统计学课件--方差分析与实验设计2
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Factorial Experiment
Illustration:
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Factorial Experiment
Illustration:
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Factorial Experiment
Motivation:
In some experiments we want to draw conclusions about more than one categorical variable or factor.
Outline
Introduction to Experimental Design ANOVA & Completely Randomized Design • Multiple Comparison Randomized Block Design Factorial Experiment
Source of Variation
Sum of Degrees of Squares Freedom k-1
Mean Square
F
Treatments SSTR Blocks Error Total
SSTR MSTR MSTR k-1 MSE SSBL b-1 SSBL MSBL b -1 SSE SSE (k – 1)(b – 1) MSE ( k 1)(b 1)
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Randomized Block Design
ANOVA Procedure：
Total sum of squares 总平方和 Sum of squares due to treatments 处理平方和 Sum of squares due to blocks 区组平方和 Sum of squares due to error 误差平方和
Formula for this partitioning SST = SSTR + SSBL + SSE
•
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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. nT (total number) = k (number of treatments) × b (number of blocks)
SST nT = kb
nT - 1
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Case 3: Crescent Oil Co.
Randomized Block Design
Crescent Oil has developed three 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.
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。
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Data Structure:
Randomized Block Design
Treatments (j) Column Factors 1 1 Blocks (i) Row Factors 2 ┇ b Treatment mean x j 2 ┅ ┅ ┅ ┇ ┅ ┅ k Block mean
SST ( xij x )2
i 1 j 1
b
k
SSTR b ( x. j x )2
j 1
k
SSBL k ( xi . x )2
i 1
b
SSE SST SSTR SSBL
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Randomized Block Design
pValue
ANOVA Table：
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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. 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. Blocking: to form homogeneous groups from heterogeneous experiment units.
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Automobile (Block) 1 2 3 4 5 Treatment Means b=5
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 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 SST = [(30 - 29)2 + . . . + (26 - 29)2] = 62 SSE = 62 - 5.2 - 51.33 = 5.47 MSE = 5.47/[(3 - 1)(5 - 1)] = .68
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Case 3: Crescent Oil Co.
Randomized Block Design Type of Gasoline (Treatment) Blend X 31 30 29 33 26 29.8 Blend Y 30 29 29 31 25 28.8 Blend Z 30 29 28 29 26 28.4 Block Means k = 3 30.333 29.333 28.667 31.000 25.667 29.000
Factorial experiment and it corresponding ANOVA computations are valuable designs in this case.
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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. Factor . . . Gasoline blend Treatments . . . Blend X, Blend Y, Blend Z Blocks . . . Automobiles Response variable . . . Miles per gallon
Source of Variation Treatments Blocks Error Total
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Case 3: Crescent Oil Co.
Randomized Block Design Rejection Rule
p-Value Approach: Reject H0 if p-value < .05 Reject H0 if F > 4.46 Critical Value Approach:
How Strong is the Relationship?
测量两个无交互作用的自变量与因变量之间的关系强度
联合效应 R 总效应 SSTR(处理平方和) SSB(区组平方和) SST (总平方和)
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SSE (误差平方和) 1 SST (总平方和)
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In Case 3, R2 = (SSTR+SSBL)/SST = (5.2+51.33)/62=0.91. • Factors of gasoline blends and automobiles explain 91 percentof the
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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 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
For = .05, F.05 (2, 8) = 4.46
