Census counts of the flows of people migrating, or travelling to work, from area to area, were released in the Origin-Destination Statistics for Output Areas in May 2004, for wards in July 2004 and for local authorities in October 2004. As with all Census output, small counts in the tables were adjusted before release to protect confidentiality. This adjustment has a greater effect on the variability of Origin-Destination statistics because of the large number of small counts compared with other output.

This note provides guidance on the level of variability which can be expected in the Origin-Destination statistics, and on how this variability can be minimised. The guidance is based on an analysis of tables MG201 and W206 for several local authorities, and a full description of this analysis is available as a pdf document HERE.

How the counts are adjusted
Small counts are adjusted or 'perturbed' to protect confidentiality. The cells with small values were adjusted independently upwards or downwards according to prescribed probabilities. The scheme was designed so that perturbations should have an expected mean of zero (that is, the adjustment does not introduce systematic biases into the counts) and a variance (a measure of how much the perturbed counts vary from the true value) that is proportional to the number of cells that were adjusted. The more cells adjusted, the larger the possible variations from the true values.

The probability mechanism that was devised for carrying out the perturbations is described in full in the linked note. The method ensures the same adjustment outcome in any future reproduction of the table. One result of this method is that a flow which appears, for example, in one table as 3 people moving from one area to another may be adjusted to zero in another table, and thus 'disappears' from the second table.

Possible bias in the adjusted data
Several matrices relating to migration flows were checked for any evidence of bias - that is, the adjustments to the data tended to be more one way than the other. Statistical tests suggest that, in some cases, there is a very small amount of bias because of the nature of the random processes used in the adjustment. The reasons relate to the adjustments in the Origin-Destination matrices, but do not apply to tother sets of output such as Standard Tables or Census Area Statistics.

Variability in the adjusted data
The adjustments made to the data to protect confidentiality mean that many counts in the Origin-Destination Matrices will differ from the underlying 'true' counts. This explains why counts produced by aggregating several values in an Origin-Destination Matrix (say, all values relating to migration into a particular Output Area) can be different from corresponding counts in the Census Area Statistics tables.

The statistical technique of regression modelling can be used to indicate the extent to which such derived totals can be expected to differ from the underlying counts. Such a regression model was produced by comparing the underlying counts with those produced by aggregating entries in the migration table MG201 for wards within two local authorities and the workplace table W206 for wards within seven local authorities. More detail on how the model was produced is available in the full linked note.

The model allowed the calculation of a predicted 'perturbation interval'. The perturbation interval is the range in which it is possible to be 90 per cent confident the underlying value occurs for any total produced by adding together entries in table MG201.

Where adjusted values are added together to produce a perturbed total T, the perturbation interval can be expressed algebraically as:

For the migration table:

• Upper bound of prediction interval = T + 5.519 T ^0.302
• Lower bound of prediction interval = T - 1.787 T ^0.530

• Upper bound of prediction interval = T + 1.717 T ^0.462
• Lower bound of prediction interval = T - 3.196 T ^0.315

Alternatively, the perturbation interval can be approximated using the tables below. Table 1 indicates, for example, that, if a total of 750 is obtained by adding together entries in table MG201, it is 90 per cent sure that the underlying count is between 691 and 791.

Table 1: 90 per cent Perturbation Interval for Specified Perturbed Totals from MG201

Perturbed Total	Prediction Interval		Perturbed Total	Prediction Interval
	Lower Bound	Upper Bound		Lower Bound	Upper Bound
5	1	14	380	339	413
10	4	21	400	357	434
15	8	28	420	376	454
20	11	34	440	395	475
30	19	45	460	414	495
40	27	57	480	433	516
50	36	68	500	452	536
60	44	79	550	500	587
70	53	90	600	547	638
80	62	101	650	595	689
90	71	111	700	643	740
100	80	122	750	691	791
120	97	143	800	738	842
140	116	165	850	786	892
160	134	186	900	834	943
180	152	207	950	883	994
200	170	227	1,000	931	1,045
220	189	248	1,500	1,414	1,550
240	207	269	2,000	1,900	2,055
260	226	290	2,500	2,387	2,559
280	245	310	3,000	2,876	3,062
300	263	331	3,500	3,366	3,565
320	282	352	4,000	3,856	4,068
340	301	372	4,500	4,346	4,570
360	320	393	5,000	4,838	5,072

Table 2: 90 per cent Perturbation Interval for Specified Perturbed Totals from W206

Perturbed Total	Prediction Interval		Perturbed Total	Prediction Interval
	Lower Bound	Upper Bound		Lower Bound	Upper Bound
5	0	9	500	477	530
10	3	15	550	527	582
20	12	27	600	576	633
30	21	38	650	625	684
40	30	49	700	675	735
50	39	60	750	724	786
60	48	71	800	774	838
70	58	82	850	823	889
80	67	93	900	873	940
90	77	104	950	922	991
100	86	114	1,000	972	1,042
120	106	136	1,250	1,220	1,296
140	125	157	1,500	1,468	1,550
160	144	178	1,750	1,716	1,804
180	164	199	2,000	1,965	2,057
200	183	220	2,250	2,214	2,311
220	202	241	2,500	2,462	2,564
240	222	262	2,750	2,711	2,816
260	242	282	3,000	2,960	3,069
280	261	303	3,500	3,458	3,574
300	281	324	4,000	3,956	4,079
320	300	345	4,500	4,455	4,583
340	320	365	5,000	4,953	5,087
360	340	386	5,500	5,452	5,591
380	359	407	6,000	5,950	6,095
400	379	427	6,500	6,449	6,599
420	399	448	7,000	6,948	7,102
440	418	468	7,500	7,447	7,606
460	438	489	8,000	7,946	8,109
480	458	510	9,000	8,944	9,115

Advice for users
Users of the Origin-Destination Statistics may find the following advice helpful:

Variation from an underlying true value exists in all Census results for a variety of reasons. Whilst the Origin-Destination Statistics are more affected than other tables by adjustments to the data to protect confidentiality, they are more reliable than similar results published in 1991 which were only based on 10 per cent of the total population. The variability in the aggregated totals can be minimised by using the highest level geography possible - for example, deriving results for a Government Office Region by aggregating counts for local authorities rather than Output Areas.

In addition, the most accurate count of the overall flow between two areas is most likely to be contained in the table with the fewest cells. So, for local authorities the most reliable tables to use to estimate the flow between two areas are MG103 (migration) and W107 (workplace). For wards, the corresponding tables are MG203 (migration) and W206 (workplace). For sub-groups of the population, choose the table that has the fewest possible cells that need to be aggregated to obtain the overall flow.

This page last revised: 18 November 2004