ECOL 425

• There are 2 types of studies:

  • Experimental
  • the researcher assigns treatments
  • can introduce artefacts
  • bias in measurements produced by unintended consequences of procedures
  • Observational
  • no influence over treatments
  • used for detecting large scale patterns
• Experiments examine the casual relationship between a predictor (x) and response (y) variable
• Strength is the effect of predictor (x) when isolated from the effects of confounding variables
• Good experiments are designed "a priori" and:

  • reduce bias
  • reduce sampling error
  • difference between sample and population result
• Bias is reduced through:

  • control groups
  • similar conditions as sample, no treatment
  • randomization
  • of individuals receiving treatment
  • cannot occur in observational studies
  • blinding
  • concealing information about treatment assigned
  • Single-blind = subjects unaware
  • Double-blind = researcher + subjects unaware
• Sampling error is reduced by:

  • replication
  • necessary due to unique individuals
  • increased sample sizes decrease error and provide more information
  • to decide sample size:
  • predetermine level of precision OR power
  • pseudoreplication = measurements are not independent, but are recorded as so
  • balance
  • equal sample sizes in treatments
  • blocking
  • reduces variance by dividing individuals into groups and randomizing within a block
  • Ensures each group is representative of population
• Confounding variables introduce bias.

  Can be minimised by:

  • pairing individuals with a control of similar characteristics
  • adjustment - categorising data based on the confounding variable and analysing the relationship between x and y
• ANOVA compares group means by comparing variance between groups.

  • Individual treatment means are fitted and their distance from their treatment mean is analysed
  • Variation within groups is compared with variation among groups
  • If residuals (SSE) < individual treatment mean (SSA) = means are different
• ANOVA is used when explanatory variables (x) are categorical
• SSA measures the among group variation and has a df of k-1
• SSE measures the within group variation has a df of N-k
• The total variation in ANOVA has a df of N-1
• Assumption of ANOVA:

  • Random sampling
  • Equal variance
  • Independence of errors
  • Normal distribution of errors
• Factorial ANOVA tests the effects of 2+ factors and their interaction on a response (y) variable.

  • Reduces Type I error and accounts for variation from crossing variables
• Factorial ANOVA compares variance of each effect to error variance using the mean square
• Factorial ANOVA table:

  A) SSA

  B) SSB

  C) SSAB

  D) SSE

  E) ab - 1

  F) a - 1

  G) b - 1

  H) (a - 1)(b - 1)

  I) ab(n - 1)

  J) N - 1

  K) SSA / df

  L) MSA / MSE

  M) MSB / MSE

  N) MSAB / MSE

  
• An interaction means that the effect of one factor on a response variable (y) is not constant and depends on the other factor
• A contrast is an interpretation of a significant Multi-way ANOVA result.

  • compares groups of means (single df comparisons)
• Contrast significance is judged by an F-test:

  $F =$$\frac{SS_{contrast}}{k(n-1)}$
• • "a priori" means "before"
  • "a posteriori" means "after the fact"
• There can only be k-1 orthogonal contrasts.

  • statistically independent comparisons = compared only once
  • Product of contrast coefficient = 0
• Contrast Coefficient: numerical description of hypothesis tested.

  Rules:

  • grouped levels get same sign
  • contrasting levels get opposite sign
  • Excluded levels get 0
  • All coefficients in contrast must sum to 0
• Fixed effects influence the mean of y
• Random effects influence the variance of y

  • Include numeric and factor levels
• Nested sampling reduces random effects by accounting for variation contributed by each factor.

  Used for:

  • studies conducted at different spatial scales
  • repeated measurements from same individual
• Split-plot analysis reduces fixed effects by splitting a sample into plots of different sizes and applying different treatments.

  • Each plot has own error variance
  • Ordered from largest plot with lowest replication to smallest plot with high replication

  Error term = error(largest/medium/smallest plot)
• Difference between PCA and RDA:

  A) Variable reduction

  B) data visualisation

  C) Regression analysis

  D) relationship exploration

  E) only x variables

  F) x and y variables

  G) Unconstrained

  H) Constrained ordination analysis

  I) Captures overall data variation

  J) Explains variation in y by looking at variation in x

  K) PCs

  L) Significance test

  
• Similarities between PCA and RDA:

  • Use loading systems
  • multi-variate
  • useful for large datasets
• Linear regression is a measure of how steeply the response (y) variable changes with a change in explanatory variable (x)

  • uses least squares regression (line of best fit)
  • both variables are continuous
  • applied mostly in observational studies
• Maximum Likelihood is applied for parameter estimation to increase the probability of observed data appearing
• Regression line is calculated by: $Y =$$a +$$bX$
• The regression slope is calculated by:

  $b =$$\frac{\sum (X_{i} - \overline{X})(Y_i - \overline{Y})}{\sum (X_i - \overline{X})^{2}}$
• The least squares regression line always goes through the means of x and y

  $a =$$\overline{Y} - b\overline{X}$
• The assumptions of linear regression:

  1. at each X, the mean of Y lies on the regression line
  2. at each X, the distribution of Y is normal
  3. at each X, Y variance is the same
  4. at each X, Y is a random sample from all possible Ys
• Variance of residuals (MSresidual) quantifies the spread of the scatter above and below a regression line

  • df = n - 2
  • estimate slope and intercept
• Outliers create non-normal distributions and affect estimates

  • Can affect slope calculations
  • Can cause equal variance assumption violations
• Residuals plots can show normality and variance of the data
• Uncertainty of estimation of the slope is measured with

  $SE_b =$$\sqrt{\frac{MS_{residual}}{\sum (X_i - \overline{X})^2}}$
• Hypothesis testing with regression is used to evaluate whether the slope is equal to null slope, or β0.

  • under null, the df = n-2
  • $t =$$\frac{b - \beta_0}{SE_b}$
• Regression takes the deviation between an observation Y and mean Y and breaks it into a:

  • Residual component
  • $Y_i - \hat{Y}$
  • Regression component
  • $\hat{Y} - \bar{Y}$
  • if H0 true, both MS will be equal
  • if H0 not true, regression MS > residual MS