Background
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Question 1: What advice can be given to public health analysts to reduce sets of correlated public health data?
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Question 2: What advice can be given to aid decisions related to selecting predictors for importance?
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Question 3: What advice can be given to decide regarding the trade-off between predictive power and interpretability?
Methods
Data preparation
Minimum | Maximum | Mean |
SD
| |
---|---|---|---|---|
2014 Suicide (age-standardised rate per 100,000 - outcome measure) | 6.12 | 18.26 | 10.13 | 2.14 |
2013 Adult social care users who have as much social contact as they would like (% of adult social care users) | 35.40 | 54.40 | 43.89 | 3.98 |
2013 Adults in treatment at specialist alcohol misuse services (rate per 1000 population) | 0.67 | 6.19 | 2.40 | 1.08 |
2013 Adults in treatment at specialist drug misuse services (rate per 1000 population) | 1.69 | 16.07 | 5.59 | 2.48 |
2013 Alcohol-related hospital admission (female) (directly standardised rate per 100,000 female population) | 498.46 | 1386.28 | 903.85 | 175.78 |
2013 Alcohol-related hospital admission (male) (directly standardised rate per 100,000 male population) | 1011.15 | 2819.52 | 1805.52 | 344.23 |
2013 Alcohol-related hospital admission (directly standardised rate per 100,000 population) | 731.04 | 2069.94 | 1318.11 | 253.33 |
2013 Children in the youth justice system (rate per 1,000 aged 10–18) | 2.91 | 17.08 | 7.75 | 2.77 |
2013 Children leaving care (rate per 10,000 < 18 population) | 9.59 | 70.62 | 28.78 | 10.32 |
2013 Depression recorded prevalence (% of adults with a new diagnosis of depression who had a bio-psychosocial assessment) | 3.09 | 10.71 | 6.48 | 1.40 |
2013 Domestic abuse incidents (rate per 1,000 population) | 4.87 | 30.38 | 19.84 | 4.74 |
2013 Emergency hospital admissions for intentional self-harm (female) (directly age-standardised rate per 100,000 women) | 76.46 | 751.06 | 257.43 | 103.75 |
2013 Emergency hospital admissions for intentional self-harm (male) (directly age-standardised rate per 100,000 men) | 45.43 | 614.41 | 166.45 | 81.79 |
2013 Emergency hospital admissions for intentional self-harm (directly age-and-sex-standardised rate per 100,000) | 60.23 | 682.62 | 211.02 | 90.47 |
2013 Looked after children (rate per 10,000 < 18 population) | 19.83 | 153.29 | 64.91 | 25.10 |
2013 Self-reported well-being - high anxiety (% of people) | 9.61 | 29.71 | 20.27 | 2.75 |
2013 Severe mental illness recorded prevalence (% of practice register [all ages]) | 0.47 | 1.47 | 0.87 | 0.19 |
2013 Social care mental health clients receiving services (rate per 100,000 population) | 67.43 | 2331.12 | 387.68 | 299.36 |
2013 Statutory homelessness (rate per 1000 households) | 0.10 | 12.55 | 2.54 | 2.18 |
2013 Successful completion of alcohol treatment (% who do not represent within 6 months) | 15.13 | 67.59 | 37.40 | 8.79 |
2013 Successful completion of drug treatment - non-opiate users (% who do not represent within 6 months) | 7.08 | 59.72 | 36.95 | 8.60 |
2013 Successful completion of drug treatment - opiate users (% who do not represent within 6 months) | 3.52 | 15.79 | 8.15 | 2.41 |
2013 Unemployment (% of working-age population) | 3.70 | 14.50 | 7.99 | 2.52 |
2012 Adult carers who have as much social contact as they would like (18+ yrs) (% of 18+ carers) | 23.90 | 58.50 | 40.95 | 7.24 |
2012 Adult carers who have as much social contact as they would like (all ages) (% of adult carers) | 23.90 | 58.50 | 40.95 | 7.24 |
2011 Estimated prevalence of opiates and/or crack cocaine use (rate per 1,000 aged 15–64) | 2.93 | 20.76 | 9.13 | 3.79 |
2011 Long-term health problems or disability (% of people whose day-to-day activities are limited by their health or disability) | 11.20 | 25.57 | 17.68 | 3.26 |
2011 Marital breakup (% of adults whose current marital status is separated or divorced) | 7.73 | 16.30 | 11.67 | 1.24 |
2011 Older people living alone (% of households occupied by a single person aged 65 or over) | 2.29 | 7.57 | 5.12 | 1.06 |
2011 People living alone (% of all households occupied by a single person) Mental Health Service users with crisis plans: % of people in contact with services with a crisis plan in place (end of quarter snapshot) Older people | 8.02 | 23.42 | 13.03 | 2.20 |
2011 Self-reported well-being - low happiness (% of people with a low happiness score) | 6.55 | 17.68 | 10.98 | 2.09 |
Data analysis
Availability of data and materials
Results
Reducing the set of indicators
Multi-collinearity analysis
Variable # | Variable name | Transformationa | Tolerance | VIF | Decision |
---|---|---|---|---|---|
2 | Adult social-care users who have as much social contact as they would like | – | 0.64 | 1.57 | Keep |
3 | Adults in treatment at specialist alcohol misuse services | Log | 0.31 | 3.24 | Keep |
4 | Adults in treatment at specialist drug misuse services | Log | 0.08 | 12.66 | Remove |
5 | Alcohol-related hospital admission (female) | Log | 0.00 | 422.64 | Remove |
6 | Alcohol-related hospital admission (male) | Log | 0.00 | 1126.14 | Remove |
7 | Alcohol-related hospital admission (all) | Log | 0.00 | 2813.20 | Keep |
8 | Children in the youth justice system | Log | 0.36 | 2.74 | Keep |
9 | Children leaving care | Log | 0.22 | 4.56 | Keep |
10 | Depression recorded prevalence | – | 0.39 | 2.59 | Keep |
11 | Domestic abuse incidents | – | 0.49 | 2.03 | Keep |
12 | Emergency hospital admissions for intentional self-harm (female) | square root | 0.08 | 13.18 | Remove |
13 | Emergency hospital admissions for intentional self-harm (male) | square root | 0.06 | 17.53 | Remove |
14 | Emergency hospital admissions for intentional self-harm (all) | square root | 0.00 | 11956.59 | Keep |
15 | Looked after children | Log | 0.19 | 5.35 | Remove |
16 | Self-reported well-being - high anxiety | – | 0.65 | 1.53 | Keep |
17 | Severe mental illness recorded prevalence | Log | 0.30 | 3.30 | Keep |
18 | Social care mental health clients receiving services | Log | 0.75 | 1.34 | Keep |
19 | Statutory homelessness | Log | 0.41 | 2.41 | Keep |
20 | Successful completion of alcohol treatment | – | 0.45 | 2.22 | Keep |
21 | Successful completion of drug treatment - non-opiate users | – | 0.41 | 2.47 | Keep |
22 | Successful completion of drug treatment - opiate users | Log | 0.49 | 2.02 | Keep |
23 | Unemployment | Log | 0.16 | 6.40 | Keep |
24 | Adult carers who have as much social contact as they would like (18+ yrs) | – | 0.00 | infinity | Remove |
25 | Adult carers who have as much social contact as they would like (all ages) | – | 0.59 | 1.71 | Keep |
26 | Estimated prevalence of opiates and/or crack cocaine use | Log | 0.09 | 10.60 | Keep |
27 | Long-term health problems or disability | – | 0.08 | 12.84 | Remove |
28 | Marital breakup | – | 0.39 | 2.59 | Keep |
29 | Older people living alone | – | 0.08 | 11.77 | Remove |
30 | People living alone | Inverse | 0.23 | 4.36 | Keep |
31 | Self-reported well-being - low happiness | – | 0.36 | 2.77 | Keep |
Principal-component analysis
Component | |||
---|---|---|---|
1 | 2 | 3 | |
Unemployment |
0.87
| −0.19 | 0.01 |
Estimated prevalence of opiates and/or crack cocaine use |
0.86
| −0.04 | − 0.08 |
Alcohol-related hospital admission |
0.83
| 0.12 | 0.06 |
Children leaving care |
0.82
| − 0.09 | − 0.03 |
Severe mental illness recorded prevalence |
0.75
| −0.37 | − 0.03 |
Self-reported well-being - low happiness |
0.71
| 0.11 | 0.21 |
Children in the youth justice system |
0.69
| 0.07 | −0.15 |
Adults in treatment at specialist alcohol misuse services |
0.63
| 0.32 | −0.06 |
inverse People living alone |
−0.48
| −0.21 | 0.19 |
Self-reported well-being - high anxiety |
0.48
| −0.01 | 0.21 |
Domestic abuse incidents |
0.46
| 0.07 | −0.02 |
Emergency hospital admissions for intentional self-harm | 0.25 |
0.76
| 0.03 |
Depression recorded prevalence | 0.04 |
0.76
| −0.03 |
Statutory homelessness | 0.22 |
−0.76
| −0.04 |
Marital breakup | 0.20 |
0.61
| 0.10 |
Adult carers who have as much social contact as they would like (all ages) | −0.07 |
0.56
| 0.04 |
Adult social-care users who have as much social contact as they would like | 0.00 |
0.52
| −0.11 |
Successful completion of drug treatment - non-opiate users | 0.12 | 0.08 |
0.89
|
Successful completion of alcohol treatment | −0.04 | 0.18 |
0.80
|
Successful completion of drug treatment - opiate users | −0.09 | −0.27 |
0.64
|
Analysing the importance of predictors
Prediction research perspective: indicators as predictors
Source | Unstan-dardised coefficients | Stan-dardised coefficients |
F/t
|
p
| 95%-confidence interval for b | Correlations | Collinearity statistics | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
b
|
SE
| β | Lower limit | Upper limit |
r
|
sr
|
sr
2
| Tolerance | VIF | |||
Model 1 | 64.07 | < 0.001 | 0.30 | |||||||||
Constant | 1.77 | 0.07 | 26.75 | < 0.001 | 1.64 | 1.91 | ||||||
Emergency hospital admissions for intentional self-harm: alla | 0.04 | 0.00 | 0.55 | 8.00 | < 0.001 | 0.03 | 0.05 | 0.55 | 0.55 | 0.55 | 1.00 | 1.00 |
Model 2 | 17.04 | < 0.001 | 0.07 | |||||||||
Constant | 1.30 | 0.13 | 9.92 | < 0.001 | 1.04 | 1.56 | ||||||
Emergency hospital admissions for intentional self-harm: alla | 0.03 | 0.00 | 0.47 | 6.93 | < 0.001 | 0.02 | 0.04 | 0.55 | 0.45 | 0.21 | 0.92 | 1.09 |
Children leaving careb | 0.17 | 0.04 | 0.28 | 4.13 | < 0.001 | 0.09 | 0.25 | 0.41 | 0.27 | 0.07 | 0.92 | 1.09 |
Model 3 | 9.09 | < 0.001 | 0.04 | |||||||||
Constant | 1.29 | 0.13 | 10.06 | < 0.001 | 1.03 | 1.54 | ||||||
Emergency hospital admissions for intentional self-harm: alla | 0.02 | 0.01 | 0.36 | 4.63 | < 0.001 | 0.01 | 0.03 | 0.55 | 0.29 | 0.09 | 0.69 | 1.46 |
Children leaving careb | 0.21 | 0.04 | 0.36 | 5.07 | < 0.001 | 0.13 | 0.30 | 0.41 | 0.32 | 0.10 | 0.79 | 1.27 |
Statutory homelessnessb | −0.06 | 0.02 | −0.23 | −3.01 | .003 | −0.09 | − 0.02 | − 0.30 | − 0.19 | 0.04 | 0.71 | 1.41 |
Model 4 | 5.98 | < 0.001 | 0.02 | |||||||||
Constant | 1.30 | 0.13 | 10.36 | < 0.001 | 1.05 | 1.55 | ||||||
Emergency hospital admissions for intentional self-harm: alla | 0.03 | 0.01 | 0.40 | 5.13 | < 0.001 | 0.02 | 0.04 | 0.55 | 0.32 | 0.10 | 0.65 | 1.53 |
Children leaving careb | 0.26 | 0.04 | 0.43 | 5.69 | < 0.001 | 0.17 | 0.34 | 0.41 | 0.36 | 0.13 | 0.68 | 1.47 |
Statutory homelessnessb | −0.05 | 0.02 | −0.21 | −2.88 | .005 | −0.09 | − 0.02 | − 0.30 | − 0.18 | 0.03 | 0.71 | 1.41 |
Self-reported well-being - low happiness | −0.02 | 0.01 | −0.18 | −2.44 | .016 | −0.03 | 0.00 | 0.13 | −0.15 | 0.02 | 0.74 | 1.34 |
Prediction approach: principal components as predictors
Source | Unstan-dardised coefficients | Stan-dardised coefficients |
F/t
|
p
| 95%-confidence interval for b | Correlations | Collinearity statistics | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
b
|
SE
| β | Lower limit | Upper limit |
r
|
sr
|
sr
2
| Tolerance | VIF | |||
Model 1 | 44.70985 | < 0.001 | 0.23 | |||||||||
Constant | 2.29 | 0.01 | 155.42 | < 0.001 | 2.26 | 2.32 | ||||||
Component 2: relatedness dysfunction | 0.10 | 0.01 | 0.48 | 6.69 | < 0.001 | 0.07 | 0.13 | 0.48 | 0.48 | 0.48 | 1.00 | 1.00 |
Model 2 | 20.74 | < 0.001 | 0.10 | |||||||||
Constant | 2.29 | 0.01 | 165.53 | < 0.001 | 2.27 | 2.32 | ||||||
Component 2: relatedness dysfunction | 0.09 | 0.01 | 0.43 | 6.32 | < 0.001 | 0.06 | 0.12 | 0.48 | 0.43 | 0.18 | 0.98 | 1.03 |
Component 1: behavioural problems and mental illness | 0.06 | 0.01 | 0.31 | 4.55 | < 0.001 | 0.04 | 0.09 | 0.38 | 0.31 | 0.10 | 0.98 | 1.03 |
Source | Unstan-dardised coefficients | Stan-dardised coefficients |
F/t
|
p
| 95%-confidence interval for b | Correlations | Collinearity statistics | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
b
|
SE
| β | Lower limit | Upper limit |
r
|
sr
|
sr
2
| Tolerance | VIF | |||
Model 1 | 27.89 | < 0.001 | ||||||||||
Constant | 1.30 | 0.13 | 10.36 | < 0.001 | 1.05 | 1.55 | ||||||
Emergency hospital admissions for intentional self-harm: alla | 0.03 | 0.01 | 0.40 | 5.13 | < 0.001 | 0.02 | 0.04 | 0.55 | 0.32 | 0.10 | 0.65 | 1.53 |
Children leaving careb | 0.26 | 0.04 | 0.43 | 5.69 | < 0.001 | 0.17 | 0.34 | 0.41 | 0.36 | 0.13 | 0.68 | 1.47 |
Statutory homelessnessb | −0.05 | 0.02 | −0.21 | −2.88 | .005 | −0.09 | −0.02 | −0.30 | −0.18 | 0.03 | 0.71 | 1.41 |
Self-reported well-being - low happiness | −0.02 | 0.01 | −0.18 | −2.44 | .016 | − 0.03 | 0.00 | 0.13 | −0.15 | 0.02 | 0.74 | 1.34 |
23.61 | < 0.001 | 0.02 | ||||||||||
Constant | 1.33 | 0.13 | 10.63 | < 0.001 | 1.09 | 1.58 | ||||||
Emergency hospital admissions for intentional self-harm: alla | 0.03 | 0.01 | 0.39 | 5.12 | < 0.001 | 0.02 | 0.04 | 0.55 | 0.32 | 0.10 | 0.65 | 1.53 |
Children leaving careb | 0.24 | 0.04 | 0.41 | 5.49 | < 0.001 | 0.16 | 0.33 | 0.41 | 0.34 | 0.12 | 0.67 | 1.48 |
Statutory homelessnessb | −0.02 | 0.01 | −0.17 | −2.44 | .016 | −0.03 | 0.00 | 0.13 | −0.15 | 0.02 | 0.74 | 1.34 |
Self-reported well-being - low happiness | −0.04 | 0.02 | −0.18 | −2.32 | .022 | −0.08 | −0.01 | − 0.30 | −0.14 | 0.02 | 0.66 | 1.51 |
Homelessness by low happiness | −0.01 | 0.01 | −0.13 | −2.03 | .045 | −0.03 | 0.00 | −0.28 | −0.13 | 0.02 | 0.90 | 1.11 |
Explanatory approach: theory-based model
Explanatory approach: intervention-based model
Source | Unstan-dardised coefficients | Stan-dardised coefficients |
F/t
|
p
| 95%-confidence interval for b | Correlations | Collinearity statistics | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
b
|
SE
| β | Lower limit | Upper limit |
r
|
sr
|
sr
2
| Tolerance | VIF | |||
Model 1 | 8.40 | < 0.001 | 0.11 | |||||||||
Constant | 1.58 | 0.19 | 8.52 | .000 | 1.21 | 1.95 | ||||||
Social-care users’ social-contact need fulfilment | 0.01 | 0.00 | 0.22 | 2.70 | .008 | 0.00 | 0.02 | 0.26 | 0.21 | 0.22 | 0.05 | 1.05 |
Social-care carers’ social-contact need fulfilment | 0.01 | 0.00 | 0.19 | 2.40 | .018 | 0.00 | 0.01 | 0.24 | 0.19 | 0.19 | 0.04 | 1.05 |