Chapter 11 Flashcards
What is ANOVA
Analysis of Variance
Def: Allows statistical comparison among samples taken from many populations
The comparison is typically the result of an experiment.
What is the basis for an ANOVA experiment
What is a “factor”
Define the levels of a factor
The levels of a factor are the groups that comprise the experiment and analysis. They provide the basis for comparison.
What are the three types of experiments that can be conducted with ANOVA?
And which one will we use?
- Completely randomized design: an experiment with only one factor
- Factoral design: An experiment in which more than one factor is considered (Two-way ANOVA)
- Randomized block design: An experiment in which the members of each group have been placed in blocks either by being matched or subjected to repeated measurements as was done the two populations of a paired t test
We will only use Completely randomized design
What is the Two Part Process of ANOVA
- Determine if there is a significant difference among the group means (if you reject H0, continue with #2)
- Identify the groups whose means are significantly different from the other group means
What is the purpose of ANOVA
To reach conclusions about possible differences month the means of each group
Define the Completely Randomized Design
The ANOVA method that analyzes a single factor.
This design is executed using the statistical method one-way ANOVA
Define SST
The sum of squares total represents total variation. It is partitioned into 2 groups
- Within-group variation (SSW) measures random variation
- Among-group variation (SSA) measures differences from group to group
Define n and c
- ‘n’* represents the number of values in all groups
- ‘c’* represents the number of groups
Review “Partitioning the Total Variation”
SST = SSA + SSW
What are the assumptions of ANOVA
The c groups represent populations who(se)
- Values are randomly and independently selected
- Follow a normal distribution
- Have equal variances
What is the H0 for ANOVA
H0 = μ1 = μ2 = . . . = μc
What is H1 for ANOVA
H1: not all μj are equal
where j = 1, 2, . . . , c
Total variation is represented by what
The sum of squares total (SST)
What is the grand mean
The mean of all the values in all the groups combined
Xdouble-bar
Define SSW
“Within-group variation”
Measures random variation
Define SSA
“Among-group” variation
Measures differences from group to group
Review the equation for Total Variation in One-Way ANOVA
Review equation for Among-group variation in One-way ANOVA
Review Within-group variation in One-way ANOVA
Review the equation for the Mean Squares in One-way ANOVA
Note: the mean square is just another term for variances that is used in ANOVA
Define the F Test for Differences Among More than Two Means
Tests to see if there is a significant difference b/t the means.
Note: It does not tell you where the differences are
Review the One-way ANOVA FSTAT Test Statistic
Review the Area of Rejection and Non-rejection
Note: start drawing these for HW and tests
Review the ANOVA Summary Table
Summarized the results of a one-way ANOVA
What are the One-way ANOVA F Test Assumptions
- Randomness and independence of the sample selected
- Normality of the c groups from which samples are selected Note: use a Normal Probability Plot to determine normality. (see image)
- Homogeneity of variance (the variances of the c groups are equal) Note: this will require the Levene test
What is the Levene Test
A Test for Homogeneity of Variance
- Compute the absolute value of the difference between each value and the median of the group
- Perform a one-way ANOVA on these absolute differences
Note: most statisticians suggest using a level of significance of α = .05 when performing the ANOVA
Define the Turkey-Kramer Procedure
Procedure to construct multiple comparisons to test H0 that the differences in the means of all pairs of “items” are equal to 0
Determines with of the c means are significantly different
What are the four steps of the Turkey-Kramer Procedure
- Compute the absolute mean differences (|x̄j - x̄j’|; where j, j’ refers to group j & j’, and j ≠ j’) among all pairs of sample means [c(c - 1) / 2 pairs]
- Compute the critical range (Using equation 11.6, attached)
- Compare each of the c*(c - 1) / 2 pairs of means against its corresponding critical range. (Declare a specific pair significantly different if the absolute difference in the sample means,|x̄j - x̄j’| , is greater than the critical range)
- Interpret the results
How many degrees of freedom are there in determining
- the among-group variation
- the within-group variation
- the total variation
Among-group variation: c - 1
Within-group variation: n - c
Total variation: n - 1
Where c = number of groups; n = total number of values in all groups
Define the Mean Squares in One-way ANOVA
Mean square among (MSA) = SSA / c-1
Mean square within (MSW) = SSW / n - c
Mean square total (MST) = SST / n - 1
How do you find the value of FSTAT
MSA / MSW
How do you find Fα
Use the Critical Values of F Table
Numerator = c - 1
Denominator = n - c
How do you find Qα
Use the Critical Values of the Sudentized Range, Q Table
Numerator = c
Denominator = n - c
