Lecture 13: Chapter 7: Analysis of Variance (ANOVA) | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

7.1. Inferences about a population variance 7.2. Inferences about two population variances 7.3. Assumptions for analysis of variance 7.4. A conceptual overview 7.5. ANOVA table 7.6. ANOVA procedure. Total Sum Square (TSS). The analysis of variance can be viewed as the process of partitioning the total sum of squares and the degrees of freedom into their corresponding sources: treatments and error. Tài liệu giúp bạn tham khảo, ôn tập và đạt kết quả cao. Mời bạn đọc đón xem!

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Môn: Applied statistics (ENEE1006IU)

Trường: Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

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lOMoARcPSD| 45903860
APPLIED STATISTICS
COURSE CODE: ENEE1006IU
Lecture 13:
Chapter 7: Analysis of Variance (ANOVA)
(3 credits: 2 is for lecture, 1 is for lab-work)
1
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2
CHAPTER 7: ANALYSIS OF VARIANCE (ANOVA)
•7.1. Inferences about a populaon variance
•7.2. Inferences about two populaon variances
•7.3. Assumpons for analysis of variance
•7.4. A conceptual overview
•7.5. ANOVA table
•7.6. ANOVA procedure
7.5. ANOVA TABLE
•Total Sum Square (TSS):
lOMoARcPSD| 45903860
SSTR (sum of squares due to
treatments)
the degrees of freedom
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4
7.5. ANOVA TABLE
•Total Sum Square (TSS):
•The analysis of variance can be viewed as the process of paroning the total sum
of squares and the degrees of freedom into their corresponding sources:
treatments and error.
•Dividing the sum of squares by the appropriate degrees of freedom provides the
variance esmates, the F value, and the p-value used to test the hypothesis of
equal populaon means.
•ANOVA TABLE:
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MULTIPLE COMPARISON PROCEDURES
•When we use analysis of variance to test whether the means of k populaons are
equal, rejecon of the null hypothesis allows us to conclude only that the
populaon means are not all equal.
•In some cases we will want to go a step further and determine where the
dierences among means occur.
•The purpose of this secon is to show how mulple comparison procedures can
be used to conduct stascal comparisons between pairs of populaon means.
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6
Fisher’s least signicant dierence (LSD) procedure can be used to determine
where the dierences occur (pairwise comparisons)
MULTIPLE COMPARISON PROCEDURES
Fisher’s least signicant dierence (LSD) procedure:
lOMoARcPSD| 45903860
MULTIPLE COMPARISON PROCEDURES
Fisher’s least signicant dierence (LSD) procedure:
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8
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EXAMPLE
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10
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MULTIPLE COMPARISON PROCEDURES
- If the condence interval includes the value zero, we cannot reject the
hypothesis that the two populaon means are equal.
- If the condence interval does not include the value zero, we conclude that
there is a dierence between the populaon means.
lOMoARcPSD| 45903860
12
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cannot reject the hypothesis that the two populaon means of A and B are
equal
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14
PAIRWISE COMPARISONS
• In discussing mulple comparison procedures we refer to this probability of a Type I error (α
= 0.05) as the comparisonwise Type I error rate;
• Comparisonwise Type I error rates indicate the level of signicance associated with a single
pairwise comparison.
• What is the probability that in making three pairwise comparisons, we will commit a Type I
error on at least one of the three tests?
• The probability that we will not make a Type I error on any of the three tests is
(0.95)(0.95)(0.95) = 0.8574
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15
the probability of making at least one Type I error is 1 − 0.8574 = 0.1426 overall or
experimentwise Type i error rate ( tu@hcmiu.edu.vn
PAIRWISE COMPARISONS
•The experimentwise Type I error rate gets larger for problems with more
populaons.
should look for alternaves that provide beer control over the
experimentwise error rate
Bonferroni adjustment: involves using a smaller comparisonwise error rate for
each test:
lOMoARcPSD| 45903860
•If we want to test C pairwise comparisons and want the maximum probability
of making a Type I error for the overall experiment to be , we simply use a
comparisonwise error rate equal to /C
tu@hcmiu.edu.vn
7.6. ANOVA PROCEDURE – RANDOMIZED BLOCK DESIGN
•Randomized block design: Its purpose is to control some of the extraneous
sources of variaon by removing such variaon from the MSE
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17
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18
7.6. ANOVA PROCEDURE – RANDOMIZED BLOCK DESIGN
•The ANOVA procedure for the randomized block design requires us to paron
the sum of squares total (SST) into three groups: sum of squares due to
treatments (SSTR), sum of squares due to blocks (SSBL), and sum of squares due
to error (SSE):
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19
•ANOVA table for the randomized block design with k treatments and b blocks:
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20
7.6. ANOVA PROCEDURE
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lOMoAR cPSD| 45903860 APPLIED STATISTICS COURSE CODE: ENEE1006IU Lecture 13:
Chapter 7: Analysis of Variance (ANOVA)
(3 credits: 2 is for lecture, 1 is for lab-work) 1 lOMoAR cPSD| 45903860
CHAPTER 7: ANALYSIS OF VARIANCE (ANOVA)
•7.1. Inferences about a population variance
•7.2. Inferences about two population variances
•7.3. Assumptions for analysis of variance •7.4. A conceptual overview •7.5. ANOVA table •7.6. ANOVA procedure 7.5. ANOVA TABLE •Total Sum Square (TSS): 2 lOMoAR cPSD| 45903860 SSTR (sum of squares due to treatments) the degrees of freedom lOMoAR cPSD| 45903860 7.5. ANOVA TABLE •Total Sum Square (TSS):
•The analysis of variance can be viewed as the process of partitioning the total sum
of squares and the degrees of freedom into their corresponding sources: treatments and error.
•Dividing the sum of squares by the appropriate degrees of freedom provides the
variance estimates, the F value, and the p-value used to test the hypothesis of equal population means. •ANOVA TABLE: 4 lOMoAR cPSD| 45903860
MULTIPLE COMPARISON PROCEDURES
•When we use analysis of variance to test whether the means of k populations are
equal, rejection of the null hypothesis allows us to conclude only that the
population means are not all equal.
•In some cases we will want to go a step further and determine where the
differences among means occur.
•The purpose of this section is to show how multiple comparison procedures can
be used to conduct statistical comparisons between pairs of population means. lOMoAR cPSD| 45903860
Fisher’s least significant difference (LSD) procedure can be used to determine
where the differences occur (pairwise comparisons)
MULTIPLE COMPARISON PROCEDURES
Fisher’s least significant difference (LSD) procedure: 6 lOMoAR cPSD| 45903860
MULTIPLE COMPARISON PROCEDURES
Fisher’s least significant difference (LSD) procedure: lOMoAR cPSD| 45903860 8 lOMoAR cPSD| 45903860 EXAMPLE lOMoAR cPSD| 45903860 10 lOMoAR cPSD| 45903860
MULTIPLE COMPARISON PROCEDURES
- If the confidence interval includes the value zero, we cannot reject the
hypothesis that the two population means are equal.
- If the confidence interval does not include the value zero, we conclude that
there is a difference between the population means. lOMoAR cPSD| 45903860 12 lOMoAR cPSD| 45903860
cannot reject the hypothesis that the two population means of A and B are equal lOMoAR cPSD| 45903860 PAIRWISE COMPARISONS
• In discussing multiple comparison procedures we refer to this probability of a Type I error (α
= 0.05) as the comparisonwise Type I error rate;
• Comparisonwise Type I error rates indicate the level of significance associated with a single pairwise comparison.
• What is the probability that in making three pairwise comparisons, we will commit a Type I
error on at least one of the three tests?
• The probability that we will not make a Type I error on any of the three tests is (0.95)(0.95)(0.95) = 0.8574 14 lOMoAR cPSD| 45903860
the probability of making at least one Type I error is 1 − 0.8574 = 0.1426 overall or
experimentwise Type i error rate ( tttu@hcmiu.edu.vn PAIRWISE COMPARISONS
•The experimentwise Type I error rate gets larger for problems with more populations.
should look for alternatives that provide better control over the experimentwise error rate
Bonferroni adjustment: involves using a smaller comparisonwise error rate for each test: 15 lOMoAR cPSD| 45903860
•If we want to test C pairwise comparisons and want the maximum probability
of making a Type I error for the overall experiment to be , we simply use a
comparisonwise error rate equal to /C tttu@hcmiu.edu.vn
7.6. ANOVA PROCEDURE – RANDOMIZED BLOCK DESIGN
•Randomized block design: Its purpose is to control some of the extraneous
sources of variation by removing such variation from the MSE lOMoAR cPSD| 45903860 17 lOMoAR cPSD| 45903860
7.6. ANOVA PROCEDURE – RANDOMIZED BLOCK DESIGN
•The ANOVA procedure for the randomized block design requires us to partition
the sum of squares total (SST) into three groups: sum of squares due to
treatments (SSTR), sum of squares due to blocks (SSBL), and sum of squares due to error (SSE): 18 lOMoAR cPSD| 45903860
•ANOVA table for the randomized block design with k treatments and b blocks: 19 lOMoAR cPSD| 45903860 7.6. ANOVA PROCEDURE 20