Chpt 12 - One-Way ANOVA Flashcards
What does ANOVA stand for?
ANalysis Of VAriance
We want to determine if changing the cotton content changes the mean tensile strength. We set up 5 different levels of cotton content.
In this experiment, what is the response variable?
Tensile strength
It is the variable to be measured in the experiment
We want to determine if changing the cotton content changes the mean tensile strength. We set up 5 different levels of cotton content.
In this experiment, what is the factor?
Cotton content
Its the variable that is going to be changed for the study
We want to determine if changing the cotton content changes the mean tensile strength. We set up 5 different levels of cotton content.
What type of test can be used to determine this?
ANOVA test
What is the response variable?
The dependent variable.
It is the variable of interest to be measured in the experiment
What are factors?
The variables whose effect on the response variable is studied in the experiment
What are the factor levels?
The values of a factor in the experiment
For cotton content, you may have 20% and 25% to check
What is a one-way ANOVA test?
Used when there is only one factor that we are changing in the experiment.
We want to determine if changing the cotton content changes the mean tensile strength. We set up 5 different levels of cotton content (A, B, C, D, E). We also have 3 different colour dyes we are going to use (1, 2, 3).
What are the treatments in this experiment?
The possible factor level combinations in the experiment.
For this example we have:
A1, B1, C1, D1, E1
A2, B2, C2, D2, E2
A3, B3, C3, D3, E3
How do we determine the overall mean of an ANOVA test?
x̄ = sum of all sample data/sum of all sample sizes
How do we determine the sample mean for each sample?
x̄1 = sum of all data in sample 1/sample size of sample 1
Which statistic measures the total variation of the data from all samples in an ANOVA test?
Total Sum of Squares (SST)
What does the total sum of squares (SST) measure?
The total variation of the data from all samples
What does the SST consist of?
Variation between groups (SST) = Variation between groups (SSTR) + variation within groups (SSE)
Which statistic measures the variation between different samples of an ANOVA test?
Sum of Squares of TReatment (SSTR) is the between group variation
Which statistic measures the variation within samples of an ANOVA test?
Sum of Squares of Error (SSE) is the within group variation
What is the equation for the SST?
Sum of Squares Total
SST = Σ(xi-x̄) squared
OR
SST = Σx squared - (Σx values) squared/n
With degrees of freedom = n-1
n=total number of samples
What is the equation for the SSTR?
Sum of Squares of TReatment
SSTR = Σni(x̄i-x̄)squared
The i is observations in a sample
So ni is the sample size of the individual sample
x̄i is the sample mean of the individual sample
x̄ is the overall mean
degrees of freedom = k-1
k - number of factors
What is the equation for the SSE?
Sum of Squares of Error
SSE = Σ all samples Σ each sample point in a given sample (xi-x̄i)
xi is the individual sample point in a given sample
x̄i is the sample mean of the individual sample
OR
SSE = SST - SSTR
degrees of freedom = n-k
n - total number of samples
k - number of factors
We have an the following numbers for a one-way ANOVA test:
x̄ = 5.5, SST = 95, SSTR = 3, SSE = 92
What does this tell us?
The between groups (SSTR) is small and the variation within groups (SSE) is large.
Therefore it is hard to tell if the variation is due to the difference between the population means or to the variation within the samples.
This means we are unable to conclude that the two population means are different.