choosing and using statistics to test ecological hypotheses botany 332 lab tutorial department of...
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Choosing and using Choosing and using statistics to test statistics to test
ecological hypothesesecological hypothesesBotany 332 Lab TutorialBotany 332 Lab Tutorial
Department of Biological SciencesDepartment of Biological SciencesUniversity of AlbertaUniversity of Alberta
November 2004November 2004
EXPERIMENTCritical test of null hypothesis
NULL HYPOTHESISLogical opposite to hypothesis
OBSERVATIONSPatterns in space or time
HYPOTHESISPredictions based on model
MODELSExplanations or theories
INTERPRETATION
Retain Ho (Null Hypothesis)
Refute hypothesis and model
Reject Ho (Null Hypothesis)
Support hypothesis and
model
Underwood (1997)
Ecological experimentsEcological experiments
1.1. OBSERVE things.OBSERVE things.2.2. Come up with MODELS (explanations or Come up with MODELS (explanations or
theories) to explain your observations.theories) to explain your observations.3.3. Based on your model, come up with a testable Based on your model, come up with a testable
HYPOTHESIS (and a NULL hypothesis).HYPOTHESIS (and a NULL hypothesis).4.4. Design an EXPERIMENT to test your null Design an EXPERIMENT to test your null
hypothesis statistically.hypothesis statistically.5.5. Conduct the experiment and collect DATA.Conduct the experiment and collect DATA.6.6. Use STATISTICS with your data to TEST the null Use STATISTICS with your data to TEST the null
hypothesis.hypothesis.7.7. INTERPRET your results. Did you accept or INTERPRET your results. Did you accept or
reject the null hypothesis?reject the null hypothesis?8.8. Repeat!Repeat!
Testing (null) hypotheses Testing (null) hypotheses statisticallystatistically
Recall we can’t prove our hypothesis, Recall we can’t prove our hypothesis, so we try to so we try to disdisprove a null prove a null hypothesis instead!hypothesis instead!
Null hypothesis = opposite of our Null hypothesis = opposite of our actual hypothesisactual hypothesis– HH00 = Null Hypothesis = Null Hypothesis
– HHAA = Alternative hypothesis = Alternative hypothesis
Testing (null) hypotheses Testing (null) hypotheses statisticallystatistically
We formally test hypotheses using statistics
Which statistical test to use? Depends on your experimental design, data and your hypotheses
It’s important to understand the basics of statistical hypothesis testing
Testing (null) hypotheses Testing (null) hypotheses statisticallystatistically
Based on assumptions about the data, Based on assumptions about the data, statistics tell us the probability that the null statistics tell us the probability that the null hypothesis is true (P-value).hypothesis is true (P-value).
If P is small enough, we can reject the null If P is small enough, we can reject the null hypothesis (result is “statistically hypothesis (result is “statistically significant”).significant”).
What’s “small enough”?What’s “small enough”?– P < 0.05P < 0.05
Reject null hypothesis (accept our hypothesis)Reject null hypothesis (accept our hypothesis)
– P > 0.05P > 0.05 Accept null hypothesis (reject our hypothesis)Accept null hypothesis (reject our hypothesis)
Testing (null) hypotheses Testing (null) hypotheses statisticallystatistically
Many statistical methods also tell us the Many statistical methods also tell us the effect size or proportion of variation in the effect size or proportion of variation in the independent variable explained by the independent variable explained by the dependent variable.dependent variable.
e.g. Regression and correlatione.g. Regression and correlation– P-valuesP-values
HH00 = No relationship between variables = No relationship between variables
HHAA = Relationship between variables = Relationship between variables
– RR22 (variation explained) (variation explained)– Can have significant P-values but very small RCan have significant P-values but very small R22
Choosing and using Choosing and using statisticsstatistics
Determine what kinds of data you Determine what kinds of data you havehave
Describe your dataDescribe your data Choose an appropriate statistical testChoose an appropriate statistical test Perform the testPerform the test Report and interpret the resultsReport and interpret the results
What kinds of data do you What kinds of data do you have?have?
CategoricalCategorical– Fertilizer addition, species identityFertilizer addition, species identity
Continuous and discreteContinuous and discrete– Biomass, height, number of bitesBiomass, height, number of bites
Independent and Dependent Independent and Dependent variablesvariables
Describe your dataDescribe your data
Measures of central tendencyMeasures of central tendency– Mean, medianMean, median
Measures of dispersionMeasures of dispersion– Variance, standard deviation, standard Variance, standard deviation, standard
error, range, quartileserror, range, quartiles
Mean # of seeds/pod
13.0
12.0
11.0
10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
In
Fre
quen
cy
8
6
4
2
0
Descriptive Statistics – Visual Descriptive Statistics – Visual AidsAids
BoxplotsBoxplots- median, upper and lower - median, upper and lower
quartiles, whiskers (fences), quartiles, whiskers (fences), outliersoutliers
HistogramsHistograms- separate, stackbar, or paired- separate, stackbar, or paired
Error Bar PlotsError Bar Plots
3744N =
Treatment
InOut
Me
an
# o
f se
ed
s/p
od
30
20
10
0
-10
54
Mean # of seeds/pod
26.0
24.0
22.0
20.0
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
Out
Fre
quen
cy
7
6
5
4
3
2
1
0
4437N =
Treatment
OUTIN
Mea
n #
of s
eeds
/pod
16
14
12
10
8
6
42
Describe your dataDescribe your data
Normal vs. non-normal distributionsNormal vs. non-normal distributions– histograms, Q-Q plots, K-S test histograms, Q-Q plots, K-S test
(significant means non-normal)(significant means non-normal)
Data transformationData transformation If your data are non-normalIf your data are non-normal
– Use non-parametric statisticsUse non-parametric statistics– Transform your dataTransform your data
square-root transformsquare-root transform log transformlog transform
Choose your statistical testChoose your statistical test
Choose statistical tests based on Choose statistical tests based on your hypothesis, experimental design your hypothesis, experimental design and the data you have collectedand the data you have collected
Parametric tests assume data are Parametric tests assume data are normal, non-parametric tests do notnormal, non-parametric tests do not
Many textbooks have recipes or Many textbooks have recipes or flowcharts for choosing statisticsflowcharts for choosing statistics
Check with your TA’sCheck with your TA’s
Common statistical testsCommon statistical tests
Chi-squared testChi-squared test t-test (Mann-Whitney U test)t-test (Mann-Whitney U test) One-way ANOVA (Kruskal-Wallis test)One-way ANOVA (Kruskal-Wallis test) Two-way ANOVATwo-way ANOVA ANCOVAANCOVA
– ANOVA with covariateANOVA with covariate Correlation and regressionCorrelation and regression
Chi-squared testChi-squared test
For analysis of tables of counts or For analysis of tables of counts or frequenciesfrequencies
Good with categorical variablesGood with categorical variables Non-parametricNon-parametric
# plants# plants GerminatedGerminated Not Not GerminatedGerminated
OutcrossedOutcrossed 1414 1010
InbredInbred 66 1010
t-testt-test
For analysis of categorical For analysis of categorical independent variable (2 categories) independent variable (2 categories) and a continuous dependent variableand a continuous dependent variable
Samples may be paired Samples may be paired (measurements on same individual) or (measurements on same individual) or independent (measurements on two independent (measurements on two sets of individuals)sets of individuals)
Assumes data are parametricAssumes data are parametric (non-parametric – Mann-Whitney U)(non-parametric – Mann-Whitney U)
ANOVAANOVA
Analysis of Variance examines Analysis of Variance examines variation within and between groupsvariation within and between groups
For analysis of categorical For analysis of categorical independent variables (2 or more independent variables (2 or more categories) and a continuous categories) and a continuous dependent variabledependent variable
Assumes data are parametricAssumes data are parametric (non-parametric – Kruskal-Wallis)(non-parametric – Kruskal-Wallis)
ANOVAANOVA
One-way ANOVAOne-way ANOVA– Single independent variableSingle independent variable– Main effectMain effect
Two-way ANOVATwo-way ANOVA– Two independent variablesTwo independent variables– Main effects and interaction termsMain effects and interaction terms
Significant result means at least one group Significant result means at least one group differed from anotherdiffered from another
Use Use post-hoc testspost-hoc tests to test for differences to test for differences among individual treatmentsamong individual treatments
ANCOVAANCOVA
Analysis of CovarianceAnalysis of Covariance For analysis of categorical For analysis of categorical
independent variables (2 or more independent variables (2 or more categories), a continuous dependent categories), a continuous dependent variable, and a covariatevariable, and a covariate
Effects of covariate removed before Effects of covariate removed before testing for effect of independent testing for effect of independent variable(s)variable(s)
Correlation and regressionCorrelation and regression
Tests for relationships between Tests for relationships between two (or more) continuous variablestwo (or more) continuous variables
Important to consider both Important to consider both significance (P-value) and effect significance (P-value) and effect size (Rsize (R22))
Report statistical resultsReport statistical results
What’s important?What’s important?– Test used and assumptions testedTest used and assumptions tested– Test statistic (t, F, RTest statistic (t, F, R22, , χχ22, etc.), etc.)– Significance (P-value)Significance (P-value)– Sample size / degrees of freedomSample size / degrees of freedom
How to report results?How to report results?– TextText– FiguresFigures– TablesTables
3946N =
Treatment
OUTIN
Nu
mF
low
ers
140
120
100
80
60
40
20
0
-20
13
ANOVA, F = 1.8, df = 1,83
P = 0.17
Nu
mb
er
of
flow
ers
per
pla
nt
Interpret your resultsInterpret your results
Remember to relate results/tests to Remember to relate results/tests to your original hypothesesyour original hypotheses
Correlation ≠ causationCorrelation ≠ causation (P > 0.05) ≠ bad(P > 0.05) ≠ bad Recognize trends even when not Recognize trends even when not
statistically significantstatistically significant Talk to your TAs if you have any Talk to your TAs if you have any
questionsquestions
SPSS walkthroughSPSS walkthrough
Data entry and transformationData entry and transformation Descriptive statisticsDescriptive statistics Creating figuresCreating figures AnalysesAnalyses
– Chi-square (inbreeding data)Chi-square (inbreeding data)– t-test / ANOVA (inbreeding data)t-test / ANOVA (inbreeding data)– ANCOVA (tomato data)ANCOVA (tomato data)– Correlation and regression (inbreeding Correlation and regression (inbreeding
data)data)