Analysis of Data Graphics Quantitative data Estimating one quantity of interest Comparing two estimates Studying the relationship between two or more variables

Analysis of Data Graphics Charts, graphs, and tables Presentation of results Analysis Interpretation

Analysis of Data Graphics Estimate and precision

Analysis of Data Graphics

Analysis of Data Graphics

Analysis of Data Graphics

Analysis of Data Quantitative Analysis Population unit vs. sample unit Experimental unit? Response (dependent) variables Explanatory (independent) variables

Analysis of Data Quantitative Analysis Variables Continuous E.g., mass, age Interval E.g., years Categorical (nominal and ordinal) Can have ≥2 values No order vs. order E.g., male/female, study sites, age (ad vs. juv), years Dichotomous or binomial Can only have 2 values E.g., male/female, dead/alive Effects statistical tests used E.g., SLR, ANOVA, Logistic Regression, chi-square

Analysis of Data Quantitative Analysis Types of error in statistics Difference between the estimate (statistic) and quantity being estimated (parameter) Bias Consistent tendency to over or underestimate the parameter Measurement bias When some samples not measured Inaccurate measurements Statistical bias Negligible with standard tests Sampling error Caused by random selection of items in the sample Small sample size = bigger problem Type I & II Error in hypothesis testing

Analysis of Data Quantitative Analysis Sampled population vs. population of interest Research or study population Proper sampling

Analysis of Data Quantitative Analysis Describing a population Mean Best guess? E.g., male height:  = 10.2 cm Confidence Intervals (CI) How good is the guess Interpretation E.g., male height: 95% CI = 8.2-11.5 cm Variation: variance, SD, SE, CV Median, range, others

Analysis of Data Quantitative Analysis Comparing 2 estimates 1 sample vs. 2 sample comparison 1 tailed vs. 2 tailed test

Analysis of Data Quantitative Analysis Comparing 2 estimates t-test Mann-Whitney U Paired t-test Wilcoxon signed-rank Friedman 1-factor ANOVA Kruskal-Wallis test Block ANOVA P ≤ 0.05*

Analysis of Data Quantitative Analysis Parametric vs. nonparametric (distribution-free) Means vs. medians/distributions Power

Analysis of Data Quantitative Analysis Parametric vs. nonparametric Deer mass data Sex Mass (kg) Mass (Ranked) M 95 5.5 90 8.5 100 1 97 2.5 96 4 F 80 12 86 11 92 7 87 10 Mass (±SE or Median) Test M F P Power 2 sample t-test 95.5 ± 2.4 88.7 ± 1.3 0.030 0.623 Mann-Whitney U test 95.5 88.5 0.053 0.592

Analysis of Data Quantitative Analysis Studying relationships between 2 or more variables Multivariate data Categorical Continuous Both

Analysis of Data Quantitative Analysis Correlation analysis – association P, r (-1 – 1), & r2 (0-1) H0: no association/relationship

Analysis of Data Quantitative Analysis Simple Linear Regression (SLR) – cause/effect Outliers P vs. r vs. r2 H0: slope = 0 Inference and interpretation Prediction Description of relationships/dependence Y = β0 + β1X + є or Y = mx + b

Analysis of Data Quantitative Analysis Multiple Regression (MR) Inference and interpretation Prediction Description of relationships/dependence P vs. R vs. R2 Y = β0 + β1X1 + β2X2 … + є

Analysis of Data Quantitative Analysis Regression model fitting Stepwise* Forward or backward Other R2, others AIC

Analysis of Data Quantitative Analysis Multifactor ANOVA ≥2 factor ANOVA Interactions Repeated measures Blocks

Analysis of Data Categorical data analysis (count data) Goodness of fit H0: observed frequencies/counts = expected frequencies/counts Σ((observed-expected)2/expected) Χ2 (Chi-square) G (Likelihood ratio statistic)*

Analysis of Data Categorical data analysis (count data) Contingency table/test for independence H0: no dependence Χ2 (Chi-square) G (Likelihood ratio statistic)* Ground Roost Tree Roost Dead 20 60 Alive 50 10

Analysis of Data Assumptions Parametric tests Nonparametric Normality Homogeneous variances Independence Nonparametric Categorical/distribution-free

Analysis of Data Violations of Assumptions Transformations Change test Data presented untransformed Change test nonparametric

Analysis of Data Multiple comparisons Within and among experiments/studies Pairs (individual tests) vs. experiment error Tukey Bonferroni Fishers LSD* others

Cut-off for Significant Effect Analysis of Data Multiple comparisons – within experiment/study Also referred to as post hoc or pairwise tests Pair (individual tests) vs. experiment error Tukey vs. Fishers LSD E.g., plant height in control (C; unburned or chopped), burned (B), and roller-chopped (R) treatments/plots 1-factor ANOVA (P = 0.050) Cut-off for Significant Effect C vs. B C vs. R B vs. R Experiment Error Fishers LSD 0.050 0.150 Tukey 0.017

Analysis of Data Multiple comparisons – among experiments/studies Single experiment vs. multiple experiments E.g., plant height in control (C; unburned or chopped), burned (B), and roller-chopped (R) treatments/plots and plant richness in control (C; unburned or chopped), burned (B), and roller-chopped (R) treatments/plots 2, 1 factor ANOVAs: Plant height (P = 0.050) and Plant richness (P = 0.050); P = 0.100 (all experiments) Fishers LSD: P = 0.300 Tukey: P = 0.100