The mean (x) of random variable x is: where n is the number of observations, the variance (s 2 ) is:

The standard deviation ( s ) is:
The coefficient of variation is:

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%

“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

• 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

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
Y = aN/(1+bN)
Number of young increases with the number of adults until nesting areas are saturated
Y = aNe -bN
Number of young is increases until the carrying capacity of nesting and rearing areas is reached

Number of shoal bass
Y = aN
Y = aN/(1+bN)
Y = aNe -bN

• 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

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 = B
0
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

• 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

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

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
45
40
35
30
25
20
15
10
5
0
0 5 10
SSE =
S deviations 2

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
Few
Heuristic interpretation
Number of parameters
Many
AIC = -2ln(likelihood) + 2*K
Measures model lack of fit Penalty for increasing model size
(enforces parsimony)

If ratio of n/K is < 40 then use AIC c
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
So….
AICc = 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

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

Best model
(lowest AICc)
Difference between lowest AIC and model
(relative distance from truth)

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

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

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).

Y: response variable (dependent variable)
X: predictor variable (independent variable)
Y = b
0
+ b
1
*X + e b
0 b
1 is the intercept is the slope (parameter) associated with X e is the residual error

When Y is a probability it is bounded by 0, 1
Y = b
0
+ b
1
*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

h = ln( ) h is the log odds p is the probability of an event

h = b
0
+ b
1
*X h is the log odds b
0 is the intercept b
1 is the slope (parameter) associated with X
Betas are on a logit scale and the log-odds needs to be back transformed

p =
1
h h is the log odds p is the probability of an event

h = b
0
+ b
1
*X b
0 b
1
= - 2.5
= 0.5
X = 2

h = -2.5 + 0.5*2 h = -1.5
1
1+exp(1.5)
= 0.18 or 18%

Betas are on a logit scale, to interpret calculate odds ratios
Using the exponential function b
1
= 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