statistics for water science: hypothesis testing: fundamental concepts and a survey of methods unite...
TRANSCRIPT
Statistics for Water Science: Hypothesis Testing: Fundamental concepts and a survey of methods
Unite 5: Module 17, Lecture 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…
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!)
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
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.
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
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 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.0 1.0 0 1.0 2.0
Tales of the normal distribution
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 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.0 1.0 0 1.0 2.0
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 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.0 1.0 0 1.0 2.0
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
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:
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:
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:
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
Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s14
AlphaLevel
Reject Null Hypothesis
P > 0.05 Not significant No
P < 0.05 1 in 20 Significant Yes
P <0.01 1 in 100 Significant Yes
P < 0.001 1 in 1000Highly
SignificantYes
Common probability levels
Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s15
Accept Ho Reject Ho
Ho is True Correct DecisionType I Error
Alpha
Ho is FalseType II Error
BetaCorrect Decision
Types of statistical errors (you could be right, you could be wrong)
Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s16
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.0 1.0 0 1.0 2.0
Type I Error
Type II Error
Examples of type I and type II errors
Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s17
Common statistical tests
Question Test
Does a single observation belong to a population of values? Z-test
Are two (or more populations) of number different? T-testF-test (ANOVA)
Is there a relationship between x and y Regression
Is there a trend in the data (special case of above Regression
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 20.30 C. Is this unusually high for June?
Medicine Lake June 2002 Temp - 7 m
0
2
4
6
8
10
12
18.00
18.25
18.50
18.75
19.00
19.25
19.50
19.75
20.00
20.25
20.50
20.75
21.00
Temperature
# o
bse
rvat
ion
s
June Temp
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
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)
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 June 2002 Temp - 7 m
0
2
4
6
8
10
12
18.00
18.25
18.50
18.75
19.00
19.25
19.50
19.75
20.00
20.25
20.50
20.75
21.00
Temperature
# o
bse
rvat
ion
s
June Temp
Medicine lake example
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
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
Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s23
Are two populations different: The t-test
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
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 variance3 - unequal variance
Test returns exact probability value
Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s26
@ttest(Pop1, Pop2, 1, 3) = 1.5 * 10-149
Example: 1-tailed temperature comparison
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
Developed by: Host Updated: Jan. 21, 2004 U5-m17b-s28
Next Time: Regression - Finding relationships among variables
Halsted Surface - August 1999
0
5
10
15
20
7.5 8 8.5 9 9.5
pH
Dis
solv
ed
Ox
yg
en
(p
pm
)
DO