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Cautionary note on interpreting the results of this paper

Posted by jhadfield on 06 Apr 2011 at 17:22 GMT

This paper presents results suggesting simultaneous increases in both the strength of natural selection on, and additive genetic variance in laying date with spring temperature in a wild bird population. We feel that the results presented here on changes in genetic variance with temperature could potentially be misinterpreted by researchers who are not familiar with quantitative genetic and mixed-effects model terminology and analysis. We hope our comments below clarify matters.

Several attempts have been made recently to determine the relative importance of genes and environment in determining individual-level variation in response to environmental variables (phenotypic plasticity) in wild vertebrate populations. Most have utilised random regression models, which estimate the variance between individuals in functions relating phenotype to the environment. Such models allow between-individual variation in responses to the environment (I x E) to be partitioned into an additive genetic (G x E) component and a non-genetic or permanent environment component.

In this paper, just such a random regression analysis is undertaken. In the results section of the paper, under "Environmental Dependence of Additive Genetic Variance and Heritability" the authors’ state: “Comparing a model in which the additive genetic and permanent environment components of variance (VA and VPE) in a given year were constant across different spring temperatures to one in which VA and VPE could vary with spring temperature gave strong support for environmental dependence of VA and VPE (χ2(4) = 74.90, P < 0.001).”

Here, the authors have simply tested a model in which individuals vary in their response to spring temperatures (I x E) against a model in which they cannot. The stated comparison of models does not provide statistical support for an additive genetic basis to those responses to spring temperature (G x E). Although support for G x E in this population for the same trait – environment relationship has been reported (Nussey et al. 2005) these results used BLUP methodology in a way that is now known not to be robust. Indeed, in a previous paper, again analysing the same population and the same trait – environment relationship, the authors do compare models capable of providing statistical support for G x E, but find it to be non-significant (P = 0.15, Husby et al. 2010).

The authors go on to use models in which additive genetic variance in responses (G x E) is included to predict changes in VA with spring temperature. Predictions of VA generated by this model are forced to change in a quadratic fashion with spring temperature, so if selection on laying date is changing with temperature or time (a result shown previously in this population: Gienapp et al. 2006; Visser et al. 1998) a relationship between predicted VA or h2 and selection is highly likely.

We have illustrated the problem with a simple simulation which, for simplicity, involves between-individual phenotypic variation (VI or repeatability) and I x E, rather than underlying genetic variation and G x E. We simulated data where individual laying dates were repeatable but there was no systematic between-individual variance in slopes. As a consequence, the true underlying model has no I x E and constant repeatability (VI) with environment. We then fitted a random intercept-slope model (i.e. random regression) to this simulated data and from this predicted repeatability as a function of spring temperature. We then simulated selection differentials based on the estimates presented in the paper and regressed these on estimates of repeatability. Although there is no true relationship between selection and repeatability in any of the simulations, the P-values associated with these regressions were significant at the 5% level in 76% of simulations rather than the expected 5%. The full R code used to run these simulations is presented with annotations at the end of this comment. The key result in the paper is a negative relationship between annual selection differentials and heritability estimates. The simulations illustrate how spurious significant relationships between predicted variance components and selection measures can arise from random regression models when there is no underlying evidence for variation in I x E and, correspondingly, G x E.

We note that the authors do state in their discussion that “the changes in VA alone were not statistically significant” and that “statistical power to detect significant changes in additive genetic variance in relation to varying environmental conditions using a random regression animal model approach may be limited”. However, they discuss changes in VA and heritability with environment throughout the paper as if these effects had robust statistical support. Evidence from previous analyses show that this is not the case (Husby et al. 2010). We therefore suggest that caution should be used when interpreting the patterns discussed in this paper.

Dan Nussey1,2, Jarrod Hadfield1

1 Institute of Evolutionary Biology, School of Biological Sciences, University of Edinburgh, Edinburgh, EH9 3JT, Scotland

2 Centre for Immunity, Infection and Evolution, School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JT, Scotland


Gienapp, P., E. Postma, and M. E. Visser. 2006. Why breeding time has not responded to selection for earlier breeding in a songbird population. Evolution 60:2381-2388.

Husby, A., D. H. Nussey, M. E. Visser, A. J. Wilson, B. C. Sheldon, and L. E. B. Kruuk. 2010. Contrasting patterns of phenotypic plasticity in reproductive traits in two great tit populations. Evolution 64:2221-2237.

Nussey, D. H., E. Postma, P. Gienapp and M.E. Visser. 2005. Selection on heritable phenotypic plasticity in a wild bird population. Science 310 5746 304-306.

Visser, M. E., A. J. van Noordwijk, J. M. Tinbergen, and C. M. Lessells. 1998. Warmer springs lead to mistimed reproduction in great tits (Parus major). Proceedings of the Royal Society of London Series B-Biological Sciences 265:1867-1870.

R code used for illustrative simulations:


nid<-100 # number of individuals
nr<-4 # number of records per individual
ns<-1000 # number of simulated data-sets

id<-gl(nid,nr) # nid individuals measured nr times

res<-1:ns # vector for storing results

for(i in 1:ns){

x<-rnorm(35) # 35 annual spring temperatures
x<-2*(x-min(x))/(min(x)-max(x))+1 # standardise

s<-rnorm(35,-1.3-x*1.6, 1.26) # selection (regression parameters inferred from Fig 1b)

y<-y+rnorm(nid, 0, sqrt(0.15))[id] # lay-dates with repeatability of 0.15<-sample(x, nid*nr, TRUE) # randomly assign records to spring temperatures

m1<-lmer(|id)) # fit "random-regression" model

P<-VarCorr(m1)$id # obtain (co)variance estimates

v<-P[1,1]+2*x*P[1,2]+(x^2)*P[2,2] # predict variances
r<-v/(v+attr(VarCorr(m1), "sc")^2) # predict repeatability

res[i]<-summary(lm(s~r))$coef[,4][2] # p-values from linear regression


# ideally 5% of simulations are significant not ~76%.
# The actual level of anti-conservatism will depend on the data and the model fitted, and this simplified simulation is only intended to demonstrate the issues.

No competing interests declared.

RE: Cautionary note on interpreting the results of this paper

arildhus replied to jhadfield on 12 May 2011 at 06:37 GMT

We would like to clarify the issues raised by Nussey and Hadfield (N&H) in response to our paper (Husby et al. 2011) regarding statistical support for a GxE interaction and the implications for the interpretation of the results of our paper. While N&H raise important points, we show here that these issues do not undermine the results from our paper. A broader-scale analysis using a more traditional character-state approach confirm our finding that additive genetic variance and heritability in laying date is higher in years with high spring temperatures compared to low and, similarly, the strength of directional selection is also higher in warm compared to cold years.

In our paper, we were interested in exploring the relationship between annual selection and genetic variance (and heritability) for laying date across a long-term study. Our estimates of the expression of Va (and hence heritability) in each year are derived from a random regression model in which we split individual birds’ responses to temperature into an additive genetic (GxE) and ‘permanent environment’ (PExE) component. We compare several such models in detail in a previous paper, Husby et al 2010. As that paper showed, the GxE + PExE model is not significantly different from one which does not separate the two components of individual response (i.e. fits only IxE). As we state in the current paper (page 5):
“the change in VA alone was not statistically significant (Husby et al. 2010), something that is reflected in the large standard errors in Figure 2a. However, the statistical power to detect significant changes in additive genetic variance in relation to varying environmental conditions using a random regression animal model approach may be limited (Husby et al. 2010; Charmantier et al. 2008). Most importantly, the increase in VA is very large and represents 81.4% of the total change in VP (Figure 2a).”
Figure 2a shows the predicted change in Va from coldest to hottest years from this model, showing a greater than fourfold increase, but with large confidence intervals. If the significance of the GxE from the random regression model was unclear, we welcome the opportunity to state this again.

Although the GxE from the random regression model is not significant we show here that analyzed with an alternative approach (‘character-state’) we find clear evidence of an increase in genetic variance with spring temperature. This is consistent with the large effect size seen in the random regression model. There are concerns about the statistical power of random regression animal models and, more generally, that significance tests that do not account for the fact that variance components can never be negative are inevitably conservative (e.g. Verbeke and Molenberghs 2000) and we therefore present an alternative approach to verifying the results from our paper. Simply defining laying date as two separate traits in colder than average years and warmer than average years, and so analyzed with a ‘character-state’ approach, we find estimates of Va to be 1.014 (±1.56 SE) in ‘cold years’ and 5.88 in ‘warm years’ (±2.73SE), a difference that is highly significant (x21 = 11.08, P = 0.00087) and constitute more than fivefold increase in Va in ‘warm’ compared to ‘cold’ years. Similarly, the heritability shows a five-fold increase from 0.033 (±0.051 SE) in cold years to 0.164 (±0.076 SE) in warm years. As expected, changes in selection are also apparent using this simpler approach: estimates of selection gradients changed from 0.254 in ‘cold’ years to -0.572 in ‘warm’ years (F1,3848 = 8.87, P = 0.002) and -0.669 in ‘cold’ years to -1.859 in ‘warm’ years (selection differentials, Wilcoxon test for difference: w= 222, P =0.015). Therefore, in this population, the 'colder' years have both lower Va and weaker selection than the 'warmer' years, confirming that Va and the heritability of laying date is higher in years with stronger directional selection.

There are some misconceptions in relation to our random regression models in N&H’s text that need to be clarified. Firstly they state that: “Here, the authors have simply tested a model in which individuals vary in their response to spring temperatures (I x E) against a model in which they cannot”. This is not correct: as stated in the methods (see the section on quantitative genetic methods in our paper) we are testing a model in which individuals vary in their response to spring temperatures on the genetic (GxE) and phenotypic level (PExE) simultaneously and compare this model to a model in which there is no GxE or PExE (i.e. no IxE). Using the notation in our paper (p. 7-8): we compare a model where n1=n2=0 with a model in which n1=n2=1. This is not the same as an IxE model (i.e. one in which n1=1 and n2=0) that N&H claim we have fitted.

Second, N&H state that: “Predictions of VA generated by this model are forced to change in a quadratic fashion with spring temperature”. However it is perfectly possible that a random regression model does not predict any change with the environmental covariate: e.g. see the relationship between Va for lay date and spring temperature in the Wytham Woods great tit population (Fig 2B, Husby et al. 2010), in which there was clearly no indication of a change (quadratic or not). Changes in Va with an environmental variable will depend on the specific structure of the G matrix.

N&H further present a simulation model (on the phenotypic level) in which a significant relationship between repeatability and selection comes up (with the simulated data) some 76% of the time when there is no true variation in the slopes. The simulation raises an interesting and important issue that users of random regression models should be very cautious about. As highlighted by N&H: if there is very small variation in slopes then a relationship between a variable that covaries with the environmental covariate will be significant in many cases. However, it is worth considering in detail why this occurs. The change in variance along an environmental predictor is caused by three components: the variation in intercepts, variation in slopes and the correlation between the two. Together these three parameters determine the projection of how the variance should change. N&H simulate very low variation in slopes (median of 0.0083, a variance of approximately 0.0006 and a range between 0.0000 and 0.1556. Note this will vary in each simulation run). The issue however, is that when the variance in slopes is very small the predicted change in variance over the environmental predictor becomes linear (as can be seen from the equation used by N&H to predict variance). Therefore, any variable that correlates with the environmental predictor will then often become significant when correlated with the variance. Also, in the simulations the correlation between elevation and slope is approximately ±1 in close to 2/3 of the simulation runs, this is a very unrealistic scenario. For example, in our case the (genetic) correlation between elevation and slope is 0.66, clearly much lower than in the majority of N&H’s simulations.

The low slope variance modeled by N&H is unrealistic in our case: in fact, the estimated slope variance is very high. At the phenotypic level, there is clearly very large variation between individuals in slopes (highly statistically significant, Husby et al. 2010), and furthermore 81.4% of that slope variation is predicted to be genetic. Because the slope variance is high (genetic estimate is 3.828 and total slope variance is 4.705) this will not generate a linear change in variance over the environmental variable (as can clearly be seen from our Figure 2 in the paper) and thus a significant relationship between the variance and our selection estimates occurring by chance is unlikely. Another issue with the simulation model is that it assigns phenotypic values randomly to individuals (with repeatability of 0.15) with respect to temperature. This is biologically unrealistic: there is a very strong effect of temperature on timing of breeding in our population (see Visser et al. 1998; Nussey et al. 2005; Husby et al. 2010) as well as in bird populations in general, i.e. an individual breeding in a warm year will breed earlier than the same individual breeding in a cold year. In the simulation presented by N&H there is no such relationship.

The use of random regression animal models fitted to data from natural populations is a relatively new approach and its suitability to examine how environmental-dependence of Va and selection may influence evolutionary dynamics needs to be thoroughly explored; see also our discussion in our paper of issues regarding a previous use of the same approach as here (Wilson et al. 2006). There is clearly much to be gained from simulations such as N&H present (see also Martin et al. 2011), and we strongly encourage research in the area. We appreciate that presenting a character-state approach analyses to support the original random regression analyses would have been helpful here and welcome the opportunity to do so in this comment. A broader-scale comparison of cold versus warm years gives the same conclusion as the random regression analyses: in this population, the estimates of Va/heritability and of selection are higher in the warmer years than the colder years. We hope this clarify the content of our paper and the reply from N&H.

Arild Husby
Marcel E. Visser
Loeske E. B. Kruuk

Charmantier, A., R. H. McCleery, L. R. Cole, C. Perrins, L. E. B. Kruuk, and B.
C. Sheldon. 2008. Adaptive phenotypic plasticity in response to climate change in a wild bird population. Science 320:800-803.
Gienapp, P., E. Postma, and M. E. Visser. 2006. Why breeding time has not responded to selection for earlier breeding in a songbird population. Evolution 60:2381-2388.
Husby, A., D. H. Nussey, M. E. Visser, A. J. Wilson, B. C. Sheldon, and L. E.
Kruuk. 2010. Contrasting Patterns of Phenotypic Plasticity in Reproductive Traits in Two Great Tit (Parus major) Populations. Evolution 64:2221-2237.
Husby A, Visser ME, Kruuk LEB, 2011 Speeding Up Microevolution: The Effects of Increasing Temperature on Selection and Genetic Variance in a Wild Bird Population. PLoS Biol 9(2): e1000585. doi:10.1371/journal.pbio.10005
Martin, J. G. A., D. H. Nussey, A. J. Wilson, and D. Réale. 2011. Measuring individual differences in reaction norms in field and experimental studies: a power analysis of random regression models. Methods in Ecology and Evolution online early DOI: 10.1111/j.2041-210X.2010.00084.x.
Nussey, D. H., E. Postma, P. Gienapp, and M. E. Visser. 2005. Selection on heritable phenotypic plasticity in a wild bird population. Science 310:304-306.Verbeke, G and Molenberghs. 2000. Linear mixed models for longitudinal data. Springer series in statistics, New York, Springer.
Wilson, A. J., J. M. Pemberton, J. G. Pilkington, D. W. Coltman, D. V. Mifsud, T. H. Clutton-Brock, and L. E. B. Kruuk. 2006. Environmental coupling of selection and heritability limits evolution. PloS Biology 4:1270-1275.

No competing interests declared.