National Records of Scotland

Preserving the past, Recording the present, Informing the future

Annex A - Full Description of Methodology

Annex A - Full Description of Methodology



A.1 Introduction
A.2 Method used for 1996-based household projections
A.3 Changes from earlier methods used
A.4 Alternative methods tried
A.5 Future Projections

A.1 Introduction

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.

A.2 Method used for 1996-based household projections

A.2.1 Population projections

The first main input to the projection of households is the 1996-based 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 (1996 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 last round of 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 1996 and the projection years, an assumption was made that the numbers living in communal establishments form the same proportion of the total population as is estimated for 1994.

A.2.2 Household composition

The second main input to the household projections method is the historic information on the number of households, from the 1971 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 1971 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


Persons who are not HoH
Total persons in age group



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 ‘non-headship’ rates to be projected to avoid the theoretical possibility of negative ‘non-headship’ 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-based projections. Previously, projected ‘non-headship’ rates were calculated as the residual for each group when projected headship rates were subtracted from 1.


These headship (and non-headship) rates were then projected forward using the modified two-point exponential model, the formula for which is as follows:





yi = k + a*(b ^ xi)



i is each projection year, from 1996 to 2010


yi = headship rate in year i


xi = (i - 1971)/20 in year i

k = ( 1 if y1991>=y1971
      ( 0 if y1991< y1971

a = y1971 - k


b = (y1991 - k)/(y1971 - k)


y1971 = known headship rate for census year 1971


y1991 = known headship rate for census year 1991

The projected headship (and non-headship) 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 (1996). 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.

A.3 Changes from earlier methods used

Prior to the production of the 1994-based household projections, the Scottish Government 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 Government.

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 1994-based and 1996-based household projections.

A.3.1 Change to the way headship rates are projected

As mentioned in section A.2.2, one improvement identified by the research was the need to project both headship and non-headship rates to avoid the theoretical possibility of negative ‘non-headship’ rates. This would also deal with the conceptual oddity inherent in projecting only headship rates.

Both headship and ‘non-headship’ rates were projected for the current set of household projections.

A.3.2 Changes to the categories of household type

The difficulty about identifying 'head of household' was illustrated in the 1992-based household projections. In these, it was evident that there were problems in projecting trends for households containing two or more people where the head of household was married.

The problems arose from the treatment of the first adult on the census form as head of household, irrespective of sex. It appears that a higher proportion of women in households of this type were recorded as first person in 1991 than in 1981. The result was a spurious increase in the projected headship rates for married women, compensated for by a decrease in households headed by married men.

Therefore, marital status was not included as a description of household type, and the sex of the head of household was restricted to one person households where the same difficulties did not occur and there was an interest in distinguishing male and female households.

The advisory group for the research agreed that where census data points were used, these should be restricted to 100 per cent census variables in the interests of robustness. This effectively precluded the consideration of relationships within households, and thus had implications for the output household type.

There was a very clear requirement from users for projections by broad household type. The main interests identified by consultation with users were one parent households, households with children, and households headed by elderly people. However, because it was already decided that relationships within households would be excluded, the categories do not include the term ‘parent’. 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.

A.4 Alternative methods tried

The following alternative methods were tried for the 1994-based projections. (The 1996-based projections used the same method as that which was finally decided upon for the 1994-based projections).

A.4.1 Two-point exponential method

The two-point exponential method was tried with the three separate pairings of available census data, that is, 1971 and 1981, 1971 and 1991, 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 1981 Censuses being comparatively low, and those from the 1981 and 1991 Censuses being comparatively high. Since there was no evidence to suggest that either of these trends would be likely to continue in the future, the method using the two outer census points (1971 and 1991) was used as this effectively smoothes out the trend over the total period to produce a set of household projections somewhere in between the two extreme cases.

A.4.2 Modified three-point exponential method

A modified three-point exponential model was tried which could take 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 two-point exponential using 1971 and 1991 census data.

However the projected households with children from this method did not follow the pattern expected when compared to the projected numbers of children aged 0 to 15 from NRS, whilst those produced using the two-point exponential method did. Having taken this into consideration, the three-point method could not be recommended for use.

A.4.3 Tanh function

The tanh function, which is part of the current Department of the Environment methodology, was intended to be tested for the 1994-based round of household projections for Scotland. This method is computationally very complex, and tests on the methodology are not yet complete. It is planned that this testing will be complete prior to the next projections round.

A.4.4 Decision taken

With no evidence to suggest that there were any better methods than the 2-point exponential method using the 1971 and 1991 census data points, this has been kept as the method for the 1994-based household projections (and used thereafter for the 1996-based projections which are presented in this bulletin). This also provides a useful continuation to the previous set of household projections (1992-based) which also used the same method.

A.5 Future Projections

A working group will shortly be set up to review the projections methodology and make recommendations on the methodologies which should be considered for the next round of household projections (likely to be the 1998-based in two years time).

All statistical publications