12 Diversity

# 12 Diversity - MULTIVARIATE ANALYSIS - UNIVARIATE REVIEW a)...

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MULTIVARIATE ANALYSIS - UNIVARIATE REVIEW a) t–test ii Y μ ε =+ H 0 : μ = μ 0 or ij i ij ij i ij YY τε με =++ = + H 0 : μ 1 = μ 2 or H 0 : τ i = 0 (t-test only) b) ANOVA ij i ij Y H 0 : all μ i equal or H 0 : Στ i = 0 , CRD ijk i ij ijk Y τγ + same H 0 ; CRD, with nested error ij i j ij Y τβε + same H 0 ; RBD, no reps ijk i j ij ijk Y τβγ + + same H 0 ; RBD, with reps ijkl i j ij ijk ijkl Y δ + + + same H 0 ; RBD, with reps and nested error etc. for others, bigger design c) REGRESSION 01 ij i ij YX β βε + H 0 : β i = 0 1 2 2 3 3 ij i i i ij X X ββ + + + H 0 : β i = 0 d) ANALYSIS OF COVARIANCE 01 11 1 02 2 12 1 2 ij i i i i ij X X X + + + H 0 : β i = 0 where X 2 is a dummy variable (values of 0 or 1) all above have something in common, there is only a single dependent variable Y many have multiple independent variables, but are not “multivariate" in terms of Y e) MANOVA or HOTELLING'S T 2 12 ... ij ij kij i ij Y ++ += + + The test is a joint test of the Y i variables on the treatments

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1) Reason for using MULTIVARIATE TECHNIQUE in a situation where we want to test for differences between 4 populations A,B,C,D we could do 6 t tests A vrs B B vrs C A vrs C B vrs D A vrs D C vrs D or do 3 tests between ranked pairs each could be done at probability of 0.05 but if we find something significant at 0.05 our experiment wide error rate is not actually 0.05 because we have done 6 tests the actual probability of testing between p means is given by (see Steele and Torrie, Duncan's procedure) ' = 1 (1 ) !!  5" so in testing between 4 means the experiment wide probability of error is ' = 1 (1 0.05) = 1 (0.95) = 1 0.8574 = 0.1426 ! %" \$ an of 0.05 implies 1 chance in 20 of error, if we do 20 tests the probability of error is not 1, ! there's always a chance of no errors, actually ' = 1 (1 0.05) = 1 0.3585 = 0.6415 ! #! in the case of testing between p means we simply use ANOVA which in the test of gives us an 7 3 experiment wide error rate of the selected value (usually 0.05 or 0.01). we could also be conservative eg. on 20 tests using 0.999 ' = 1 (1 0.001) = 1 (0.9802) = 0.0198 ! #! there is little power in this approach
When we wish to do an ANOVA or REGRESSION or ANACOV for a number of different Y values 3 we run into a similar problem each analysis is tested for significance at = 0.05, but if we have 20 species we do 20 ! analyses with the same problem. even in examining simple correlation between 20 species we come up with 190 correlations or S.L.R. so, we turn to either. .. a) indices reduce 20 or 100 species to 1 or 2 or 3 index values, considerably diminishing multivariate problems b) multivariate analysis techniques which "guarantee" an experiment wide error rate of 0.05 (or some other selected value) c) or a multivariate data reduction technique with objectives similar to using indices START with an Analysis of Variance testing for differences between t Treatments Y = 34 3 34 .7 %  except that we have K species, Then use a) Species diversity (or some other) indices X, Y and Z X , Y , Z = 34 34 34 3 34 .7% have only 3 analyses, with a little conservative analysis I wouldn't worry, this is relatively simple

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## This note was uploaded on 12/29/2011 for the course EXST 7025 taught by Professor Geaghan,j during the Spring '08 term at LSU.

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12 Diversity - MULTIVARIATE ANALYSIS - UNIVARIATE REVIEW a)...

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