Lecture 12: Chapter 7: Analysis of Variance (ANOVA) | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE;completely randomized design;randomized block design;factorial experiment;Three assumptions are required to use analysis of variance: For each population, the response variable is normally distributed. The variance of the response variable, denoted σ2, is the same for all of the populations. The observations must be independent. 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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Profile Nhi Nhi
lOMoARcPSD| 45903860
APPLIED STATISTICS
COURSE CODE: ENEE1006IU
Lecture 12:
Chapter 7: Analysis of Variance (ANOVA)
(3 credits: 2 is for lecture, 1 is for lab-work)
1
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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
7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE
•completely randomized design
•randomized block design
•factorial experiment
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3
Three assumpons are required to use analysis of variance:
-For each populaon, the response variable is normally distributed
-The variance of the response variable, denoted σ
2
, is the same for all of the
populaons
-The observaons must be independent
7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS
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OF VARIANCE
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7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS
OF VARIANCE
7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE
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- If H
0
is true:
(between-treatments)
-If H
0
is false: between-treatments esmate of σ
2
will be overstated
(pooled esmate)
If the null hypothesis is true, the two esmates will be similar and their rao will
be close to 1.
If the null hypothesis is false, the between- treatments esmate will be larger than
the within-treatments esmate, and their rao will be large.
•By comparing these two esmates of σ
2
, we will be able to determine whether the
populaon means are equal. ANOVA
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7.4. A CONCEPTUAL OVERVIEW
•ANOVA and the Completely Randomized Design
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7.4. A CONCEPTUAL OVERVIEW
•Between-Treatments Esmate of Populaon Variance
- In between-treatments, the esmate of σ
2
is called the mean square due to
treatments (MSTR):SSTR (sum of squares
due to treatments)
If H
0
is true, MSTR provides an unbiased esmate of σ
2
however, if the means of the k populaons are not equal,
MSTR is not an unbiased esmate of σ
2
; in fact, in that case,
MSTR should overesmate σ
2
.
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9
7.4. A CONCEPTUAL OVERVIEW
•Within-Treatments Esmate of Populaon Variance
- In within-treatments, the esmate of σ
2
is called the mean square due to
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10
7.4. A CONCEPTUAL OVERVIEW
•Comparing the Variance Esmates: The F Test
•If the null hypothesis is true, MSTR and MSE provide two independent, unbiased
esmates of σ
2
•If the null hypothesis is false, the value of MSTR/MSE will be inated because
MSTR overesmates σ
2
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7.4. A CONCEPTUAL OVERVIEW
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7.4. A CONCEPTUAL OVERVIEW
On the other hand, when the null hypothesis is false, then MSTR will tend to be
larger than MSE.
So the rao of MSTR and MSE can be used as an indicator of the equality or
inequality of the r populaon means.
This rao (MSTR/MSE) will tend to be near to 1 if the null hypothesis is true, and
greater than 1 if the null hypothesis is false.
The ANOVA test is a test of whether (MSTR/MSE) is equal to, or greater than, 1.
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EXAMPLE
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EXAMPLE
REVIEW HOMEWORK – WEEK13
•Group 6 did not submit
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•All groups met the requirements, although there are dierences in the answers
Group
Link
1
link
2
link
3
link
5
link
6
7
link
| 1/17
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lOMoAR cPSD| 45903860 APPLIED STATISTICS COURSE CODE: ENEE1006IU Lecture 12:
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
7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE
•completely randomized design •randomized block design •factorial experiment 2 lOMoAR cPSD| 45903860
Three assumptions are required to use analysis of variance:
-For each population, the response variable is normally distributed
-The variance of the response variable, denoted σ2, is the same for all of the populations
-The observations must be independent
7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS 3 lOMoAR cPSD| 45903860 OF VARIANCE 4 lOMoAR cPSD| 45903860
7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE
7.3. AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE 5 lOMoAR cPSD| 45903860 - If H0 is true: (between-treatments)
-If H0 is false: between-treatments estimate of σ2 will be overstated (pooled estimate)
If the null hypothesis is true, the two estimates will be similar and their ratio will be close to 1.
If the null hypothesis is false, the between- treatments estimate will be larger than
the within-treatments estimate, and their ratio will be large.
•By comparing these two estimates of σ2, we will be able to determine whether the
population means are equal. ANOVA 6 lOMoAR cPSD| 45903860 7.4. A CONCEPTUAL OVERVIEW
•ANOVA and the Completely Randomized Design 7 lOMoAR cPSD| 45903860 7.4. A CONCEPTUAL OVERVIEW
•Between-Treatments Estimate of Population Variance
- In between-treatments, the estimate of σ2 is called the mean square due to
treatments (MSTR):SSTR (sum of squares due to treatments)
If H0 is true, MSTR provides an unbiased estimate of σ2
however, if the means of the k populations are not equal,
MSTR is not an unbiased estimate of σ2 ; in fact, in that case,
MSTR should overestimate σ2 . 8 lOMoAR cPSD| 45903860 7.4. A CONCEPTUAL OVERVIEW
•Within-Treatments Estimate of Population Variance
- In within-treatments, the estimate of σ2 is called the mean square due to 9 lOMoAR cPSD| 45903860 7.4. A CONCEPTUAL OVERVIEW
•Comparing the Variance Estimates: The F Test
•If the null hypothesis is true, MSTR and MSE provide two independent, unbiased estimates of σ2
•If the null hypothesis is false, the value of MSTR/MSE will be inflated because MSTR overestimates σ2 10 lOMoAR cPSD| 45903860 7.4. A CONCEPTUAL OVERVIEW 11 lOMoAR cPSD| 45903860 7.4. A CONCEPTUAL OVERVIEW
On the other hand, when the null hypothesis is false, then MSTR will tend to be larger than MSE.
So the ratio of MSTR and MSE can be used as an indicator of the equality or
inequality of the r population means.
This ratio (MSTR/MSE) will tend to be near to 1 if the null hypothesis is true, and
greater than 1 if the null hypothesis is false.
The ANOVA test is a test of whether (MSTR/MSE) is equal to, or greater than, 1. 12 lOMoAR cPSD| 45903860 EXAMPLE 13 lOMoAR cPSD| 45903860 14 lOMoAR cPSD| 45903860 15 lOMoAR cPSD| 45903860 EXAMPLE REVIEW HOMEWORK – WEEK13 •Group 6 did not submit 16 lOMoAR cPSD| 45903860
•All groups met the requirements, although there are differences in the answers Group Link 1 link 2 link 3 link 5 link 6 7 link 17