0371862 | 0.0411653 | 0.0368545 | 0.0393848 | 0.0240486 | 0.0253664 | 0.0760935 |
| Sydney - Parramatta | 0.0948854 | 0.0289294 | 0.0304350 | 0.0304295 | 0.0333732 | 0.0163348 | 0.0186995 | 0.0912926 |
| Sydney - Outer South West | 0.0641231 | 0.0335453 | 0.0374312 | 0.0347042 | 0.0395860 | 0.0233547 | 0.0240668 | 0.0582049 |
| Illawarra | 0.0627082 | 0.0308878 | 0.0360655 | 0.0326938 | 0.0368623 | 0.0220838 | 0.0228470 | 0.0571420 |
| Mid North Coast | 0.0589684 | 0.0420058 | 0.0465580 | 0.0407030 | 0.0434201 | 0.0270556 | 0.0288241 | 0.0456364 |
| Brisbane - East | 0.0759217 | 0.0359379 | 0.0405363 | 0.0355779 | 0.0395151 | 0.0255130 | 0.0265588 | 0.0710720 |
| Bendigo | 0.1167441 | 0.0466095 | 0.0534867 | 0.0466724 | 0.0522857 | 0.0321961 | 0.0336896 | 0.1092190 |
| Central West | 0.0585612 | 0.0416453 | 0.0466182 | 0.0416507 | 0.0461565 | 0.0281045 | 0.0297648 | 0.0451305 |
| Sunshine Coast | 0.0629232 | 0.0302889 | 0.0336644 | 0.0295708 | 0.0315223 | 0.0202811 | 0.0209616 | 0.0584560 |
| Australian Capital Territory | 0.0720463 | 0.0299648 | 0.0338948 | 0.0320779 | 0.0363560 | 0.0209233 | 0.0242657 | 0.0674891 |
| Adelaide - South | 0.0830518 | 0.0287172 | 0.0317852 | 0.0297831 | 0.0325370 | 0.0195464 | 0.0201209 | 0.0795326 |
| Moreton Bay - North | 0.0569202 | 0.0321802 | 0.0361420 | 0.0329029 | 0.0369992 | 0.0224096 | 0.0230274 | 0.0511395 |
| Logan - Beaudesert | 0.0521286 | 0.0269115 | 0.0300302 | 0.0288345 | 0.0329718 | 0.0188779 | 0.0194353 | 0.0476282 |
| Moreton Bay - South | 0.0824004 | 0.0357261 | 0.0406403 | 0.0362737 | 0.0423633 | 0.0259596 | 0.0272857 | 0.0776847 |
| Mandurah | 0.1024272 | 0.0479828 | 0.0533998 | 0.0469203 | 0.0526149 | 0.0321386 | 0.0332992 | 0.0944167 |
| Melbourne - West | 0.0721730 | 0.0207398 | 0.0220501 | 0.0217253 | 0.0239272 | 0.0125373 | 0.0135833 | 0.0699207 |
| Murray | 0.0980359 | 0.0544230 | 0.0619971 | 0.0539824 | 0.0605761 | 0.0372918 | 0.0397018 | 0.0837206 |
| Adelaide - West | 0.0840870 | 0.0359008 | 0.0395517 | 0.0366196 | 0.0404745 | 0.0227382 | 0.0240798 | 0.0781095 |
| Melbourne - South East | 0.0923716 | 0.0199621 | 0.0218850 | 0.0214741 | 0.0236256 | 0.0129608 | 0.0141085 | 0.0909422 |
| Brisbane Inner City | 0.0971106 | 0.0344027 | 0.0389527 | 0.0370395 | 0.0424314 | 0.0218555 | 0.0294934 | 0.0926042 |
| Darling Downs - Maranoa | 0.0785077 | 0.0530914 | 0.0602599 | 0.0526746 | 0.0595066 | 0.0365698 | 0.0388629 | 0.0632652 |
| Adelaide - North | 0.0557659 | 0.0246284 | 0.0266864 | 0.0260322 | 0.0288484 | 0.0160722 | 0.0163685 | 0.0514440 |
| Melbourne - North East | 0.0886391 | 0.0259695 | 0.0284627 | 0.0272244 | 0.0304777 | 0.0167860 | 0.0189384 | 0.0856854 |
| Coffs Harbour - Grafton | 0.0711772 | 0.0475131 | 0.0531990 | 0.0463726 | 0.0507926 | 0.0319394 | 0.0337018 | 0.0574185 |
| Far West and Orana | 0.0663484 | 0.0548768 | 0.0618826 | 0.0547819 | 0.0615754 | 0.0359215 | 0.0396765 | 0.0413966 |
| Wide Bay | 0.0487323 | 0.0320455 | 0.0348705 | 0.0316949 | 0.0336891 | 0.0207799 | 0.0217989 | 0.0411276 |
| Launceston and North East | 0.0859485 | 0.0448077 | 0.0518136 | 0.0448073 | 0.0497191 | 0.0306924 | 0.0319274 | 0.0764824 |
| Central Queensland | 0.0493783 | 0.0348106 | 0.0377625 | 0.0348854 | 0.0384286 | 0.0226189 | 0.0233706 | 0.0400504 |
| Sydney - Blacktown | 0.0590544 | 0.0298324 | 0.0321822 | 0.0308275 | 0.0349151 | 0.0195828 | 0.0208661 | 0.0533289 |
| Adelaide - Central and Hills | 0.1192940 | 0.0337752 | 0.0377744 | 0.0347019 | 0.0388092 | 0.0220790 | 0.0257083 | 0.1159487 |
| North West | 0.0917915 | 0.0489650 | 0.0553450 | 0.0484450 | 0.0539048 | 0.0332473 | 0.0357474 | 0.0795743 |
| Barossa - Yorke - Mid North | 0.1106008 | 0.0554036 | 0.0630386 | 0.0544294 | 0.0598820 | 0.0372246 | 0.0396457 | 0.0992008 |
| Brisbane - North | 0.0780780 | 0.0378332 | 0.0420886 | 0.0382363 | 0.0434172 | 0.0254849 | 0.0282885 | 0.0716341 |
| Ipswich | 0.0475622 | 0.0271930 | 0.0300767 | 0.0286901 | 0.0328065 | 0.0187823 | 0.0194325 | 0.0423751 |
| Brisbane - South | 0.0762209 | 0.0279241 | 0.0314131 | 0.0302506 | 0.0344835 | 0.0184331 | 0.0225238 | 0.0726774 |
| Toowoomba | 0.0718700 | 0.0410332 | 0.0467056 | 0.0422043 | 0.0493028 | 0.0295572 | 0.0310939 | 0.0631058 |
| Perth - South West | 0.0611434 | 0.0259638 | 0.0278569 | 0.0263350 | 0.0289028 | 0.0167144 | 0.0171486 | 0.0568783 |
| Melbourne - Inner | 0.1054754 | 0.0252369 | 0.0280198 | 0.0279244 | 0.0311598 | 0.0142234 | 0.0201330 | 0.1031358 |
| Mackay - Isaac - Whitsunday | 0.0594576 | 0.0407609 | 0.0447984 | 0.0404853 | 0.0447078 | 0.0264405 | 0.0276297 | 0.0487990 |
| Sydney - Eastern Suburbs | 0.1194127 | 0.0427711 | 0.0482173 | 0.0441773 | 0.0495548 | 0.0255863 | 0.0358381 | 0.1137951 |
| New England and North West | 0.0560107 | 0.0437969 | 0.0492745 | 0.0439530 | 0.0488976 | 0.0295841 | 0.0314148 | 0.0390452 |
| Perth - Inner | 0.1491564 | 0.0441626 | 0.0492126 | 0.0452462 | 0.0507363 | 0.0271671 | 0.0354570 | 0.1443834 |
| Perth - North East | 0.0667752 | 0.0325463 | 0.0355002 | 0.0330577 | 0.0367716 | 0.0212229 | 0.0218540 | 0.0606342 |
| Western Australia - Wheat Belt | 0.0844141 | 0.0557302 | 0.0619187 | 0.0538975 | 0.0590164 | 0.0359667 | 0.0375530 | 0.0664616 |
| Perth - North West | 0.0630614 | 0.0228777 | 0.0244871 | 0.0233894 | 0.0256473 | 0.0148320 | 0.0154863 | 0.0599031 |
| Darwin | 0.0715119 | 0.0514275 | 0.0576077 | 0.0519040 | 0.0602546 | 0.0346801 | 0.0362333 | 0.0543159 |
| Perth - South East | 0.0513661 | 0.0234317 | 0.0250366 | 0.0245927 | 0.0270033 | 0.0146319 | 0.0154184 | 0.0471200 |
| Sydney - City and Inner South | 0.0827217 | 0.0386104 | 0.0425408 | 0.0407628 | 0.0454639 | 0.0203672 | 0.0272697 | 0.0749264 |
| South Australia - Outback | 0.0809312 | 0.0595431 | 0.0678946 | 0.0589876 | 0.0657281 | 0.0386340 | 0.0416312 | 0.0550700 |
| Cairns | 0.0439531 | 0.0369253 | 0.0402121 | 0.0360878 | 0.0390527 | 0.0233270 | 0.0240172 | 0.0307867 |
| Townsville | 0.0439444 | 0.0324764 | 0.0360087 | 0.0332404 | 0.0371423 | 0.0221740 | 0.0230135 | 0.0346127 |
| Western Australia - Outback (South) | 0.0692522 | 0.0524842 | 0.0586290 | 0.0514518 | 0.0571172 | 0.0338000 | 0.0357934 | 0.0473424 |
| Queensland - Outback | 0.0685434 | 0.0615289 | 0.0696990 | 0.0617849 | 0.0704670 | 0.0396127 | 0.0427332 | 0.0439218 |
| Western Australia - Outback (North) | 0.0745273 | 0.0649203 | 0.0727398 | 0.0657756 | 0.0757019 | 0.0395106 | 0.0421213 | 0.0434161 |
| Northern Territory - Outback | 0.0709484 | 0.0604832 | 0.0669884 | 0.0609630 | 0.0706320 | 0.0361480 | 0.0431131 | 0.0498306 |
Direction of estimates here is defined as being positive if \(> 1\), negative if \(< 1\) and null if \(=1\)↩
Random effects for interaction terms are out of scope for this article↩
Classification metrics are intended to evaluate uniquely the goodness of fit. Prediction power on new data is out of scope for this article↩
For methodologies to calculate marginalised coefficients please refer to https://drizopoulos.github.io/GLMMadaptive/articles/Methods_MixMod.html#marginalized-coefficients↩
Coefficients isolated - not considering interactions↩