Short-term biological changes in response to particular environmental stimuli are referred to as:

Phenotypic Plasticity

Phenotypic plasticity is the ability of an organism to change in response to stimuli or inputs from the environment. Synonyms are phenotypic responsiveness, flexibility, and condition sensitivity. The response may or may not be adaptive, and it may involve a change in morphology, physiological state, or behavior, or some combination of these, at any level of organization, the phenotype being all of the characteristics of an organism other than its genes.

There is a great potential for confusion regarding definitions of phenotypic plasticity, including this one. Even though the phenotype is defined here to exclude the genome, in fact phenotypic plasticity always involves a change in gene expression or gene-product use (morphological, physiological, and behavioral traits always being products, in part, of gene expression). Some definitions of phenotypic plasticity refer to the environmental sensitivity of a genotype, a potentially confusing terminology because it uses the word ‘genotype’ to mean ‘organism bearing a particular gene or set of genes’ and may be mistakenly understood to imply that the organism’s genome, rather than its phenotype (whose nature has been influenced by both environment and genome), responds to the environment. Such definitions attempt to convey the correct idea that phenotypic plasticity involves a change in some aspect of the phenotype without a change in the individual’s genes, or the genetic underpinnings of a particular trait. Thus, one could examine the phenotypes of genetically identical individuals and find that they differ phenotypically in different environments, indicating phenotypic plasticity for particular traits. Or, conversely, one could begin with individuals from phenotypically different populations found in different environments, and subject them to the same environment in a ‘common garden’ experiment designed to control environmental variables, and see to what degree the phenotypic differences are maintained, indicating the degree to which genetic differences between the populations, rather than plasticity, account for the phenotypic differences between them.

Phenotypic plasticity can be a source of ‘noise’, or confounding variation, in genetic experiments. Such experiments are therefore often designed to control environmental variation and reduce the effects of plasticity. But research in behavioral ecology, rather than eliminating plasticity, often focuses on it. Behavioral phenotypes are eminently plastic, often in adaptively appropriate ways. Plasticity of behavioral responses – the occurrence of complex, condition-sensitive behavioral repertoires – can increase the diversity of phenotypes within populations. But behavioral plasticity can also reduce phenotypic variation, as when behavioral responses are stability increasing or homeostatic in their effects. For example, individuals may adopt postures or move to locations that help reduce extremes of variation in body temperature. Homeostatic behavior can be quite elaborate: some social insects engage in behaviors (water transport and application to nest surfaces, followed by fanning wing movements that promote evaporative cooling) and effectively lower the temperature of nest and brood.

An Adaptive Strategy in Heterogeneous Environments

Phenotypic plasticity refers to the observation that a given genotype expresses different phenotypes in different ecological settings. At its most basic level, the concept applies to any differences in trait means between environments. For example, plants may show stunted growth in habitats other than the one to which they are adapted. However, when trait expression in different habitats has been shaped by natural selection, phenotypic plasticity represents an ecological strategy that adapts organisms to heterogeneous environments and forms an integral part of species interactions (Agrawal, 2000). Such adaptive plasticity can range from repeated ontogenetic adjustments in behavior, physiology, or life history to the expression of distinct, irreversible morphologies (polyphenism) such as fully winged and flightless forms in insects (Harrison, 1980; Roff, 1986), the shade avoidance growth form of plants (Schmitt et al., 2003) carnivorous and omnivorous tadpole morphs of spadefood toads (Pfennig et al., 2007) inducible defensive morphs of Daphnia water fleas (Tollrian and Dodson, 1999) and the seasonally alternating morphs of Bicyclus anynana butterflies (Lyytinen et al., 2004).

Phenotypic Plasticity

Phenotypic plasticity refers to the ability of a genotype to express different phenotypes depending on the environment in which it resides. For example, genetically identical water flea (Daphnia) clones can differ in their morphology depending on whether reared in the absence or presence of a potential predator. Similarly, many species adjust their phenology, the seasonal timing of life-cycle events, to ambient temperatures, with phenology being earlier under warm than cool conditions. Such phenotypic plasticity in morphology or behavior has generally evolved as a response to varying environmental conditions and allows individuals or genotypes to adjust their phenotypes so that they can track the varying environmental optimum.

The role of phenotypic plasticity in the context of climate change can be nicely illustrated with the well-studied example of the winter moth-great tit predator–prey system. The development of the caterpillars (mainly winter moth, Operophtera brumata), including the timing of egg hatching, is temperature dependent, which means that the period of highest caterpillar abundance occurs early in years with warm springs. Because these caterpillars constitute an important food source for their nestlings, the great tits (Parus major) need to synchronise the nestling period with the time of maximum caterpillar availability, and breed early in warm springs. This phenotypically plastic response to warmer spring temperatures allows the birds to track the annually variable optimal breeding time.

Climate change has lead to increasing spring temperatures and has thereby advanced the timing of maximum caterpillar abundance. However, long-term data collected over the last three decades show that the phenology of the caterpillars has advanced at a faster rate than that of the birds. As such, the question remains why has phenotypic plasticity not (sufficiently) allowed birds to track this climate-induced change in their environment? This is because the environmental variable that determines the breeding time of the birds (here: temperature from mid March to mid April) is not the same as the environmental variable that determines maximum caterpillar peak abundance (temperature from mid-March to mid-May). Because these two temperatures are not perfectly correlated and the birds cannot therefore “predict” caterpillar phenology perfectly, it is optimal for the birds to respond slightly less strongly to temperatures than the caterpillars (Gienapp et al., 2014). This means that the phenology of the birds will advance at a slower rate than that of the caterpillars, which explains the observed mismatch in their phenologies.

The situation in which phenotypic plasticity does not allow the environment to be perfectly tracked appears to be a fairly general one. This is because the “cue” used for trait expression (e.g., temperature during egg laying in the great tit-caterpillar example above) is in most cases an imperfect predictor of conditions at the time when the trait is actually expressed. For example, a Daphnia that has perceived the presence of a predator and developed its defensive morphology may not be faced with predation because the predator threat may have disappeared in the meanwhile. This imperfect correlation between the perceived environment (cue) and the environment exerting selection means that tracking on continuously changing environment will rarely, if ever, be possible (Gienapp et al., 2014). In fact, a large meta-analysis of terrestrial ecosystems shows that the phenology at lower trophic levels has advanced consistently at faster rates than that at higher trophic levels (Thackeray et al., 2010). These mismatches will lead to selection on phenology of the higher trophic level, and staying adapted will require genetically based microevolutionary responses.

To sum up, rapid phenotypic adjustments via phenotypic plasticity can alleviate immediate negative fitness consequences of environmental changes, but these adjustments are hardly ever perfect as there may be limits and costs to expression of plasticity. Hence, the prevailing consensus is that phenotypic plasticity is an important mechanism providing the first line of defence against fitness loss in changing climatic conditions, but long-term persistence depends critically on populations’ ability to adapt through microevolutionary responses.

B REACTION NORMS

Phenotypic plasticity can be studied by means of several types of phenotypic response data or “norms of reaction.” Here we build up an array of plasticity concepts starting with the most elemental, the “genotypic reaction norm” (Woltereck, 1909; Schmalhausen, 1949; examples in Sultan and Bazzaz, 1993), then passing through a series of classes of reaction norms that integrate increasingly complex effects, and ending with a comment on the relationship of reaction norms to behavior.

The reaction norm is most precisely conceptualized as the phenotypic expression of a given genotype for a single trait at several specified levels of a particular environmental factor, for example, the reaction norm of a particular clone of Daphnia pulex for head spine length as a function of a defined range of ambient temperatures. As with all reaction norms, both the trait of interest and the environmental variable must be precisely defined, so that comparable measurements can be made on the different phenotypes produced by genetic replicates in different experimental environments. For instance, because developmental rates often vary with environment, traits may be defined with respect to either absolute age or specified ontogenetic stages (Gedroc et al., 1996). In the preceding example, head spine would be measured in Daphnia individuals at a particular instar at each temperature. Environmental factors should be carefully controlled so as to avoid covarying changes that can confound interpretation.

From this concept we can build up to several others that are experimentally useful, particularly in organisms for which genotypic replicates cannot be readily obtained. If the study organism has many offspring, we can estimate “family mean reaction norms” by measuring several sibling offspring at each environmental level (Gebhardt and Stearns, 1992). Such family mean data provide the best estimate of reaction norms for organisms that cannot be cloned and are particularly robust when siblings are inbred full sibs (e.g., Gupta and Lewontin, 1992).

When familial relationship cannot be determined in the sample, one may still wish to compare the plastic reactions of two populations based on a random sample of individuals from each population split into subgroups and measured at a series of environmental levels. Such data represent “undefined population mean reaction norms.” It is also possible to build up population mean reaction norms from a collection of genotypic reaction norms, then termed “genotype-based population mean reaction norms,” or from a collection of family mean reaction norms, then termed “family-based population mean reaction norms.”

Such distinctions, though cumbersome, are critical, because the essential aim of reaction norm analysis is to establish clarity about the genotype–phenotype relationship. Another approach taken in the literature is to reserve the term reaction norm for genotypic data and refer to “plasticity patterns” when the sample is genetically undefined. Whatever approach is taken, it is essential to make clear the type of data being presented.

The explanation

The idea that phenotypic plasticity could be molded by natural selection had been made earlier than Bradshaw's 1965 review article by Gause (1947) and Schmalhausen (1949). However, Bradshaw's paper brought special attention to plants. He notes that unlike animals plants do not have the option of moving if the environmental conditions change.

Bradshaw reviews several conditions under which phenotypic plasticity might be expected to evolve. Bradshaw notes that Clausen et al. (1940) have shown that there is variation in phenotypic plasticity response between populations, suggesting some tuning by natural selection. Perhaps the most important type of selection would be what Bradshaw called disruptive selection. By this Bradshaw suggests that over time the conditions in the same location may vary and thus favor different phenotypes. Alternatively, conditions may vary over space and thus favor different phenotypes within the same population.

Bradshaw also suggests that even if selection is directional for some phenotype, phenotypic plasticity could be advantageous. That is, after genetic variation has pushed the phenotype as far as possible, further gains might be made from a phenotypically plastic response. One of Bradshaw's less convincing arguments is that multiple genotypes may show the same phenotype due to plasticity of the trait. He suggests that this may be a way of maintaining genetic variation in populations but provides no quantitative support for this claim. Nevertheless, Bradshaw produces strong arguments that many phenotypically plastic traits in plants may have been fine tuned by selection.

6 Growth and age

As characteristic of the phenotypic plasticity in sea urchins (Ebert, 1996; Russell, 1998), C. rodgersii adjusts body growth with food availability (Blount, 2004; Ling and Johnson, 2009). This species reaches a maximum size in macroalgal habitats compared with barrens urchins and different ecomorphs occur in these habitats (Blount, 2004; Ling and Johnson, 2009). Sea urchins in macroalgal habitat have thicker tests, shorter spines, and faster growth rates, than while those from barrens (Blount, 2004; Ling and Johnson, 2009). Barrens C. rodgersii also have relatively larger jaws (Ling and Johnson, 2009), an indication of food limitation (Ebert, 1980). In New South Wales, a study of 32 sites along the coast indicated that C. rodgersii in barrens were generally ≤ 80-mm test diameter, while those from nearby macroalgal habitats were generally ≥ 90-mm test diameter (Blount, 2004). Experimental reductions in density by as little as 33% cause growth rates of the remaining C. rodgersii to double (Blount, 2004).

Growth of C. rodgersii in New South Wales, Tasmania, and New Zealand has been estimated using mark recapture techniques and direct aging using validated growth lines visible on the demipyramids and tetracycline injections to tag the skeleton (Ebert, 1982; Blount, 2004; Ling et al., 2009a; Pecorino et al., 2012) and by cohort analysis (Andrew, 1991, 1993). Despite variability between studies in different regions, overall trends are similar.

In New South Wales, following recruitment, C. rodgersii grows quickly reaching 35–40-mm test diameter after 1 year with a growth rate estimated to be 4–6 mm every 67 days (Andrew, 1991, 1993) and with a 50-mm test diameter estimated by year 2. A mark-recapture studies indicated that growth rates are variable depending on habitat, site, and food availability, with 80- and 100-mm test diameter sea urchins being 5–9 years old and 12–25 years old, respectively (Blount, 2004). Modal size C. rodgersii (test diameter = 70–90 mm) are estimated to be 4–10 years old (C. Blount and N. Andrew, unpublished data). The largest sea urchins found (test diameter > 110-mm test diameter) may be up to 20 years old. Longevity is estimated to be 25–30 years. Although growth is faster at Sydney than at Eden (~ 350 km to the south), maximum age shows little difference between these localities (C. Blount and N. Andrew, unpublished data) nor between fringe and barrens habitats (Blount, 2004).

In Tasmania, mark-recapture studies indicate that C. rodgersii reaches a test diameter of ca. 50 mm in 4–5 years followed by slower growth with a maximum test diameter of 114 mm within 25–35 years (analysis of data in Ling et al., 2009a by Pecorino et al., 2012).

The mark-recapture study of C. rodgersii in New Zealand indicated the sea urchins were 3–10 years old (Pecorino et al., 2013a). In this study, depending on the model used, C. rodgersii is estimated to reach a maximum growth rate of 17.7–23.8 mm year− 1 at 1.5–3.0 years old followed by a decrease in growth rate to reach a test diameter of 10–13 mm in their fifth year, reaching 85-mm test diameter. This is followed by growth to a test diameter of 126 mm between 15 and 20 years. Comparison of estimated growth rates in Tasmania and New Zealand indicate similar growth to year 1. Thereafter, growth is faster in the New Zealand population, a difference suggested to be influenced by the warmer (3–4°C) water temperature in New Zealand (Pecorino et al., 2012).

1 Introduction

Morphological plasticity, also called phenotypic plasticity, refers to the potential of organisms to change specific anatomical traits in response to different environments independent of their genotype (Richards et al., 2006; Sultan, 2003, 2000). Variation in phenotypes can occur as a response to neighboring individuals (e.g., biotic interactions) or changes in abiotic conditions (e.g., nutrients, water, light, temperature, wind, etc.). Plasticity generally allows organisms to optimize their performance, to cope with stress and increase their survival (Callaway et al., 2003; Grime and Mackey, 2002; Miner et al., 2005; Wolf et al., 2011). An example of morphological plasticity in plants is the differential allocation of biomass to roots when nutrients are limited, or to branches and leaves if light is the growth-limiting resource.

Plastic responses that increase individual fitness are considered adaptive (Longuetaud et al., 2013; Sultan, 2003), potentially contributing to species persistence in changing environments (Burggren, 2018). This chapter focuses on our current understanding of mangrove plastic traits that provide an advantage in withstanding abiotic forcing of active coastal sedimentary environments from seedlings to fully grown trees. We will identify knowledge gaps that need to be filled in order to fully appreciate the biophysical interactions along mangrove coasts that determine mangrove resilience and resistance in a changing climate.

Mangroves show not only a range of plastic responses to environmental variability such as salinity and soil anoxia (Ball, 1988; Feller et al., 2010; Tomlinson, 2016) but also in response to coastal physical processes (erosion, sedimentation, hydrodynamic forcing, and wind) and biotic interactions like tree-to-tree competition (Balke et al., 2013; Dahdouh-Guebas et al., 2007, 2004; Vovides et al., 2018). Equipped with mechanisms for salt exclusion or excretion, aerial structures for oxygen exchange, and different root architectures to colonize unstable sediments, the morphological traits of mangrove trees respond to the magnitude of the experienced stress (e.g., pneumatophore height increases with flooding frequency) (Toma et al., 1991; Dahdouh-Guebas et al., 2007) and play critical roles in the survival of individuals throughout their ontogeny.

One of the most crucial life history stages of mangrove trees is the seedling stage. Seedlings not only are most susceptible to stress and disturbance but also show plastic traits that aid rapid anchorage in sedimentary environments (Augustinus, 1996; Balke et al., 2014, 2013, 2011; Krauss et al., 2014). This is especially important on bare tidal flats, where the mangrove canopy cannot provide shelter from wave attack and erosion (Van Santen et al., 2007) to establishing individuals. Thus, the ability of seedlings to modify their morphological traits can determine their survival and fitness (Burggren, 2018; Simpson et al., 2017). In fully established and mature trees, morphological plasticity still occurs and remains a key factor in reducing competition, promoting species coexistence, and in the maintenance of mechanical integrity in unstable sediments (Grueters et al., 2014; Peters et al., 2018; Vovides et al., 2018).

Plant traits can also shape biophysical and biogeomorphic interactions. For example, stilt roots and stems attenuate wave action, while pneumatophores create bottom friction that can further reduce current velocity and promote sedimentation (Horstman et al., 2016; Le Minor et al., 2019; Mazda et al., 2006; Mullarney et al., 2017). Pneumatophore density can also prevent sediment burial in the fringe zones by creating localized scour (Mullarney et al., 2017; Spenceley, 1977). This shows that phenotypic plasticity of mangroves in sedimentary environments is not only an exclusive response to abiotic stressors but is, at least partially, also influenced by the individual species' ecosystem engineering capacity (Jones et al., 1994). The environment can, thus, modulate the response of a plastic trait, whereas effect traits determine the ecosystem engineering capacity of plants to change their abiotic environment (Jones et al., 1994). A given trait can be both a response and an effect trait; pneumatophore height, for example, is a response trait that not only changes with elevation but can also act as an effect trait by increasing sedimentation rates or promoting scouring depending on density and diameter (Dahdouh-Guebas et al., 2007; Mullarney et al., 2017), as shown in Fig. 5.1.

Environmental drivers of mangrove establishment and development are well studied from an ecophysiological perspective (reviewed by Krauss et al., 2008). Changes of anatomical and physiological traits, including dwarfing, can be associated with coping mechanisms for nutrient limitations (Feller, 1995; Feller et al., 2010), salt stress (Ball and Farquhar, 1984; López-Portillo et al., 2014; Méndez-Alonzo et al., 2013), and low temperatures (Cook-Patton et al., 2015; Osland et al., 2014). Such ecophysiological responses are critical as they determine plant architecture and, hence, forest structure and function (Ewel et al., 1998). Reduced growth rates and plant height increase vessel density and reduce vessel size in response to salt stress (Ball, 1988; Robert et al., 2009; Schmitz et al., 2006; Sobrado, 2007), for example, to ensure a safe hydraulic architecture that reduces the risk of embolisms (Robert et al., 2009).

Other expressions of plastic responses to salt stress include reduced slenderness and a shift from above- to below-ground biomass allocation (Castañeda-Moya et al., 2011; Peters et al., 2018, 2014). These responses create changes in allometric relationships and also modulate local interactions. With smaller trees, reduced size asymmetry, and increased distance between neighbors, above-ground competition is reduced with salinity increasing up to 60 ppt (Méndez-Alonzo et al., 2012; Vovides et al., 2014, 2018) within tropical coasts. In sites with extreme droughts, where salinity can reach >80 ppt (Vovides et al., 2011), or under nutrient-limiting conditions (Lovelock et al., 2004), trees develop a shrub-like architecture and facilitate each other by changing microclimatic conditions or species associations (Pranchai et al., 2018; Vogt et al., 2014).

Environmental versus Evolutionary Response

Environmental response of organismal traits, or phenotypic plasticity, reflects the direct dependence of organisms' physiology, growth, development, and behavior on the environmental conditions they experience through their lifetimes (Bradshaw, 1965). Plastic responses do not involve changes in the genetic composition of a population but manifest environmental influences on the expression of genes (e.g., flowering time, Aikawa et al., 2010). In contrast, evolutionary change is the change over generations in the genetic composition of a population. All populations are continually subject to evolutionary change. It results from several processes, including natural selection, i.e., variation among genotypes in survival or reproductive success. Natural selection is the basis for adaptation. Genotypes that express adaptive phenotypes make a greater genetic contribution to the subsequent generation; in this way, alleles that confer higher fitness in a given generation gain in frequency in the population. Natural selection may also favor plasticity that maintains high fitness in a variable environment. For example, a genotype that produces thin leaves in mesic conditions but thicker leaves when it develops in more arid conditions may have a survival or reproductive advantage over genotypes that do not respond in this way. If there is variation among genotypes in their phenotypic plasticity and this variation is under selection, plasticity itself can evolve (Via, 1993).

In the context of climate change, phenotypic changes such as timing of flowering and breeding that correspond statistically with recent climatic trends are well documented (Parmesan and Yohe, 2003). Observation of such changes does not reveal the basis of these responses, whether plasticity in response to immediate environmental condition, adaptive evolution in response to selection mediated by environmental change, or a combination of these. Phenotypic plasticity may buffer populations against natural selection in the short term. However, the limits of phenotypic plasticity in maintaining fitness over a range of environments are poorly understood, and, over the long term, ongoing climate change is expected to exceed those limits (DeWitt et al., 1998). Thus, as climate continues to change, it is likely that natural selection will alter plasticity of traits as well as their expression in particular environments, given climate model predictions that include both directional change and increased variability.

Permanent versus Inducible Defenses

Defenses can be permanent (constitutive) or inducible. Phenotypic plasticity in defensive traits enables prey organisms to express a particular defense only if a reliable cue for a future attack is present. Thereby, the organisms can minimize costs affiliated with the formation or maintenance of a defense when predation risk is low. Inducible defenses are an appropriate mechanism to cope with the variable hazard of a frequently changing predator spectrum. In the animal kingdom, inducible defenses cover a taxonomic range from protozoans to vertebrates. The defensive traits range from behavior, morphology, and life-history adaptations to the activation of the specific immune system of vertebrates. Daphnia show the most prominent examples of morphological plasticity triggered by chemical cues, so-called kairomones, released by predatory invertebrates and fish. For example, elongated helmets, tail spines, or crests have been shown to reduce predator-caused mortality (Figure 2).

Several factors have been identified that favor the evolution of inducible as opposed to permanent defenses: (1) The attacker has to have a variable but sometimes relevant impact; (2) the defense must be effective within a relatively short time, so lag phases can be avoided; (3) a reliable cue has to indicate the danger; and (4) costs or tradeoffs have to outweigh the benefit of the defense during relevant periods of time.

X ANALYSIS FOR THE RxNM APPROACH

When canalization is viewed as the “opposite” to phenotypic plasticity, then the framework for analysis is somewhat clearer. Given k lines (L) and j environments (E), we can start out within the framework of an ANOVA. With some variant of the model

Yijk=μ+E+L+E×L+ε

As with the analysis of phenotypic plasticity, we can begin by examining the significance of these model terms. Evidence for genetic variation for “plasticity” can be inferred if there is a significant E × L term for the model. If this term is not significant, but both E and L are, then there is evidence for plasticity of the trait and genetic variation for the trait itself, but not for genetic variation for plasticity of the trait (Figure 8-4).

If there is a significant E × L term, then we need to determine whether there is in fact evidence for canalization. It is important to recognize that a significant E × L term can arise from different processes (Robertson, 1959; Gibson and van Helden, 1997; Lynch and Walsh, 1998; Gibson and Wagner, 2000), but not all are evidence for canalization. Specifically, we want to separate out cases where the E × L term arose because of significant line crossing (change in relative ranks) across environment (Figure 8-1A), as opposed to a change in the scaling of the line means across environments (Figure 8-1B). For this latter case (i.e., Figure 8-1B), this will result in a perfect correlation across environments for the line means. For studies on phenotypic plasticity, the E × L term is usually partitioned into the two components (Robertson, 1959; see Lynch and Walsh, 1998, for review). However, knowing what proportion of the variation results from line crossing versus scaling effects is not itself of interest for canalization. We are specifically interested in whether this scaling effect (i.e., the increase in between-line variance) is significant.

One line of evidence that is consistent with canalization of a trait is a release of cryptic genetic variation in the “stressful” environment. This is inferred by the increase in between-line variation in the stressful environment (Gibson and van Helden, 1997; Gibson and Wagner, 2000). Thus, a priori, we recognize the need for some supplementary tests to determine whether there is a release of cryptic genetic variation. Here we return to many of the same statistical issues (and variety of tests) that we used to examine patterns of variation in the preceding section.

Gibson and van Helden (1997) used the F-test approach, by comparing phenotypic variance across environments, as well as comparing the variances from between-line means (an estimate of genetic variation, but see Lynch and Walsh, 1998, for concerns with this approach). If using this approach, then we suggest the use of the CV2 as opposed to the variances to cast the F-tests (Lewontin, 1966; Schultz, 1985). If this test is significant, then there is evidence for an increase in phenotypic variation in one environment over the other. The F-test used in this context suffers from all of the same inadequacies as discussed earlier. Demonstrating an increase in phenotypic variation is not sufficient to conclude canalization for the trait. It is the test of the variation of between-line means (again using CV2) that will (if significant) infer a release in cryptic genetic variation (and thus provide evidence for canalization). It should be pointed out that, unless a large number of lines are used, this criterion may be hard to meet as formal significance will be unlikely (even if there is indeed cryptic genetic variation). Therefore, one of the other tests described in the preceding section may in fact be preferred (such as Levene's test). Of course all the tests can be employed, but formal significance must be adjusted (by Bonferroni or other such approaches) to control for multiple tests. In the interim, we again suggest the use of Levene's test to test for an increase in the between-line variation. In this case, the test is perhaps best framed as follows:

LSjk=|Log( μjk)−E[Log(μk)]|

where LSjk is the Levene's statistic for line j in environment k. Log(μjk) is the log-transformed line mean for j in environment k, and E[Log(μk)] is the mean of the log-transformed line means in environment k. If this approach is used, and there are two environments (k = E1, E2), then a paired t test may in fact be the most logical test using the pairs (LSj, E1 and LSj, E2).