Curve fitting. Session 2

Curve fitting Session 2 Method background • Disability rates are strongly linked to age • However HSE disability rates for single years of age are u...
Author: Darleen King
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Curve fitting Session 2

Method background • Disability rates are strongly linked to age • However HSE disability rates for single years of age are unstable • We can fit a curve to the disability schedule to smooth the fluctuations • Model rates (national or regional)*local population totals

0

Prevalence rate .2 .4

.6

Mobility disability – England (Males)

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Age

Source: HSE 2001

0

Prevalence rate .1 .2

.3

Personal care disability – England (males)

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Age

Source: HSE 2001

Dealing with sampling variability Mobility disability schedule .6

HSE 2000/01

Proportion

.4

Rates are unreliable particularly where sample sizes are small

0

.2

Smooth fluctuations by fitting a curve

0

20

40

60 Age

80

Dealing with sampling variability

0

.2

Weight

.4

.6

Mobility schedules - observed and modelled

0

20

40

60

80

Age Observed survey rates Source: Health Survey for England 2000/01

Modelled rates

What function?

• Lots of choices • Quadratic (y=b0+b1x+b2x3+b3x3 • Exponential functions • Estimation of mortality schedules • Statistics Canada use an exponential curve to model disability schedules in Canadian territories

Exponential curve

D( x) = e

a + bx

Where: D(x)= the proportion of people with a disability at age x

Practical structure • Task 3 – Fit an exponential curve to (England) mobility schedules (with and without weights). Uses saved data from task 2 • Task 4 – Fit curves to regional mobility schedules • Task 5 – Use your model rates to calculate the number of people with a mobility disability in six districts. (Data provided)

Fitting a curve in stata

D( x ) = e

a +bx

nl (MO_OBS_RT=exp({a}+{b}*age)) predict pred_MO_UK

Exponential curve – parameter estimates (males) Confidence interval

a

-4.4

-4.79

-4.09

b

0.04

0.04

0.05

Mobility disability schedules – observed and modelled

0

.2

Weight

.4

.6

Mobility proportions - observed and modelled

0

20

40

60

80

Age Observed survey rates Source: Health Survey for England 2000/01

Modelled rates

Analytic weights • Stata treats the rates at each age as being equally reliable. • Can use weights to relax this assumption • If we assume our rates stem from a binomial process then:

Nx wx ( p) = p x (1 − p x )

Where px = proportion with a disability at age x and Nx equals the number of people sampled at age x.

Calculating weights (task 3)

• Re-open the HSE data • Re-calculate age specific rates (MO_OBS_RT) (as in task 2)

Nx wx ( p ) = p x (1 − p x ) egen mobilitycount=count(MO_OBS_RT), by (age sex) gen mobilityweight=mobilitycount/(MO_OBS_RT*(1*MO_OBS_RT))

Model weights – mobility disability

0

Weight 10000 20000

30000

Weights associated with locomotor proportions

0

20

40

Age Source: Health Survey for England 2000/01

60

80

Fitting a curve in stata

D( x) = e

a + bx

nl (MO_OBS_RT=exp({a}+{b}*age)) [aweight=mobilityweight] predict pred_MO_UK

Mobility schedules – observed and modelled (with weights)

Weight

.4

.6

Mobility schedules - observed and modelled

0

.2

Better fit at youngest ages

0

20

40

60

80

Age Observed survey rates Source: Health Survey for England 2000/01

Modelled rates

Task 4 – regional curves

• Open HSE data • Drop institutional residents (no gora) • Are differences in regional rates of mobility disability significant? (1.4.2-1.4.3) •

Task 4 - regional curves

• Calculate regional schedules of mobility disability rates by sex age gora: egen MO_num=total(mobility_w) by sex age gora: egen MO_denom=total(count_w)

gen MO_OBS_RT=MO_num/MO_denom

Task 4 – regional curves

• Weights are the same as used for national data (task 3) • Regional age patterns of weight very unstable • After calculating regional rates and weights: • Duplicates drop age sex gora, force

Task 4 –regional curves Fit curves for each region (males and females) nl (MO_OBS_RT=exp({a}+{b}*age)) if sex==1&gora==1 [aweight=mobilityweight] predict pred_MO1_M nl (MO_OBS_RT=exp({a}+{b}*age)) if sex==1&gora==2 [aweight=mobilityweight] predict pred_MO2_M

Regional mobility disability schedules

0

.2

Proportion .4

.6

.8

Males

0

20

40

60 Age

North East Source: Health Survey for England

South East

80

Task 5 • Aim - generate district estimates of the numbers of people with mobility disabilities • Practical 1 task 5 dataset.dta • a row for each single year of age (10, 11,….84,88) for males and females in each of the six districts • Contains the national and regional model rates from tasks 3 and 4 • Population counts