A.1 Introduction [1]
A.2 Method used for 1998based household projections [2]
A.3 Changes from earlier methods used [3]
A.4 Alternative methods tried [4]
Household projections are produced approximately every 2 years, in line with the biennial population projections. Their purpose is to give an indication of possible future numbers of households if trends observed in the past continue. It is important to realise that projections are not forecasts. Household projections are the outcome of calculating what happens to the number of households if certain assumptions are made about future population growth and patterns of household formation.
The first main input to the projection of households is the 1998based population projections for Scotland produced by the National Records of Scotland (NRS). These are produced for Scotland by local authority and age group using assumptions about births, deaths and migrations. Refer to the publication ‘Population Projections Scotland (1998 Based)’ for further information on the assumptions used. The relevant population for household formation is taken to be the adult population, aged 16 or over.
The projected number of households is derived from projections of the numbers of adults living in private households. To obtain this from the population data, estimates of the numbers living in communal establishments have been subtracted from the total adult population for each projections year. Work was carried out prior to the 1994based projections to estimate the number of people living in communal establishments in 1994. These estimates were amended following comments from Local Authorities. To obtain estimates of the numbers in communal establishments for 1998 and the projection years, an assumption was made that the numbers living in communal establishments form the same proportion of the total population as was estimated for 1994.
The second main input to the household projections is the historic information on the number of households, from the 1981 and 1991 population Censuses, for each combination of age group, household type and area within Scotland.
Household types were classified in terms of household composition, that is, the number of adults and children in a household (for example, one adult with two or more children). Relationships between persons in the household were not distinguished.
The proportions of households by household type, age group of head of household and local authority area are known for the census years of 1981 and 1991. These proportions sum to one within an area and age group over the households types.
An example, for Dumfries and Galloway 1991 census data for persons aged 35 to 44 years old, is given below.
Number and proportion of heads of households (HoH) by household type



1 person male

1 person female

2 adults

1 adult with 1 child

3 or more adults

1 adult, 2 or more children

2 or more adults with 1 or more children

Number

730

404

1,345

276

840

301

6,456

Proportion

0.03686

0.02040

0.06792

0.01394

0.04242

0.01520

0.32603


Persons who are not HoH

Total persons in age group

Number

9,450

19,802

Proportion

0.47722

1.000

These proportions are known as the headship rates, since if we know the number of persons who head particular household types, this will be the same as the number of households.
We also know the residual number of persons who are in each area and age group but who are not the head of household. This allows both the ‘headship’ and ‘nonheadship’ rates to be projected to avoid the theoretical possibility of negative ‘nonheadship’ rates. This improvement was recommended from the research done by CHRUS, University of Glasgow, on viable alternatives for the method of projecting headship rates (refer to section A.3) and was also used in the 1994 and 1998based projections. Previously, projected ‘nonheadship’ rates were calculated as the residual for each group when projected headship rates were subtracted from 1.
These headship (and nonheadship) rates were then projected forward using the modified twopoint exponential model, the formula for which is as follows:
y_{i} = k + a*(b ^ x_{i}) 

where 
i is each projection year, from 1998 to 2012 
y_{i} = headship rate in year i 

x_{i} = (i  1981)/20 in year i 

k = ( 1 if y_{1991}>=y_{1981} 

( 0 if y_{1991}< y_{1981} 

a = y_{1981}  k 

b = (y_{1991}  k)/(y_{1981}  k) 

y_{1981 }= known headship rate 

y_{1991 }= known headship rate 
The projected headship (and nonheadship) rates are constrained in two ways, firstly so that they cannot individually go above 1 or below 0, and secondly so that they sum to 1 within an area and household type.
The household projections are then calculated by applying the projected headship rates to the population projections to give an estimate of the number of heads of household in the projection years for each household type, age group of head of household and area.
The projected households are controlled, so that figures for the structure plan areas sum to the figure for Scotland, and figures for local authorities sum to the total for their structure plan area.
The projected households are then adjusted, so that the number of households for the base projection year equals the estimate produced by the Scottish Government for the number of households by local authority that year (1998). Any adjustments required to bring the figures for the base projection year into line with the household estimate for that year are then applied to the figures for the other projection years.
Prior to the production of the 1994based household projections the, then, Scottish Office commissioned research from the Centre for Housing Research and Urban Studies (CHRUS) at the University of Glasgow to review the alternative methods for projecting household headship rates. They considered whether or not there were viable alternatives to the cross sectional, headship rate based approach used by The Scottish Office.
They concluded that although there was no alternative due to the absence of suitable information on which to base dynamic models which take into account transitions between different household types, there was scope for improvement within the existing approach. The following improvements recommended were incorporated in the 1994based and subsequently also the 1996 and 1998based household projections.
Firstly, as mentioned in section A.2.2, both headship and nonheadship rates were projected to avoid the theoretical possibility of negative ‘nonheadship’ rates. Other changes related to changes in the categories of household type, largely due to lack of census information on relationships within households. The decision was taken to only identify the numbers of adults and children within a household, so a ‘lone parent’ would not be identified as such, but would be included in the categories which have one adult with children. In addition, the sex of the head of household would be restricted to one person households. More detailed information on these changes can be found in the publication, ‘1996based Household Projections for Scotland’ (HSG/1998/5).
The following alternative methods were tried for the 1998based projections.
The twopoint exponential method was tried with two pairings of available Census data: 1971 and 1991 and 1981 and 1991.
There were distinctly different trends in the numbers of households between 1971 and 1981, where the increase was relatively small, and between 1981 and 1991 when there was a sharper increase.
This resulted in the test set of projections produced from the 1971 and 1991 Censuses being lower than those from the 1981 and 1991 Censuses. However, differences in the compositions of the households projected meant they effectively projected different numbers of adults. Projections of the minimum number of adults were produced by multiplying projected numbers of households within each household type by the minimum number of adults in the household type (e.g. for household type '2 or more adults and one or more children' the minimum number adults would be 2). The minimum number of adults projected using the 1971 and 1991 census data were higher than those projected using the 1981 and 1991 Census data  refer to charts 6 and 7.
Using the 1971 and 1991 data resulted in projections of the minimum number of adults which were higher than the National Records of Scotland (NRS) 16+ population projections for 8 authorities. However, only two authorities had a higher projected minimum number of adults using the 1981 and 1991 census data, than the 16+ population projections.
A modified threepoint exponential model was tried which took all three census data points for 1971, 1981 and 1991 as input to the projections. The resulting total households projected from this method were, in fact, very similar to that from the twopoint exponential method using 1971 and 1991 census data.
The minimum number of adults projected using this method could not be met by NRS 16+ population projections for a number of authorities, one authority for most of the projection period.
Note: This chart shows that the minimum number of adults projected by all methods tried can be met by NRS's 16+ population projections at Scotland level. For some authorities, however, the graph would show that the minimum number of adults projected by some methods could not be met by NRS's population projections.
The decision to use the twopoint exponential method with 1981 and 1991 Census data, was made as the minimum number of adults projected from this method could be satisfied by NRS's 16+ population projections for all but two authorities. There were shortfalls for 8 authorities using the 1971 and 1991 Census data and for 3 authorities using the threepoint exponential method. A similar check on the number of children was also undertaken and was satisfied for all three methods tried.
None of the methods, however, fully satisfied the condition of projecting a minimum number of adults less than the 16+ population projection for every authority for the later years. Although the 2 point exponential method with 1981 and 1991 Census data satisfied the condition for most authorities, there were two authorities where a very minor adjustment was required. The adjustment involved a slight reduction in the number of larger households projected for the two authorities, offset by an increase, of the same amount, in the number of smaller households. This had the effect of reducing the projected minimum number of adults whilst not affecting the total projected number of households for each authority. The adjustment was made proportionately across the different age groups.
Links
[1] https://www.nrscotland.gov.uk/statisticsanddata/statistics/statisticsbytheme/housholds/householdprojections/archive/1998basedhouseholdprojections/fulldescriptionofmethodology#a1
[2] https://www.nrscotland.gov.uk/statisticsanddata/statistics/statisticsbytheme/housholds/householdprojections/archive/1998basedhouseholdprojections/fulldescriptionofmethodology#a2
[3] https://www.nrscotland.gov.uk/statisticsanddata/statistics/statisticsbytheme/housholds/householdprojections/archive/1998basedhouseholdprojections/fulldescriptionofmethodology#a3
[4] https://www.nrscotland.gov.uk/statisticsanddata/statistics/statisticsbytheme/housholds/householdprojections/archive/1998basedhouseholdprojections/fulldescriptionofmethodology#a4