240 9 Variance Power

8/19/2019 240 9 Variance Power

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Distribution of British birds mapped very closely to pubs – sampling bias.

Likewise, size of net mesh greatly affects the size of sampled fish


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Model = analysis

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Each requires its own set of statistical techniques therefore it is important toknow what you are dealing with.


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Continuous, Discrete or Categorical data? The type of data determines the thetype of test – i.e. can’t talk in terms of “average:” slug colour.


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We’ll now focus on this last point.


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Growth rate = “realized r” (which we have already discussed)

Here in a statistical analysis context, “R” is entirely different. Here we are

referring to the amount of variation explained by the BEST FIT LINE betweenof the two variables – which in this case is 23% (23% of the variation in moosepopulation growth rate is attributable to the mean age of moose in the

population while the remaining 77% of variation is caused by other factors(which includes random chance). If age, and only age, explained the

differences in population growth rate, all the data points would be on the lineand R2=1.0 (or 100%).

Define “best fit line” in simple terms: A line that minimizes the average distance

of each data point to the line. In this example we have chosen a “linearrelationship” (a straight line). Best fit lines can be a number of different shapes

(logistic, exponential, normal etc.). Linear relationships are the simplest todescribe.

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Confidence in the relationship is the degree to which points fall on the “best fit”line. Here 81% of the variation in fish abundance is explained by the size ofthe pothole.

The average distance of points from the line is the “variation”

Why are all points not on the line?Real, biological reasons but also artifacts of how we measure. We want to

eliminate the latter as best we can, leaving only the former. We do this byoptimizing our experimental design (measure as many of the biological driers

as possible) and use analysis tools that can tease these various drivers apart.


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All of these plots have low variation but plots B,C,and D are not good modelsof the data.

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Was every grizzly bear enumerated? No. Therefore we must use samples toestimate the actual (aka parameter ) abundance of the population.

This example is a simplified interpretation of the Artelle et al paper.

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Since these are estimates, we must also indicate the amount of likely variation(i.e. the error) associated with each.

The line around the estimate of the mean (the dots) may be standard errors,

standard deviations, or confidence intervals –each mean very different thingsand knowing those differences is critical in interpreting scientific data.

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Interpretation: There is a 95% chance the parameter (the actual, realabundance) lays within this interval. Conversely, there is a 5% chance theparameter value lays outside this interval (i.e. there is a 5% of being wrong)

Another way to look at this, if the experiment were run 100 times, theabundance statistic would fall within this interval at least 95 times. The CI is an

estimate of confidence in the statistic.

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Overlapping CIs indicate the real parameter value is likely captured by bothintervals and therefore statistically they cannot be said to be significantlydifferent. In other words, because each interval defines the range of values

within which we are 95% certain the real value lies within, we have to acceptthe possibility that the real population abundance for both GOV and NGO are

the same value.

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The far right estimate – it has the narrowest interval within which we are 95%certain the real value exists.

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!.and therefore reflects the confidence in your parameter estimates!


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Here the mean is 10 but of equal interest is the dispersion of points around“10”. The greater the dispersion (aka variance) the less meaningful the meanof 10 is.

This plot shows the “bell curve” or more correctly, “the normal distribution”. In anormal distribution the dispersion of data points around the mean (i.e. in the

positive and negative direction) is roughly equal and symmetric. As a result,when data are “normally distributed”, the mean, mode and median are thesame value (or very nearly so).

Put another way: the number or extremely small individuals is roughly

equivalent to the number of really large individuals.


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These are all “normal” distributions.

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The number of species plotted for different abundance intervals, each intervalbeing twice the preceding one. The overall pattern is normally distributed. Aninteresting twist: The portion of the graph (red) left of Preston's veil line is

theoretical, depicting those species that are expected to be present but theirlow abundance prevents them from being represented in the sample. Note the

x-axis is on a log scale.

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From a very interesting paper that is directly relevant to your tutorial projects.

The relevance here is that various factors that affect detectability are all

normally distrusted around a mean value.

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The power of the normal (aka Gaussian) distribution.

With standard deviation we can estimate how dispersed the data are withoutactually sampling all individuals.


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Based on our samples, 95% of individuals from the sampled population will bebetween (mean-2SD) and (mean+2SD)


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Here we can sample every individual in our population (i.e. we are not trying toinfer anything about ALL dogs, just the 5 dogs we have here). In this special,rare, case, we divide by n (which here is 5). In future examples, where we use

samples to make inferences about a larger population (almost always thesituation we face), we would use n-1 as the denominator in the variation

calculation (here would be 4).

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Having an estimate is only the start of an analysis. What we also need to knowis how precise that estimate is – this is an estimate of the “power” of theestimate or put another way, the confidence in the estimate.

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If two researchers measure the abundance of a particular bird species,

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The simplest of examples.

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What is an example hypothesis here?

What is the null hypothesis (Ho)

What is the alternative hypothesis (H A)

What is the prediction?

How many tosses are necessary? Lets say each toss (‘replicate’) costsmoney!here we introduce a real life consideration – data costs money (and/

or time, risk etc).

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The answer is, the ability to detect a fixed coin increases with the number oftosses!

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As sample size increases, the test becomes more precise (in other words, ourpower to detect a fixed coin increases)

By increasing sample size you can overcome random runs of heads or tails

because the confidence limits shrink. Low sample sizes may not allowconfidence limits small enough to differentiate overlapping distributions.

Confidence interval is telling us that we are 95% certain that the real

population mean is within these bounds. Recall that we are using a sample toinfer facts of the population.

Deviation from 1:1 in heads:tails we can expect in a fair coin with 5% error

2 tosses = 0 to 2

10 tosses = 2 to 8

50 tosses = 18 to 32

100 tosses = 40 to 601000 tosses = 470 to 530

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Therefore the mean estimate is meaningless unless the variance around thatmean is also reported. Gov and NGO estimates may be dramatically differentor the same number !.depending on the variance. The take-home here that it

is NOT the mean that should be focus, but the confidence interval because thereal value (parameter) can be anywhere in this interval.

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Remember that we are not talking aboutbiologically significanthere. Atissue is NOT whether the size of the difference is significant or not. It is thatsmaller average differences in size will be harder to detect (given there will be

variation around the means) when the means are not greatly different.

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When the effect size is large enough and the experiment is well designed, youdon’t need statistics!.

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but more often the differences are more subtle and statistics are used todetermine if differences are large enough to be considered “significant” akascientifically valid.

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The two means could be VERY similar and the difference could still beconfidently detected even with modest sample size if the variance is very low.

Think how low vs. high sample size may hurt or help differentiating the two

populations on the lower panel

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Imagine one coin always landed “heads” 100% o the time. Ability to spot itrelative to a “normal” coin would be high (fewer tosses needed)

Imagine one coin always landed “heads” and the other always landed “tails” –

would need very few tosses to confidently declare both as “fixed”

Now imagine one coin is fixed, but less so – it lands “heads” 60% of the time.

How many tosses needed to detect it!?

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A priori or prospective power analysis – how many samples do I need to detecta difference at a given effect size

post hoc or after the fact – can I detect a difference at a particular threshold

given my sample size

These considerations have very real economic implications in terms of time

and money required to gather data.

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" “equal to or less than”


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Recall that standard deviation (SD) is the square root of variance.


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95% Confidence Limits – The interval in which we are 95% certain the realmean resides.

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240 9 Variance Power

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