Nothing
inst/shiny-squid/source/text/modules/txt_module3.R
In squid: Statistical Quantification of Individual Differences
module3_txt <- list(
"title" = "Exploring the impact of sampling",
"goal" = "<b>Goal:</b> Real studies encounter a variety of issues that affects the conclusions
that can be drawn from data analysis. Here, we explore the real world problem of
biases in sampling and what happens when the researcher does not know about the
reason for the bias. We also then investigate how the results from data analysis
improve as the researcher knows more about the underlying mechanisms leading to biased sampling."
)
# Step 1 --------------
Mod3Step1_txt <- list(
"title" = "Step 1: Sampling and non-stochastic environment",
"subgoal" = "<b>Sub-goal:</b> to explore how hidden patterns in environment combined with variance in sampling
affect estimates of variance parameters and their interpretation.",
"intro" = paste0("<b>Introduction:</b> In the previous module (<i>",Module_titles$mod1,"</i>),
we partitioned phenotypic variance into several components
(the variance among individuals, $V_",NOT$devI,"$, the variance caused by measurement error, $V_",NOT$mError,"$,
and variance caused by the environment, $V_",NOT$envEffect,"$). In the final step of that module,
we illustrated how measurement of the environment could help explain some of the variance.
Often, when we study phenotypes in natural populations, many aspects of the environment
that could affect phenotypes will be unknown and so not measured. In Step 3,
this unmeasured environmental variance ended up as “residual” variance,
and it had no effect on the estimate of among-individual variance because
the environment was randomly determined from one sampling period to another
and all individuals were sampled at the same time and experienced the same environment.
In the present module, we explore what happens when we relax this obviously simplified assumption.
For example, suppose the environment changes steadily over the sampling period.
What happens when the pattern of how an investigator measures individuals varies,
such as if the timing of measurement is different for different individuals?"),
"exercise" = paste0("<b>Exercise:</b> As in previous simulations, we will generate a new group of individuals,
with phenotypic variation generated by measurement error $(V_",NOT$mError,")$, individual differences $(V_",NOT$devI,")$,
and the impact of the a specified environmental variable $",NOT$env,"$ which produces variance due
to the environment $(V_",NOT$envEffect,")$. We have shifted to using the notation $V_",NOT$envEffect,"$ here instead of $V_{",NOT$mean," ",NOT$env,"}$
which we used in Step 3 and 4 of the basic module “<i>",Module_titles$mod1,"</i>”.
We do this because we will soon explore what happens when only some of the environmental
variance is known, and we will use $V_{",NOT$mean," ",NOT$env,"}$ for that known variance."),
"para1" = paste0("As before, you can set $V_",NOT$mError,"$, $V_",NOT$devI,"$ and $V_{",NOT$envEffect,"}$,
and for this module, $V_{",NOT$envEffect,"}$ must be greater than 0."),
"note1" = paste0("Note that from now on, the total variance ($V_",NOT$total,"$) is not restrained to 1 anymore
and the proportion of each variance component is shown next to the input element."),
"note2" = "Also, the number of individuals will be set to 100 all along this module.",
"para2" = paste0("The environment for this simulation is, for convenience, set as being linear over time,
affecting all individuals similarly (i.e., it is “shared”).
The environment is also expressed in unit variance (i.e., $Var(",NOT$env,")=1$)
and mean-centered (i.e., $E(",NOT$env,")=0$)."),
"para3" = "You also must enter parameters for variance in the sampling timing within
and among individuals. For this simulation, the total number of expressions of
the phenotype from which you can sample is fixed at 100. While you can vary the
number of individual samples taken, for this module to effectively illustrate the
issues with sampling, the number of samples must be much less than 100.
The key parameter to be entered by you will be the among-individual variance in timing
of those records. To illustrate, below are examples of sampling records for a small
number of individuals when the among-individual variance in sampling timing is 0,
and when it is 0.9.",
"para4" = "Now you can input your own values.",
"para5" = "The figure below shows time of sampling of a subset of individuals according to the values entered.",
"results" = "<b>Results</b>",
"para6" = "If we have no information about the environment, the model we incorrectly assume to be true is:",
"RCode" = "# install.packages("lme4")<br>
LMM <- lme4::lmer(Phenotype ~ 0 + (1|Individual), data = sampled_data)",
"para7" = "A mixed-effects statistical model can then estimate these model parameters:",
"para8" = paste0("The above should show that if the unmeasured environment changes over
time AND there is among-individual variance in sampling, then some of the unknown
$V_",NOT$envEffect,"$ is placed into residual variance (making residual variance larger
than just measurement variance $V_",NOT$mError,"$), and some ends up in the estimated
$V_",NOT$devI,"$, also making it bigger than it should be."),
"conclusion" = "<b>Conclusion</b>",
"para9" = paste0("This exercise demonstrates that if there is among-individual variance in timing of sampling,
then estimates of $V_",NOT$devI,"$ will be incorrect since inevitably there are systematic differences
in environments over time. Sampling biases thus can produce “pseudo-personality” or
“pseudo-repeatability” (see also
<a href='http://onlinelibrary.wiley.com/doi/10.1111/1365-2656.12013/abstract' target='_blank'>Dingemanse & Dochtermann 2013</a>)
and could mislead a researcher into believing there are consistent differences between individuals
when there are none (or they are much smaller than it appears)."),
"para10" = paste0("Because among-individual variance in sampling and systematic changes in environment
are extremely likely in real systems, how can we get accurate estimates of $V_",NOT$devI,"$?"),
"para11" = "We explore one solution to this problem:
<ol>
<li>Adjust sampling regime to minimize it (go to Step 2)</li>
<li>Accounting for biases in your analysis (go to Step 3)</li>
</ol>"
)
# Step 2 --------------
Mod3Step2_txt <- list(
"title" = "Step 2: Sampling to reduce effects of non-stochastic environment",
"subgoal" = "<b>Sub-goal:</b> Using simulations to generate sampling regimes
that limit the effects of non-stochastic environments.",
"intro" = "<b>Introduction:</b> Step 1 revealed a problem—non-stochastic environments
through time and variability in the timing of sampling can create biases
in estimates of among-individual variation. In this step we encourage you
to adjust the sampling regime to minimize this problem. It should be obvious
that if all individuals are sampled with the same timing, then the bias
in the estimates of among-individual variance disappears,
but it is worthwhile assessing how close one has to be to identical sampling
and whether there are biases in other parameters that remain.
So, in this step we will allow you to simulate several types of
non-stochastic environments and adjust the sampling regime.",
"exercise" = paste0("<b>Exercise:</b> As in Step 1, we will generate a new group of individuals,
with phenotypic variance caused by measurement error $(V_",NOT$mError,")$, individual differences $(V_",NOT$devI,")$,
and the impact of the environment $(V_",NOT$envEffect,")$."),
"para1" = "You now get to set the environment. In Step 1 of this module, we used an environment that w
as experienced similarly by all individuals (“shared”) and which changed systematically over time.
Below, you can change these settings to have environments that each individual experiences uniquely
(“unshared”), and which changes over time as some other function (e.g., stochastically or
as a regressive autocorrelated decay function).",
"para2" = "As in step 1, you also must enter parameters for variance in the sampling timing
within and among individuals. As before, the number of expressions of the
phenotype will be set by us at 100, so keep this in mind as you enter values here.",
"para3" = "The figure below shows time of sampling of a subset of individuals according to the values entered.",
"results" = "<b>Results</b>",
"para4" = "As before, the model we assume to be true (but which is not since the environmental
effect is not included) is:",
"RCode" = "# install.packages("lme4")<br>
LMM <- lme4::lmer(Phenotype ~ 0 + (1|Individual), data = sampled_data)",
"para5" = "A mixed statistical model estimates the parameters which we can compare with the true values:",
"conclusion" = "<b>Conclusion</b>",
"para6" = paste0("The results of any given simulation may vary, but the overall picture that emerges
if you do several simulations should be that your estimates are better when $V_",NOT$envEffect,"$ is small,
as you measure each individual more often, and your sampling time is increasingly similar
among individuals."),
"para7" = paste0("Did you simulate a population where the environment is not shared among individuals?
If not, try it now. What you should find is that no matter what the sampling regime,
your estimate of $V_",NOT$devI,"$ is too high. To understand, let's return to the definitions
of the variance components: We defined $V_",NOT$devI,"$ as the variance among individuals that permanently
affected their phenotype throughout the sampling period. Biologically,
this can be ascribed to genetic differences or environments acting during development
(e.g., before measurements started). When environments are unshared during sampling,
the environment is affecting the phenotype each time it is expressed. However, because
the environment is autocorrelated across sampling episodes and differs among individuals,
apparent individual differences arise because individuals are in different environments
not because they entered the time period of phenotypic expression differing in their
phenotype (note: You may be thinking that since individuals in the real world partially
choose their environment then their phenotype is not solely due to the environment.
That is true but does not change the fact that for the focal trait it is sensitive
to the environment the individual is in each time it is expressed. We will get to
the issue of multiple phenotypic characters and how they might integrate in the
“<i>",Module_titles$mod4,"</i>” module)."),
"para8" = "To conclude for this step, if you do not know what environments are affecting
trait expression, sampling in parallel for all individuals is a possible
solution to potential biases created by non-stochastic environments.
But, because unshared environments can create biases even with identical
sampling (and often identical sampling will be nearly impossible to achieve),
the only other solution is to measure the environment and account for
possible biases explicitly. This is explored next in Step 3."
)
# Step 3 --------------
Mod3Step3_txt <- list(
"title" = "Step 3: Biased sampling and known and unknown environments",
"subgoal" = "<b>Sub-goal:</b> Accounting for the environment to control for environmental biases.",
"intro" = paste0("<b>Introduction:</b> Step 1 of this module illustrated that environmental
effects on phenotypes can produce biases in estimates of among-individual variance $(V_",NOT$devI,")$.
Step 2 explored how altering sampling regimes could reduce this problem but also
revealed that in some circumstances no sampling regime would work.
Sometimes individuals experience different environments, and no sampling regime can adjust for that.
However, if investigators can measure the environment, then such differences could be accounted for.
Environmental variance was accounted for using linear regression in step 4 of the
“<i>",Module_titles$mod1,"</i>” module. Here we demonstrate that this can,
under some circumstances, solve the bias in sampling problem."),
"exercise1" = paste0("<b>Exercise 1:</b> This exercise follows the same structure as all of our other simulations so far.
We will generate a group of individuals, with phenotypic variance caused by measurement error $(V_",NOT$mError,")$,
individual differences $(V_",NOT$devI,")$, and the impact of the environment $(V_",NOT$envEffect,")$.
So, first set the true values of these variances:"),
"para1" = "The environment can be chosen as in Step 2.
It, combined with the sampling regime, will affect within- and among-individual
variance in the environment.",
"para3" = "Finally, we will have you set how much of the environmental variance has
been measured and is therefore known. You will select a proportion,
from 0 to 1 of this variance. This proportion along with the proportion
of total variance that is environmental will determine the correlation
between phenotype and the known environment. The results of Step 1 should
have shown you what happens when all the environmental variance is unknown
(or not included in your statistical model). Here, let’s start with all
the environmental variance being known and measurable.",
"results" = paste0("<b>Results:</b> In the module “<i>",Module_titles$mod1,"</i>”,
Step 4, we said the statistical model was"),
"RCode1" = "# install.packages("lme4")<br>
LMM1 <- lme4::lmer(Phenotype ~ 1 + X1 + (1|Individual), data = sampled_data)",
"para4" = "This is the model we will investigate here.
We will compare it to a model in which all of the environmental
variance is unknown, e.g.,",
"RCode2" = "LMM2 <- lme4::lmer(Phenotype ~ 1 + (1|Individual), data = sampled_data)",
"para5" = "A mixed effects statistical model estimates the parameters,
which we can compare with the true values:",
"para6" = paste0("This should show you that when there is among-individual variance in sampling
and you can account for all the environmental variance with an x variable,
any bias in $V_",NOT$devI,"$ caused by the biased sampling disappears."),
"reminder" = paste0("A brief reminder about notation: When unknown environments affect phenotypic variance,
we have referred to that variance as $V_",NOT$envEffect,"$. In the model where the environment is known $(",NOT$env,")$,
there now is a specific component of variance due to that known environmental factor,
$V_{",NOT$mean," ",NOT$env,"}$. In the case above, $V_",NOT$envEffect,"=V_{",NOT$mean," ",NOT$env,"}$,
but in the real world with many environmental variables,
$V_{",NOT$mean," ",NOT$env,"}$ will be only a fraction of $V_",NOT$envEffect,"$."),
"exercise2" = "<b>Exercise 2:</b> Now, let's repeat the same simulation as above,
expect this time explore what happens as you change the proportion of the environmental
variance that is known. Below is the bar that allows you to adjust this.",
"para8" = "If you want, you can also change the level of bias in sampling.",
"para9" = "<b>Results:</b> As above, we will show you the true values you entered,
the values estimated when the environment is unknown, and those estimated
when some portion of the environment is known and included in the model.",
"conclusion" = "<b>Conclusion:</b> There are two lessons that emerge from this exercise.
First, biases in sampling are usually inevitable, but measuring the underlying
environments that differ among individuals can reduce them.
Thus, if you want to measure among-individual variance,
you must think carefully about potential biases in environments,
and measure those environments.
That will give you a better estimate of among-individual variance.",
"conclusion2" = paste0("The second lesson is that bias in sampling may occur without
you being aware of it. This unknown environment will affect
your estimate of among-individual variance. Put another way,
any among-individual variance estimated from real data could
be due to unknown biased environments. One cannot be sure that
you have accounted for all of the environmental variance.
The $V_",NOT$devI,"$ that is found from real data must therefore be interpreted cautiously."),
"finalcaveat" = paste0("<b>A final caveat:</b> An interesting consequence of having variance
in sampling among individuals is that it produces variance in the experienced
environment that exists both within and among individuals. We have assumed
that the impact of the environmental variance that exists among individuals
is the same as that of the variance in environment within-individuals.
As an example, individuals may be on territories with different average
levels of resources through the whole period of time you are taking measurements,
and those resources may fluctuate some from day to day as well. Thus in your population,
there is both among-individual variance in environment
(e.g., differences between territories) and within-individual variance in environment
(differences between days within a territory). We have assumed these have
the same effect on phenotype. It is possible that this is not the case.
If so, the method we have demonstrated here will not give accurate estimates of $V_",NOT$devI,"$.
We discuss one solution to this in a module on within and among-subject centering.
The issues related to centering are complex, so we recommend this module be done
after the module on random regression.")
)
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module3_txt <- list(
"title" = "Exploring the impact of sampling",
"goal" = "<b>Goal:</b> Real studies encounter a variety of issues that affects the conclusions
that can be drawn from data analysis. Here, we explore the real world problem of
biases in sampling and what happens when the researcher does not know about the
reason for the bias. We also then investigate how the results from data analysis
improve as the researcher knows more about the underlying mechanisms leading to biased sampling."
)
# Step 1 --------------
Mod3Step1_txt <- list(
"title" = "Step 1: Sampling and non-stochastic environment",
"subgoal" = "<b>Sub-goal:</b> to explore how hidden patterns in environment combined with variance in sampling
affect estimates of variance parameters and their interpretation.",
"intro" = paste0("<b>Introduction:</b> In the previous module (<i>",Module_titles$mod1,"</i>),
we partitioned phenotypic variance into several components
(the variance among individuals, $V_",NOT$devI,"$, the variance caused by measurement error, $V_",NOT$mError,"$,
and variance caused by the environment, $V_",NOT$envEffect,"$). In the final step of that module,
we illustrated how measurement of the environment could help explain some of the variance.
Often, when we study phenotypes in natural populations, many aspects of the environment
that could affect phenotypes will be unknown and so not measured. In Step 3,
this unmeasured environmental variance ended up as “residual” variance,
and it had no effect on the estimate of among-individual variance because
the environment was randomly determined from one sampling period to another
and all individuals were sampled at the same time and experienced the same environment.
In the present module, we explore what happens when we relax this obviously simplified assumption.
For example, suppose the environment changes steadily over the sampling period.
What happens when the pattern of how an investigator measures individuals varies,
such as if the timing of measurement is different for different individuals?"),
"exercise" = paste0("<b>Exercise:</b> As in previous simulations, we will generate a new group of individuals,
with phenotypic variation generated by measurement error $(V_",NOT$mError,")$, individual differences $(V_",NOT$devI,")$,
and the impact of the a specified environmental variable $",NOT$env,"$ which produces variance due
to the environment $(V_",NOT$envEffect,")$. We have shifted to using the notation $V_",NOT$envEffect,"$ here instead of $V_{",NOT$mean," ",NOT$env,"}$
which we used in Step 3 and 4 of the basic module “<i>",Module_titles$mod1,"</i>”.
We do this because we will soon explore what happens when only some of the environmental
variance is known, and we will use $V_{",NOT$mean," ",NOT$env,"}$ for that known variance."),
"para1" = paste0("As before, you can set $V_",NOT$mError,"$, $V_",NOT$devI,"$ and $V_{",NOT$envEffect,"}$,
and for this module, $V_{",NOT$envEffect,"}$ must be greater than 0."),
"note1" = paste0("Note that from now on, the total variance ($V_",NOT$total,"$) is not restrained to 1 anymore
and the proportion of each variance component is shown next to the input element."),
"note2" = "Also, the number of individuals will be set to 100 all along this module.",
"para2" = paste0("The environment for this simulation is, for convenience, set as being linear over time,
affecting all individuals similarly (i.e., it is “shared”).
The environment is also expressed in unit variance (i.e., $Var(",NOT$env,")=1$)
and mean-centered (i.e., $E(",NOT$env,")=0$)."),
"para3" = "You also must enter parameters for variance in the sampling timing within
and among individuals. For this simulation, the total number of expressions of
the phenotype from which you can sample is fixed at 100. While you can vary the
number of individual samples taken, for this module to effectively illustrate the
issues with sampling, the number of samples must be much less than 100.
The key parameter to be entered by you will be the among-individual variance in timing
of those records. To illustrate, below are examples of sampling records for a small
number of individuals when the among-individual variance in sampling timing is 0,
and when it is 0.9.",
"para4" = "Now you can input your own values.",
"para5" = "The figure below shows time of sampling of a subset of individuals according to the values entered.",
"results" = "<b>Results</b>",
"para6" = "If we have no information about the environment, the model we incorrectly assume to be true is:",
"RCode" = "# install.packages("lme4")<br>
LMM <- lme4::lmer(Phenotype ~ 0 + (1|Individual), data = sampled_data)",
"para7" = "A mixed-effects statistical model can then estimate these model parameters:",
"para8" = paste0("The above should show that if the unmeasured environment changes over
time AND there is among-individual variance in sampling, then some of the unknown
$V_",NOT$envEffect,"$ is placed into residual variance (making residual variance larger
than just measurement variance $V_",NOT$mError,"$), and some ends up in the estimated
$V_",NOT$devI,"$, also making it bigger than it should be."),
"conclusion" = "<b>Conclusion</b>",
"para9" = paste0("This exercise demonstrates that if there is among-individual variance in timing of sampling,
then estimates of $V_",NOT$devI,"$ will be incorrect since inevitably there are systematic differences
in environments over time. Sampling biases thus can produce “pseudo-personality” or
“pseudo-repeatability” (see also
<a href='http://onlinelibrary.wiley.com/doi/10.1111/1365-2656.12013/abstract' target='_blank'>Dingemanse & Dochtermann 2013</a>)
and could mislead a researcher into believing there are consistent differences between individuals
when there are none (or they are much smaller than it appears)."),
"para10" = paste0("Because among-individual variance in sampling and systematic changes in environment
are extremely likely in real systems, how can we get accurate estimates of $V_",NOT$devI,"$?"),
"para11" = "We explore one solution to this problem:
<ol>
<li>Adjust sampling regime to minimize it (go to Step 2)</li>
<li>Accounting for biases in your analysis (go to Step 3)</li>
</ol>"
)
# Step 2 --------------
Mod3Step2_txt <- list(
"title" = "Step 2: Sampling to reduce effects of non-stochastic environment",
"subgoal" = "<b>Sub-goal:</b> Using simulations to generate sampling regimes
that limit the effects of non-stochastic environments.",
"intro" = "<b>Introduction:</b> Step 1 revealed a problem—non-stochastic environments
through time and variability in the timing of sampling can create biases
in estimates of among-individual variation. In this step we encourage you
to adjust the sampling regime to minimize this problem. It should be obvious
that if all individuals are sampled with the same timing, then the bias
in the estimates of among-individual variance disappears,
but it is worthwhile assessing how close one has to be to identical sampling
and whether there are biases in other parameters that remain.
So, in this step we will allow you to simulate several types of
non-stochastic environments and adjust the sampling regime.",
"exercise" = paste0("<b>Exercise:</b> As in Step 1, we will generate a new group of individuals,
with phenotypic variance caused by measurement error $(V_",NOT$mError,")$, individual differences $(V_",NOT$devI,")$,
and the impact of the environment $(V_",NOT$envEffect,")$."),
"para1" = "You now get to set the environment. In Step 1 of this module, we used an environment that w
as experienced similarly by all individuals (“shared”) and which changed systematically over time.
Below, you can change these settings to have environments that each individual experiences uniquely
(“unshared”), and which changes over time as some other function (e.g., stochastically or
as a regressive autocorrelated decay function).",
"para2" = "As in step 1, you also must enter parameters for variance in the sampling timing
within and among individuals. As before, the number of expressions of the
phenotype will be set by us at 100, so keep this in mind as you enter values here.",
"para3" = "The figure below shows time of sampling of a subset of individuals according to the values entered.",
"results" = "<b>Results</b>",
"para4" = "As before, the model we assume to be true (but which is not since the environmental
effect is not included) is:",
"RCode" = "# install.packages("lme4")<br>
LMM <- lme4::lmer(Phenotype ~ 0 + (1|Individual), data = sampled_data)",
"para5" = "A mixed statistical model estimates the parameters which we can compare with the true values:",
"conclusion" = "<b>Conclusion</b>",
"para6" = paste0("The results of any given simulation may vary, but the overall picture that emerges
if you do several simulations should be that your estimates are better when $V_",NOT$envEffect,"$ is small,
as you measure each individual more often, and your sampling time is increasingly similar
among individuals."),
"para7" = paste0("Did you simulate a population where the environment is not shared among individuals?
If not, try it now. What you should find is that no matter what the sampling regime,
your estimate of $V_",NOT$devI,"$ is too high. To understand, let's return to the definitions
of the variance components: We defined $V_",NOT$devI,"$ as the variance among individuals that permanently
affected their phenotype throughout the sampling period. Biologically,
this can be ascribed to genetic differences or environments acting during development
(e.g., before measurements started). When environments are unshared during sampling,
the environment is affecting the phenotype each time it is expressed. However, because
the environment is autocorrelated across sampling episodes and differs among individuals,
apparent individual differences arise because individuals are in different environments
not because they entered the time period of phenotypic expression differing in their
phenotype (note: You may be thinking that since individuals in the real world partially
choose their environment then their phenotype is not solely due to the environment.
That is true but does not change the fact that for the focal trait it is sensitive
to the environment the individual is in each time it is expressed. We will get to
the issue of multiple phenotypic characters and how they might integrate in the
“<i>",Module_titles$mod4,"</i>” module)."),
"para8" = "To conclude for this step, if you do not know what environments are affecting
trait expression, sampling in parallel for all individuals is a possible
solution to potential biases created by non-stochastic environments.
But, because unshared environments can create biases even with identical
sampling (and often identical sampling will be nearly impossible to achieve),
the only other solution is to measure the environment and account for
possible biases explicitly. This is explored next in Step 3."
)
# Step 3 --------------
Mod3Step3_txt <- list(
"title" = "Step 3: Biased sampling and known and unknown environments",
"subgoal" = "<b>Sub-goal:</b> Accounting for the environment to control for environmental biases.",
"intro" = paste0("<b>Introduction:</b> Step 1 of this module illustrated that environmental
effects on phenotypes can produce biases in estimates of among-individual variance $(V_",NOT$devI,")$.
Step 2 explored how altering sampling regimes could reduce this problem but also
revealed that in some circumstances no sampling regime would work.
Sometimes individuals experience different environments, and no sampling regime can adjust for that.
However, if investigators can measure the environment, then such differences could be accounted for.
Environmental variance was accounted for using linear regression in step 4 of the
“<i>",Module_titles$mod1,"</i>” module. Here we demonstrate that this can,
under some circumstances, solve the bias in sampling problem."),
"exercise1" = paste0("<b>Exercise 1:</b> This exercise follows the same structure as all of our other simulations so far.
We will generate a group of individuals, with phenotypic variance caused by measurement error $(V_",NOT$mError,")$,
individual differences $(V_",NOT$devI,")$, and the impact of the environment $(V_",NOT$envEffect,")$.
So, first set the true values of these variances:"),
"para1" = "The environment can be chosen as in Step 2.
It, combined with the sampling regime, will affect within- and among-individual
variance in the environment.",
"para3" = "Finally, we will have you set how much of the environmental variance has
been measured and is therefore known. You will select a proportion,
from 0 to 1 of this variance. This proportion along with the proportion
of total variance that is environmental will determine the correlation
between phenotype and the known environment. The results of Step 1 should
have shown you what happens when all the environmental variance is unknown
(or not included in your statistical model). Here, let’s start with all
the environmental variance being known and measurable.",
"results" = paste0("<b>Results:</b> In the module “<i>",Module_titles$mod1,"</i>”,
Step 4, we said the statistical model was"),
"RCode1" = "# install.packages("lme4")<br>
LMM1 <- lme4::lmer(Phenotype ~ 1 + X1 + (1|Individual), data = sampled_data)",
"para4" = "This is the model we will investigate here.
We will compare it to a model in which all of the environmental
variance is unknown, e.g.,",
"RCode2" = "LMM2 <- lme4::lmer(Phenotype ~ 1 + (1|Individual), data = sampled_data)",
"para5" = "A mixed effects statistical model estimates the parameters,
which we can compare with the true values:",
"para6" = paste0("This should show you that when there is among-individual variance in sampling
and you can account for all the environmental variance with an x variable,
any bias in $V_",NOT$devI,"$ caused by the biased sampling disappears."),
"reminder" = paste0("A brief reminder about notation: When unknown environments affect phenotypic variance,
we have referred to that variance as $V_",NOT$envEffect,"$. In the model where the environment is known $(",NOT$env,")$,
there now is a specific component of variance due to that known environmental factor,
$V_{",NOT$mean," ",NOT$env,"}$. In the case above, $V_",NOT$envEffect,"=V_{",NOT$mean," ",NOT$env,"}$,
but in the real world with many environmental variables,
$V_{",NOT$mean," ",NOT$env,"}$ will be only a fraction of $V_",NOT$envEffect,"$."),
"exercise2" = "<b>Exercise 2:</b> Now, let's repeat the same simulation as above,
expect this time explore what happens as you change the proportion of the environmental
variance that is known. Below is the bar that allows you to adjust this.",
"para8" = "If you want, you can also change the level of bias in sampling.",
"para9" = "<b>Results:</b> As above, we will show you the true values you entered,
the values estimated when the environment is unknown, and those estimated
when some portion of the environment is known and included in the model.",
"conclusion" = "<b>Conclusion:</b> There are two lessons that emerge from this exercise.
First, biases in sampling are usually inevitable, but measuring the underlying
environments that differ among individuals can reduce them.
Thus, if you want to measure among-individual variance,
you must think carefully about potential biases in environments,
and measure those environments.
That will give you a better estimate of among-individual variance.",
"conclusion2" = paste0("The second lesson is that bias in sampling may occur without
you being aware of it. This unknown environment will affect
your estimate of among-individual variance. Put another way,
any among-individual variance estimated from real data could
be due to unknown biased environments. One cannot be sure that
you have accounted for all of the environmental variance.
The $V_",NOT$devI,"$ that is found from real data must therefore be interpreted cautiously."),
"finalcaveat" = paste0("<b>A final caveat:</b> An interesting consequence of having variance
in sampling among individuals is that it produces variance in the experienced
environment that exists both within and among individuals. We have assumed
that the impact of the environmental variance that exists among individuals
is the same as that of the variance in environment within-individuals.
As an example, individuals may be on territories with different average
levels of resources through the whole period of time you are taking measurements,
and those resources may fluctuate some from day to day as well. Thus in your population,
there is both among-individual variance in environment
(e.g., differences between territories) and within-individual variance in environment
(differences between days within a territory). We have assumed these have
the same effect on phenotype. It is possible that this is not the case.
If so, the method we have demonstrated here will not give accurate estimates of $V_",NOT$devI,"$.
We discuss one solution to this in a module on within and among-subject centering.
The issues related to centering are complex, so we recommend this module be done
after the module on random regression.")
)
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