1. Statistics for Water Science: Hypothesis Testing: Fundamental concepts and a survey of methods Unite 5: Module 17, Lecture 2

2. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s2 Statistics  A branch of mathematics dealing with the collection, analysis, interpretation and presentation of masses of numerical data:  Descriptive Statistics (Lecture 1)  Basic description of a variable  Hypothesis Testing (Lecture 2)  Asks the question – is X different from Y?  Predictions (Lecture 3)  What will happen if…

3. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s3 Objectives  Introduce the basic concepts and assumptions of significance tests  Distributions on parade  Developing hypotheses  What is “true”?  Survey statistical methods for testing for differences in populations of numbers  Sample size issues  Appropriate tests  What we won’t do:  Elaborate on mathematical underpinnings of tests (take a good stats course for this!)

4. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s4  The mean:  A measure of central tendency  The Standard Deviation:  A measure of the ‘spread’ of the data From our last lecture

5. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s5 Tales of the normal distribution  Many kinds of data follow this symmetrical, bell-shaped curve, often called a Normal Distribution.  Normal distributions have statistical properties that allow us to predict the probability of getting a certain observation by chance.

6. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s6  When sampling a variable, you are most likely to obtain values close to the mean  68% within 1 SD  95% within 2 SD 2.0 1.0 0 1.0 2.0 Tales of the normal distribution

7. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s7  Note that a couple values are outside the 95th (2 SD) interval  These are improbable Tales of the normal distribution 2.0 1.0 0 1.0 2.0

8. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s8  The essence of hypothesis testing:  If an observation appears in one of the tails of a distribution, there is a probability that it is not part of that population. Tales of the normal distribution 2.0 1.0 0 1.0 2.0

9. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s9 “Significant Differences”  A difference is considered significant if the probability of getting that difference by random chance is very small.  P value:  The probability of making an error by chance  Historically we use p < 0.05

10. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s10  The magnitude of the effect  A big difference is more likely to be significant than a small one The probability of detecting a significant difference is influenced by:

11. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s11  The spread of the data  If the Standard Deviation is low, it will be easier to detect a significant difference The probability of detecting a significant difference is influenced by:

12. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s12  The number of observations  Large samples more likely to detect a difference than a small sample The probability of detecting a significant difference is influenced by:

13. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s13 Hypothesis testing  Hypothesis:  A statement which can be proven false  Null hypothesis HO:  “There is no difference”  Alternative hypothesis (HA):  “There is a difference…”  In statistical testing, we try to “reject the null hypothesis”  If the null hypothesis is false, it is likely that our alternative hypothesis is true  “False” – there is only a small probability that the results we observed could have occurred by chance

14. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s14 Alpha Level Reject Null Hypothesis P > 0.05Not significantNo P < 0.051 in 20SignificantYes P <0.011 in 100SignificantYes P < 0.0011 in 1000 Highly Significant Yes Common probability levels

15. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s15 Accept HoReject Ho Ho is TrueCorrect Decision Type I Error Alpha Ho is False Type II Error Beta Correct Decision Types of statistical errors (you could be right, you could be wrong)

16. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s16 2.0 1.0 0 1.0 2.0 Type I Error Type II Error Examples of type I and type II errors

17. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s17 Common statistical tests QuestionTest Does a single observation belong to a population of values?Z-test Are two (or more populations) of number different?T-test F-test (ANOVA) Is there a relationship between x and yRegression Is there a trend in the data (special case of aboveRegression

18. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s18  On June 26, 2002, a temperature probe reading at 7 m depth in Medicine Lake was 20.3 0 C. Is this unusually high for June? Note: this is a “one-tailed test”, we just want to know if it’s high We’re not asking if it is unusually low or high (2- tailed) Does a single observation belong to a population of values: The Z-test

19. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s19  The Z-distribution is a Normal Distribution, with special properties:  Mean = 0 Variance = 1  Z = (observed value – mean)/standard error  Standard error = standard deviation * sqrt(n) The Z distribution The z distribution: Standard normal distribution)

20. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s20  Calculate the Z-score for the observed data  Compare the Z score with the significant value for a one tailed test (1.645) Medicine lake example

21. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s21 The Deep Math…  Since 6.89 > the critical Z value of 1.64  Our deep temperature is significantly higher than the June average temperature.  Further exploration shows that a storm the previous day caused the warmer surface waters to mix into the deeper waters. Z = (observed value – mean)/standard error Standard error = standard deviation * sqrt(n) Z = (20.3 – 19.7) 0.08 = 6.89

22. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s22 Are two populations different: The t-test  Also called Student’s t-test. “Student” was a synonym for a statistician that worked for Guinness brewery  Useful for “small” samples (<30)  One of the most basic statistical tests, can be performed in Excel or any common statistical package

23. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s23 Are two populations different: The t-test

24. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s24 Are two populations different: The t-test  One of the most basic statistical tests, can be performed in Excel or any common statistical package  Same principle as Z-test – calculate a t value, and assess the probability of getting that value

25. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s25 In Excel  Formula:  @ttest(Pop1, Pop2, #Tails, TestType)  Tailed tests: 1 or 2  TestType  1 - paired (if there is a logical pairing of XY data)  2 - equal variance  3 - unequal variance  Test returns exact probability value

26. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s26  @ttest(Pop1, Pop2, 1, 3) = 1.5 * 10-149 Example: 1-tailed temperature comparison

27. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s27 ANOVA: Tests of multiple populations  ANOVA – analysis of variance  Compare 2 or more populations  Surface temperatures for 3 lakes  Can handle single or multiple factors  One way ANOVA – comparing lakes  Two-way ANOVA – compare two factors  Temperature x Light effects on algal populations  Repeated measures ANOVA – compare factors over time

28. Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s28 Next Time: Regression - Finding relationships among variables