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BRIEF REVIEW OF STATISTICAL CONCEPTS AND METHODS

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BRIEF REVIEW OF STATISTICAL

CONCEPTS AND METHODS

Mathematical expectation

The mean (x) of random variable x is:

where n is the number of observations, the variance (s2) is:

Mathematical expectation

The standard deviation (s) is:

The coefficient of variation is:

Precision, bias, and accuracy

Basic probability

The probability of a event occurring is expressed as: P(event)

The probability of the event not occurring, 1- P(event) or P(~event).

If events are independent, the probability of events A and B occurring is estimated as: p(A) * p(B).

Detection and capture probabilities

The probability of capturing a single fish given 1 is present: p(capture)

capturing 2 fish given 2 are present:

p(capture)*p(capture) = p(capture)2,

the probability of catching at least 1 given 2 present:

p(capture)*(1-p(capture )) + (1- p(capture))*p(capture) + p(detect)2

or:

1- (1- p(capture))N

where N = number of fish present

Probability example

The probability of detecting a fish during a single event: p(detect)

On all three sampling occasions is:

p(detect)*p(detect)*p(detect) = p(detect)3,

the probability of not catching it during any of the 3 occasions is:

(1- p(detect))*(1- p(detect))*(1- p(detect)) = (1- p(detect))3,

and the probability of catching it on at least 1 occasion is the complement of not catching it during any of the occasions:

1- (1- p(detect))3.

Conditional probability

The probability a fish is present: p(present)

The probability detecting a fish, given it is present : p(detect | present)

The probability detecting a fish given it is not present?

The probability a fish is present and detected:

p(detect | present) * p(present)

Conditional probability

The probability N fish are present: p(N)

The probability detecting at least 1 fish, given N are present : p(detect | N)

The probability N fish are present and at least 1 is detected:

p(detect | N) * p(N)

Question: if we sampled but did not detect a fish species, what are the chances it was present?

p(present | not detected)

The probability fish species present: p(present)

not present: 1- p(present)

The probability detection, given present : p(detect | present)

probability detection, given not present : p(detect | not present) = 1

Total probability of the event of not detecting the species:

Two possibilities: (1) present but not detected and (2) not present

P(not detected| present)*P(present) + P(not detected| not present)*P(not present)

Bayes rule

p(present | not detected) =

p(not detected| present)*p(present)

p(not detected| present)*p(present) + p(not detected| not present)*p(not present)

Assume 80% probability of detection:

p(not detected| present) = 1- 0.80 = 0.20

Assume 40% probability of bull trout present:

p(present) = 0.40, p(not present) = 0.60

p(not detected| not present) = 1

Now calculate:

0.20*0.40

0.20*0.40 + 1*0.60 = 0.118 or 11.8%

Models and fisheries management

“True” Models

- Fundamental assumption: there is no “true” model that generates biological data

- Truth in biological sciences has essentially infinite dimension; hence,
- full reality cannot be revealed with finite samples.

- Biological systems are complex with many small effects, interactions, individual
- heterogeneity, and environmental covariates.

- Greater amounts of data are required to model smaller effects.

- Thus all models are approximations of reality

Models = hypotheses

Models and hypotheses

- Hypotheses are unproven theories, suppositions that are tentatively
- accepted to explain facts or as the basis for further investigation

- Models are very explicit representations of hypotheses

- Several models can represent a single hypotheses

- Models are tools for evaluating hypotheses

Models and hypotheses: example

Hypothesis: shoal bass reproduction success is greater when there are more reproductively active adults

Y = aN

Number of young is proportional to the number of adults

Number of young increases with the number of adults

until nesting areas are saturated

Y = aN/(1+bN)

Number of young is increases until the carrying capacity of

nesting and rearing areas is reached

Y = aNe-bN

Y = aN

Y = aN/(1+bN)

Number of YOY

Y = aNe-bN

Number of shoal bass

Tapering Effect Sizes

- Biological systems there are often large important effects, followed by smaller
- effects, and then yet smaller effects.

- These effects might be sequentially revealed as sample size increases
- because information content increases

- Rare events yet are more difficult to study (e.g. fire, flood, volcanism)

Big

effects

small

effects

- Determine what is the best explanation given the data

- Determine what is the best model for predicting the response

- Two approaches in fisheries/ecology
- Null hypothesis testing
- Information theoretic approaches

Null hypothesis testing

Develop an a priori hypothesis

Deduce testable predictions (i.e., models)

Carry out suitable test (experiment)

Compare test results with predictions

Retain or reject hypothesis

Hypothesis testing example:

Density independence for lake sturgeon populations

Hypothesis: lake sturgeon reproduction is density independent

Prediction: there is no relation between adult density and age 0 density

Model: Y = B0

Test: measure age 0 density for various adult densities over time

Compare:

Linear regression between age 0 and adult sturgeon densities, P value = 0.1839

Using a critical a-level = 0.05, we conclude no significant relationship

Result: Retain hypothesis lake sturgeon reproduction is density

independent

Model selection based on p-values

- No theoretical basis for model selection
- P-values ~ precision of estimate
- P-values strongly dependent on sample size

P(the data (or more extreme data)| Model) vs. L(model | the data)

JUST SAY NO TO STATISTICAL SIGNIFICANCE TESTING

FOR MODEL SELECTION

Information theory

If full reality cannot be included in a model, how do we tell how close we are to truth.

truth

Kullback-Leibler distance based on information theory

The measures how much information is in accounted for in a model

Entropy is synonymous with uncertainty

Information theory

K,L distance (information) is represented by: I(truth| model)

It represents information lost when the candidate model is used to

Approximate truth thus SMALL values mean better fit

AIC is based on the concept of minimizing K-L distance

Akaike noticed that the maximum log likelihood

Log( L (model or parameter estimate | the data) ) was related to K-L distance

45

40

35

30

25

20

15

10

5

0

0

5

10

What a maximum likelihood estimate?

It is those parameter values that maximize the value of the likelihood,

given the data

Sums of squares in regression also is a measure of the relative fit of a model

SSE = Sdeviations2

The maximum log likelihood (and SSE) is a biased estimate of K-L distance

Akaike’s contribution was that he showed that:

AIC = -2ln(L (model | the data)) + 2K

It is based on the principle of parsimony

Bias2

Variance

Few

Many

Number of parameters

Heuristic interpretation

AIC = -2ln(likelihood) + 2*K

Measures model lack of fit

Penalty for increasing model size

(enforces parsimony)

AIC: Small sample bias adjustment

If ratio of n/K is < 40 then use AICc

AICc = -2*ln(likelihood | data) + 2*K + (2*K*(K+1))/(n-K-1)

As n gets big….

(2*K*(K+1))/(n-K-1) = 1/very large number

(2*K*(K+1))/(n-K-1) = 0

AICc = AIC

So….

Model selection with AIC

What is model selection?

AIC by itself is relatively meaningless.

Recall that we find the best model by comparing various models and examining

Their relative distance to the “truth”

We do this by calculating the difference between the best fitting model (lowest AIC) and the other models.

Model selection uncertainty

Which model is the best?

What about if you collect data at the same spot next year,

next week, next door?

AIC weights-- long run interpretation vs. Bayesian.

Confidence set of models analogous to confidence intervals

Where do we get AIC?

K

-2ln(L (model | the data))

Interpreting AIC

Best model

(lowest AICc)

Difference between lowest AIC and model

(relative distance from truth)

Interpreting AIC

AICc weight, ranges 0-1 with 1 = best model

Interpreted a relative likelihood that model is best, given the data and the other models in the set

Interpreting AIC

Ratio of 2 weights interpreted as the strength of evidence for one model over another

Here the best model is 0.86748/0.13056 = 6.64 times more likely to be

The best model for estimating striped bass population size

Confidence model set

Analogous to a confidence interval for a parameter estimate

Using a 1/8 (0.12) rule for weight of evidence, my confidence set includes the

top two models (both model likelihoods > 0.12).

Linear models review

Y: response variable (dependent variable)

X: predictor variable (independent variable)

Y = b0 + b1*X + e

b0 is the intercept

b1 is the slope (parameter) associated with X

e is the residual error

Linear models review

When Y is a probability it is bounded by 0, 1

Y = b0 + b1*X

Can provide values <0 and > 1, we need to transform

or use a link function

For probabilities, the logit link is the most useful

Logit link

p

h = ln( )

1- p

h is the log odds

p is the probability of an event

Log linear models

(logistic regression)

h = b0 + b1*X

h is the log odds

b0 is the intercept

b1 is the slope (parameter) associated with X

Betas are on a logit scale and the log-odds needs to be back transformed

Back transformation:

Inverse logit link

1

p =

-h

1+exp( )

h is the log odds

p is the probability of an event

Back transformation example

h = b0 + b1*X

b0 = - 2.5

b1 = 0.5

X = 2

Back transformation example

h = -2.5 + 0.5*2

h = -1.5

1

= 0.18 or 18%

1+exp(1.5)

Interpreting beta estimates

Betas are on a logit scale, to interpret calculate odds ratios

Using the exponential function

b1 = 0.5

exp(0.5) = 1.65

Interpretation: for each 1 unit increase in X, the event is 1.65 times more likely to occur

For example, for each 1 inch increase in length, a fish is 1.65 times more likely to be

caught