Explorations in statistics: regression

Learning about statistics is a lot like learning about science: the learning is more meaningful if you can actively explore. This seventh installment of Explorations in Statistics explores regression, a technique that estimates the nature of the relationship between two things for which we may only surmise a mechanistic or predictive connection. Regression helps us answer three questions: does some variable Y depend on another variable X; if so, what is the nature of the relationship between Y and X; and for some value of X, what value of Y do we predict? Residual plots are an essential component of a thorough regression analysis: they help us decide if our statistical regression model of the relationship between Y and X is appropriate.     

Regression of calculated variables in the presence of shared measurement error

To test hypotheses regarding relations between meaningful parameters, it is often necessary to calculate these parameters from other directly measured variables. For example, the relationship between O2 consumption and O2 delivery may be of interest, although these may be computed from measurements of cardiac output and blood O2 contents. If a measured variable is used in the calculation of two derived parameters, error in the measurement will couple the calculated parameters and introduce a bias, which can lead to incorrect conclusions. This paper presents a method of correcting for this bias in the linear regression coefficient and the Pearson correlation coefficient when calculations involve the nonlinear and linear combination of the measured variables. The general solution is obtained when the first two terms of a Taylor series expansion of the function can be used to represent the function, as in the case of multiplication. A significance test for the hypothesis that the regression coefficient is equal to zero is also presented. Physiological examples are provided demonstrating this technique, and the correction methods are also applied in simulations to verify the adequacy of the technique and to test for the magnitude of the coupling effect. In two previous studies of O2 consumption and delivery, the effect of coupled error is shown to be small when the range of O2 deliveries studied is large, and measurement errors are of reasonable size.     

The Squid Giant Axon: Mathematical Models

The voltage clamp results of Hodgkin and Huxley have been reanalyzed in terms of alternative mathematical models. The model used for the potassium conductance changes is similar to that of the HH model except that an empirical functional relationship replaces the fourth power Law used by HH and the twenty-fifth power law used by Cole and Moore. The model used for the sodium conductance changes involves the explicit use of one variable only rather than the two variables m and h of HH. The rise and fall of the sodium conductance during a depolarizing voltage clamp is obtained by specifying that this one variable satisfies a second order differential equation which results from the coupling of two first order equations. Not only can the adjustable parameters of these models be made to give good fit to the clamp conductance data but the models can also then be used to compute action potential curves. Theoretical interpretations can also be given to these mathematical models.     

Predictors of Views about Punishing Animal Abuse

The current study used scenarios of animal cruelty based on actual events to explore what factors predict perceptions of punishments for animal cruelty. Five hundred thirty-eight university undergraduates participated in the study. Participants read scenarios based on an actual event of animal cruelty and with information on statutes addressing animal cruelty in their state. Participants were then asked to give ratings for punishment of the animal cruelty perpetrator. A canonical correlation analysis was used to test the multivariate shared relationship between the set of eight predictor variables and the criterion variable set of 15 punishment ratings. The response distributions were highly skewed for seven of the 15 criterion variables, highlighting the strong sentiments participants expressed against animal cruelty. The canonical correlation analysis identified significant criterion variables that were predicted by two of the empirically manipulated predictor variables (Perpetrator Age and Location of Crime) along with one subject variable (Participant Femininity). This set of predictor variables correlated with the criterion variable set that included Allowed with Children and Amount of Fine for Suffering. There was a main effect for Perpetrator Age, such that the degree to which participants thought a perpetrator of animal cruelty should be allowed alone with children depends on the age of the perpetrator. Mean responses were stronger for scenarios with perpetrators aged 28 than with perpetrators aged 12. The average response to Amount of Fine for Suffering was larger when the animal cruelty happened at a pet kennel than if the crime occurred at an animal shelter. Implications for these significant predictor and criterion variables are discussed in the context of interdisciplinary theory and practical implications for public policy.     

Monitoring population‐level responses of marine mammals to human activities

We provide guidance for monitoring whether human activities affect the physiology or behavior of marine mammals and, if so, whether those effects may lead to changes in survival and reproduction at the population level. We suggest that four elements be included in designing and implementing such a monitoring program. The first is development of a theory of change: a set of mechanistic hypotheses that outline why a given activity might be expected to have one or more measurable effects on individuals and populations, and ideally the magnitude, timing, and duration of the effects. The second element, definition of biologically meaningful effect sizes, ultimately facilitates the development of a monitoring program that can detect those magnitudes of effect with the desired levels of precision. The third element, selection of response variables for monitoring, allows inference to whether observed changes in the status of individuals or populations are attributable to a given activity. Visual observations, passive acoustic and tagging instruments, and direct physical measurements all can provide data that facilitate quantitative hypothesis testing. The fourth element is specification of the temporal sequence of monitoring. These elements also can be used to inform monitoring of the responses of other taxonomic groups to human activities.     

The relationship between specific dynamic action and otolith growth in pike

The hypothesis was tested that the daily increment width (IW) of the otolith comprises two components, one that correlates with basal metabolic rate (as has been demonstrated previously) and the other that correlates with apparent specific dynamic action ( R _sda)(the post‐prandial elevation in metabolism). Simultaneous measurements of IW and metabolic rate before and after a meal were collected from individual pike Esox lucius . After feeding, IW and metabolic rate increased above basal levels for 5–6 days. There was no correlation between daily IW and R _sda, reflecting within‐individual difference in the shapes of the post‐prandial responses of the two variables. There was a significant relationship between the total changes of IW and metabolic rate integrated following meals. The magnitude of the post‐prandial response as a proportion of the basal level was larger for metabolic rate than IW, mirroring the previously reported responses of these variables to acute temperature change. This study suggests that analysis of IW has the potential to provide a historic record of energy intake but only when integrated over a period equivalent to the digestion time. Consideration of energy budget theory indicates that IW is unlikely to provide a robust record of short‐term somatic growth if activity metabolism is significant and variable.     

On testing simultaneously non-inferiority in two multiple primary endpoints and superiority in at least one of them

In a clinical trial with an active treatment and a placebo the situation may occur that two (or even more) primary endpoints may be necessary to describe the active treatment's benefit. The focus of our interest is a more specific situation with two primary endpoints in which superiority in one of them would suffice given that non-inferiority is observed in the other. Several proposals exist in the literature for dealing with this or similar problems, but prove insufficient or inadequate at a closer look (e.g. Bloch et al. (2001, 2006) or Tamhane and Logan (2002, 2004)). For example, we were unable to find a good reason why a bootstrap p-value for superiority should depend on the initially selected non-inferiority margins or on the initially selected type I error alpha. We propose a hierarchical three step procedure, where non-inferiority in both variables must be proven in the first step, superiority has to be shown by a bivariate test (e.g. Holm (1979), O'Brien (1984), Hochberg (1988), a bootstrap (Wang (1998)), or L?uter (1996)) in the second step, and then superiority in at least one variable has to be verified in the third step by a corresponding univariate test. All statistical tests are performed at the same one-sided significance level alpha. From the above mentioned bivariate superiority tests we preferred L?uter's SS test and the Holm procedure for the reason that these have been proven to control the type I error strictly, irrespective of the correlation structure among the primary variables and the sample size applied. A simulation study reveals that the performance regarding power of the bivariate test depends to a considerable degree on the correlation and on the magnitude of the expected effects of the two primary endpoints. Therefore, the recommendation of which test to choose depends on knowledge of the possible correlation between the two primary endpoints. In general, L?uter's SS procedure in step 2 shows the best overall properties, whereas Holm's procedure shows an advantage if both a positive correlation between the two variables and a considerable difference between their standardized effect sizes can be expected.     

Assessing ecological integrity: A multi-scale structural and functional approach using Structural Equation Modeling

Facing increasing levels of ecosystem degradation, scientists and practitioners aim to preserve ecological integrity — to maintain structures and functions expected of ecosystems in a region. This requires an understanding of the relationship between structural components and functional integrity. In this paper we focused on the study forests of the Credit River Watershed (Southern Ontario, Canada). For this ecosystem we consider one of the major contributors to functional integrity: habitat functions which are defined as the capacity of the ecosystem to provide refuge and reproduction habitat to wild species of plants and animals. We define these ‘habitat functions’ as a latent variable in Structural Equation Modeling, which allows us to examine its relationship with a number of candidate indicators. We first determined two community-level structural indicators to represent the latent variable: native plant cover and forest bird abundance. We then found underlying causal relationships between multi-scale structural components of the ecosystem and the provision of habitat functions. Three variables at the local scale explain native plant cover — soil nitrogen, soil organic matter, and soil pH. A significant landscape-level variable, patch area, explained native plant cover. Percent natural land cover in a 500 m radius explained forest bird abundance. From a theoretical point of view, this modeling technique allows us to explore complex and simultaneous interactions between structures and functions of ecosystems. As for its practical applications, it can be used to improve ecological integrity monitoring programs by contributing to the selection of meaningful indicators.     

An analysis of density responses of forest and woodland birds to composite physiognomic variables

Habitat ordinations were performed on 37 sites in forests and woodlands along a latitudinal transect that spanned over 250 km in central Victoria. Australia. Multivariate analyses of these data by using nonmetric. multidimensional scaling (AIDS) were used to generate composite variables. The relationships of these composite variables and densities (by area) of 58 species of forest and woodland birds were assessed by using linear and polynomial regressions. Only seven of the 58 species of birds did not display a significant relationship to one or other of the five composite variables. Approximately 50% of the avian species showed significant relationships with each of the first two composite variables, but lower percentages were observed for the other three variables. About a third of all significant correlations were either of second or third order, ft appears that marked curvilinearities associated with the first composite variable can be interpreted as linear responses with respect to the secondary composite variable. These results suggest that, although composite variables derived from multivariate classifications are statistically independent, there often may be substantial biological dependence between the composite variables. Therefore, for biological interpretation, it would be appropriate to regard composite variables derived from multivariate classifications as suites of related variables. Some authors have traced sharp discontinuities in distributions of species with respect to habitat structure by using presence/absence data alone. This approach appears to be sensitive to sampling of habitat types, and densities should be used wherever possible.     

Testing the ability of species distribution models to infer variable importance

Models of species’ distributions and niches are frequently used to infer the importance of range- and niche-defining variables. However, the degree to which these models can reliably identify important variables and quantify their influence remains unknown. Here we use a series of simulations to explore how well models can 1) discriminate between variables with different influence and 2) calibrate the magnitude of influence relative to an ‘omniscient’ model. To quantify variable importance, we trained generalized additive models (GAMs), Maxent and boosted regression trees (BRTs) on simulated data and tested their sensitivity to permutations in each predictor. Importance was inferred by calculating the correlation between permuted and unpermuted predictions, and by comparing predictive accuracy of permuted and unpermuted predictions using AUC and the continuous Boyce index. In scenarios with one influential and one uninfluential variable, models failed to discriminate reliably between variables when training occurrences were < 8–64, prevalence was > 0.5, spatial extent was small, environmental data had coarse resolution and spatial autocorrelation was low, or when pairwise correlation between environmental variables was |r| > 0.7. When two variables influenced the distribution equally, importance was underestimated when species had narrow or intermediate niche breadth. Interactions between variables in how they shaped the niche did not affect inferences about their importance. When variables acted unequally, the effect of the stronger variable was overestimated. GAMs and Maxent discriminated between variables more reliably than BRTs, but no algorithm was consistently well-calibrated vis-à-vis the omniscient model. Algorithm-specific measures of importance like Maxent's change-in-gain metric were less robust than the permutation test. Overall, high predictive accuracy did not connote robust inferential capacity. As a result, requirements for reliably measuring variable importance are likely more stringent than for creating models with high predictive accuracy.     

Understanding relationship between sequence and functional evolution in yeast proteins

The underlying relationship between functional variables and sequence evolutionary rates is often assessed by partial correlation analysis. However, this strategy is impeded by the difficulty of conducting meaningful statistical analysis using noisy biological data. A recent study suggested that the partial correlation analysis is misleading when data is noisy and that the principal component regression analysis is a better tool to analyze biological data. In this paper, we evaluate how these two statistical tools (partial correlation and principal component regression) perform when data are noisy. Contrary to the earlier conclusion, we found that these two tools perform comparably in most cases. Furthermore, when there is more than one ‘true’ independent variable, partial correlation analysis delivers a better representation of the data. Employing both tools may provide a more complete and complementary representation of the real data. In this light, and with new analyses, we suggest that protein length and gene dispensability play significant, independent roles in yeast protein evolution. Electronic supplementary material Supplementary material is available in the online version of this article at and is accessible for authorized users.     

A computer program for testing average partial association in three-way contingency tables (PARCAT).

PARCAT is a computer program which implements alternative tests for average partial association in three-way contingency tables within the framework of the product multiple hypergeometric probability model. Primary attention is directed at the relationship between two of the variables, controlling for the effects of a covariable. This approach is essentially a multivariate extension of the Cochran/Mantel-Haenszel test to sets of (s x r) tables. A set of scores such as uniform, ridits, or probits can be assigned to categories which are ordinally scaled. In particular, if ridit scores with midranks assigned for ties are utilized, this procedure is equivalent to a partial Kruskal-Wallis test when one variable is ordinally scaled, and is equivalent to a partial Spearman rank correlation test when both variables are ordinally scaled.     

Variability in mammalian dentitions: size-related bias in the coefficient of variation

The coefficient of variation has been used in many evolutionary studies. However, a strong negative correlation between this index and size may artificially inflate the apparent variability of small traits. This is most pronounced when variables whose size differs by more than an order of magnitude are compared or when the index is applied to variables whose size is within an order of magnitude of their measurement error. When this is likely to affect conclusions, other indexes of variability should be considered. One alternative is to use the standard deviation of log-transformed data; however, this index is sometimes still correlated with variable size so care should be exercised in its use. Another alternative is to regress the standard deviation onto mean variable size; however, this method may also be misleading if variables are not randomly distributed about the regression line. As an example of the effect of bias in the coefficient of variation, previous studies of mammalian dental variability profiles were re-evaluated. It was found that variation among teeth is relatively homogenous, both within and among species. The exception is that the canines of some mammalian species have variability that is considerably higher than would be expected from tooth size. Previous explanations of variability patterns that invoked developmental fields are incompatible with the new data.     

Rate of excitation propagation in a reduced Hodgkins-Huxley model. III. Integrodifferential equations]

The Hodgkin - Huxley system of equations is reduced to single integral-differential equation in neglection of slow variables dynamics. Two limiting cases of fast and slow sodium activation processes are considered. The first case leads to a nonlinear differential equation for the potential, the second one - to an ordinary differential equation with a known source as a function of coordinate. Such a simplification is due to approximation of steady-state sodium activation variable with the help of Heviside function. The validity of this approximation is discussed; the corresponding error is estimated by calculation of the second approximation for the source function.     

Specificity of sensorimotor learning and the neural code: Neuronal representations in the primary motor cortex

Human studies show that the learning of a new sensorimotor mapping that requires adaptation to directional errors is local and generalizes poorly to untrained directions. We trained monkeys to learn new visuomotor rotations for only one target in space and recorded neuronal activity in the primary motor cortex before, during and after learning. Similar to humans, the monkeys showed poor transfer of learning to other directions, as observed by behavioral aftereffects for untrained directions. To test for internal representations underlying these changes, we compared two features of neuronal activity before and after learning: changes in firing rates and changes in information content. Specific elevations of firing rate were only observed in a subpopulation of cells in the motor cortex with directional properties corresponding to the locally learned rotation; namely cells only showed plasticity if their preferred direction was near the training one. We applied measures from information theory to probe for learning-related changes in the neuronal code. Single cells conveyed more information about the direction of movement and this specific improvement in encoding was correlated with an increase in the slope of the neurons' tuning curve. Further, the improved information after learning enabled a more accurate reconstruction of movement direction from neuronal populations. Our findings suggest a neural mechanism for the confined generalization of a newly acquired internal model by showing a tight relationship between the locality of learning and the properties of neurons. They also provide direct evidence for improvement in the neural code as a result of learning.     

Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion

This article introduces a two-strain spatially explicit SIS epidemic model with space-dependent transmission parameters. We define reproduction numbers of the two strains, and show that the disease-free equilibrium will be globally stable if both reproduction numbers are below one. We also introduce the invasion numbers of the two strains which determine the ability of each strain to invade the single-strain equilibrium of the other strain. The main question that we address is whether the presence of spatial structure would allow the two strains to coexist, as the corresponding spatially homogeneous model leads to competitive exclusion. We show analytically that if both invasion numbers are larger than one, then there is a coexistence equilibrium. We devise a finite element numerical method to numerically confirm the stability of the coexistence equilibrium and investigate various competition scenarios between the strains. Finally, we show that the numerical scheme preserves the positive cone and converges of first order in the time variable and second order in the space variables.     

Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion

This article introduces a two-strain spatially explicit SIS epidemic model with space-dependent transmission parameters. We define reproduction numbers of the two strains, and show that the disease-free equilibrium will be globally stable if both reproduction numbers are below one. We also introduce the invasion numbers of the two strains which determine the ability of each strain to invade the single-strain equilibrium of the other strain. The main question that we address is whether the presence of spatial structure would allow the two strains to coexist, as the corresponding spatially homogeneous model leads to competitive exclusion. We show analytically that if both invasion numbers are larger than one, then there is a coexistence equilibrium. We devise a finite element numerical method to numerically confirm the stability of the coexistence equilibrium and investigate various competition scenarios between the strains. Finally, we show that the numerical scheme preserves the positive cone and converges of first order in the time variable and second order in the space variables.     

Red-shifts and red herrings in geographical ecology

I draw attention to the need for ecologists to take spatial structure into account more seriously in hypothesis testing. If spatial autocorrelation is ignored, as it usually is, then analyses of ecological patterns in terms of environmental factors can produce very misleading results. This is demonstrated using synthetic but realistic spatial patterns with known spatial properties which are subjected to classical correlation and multiple regression analyses. Correlation between an autocorrelated response variable and each of a set of explanatory variables is strongly biased in favour of those explanatory variables that are highly autocorrelated - the expected magnitude of the correlation coefficient increases with autocorrelation even if the spatial patterns are completely independent. Similarly, multiple regression analysis finds highly autocorrelated explanatory variables "significant" much more frequently than it should. The chances of mistakenly identifying a "significant" slope across an autocorrelated pattern is very high if classical regression is used. Consequently, under these circumstances strongly autocorrelated environmental factors reported in the literature as associated with ecological patterns may not actually be significant. It is likely that these factors wrongly described as important constitute a red-shifted subset of the set of potential explanations, and that more spatially discontinuous factors (those with bluer spectra) are actually relatively more important than their present status suggests. There is much that ecologists can do to improve on this situation. I discuss various approaches to the problem of spatial autocorrelation from the literature and present a randomisation test for the association of two spatial patterns which has advantages over currently available methods.     

Testing evolutionary hypotheses about human biological adaptation using cross-cultural comparison

Physiological data from a range of human populations living in different environments can provide valuable information for testing evolutionary hypotheses about human adaptation. By taking into account the effects of population history, phylogenetic comparative methods can help us determine whether variation results from selection due to particular environmental variables. These selective forces could even be due to cultural traits-which means that gene-culture co-evolution may be occurring. In this paper, we outline two examples of the use of these approaches to test adaptive hypotheses that explain global variation in two physiological traits: the first is lactose digestion capacity in adults, and the second is population sex-ratio at birth. We show that lower than average sex ratio at birth is associated with high fertility, and argue that global variation in sex ratio at birth has evolved as a response to the high physiological costs of producing boys in high fertility populations.