1. Data Analysis

2. A Few Necessary Terms
Categorical Variable: Discrete groups, such as Type of Reach (Riffle, Run, Pool)
Continuous Variable: Measurements along a continuum, such as Flow Velocity
What type of variable would “Mottled Sculpin /meter2” be?
What type of variable is “Substrate Type”?
What type of variable is “% of bank that is undercut”?

3. A Few Necessary Terms
Explanatory Variable: Independent variable. On x-axis. The variable you use as a predictor.
Response Variable: Dependent variable. On y-axis. The variable that is hypothesized to depend on/be predicted by the explanatory variable.

4. Statistical Tests: Appropriate Use
For our data, the response variable will always be continuous.
T-test: A categorical explanatory variable with 2 options.
ANOVA: A categorical explanatory variable with >2 options.
Regression: A continuous explanatory variable

5. Statistical Tests
Hypothesis Testing: In statistics, we are always testing a Null Hypothesis (Ho) against an alternate hypothesis (Ha).
Test Statistic:
p-value: The probability of observing our data or more extreme data assuming the null hypothesis is correct
Statistical Significance: We reject the null hypothesis if the p-value is below a set value, usually 0.05.

6. Student’s T-Test
Tests the statistical significance of the difference between means from two independent samples

7. Compares the means of 2 samples of a categorical variable
Mottled Sculpin/m2
Cross Plains Salmo Pond

8. Precautions and Limitations
Meet Assumptions
Observations from data with a normal distribution (histogram)
Samples are independent
Assumed equal variance (boxplot)
No other sample biases
Interpreting the p-value

9. Analysis of Variance (ANOVA)
Tests the statistical significance of the difference between means from two or more independent samples
Grand Mean
Mottled Sculpin/m2
Riffle Pool Run
ANOVA website

10. Precautions and Limitations
Meet Assumptions
Observations from data with a normal distribution
Samples are independent
Assumed equal variance
No other sample biases
Interpreting the p-value
Pairwise T-tests to follow

11. Simple Linear Regression
What is it? Least squares line
When is it appropriate to use?
Assumptions?
What does the p-value mean? The R-value?
How to do it in excel

12. Simple Linear Regression
Tests the statistical significance of a relationship between two continuous variables, Explanatory and Response

13. Precautions and Limitations
Meet Assumptions
Observations from data with a normal distribution
Samples are independent
Assumed equal variance
Relationship is linear
No other sample biases
Interpret the p-value and R-squared value.

14. Residual Plots
Residuals are the distances from observed points to the best-fit line
Residuals always sum to zero
Regression chooses the best-fit line to minimize the sum of square-residuals. It is called the Least Squares Line.

15. Residuals

16. Residual vs. Fitted Value Plots
Observed Values (Points)
Model Values (Line)

17. Residual Plots Can Help Test Assumptions
“Normal” Scatter
Curve (linearity)
Fan Shape: Unequal Variance

18. Have we violated any assumptions?

19. R-Squared and P-value High R-Squared
Low p-value (significant relationship)

20. R-Squared and P-value Low R-Squared
Low p-value (significant relationship)

21. R-Squared and P-value High R-Squared
High p-value (NO significant relationship)

22. R-Squared and P-value Low R-Squared
High p-value (No significant relationship)

23. P-value indicates the strength of the relationship between the two variables
You can think of this as a measure of predictability
R-Squared indicates how much variance is explained by the explanatory variable.
If this is low, other variables likely play a role. If this is high, it DOES NOT INDICATE A SIGNIFICANT RELATIONSHIP!