统计学课件方差分析与实验设计
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Factor . . . Gasoline blend Treatments . . . Blend X, Blend Y, Blend Z Blocks . . . Automobiles Response variable . . . Miles per gallon
9
Case 3: Crescent Oil Co.
8
Case 3: Crescent Oil Co.
Randomized Block Design Five automobiles have been
tested using each of the three gasoline blends and the miles per gallon ratings are shown on the next slide.
The term factorial is used because the experimental conditions include all possible combinations of the factors.
法效果大，以某牌子的巧克力做一实验，共有4种处理：
• A：在进口处摆设该巧克力的广告牌 • B：按原价减价 5% • C：送增券 • D：油印广告，放在进口处由购买者自取
该公司决定以三个区域的超市作为实验单位，实验期 为四个星期。至于何种促销方法在某区域何超市采 用，则由随机抽样方法决定。
应考虑销售区域在消费倾向方面的差异。
x1
x 2
┅
x k
Block mean
xi x1 x2
xb x
4
ANOVA Procedure：
Randomized Block Design
ANOVA procedure
• to partition sum of squares total (SST) into three groups: sum of squares due to treatments (SSTR), sum of squares due to blocks (SSBL) 区组平方和, and sum of squares due to error (SSE).
Blocking: to form homogeneous groups from heterogeneous experiment units.
2
Illustration:
Randomized Block Design
两个因子：促销方法 / 地区消费倾向 某经营超级市场的集团公司，欲了解何种销售促销方
Test Statistic F = MSTR/MSE = 2.6/.68 = 3.82 Conclusion
p-value Approach : The p-value (.07) is larger than .05 where F = 4.46
Critical Value Approach : The test statistic (3.82) is smaller than F.05 = 4.46.
Outline
Introduction to Experimental Design ANOVA & Completely Randomized Design
• Multiple Comparison Randomized Block Design Factorial Experiment
i1 j1
Sum of squares due to treatments
处理平方和
k
SSTR b (x.j x )2 j1
Sum of squares due to blocks
区组平方和
b
SSBL k (xi. x)2 i1
Sum of squares due to error
Randomized Block Design Rejection Rule
p-Value Approach: Reject H0 if p-value < .05
Critical Value Approach: Reject H0 if F > 4.46
For = .05, F.05 (2, 8) = 4.46
4
33
31
29
31.000
5
26
25
26
25.667
Treatment Means
29.8
b=5
28.8
28.4 29.000
10
Case 3: Crescent Oil Co.
Randomized Block Design Mean Square Due to Treatments
The overall sample mean is 29. Thus, SSTR = 5[(29.8 - 29)2 + (28.8 - 29)2 + (28.4 - 29)2] = 5.2
测量两个无交互作用的自变量与因变量之间的关系强度
R2
联合效应 总效应
SSTR(处理平方和) SSB(区组平方和) SST (总平方和)
1
SSE(误差平方和) SST (总平方和)
• In Case 3, R2 = (SSTR+SSBL)/SST = (5.2+51.33)/62=0.91.
Source of Sum of Degrees of Mean Variation Squares Freedom Square
F p-Value
Treatments Blocks Error
5.20 51.33
5.47
2
2.60 3.82 .07
4
12.80
8
.68
Total
62.00
14
12
Case 3: Crescent Oil Co.
Randomized Block Design
Automobile Type of Gasoline (Treatment) Block (Block) Blend X Blend Y Blend Z Means k = 3
1
31
30
30
30.333
2
30
29
29
29.333
3
29
29
28
28.667
• Factors of gasoline blends and automobiles explain 91 percentof the
total variation in miles per gallon.
14
Outline
Introduction to Experimental Design ANOVA & Completely Randomized Design
1
Motivation:
Randomized Block Design
F = MSTR/MSE
= (SSTR/(k-1))/(SSE/nT – k) ~ F(k – 1, nT - k)
Differences due to extraneous factors (such as heterogeneous experiment units) cause MSE to be large, and F test will make a Type Ⅱ error.
new blends of gasoline and must decide which blend or blends to produce and distribute. A study of the miles per gallon ratings of the three blends is being conducted to determine if the mean ratings are the same for the three blends.
• nT (total number) = k (number of treatments) × b (number of blocks)
5
ANOVA Procedure：
Randomized Block Design
Total sum of squares 总平方和
bk
SST
(xij x )2
数据间的差异可能不只受到一个因子的影响。
In this design, one needs to control some of these extraneous sources of variation by removing such variation from the MSE term.
k-1
MSTR SSTR MSTR
k - 1 MSE
Blocks
SSBL
b - 1 MSBL SSBL b-1
Error
SSE
(k – 1)(b – 1)
MSE
(k
SSE 1)(b
1)
Total
SST
nT - 1
nT = kb
7
Case 3: Crescent Oil Co.
Randomized Block Design Crescent Oil has developed three
SST = [(30 - 29)2 + . . . + (26 - 29)2] = 62
SSE = 62 - 5.2 - 51.33 = 5.47
MSE = 5.47/[(3 - 1)(5 - 1)] = .68
11
Case 3: Crescent Oil Co.
Randomized Block Design ANOVA Table
Formula for this partitioning
SST = SSTR + SSBL + SSE
• Total d.f., nT - 1, are partitioned such that k - 1 d.f. go to treatments, b - 1 go to blocks, and (k - 1)(b - 1) go to the error term.
Therefore, we cannot reject H0. There is insufficient evidence to conclude that miles per gallon13 ratings differ for the three gasoline blends.
How Strong is the Relationship?
MSTR = 5.2/(3 - 1) = 2.6 Mean Square Due to Blocks
SSBL = 3[(30.333 - 29)2 + . . . + (25.667 - 29)2] = 51.33 MSBL = 51.33/(5 - 1) = 12.8
Mean Square Due to Error
Factorial Experiment
In some experiments we want to draw conclusions about more than one categorical variable or factor.
Factorial experiment and it corresponding ANOVA computations are valuable designs in this case.
• Multiple Comparison Randomized Block Design Factorial Experiment
15
Illustration:
Factorial Experiment
16
Illustration:
Factorial Experiment
17
Motivation:
误差平方和
SSE SST SSTR SSBL
6
ANOVA Table：
Randomized Block Design
Source of Sum of Degrees of Variation Squares Freedom
Mean Square
pF Value
Treatments SSTR
3
DataBaidu NhomakorabeaStructure:
Randomized Block Design
1
Blocks (i)
2
Row Factors
┇
b
Treatment mean x j
Treatments (j) Column Factors
1
2
┅
k
x11
x12
┅
x1k
x21
x22
┅
x2k
┇
┇
┇
┇
xb1
xb2
┅
xbk
9
Case 3: Crescent Oil Co.
8
Case 3: Crescent Oil Co.
Randomized Block Design Five automobiles have been
tested using each of the three gasoline blends and the miles per gallon ratings are shown on the next slide.
The term factorial is used because the experimental conditions include all possible combinations of the factors.
法效果大，以某牌子的巧克力做一实验，共有4种处理：
• A：在进口处摆设该巧克力的广告牌 • B：按原价减价 5% • C：送增券 • D：油印广告，放在进口处由购买者自取
该公司决定以三个区域的超市作为实验单位，实验期 为四个星期。至于何种促销方法在某区域何超市采 用，则由随机抽样方法决定。
应考虑销售区域在消费倾向方面的差异。
x1
x 2
┅
x k
Block mean
xi x1 x2
xb x
4
ANOVA Procedure：
Randomized Block Design
ANOVA procedure
• to partition sum of squares total (SST) into three groups: sum of squares due to treatments (SSTR), sum of squares due to blocks (SSBL) 区组平方和, and sum of squares due to error (SSE).
Blocking: to form homogeneous groups from heterogeneous experiment units.
2
Illustration:
Randomized Block Design
两个因子：促销方法 / 地区消费倾向 某经营超级市场的集团公司，欲了解何种销售促销方
Test Statistic F = MSTR/MSE = 2.6/.68 = 3.82 Conclusion
p-value Approach : The p-value (.07) is larger than .05 where F = 4.46
Critical Value Approach : The test statistic (3.82) is smaller than F.05 = 4.46.
Outline
Introduction to Experimental Design ANOVA & Completely Randomized Design
• Multiple Comparison Randomized Block Design Factorial Experiment
i1 j1
Sum of squares due to treatments
处理平方和
k
SSTR b (x.j x )2 j1
Sum of squares due to blocks
区组平方和
b
SSBL k (xi. x)2 i1
Sum of squares due to error
Randomized Block Design Rejection Rule
p-Value Approach: Reject H0 if p-value < .05
Critical Value Approach: Reject H0 if F > 4.46
For = .05, F.05 (2, 8) = 4.46
4
33
31
29
31.000
5
26
25
26
25.667
Treatment Means
29.8
b=5
28.8
28.4 29.000
10
Case 3: Crescent Oil Co.
Randomized Block Design Mean Square Due to Treatments
The overall sample mean is 29. Thus, SSTR = 5[(29.8 - 29)2 + (28.8 - 29)2 + (28.4 - 29)2] = 5.2
测量两个无交互作用的自变量与因变量之间的关系强度
R2
联合效应 总效应
SSTR(处理平方和) SSB(区组平方和) SST (总平方和)
1
SSE(误差平方和) SST (总平方和)
• In Case 3, R2 = (SSTR+SSBL)/SST = (5.2+51.33)/62=0.91.
Source of Sum of Degrees of Mean Variation Squares Freedom Square
F p-Value
Treatments Blocks Error
5.20 51.33
5.47
2
2.60 3.82 .07
4
12.80
8
.68
Total
62.00
14
12
Case 3: Crescent Oil Co.
Randomized Block Design
Automobile Type of Gasoline (Treatment) Block (Block) Blend X Blend Y Blend Z Means k = 3
1
31
30
30
30.333
2
30
29
29
29.333
3
29
29
28
28.667
• Factors of gasoline blends and automobiles explain 91 percentof the
total variation in miles per gallon.
14
Outline
Introduction to Experimental Design ANOVA & Completely Randomized Design
1
Motivation:
Randomized Block Design
F = MSTR/MSE
= (SSTR/(k-1))/(SSE/nT – k) ~ F(k – 1, nT - k)
Differences due to extraneous factors (such as heterogeneous experiment units) cause MSE to be large, and F test will make a Type Ⅱ error.
new blends of gasoline and must decide which blend or blends to produce and distribute. A study of the miles per gallon ratings of the three blends is being conducted to determine if the mean ratings are the same for the three blends.
• nT (total number) = k (number of treatments) × b (number of blocks)
5
ANOVA Procedure：
Randomized Block Design
Total sum of squares 总平方和
bk
SST
(xij x )2
数据间的差异可能不只受到一个因子的影响。
In this design, one needs to control some of these extraneous sources of variation by removing such variation from the MSE term.
k-1
MSTR SSTR MSTR
k - 1 MSE
Blocks
SSBL
b - 1 MSBL SSBL b-1
Error
SSE
(k – 1)(b – 1)
MSE
(k
SSE 1)(b
1)
Total
SST
nT - 1
nT = kb
7
Case 3: Crescent Oil Co.
Randomized Block Design Crescent Oil has developed three
SST = [(30 - 29)2 + . . . + (26 - 29)2] = 62
SSE = 62 - 5.2 - 51.33 = 5.47
MSE = 5.47/[(3 - 1)(5 - 1)] = .68
11
Case 3: Crescent Oil Co.
Randomized Block Design ANOVA Table
Formula for this partitioning
SST = SSTR + SSBL + SSE
• Total d.f., nT - 1, are partitioned such that k - 1 d.f. go to treatments, b - 1 go to blocks, and (k - 1)(b - 1) go to the error term.
Therefore, we cannot reject H0. There is insufficient evidence to conclude that miles per gallon13 ratings differ for the three gasoline blends.
How Strong is the Relationship?
MSTR = 5.2/(3 - 1) = 2.6 Mean Square Due to Blocks
SSBL = 3[(30.333 - 29)2 + . . . + (25.667 - 29)2] = 51.33 MSBL = 51.33/(5 - 1) = 12.8
Mean Square Due to Error
Factorial Experiment
In some experiments we want to draw conclusions about more than one categorical variable or factor.
Factorial experiment and it corresponding ANOVA computations are valuable designs in this case.
• Multiple Comparison Randomized Block Design Factorial Experiment
15
Illustration:
Factorial Experiment
16
Illustration:
Factorial Experiment
17
Motivation:
误差平方和
SSE SST SSTR SSBL
6
ANOVA Table：
Randomized Block Design
Source of Sum of Degrees of Variation Squares Freedom
Mean Square
pF Value
Treatments SSTR
3
DataBaidu NhomakorabeaStructure:
Randomized Block Design
1
Blocks (i)
2
Row Factors
┇
b
Treatment mean x j
Treatments (j) Column Factors
1
2
┅
k
x11
x12
┅
x1k
x21
x22
┅
x2k
┇
┇
┇
┇
xb1
xb2
┅
xbk