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

0.2

.4.6

Pre

vale

nce

rat

e

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

Source: HSE 2001

Mobility disability – England (Males)

Personal care disability – England (males)

0.1

.2.3

Pre

vale

nce

rate

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

Source: HSE 2001

Dealing with sampling variability0

.2.4

.6P

rop

ortio

n

0 20 40 60 80Age

HSE 2000/01Mobility disability schedule

Rates are unreliable particularly where sample sizes are small

Smooth fluctuations by fitting a curve

Dealing with sampling variability

0.2

.4.6

We

ight

0 20 40 60 80Age

Observed survey rates Modelled rates

Source: Health Survey for England 2000/01

Mobility schedules - observed and modelled

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

bxaexD )(

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

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

predict pred_MO_UK

bxaexD )(

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

.4.6

We

ight

0 20 40 60 80Age

Observed survey rates Modelled rates

Source: Health Survey for England 2000/01

Mobility proportions - observed and modelled

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:

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

)1()(

xx

xx pp

Npw

Calculating weights (task 3)

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

in task 2)

egen mobilitycount=count(MO_OBS_RT), by (age sex)

gen mobilityweight=mobilitycount/(MO_OBS_RT*(1*MO_OBS_RT))

)1()(

xx

xx pp

Npw

Model weights – mobility disability

010

000

2000

030

000

We

ight

0 20 40 60 80Age

Source: Health Survey for England 2000/01

Weights associated with locomotor proportions

Fitting a curve in stata

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

predict pred_MO_UK

bxaexD )(

Mobility schedules – observed and modelled (with weights)

0.2

.4.6

We

ight

0 20 40 60 80Age

Observed survey rates Modelled rates

Source: Health Survey for England 2000/01

Mobility schedules - observed and modelled

Better fit at youngest ages

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

nl (MO_OBS_RT=exp({a}+{b}*age)) ifsex==1&gora==1 [aweight=mobilityweight]

predict pred_MO1_M

nl (MO_OBS_RT=exp({a}+{b}*age)) ifsex==1&gora==2 [aweight=mobilityweight]

predict pred_MO2_M

Fit curves for each region (males and females)

0.2

.4.6

.8P

ropo

rtio

n

0 20 40 60 80Age

North East South East

Source: Health Survey for England

MalesRegional mobility disability schedules

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