1. Introduction
Much oceanographic field research seeks to establish a functional relation, that is, find the slope and intercept of the line that best fits the bivariate scatter between two field-observed variables. A good example of this is found in Tett et al. (1975), where particulate carbon and particulate phosphorus are compared. Model II regression has often been suggested in this type of situation to estimate the functional relationship between two variables (Ricker 1973; Sokal and Rohlf 1995; Warton et al. 2006). The literature is substantial regarding estimating a linear functional relationship, dating back to at least Pearson (1901), and several different methods have been proposed. A model II analysis was applied by Sverdrup (1916) in analyzing meteorological variables as early as 1916. Ricker (1973) proposed the use of model II regression in fishery studies. Laws and Archie (1981) asserted the advisability of using model II regression for various field studies, such as the investigation of oxygen consumption per body mass, that is, the metabolic rate, by oceanic zooplankton in the study of Ikeda (1970). Laws and Archie (1981) pointed out that studies of metabolic rate per body mass, phytoplankton ratios of carbon to chlorophyll, and morphometric analyses, all of which are determined from regression slopes, lend themselves to more efficient analyses by model II regression. Further studies of the variations of particulate carbon body mass regressed against chlorophyll concentration of phytoplankton as discussed by Banse (1977) are also amenable to model II analysis.
In virtually all of the studies mentioned above, arbitrary bivariate pairings of variables were modeled, even though the situations investigated actually involved three variables. For example, the studies of Ikeda (1970) involved oxygen uptake, body mass, and environmental temperature, and Banse (1977) considered the variations in cellular particulate phosphorus content and other environmental variables in addition to chlorophyll concentration and particulate organic carbon mass. The multivariate nature of oceanographic investigations is becoming increasingly recognized, as in the studies of Fichot et al. (2008), while Gallie and Murtha (1992) applied a multiple regression analysis to the biogeo-optical properties of inland waters that will be instructive to oceanographers. Stavn and Richter (2008) applied model II multiple regression to determine the mass-specific scattering cross sections of suspended matter in the ocean, both inorganic and organic, to allow more accurate determinations of these materials by satellite remote sensing. All of these situations come under the subject of the interaction of physical properties with coupled biogeochemical cycles (Schlesinger et al. 2011).
In this paper a method for determining model II multiple regression estimates is presented. The method is applied to the determination of optimum mass-specific optical scattering cross sections for suspended matter in the coastal ocean. Section 2 contains a brief background on the development of model II regression methods. Section 3 introduces the notation for the bivariate case, applied to estimate the relation between fish mass and egg production. Section 4 extends the method to more than two variables and provides an example for the three-variable case: determining optical scattering cross sections from oceanographic field measurements. Section 5 contains a discussion of theoretical issues of modeling relations and suggests other possible areas of application of model II multiple regression in oceanography.
2. Model II regression methods
The most widely recommended methods for model II regression are the major axis (MA) method and the standardized (or reduced) major axis (SMA), also referred to as the geometric mean method. These methods are well developed for the case of two variables (e.g., Laws 1997; Sokal and Rohlf 1995; Warton et al. 2006), but not for three or more variables. Thus, the primary goal of this paper is to present a method of estimating the functional relationship between any number of variables. Many authors have discussed aspects of regression for describing functional relationships, often in a broader context (see Anderson 1984; Kendall and Stuart 1977). However, we do not intend for this to be a general discussion of the various issues of estimation arising in those contexts. Instead, our goal is to outline a straightforward procedure for estimating the parameters in a multivariate functional relationship.
Pearson (1901) showed that the best-fitting line to a system of points is in the direction of the major axis of the corresponding ellipse. Kermack and Haldane (1950) later showed that Pearson’s method was not invariant to changes in scale of the variables and proposed the use of a “reduced” major axis, where the major axis method is applied to standardized variables. Ricker (1973) showed that geometric mean regression is identical to reduced major axis regression but far easier to compute. Jolicoeur (1975), however, argued against the reduced (or “standard”) major axis method, and Sprent and Dolby (1980) suggested using the line that bisects the minor angle between the two model I (ordinary least squares) regressions as the estimate of the functional relationship. Warton et al. (2006) argue that the term “standardized major axis” is best used to describe the major axis method applied to standardized data, and this is the term that will be used in this paper.
With the exception of Pearson (1901), who also considered the three-variable case, all of the previously mentioned references address only the case of relating two variables. Pearson’s 1901 paper is considered the foundation of principal component analysis (PCA), and PCA provides a framework for estimating functional relationships that can easily be extended to more than two variables. While the relationship between PCA and model II methods is generally appreciated, rarely are model II estimates presented using the results of a PCA analysis. Rencher (2002) described the relation between PCA and model II estimates for the two-variable case. In section 3, we present the notation for the two-variable case, and in section 4 we derive estimates for models with three or more variables.
3. Model II regression using PCA
a. Two-variable case
The two-variable case is considered here to introduce the notation required for extension to the multivariate case. One advantage of using PCA is that most statistical software packages (e.g., SAS, R) contain procedures to perform PCA, thus allowing model II estimates to be calculated without special programs.
The following example of biological oceanographic data provides an illustration of the ideas described above:
Example 1—Sokal and Rohlf (1995) reported weights (Y1) of unspawned female cabezon (a California marine fish, Scorpaenichthys marmoratus) and the number of eggs subsequently produced (Y2) for 11 fish. The data are given in Table 1.
Weights (Y1) of unspawned female cabezon (a California fish, Scorpaenichthys marmoratus) and the number of eggs they subsequently produce (Y2) for 11 fish.
For the data from Table 1, 
Scatterplot of Y2 (number of eggs from each female after spawning) vs Y1 (weight of unspawned female Cabezon) from Table 1, with MA and SMA fitted lines superimposed. Note the similarity of the scales of the two axes.
Citation: Journal of Atmospheric and Oceanic Technology 31, 7; 10.1175/JTECH-D-13-00210.1
Scatterplot of Y2 (number of eggs from each female after spawning) vs Y1 (weight of unspawned female Cabezon) from Table 1, with MA and SMA fitted lines superimposed. Note the similarity of the scales of the two axes.
Citation: Journal of Atmospheric and Oceanic Technology 31, 7; 10.1175/JTECH-D-13-00210.1
Scatterplot of Y2 (number of eggs from each female after spawning) vs Y1 (weight of unspawned female Cabezon) from Table 1, with MA and SMA fitted lines superimposed. Note the similarity of the scales of the two axes.
Citation: Journal of Atmospheric and Oceanic Technology 31, 7; 10.1175/JTECH-D-13-00210.1
b. Using standardized data
If standardized data are used for the Cabezon weight and egg production data in example 1, then the eigenvectors of the correlation matrix are 
4. Extension to more than two variables
a. PCA with three variables
Example 2—Oceanographic field data were collected on the optical scattering coefficient at a wavelength of 555 nm (b555), the particulate inorganic matter concentration (PIM), and the particulate organic matter concentration (POM) of surface samples, from 24 field stations at Mobile Bay, Alabama (see Stavn and Richter 2008). The data are given in Table 2.
Data on the total scattering coefficient at a wavelength of 555 nm [b555 (m−1)], the concentration of particulate inorganic matter [PIM (g m−3)], and particulate organic matter [POM (g m−3)] of surface samples from 24 field stations at Mobile Bay, Alabama.
For the data from Table 2, 
b. Using standardized data
Returning to example 2:
These values of b555 are compared with values determined in the same way from the Southwest Pass, Mississippi River, a theoretical calculation for suspended coccolithophore plates, and values of a claimed empirical proxy for the slope coefficient in Table 3. The R code (R Core Team 2012) used to perform the calculations in sections 4a and 4b is included in the appendix.
Mass-specific scattering cross sections at 555 nm compared with mass-specific scattering coefficients at 555 nm (Babin et al. 2003). The asterisk represents the calculation that is from theoretical data at 532 nm reported in Stavn and Richter (2008).
c. Interpretation of slope coefficients
As in model I multiple regression, each of the coefficients of 
Three-dimensional plot of suspended mineral (PIMreal; mg L−1) concentration (Y1), organic matter (POMreal; mg L−1) concentration (Y2), and optical scattering coefficient (Y3) at 412 nm [bp(412); m−1]. Plane represents predicted bp(412) values from first two principal components of standardized MA analysis. Observed bp(412) values are black above plane and red below plane.
Citation: Journal of Atmospheric and Oceanic Technology 31, 7; 10.1175/JTECH-D-13-00210.1
Three-dimensional plot of suspended mineral (PIMreal; mg L−1) concentration (Y1), organic matter (POMreal; mg L−1) concentration (Y2), and optical scattering coefficient (Y3) at 412 nm [bp(412); m−1]. Plane represents predicted bp(412) values from first two principal components of standardized MA analysis. Observed bp(412) values are black above plane and red below plane.
Citation: Journal of Atmospheric and Oceanic Technology 31, 7; 10.1175/JTECH-D-13-00210.1
Three-dimensional plot of suspended mineral (PIMreal; mg L−1) concentration (Y1), organic matter (POMreal; mg L−1) concentration (Y2), and optical scattering coefficient (Y3) at 412 nm [bp(412); m−1]. Plane represents predicted bp(412) values from first two principal components of standardized MA analysis. Observed bp(412) values are black above plane and red below plane.
Citation: Journal of Atmospheric and Oceanic Technology 31, 7; 10.1175/JTECH-D-13-00210.1
d. Extension to more than three variables
e. Adjusted versus unadjusted coefficients
When considering the analysis of environmental data, collected over various nonconstant conditions of environment, it becomes necessary to factor in other relevant variables in addition to a single variable pair of interest. Therefore, most models for environmental data are actually multivariate in nature. Coefficients adjusted for other variables are easy to obtain using model I regression, but in situations where model II regression is more appropriate, researchers have usually resorted to fitting separate one-predictor models to obtain coefficient estimates (Banse 1977; Ikeda 1970; Steele and Baird 1961). In such a situation, if the explanatory variables are related, then regression applied in turn to variable pairs will result in incorrect estimates of the regression coefficients. Multiple predictor models provide more appropriate estimates than separate one-predictor models, especially when there is substantial dependence between the predictors.
Consider the coefficients obtained from Eq. (4) in section 4b, using standardized data. If separate one-predictor model II regression models, using Eq. (2), are fit to the data of Table 2, then the unadjusted slope coefficients for PIM and POM are 0.841 and 3.934, respectively. Recall that the estimates obtained from the SMA method were 0.733 and 0.764, respectively, which are substantially different. This is not surprising, since the variables PIM and POM are moderately correlated (r = 0.57). Note also that the same phenomenon occurs using model I regression, where the multiple model I regression model estimates the coefficients to be 0.690 and 0.846 for PIM and POM, respectively, compared to 0.793 and 2.686, respectively, for the separate model I univariate regression models.
f. Confidence intervals for slope coefficients
Once slope estimates are obtained, it might be desirable to provide confidence limits on the true parameter values. Anderson (1963) discussed approximate confidence intervals for eigenvalues under asymptotic multivariate normality. Jolicoeur (1968) proposed a small sample interval for the two-variable case under bivariate normality. Clarke and Van Gorder (2013) derived the density of the ratio of the true regression coefficient to that of the true geometric mean coefficient and use it to construct confidence intervals for the true regression coefficient, also for the two-variable case. Alternatively, nonparametric multivariate bootstrapping may be used to construct confidence limits for the slope coefficients. Advantages of the bootstrap technique are that no assumptions regarding the form of the joint distribution of the variables are necessary, and it can be used to calculate confidence limits using estimates obtained from both standardized and unstandardized data.
Consider the two-variable case discussed in section 3a, and let the true slope coefficient be denoted by 
For the multivariate case, the same process can be used. For example, for the three-variable case described in example 2 of section 4a random triplets (PIM, POM, b555) are selected, with replacement, and the slope coefficients for PIM and POM calculated each time. This results in two bootstrap distributions, one for each of the coefficients. Then the appropriate quantiles of the bootstrap sampling distributions provide confidence limits. This process can be extended to any number of variables. In example 2, the estimates using the MA method [Eq. (3)] were 0.558 and 2.000 for PIM and POM, respectively. Again, using 10 000 multivariate bootstrap samples, the 95% confidence intervals are 
Note that in example 2, the variability in the observed values of PIM was substantially higher than that of POM (the variance of PIM is almost 24 times as large). The result is a much larger slope estimate for POM compared to PIM, and a great deal of variability in the confidence interval estimates. As was discussed in section 3b, when the variability of the variables is substantially different, calculations on standardized values may yield more useful results. Using the SMA method [Eq. (4)], the slope for PIM is estimated as 0.733, with 95% confidence interval 
5. Discussion
The importance of multivariate models in oceanography and environmental studies is becoming increasingly recognized, and more studies are taking advantage of this point of view. In situations where determining the best functional relation is the goal, the results in section 4 provide straightforward extensions of the MA and SMA methods that can be used to find slope coefficients to estimate a functional relation in a multivariate system. There are many examples in the literature where the parameters of a multivariate model have been estimated using estimates from a series of bivariate models. However, this can result in incorrect estimates, since when the variables in the system are correlated, the slope coefficient obtained in a bivariate model is affected by the omitted variables.
There have been attempts to perform analyses of the ocean environment similar to that possible with model II multiple regression by a bivariate approximation to the mass-specific scattering cross section called the mass-specific scattering coefficient, which is equal to bλ/TSS, the ratio of the total volume scattering coefficient to the total suspended solids (the sum of suspended mineral matter and suspended organic matter) (Babin et al. 2003). There are two problems with this empirical ratio: the volume total scattering coefficient, 
The assertion that the approximation to the mass-specific scattering cross section provided by the coastal average of b555/TSS is 0.5 m2 g−1 simply does not stand up to scrutiny. Stavn and Richter (2008) used multivariate model II methods to determine the functional relations of the optical volume scattering coefficient of the ocean hydrosol and the concentration of suspended mineral and organic matter in the ocean. The model II multivariate analysis yields the mass-specific optical scattering cross sections of suspended mineral and organic matter. These optical cross sections allow the inversion of the suspended mass of mineral and organic matter by their separate contributions to the remote sensing reflectance signal detected by satellites and aircraft. Their magnitude and spectral slope are related to the modal size of the suspended matter size distribution and their fractal dimensions, respectively (Stavn 2012). Furthermore, these cross sections allow the partitioning of the hydrosol scattering coefficient into the scattering coefficient due to suspended mineral matter and the scattering coefficient due to suspended organic matter. From these relations it is possible to derive algorithms and models of photon penetration and reflectance, determine a photon budget, and quantitatively account for the components of the remote sensing signal. The values of σ(PIM) reported in Stavn and Richter (2008) vary from 0.57 to 0.76 m2 g−1 for the northern Gulf of Mexico in a spectral range comparable to the results reported by Babin et al. (2003). Furthermore, the organic values in Table 3 are significantly less than the organic value of 1.0 m2 g−1 reported for the empirical ratio, while the Southwest Pass organic value actually approaches the value of the empirical ratio reported for suspended minerals. The closest value of Table 3 to that reported for suspended organic matter (Babin et al. 2003) is the 0.98 m2 g−1 calculated for lith plates by Stavn and Richter (2008), except that the lith plates are essentially pure mineral, CaCO3, that would be expected from coccolithophores growing in the open ocean. We see, then, that the empirical ratio bλ/TSS does not supply unequivocal information about the optical nature nor the type of suspended matter in the coastal ocean.
Model II methods are often recommended when all variables are measured subject to error (Sokol and Rholf 1995). However, as Warton et al. (2006) point out, this is not the most effective justification for using model II methods. Rather, model II methods should be preferred when the goal is to estimate the linear relation that best describes the scatter of the variables. An example of this use was the estimation of scattering coefficients discussed earlier in this section. Alternatively, if the goal is prediction of one variable value based on the values of one or more predictors, then model I regression is usually preferred.
It should also be noted that in addition to potential measurement error, which also may include sampling error, equation error may also exist. If the equation is misspecified, then estimates for both model I and model II methods will generally be biased. Further research is needed to better understand properties of multiple model II estimates.
6. Conclusions
A straightforward procedure for estimating parameters in a model II multiple regression has been presented. The utilization of MA and SMA and where each may be most appropriate was investigated. The procedure was applied to determine the mass-specific scattering cross section of suspended mineral and organic matter in the ocean. Many field situations presently investigated or analyzed as bivariate regressions are actually embedded in a multivariate system. Multivariate estimation is preferred in these situations to obtain more accurate estimates.
Acknowledgments
SJR wishes to acknowledge support, in part, by a University of North Carolina at Greensboro Research Assignment. RHS wishes to acknowledge the valuable discussions on regression from Charles C. Trees, Center for Hydro-Optics and Remote Sensing, San Diego State University, San Diego California. In addition there was the valuable support of a Research Assignment Leave from the University of North Carolina at Greensboro; a National Research Council Research Associateship through NASA, Applied Sciences Directorate, Stennis Space Center, Mississippi; a summer fellowship from the American Society for Engineering Education through the Naval Research Laboratory, Code 7323, Coastal and Marginal Seas Section, Stennis Space Center, Mississippi; early support from ONR Grant N00014-97-0812; and the Naval Research Lab Project to Richard W. Gould Jr., “Predicting Coastal Bio-Optical Response to Atmospheric/Oceanographic Forcing,” Stennis Space Center, Mississippi.
APPENDIX
R Code
#Analysis of Table 2 data
#Enter data vectors. (Each variable entered as a separate vector)
PIM=c(11.36055,6.98143,6.89490,14.60099,12.52414,5.40229,6.45010,1.56994,2.15346,22.30509,4.67338,5.00835,5.32663,5.45830,9.98043,5.67296,6.88813,3.20948,4.56163,6.55727,4.62544,5.48337,3.87877,2.80227)
POM=c(2.35529,1.48723,1.15241,3.00136,1.59180,2.53472,2.20666,0.17953,0.45311,3.28352,2.05125,0.52108,1.94375,2.16183,2.86684,1.80909,3.10551,3.32273,1.69203,1.06135,1.05097,0.69146,0.71080,0.48471)
b555 = c(8.40735,5.85045,6.90855,11.30800,10.02730,3.39939,5.43069,1.08864,1.84174,17.94280,4.85276,1.19745,4.99449,5.05150,9.11405,8.39094,7.57381,4.38397,3.77923,4.91364,3.09047,3.66245,2.40817,1.76638)
#Combine vectors into single data matrix
combined=cbind(PIM,POM,b555)
#Compute sample size
n=nrow(combined)
#Compute means and standard deviations and display results
ybar_PIM=mean(PIM)
ybar_POM=mean(POM)
ybar_b555 = mean(b555)
sd_PIM=sd(PIM)
sd_POM=sd(POM)
sd_b555 = sd(b555)
ybar_PIM
ybar_POM
ybar_b555
sd_PIM
sd_POM
sd_b555
#Compute covariance (
S=cov(combined)
R=cor(combined)
S
R
#Calculate eigenvalues and eigenvectors and display results
PCA_S=eigen(S)
PCA_R=eigen(R)
PCA_S
PCA_R
#Compute intercept and slope estimates presented in section 4a, and display results
intercept_S=PCA_S$vectors[1,3]/PCA_S$vectors[3,3]*mean(PIM)+PCA_S$vectors[2,3]/PCA_S$vectors[3,3]*mean(POM)+mean(b555)
slope_S.PIM=-PCA_S$vectors[1,3]/PCA_S$vectors[3,3]
slope_S.POM=-PCA_S$vectors[2,3]/PCA_S$vectors[3,3]
intercept_S
slope_S.PIM
slope_S.POM
#Compute intercept and slope estimates presented in section 4b, and display results
intercept_R=PCA_R$vectors[1,3]*sd_b555/PCA_R$vectors[3,3]/sd_PIM*mean(PIM)+-PCA_R$vectors[2,3]*sd_b555/PCA_R$vectors[3,3]/sd_POM*mean(POM)+mean(b555)
slope_R.PIM=-PCA_R$vectors[1,3]*sd_b555/PCA_R$vectors[3,3]/sd_PIM
slope_R.POM=-PCA_R$vectors[2,3]*sd_b555/PCA_R$vectors[3,3]/sd_POM
intercept_R
slope_R.PIM
slope_R.POM
#Generate bootstrap percentile intervals, presented in section 4f, for slope estimates and display #results.
set.seed(1234)
boots=10000
slopePIM_S.boot <- numeric(boots)
slopePOM_S.boot <- numeric(boots)
slopePIM_R.boot <- numeric(boots)
slopePOM_R.boot <- numeric(boots)
for(i in 1:boots)
{
index = 1:n
bootindex = sample(index, n, replace=T)
bootsample = combined[bootindex,]
Sb=cov(bootsample)
Rb=cor(bootsample)
PCA_Sb=eigen(Sb)
PCA_Rb=eigen(Rb)
slopePIM_S.boot[i]=-PCA_Sb$vectors[1,3]/PCA_Sb$vectors[3,3]
slopePOM_S.boot[i]=-PCA_Sb$vectors[2,3]/PCA_Sb$vectors[3,3]
slopePIM_R.boot[i]=-PCA_Rb$vectors[1,3]*sd(b555)/PCA_Rb$vectors[3,3]/sd(PIM)
slopePOM_R.boot[i]=-PCA_Rb$vectors[2,3]*sd(b555)/PCA_Rb$vectors[3,3]/sd(POM)
}
#Compute lower and upper confidence limits and display results
L95_S.PIM=quantile(slopePIM_S.boot,0.025)
U95_S.PIM=quantile(slopePIM_S.boot,0.975)
L95_S.POM=quantile(slopePOM_S.boot,0.025)
U95_S.POM=quantile(slopePOM_S.boot,0.975)
slope_S.PIM
L95_S.PIM
U95_S.PIM
slope_S.POM
L95_S.POM
U95_S.POM
L95_R.PIM=quantile(slopePIM_R.boot,0.025)
U95_R.PIM=quantile(slopePIM_R.boot,0.975)
L95_R.POM=quantile(slopePOM_R.boot,0.025)
U95_R.POM=quantile(slopePOM_R.boot,0.975)
slope_R.PIM
L95_R.PIM
U95_R.PIM
slope_R.POM
L95_R.POM
U95_R.POM
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