1. Final Review Session

2. 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, 12-1:30 pm
Location: Osborne Centre, Unit 1 (”A”)

4. Things to Review
Concepts
Basic formulae
Statistical tests

5. Things to Review
Concepts
Basic formulae
Statistical tests

6. First Half Populations Samples Random sample 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

7. Second Half Normal distribution Simulation Quantile plot Randomization
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

8. Example Conceptual Questions
(you’ve 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.

9. Calculate the same test statistic on the randomized data
Randomization test
Null hypothesis
Randomized data
Sample
Calculate the same test statistic on the randomized data
Test statistic
Null distribution
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

10. Things to Review
Concepts
Basic formulae
Statistical tests

12. Things to Review
Concepts
Basic formulae
Statistical tests

13. Sample
Null hypothesis
Test statistic
Null distribution
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

14. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

15. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

16. 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 * Pr[xX]
P(x) = probability of a total of x successes
p = probability of success in each trial
n = total number of trials

17. Binomial test
Null hypothesis
Pr[success]=po
Sample
Test statistic
x = number of successes
Null distribution
Binomial n, po
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

18. Binomial test

19. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

20. 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 < 1; no more than 20% of expected values < 5
Test statistic: 2
Distribution under Ho: 2 with
df=# categories - # parameters - 1
Formula:

21. Discrete distribution
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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

22. 2 Goodness-of-Fit test

23. Possible distributions
Pr[x] = n * frequency of occurrence

24. Proportional Binomial Poisson 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 p=0.5 examples
Binomial
Number of events in interval of space or time
n not fixed, not given p
Car wrecks, flowers in a field
Poisson

25. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

26. 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 < 1; no more than 20% of expected values < 5
Test statistic: 2
Distribution under Ho: 2 with
df=(r-1)(c-1) where r = # rows, c = # columns
Formulae:

27. 2 Contingency Test Null hypothesis: Sample 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

28. 2 Contingency test

29. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

30. 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:

31. 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

32. One-sample t-test Ho: The population mean is equal to o
Ha: The population mean is not equal to o

33. Paired vs. 2 sample comparisons

34. 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:

35. *n is the number of pairs
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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

36. Paired t-test Ho: The mean difference is equal to 0
Ha: The mean difference is not equal 0

37. 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 n1+n2-2 degrees of freedom.
Formulae:

38. Two-sample t-test
Null hypothesis
The two populations
have the same mean
12
Sample
Null distribution
t with n1+n2-2 df
Test statistic
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

39. Two-sample t-test Ho: The means of the two populations are equal
Ha: The means of the two populations are not equal

40. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

41. F-test for Comparing the variance of two groups

42. 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

43. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

44. Welch’s 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

45. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

46. Parametric
Nonparametric
One-sample and
Paired t-test
Sign test
Mann-Whitney
U-test
Two-sample t-test

47. 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= Uses the number of subjects with values greater than and less than a hypothesized median as the test statistic.
P = 2 * Pr[xX]
P(x) = probability of a total of x successes
p = probability of success in each trial
n = total number of trials

48. 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

49. Sign Test Ho: The median is equal to some value mo
Ha: The median is not equal to mo

50. 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

51. 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

52. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

53. 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:

54. 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:

55. 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

56. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

57. Quick Reference Guide - Spearman’s 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 n>100, use t-distribution
Formulae: Same as linear correlation but based on ranks

58. Spearman’s rank correlation
Null hypothesis
=0
Sample
Test statistic rs
Null distribution
Spearman’s rank
Table H
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

59. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

60. 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

61. OK
Non-normal
Unequal
variance
Non-linear

62. 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:

64. 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:

65. 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

66. 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
Welch’s t-test
Sign test
Mann-Whitney U
Correlation
Spearman’s r
Regression
ANOVA

67. 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

68. Quick Reference Summary: ANOVA (analysis of variance)
Formulae:
= mean of group i
= overall mean
ni = size of sample i
N = total sample size

69. 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 > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

70. 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

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

72. Picture of ANOVA Terms
SSTotal
MSTotal
SSGroup
MSGroup
SSError
MSError

73. Two-factor ANOVA Table
Source of variation
Sum of Squares df
Mean Square
F ratio P
Treatment 1
SS1
k1 - 1
MS1
MSE
Treatment 2
SS2
k2 - 1
MS2
Treatment 1 * Treatment 2
SS1*2
(k1 - 1)*(k2 - 1)
MS1*2
Error
SSerror
XXX
Total
SStotal
N-1

74. Interpretations of 2-way ANOVA Terms

75. Interpretations of 2-way ANOVA Terms
Effect of Temperature,
Not pH

76. Interpretations of 2-way ANOVA Terms
Effect of pH,
Not Temperature

77. Interpretations of 2-way ANOVA Terms
Effect of pH and Temperature,
No interaction

78. Interpretations of 2-way ANOVA Terms
Effect of pH and Temperature,
with interaction

79. 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

80. Quick Reference Summary: 2-Way ANOVA
Formulae:
Just need to know how to
fill in the table

81. 2-way ANOVA
Null hypotheses
(three of them)
Samples
Test statistic
Null distribution F
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

82. Treatment 1 2-way ANOVA Null hypotheses (three of them) Samples
Test statistic
Null distribution F
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

83. Treatment 2 2-way ANOVA Null hypotheses (three of them) Samples
Test statistic
Null distribution F
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

84. Interaction 2-way ANOVA Null hypotheses (three of them) Samples
Test statistic
Null distribution F
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

85. General Linear Models
First step: formulate a model statement
Example:

86. General Linear Models Second step: Make an ANOVA table Example: k-1
Source of variation
Sum of squares df
Mean Squares
F ratio P
Treatment
k-1
Error
N-k
Total
N-1 *

87. Calculate the same test statistic on the randomized data
Randomization test
Null hypothesis
Randomized data
Sample
Calculate the same test statistic on the randomized data
Test statistic
Null distribution
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

88. Which test do I use?

89. Methods for a
single variable 1
How many variables
am I comparing? 2
Methods for
comparing two
variables

90. Methods for a
single variable 1
How many variables
am I comparing? 2
Methods for
comparing two
variables 3
Methods for
comparing three
or more variables

91. 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

92. Methods for two variablesX
Contingency
analysis
Logistic
regression Y
t-test
ANOVA
Correlation
Regression

93. How many variables am I comparing? 1 2 Categorical Numerical
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