1 K-sample Test for 3 or More Independent Sample Means: The Analysis of Variance (ANOVA)
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
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2 Case 5: A researcher randomly assigned newborn rats to one of three diets: (1) Unlimited access to food, (2) 90% of the amount of food that a rat of that size would normally eat. (3) 80% of the amount of food that a rat of that size would normally eat. She maintained the rats throughout their lives. Is there evidence that diet affected life span in this study. Let Îą= 0.05.
Unlimited
90% diet
80% diet
2.5
3.7
3.1
3.1
3.1
2.9
2.3
2.9
3.8
1.9
3.7
3.9
2.4
3.5
4.0
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3 Step 1. Hypothesis and Descriptive Statistics • H0: µunlimited= µ90= µ80 HA: at least one pair of µ ‘s are not equal.
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4 Step 2. Calculate the F statistic (ANOVA table)
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
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5 Step 3. Formulate the Decision Rules: Fcritical value •
Degrees of freedom are : df treatment = k-1 = 3-1 = 2 df error = N – k = 15-3 = 12
•
α= 0.05 for a one tailed test.
•
Reject H0 if the F statistic is >3.89. (Refer F distribution table in the next slide)
• Since the additional term in the numerator must be positive, we expect the F statistics to be bigger than 1 (one) if H0 is false. • So the F is always righttailed.
test in ANOVA one-tailed and
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
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6
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
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7 Step 4. Interpretation of Results • •
Since F (2,12) = 7.76> 3.89, is rejected. The mean squares diets (between groups) is significantly bigger than the error mean squares (within groups), indicating at least one pair of the diets are not equal.
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
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8 Mean Separation Techniques For ANOVA: Multiple Comparisons
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
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9 Which Means is Different? • When H0 was rejected as in the previous example, the conclusion at that point was at least one pair of means are not equal. • The task at hand is to discover exactly which means are different. • The techniques to do this are called mean separation techniques or multiple comparisons. • Most commonly used tests: Bonferroni t tests and Duncan’s multiple range test.
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10 Pairwise Comparisons: 3 possibilities
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11 Test Statistics
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12 Comparisons 1: Âľunlimited vs Âľ90
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13
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
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14 How to estimate the Critical Value for α= 0.01? Step 1: Calculate the difference between α=0.01 and α= 0.02 = 3.055-2.681= 0.374 Step 2: Divide it by 10. 0.374/10= 0.0374. Meaning that 0.0374 is weighed as α= 0.001 Step 3: Thus α= 0.017 = 3.055 – (0.0374x7) = 2.7932 (Critical Value)
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15 Comparisons 2: Âľunlimited vs Âľ90
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16 Comparisons 3: µ90 vs µ80
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
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17 Interpretation of the Results • •
•
Bonferroni t test, t(12)=-3.86<-3.055, indicates that the mean for unlimited diet is significantly different from the 80% group. The unlimited diet is not significantly different from the 90% group. The 90% and 80% diets are not significantly different from each other.
Online Course Pharmaceutical Biostatistics Chapter 4: Parametric Techniques
18/7/2017