the perception of distance in migration on merseyside

The Perception of Distance in Migration on MerseysideAuthor(s): M. MaddenSource: Area, Vol. 10, No. 3 (1978), pp. 167-173Published by: The Royal Geographical Society (with the Institute of British Geographers)Stable URL: http://www.jstor.org/stable/20001341 .

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The perception of distance in migration on Merseyside

M. Madden, University of Liverpool

Summary. The differences in the closenesses of fit between observed and predicted values of dependent variables produced by submitting original and transformed data to log transform regression models is noted, and is demonstrated on inter-district migration

data for Merseyside. Residuals are absorbed into the distance variable, a new model produced by regression on the modified data set, and the process repeated until maximum explanation is achieved. Sets of 'perceived' distances.are obtained, and an indication of their use in forecasting made.

One of the most difficult problems in the use of multiple regression analysis lies in the initial choice of variables. As Beals (1972) so rightly states: ' . . . theories simplify.

Many variables expected to have only small effect may be omitted. The omitted vari ables show up in the error term'. Obviously, the more complex the phenomenon under investigation, the larger the number of omitted variables is likely to be, and the greater the error term may be. A phenomenon such as human migration is very complex, and the number of variables possibly affecting such migration evidently large: nonetheless, the application of multiple regression in studies of migration has a fairly long history. Willis (1974) lists most of the important publications in this field, and highlights many deficiencies that have occurred when regression methods are applied to migration.

Perhaps the widest used and best known of the models developed in these studies is that of Rogers (1968), where an earlier model by Lowry (1966) is refined, that involves the regression of migration on to unemployment, labour force and salary in both producing and receiving regions, and on to some measure of distance between pairs of such regions. The model is rendered non-linear by taking log-transforms of the data, and high values for R2, the coefficient of determination, are obtained. However, these values of R2 are calculated for the transformed data: if models of this type are tested using original, untransformed data, then the closenesses of fit are revealed to be very much lower. Curiously, this exercise seems to be very rarely carried out,

Masser (1970) presenting probably the most thorough example of its execution in his comparative study of several log-transform models of inter-metropolitan popula tion movements in England and Wales. This study indicates that very high values of R2 for models developed from log-transformed data may not preclude substantial errors appearing when actual numbers of migrants are compared with those predicted using untransformed data in the models.

A possible method of minimizing this error is to artificially subsume each residual in one of the independent variables in the model by altering the observed values of the variables to produce perfect fit for each observation. This can be very easily done by solving the regression equation for each observation of the unknown ' observed' independent variable, after substituting the observed dependent variable values for those predicted by the model. Effectively, then, a new set of ' observed' values of the variable selected to absorb the residuals is obtained. If a further regression is now carried out on the amended data, a new model can be derived which will have higher explanatory power than the previous one. This process can be repeated until maximum power is obtained.

The method outlined here was applied to a data set associated with gross migration between districts of the metropolitan county of Merseyside. A model based partly

167



168 Migration on Merseyside

on that of Rogers, but incorporating a number of variables considered more relevant to inter-district migration on what is essentially a sub-regional scale, was developed:

Mij = f(D i, L,, Lj, Dw,, Dwj, Comrn, Demrn, Comj, Demj) where

M,; is the migration between districts i and j D ij is the distance between districts i and j, in miles Li, Lj are the labour forces in districts i and j Dwi, DwJ are the housing stocks in districts i and j Comrn, Comj are the housing completions in districts i and j Demr,, Demj are the housing demolitions in districts i and j

The data were arranged quinquennially, and covered the periods 1961-66 and 1966-71. Mij data spanned each five-year period, and Li, Lj referred to the beginning of each quinquennium, whilst the housing variables referred, in the case of Dwi, Dwj to the end of the quinquennium, and in the case of demolition and completion variables to the whole five-year period. The initial regression, on logarithmic trans forms of this data, gave the results shown in Table 1. The coefficient of determination indicates that the model explains about 860% of the variation: three of the variables have coefficients significantly different from zero at the 0-01 level, and inspection of the residuals yields no significant patterning.

Table 1. First regression on complete data set

Variable Coefficient

Constant -0-34033 In Dij -2-34053* In Li 6.42268* In Lj -0-77846 In Dwi -6-57806* In Dwj 0-70583 In Demr 0 06095 In Comi 1-14740 In Demj 0 17093 In Comj 0-12003

R 2 0-86082

*Significant at the 0 01 level.

The predictions made on untransformed data, using the model for the period 1961 66, and presented in Table 2, demonstrate deviations from the observed values that are much greater than those implied by the fairly high coefficient of determination in Table 1. In particular, the estimation of migration from Liverpool to Wirral is nearly twice the actual movement, and in the reverse direction over one and a half times the observed value. Discrepancies of under-estimation of a similar order are particularly evident in the Sefton-Liverpool and Liverpool-Sefton predictions, and in general the match between the predicted and observed values is not close. Obviously, the 1961-66 interaction matrix presents only half the picture of application of the model to the data: the other half is shown in Table 3, where 1966-71 data are fed into the

model, and predicted values compared with those observed. Here certain discrepancies are even more marked. The Liverpool-Wirral movement is overestimated by a factor of about three, and its reverse by one of over two. Similar discrepancies of under estimation are particularly noticeable in this quinquennium in the flows from Sefton to Liverpool and St Helens to Knowsley. It is interesting to note that the Liverpool

Wirral connection, in both directions, is considerably more poorly estimated in the second quinquennium, the Sefton-Liverpool connection much more accurately, and



Migration on Merseyside 169

Table 2. First regression: comparison of predicted and actual values (1961-66)

Liverpool Sefton St Helens Knowsley Wirral

a* 13582 6406 34880 8131 Liverpool p** 5687 4779 29489 14340

A% --58-13 -25-40 -15-46 76 36

a 5647 380 1484 1518 Sefton p 1595 572 1261 838

A% -71-75 50 53 -15-03 44-80

a 552 460 4784 196 St Helens p 1225 523 3259 368

A% 121 92 13-70 -31 88 87-76

a 9419 1622 3485 932 Knowsley p 8299 1265 3576 980

A% -11-89 -22-01 2-61 5 15

a 2185 1288 207 483 Wirral p 3554 740 356 863

A% 62 65 -42-55 71*98 78 67

Theil's coefficient of inequality= 0 16379

* a indicates actual migration **p indicates predicted migration A% indicates p-a as a percentage of a

Table 3. First regression: comparison of predicted and actual values (1 966-71)


a 12963 9625 29359 7722 Liverpool p 9419 8376 32572 22348

A% --27 34 -12 98 10 94 189-41

a 4983 1243 957 1548 Sefton p 2226 771 1071 1005

A% -55-33 -37 97 11*91 -36-55

a 1628 484 4070 308 St Helens p 1316 512 2131 339

A% -19-16 5 79 -47-64 10 06

a 8228 2299 5060 1133 Knowsley p 12734 1772 5301 1292

A% 54-76 -22-92 4 76 14 03

a 2024 1078 253 319 Wirral p 4125 784 399 610

A% 103-80 -27 27 57-71 91*22

Theil's coefficient of inequality: 0-20577

error signs reversed in some connections (St Helens-Liverpool, for example). Overall, the closeness of fit (measured in these experiments by Theil's (1958) inequality coeffi cient) falls from 1961-66 to 1966-71.

Bearing these facts in mind, it is clear that the initial model is quite unsuitable for forecasting purposes. The selection of variables made is either not accurate or not

wide enough to produce a regression model which explains a sufficiently high propor tion of the variance to make forecasting reliable. However, if the error term is absorbed into one of the variables, and a new model obtained from the revised data set, higher forecasting accuracy should result.




Deriving ' perceived distance'

In these investigations, the errors were subsumed into the distance variable, producing what may intuitively be described as a 'perceived distance' measure. For example, in the case of Wirral-Liverpool migration in 1966-71, the actual (airline) distance is 5-80 miles. If the distance is adjusted, to give 100% fit in the initial regression model, the ' perceived distance' is 7-86 miles. If each distance is adjusted to give 100% fit, and a new model obtained from regression, an increase in the overall closeness of fit is to be expected.

The distance measure was the variable that lent itself most readily to adjustment in this study. The other variables in the model are all likely to vary with time: indeed, if the model is to be used as a policy-testing instrument, they are bound to vary as they reflect the implementation of house building and clearance schemes, for example. Since the distance measure is partly a surrogate anyway, in that it represents straight line distance between notional centroids of the metropolitan districts, rather than network distance, it is conceptually easier to alter artificially: coupling this with the difficulty of combining an already altered housing variable with a policy phrased in actual numbers of housing units, or an altered labour force variable with a change in

working population due to natural growth and perhaps change in activity rates, indicates strongly that the distance measure should be the variable to be altered.

The process of re-regression, then, was carried out in three ways: first, the 1961-66 'perceived distances' were evaluated, and substituted in each iteration across the

complete data set; secondly, the whole process was executed using the 1966-71 migra tion flows to obtain the new distance measures; and thirdly the exercise was carried out manipulating the distances in each quinquennium together, and regressing over the whole data set. Each regression series was iterated until there was no improvement in the value of Theil's inequality coefficient. The final results of the three series are shown in Table 4, and their predictions for the migration flows in Table 5. The first point to notice is the markedly improved values for the coefficients of determination, particularly in the case of the third model, which are paralleled by the much smaller values for Theil's coefficients. The number of variables with coefficients different from zero at the 0 01 level of significance shows a great improvement as well: the first and second models have seven out of nine variables and the third model all its variables with coefficients significant at this level.

Notice too how accurately the predicted values for migration (given in the second two rows of each triplet in Table 5) match the observed values. Although all three

models demonstrate marked improvements on the predictive capacity of the initial

Table 4. Final regression models

1961-66 data 1966-71 data 1961-71 data manipulation manipulation manipulation

Variable Coefficient Coefficient Coefficient

Constant 11 11120 -15-28238 -0-74642 In Dij -1*79450 -3-66508* -2-35140* In Li 5-48280* 0-24580 6.34869* In Lj 0 06472 -8-32939* -0-81378* In Dw, -6-26482* 1 99429 -6-46744* In Dwj -0-73766 9-54227* 0.74029* In Demr 0-21035* -0.35574* 0-04973* In Com, 0-87890* 0.92388* 1-17569* In Demj 0-24457* 0.15171* 0-17353* In Comj 0-36328* -1 35700* 0-10525*

R2 0096467 0-97026 0 99995

*Significant at the 0-01 level




Table 5. Final regressions: comparisons of predicted and actual figures


61-66 66-71 61-66 66-71 61-66 66-71 61-66 66-71 61-66 66-71

a 13582 12963 6406 9625 34880 29359 8131 7722 Liverpool p1 * 10933 13557 5963 10507 28599 29765 6700 8356

p2* * 13536 12971 6380 9651 34747 29402 8072 7721

a 5647 4983 380 1243 1484 954 1518 1584 Sefton p1 4767 4832 472 1536 1677 997 1490 2012

p2 5615 5003 376 1253 1471 958 1509 1594

a 552 1628 460 484 4784 4070 196 308 St Helens p1 643 1322 572 477 5319 4265 259 315

p2 532 1589 444 474 4656 4020 188 302

a 9419 8228 1622 2299 3485 5060 932 1133 Knowsley p1 8263 7228 1774 2460 3820 6083 1050 1232

p2 9463 8187 1622 2287 3501 5076 927 1128

a 2185 2024 1288 1078 207 253 483 319 Wirral p1 1947 2038 1268 1381 262 311 586 341

p2 2189 2038 1296 1089 206 255 481 320

*p1 indicates predicted values derived from regression equation obtained from altering data set for relevant quinquennium only

**p2 indicates predicted values derived from regression equation from altering data set for both quinquennia

Theil's coefficient of inequality: 61-66 0-09539680 66-71 0-02801188 61-71 0 00212588

Table 6. Perceived distances associated with the final regression models


61-66 66-71 61-66 66-71 61-66 66-71 61-66 66-71 61-66 66-71

a 8-30 8-90 3 90 5 80 Liverpool p1 4-58 8-13 7 31 8 67 2-96 4-63 6 03 10-60

p2 5-72 7-24 7-85 8-39 3-63 4-08 7 39 9-13

a 8-30 12-20 8-30 10-80 Sefton p1 4-22 6-32 17 68 11-05 8-46 8-82 8-20 11-79

p2 4-84 5-88 14-53 9 95 7-74 8-71 8-38 8-89

a 8-90 12-20 5 30 14-70 St Helens p1 14-69 6-31 15-90 10-71 5 07 4-16 24-79 13-70

p2 12-51 8-13 12-89 12-50 4-50 4-02 19-24 15-31

a 3 90 8-30 5 30 9-60 Knowsley p1 3-25 4-55 7-77 7-85 5-77 6-10 10-43 10-83

p2 3-69 4 70 7 46 7-43 5-36 5-41 9-81 10-15

a 5 80 10-80 14-70 9-60 Wirral p1 6-49 8-75 8-36 12-51 22-90 18-69 14-19 12-93

p2 7-14 7-86 8-52 9 43 18-53 17-86 12*30 12-66

a, pl, p2 as in Table 5

model, it is immediately obvious that the third model produces an immensely more satisfactory fit to the data than either of the other two. This greater accuracy is of course reflected in both the coefficient of determination and the inequality coefficient.




Table 6 demonstrates the differences between actual inter-district distances, and those 'perceived' distances produced by the iterative regression. Perhaps the most interesting facet of this table is the change in perceived distances that occurs over time, coupled with the differences that occur between pairs of districts, and amongst sets of districts observed from one base district. For example, the perceived distance from Liverpool to St Helens is noticeably lower than the actual distance, particularly in the first quinquennium, while its reverse, St Helens to Liverpool, is substantially greater in the first quinquennium, and smaller in the second. This indicates that inhabitants of St Helens have in the past tended to perceive Liverpool as being further away in terms of migration than it actually is, but that this tendency has reversed itself over the time periods considered. Inhabitants of Liverpool, however, have perceived St Helens as being closer than it in fact is, but are tending to a more accurate assessment of its proximity.

Figure 1, which should be interpreted in conjunction with Table 6 and is based on distances in the third regression model, represents in diagrammatic form the changes in the perception of the rest of Merseyside by the inhabitants of St Helens that have occurred over the two five-year periods. In the period 1961-66 this meant that less St Helens migrants located themselves in other parts of Merseyside than might have been expected, while during 1966-71 a migration pattern from St Helens developed that in two cases produced more migrants than might have been expected. The other districts show trends less coherent than this, with the exception of Liverpool, which displays an increase in all perceived distances originating from itself over the quin quennia investigated.

KSH S H

L

w

Figure 1. Perceived distance from St Helens, 1961-66 (left) and 1966-71 (right).

For predictive purposes it is necessary to estimate some assumed measures for the perceived distances between the districts. In the absence of any data for migration in other quinquennia, a first attempt at this involved linear extrapolation. This pro duced some fairly unlikely estimates for perception of distance: for example, the distance from St Helens to Liverpool became only very slightly greater than that to Knowsley. Inspection of Table 6 reveals why this should be, if linear extrapolation is used. However, application of the third regression model, using linearly extrapolated perceived distances, actual labour forces and policy-generated housing variables for the period 1971-76, produced results that intuitively appeared reasonable.

Further development of this particular technique, apart from the customary refine ment of variables, and removal of those displaying multicollinearity (although this need not matter if the multicollinearity is expected to continue into the prediction




period) would involve the more accurate estimation of expected perceived distance. Obviously, the availability of migration data for more than two time periods would help, as would careful study of the reasons why perceived distances should differ from actual distances. For example, Wirral is separated from the rest of Merseyside by the physical barrier of the River Mersey; it is reasonable to expect from this fact alone that perceived distances from Wirral to all other districts will be greater than actual distances. This is true, with the exception of the Wirral-Sefton perceived distance, which implies that there are forces of attraction between these two districts that overcome the inhibiting effect of the river barrier. Possibly this is an indication of some homogeneity between Wirral and Sefton which further indicates that there are variables which should be included in the regression equation to distinguish between Sefton and Wirral and the rest of the county. Of course, this kind of conclusion can be drawn from straightforward comparison of the actual and predicted migration figures: the value of obtaining perceived distance measures is that the discrepancies between observed and predicted figures can be represented in one variable and, in the absence of further sophistication of the regression model, forecast changes in these discrepancies can be incorporated into data sets for future prediction.

References Beals, R. E. (1972) Statistics for economists: an introduction (Chicago: Rand McNally) Lowry, I. S. (1966) Migration and metropolitan growth: two analytical models (San Francisco:

Chandler) Masser, I. (1970) A test of some models for predicting intermetropolitan movement in population

in England and Wales (London: Centre for Environmental Studies) Rogers, A. (1968) Matrix analysis of interregional population growth and distribution (Berkeley:

Univ. of California) Theil, H. (1958) Economic forecasts and policy (Amsterdam: North-Holland) Willis, K. G. (1974) Problems in migration analysis (Farnborough: Saxon House)

The world of words

Frontiers Territory is of fundamental importance to human beings, and it is a fundamental aspect of knowledge to show how one piece differs from another. Hence, both the origin and the importance of geography. In order to distinguish one territory from another, it is necessary for there to be terms describing the places where territories adjoin; and the large number of such words indicates the importance men attach to the description and differentiation of the landscape. A list of these words provides a catalogue of rich cultural variety: belt; boundary; bounds, an English version of the Norman-French bourne; discontinuity; divide, which, as with many of these words, has a rich metaphorical usage in social, political and economic language; edge; fringe; frontier; littoral; limit; line; march; margin; mete; pale; parting; perimeter; periphery; shore; verge, from the name of the staff of office used by the King's

Marshall to show the extent of the King's Peace; watershed; and zone. After ' boundary ', the most commonly used word is probably ' frontier '. It is

derived from the Latin frons, meaning 'forehead', and is used in this sense by Shakespeare in Henry IV's rebuke:

And majesty might never yet endure The moody frontier of a servant brow.

It was during the fifteenth century that it began to be used to mean the boundary between countries or kingdoms. De Guilleville's Pylgremage of the sowle (printed by Caxton in 1483) refers to keeping ' the frounters of the realme fro perille of enemyes ',



the perception of distance in migration on merseyside

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