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Stat 470-3
• Today: Will consider the one-way ANOVA model for comparing means of several treatments
Example
• Issue – shelf-life of pre-packaged meat• Objective – Compare four different packaging methods. Are there
differences? Which packaging is best? :1. T1. commercial plastic wrap2. T2. vacuum package
3. T3. 1%CO, 40%O2, 59%N
4. T4. 100% CO2
– Factor: Packaging– Experimental units: 12 steaks– Experimental Design: randomly assign 3 steaks to each packaging condition
balanced completely randomized design with a = 4, n = 3– Response: count of bacteria after 9 days at 4oC (39oF);
y = log(bacteria count/cm2)
DataAnalysis 1: Plot the Data
T1
T2
T3
T4
3
4
5
6
7
8
Treatment
y
Dotplots of y by Treatmen(group means are indicated by lines)
Notation:
k = 4 Treatments
ni = 3 reps per Treatment
N = 12 total observations
Eyeball Analysis: Does it look like all of these data could come from the same distribution? Or from four different distributions?
Package Rep. 1 Rep. 2 Rep 3. T1 7.66 6.98 7.80 T2 5.26 5.44 5.80 T3 7.41 7.33 7.04 T4 3.51 2.91 3.66
Experiments with a Single Factor: Completely Random Design
• Objective: – Determine if the mean response of a factor is the same at all levels
– If there is a difference, which levels differ?
• Method:– Have a single factors with k levels
– N experimental units available for the experiment
– N = n1 + n2+…+nk
– Randomly assign treatments to different experimental units
– Conduct experiment
– Results: yij, i=1,…,k; j=1,…,ni
Experiments with a Single Factor: Completely Random Design
• Model:
Sums of Squares
Test Statistic
ANOVA Table
Source Degrees of Freedom
Sum of Squares Mean Squares
F-Statistic P-Value
Treatments k-1
k
iii yyn
1
2... )(
Residual N-k
k
i
n
jiij
i
yy1 1
2.)(
Total N-1
k
i
n
jij
i
yy1 1
2..)(
Summary
• We have found a statistic (F) which:– compares the variance among treatment means to the variance within
treatments
– has a known distribution when all the treatment means are equal
• By comparing this F statistic to the F(k-1, N-k) distribution, we evaluate the strength of the evidence against the assumption of equal underlying treatment means
Back to the Example
Descriptive Statistics
Dependent Variable: RESPONSE
7.4800 .43863 3
5.5000 .27495 3
7.2600 .19468 3
3.3600 .39686 3
5.9000 1.75291 12
PACKAGET1
T2
T3
T4
Total
Mean Std. Deviation N
Tests of Between-Subjects Effects
Dependent Variable: RESPONSE
a
32.873 3 10.958 94.584 .000
.927 8 .116
33.800 11
Source
PACKAGE
Error
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .973 (Adjusted R Squared = .962)a.
Back to the Example
• Interpretation:
Comment
• When a = 2 (two treatments), F for testing for no difference among treatments is equal to t2 in the two-sample (unpaired) t-test
– Out-of-Class Exercise. • Demonstrate this equality by doing an ANOVA on the data in tomato
plant problem.
• Compare percentiles in F and t tables…what do you observe?
– For the mathematically inclined, demonstrate this equality algebraically
NOTE: It All Adds Up!
• It can be shown algebraically thatTotal SS = Treatment SS + Error SS
• Also, the degrees of freedom add up:N-1 = (k-1) + (N-k)
Exercise: Out-of-Class
• By using the formulas in the ANOVA table, verify the above ANOVA table for the meat packaging data
Estimation of Model Parameters: Constraints
• The model is over-parameterized
• Have k types of observation
• Have (k+1) parameters in the model– k for the treatment effects
– 1 for the grand mean
• Need to impose constraints to get solution
Constraints
• Sum to Zero Constraint:
• Interpretation:
Constraints
• Baseline Constraint:
• Interpretation:
Multiple Comparisons
• In previous example, we saw that there was a significant treatment effect…so what?
• If an ANOVA is conducted and the analysis suggests that there is a significant treatment effect, then a reasonable question to ask is
Multiple Comparisons
• Would like to see if there is a difference between treatments i and j
• Can use two-sample t-test statistic to do this
• For testing reject if
• Perform many of these tests
jiAji HH : versus:0
Multiple Comparisons
• Perform many of these tests
• Error rate must be controlled
Tukey Method
• Tests:
• Confidence Interval:
Back to Example
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