Final Review Session

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Final Review Session
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Exam details

• Short answer, similar to book problems
• Formulae and tables will be given
• You CAN use a calculator
• Date and Time Dec. 7, 2006, 12-130 pm
• Location Osborne Centre, Unit 1 (A)

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Things to Review

• Concepts
• Basic formulae
• Statistical tests

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Things to Review

• Concepts
• Basic formulae
• Statistical tests

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Populations Samples Random sample
First Half
Null hypothesis Alternative hypothesis P-value
Parameters Estimates
Mean Median Mode
Type I error Type II error
Sampling distribution Standard error
Variance Standard deviation
Central limit theorem
Categorical Nominal, ordinal Numerical Discrete,
continuous
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Second Half
Normal distribution Quantile plot Shapiro-Wilk
test Data transformations
Simulation Randomization Bootstrap Likelihood
Nonparametric tests
Independent contrasts
Observations vs. experiments Confounding
variables Control group Replication and
pseudoreplication Blocking Factorial design Power
analysis
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Example Conceptual Questions

• (youve just done a two-sample t-test comparing
  body size of lizards on islands and the mainland)
• What is the probability of committing a type I
  error with this test?
• State an example of a confounding variable that
  may have affected this result
• State one alternative statistical technique that
  you could have used to test the null hypothesis,
  and describe briefly how you would have carried
  it out.

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Randomization test
Null hypothesis Randomized data
Sample
Calculate the same test statistic on the
randomized data
Null distribution
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Things to Review

• Concepts
• Basic formulae
• Statistical tests

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Things to Review

• Concepts
• Basic formulae
• Statistical tests

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Sample
Null hypothesis
Test statistic
Null distribution
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Quick reference summary Binomial test

• What is it for? Compares the proportion of
  successes in a sample to a hypothesized value, po
• What does it assume? Individual trials are
  randomly sampled and independent
• Test statistic X, the number of successes
• Distribution under Ho binomial with parameters n
  and po.
• Formula

P 2 Prx?X
P(x) probability of a total of x successes p
probability of success in each trial n total
number of trials
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Binomial test
Null hypothesis Prsuccesspo
Sample
Test statistic x number of successes
Null distribution Binomial n, po
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Binomial test
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Quick reference summary ?2 Goodness-of-Fit test

• What is it for? Compares observed frequencies in
  categories of a single variable to the expected
  frequencies under a random model
• What does it assume? Random samples no expected
  values lt 1 no more than 20 of expected values lt
  5
• Test statistic ?2
• Distribution under Ho ?2 with
• df categories - parameters - 1
• Formula

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?2 goodness of fit test
Null hypothesis Data fit a particular Discrete
distribution
Sample
Calculate expected values
Test statistic

• Null distribution
• 2 With
• N-1-param. d.f.

compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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?2 Goodness-of-Fit test
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Possible distributions
Prx n frequency of occurrence
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Given a number of categories Probability
proportional to number of opportunities Days of
the week, months of the year
Proportional
Number of successes in n trials Have to know n, p
under the null hypothesis Punnett square, many
p0.5 examples
Binomial
Number of events in interval of space or time n
not fixed, not given p Car wrecks, flowers in a
field
Poisson
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Quick reference summary ?2 Contingency Test

• What is it for? Tests the null hypothesis of no
  association between two categorical variables
• What does it assume? Random samples no expected
  values lt 1 no more than 20 of expected values lt
  5
• Test statistic ?2
• Distribution under Ho ?2 with
• df(r-1)(c-1) where r rows, c columns
• Formulae

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?2 Contingency Test
Null hypothesis No association between variables
Sample
Calculate expected values
Test statistic

• Null distribution
• 2 With
• (r-1)(c-1) d.f.

compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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?2 Contingency test
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Quick reference summary One sample t-test

• What is it for? Compares the mean of a numerical
  variable to a hypothesized value, µo
• What does it assume? Individuals are randomly
  sampled from a population that is normally
  distributed.
• Test statistic t
• Distribution under Ho t-distribution with n-1
  degrees of freedom.
• Formula

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One-sample t-test
Null hypothesis The population mean is equal to
?o
Sample
Null distribution t with n-1 df
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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One-sample t-test

• Ho The population mean is equal to ?o
• Ha The population mean is not equal to ?o

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Paired vs. 2 sample comparisons
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Quick reference summary Paired t-test

• What is it for? To test whether the mean
  difference in a population equals a null
  hypothesized value, µdo
• What does it assume? Pairs are randomly sampled
  from a population. The differences are normally
  distributed
• Test statistic t
• Distribution under Ho t-distribution with n-1
  degrees of freedom, where n is the number of
  pairs
• Formula

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Paired t-test
Null hypothesis The mean difference is equal to ?o
Sample
Null distribution t with n-1 df n is the number
of pairs
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Paired t-test

• Ho The mean difference is equal to 0
• Ha The mean difference is not equal 0

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Quick reference summary Two-sample t-test

• What is it for? Tests whether two groups have the
  same mean
• What does it assume? Both samples are random
  samples. The numerical variable is normally
  distributed within both populations. The
  variance of the distribution is the same in the
  two populations
• Test statistic t
• Distribution under Ho t-distribution with
  n1n2-2 degrees of freedom.
• Formulae

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Two-sample t-test
Null hypothesis The two populations have the
same mean ?1??2
Sample
Null distribution t with n1n2-2 df
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Two-sample t-test

• Ho The means of the two populations are equal
• Ha The means of the two populations are not equal

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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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F-test for Comparing the variance of two groups
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F-test
Null hypothesis The two populations have the
same variance ?21? ?22
Sample
Null distribution F with n1-1, n2-1 df
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Welchs t-test
Null hypothesis The two populations have the
same mean ?1??2
Sample
Null distribution t with df from formula
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Parametric
Nonparametric
One-sample and Paired t-test
Sign test
Mann-Whitney U-test
Two-sample t-test
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Quick Reference Summary Sign Test

• What is it for? A non-parametric test to compare
  the medians of a group to some constant
• What does it assume? Random samples
• Formula Identical to a binomial test with po
  0.5. Uses the number of subjects with values
  greater than and less than a hypothesized median
  as the test statistic.

P 2 Prx?X
P(x) probability of a total of x successes p
probability of success in each trial n total
number of trials
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Sign test
Null hypothesis Median mo
Sample
Test statistic x number of values greater than
mo
Null distribution Binomial n, 0.5
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Sign Test

• Ho The median is equal to some value mo
• Ha The median is not equal to mo

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Quick Reference Summary Mann-Whitney U Test

• What is it for? A non-parametric test to compare
  the central tendencies of two groups
• What does it assume? Random samples
• Test statistic U
• Distribution under Ho U distribution, with
  sample sizes n1 and n2
• Formulae

n1 sample size of group 1 n2 sample size of
group 2 R1 sum of ranks of group 1
Use the larger of U1 or U2 for a two-tailed test
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Mann-Whitney U test
Null hypothesis The two groups Have the same
median
Sample
Test statistic U1 or U2 (use the largest)
Null distribution U with n1, n2
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Quick Reference Guide - Correlation Coefficient

• What is it for? Measuring the strength of a
  linear association between two numerical
  variables
• What does it assume? Bivariate normality and
  random sampling
• Parameter ?
• Estimate r
• Formulae

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Quick Reference Guide - t-test for zero linear
correlation

• What is it for? To test the null hypothesis that
  the population parameter, ?, is zero
• What does it assume? Bivariate normality and
  random sampling
• Test statistic t
• Null distribution t with n-2 degrees of freedom
• Formulae

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T-test for correlation
Null hypothesis ?0
Sample
Test statistic
Null distribution t with n-2 d.f.
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Quick Reference Guide - Spearmans Rank
Correlation

• What is it for? To test zero correlation between
  the ranks of two variables
• What does it assume? Linear relationship between
  ranks and random sampling
• Test statistic rs
• Null distribution See table if ngt100, use
  t-distribution
• Formulae Same as linear correlation but based on
  ranks

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Spearmans rank correlation
Null hypothesis ?0
Sample
Test statistic rs
Null distribution Spearmans rank Table H
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Assumptions of Regression

• At each value of X, there is a population of Y
  values whose mean lies on the true regression
  line
• At each value of X, the distribution of Y values
  is normal
• The variance of Y values is the same at all
  values of X
• At each value of X the Y measurements represent a
  random sample from the population of Y values

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OK
Non-normal
Unequal variance
Non-linear
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Quick Reference Summary Confidence Interval for
Regression Slope

• What is it for? Estimating the slope of the
  linear equation Y ? ?X between an explanatory
  variable X and a response variable Y
• What does it assume? Relationship between X and
  Y is linear each Y at a given X is a random
  sample from a normal distribution with equal
  variance
• Parameter ?
• Estimate b
• Degrees of freedom n-2
• Formulae

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Quick Reference Summary t-test for Regression
Slope

• What is it for? To test the null hypothesis that
  the population parameter ? equals a null
  hypothesized value, usually 0
• What does it assume? Same as regression slope
  C.I.
• Test statistic t
• Null distribution t with n-2 d.f.
• Formula

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T-test for Regression Slope
Null hypothesis ?0
Sample
Test statistic
Null distribution t with n-2 df
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Statistical tests

• Binomial test
• Chi-squared goodness-of-fit
• Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
• One-sample t-test
• Paired t-test
• Two-sample t-test

• F-test for comparing variances
• Welchs t-test
• Sign test
• Mann-Whitney U
• Correlation
• Spearmans r
• Regression
• ANOVA

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Quick Reference Summary ANOVA (analysis of
variance)

• What is it for? Testing the difference among k
  means simultaneously
• What does it assume? The variable is normally
  distributed with equal standard deviations (and
  variances) in all k populations each sample is a
  random sample
• Test statistic F
• Distribution under Ho F distribution with k-1
  and N-k degrees of freedom

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Quick Reference Summary ANOVA (analysis of
variance)

• Formulae

mean of group i overall mean
ni size of sample i N total sample size
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ANOVA
Null hypothesis All groups have the same mean
k Samples
Test statistic
Null distribution F with k-1, N-k df
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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ANOVA

• Ho All of the groups have the same mean
• Ha At least one of the groups has a mean that
  differs from the others

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ANOVA Tables
Source of variation Sum of squares df Mean Squares F ratio P
Treatment k-1
Error N-k
Total N-1
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Picture of ANOVA Terms
SSTotal MSTotal
SSGroup MSGroup
SSError MSError
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Two-factor ANOVA Table
Source of variation Sum of Squares df Mean Square F ratio P
Treatment 1 SS1 k1 - 1 SS1 k1 - 1 MS1 MSE
Treatment 2 SS2 k2 - 1 SS2 k2 - 1 MS2 MSE
Treatment 1 Treatment 2 SS12 (k1 - 1)(k2 - 1) SS12 (k1 - 1)(k2 - 1) MS12 MSE
Error SSerror XXX SSerror XXX
Total SStotal N-1
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Interpretations of 2-way ANOVA Terms
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Interpretations of 2-way ANOVA Terms
Effect of Temperature, Not pH
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Interpretations of 2-way ANOVA Terms
Effect of pH, Not Temperature
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Interpretations of 2-way ANOVA Terms
Effect of pH and Temperature, No interaction
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Interpretations of 2-way ANOVA Terms
Effect of pH and Temperature, with interaction
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Quick Reference Summary 2-Way ANOVA

• What is it for? Testing the difference among
  means from a 2-way factorial experiment
• What does it assume? The variable is normally
  distributed with equal standard deviations (and
  variances) in all populations each sample is a
  random sample
• Test statistic F (for three different
  hypotheses)
• Distribution under Ho F distribution

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Quick Reference Summary 2-Way ANOVA

• Formulae

Just need to know how to fill in the table
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2-way ANOVA
Null hypotheses (three of them)
Samples
Test statistic
Null distribution F
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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2-way ANOVA
Null hypotheses (three of them)
Samples
Treatment 1
Test statistic
Null distribution F
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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2-way ANOVA
Null hypotheses (three of them)
Samples
Treatment 2
Test statistic
Null distribution F
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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2-way ANOVA
Null hypotheses (three of them)
Samples
Interaction
Test statistic
Null distribution F
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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General Linear Models

• First step formulate a model statement
• Example

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General Linear Models

• Second step Make an ANOVA table
• Example

Source of variation Sum of squares df Mean Squares F ratio P
Treatment k-1
Error N-k
Total N-1

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Randomization test
Null hypothesis Randomized data
Sample
Calculate the same test statistic on the
randomized data
Null distribution
Test statistic
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Which test do I use?
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Methods for a single variable
1
How many variables am I comparing?
Methods for comparing two variables
2
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Methods for a single variable
1
How many variables am I comparing?
Methods for comparing two variables
2
3
Methods for comparing three or more variables
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Methods for one variable
Is the variable categorical or numerical?
Categorical
Comparing to a single proportion po or to a
distribution?
Numerical
po
distribution
One-sample t-test
?2 Goodness- of-fit test
Binomial test
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Methods for two variables
X
Contingency analysis
Logistic regression
Y
Correlation Regression
t-test ANOVA
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How many variables am I comparing?
1
2
Is the variable categorical or numerical?
Categorical
Contingency analysis
Logistic regression
Numerical
Comparing to a single proportion po or to a
distribution?
t-test ANOVA
Correlation Regression
One-sample t-test
po
distribution
Contingency analysis
?2 Goodness- of-fit test
Binomial test