Transcript Document

Final Review Session
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”)
QuickTime™ and a
TIFF (Uncompressed) decompressor
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Things to Review
• Concepts
• Basic formulae
• Statistical tests
Things to Review
• Concepts
• Basic formulae
• Statistical tests
Populations
Samples
Random sample
First Half
Parameters
Estimates
Null hypothesis
Alternative hypothesis
P-value
Mean
Median
Mode
Type I error
Type II error
Variance
Standard deviation
Categorical
Nominal, ordinal
Numerical
Discrete, continuous
Sampling distribution
Standard error
Central limit theorem
Second Half
Normal distribution
Quantile plot
Shapiro-Wilk test
Data transformations
Nonparametric tests
Independent contrasts
Observations vs. experiments
Confounding variables
Control group
Replication and pseudoreplication
Blocking
Factorial design
Power analysis
Simulation
Randomization
Bootstrap
Likelihood
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.
Randomization test
Null hypothesis
Randomized data
Sample
Calculate the same test statistic on the
randomized data
Test statistic
compare
Null distribution
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
Things to Review
• Concepts
• Basic formulae
• Statistical tests
Things to Review
• Concepts
• Basic formulae
• Statistical tests
Sample
Test statistic
Null hypothesis
compare
Null distribution
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
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
Statistical tests
• Binomial test
• Chi-squared goodnessof-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
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:
n x
nx
P(x)   p 1 p
x 
P(x) = probability of a total of x successes
p = probability of success in each trial
n = total number of trials
P = 2 * Pr[xX]
Binomial test
Null hypothesis
Pr[success]=po
Sample
Test statistic
x = number of successes
compare
Null distribution
Binomial n, po
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
Binomial test
H0: The relative frequency of successes in the population is p0
HA: The relative frequency of successes in the population is not p0
Statistical tests
• Binomial test
• Chi-squared goodnessof-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
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:
 
2

all classes
Observedi  Expectedi 
2
Expectedi
2 goodness of fit test
Null hypothesis:
Data fit a particular
Discrete distribution
Sample
Calculate expected values
Test statistic
 
2
Observedi  Expectedi 
2

all classes
Expectedi
compar
e
Null distribution:
2 With
N-1-param. d.f.

How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
2 Goodness-of-Fit test
H0: The data come from a certain distribution
HA: The data do not come from that distrubition
Possible distributions
n x
nx
Pr[x]   p 1 p
x 


e 
Pr X  
X!
X
Pr[x] = n * frequency of occurrence
Proportional
Given a number of categories
Probability proportional to number of opportunities
Days of the week, months of the year
Binomial
Number of successes in n trials
Have to know n, p under the null hypothesis
Punnett square, many p=0.5 examples
Poisson
Number of events in interval of space or time
n not fixed, not given p
Car wrecks, flowers in a field
Statistical tests
• Binomial test
• Chi-squared goodnessof-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
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:
RowTotal*ColTotal
Expected
GrandTotal
2 

all classes
Observedi  Expectedi 
2
Expectedi
2 Contingency Test
Null hypothesis:
No association
between variables
Sample
Calculate expected values
Test statistic
 
2
Observedi  Expectedi 
2

all classes
Expectedi
compar
e
Null distribution:
2 With
(r-1)(c-1) d.f.

How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
2 Contingency test
H0: There is no association between these two variables
HA: There is an association between these two variables
Statistical tests
• Binomial test
• Chi-squared goodnessof-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
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:
Y  o
t
SEY
One-sample t-test
Null hypothesis
The population mean
is equal to o
Sample
Test statistic
Y  o
t
s/ n
compare
Null distribution
t with n-1 df
How unusual is this test statistic?

P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
One-sample t-test
Ho: The population mean is equal to o
Ha: The population mean is not equal to o
Paired vs. 2 sample
comparisons
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
d  do
• Formula:
t
SEd

Paired t-test
Null hypothesis
The mean difference
is equal to o
Sample
Test statistic
d  do
t
SEd
compare
Null distribution
t with n-1 df
*n is the number of pairs
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
Paired t-test
Ho: The mean difference is equal to 0
Ha: The mean difference is not equal 0
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.

1 
2 1
SE Y Y  s p   
Y

Y
1
2
n1 n 2 
• Formulae:
t
1
SE Y Y
1
2
df1s12  df2 s22
s 
df1  df2
2
p

2
Two-sample t-test
Null hypothesis
The two populations
have the same mean
Sample
12
Test statistic
Y1  Y2
t
SE Y Y
1
compare
Null distribution
t with n1+n2-2 df
2
How unusual is this test statistic?

P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
Two-sample t-test
Ho: The means of the two populations are
equal
Ha: The means of the two populations are
not equal
Statistical tests
• Binomial test
• Chi-squared goodnessof-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
F-test for Comparing the
variance of two groups
H0 :   
2
1
2
2
HA :   
2
1

2
2
F-test
Null hypothesis
The two populations
have the same variance
Sample
 2 1   22
Test statistic
2
1
2
2
s
F
s
compare
Null distribution
F with n1-1, n2-1 df
How unusual is this test statistic?

P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
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
Welch’s t-test
Null hypothesis
The two populations
have the same mean
Sample
12
Test statistic
t

Y1  Y2
compare
Null distribution
t with df from formula
s12 s22

n1 n2
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
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
Parametric
One-sample and
Paired t-test
Two-sample t-test
Nonparametric
Sign test
Mann-Whitney
U-test
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.
n x
nx
P(x)   p 1 p
P = 2 * Pr[xX]
P(x) = probability of a total of x successes
x 
p = probability of success in each trial
n = total number of trials
Sign test
Null hypothesis
Median = mo
Sample
Test statistic
x = number of values
greater than mo
compare
Null distribution
Binomial n, 0.5
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
Sign Test
• Ho: The median is equal to some value
mo
• Ha: The median is not equal to mo
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:
n = sample size of group 1
n1n1 1
U1  n1n2 
 R1
2
U2  n1n2  U1
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
Mann-Whitney U test
Null hypothesis
The two groups
Have the same
median
Sample
Test statistic
U1 or U2
(use the largest)
compare
Null distribution
U with n1, n2
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
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
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
X  X Y  Y 
• Formulae:
1 r 2
i
i
r
X i  X  Yi  Y 
2
2
SEr 
n 2
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
r
freedom
t
• Formulae:
SE
r
T-test for correlation
Null hypothesis
=0
Sample
Test statistic
r
t
SEr

compare
Null distribution
t with n-2 d.f.
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
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
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 tdistribution
• Formulae: Same as linear correlation but
based on ranks
Spearman’s rank correlation
Null hypothesis
=0
Sample
Test statistic
rs
compare
Null distribution
Spearman’s rank
Table H
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
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
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
OK
Non-linear
Non-normal
Unequal
variance
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:
b  t(2),df SEb    b  t (2),df SEb
SEb 


MSresidual 
MSresidual
 X
 X
2
i
2
(Y
Y
)
 b(X i  X )(Yi Y )
 i
n 2
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:
b
t
SEb
T-test for Regression Slope
Null hypothesis
=0
Sample
Test statistic
b
t
SEb

compare
Null distribution
t with n-2 df
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
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
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
Quick Reference Summary:
ANOVA (analysis of variance)
• Formulae:
MSgroup
MSgroup
F
MSerror
SSgroup SSgroup


dfgroup
k 1
MSerror

SSgroup   ni (Y i Y)

2
Y i = mean of group i
Y = overall mean
SSerror SSerror


dferror
Nk
SSerror   si2 (ni 1)
ni = size of sample i
N = total sample size
ANOVA
k Samples
Test statistic
MSgroup
F
MSerror

compare
Null hypothesis
All groups have
the same mean
Null distribution
F with k-1, N-k df
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
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
ANOVA Tables
Source of
variation
Treatment
Error
Total



Sum of
squares
SSgroup   ni (Y i Y)2
df
k-1
SSerror   si2 (ni 1)
N-k
SSgroup  SSerror
N-1


Mean
Squares
MSgroup 
SSgroup
dfgroup
MSerror 
SSerror
dferror
F ratio
P
Picture of ANOVA Terms
SSTotal
MSTotal
SSGroup
MSGroup
SSError
MSError
Two-factor ANOVA Table
Source of
variation
Sum of
Squares
df
Mean Square
F ratio
Treatment 1
SS1
k1 - 1
SS1
k1 - 1
MS1
MSE
Treatment 2
SS2
k2 - 1
SS2
k2 - 1
MS2
MSE
Treatment 1 *
Treatment 2
SS1*2
(k1 - 1)*(k2 - 1) SS1*2
MS1*2
(k1 - 1)*(k2 - 1) MSE
Error
SSerror
XXX
Total
SStotal
N-1
SSerror
XXX
P
Interpretations of 2-way
ANOVA Terms
70
pH 5.5
pH 6.5
pH 7.5
60
Growth Rate
50
40
30
20
10
0
25
30
35
Temperature
40
Interpretations of 2-way
ANOVA Terms
45
Effect of Temperature,
Not pH
40
pH 5.5
pH 6.5
pH 7.5
35
Growth Rate
30
25
20
15
10
5
0
25
30
35
Temperature
40
Interpretations of 2-way
ANOVA Terms
35
pH 5.5
pH 6.5
pH 7.5
30
Effect of pH,
Not Temperature
Growth Rate
25
20
15
10
5
0
25
30
35
Temperature
40
Interpretations of 2-way
ANOVA Terms
70
Effect of pH and Temperature,
No interaction
60
pH 5.5
pH 6.5
pH 7.5
Growth Rate
50
40
30
20
10
0
25
30
35
Temperature
40
Interpretations of 2-way
ANOVA Terms
45
40
Effect of pH and Temperature,
with interaction
35
pH 5.5
pH 6.5
pH 7.5
Growth Rate
30
25
20
15
10
5
0
25
30
35
Temperature
40
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
Quick Reference Summary: 2Way ANOVA
• Formulae:
Just need to know how to
fill in the table
2-way ANOVA
Null hypotheses
(three of them)
Samples
Test statistic
MSgroup
F
MSerror

compare
Null distribution
F
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
2-way ANOVA
Samples
Null hypotheses
(three of them)
Treatment
1
Null
distribution
compare
Test statistic
MSgroup
F
MSerror

F
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
2-way ANOVA
Samples
Null hypotheses
(three of them)
Treatment
2
Null
distribution
compare
Test statistic
MSgroup
F
MSerror

F
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
2-way ANOVA
Samples
Null hypotheses
(three of them)
Interaction
Null distribution
compare
Test statistic
MSgroup
F
MSerror

F
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
General Linear Models
• First step: formulate a model statement
• Example:
Y    TREATMENT
General Linear Models
• Second step: Make an ANOVA table
• Example:
Source
of
variation
Treatme
nt
Error

Total

Sum of
squares
df
Mean
Squares
k-1
MSgroup 
SSgroup
dfgroup
SSerror   si2 (ni 1)
N-k
MSerror 
SSerror
dferror
SSgroup  SSerror
N-1
SSgroup   ni (Y i Y)2



F ratio
F
MSgroup
MSerror
P
*
Randomization test
Null hypothesis
Randomized data
Sample
Calculate the same test statistic on the
randomized data
Test statistic
compare
Null distribution
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
Which test do I use?
1
Methods for a
single variable
How many variables
am I comparing?
2
Methods for
comparing two
variables
1
Methods for a
single variable
How many variables
am I comparing?
2
3
Methods for
comparing two
variables
Methods for
comparing three
or more variables
Methods for one variable
Categorical
Is the variable
categorical
or numerical?
Comparing to a
single proportion po
or to a distribution?
po
Binomial test
Numerical
distribution
2 Goodnessof-fit test
One-sample t-test
Methods for two variables
Y
X
Explanatory variable
Response variable
Categorical
Numerical
Contingency table
Contingency
Logistic
Grouped bar graph
Categorical
analysis
regression
Mosaic plot
Multiple histograms
t-test
Correlation
Scatter plot
Cumulative frequency distributions
Numerical
ANOVA
Regression
How many variables
am I comparing?
1
2
Is the variable
categorical
or numerical?
Categorical
Explanatory variable
Response variable
Categorical
Numerical
Contingency table
Logistic
Contingency
Grouped
bar graph
Categorical
analysis
regression
Mosaic plot
Multiple
histograms
t-test
Correlation
Scatter plot
Cumulative frequency distributions
Numerical
ANOVA
Regression
Numerical
Comparing to a
single proportion po
or to a distribution?
One-sample t-test
po
Binomial test
distribution
2 Goodnessof-fit test
Contingency
analysis