Overview of Statistical Tests Available BIOL457/657 5 February 2014
Deciding Which Test to Use • Is DV meristic, mensural, or categorical? • Is IV continuous or discrete in its variation? • If IV is discrete, are two groups being compared, or more than two? • Are there multiple IVs?
No-difference H0 or no-correlation H0?Is DV meristic/mensural or categorical?
Student’s t-Test • Parametric test • Compare two independent samples • One version for equal variances • Less powerful version for unequal variances • H0: There is no difference in [DV] between two groups or treatments [IV]
Paired t-test • Parametric • Data paired rather than drawn from independent populations • Assumes equal variances
Mann-Whitney U-test • aka Rank-Sum Test • Nonparametric comparison of two samples of meristic or mensural DV
z-test • aka Binomial Test of Proportions • Two samples • DV occurs in two categories for each sample • H0: There is no difference in binomial outcomes [DV] between two samples [IV]
Chi-square (χ2) Tests • χ2 Goodness-of-Fit Test • Sample distribution vs. hypothesized/theoretical distribution • χ2 Test for Independence • Sample distribution vs. sample distribution • Mantel-Haenszelχ2 Test for Independence • Compares distributions of samples with two or more IVs that may influence DV distribution
One-way Analysis of Variance (ANOVA) • Parametric • H0: There is no difference in [DV] among three or more groups or treatments [IV] • Rejection of H0 requires post hoc testing to identify which samples differ significantly from which other samples • Fisher’s Least Significant Difference (LSD) test • Tukey’s test • Student-Newman-Keuls test • Bonferroni t-tests • Duncan’s Multiple Range test • Sheffé comparisons
HANDOUTGodfrey and Bryant, 2000European Robin Energy Expenditure Note: For both tests, post hoc Scheffé comparisons show CHILLED was significantly greater than other two treatments, which did not differ from one another.
Kruskal-Wallis Test • Nonparametric equivalent of ANOVA • Rejection of H0 requires nonparametric analog of post hoc testing • Dunn’s Multiple Comparisons test
Two-Way (or More) ANOVA • Two IVs analyzed simultaneously • Multiple H0s: • No difference in [DV] between/among groups as established by IV1 • No difference in [DV] between/among groups as established by IV2 • No difference in [DV] attributable to the interaction of IV1 and IV2 • More complex models for three or more IVs
Correlation Analysis • H0: There is no correlation of DV with IV • Positive correlation: Direct relationship • Negative correlation: Indirect relationship • No correlation: Variation in DV not related to variation in IV
Simple Linear Regression • If they are correlated, fit DV and IV data to a best-fitting line y = ax + b DV = a(IV) + b a = slope b = y-intercept
Multiple Linear Regression • >1 continuous IV • Multidimensional best-fitting line: DV = a(IV1) + b(IV2) + c(IV3) … + z (ac are slopes in separate dimensions; z is intercept) • With each added IV, partial correlation coefficient assesses whether it significantly improves fit to DV • H0: There is no correlation of DV with IVx, after correction for correlation of DV with other IVs in model
Analysis of Covariance • Two IVs: IVc continuous, IVd discrete • Typically, IVc regarded as a confounding variable, known/suspected to be correlated with DV • H0: No difference in DV between/among treatments or groups (IVd), after correction for correlation of DV with IVc.
The General Linear Model (GLM) • Mix of multiple linear regression (continuous IVs) and ANCOVA (allows for discrete IVs) • Number of IVs limited only by sample size • H0 for each IV states that its variation is not significantly related to variation in DV (given relationship of other IVs to DV) • Interaction of IVs can also be examined in additional H0s
Logistic Regression • IV is continuous, but DV is discrete and categorical • H0: Probability of categorization of DV is not correlated with IV
Multivariate Analyses • Variety of techniques that take large data sets with numerous variables and… • …allow relatively simple graphical representations • …identify interrelationships of variables
Phylogenetic Comparative Methods • Used for analyzing data from related species • Necessary because phylogenetic relationships cause violation of the assumption of independence of data points • Primarily used for correlation analysis, but can be adapted for use in other types of analysis
Monte Carlo Randomizations • Uses 100s to 1000s of randomizations of the sample data to create a null distribution for the H0 • Parametric assumption thus not necessary • Used to create highly specific H0s for particular scenarios one may wish to analyze