Download Lecture 8 - Notes - for Dr. Jason P. Turner

Analysis of Variance
(ANOVA)
MARE 250
Dr. Jason Turner
First name: Mister, Last name: T
“I pity myself that I got to be with these fools!”– Mr. T
Hypothesis Testing for Two Means
Test for comparing the means of two populations
One of the most common (and simple?) tests in statistics
What is the procedure if there are more than two populations?
Mr. T – Part 2: Electric Boogaloo
Why not run multiple T-test?
μ1
μ2
μ3
1. Number of t-tests increases with # of groups
becomes cognitively difficult
2.
↑ Number of analyses = ↑ probability of committing
Type I error
Probability of committing at least one type I error
= experiment-wise error rate
Analysis of Variance (ANOVA)
Method for comparing multiple sample means
Compares the means of a variable for populations that
result from a classification by one other variable –
Factor
Levels of the Factor – the possible values of the factor
At this point dealing with: One-Way AVOVA
Analysis of Variance (ANOVA)
Want to compare egg production in four populations of
Roi in Hawaii
Variable of interest = egg production
Four populations result from classifying Roi in Hawaii
by the Factor “Beach locations” whos Levels are Ka
Lae, Kapoho, Richardson’s, Keauhou Bay
Assumptions for One-Way ANOVA
One-Way ANOVA
Four assumptions for t-test hypothesis testing:
1.
2.
3.
4.
Random Samples
Independent Samples
Normal Populations (or large samples)
Variances (std. dev.) are equal
Other similarities T & ANOVA
“I pity the fools that think T and AVONA are similar!”– Mr. T
A one-way analysis of variance (ANOVA) tests the
hypothesis that the means of several populations are
equal.
The method is an extension of the two-sample t-test,
specifically for the case where the population variances
are assumed to be equal.
Other similarities T & ANOVA
“That sucka stole my van!”– Mr. T
A one-way analysis of variance requires the following:
Response - measurement taken from the units sampled.
Factor - discrete variable that is altered systematically.
The different values chosen for the factor variable are
called levels of the factor.
Each level of the factor in the analysis corresponds to a
larger population with its own mean.
The sample mean is an estimate of the level mean for the
whole population.
Other similarities T & ANOVA
A one-way ANOVA can be used to tell you if there are
statistically significant differences among the level
means.
The null hypothesis for the test is that all population
means (level means) are the same.
The alternative hypothesis is that one or more
population means differ from the others.
Analysis of Variance (ANOVA)
Method for comparing multiple sample means
Compares the means of a variable for populations that result from
a classification by one other variable – Factor
Levels of the Factor – the possible values of the factor
At this point dealing with: One-Way AVOVA
Mr. T Says…
“I pity the fools that think T and ANOVA are similar!”– Mr. T
also “Stay in School fool!” and the popular “Drugs are for Chumps!”
A one-way analysis of variance (ANOVA) tests the hypothesis that
the means of several populations are equal
The method is an extension of the two-sample t-test, specifically
for the case where the population variances are assumed to be
equal
The null hypothesis for the test is that all population means (level
means) are the same H0: μ1 = μ2 = μ3 = μ4
The alternative hypothesis is that one or more population means
differ from the others Ha: Not all means are equal
One-Way ANOVA : Analysis of Variance Table
One-way ANOVA: Egg production versus Location
Source DF
SS MS
F
P
Paint
3 281.7 93.9 6.02 0.004
P value
Error 20 312.1 15.6
F statistic
Total 23 593.8
Mean Squares
Sum of Squares
Degrees of Freedom
F
F is the statistic used to test the hypothesis that all the factor
level means are equal
It is calculated as the mean square for the factor divided by the
mean squares for error
F is used to determine the p-value
Like the T statistic for a t-test, or an R-J value for a
Ryan-Joiner
Four Sample Means, Only One Result?
“I pity the fools that has only one result for 4 sample means!”– Mr. T
A one-way analysis of variance (ANOVA) tests the hypothesis that
the means of several populations are equal
The null hypothesis for the test is that all population means (level
means) are the same H0: μ1 = μ2 = μ3 = μ4
The alternative hypothesis is that one or more population means
differ from the others Ha: Not all means are equal
Source DF
SS MS
F
P We reject the null
Pop
3 281.7 93.9 6.02 0.004 = all means not equal
Error 20 312.1 15.6
Total 23 593.8
Is that all?
Confidence Intervals
Confidence Interval – (CI) provides a range of numbers along
with a percentage confidence that the parameter lies in that range
Individual 95% CIs For Mean
Based on Pooled StDev
Level
Pop 1
Pop 2
Pop 3
Pop 4
N Mean StDev
6 14.733 3.363
6 8.567 5.500
6 12.983 3.730
6 18.067 2.636
Pooled StDev = 3.950
(-----*------)
(------*------)
(------*------)
(------*------)
95% confidence intervals for each level of the factor
When the p-value in the analysis of variance table indicates there
is a difference among the factor level means, you can use the table
of individual confidence intervals to explore the differences:
Each asterisk represents a sample mean
Each set of parentheses encloses a 95% confidence interval for the
mean of a population. You can be 95% confident that the
population mean for each level is within the corresponding
interval
Level N Mean StDev
Pop 1 6 14.733 3.363
(-----*------)
Pop 2 6 8.567 5.500
(------*------)
Pop 3 6 12.983 3.730
(------*------)
Pop 4 6 18.067 2.636
(------*------)
95% confidence intervals for each level of the factor
If the intervals do not overlap; suggests population means are
different
Interpret with caution; rate of type I error increases when making
multiple comparisons
Should use one of the four available methods for controlling the
rate of type I error
Intervals for Pop 2 and Pop 4 do not overlap; suggests population
means for these levels are different
Level N Mean StDev
Pop 1 6 14.733 3.363
(-----*------)
Pop 2 6 8.567 5.500
(------*------)
Pop 3 6 12.983 3.730
(------*------)
Pop 4 6 18.067 2.636
(------*------)
Multiple Comparisons
Allow you to determine the relations among all the
means
Several methods: Tukey, Fisher’s LSD, Dunnett’s,
Bonferroni, Scheffe, etc
Most focus on Tukey
Tukey's method
Tukey's method compares the means for each pair of factor
levels using a family error rate to control the rate of type I
error
Results are presented as a set of confidence intervals for the
difference between pairs of means
Use the intervals to determine whether the means are
different:
If an interval does not contain zero, there is a statistically
significant difference between the corresponding means
If the interval does contain zero, the difference between the
means is not statistically significant
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Ahi
Individual confidence level = 98.89%
Ahi = Pop1 subtracted from:
Ahi Lower Center Upper
Pop 2 -12.553 -6.167 0.219
Pop 3 -8.136 -1.750 4.636
Pop 4 -3.053 3.333 9.719
Ahi = Pop 2 subtracted from:
Ahi Lower Center Upper
Pop 3 -1.969 4.417 10.803
Pop 4 3.114 9.500 15.886
Ahi = Pop 3 subtracted from:
Ahi Lower Center Upper
Pop 4 -1.303 5.083 11.469
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
(-------*-------)
Which multiple comparisons to use
At this point, unless otherwise advised – Tukey’s is fine
Bonferroni, Fisher’s LSD, Dunnett’s, Bonferroni,
Scheffe all used heavily as well
Statistical versus practical significance
Even if the level means are significantly different from a
statistical standpoint, the difference may not be of any practical
importance
In the Ahi data, the smallest mean is 8.567 and the largest is
18.067. Is this 9.5-unit difference of any practical consequence?
Only knowledge of the subject area, not statistics,
can be used to answer this question
Assumptions for One-Way ANOVA
One-Way ANOVA
Four assumptions for t-test hypothesis testing:
1.
2.
3.
4.
Random Samples
Independent Samples
Normal Populations (or large samples)
Variances (std. dev.) are equal
Non-Parametric Version of ANOVA
Kruskal-Wallis
If samples are independent, similarly distributed data
Use nonparamentric test regardless of normality or
sample size
Is based upon mean of ranks of the data – not the mean
or variance (Like Mann-Whitney)
If the variation in mean ranks is large – reject null
Uses p-value like ANOVA
Last Resort/Not Resort –low sample size, “bad” data
When Do I Do the What Now?
“Well, whenever I'm confused, I just check my underwear. It holds the answer
to all the important questions.” – Grandpa Simpson
If you are reasonably sure that the distributions
are normal –use ANOVA
Otherwise – use Kruskal-Wallis