Data-Based Models, How to Analyse Data and Which Test to Use

Statistical packages like R work by fitting models to data

• Require you to use an appropriate model for samples and variables under investigation before they estimate parameter values that best fit data

Standard convention for presenting statistical models - response variable(s) = explanatory variable(s)

• = sign is statment of hypothesised relationship between variables

Chosen statistic quantifies the relationship of response variable to explanatory variables

3 main types of data:

• One variable, one sample - chi-squared, G-test, Kolmogorov-Smirnov
• Two variables, one sample
  • Categorical responses (contingency tables) - chi-squared, G-test for independence
  • Continous response and predictor - linear regression/correlation
• One or more predictors, two or more samples - ANOVA or GLM

Look for goodness-of-fit frequencies (observed compared to expected)

• Chi-squared or G-test of association
• For continuous data, use Kolmogorov-Smirnov

Assumptions:

• Data are nominal (not continous)
• Frequencies are independent from each other
• No cell has expected values <5

For data of this kind, look for a dependent relationship between variables

Contingency tables used to look for interaction between variables

• Ch-squared or G-test
• For cells with expected values <5, use Fisher's exact test

Model formula: colour:behaviour ~ response

Assumptions:

• Categorical data
• Frequencies independent 
• No cell with expected values <5 (if not Fisher's exact test)
• Correction for continuity

Plot response variable on y-axis and explanatory variable on x-axis

Linear regression should be used

If no clear functional relationship, use correlation to calculate r

Mdel formula: Response ~ Explanatory

Assumptions:

• Random sampling
• Independent errors
• Homogeneity of variances
• Normal distribution of errors
• Linearity

If variance increases with response there is no linearity and data must be transformed

Look for a difference between sample means

With one categorical predictor:

• t-test for two groups
• ANOVA for more than two groups
• Repeated measures ANOVA for repeated measures on subjects
• Transform data that violate asumptions
• Kruskal Wallis for non-parametric ANOVA
• Mann-Whitney for non-parametric t-test

Assumptions:

• Random sampling
• Independent errors
• Homogeneity of variances
• Normal distribution of errors

Model: Response ~ Explanatory

R offers alternative commands for ANOVA

• aov suits mode straightforward analyses with normally distributed residuals
• glm = General Linear Model - accomodate ANOVA on data with inherently non-normal distributions e.g. proportions (binomial) or frequencies of rare events (Poisson)

Look for differences between regression slops

ANOVA should be used with regression analysis on different slopes

Model formula: Response ~ Explanatory 1 + Explanatory 2 + Explanatory 1:Explanatory 2

Assumptions:

• Random sampling
• Independent errors
• Homogeneity of variances
• Normal distribution of errors
• Linearity

If regression plot shows two lines cross over = interaction between variables

Look for two-way differences between means

ANOVA or GLM (in non-normal error structures) should be used

Model formula: Response ~ Explanatory 1 + Explanatory 2 + Explanatory 1:Explanatory 2

Assumptions:

• Random sampling
• Independent errors
• Homogeneity of variances
• Normal distribution of errors

If data is unbalanced (samples have different numbers in them) use a GLM

Method depends entirely on test statistic

d.f. = no. pieces of information had - no. required to calculate variation

Chi-squared test:

• Theoretical distributions n - 2 (usually)
  • n = no. cateogries for explanatory variable
  • 2 OR no. bits information needed to calculate expected distribution
• Contingency table = (c -1) x (r -1)
  • c = no. columns
  • r = no. rows

ANOVA:

• Test = a - 1
  • a = no. sample means
• Error = n - a
  • n = no. observations
  • a = no. sample means

Linear regression:

• Test = 1
  • (Slope and intercept) 2 - 1 grand mean
• Error = n - 2
  • n = sample size
  • 2 = slope and intercept

• Define test hypothesis
• Identify model components
  • Response
  • Explanatory factor and levels
  • Sampling unit
  • Population samples
• Define model
• Degrees of freedom
• Collect data
• Input to R
• Run model and check assumptions

Always plot data first to check it meets model assumptions

Significance tells nothing about size or precision of effect

For all analyses:

• Significance (p-value) - identifies evidence of pattern
• Effect size (difference between sample means/regression slope) - gives magnitude
• Error bars/coefficient of determination (r) - gives precision

Shape of pattern depends on parameters

Theoretical mathematical models - used to work out how to transform data

• Use biology of species to help understanding

Once collected, data only suits one model - R can run any model on data

• Each model produces a unique set of results pertinent to particular design
• Only one model will represent experiment design - must know what it is before collecting data

Seek difference between averages of 2+ samples

• Parametric ANOVA
• Parametric t-test for two samples
• Non-parametric Kruskal-Wallis
• Non-parametric Mann-Whitney U for two samples

Identify trends between two continuous variables in 1+ samples

• Parametric regression
• Polynomial regressin on non-linear data
• Parametric Pearson product-moment correlation on data that you're not looking for regression with
• Non-parametric Spearman's rank for correlation

Identify a relation between frequencies in categorical classes of one sample

• Chi-square/G-test on frequencies
• Any expected frequencies <3, pool classes or Fisher exact test