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The micro-erosion meter technique in a littoral environment
Marine Geology. 22 (1976) MSl-M58 o Elsevier Scientific Publishing Company, Amsterdam - Printed
in The Netherlands
Letter Section THE MICRO-EROSION METER TECHNIQUE IN A LITTORAL ENVIRONMENT L.A. ROBINSON
Wymondham College, Wymondham,
{Great Britain)
(Revised version accepted August 19,1976)
ABSTRACT Robinson, L.A., 1976. The micro-erosion meter technique in a littoral environment. Mar. Geol., 22: M51-M58. The micro-erosion meter is used to measure erosion rates on bare rock surfaces. Several modifications to its published design are recommended to make it more suitable to coastal studies. The previously unassessed error of the technique is negligible but basic questions about the meaning of erosion rates are raised. The use of non-parametric methods is advised for statistical analysis of the data but, for multivariate studies, betacoefficient analysis using the linear regression model is recommended.
INTRODUCTION
The micro-erosion meter (MEM) is an instrument developed by High and Hanna (1970) to measure directly the rate of erosion on bare surfaces of lithified rock. Though such surfaces are abundant in coastal environments, no study has yet been published which has used the technique. Nor has any assessment of the accuracy of the technique been made. In order to make the MEM more suitable for the high erosion rates common on coasts, several modifications in the published design of the instrument are recommended based on the experience of the author. The amounts of inaccuracy caused by several factors are also examined. It is usual to draw general conclusions from samples of measurements by using statistical methods. However, samples of erosion rates tend to have unusual characteristics which are described and the suitabilities of various statistical tests arediscussed. THE MICRO-EROSION
METRR
(MRM\
M52
plate with a leg near each corner and a pillar placed off-centre on the upper surface (Fig. 1). To this pillar is attached an engineers' dial gauge whose probe passes through a hole in the base-plate so that its tip can rest on the rock surface. Exact relocation of the measurement point on the rock surface is achieved b y the use of the Kelvin Clamp Principle. When readings are being taken, the MEM rests on three metal studs permanently emplaced in the rock and the tip of the probe rests on the rock surface. Three readings can be taken within the triangle formed b y the three studs; this triangle will be
Fig. 1. The micro-erosion m e t e r with the bases of the legs fitted into a plastic plate for transport
M53
termed a "unit". The difference b e t w e e n t w o readings at the same point b u t taken at different times, is the a m o u n t of erosion that has occurred so the erosion rate can be calculated. MODIFICATIONS TO THE MEM
The following modifications are r e c o m m e n d e d on the basis of experience gained in using an MEM on the coast of northeast Yorkshire, England, an area of relatively unresistant shales which allow high erosion rates. (1) An engineers' dial gauge was used with a capacity of 10.160 cm, measurement increments being 0.0025 cm. This allowed most high erosion rates to be measured, b u t the removal of blocks of rock b y the sea did occasionally destroy measurement sites. This was partly because there were practical difficulties in the installation and cleaning of studs placed deeper than a b o u t 6.5 cm below the rock surface. Thus 6.5 cm was the actual capacity of the MEM. The extra 3.5 cm was n o t totally wasted capacity because the space allowed the operator's hand to hold the probe and to lower it gently on to the rock surface since no probe-lowering mechanism was affixed to the dial gauge. Also, when the rock surface was sharply convex upwards between the studs, the gauge registered b e t w e e n 6.5 and 10.16 cm although the studs were not that far below the rock surface. (2) The tip of the probe was convex, not pointed. This modification might introduce some inaccuracy b u t this was thought to be minor compared with the damage to the easily scratched surface of the shales in the study area resulting from the use of a pointed tip. (3) The longer dial gauge necessitated the fitting of longer legs than in the published design. In order to ensure that the legs were firmly fixed in the base plate, the tops of the legs were _riveted in addition to being pressure fitted. During transport the bases of the legs were fitted into holes in a plastic plate. (4) The Kelvin Clamp Principle allows exact relocation of the MEM on the three studs placed in the rock each time the unit is visited. A cone-shaped depression cut in the base of one leg of the MEM acts as a fulcrum when resting on a stud. A wedge-shaped depression aligned with the cone then fixes the instrument in t w o dimensions when it t o o rests on a stud. The third leg has a fiat base and fixes the third dimension. The standard MEM design employs angles of 120 ° for the cone and wedge depressions. Such instruments do not rest tightly on studs placed in vertical or highly inclined rock surfaces so these angles were reduced to 90 ° . (5) The probe projects for half its length below the bases of the legs of the standard MEM. This aids the siting of the instrument on h u m m o c k y and sharply concave surfaces. In the modified instrument the dial gauge was positioned on the pillar so that its tip lay a b o u t I cm above the bases of the legs. This was done for three reasons:
M54
(a) h u m m o c k y surfaces should be avoided because of the errors they produce (see below); (b) there is a danger of damage to the probe when itprojects beyond the bases of the legs; (c) the measurement capacity of the M E M m a y be reduced. (6) It is necessary to have some means of checking the accuracy of readings. A strong datum was made by fixing studs in concrete paving stones, pieces of glass being glued to the concrete to provide the measurement points. This datum was made of materials with low coefficients of thermal expansion and allowed the checking of readings at two bands within the range of the instrument at about 1.25 and 5.0 cm. (7) In the littoralenvironment, sediment soon fillsthe holes in the rock surface in which the studs are placed. This was prevented by fillingthe holes with black plasticine. By freeing the plugs from the walls of their holes with a scooped spatula, they can be lifted out whole without damage to the uppermost bedding laminae around the holes. The use of suitably coloured plasticine has the additional advantage of camouflaging the holes against vandalism. O n the other hand, brightly coloured plasticine enables units located on featureless parts of a shore platform or cliffto be found quickly. A C C U R A C Y OF THE M E M
Length of the legs There is an error inherent in the use of legs of equal length as in the standard design because the leg with the flat base rests on the top of its stud while the depressions in the bases of the other two legs produce contacts at points highe~ than the bases of the legs. The result is t h a t with the tops of the studs in a plane parallel to the rock surface the MEM is tilted. Readings are under-measured by an a m o u n t which can be calculated from:
pr
error ..... s
(1 -- cosec A) +
pw 2s
cot A
(1)
where: p = distance of the probe f r o m the centre of the leg with the flat base s = perpendicular distance o f the fiat leg from t h e l i n e joining the other two legs w -- diameter of legs A = half the angle of the cone or the wedge r = radius of the curved surface forming the top of the stud. In the case of an instrument cor~tructed according to the dimensions suggested by High and Hanna (1970) with W = 1.27 cm, A = 60 °, r = 0.317 cm and the ratio p/s a b o u t 0,3938, the error in readings is --0.125 cm, a
M55 large and significant a m o u n t if the gauge is deemed to read to an accuracy of 2 . 5 . 1 0 -3 cm. However, since the error is constant for all readings, t h e a m o u n t of erosion measured is unaffected. The error in the readings can be eliminated b y making the leg with the fiat base shorter than the other t w o b y an a m o u n t which can be evaluated from the relationship: correction = r (1 -- cosec A) + 0.5 w cot A
(2)
Tilting of the MEM Tilting of the probe so that it is no longer normal to the rock surface can be produced b y the design error examined above. It causes the a m o u n t of erosion to be overestimated b y an amount proportional to the a m o u n t of erosion and to the angular deviation (B) of the probe from normality. This positive error can be calculated from: error = E(1
-
-
cos B) where E = measured erosion
(3)
In the case of a standard MEM which is capable of measuring a maximum of 2.540 cm of erosion, the error due to the design amounts to 7 . 6 2 0 . 1 0 -4 cm. This is clearly minor and indicates that this source of error can be di~egarded. However, if the standard design was used with greater measurement capacity, the error might b e c o m e significant. The same error can be produced if the studs are installed in a plane which is not parallel to the rock surface. The a m o u n t of error is then more difficult to evaluate and counteract. Where the rock surface is smooth and planar, use of a tyre-tread depth-gauge helps to ensure that the tops of the studs are the same distance below the surface. However, where it is rough and the plane is not obvious, subjective judgement is the best that can be achieved, so such h u m m o c k y surfaces should be avoided.
Differential erosion Even when studs are emplaced so that the MEM p r o b e is normal to the rock surface, the same t y p e of error can arise through differential erosion which later makes the MEM tilt relative to the surface. Where such differential erosion is recorded and is thought to be significant, the angle B can be found b y calculation of the angle between the planes fitted through the three readings taken on each occasion. Alternatively all studs could be placed in the horizontal or vertical planes and the slope of the rock surface with respect to the plane measured accurately whenever readings are taken, so that the measured amounts of erosion can be corrected. Because an erosion rate is a vector quantity rather than a scalar, as has been previously assumed, differential erosion raises the question of what is actually meant when an erosion rate is quoted. Is it the depth of material removed in the direction normal to the initial rock surface or in the direction
M56
normal to the final surface -- these are n o t the same if the surfaces are n o t parallel. A third alternative is that it refers to the direction which is incident at the same angle to b o t h surfaces. The first of these three directions is preferred b u t whichever is used, error (3) might have to be calculated. It would not, however, if erosion rates were compared in fixed directions such as the horizontal and vertical directions. In this case though, t w o surfaces at different inclinations but suffering the same amount of erosion would seem to be very different. STATISTICAL ANALYSIS OF DATA
Testing the significance of differences Erosion rates in the coastal zone are highly variable in magnitude and m a y be caused b y sporadic events. These characteristics produce samples of erosion rates which are highly positively skewed and there is no reason to suppose that the population distribution from which the samples are derived are any different. Wherever there is a beach resting on the shore platform, erosion rates are more continuous but are controlled by the a m o u n t of incident wave energy (itself a positively skewed distribution) and the characteristics of the beach. Markedly skewed distribution for erosion rates are the result, so again normal population distributions cannot be assumed. Erosion rates at the same MEM unit also vary through time and on the shore platform in particular, erosion rates are highest in summer and, therefore, the variance of erosion rates for this season is much greater than winter variance. Hence, in the assessment of the significance of observed differences b e t w e e n two samples of erosion rates, assumptions of the normality of the population density functions and of their homoscedasticity cannot be met and consequently the use of parametric statistical techniques is not valid. Non-parametric methods such as the Mann-Whitney U test for t w o independent samples, the Wilcoxon matched-pairs signed-ranks test for t w o related samples and the Kruskal-Wallis one-way analysis of variance for more than t w o independent samples (Siegel, 1956) are r e c o m m e n d e d t o replace parametric techniques such as Student's t test and analysis of variance. Although non.parametric techniques are based on ordinal scale data and, therefore, the extra information contained in ratio and interval scale data is not fully used, it is preferable to underutilise information than to add faulty information b y the adoption of invalid statistical assumptions. It can also be noted that these techniques have p o w e r efficiencies of a b o u t 95% when compared with parametric techniques in situations in which the latter are valid.
M57
Relationships between variables Non-parametric techniques, such as the Spearman rank correlation coefficient (Siegel, 1956, pp. 202-213) can also be used to test the signific== of the relationship between two variables. However, if a multivariate approach is adopted, the establishment of relationships between variables is more difficult because of the paucity of appropriate techniques. The correlation model is one of the most powerful and widely used statistical tools, but one of its fundamental assumptions is that of bivariate normality (Binder, 1959). Norris and Hjelm (1961) showed by an empirical study that where there is essentially no correlation in a population, the shape of the sampling distribution for the product moment correlation coefficient is very close to the theoretical distribution, irrespective of the nature or extent of non-normality in the bivariate distribution. However, where there is substantial correlation in the population, sampling distributions of the correlation coefficient calculated from several different non-normal populations deviate markedly from their theoretical distributions and these deviations are highly unpredictable. Furthermore, they are made worse by increased sample sizes or the use of higher levels of significance. Therefore, the bivariate normal correlation model cannot be readily employed for erosion data. The only alternative non-parametric means of assessing multivariate correlation is provided by the Kendall partial rank correlation coefficient (Kendall, 1962, p. 117) but there is no firm theoretical basis for validly extending this coefficient beyond the first order and there is, as yet, no means of testing the significance of the first order partial correlation coefficient (M.G. Kendall, personal communication, 1974). The linear regression model is a more favourable parametric means of examining the relative importance of variables than is the bivariate normal correlation model. The axioms of the linear regression model were summarised by Binder (1959) as: (1) Y=a+BX+e. (2) A total of nX’s are selected and the corresponding values of Y determined. (3) The nX’s are considered fixed - that is, they are chosen prior to making experimental observations and do not vary from sample to sample. (4) The E’Sare normally and independently distributed with means equal to zero and equal variances. The third assumption can be met by measuring erosion at predetermined intervals and at approximately the same time at all sites. The fourth axiom refers to the absence of autocorrelation; this assumption and the first two are satisfied. The important characteristic of the linear regression model is that no assumption of the normality of the marginal distributions of the Y and X variables is made. This implies that it is suitable for the analysis of erosion rates. The importance of a predictor variable in a multiple regression equation can be assessed by its beta coefficient (standardised partial regression coefficient) (Yeomans, 1968, pp. 199-201). However, care must be taken since
M58
this Statistic is usually heavily influenced by the nature of the other variables in the regression equation (Darlington, 1968). This method assumes: (1) All variables which might affect the dependent variable are either included in the regression equation or are uncorrelated with the variables which are included. (2) Terms are included in the regression equation to handle any curvilinear or interactive effects. (3) The dependent variable has no effect on the independent variables.
These assumptions are not usually difficult to satisfy. By testing the significance of the partial regression coefficients from which the beta coefficients are derived it is possible to assess their significance. CONCLUSIONS
The MEM technique is potentially a very powerful tool for analysing marine erosive processes, but it is necessary to be aware of its limitations and to be wary of placing absolute confidence in its measurements. The principal source of error is caused by the installation of studs in a plane which is not parallel to the rock surface. This argues against the placing of measurement sites on rough surfaces. More insidious are the errors produced by differential erosion though they can be calculated. They do, however, depend partly on the meaning attached to the term "erosion rate." The analysis of MEM data presents few problems since the recommended non-parametric techniques are uncomplicated. For multivariate analysis, however, the linear regression model, particularly beta coefficient analysis, is advised and several assumptions must be satisfied. REFERENCES Binder, A., 1959. Considerations of the place of assumptions in correlational analysis. Am. Psychol. 14: 504--10. High, C.J. and Hanna, F.K., 1970. A method for the direct measurement of erosion on rock surfaces. Br. Geomorphol. Kes. Group Tech. Bull., 5: 1--25. Kendall, M.G., 1962, Rank Correlation Methods. Charles Griffin, London, 3rd ed., 148 pp. Norris, R.C. and Hjelm, H.F., 1961. Non-normality and product m o m e n t correlation. J. Exp. Edue., 29: 261--70. Siegel, S., 1956. Non-parametric statistics for the Behavioral Sciences. McGraw Hill, New York, N.Y., 312 pp. Yeomans, K.A., 1968. Applied Statistics. Statistics for the Social Scientist, 2. Penguin Harmondsworth, 397 pp.
in The Netherlands
Letter Section THE MICRO-EROSION METER TECHNIQUE IN A LITTORAL ENVIRONMENT L.A. ROBINSON
Wymondham College, Wymondham,
{Great Britain)
(Revised version accepted August 19,1976)
ABSTRACT Robinson, L.A., 1976. The micro-erosion meter technique in a littoral environment. Mar. Geol., 22: M51-M58. The micro-erosion meter is used to measure erosion rates on bare rock surfaces. Several modifications to its published design are recommended to make it more suitable to coastal studies. The previously unassessed error of the technique is negligible but basic questions about the meaning of erosion rates are raised. The use of non-parametric methods is advised for statistical analysis of the data but, for multivariate studies, betacoefficient analysis using the linear regression model is recommended.
INTRODUCTION
The micro-erosion meter (MEM) is an instrument developed by High and Hanna (1970) to measure directly the rate of erosion on bare surfaces of lithified rock. Though such surfaces are abundant in coastal environments, no study has yet been published which has used the technique. Nor has any assessment of the accuracy of the technique been made. In order to make the MEM more suitable for the high erosion rates common on coasts, several modifications in the published design of the instrument are recommended based on the experience of the author. The amounts of inaccuracy caused by several factors are also examined. It is usual to draw general conclusions from samples of measurements by using statistical methods. However, samples of erosion rates tend to have unusual characteristics which are described and the suitabilities of various statistical tests arediscussed. THE MICRO-EROSION
METRR
(MRM\
M52
plate with a leg near each corner and a pillar placed off-centre on the upper surface (Fig. 1). To this pillar is attached an engineers' dial gauge whose probe passes through a hole in the base-plate so that its tip can rest on the rock surface. Exact relocation of the measurement point on the rock surface is achieved b y the use of the Kelvin Clamp Principle. When readings are being taken, the MEM rests on three metal studs permanently emplaced in the rock and the tip of the probe rests on the rock surface. Three readings can be taken within the triangle formed b y the three studs; this triangle will be
Fig. 1. The micro-erosion m e t e r with the bases of the legs fitted into a plastic plate for transport
M53
termed a "unit". The difference b e t w e e n t w o readings at the same point b u t taken at different times, is the a m o u n t of erosion that has occurred so the erosion rate can be calculated. MODIFICATIONS TO THE MEM
The following modifications are r e c o m m e n d e d on the basis of experience gained in using an MEM on the coast of northeast Yorkshire, England, an area of relatively unresistant shales which allow high erosion rates. (1) An engineers' dial gauge was used with a capacity of 10.160 cm, measurement increments being 0.0025 cm. This allowed most high erosion rates to be measured, b u t the removal of blocks of rock b y the sea did occasionally destroy measurement sites. This was partly because there were practical difficulties in the installation and cleaning of studs placed deeper than a b o u t 6.5 cm below the rock surface. Thus 6.5 cm was the actual capacity of the MEM. The extra 3.5 cm was n o t totally wasted capacity because the space allowed the operator's hand to hold the probe and to lower it gently on to the rock surface since no probe-lowering mechanism was affixed to the dial gauge. Also, when the rock surface was sharply convex upwards between the studs, the gauge registered b e t w e e n 6.5 and 10.16 cm although the studs were not that far below the rock surface. (2) The tip of the probe was convex, not pointed. This modification might introduce some inaccuracy b u t this was thought to be minor compared with the damage to the easily scratched surface of the shales in the study area resulting from the use of a pointed tip. (3) The longer dial gauge necessitated the fitting of longer legs than in the published design. In order to ensure that the legs were firmly fixed in the base plate, the tops of the legs were _riveted in addition to being pressure fitted. During transport the bases of the legs were fitted into holes in a plastic plate. (4) The Kelvin Clamp Principle allows exact relocation of the MEM on the three studs placed in the rock each time the unit is visited. A cone-shaped depression cut in the base of one leg of the MEM acts as a fulcrum when resting on a stud. A wedge-shaped depression aligned with the cone then fixes the instrument in t w o dimensions when it t o o rests on a stud. The third leg has a fiat base and fixes the third dimension. The standard MEM design employs angles of 120 ° for the cone and wedge depressions. Such instruments do not rest tightly on studs placed in vertical or highly inclined rock surfaces so these angles were reduced to 90 ° . (5) The probe projects for half its length below the bases of the legs of the standard MEM. This aids the siting of the instrument on h u m m o c k y and sharply concave surfaces. In the modified instrument the dial gauge was positioned on the pillar so that its tip lay a b o u t I cm above the bases of the legs. This was done for three reasons:
M54
(a) h u m m o c k y surfaces should be avoided because of the errors they produce (see below); (b) there is a danger of damage to the probe when itprojects beyond the bases of the legs; (c) the measurement capacity of the M E M m a y be reduced. (6) It is necessary to have some means of checking the accuracy of readings. A strong datum was made by fixing studs in concrete paving stones, pieces of glass being glued to the concrete to provide the measurement points. This datum was made of materials with low coefficients of thermal expansion and allowed the checking of readings at two bands within the range of the instrument at about 1.25 and 5.0 cm. (7) In the littoralenvironment, sediment soon fillsthe holes in the rock surface in which the studs are placed. This was prevented by fillingthe holes with black plasticine. By freeing the plugs from the walls of their holes with a scooped spatula, they can be lifted out whole without damage to the uppermost bedding laminae around the holes. The use of suitably coloured plasticine has the additional advantage of camouflaging the holes against vandalism. O n the other hand, brightly coloured plasticine enables units located on featureless parts of a shore platform or cliffto be found quickly. A C C U R A C Y OF THE M E M
Length of the legs There is an error inherent in the use of legs of equal length as in the standard design because the leg with the flat base rests on the top of its stud while the depressions in the bases of the other two legs produce contacts at points highe~ than the bases of the legs. The result is t h a t with the tops of the studs in a plane parallel to the rock surface the MEM is tilted. Readings are under-measured by an a m o u n t which can be calculated from:
pr
error ..... s
(1 -- cosec A) +
pw 2s
cot A
(1)
where: p = distance of the probe f r o m the centre of the leg with the flat base s = perpendicular distance o f the fiat leg from t h e l i n e joining the other two legs w -- diameter of legs A = half the angle of the cone or the wedge r = radius of the curved surface forming the top of the stud. In the case of an instrument cor~tructed according to the dimensions suggested by High and Hanna (1970) with W = 1.27 cm, A = 60 °, r = 0.317 cm and the ratio p/s a b o u t 0,3938, the error in readings is --0.125 cm, a
M55 large and significant a m o u n t if the gauge is deemed to read to an accuracy of 2 . 5 . 1 0 -3 cm. However, since the error is constant for all readings, t h e a m o u n t of erosion measured is unaffected. The error in the readings can be eliminated b y making the leg with the fiat base shorter than the other t w o b y an a m o u n t which can be evaluated from the relationship: correction = r (1 -- cosec A) + 0.5 w cot A
(2)
Tilting of the MEM Tilting of the probe so that it is no longer normal to the rock surface can be produced b y the design error examined above. It causes the a m o u n t of erosion to be overestimated b y an amount proportional to the a m o u n t of erosion and to the angular deviation (B) of the probe from normality. This positive error can be calculated from: error = E(1
-
-
cos B) where E = measured erosion
(3)
In the case of a standard MEM which is capable of measuring a maximum of 2.540 cm of erosion, the error due to the design amounts to 7 . 6 2 0 . 1 0 -4 cm. This is clearly minor and indicates that this source of error can be di~egarded. However, if the standard design was used with greater measurement capacity, the error might b e c o m e significant. The same error can be produced if the studs are installed in a plane which is not parallel to the rock surface. The a m o u n t of error is then more difficult to evaluate and counteract. Where the rock surface is smooth and planar, use of a tyre-tread depth-gauge helps to ensure that the tops of the studs are the same distance below the surface. However, where it is rough and the plane is not obvious, subjective judgement is the best that can be achieved, so such h u m m o c k y surfaces should be avoided.
Differential erosion Even when studs are emplaced so that the MEM p r o b e is normal to the rock surface, the same t y p e of error can arise through differential erosion which later makes the MEM tilt relative to the surface. Where such differential erosion is recorded and is thought to be significant, the angle B can be found b y calculation of the angle between the planes fitted through the three readings taken on each occasion. Alternatively all studs could be placed in the horizontal or vertical planes and the slope of the rock surface with respect to the plane measured accurately whenever readings are taken, so that the measured amounts of erosion can be corrected. Because an erosion rate is a vector quantity rather than a scalar, as has been previously assumed, differential erosion raises the question of what is actually meant when an erosion rate is quoted. Is it the depth of material removed in the direction normal to the initial rock surface or in the direction
M56
normal to the final surface -- these are n o t the same if the surfaces are n o t parallel. A third alternative is that it refers to the direction which is incident at the same angle to b o t h surfaces. The first of these three directions is preferred b u t whichever is used, error (3) might have to be calculated. It would not, however, if erosion rates were compared in fixed directions such as the horizontal and vertical directions. In this case though, t w o surfaces at different inclinations but suffering the same amount of erosion would seem to be very different. STATISTICAL ANALYSIS OF DATA
Testing the significance of differences Erosion rates in the coastal zone are highly variable in magnitude and m a y be caused b y sporadic events. These characteristics produce samples of erosion rates which are highly positively skewed and there is no reason to suppose that the population distribution from which the samples are derived are any different. Wherever there is a beach resting on the shore platform, erosion rates are more continuous but are controlled by the a m o u n t of incident wave energy (itself a positively skewed distribution) and the characteristics of the beach. Markedly skewed distribution for erosion rates are the result, so again normal population distributions cannot be assumed. Erosion rates at the same MEM unit also vary through time and on the shore platform in particular, erosion rates are highest in summer and, therefore, the variance of erosion rates for this season is much greater than winter variance. Hence, in the assessment of the significance of observed differences b e t w e e n two samples of erosion rates, assumptions of the normality of the population density functions and of their homoscedasticity cannot be met and consequently the use of parametric statistical techniques is not valid. Non-parametric methods such as the Mann-Whitney U test for t w o independent samples, the Wilcoxon matched-pairs signed-ranks test for t w o related samples and the Kruskal-Wallis one-way analysis of variance for more than t w o independent samples (Siegel, 1956) are r e c o m m e n d e d t o replace parametric techniques such as Student's t test and analysis of variance. Although non.parametric techniques are based on ordinal scale data and, therefore, the extra information contained in ratio and interval scale data is not fully used, it is preferable to underutilise information than to add faulty information b y the adoption of invalid statistical assumptions. It can also be noted that these techniques have p o w e r efficiencies of a b o u t 95% when compared with parametric techniques in situations in which the latter are valid.
M57
Relationships between variables Non-parametric techniques, such as the Spearman rank correlation coefficient (Siegel, 1956, pp. 202-213) can also be used to test the signific== of the relationship between two variables. However, if a multivariate approach is adopted, the establishment of relationships between variables is more difficult because of the paucity of appropriate techniques. The correlation model is one of the most powerful and widely used statistical tools, but one of its fundamental assumptions is that of bivariate normality (Binder, 1959). Norris and Hjelm (1961) showed by an empirical study that where there is essentially no correlation in a population, the shape of the sampling distribution for the product moment correlation coefficient is very close to the theoretical distribution, irrespective of the nature or extent of non-normality in the bivariate distribution. However, where there is substantial correlation in the population, sampling distributions of the correlation coefficient calculated from several different non-normal populations deviate markedly from their theoretical distributions and these deviations are highly unpredictable. Furthermore, they are made worse by increased sample sizes or the use of higher levels of significance. Therefore, the bivariate normal correlation model cannot be readily employed for erosion data. The only alternative non-parametric means of assessing multivariate correlation is provided by the Kendall partial rank correlation coefficient (Kendall, 1962, p. 117) but there is no firm theoretical basis for validly extending this coefficient beyond the first order and there is, as yet, no means of testing the significance of the first order partial correlation coefficient (M.G. Kendall, personal communication, 1974). The linear regression model is a more favourable parametric means of examining the relative importance of variables than is the bivariate normal correlation model. The axioms of the linear regression model were summarised by Binder (1959) as: (1) Y=a+BX+e. (2) A total of nX’s are selected and the corresponding values of Y determined. (3) The nX’s are considered fixed - that is, they are chosen prior to making experimental observations and do not vary from sample to sample. (4) The E’Sare normally and independently distributed with means equal to zero and equal variances. The third assumption can be met by measuring erosion at predetermined intervals and at approximately the same time at all sites. The fourth axiom refers to the absence of autocorrelation; this assumption and the first two are satisfied. The important characteristic of the linear regression model is that no assumption of the normality of the marginal distributions of the Y and X variables is made. This implies that it is suitable for the analysis of erosion rates. The importance of a predictor variable in a multiple regression equation can be assessed by its beta coefficient (standardised partial regression coefficient) (Yeomans, 1968, pp. 199-201). However, care must be taken since
M58
this Statistic is usually heavily influenced by the nature of the other variables in the regression equation (Darlington, 1968). This method assumes: (1) All variables which might affect the dependent variable are either included in the regression equation or are uncorrelated with the variables which are included. (2) Terms are included in the regression equation to handle any curvilinear or interactive effects. (3) The dependent variable has no effect on the independent variables.
These assumptions are not usually difficult to satisfy. By testing the significance of the partial regression coefficients from which the beta coefficients are derived it is possible to assess their significance. CONCLUSIONS
The MEM technique is potentially a very powerful tool for analysing marine erosive processes, but it is necessary to be aware of its limitations and to be wary of placing absolute confidence in its measurements. The principal source of error is caused by the installation of studs in a plane which is not parallel to the rock surface. This argues against the placing of measurement sites on rough surfaces. More insidious are the errors produced by differential erosion though they can be calculated. They do, however, depend partly on the meaning attached to the term "erosion rate." The analysis of MEM data presents few problems since the recommended non-parametric techniques are uncomplicated. For multivariate analysis, however, the linear regression model, particularly beta coefficient analysis, is advised and several assumptions must be satisfied. REFERENCES Binder, A., 1959. Considerations of the place of assumptions in correlational analysis. Am. Psychol. 14: 504--10. High, C.J. and Hanna, F.K., 1970. A method for the direct measurement of erosion on rock surfaces. Br. Geomorphol. Kes. Group Tech. Bull., 5: 1--25. Kendall, M.G., 1962, Rank Correlation Methods. Charles Griffin, London, 3rd ed., 148 pp. Norris, R.C. and Hjelm, H.F., 1961. Non-normality and product m o m e n t correlation. J. Exp. Edue., 29: 261--70. Siegel, S., 1956. Non-parametric statistics for the Behavioral Sciences. McGraw Hill, New York, N.Y., 312 pp. Yeomans, K.A., 1968. Applied Statistics. Statistics for the Social Scientist, 2. Penguin Harmondsworth, 397 pp.