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

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

3. Analysis of Data Graphics
Estimate and precision

4. Analysis of Data Graphics

5. Analysis of Data Graphics

6. Analysis of Data Graphics

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

8. 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

9. 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

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

11. 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 = cm
Variation: variance, SD, SE, CV
Median, range, others

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

13. 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*

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

15. 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

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

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

18. 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

19. 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

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

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

22. 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)*

23. 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

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

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

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

27. 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

28. 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 = (all experiments)
Fishers LSD: P = 0.300
Tukey: P = 0.100