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Keywords:

  • measurement error;
  • prediction variance;
  • photogrammetry;
  • body mass;
  • Weddell seal;
  • Leptonychotes weddellii

Abstract

  1. Top of page
  2. Methods
  3. Results
  4. Discussion
  5. Acknowledgments
  6. Literature Cited

Development and application of photogrammetric mass-estimation techniques in marine mammal studies is becoming increasingly common. When a photogrammetrically estimated mass is used as a covariate in regression modeling, the error associated with estimating mass induces bias in regression statistics and decreases model explanatory power. Thus, it is important to understand and account for prediction variance when addressing ecological questions that require use of estimated mass values. In a simulation study based on data collected from Weddell seals, we developed regression models of pup weaning mass as a function of maternal postparturition mass where maternal mass was directly measured and second where maternal mass was photogrammetrically estimated. We demonstrate that when estimated mass was used, the regression coefficient was biased toward zero and the coefficient of determination was 30% less than the value obtained when using maternal postparturition mass obtained from direct measurement. After applying bias correction procedures, however, the regression coefficient and coefficient of determination were within 2% of their true values. To effectively use photogrammetrically estimated masses, prediction variance should be understood and accounted for in all analyses. The methods presented in this paper are effective and simple techniques to explore and account for prediction variance.

There is increasing interest in body mass dynamics of marine mammals (Arnbom et al. 1997; Mellish et al. 1999; Bowen et al. 2001; Crocker et al. 2001; Beck et al. 2003; Schultz and Bowen 2004) as body mass may be correlated with life history traits (Peters 1983; Partridge and Harvey 1988; Roff 1992) and may be a sensitive indicator of resource availability (Croxall et al. 1988; LeBouef and Crocker 2005). Although the body mass of large mammals can be a difficult attribute to directly measure in the field, photogrammetric techniques allow indirect estimation of body mass and has been used in many cetacean and pinniped species to make indirect measurements of animal morphometrics (Cubbage and Calambokids 1987; Best and Ruther 1992; Castellini and Caulkins 1993; Ratnaswamy and Winn 1993) and to estimate body mass in several large phocid species (northern elephant seals, Mirounga angustirostris, Haley et al. 1991; southern elephant seals, Mirounga leonina,Bell et al. 1997; Weddell seals, Leptonychotes weddellii,Ireland et al. 2006). Photogrammetric mass-estimation techniques are desirable because they are less intrusive than traditional weighing procedures and allow a large number of animals to be sampled. Although a small number of investigators studying large terrestrial mammals have employed photogrammetric techniques to estimate body attributes (e.g., bison, Bison bison, Berger and Cunningham 1994), marine mammal ecologists are aggressively experimenting and employing the technique for a wide range of species and research questions (Cubbage and Calambokids 1987; Haley et al. 1991; Best and Ruther 1992; Castellini and Caulkins 1993; Ratnaswamy and Winn 1993; Deutsch et al. 1994; Haley et al. 1994; Bell et al. 1997; Engelhard et al. 2001; Gordon 2001; Ireland et al. 2006).

Photogrammetric mass-estimation techniques first require development of a regression model to predict animal mass from morphometric measurements. Assuming that the model is appropriate for new animals, the regression model is used to predict mass for animals whose mass is unknown, but whose morphometrics are known. Mass predictions can be used (1) to estimate the mean mass of a population of animals with a given set of morphometrics or (2) to predict the mass of an individual animal with a given set of morphometrics. The expected mass (E{Yh}) of the population of animals with a given set of morphometrics (Xh) is the mean of the distribution of Y. When interest is focused on E{Yh}, sampling error (variation in the possible location of E{Yh} that is present because the regression coefficients are estimated) must be accounted for in the estimation (Neter et al. 1996). In contrast, when predicting mass of a new individual, uncertainty in the predicted mass, referred to as prediction variance, is composed of two sources of variation: (1) sampling error and (2) process variation (variation of the distribution of mass for animals with a given set of morphometrics). Accordingly, the mass of an individual animal with a given set of morphometrics cannot be estimated as precisely as can the mean mass of a population of animals with a given set of morphometrics (Fig. 1). Furthermore, the uncertainty associated with predicting an animal's mass must be incorporated into subsequent analyses that use predicted mass.

image

Figure 1. The contrast between the confidence interval (inner dashed line), which represents the range in which we are 95% confident the mean mass of animals with a given morphometric measurement is located, and the prediction interval (outer dashed line), which represents the range in which we are 95% confident the mass of a new individual with a given morphometric measurement is located, around the regression line (thick solid line) for a regression model of body mass on a single morphometric measurement.

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Regression modeling is perhaps one of the most common analysis techniques employed to investigate biological questions, and below we provide an example of how predicted masses are used in regression modeling, with a goal of clarifying that predicted mass is now a covariate measured with uncertainty. The goal is to estimate the expected value of Y given X; however, in this case X is measured with error so we must estimate the mean of Y with Z. Because Z comes from a linear regression model with additive error and the expected value of Z given X is unbiased for X, our estimate of the mean of Y given Z is a classical error model (Fuller 1987; Buzas et al. 2003). Uncertainty in the covariate value, commonly referred to as measurement error, has received considerable attention in biostatistical literature because errors in covariates induce bias in estimated regression coefficients (Jaech 1985; Stefanski 1985; Fuller 1987). Furthermore, the statistical power of studies based on imprecise measurements may be reduced (Cook and Stefanski 1994). Here, we describe measurement error associated with photogrammetric mass-estimation procedures; however, measurement error is also a more general issue that occurs in a variety of investigations, for example, diet analysis (Santos et al. 2001; Boyd 2002). Although multiple error models exist, we focus this paper on the classical error model (see Buzas et al. 2003 for a discussion of other error models), recognizing that the Berkson error model may be relevant in some experimental situations where the explanatory variables are fixed, while their true values vary for each observation (Berkson 1950).

Many investigations of variability in animal mass will likely use measures of mass as explanatory variables to investigate hypotheses of interest. When doing so, researchers must choose between measuring mass directly (time intensive and perhaps impossible depending on species; minimal measurement error) or indirectly (quick and feasible; measurement error involved). To gain insight into the tradeoffs involved when choosing between direct and indirect mass measurements, we used Monte-Carlo simulation techniques to investigate the statistical properties of regression analyses that used direct and predicted mass measurements as explanatory variables. We used simulation techniques because they are illustrative and transparent for identifying bias created by measurement errors and seeing how bias can be corrected. Using data collected from Weddell seals as an example, this paper investigates how prediction variance resulting from estimating animal mass using photogrammetric techniques affects regression statistics. We first provide an example illustrating that error in a predictor variable induces bias in regression coefficients, then introduce methods that incorporate measurement error and thereby reduce estimation bias, and finally discuss available analysis options for more complex measurement-error scenarios. Through these exercises we demonstrate that understanding prediction variance and the consequences of prediction variance in applications of photogrammetric methods are crucial to sampling design, analysis, and interpretation of results. Investigators who have been employing photogrammetric techniques to predict mass have overlooked prediction variance (Deutsch et al. 1994; Haley et al. 1994; Engelhard et al. 2001), and here we demonstrate how appropriate treatment of prediction variance can enhance biological insights by correcting for biases in coefficient estimates. The statistical properties of the scenarios considered in this paper, which are based on data collected from the Weddell seal, are likely representative of analyses applicable to other marine mammal researchers employing photogrammetric mass-estimation techniques.

Methods

  1. Top of page
  2. Methods
  3. Results
  4. Discussion
  5. Acknowledgments
  6. Literature Cited

Field data from Weddell seals were at the core of our simulation work. Therefore, we first briefly describe the field protocols for directly measuring masses, then the simulation work conducted with these data. We refer the reader to Ireland et al. (2006) for additional information regarding development and applications of the mass-estimation model. During the 2004 pupping season, the maternal postparturition and pup weaning masses of 25 Weddell seal mother–pup pairs were directly measured in Erebus Bay, Antarctica. Maternal postparturition masses were measured by coercing animals onto a digital weight platform (Maxey Manufacturing, Fort Collins, CO), and pup weaning masses were measured by rolling pups into a sling and weighing using a spring scale. The animal mass data used in our simulations were generated based on these directly measured mother–pup masses. Using bootstrap sampling, the mean, variance, and covariance of maternal postparturition and pup weaning masses were calculated, and from these calculations we simulated masses of 75 mother–pup pairs (MATLAB, The MathWorks, Inc., v6.5, Natick, MA. For the purposes of our simulation study, we considered these data as the true maternal and pup masses (measured without errors).

Regression Models and Simulations

To demonstrate the level of bias in estimated regression statistics that results from ignoring measurement error, we developed regression models that regressed pup weaning mass (Y) on maternal parturition mass (X), and we compared the results of analyses employing true mass (measured without error) with those using simulated photogrammetrically predicted mass data (measured with error). Using the linear models function in R (R Development Core Team 2004), we first developed a regression model of pup weaning mass that used the directly measured maternal mass (X) as the explanatory variable, and we considered these regression statistics the true values. Next, we developed a regression model of pup weaning mass that used photogrammatrically estimated maternal masses incorporating prediction variance (Xe, now measured with error) as the explanatory variable and compared these naive regression statistics to the true values.

Prediction variance was calculated from photogrammetrically estimating the postparturition mass of thirty-three animals (Ireland et al. 2006), calculating the mean prediction variance, then introducing the mean variance into each true maternal mass. The variance of each photogrammetrically predicted mass (σ2pred) was calculated as

  • image(1)

where MSE is the mean square error from the mass-estimation model and inline image is the estimated variance of the sampling distribution of predicted mass for a seal with a given set of photogrammetric measurements denoted as Xh (Neter et al. 1996). Based on the standard regression assumptions, the prediction error is normally distributed with mean zero and variance given by the prediction variance. We introduced the mean prediction error into each true mass by shifting each mass by a random number of standard prediction errors, where each random deviate was drawn from a normal distribution with mean 0 and variance 1. The new maternal postparturition mass (that included prediction error) for the ith individual was computed as

  • image(2)

where Xi is the true maternal mass and Xe, i is the maternal mass estimated with error.

Bias Correction Techniques

In the case of simple linear regression, when the explanatory variable has been measured with error, parameter estimates can be bias corrected using the equations,

  • image(3)
  • image(4)
  • image(5)

where β1c, β0c, and R2c are the bias-corrected slope, intercept, and coefficient of determination, mXY is the covariance between the observed X and response Y, mXX is the variance in the observed Xs, and σuu is the error variance in the predictor variable X (Fuller 1987).

Using the data described above, we applied bias-correction techniques to the model of pup weaning mass as a function of maternal postparturition mass, which incorporated prediction errors, and used a simulation approach to estimate β1c, β0c, and R2c. The original data set (the directly measured masses) was re sampled using bootstrapping techniques, and 500 sets of 75 true mother–pup masses were generated. For each data set, the data were first fit to a model regressing pup weaning mass on maternal postparturition mass, and the true regression statistics were calculated (β0T, β1T, R2T). Next, prediction variance was introduced into maternal mass estimates (equation 2), and the naive (β0N, β1N, R2N) and bias-corrected regression statistics (β0c, β1c, R2c) were calculated (equations 3–5). These three sets of regression statistics (true, naive, and bias-corrected) were estimated for each of the 500 data sets, and the average regression statistics were calculated from the replicates.

Effects of Sample Size

We first explored how the bias in regression statistics was influenced by sample size with a goal of determining if the benefits of increasing sample size using photogrammetric techniques could overcome the bias resulting from prediction variance in the photogrammetrically estimated masses. We repeated the regression modeling and bias-correction procedures described in the previous section using sample sizes of 10, 25, 50, 75, 100, 150, and 250, and investigated how the magnitude of the bias and bias-correction procedure performed as sample size changed. Five hundred data sets at each sample size were generated, and the true, naive, and bias-corrected regression statistics were averaged over all replicates.

We next explored how the magnitude of prediction errors influenced the bias in regression statistics. First, we generated twenty-five pairs of mother–pup masses and incorporated prediction variance in maternal postparturition mass, using only half of what we calculated to be our prediction variance. Five hundred data sets were then generated, and the averaged true, naive, and bias-corrected regression statistics were calculated. We repeated these procedures, varying the level of prediction error. We selected levels of prediction errors that would be realistic results of photogrammetric mass-estimation procedures (the error observed in our field study was multiplied by 0.5, 1, 1.5, and 2). Finally, we repeated these procedures for sample sizes of 50, 75, 100, and 250 pairs of animals. Five hundred data sets of each sample size were generated for each of the four levels of prediction error.

Results

  1. Top of page
  2. Methods
  3. Results
  4. Discussion
  5. Acknowledgments
  6. Literature Cited

When simulated pup weaning masses were regressed onto simulated maternal postparturition masses that were measured with error, the naïve regression coefficient for maternal mass (β1N) was biased toward zero, and the naïve coefficient of determination was lower (by 30%) than the true value. The true regression statistics (with standard errors in parentheses) were β0T=–1.32(10.30), β1T= 0.23(0.02), and R2T= 0.58, whereas the naïve regression statistics were β0N= 30.17(10.21), β1N= 0.16(0.02), and R2N= 0.41. In contrast, the bias-corrected regression coefficient approximated the true value and the bias-corrected coefficient of determination was within 2% of the true value (β0C=−2.70(11.64), β1C= 0.23(0.06), and R2C= 0.59).

At all sample sizes, the bias-corrected regression coefficients were nearer to the true values than the naive estimates (Fig. 2). At the smallest sample size tested, n= 10, a modest amount of bias remained in the bias-corrected value (β1T= 0.23[0.07], β1C= 0.19[0.90]), but it was an improvement over the naive estimate (β1N= 0.16[0.07]). At this small sample size, the precision of the bias-correction procedure was low. As sample sizes increased, the bias-corrected estimates approached the true values, whereas the naive estimates were consistently biased low. At sample sizes of 50 and above, the bias-corrected values approximated the true values very well and the standard error around the coefficient estimate decreased. Given the level of error tested here, the sample size of photogrammetrically estimated masses had to be doubled to obtain the same explanatory power as would have been realized if the maternal masses had been directly measured. The precision of the bias-corrected regression statistics increased as sample size increased, and for this data set the standard error around the coefficient estimate stabilized at a sample size of approximately seventy-five (Fig. 2).

image

Figure 2. The effect of sample size on the regression of pup weaning mass on maternal postparturition mass when the predictor variable maternal mass was directly measured (true), estimated with errors (naïve), and estimated with errors and corrected for bias (bias-corrected). Panel A shows the relationship between sample size and the coefficient estimate and panel B shows the relationship between sample size and precision of the coefficient estimate. In panel B, the true and naïve standard errors were similar and are displayed as one value.

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As the magnitude of the measurement error increased, the bias-correction procedure required larger sample sizes to approximate the true regression statistics. The sample size of photogrammetrically estimated masses needed to obtain explanatory power equivalent to that of directly measured masses varied with the level of prediction error. At the original level of error, the bias-corrected regression coefficient approximated the true value at a sample sizes of fifty and above. However, when prediction error was doubled, a minimum sample size of 250 was required for the bias-corrected regression coefficient to approximate the true value (Fig. 3).

image

Figure 3. The effects of sample size and level of prediction error on the bias in the corrected regression coefficient, calculated as β1T−β1c, for the regression of pup weaning mass on maternal postparturition mass. As the magnitude of prediction error increased, larger sample sizes were necessary for the bias-corrected regression coefficient to approximate the true value.

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Discussion

  1. Top of page
  2. Methods
  3. Results
  4. Discussion
  5. Acknowledgments
  6. Literature Cited

Estimating animal mass using photogrammetric methods is a logistically efficient and nonintrusive data collection technique. However, if mass estimates are to be used as explanatory variables in subsequent analyses, it is imperative that the prediction variance introduced through the mass-estimation procedure be considered and properly accounted for. We have demonstrated that the use of photogrammetrically estimated masses (which include prediction error) as covariates in regression analysis causes the associated regression coefficient to be biased low. It is also important to understand that including a covariate that was measured with error in a multiple regression model not only biases the regression coefficient for this covariate but also introduces error into the coefficient estimates for each of the other covariates, even if these other covariates are measured without error (Fuller 1987). Thus, the measurement error introduced when estimated masses are used significantly reduces the explanatory power of a regression model (Cook and Stefanski 1994). In the example provided, modeling pup weaning mass as a function of maternal postparturition mass, the explanatory power of the naive regression model was 30% less than the explanatory power of the true and bias-corrected models. The mass-estimation model employed in these analyses had a high coefficient of determination (r2= 0.90), which is similar to what was found for mass-estimation models that were developed for other species (northern elephant seal, r2= 0.93–0.95, Haley et al. 1991; southern elephant seal, r2= 0.84–0.99, Bell et al. 1997). However, even with such high predictive precision, masses estimated from our model, when used as covariates in subsequent modeling, still significantly biased regression coefficients and decreased the explanatory power of the subsequent modeling. We expect researchers employing similar photogrammetric mass-estimation techniques to also experience bias and reduced explanatory power if not correcting for prediction error, and the simulation-based framework presented here is an applicable method for exploring prediction variance and tradeoffs between sample size and level of precision.

We successfully employed Fuller's (1987) methodology to account for measurement error in our simple linear regression example. However, for more complex situations, alternative methodologies of handling measurement error may be needed. Alternatives include regression calibration, simulation extrapolation, structural equation modeling, and Bayesian methods (Carroll and Stefanski 1990; Cook and Stefanski 1994; Maruyama 1998; Congdon 2001). The regression calibration method relates a dependent variable, Y, by a regression model to covariate X, when X is not directly observed, but W (X with error) is observed in its place. To estimate the regression coefficients, X is replaced with an estimate that is a function of W, X(W) = E(X|W) (Carroll and Stefanski 1990; Carroll et al. 1995). The simulation-extrapolation method first incrementally adds additional measurement error to the data and iteratively computes the regression statistics; it then fits a model to these estimates and the variance of the added errors and extrapolates back to the case of no measurement error (Cook and Stefanski 1994; Stefanski and Cook 1995; Polzehl and Zwanzig 2004). To use this method, the error variance must be known or at least well estimated. The advantage of simulation extrapolation is that the method is general, making it useful in applications where the model of interest is novel and conventional methods have not been developed. Structural equation modeling is an extension of the generalized linear model that has more flexible assumptions, and can account for measurement error by having multiple indicators for the latent (true, unobservable variable). Unlike regression analysis, structural equation models test the conceptual and measurement models simultaneously, and measurement errors are identified in the measurement model (Maruyama 1998). The Bayesian state-space approach, which models the observation process and the true process of interest simultaneously, can also be applied to measurement error problems (Richardson 2002). Furthermore, Bayesian methods that allow the measurement error to combine classical and Berkson components have recently been developed (Mallick et al. 2002). Selection of methodology to incorporate measurement error should depend on the nature of the questions being addressed and the applicability of the various techniques. In this paper, we used Fuller's (1987) methods because they are straightforward and applicable to the simple linear regression model of interest. When estimated masses are used in more complex calculations, such as to estimate the ratio of pup mass gained to maternal mass loss (mass transfer efficiency) or multiple regression analysis, correcting for measurement error is more complex and novel methods to account for prediction variance must be employed.

Careful sampling design and thorough planning are necessary to effectively utilize estimated masses to address biological questions. First, a thorough understanding of the sources of variance is necessary and allows the investigator to proactively take measures toward reducing this variance. The largest source of prediction variance in the data we utilized was associated with the unexplained variation in the original mass-estimation model (MSE). This unexplained variance may be reduced by rigorous standardization of data collection and image analysis protocols. Additionally, increasing sample size of the mass-estimation model and collecting data points at the extremes of the predictive range can further reduce this variance. Second, methods of accounting for prediction variance in estimated masses should be developed when evaluating biological hypotheses that incorporate predicted masses. In regression analysis, failure to account for prediction error in covariates reduces the explanatory power of models and results in biased coefficient estimates. The simple methods described in this paper can correct for this bias. Finally, the precision of photogrammetric techniques must be quantified and considered in defining objectives and sampling protocols. Annual mean parturition or weaning masses can be well estimated using photogrammetrically estimated masses (Ireland et al. 2006); however, using estimated masses with large prediction variances in individual-based analyses may be more difficult and will likely require at least a doubling of sample sizes for relationships to be detected with the same power that would be realized if direct mass measurements were obtained.

Acknowledgments

  1. Top of page
  2. Methods
  3. Results
  4. Discussion
  5. Acknowledgments
  6. Literature Cited

Funding for this project was provided by the National Science Foundation, OPP-0225110. Research was conducted under marine mammal permit 1032-1679-00 and Montana State University Animal Care and Use Committee protocol 1093.

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  1. Top of page
  2. Methods
  3. Results
  4. Discussion
  5. Acknowledgments
  6. Literature Cited
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