Basic Statistical Concepts  M. Burgman & J. Carey 2002

Statistical Population The entire underlying set of individuals from which samples are drawn. e.g. 0.25m 2 quadrats are used to count barnacles on a sea shore. The population is defined implicitly by the sampling frame.

Strategies Define survey objectives Define population parameters to estimate Implement sampling strategy i) measure every individual (cost, time, practicality especially if destructive) ii) measure a representative portion of the population (a sample)

Statistical Sample An aggregate of objects from which measurements are taken. A representative subset of a population.

Simple Random Sampling Every unit and combination of units in the population has an equal chance of selection. a) with replacement b) without replacement c) finite and infinite populations

Sampling Objectives To obtain an unbiased estimate of a population mean To assess the precision of the estimate (i.e. calculate the standard error of the mean) To obtain as precise an estimate of the parameters as possible for time and money spent

 (x i -  ) 2 n  (x i - x) 2 n - 1  (x i - x) 2 n - 1 Statistics of Dispersion Population variance  2 = Sample variance s 2 = Sample standard deviation s = 

s 2 n s x  (x i - x ) (y i - y ) n - 1 Statistics of Dispersion Standard error of the mean s x = Coefficient of variation CV = Covariance s xy = 

Expectations and Variances E(X+b)= E(X) + b E(aX)=aE(X) E(X+Y)=E(X) + E(Y) V(X+b)=V(X) V(aX)=a 2 V(X) V(X+Y)=V(X) + V(Y) + 2Cov(X,Y)

Confidence Limits For the mean= x  t [ , n-1] This formula sets confidence limits to means of samples from a normally distributed population. s  n

Confidence Limits Confidence limits of the mean define a region that we expect will enclose the true mean. The likelihood that this is true is determined by . If we set  at 5% (hence specifying 95% confidence intervals), then the region enclosed by the confidence intervals will capture the true mean 95 times out of 100.

Confidence Limits The same formula may be used to set confidence limits to any statistic as long as it follows the normal distribution, e.g. the median, the average (absolute) deviation, standard deviation (s), coefficient of variation, or skewness.

How many samples? where : CV is coefficient of variation (expressed as a %) of samples in a pilot survey t is Student's t value for a specified degree of certainty and the number of samples used to estimate the parameters E is specified error limits (expressed as a % of the mean) n = t 2 CV 2 E 2

Measurement Error Measured variation may be decomposed into natural variation + measurement error Measurement error may be reduced by improving sampling protocols and instrumentation Reducing measurement error increases confidence in estimates without increasing the number of samples. Precision (variation) v. accuracy (bias)

Components of Measurement Error Systematic errors Random errors Causes Measurement assumptions (shape, size, allometry) Instrument error Operator error

Kinds of Uncertainty 1. Epistemic Uncertainty inherent environmental variation variation in population responses due to demographic structure imperfect knowledge model mis-specification measurement error (assessment error) ignorance

Kinds of Uncertainty 2. Semantic Uncertainty Ambiguity - interpretation of a phrase in two or more distinct ways. “Juvenile Court to Try Shooting Defendant” “Local High School Dropouts Cut in Half” Vagueness - leads to borderline cases. e.g. tall; endangered; adult

Kinds of Uncertainty More examples of vagueness: Tree crown tree foliage bounded by the first healthy branch forming part of the main crown and extending as far or further than any branch above it. forked trees?dead branches?

Kinds of Uncertainty More examples of vagueness: Epilimnion the upper layer of water in a lake, bounded by a thermocline Soil horizon a relatively uniform soil layer, differentiated by contrasts in mineral or organic properties.

Sampling Design Criteria Operational simplicity Unambiguous interpretation

Null-Hypothesis Tests An example of hypothesis testing in which management alternatives are judged on the basis of the outcome of the test. HypothesisSymbolDescription Null H 0 The strategy has no hypothesis effect. Alternative H 1 The strategy is hypothesiseffective

Statistical Outcomes in Null Hypothesis Testing Test Result Significant Not significant (H 0 rejected) (H 0 not rejected) Difference correct Type II error (H 0 false) (  ) No difference Type I error correct (H 0 true) (  ) Reality

The Character of Error Types Type I errors Alarmism/Over-reaction Incorrectly accepting a (false) alternative hypothesis Concluding (incorrectly) that there is an impact Type II errors False confidence/Cornucopia Incorrectly "accepting" a (false) null hypothesis Concluding (incorrectly) that there is no impact

t-tests A t-test of the hypothesis that two sample means come from a population with equal  i.e. H 0 :  1 =  2 t = Y 1 - Y 2 1n1n (s s 2 2 ) 

Distributions of Test Statistics distribution of mean of actual population distribution of the null hypothesis, assumed to be true until rejected P(statistic)  critical value

Assumptions The assumption of independence: correlation and autocorrelation 1. if error in one object is related to error in others, there will be bias eg. measure one and compare others. 2. the effective sample size may be less than the number of samples if measurements are correlated in space or time.

The effects of the non-independence of data on errors of interpretation of statistical tests Non-independence Among Within treatments treatments Positive Increased Increased Type II Type I Negative Increased Increased Type I Type II Correlation

Randomization Tests Jaw lengths of Golden Jackals: Males: 120, 107, 110, 116, 114, 111, 113, 117, 114, 112 Females: 110, 111, 107, 108, 110, 105, 107, 106, 111, 111 Is there a difference in jaw length between males and females?

1.Calculate means for males and for females. 2.Calculate the difference between the means D 0 = x m - x f = Randomly allocate 10 sample lengths to each of 2 groups 4.Calculate D i, the difference between means for these 2 groups 5.Repeat Steps 3 & 4 many times Randomization Tests

If D 0 is unusually large, the observed data are unlikely to have arisen if there was no difference between males and females. Randomization Tests Difference in jaw length (mm) Frequency D 0 =

Randomization Tests From 5000 runs, only 9 D i s were greater than or equal to /5000 = (t-test: p Ho = )

Confidence Limits by Randomization For 95% confidence limits, the upper and lower limits, U and L, are such that they enclose 95% of the randomization distribution. For 99% confidence, L and U must give values at the 0.5% and 99.5% points on the distribution.

Can do randomization tests in lieu of: paired comparisons ANOVA multiple regression Randomization Tests