Oneway ANOVA. Motivating Example Analysis of Variance Model & Assumptions Data Estimates of the Model Analysis of Variance Multiple Comparisons Checking Assumptions Oneway ANOVA Transformations. Motivating Example: Treating Anorexia Nervosa. Analysis of Variance.
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H0: 1= 2= = k
against:
Ha:Not all population means are the same
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Model & AssumptionsHere we would almost certainly reject the null hypothesis.
3j= y3j3
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Analysis of VarianceBetweengroup variation is large compared to the Withingroup variation
If we sampled from these populations, we would expect to reject H0
Here we would fail to reject the null hypothesis.
If we sampled from these populations, we would not expect to reject H0
All i = 0
= 1 = 2 = 3
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Analysis of Variancen
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Measure variation due to the fact different treatments are used.
Measures error variation, variation in response when same treatment is applied.
Analysis of Variance
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Treatment Sum of Squares (SSTreat) or Between Group Sum of Squares
Error Sum of Squares (SSError) or Within Group Sum of Squares
Analysis of Variance
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MSTreat = SSTreat / dfTreat= SSTreat / (k – 1)
MSError = SSError / dfError = SSError / (N – k)
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Analysis of VarianceE(MSTreat)=2+i2 / (k1)
E(MSError)= 2
within group variance:
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Error
Analysis of Variance XINote that the greater the natural variability within the groups, the larger the effects (i)will need to be (as estimated by MSTreat) for us to detect any significant differences.
SS/df
Source of
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Ratio
Pvalue
Variation
Freedom
Squares
Square
/MS
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Tail Area
Treatment
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T
Analysis of VarianceF0 = 307.32/57.68 = 5.42
0.8
0.6
Then the Pvalue = 0.06
0.4
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Let’s say our observed value for F was F0 = 2.5
Analysis of VarianceWhen H0 is true, F0 ~ F(df1,df2)
For example, consider the Fdistribution with 4 and 30 df
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Multiple Comparisonspossible pairwise comparisons.
k choose 2
simultaneous confidence intervals OR
multiple comparison procedures
(i.e. confidence intervals that are too narrow),
and the Bonferroni correction finding too few significant differences
(i.e. confidence intervals that are too wide).
We will use Tukey Intervals
SelectCompare Means > All Pairs, Tukey HSD
SelectCompare Means > All Pairs, Tukey HSD
Here we see that only Behavioral and Standard therapies differ in terms of mean weight gain. We estimate those in behavioral therapy will gain between 2 lbs. and 13 lbs. more on average.
Add Normal Quantile Plots to Assess Normality