第八章 单因素方差分析
– 其中εij是随机变量,且服从正态分布 – 固定效应(fixed effect)是由固定因素(fixed factor) 所引起的效应 – 随机效应(random effect)是由随机因素(random factor)所引起的效应 – 固定因素和随机因素区别:固定因素的水平可以严格 地人为控制,在水平确定之后,它的效应也是固定的, 是可以重复的 ;固定效应通常会被估计
作业讲解
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• 7.16 用杂合基因型Wvwv的小 鼠为父本,与纯合基因型wvwv 的小鼠为母本杂交(wv:波浪 毛,Wv:正常直毛)。其后代 得到wvwv的数量与Wvwv的数 量应该各占一半(φ = 1 – φ = 0.5)。实验只选择每窝8只的 ,多于8只的和少于8只的都被 淘汰。实验结果是否符合二项 分布?
个体号 1 2 3 4
窝号 I 34.7 33.3 26.2 31.6 31.450a 3.7225 II 33.2 26.0 28.6 32.3 30.025ab 3.3410 III 27.1 23.3 27.8 26.7 26.225b 2.0023 IV 32.9 31.4 25.7 28.0 29.500ab 3.2588
不同处理效应与不同模型
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• 线性统计模型(linear statistical model)
–
x ij = µ + α i + ε ij
i = 1, 2, , a j = 1, 2, , n
i =1 k =1
a
n
1 yij . = yij . n
a b
1 yi.. = yi.. bn
n
y... = ∑∑∑ yijk
i =1 j =1 k =1
1 y. j . = y. j . an 1 y... = y... abn
方差分析的直观理解
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• 对平均数的检验
– 检验2个平均数的差是否可以用于随机误差解释 – 检验几个样本平均数的方差是否足够大
• 组间方差和组内方差
– 组间方差:样本平均数产的方差
1 a 2 (xi . − x.. ) ∑ a − 1 i =1
a n 1 2 x − x ( ) – 组内方差:样本内的方差 ∑∑ ij i. a (n − 1) i =1 j =1
正常直毛 观察值 0 1 2 3 4 5 6 7 8 0 1 2 4 12 6 5 2 0
作业讲解
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正常 直毛
合并 观察值 观察值
理论 频率
理论值
合并 理论值
χ2
0 1 2 3 4 5 6 7 8 总计
0 1 2 4 12 6 5 2 0 32
7 12 13 32
0.0039 0.0313 0.1094 0.2188 0.2734 0.2188 0.1094 0.0313 0.0039 1
0.125 1.000 3.500 7.000 8.750 7.000 3.500 1.000 0.125 32
固定效应模型
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• 线性统计模型
– 在固定效应模型中,处理平均数与总体平均数 n 的离差,是一个常量,因而 ∑ α i = 0
i =1
– 假设: H 0 : α 1 = α 2 = = α a = 0 ,H A : α i ≠ 0
两样本平均数差异性检验
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t
多样本平均数差异t检验
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• 可以在平均数的所有对之间做t检验
品系 III 67.8 66.3 67.1 66.8 68.5 67.3b 0.8631
IV 71.8 72.1 70.0 69.1 71.0 70.8c 1.2510
V 69.2 68.2 69.8 68.3 67.5 68.6d 0.9028
四窝动物的出生重(g)
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– 每个实验都只有一个因素(factor),该因素有a个处理 (treatment)或水平(level)
简略的表示方法
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yi. = ∑ yij
j =1 a
n
பைடு நூலகம்
y.. = ∑∑ yij
i =1 j =1
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• 同时判断多样本平均数的差异显著性 • 原理
– 若各样本平均数相等,则平均数的方差等于0 – 若各样本平均数存在差异,则平均数的方差大 于0;平均数的方差越大,各样本平均数存在 的差异也就越大 – 当这种差异大于随机性时,则认为样本平均数 之间存在差异
SAS程序
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data a; input a $ b $ count @@; cards; A1 B1 8 A1 B2 3 A1 B3 0 A2 B1 13 A2 B2 8 A2 B3 3 run; proc freq data=a; weight count; table a*b/exact; run;
SAS 运行结果
Yuanmei Guo Yuanmei Guo Yuanmei Guo Yuanmei Guo Yuanmei Guo Statistics for Table of a by b Statistic DF Value Prob -----------------------------------------------------------Chi-Square 2 1.8962 0.3875 Likelihood Ratio Chi-Square 2 2.7729 0.2500 Mantel-Haenszel Chi-Square 1 1.6782 0.1952 Phi Coefficient 0.2328 Contingency Coefficient 0.2267 Cramer's V 0.2328 WARNING: 50% of the cells have expected counts less than 5. Chi-Square may not be a valid test. Sample Size = 35
5个小麦品系株高(cm)调查结果
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株号 1 2 3 4 5
x
S
I 64.6 65.3 64.8 66.0 65.8 65.3a 0.6083
II 64.5 65.3 64.6 63.7 63.9 64.4a 0.6325
作业讲解
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• 7.11 拉菲和舒巴酮是两种治疗呼吸系统和泌 尿系统感染的药物,下表给出了这两种药物治 疗淋菌性尿道炎的结果。推断这两种药物治 疗淋菌性尿道炎的疗效差异是否显著? 药物 拉菲 舒巴酮 人数 显效 3 8
2 i =1 j =1 a n
a
n
= ∑∑ ( xij − xi . ) + 2∑∑ ( xij − xi . )( xi . − x⋅⋅ ) + ∑∑ ( xi . − x.. ) 2
n
1 yi . = yi . n 1 y.. y.. = an
简略的表示方法
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yij . = ∑ yijk
k =1
n
yi.. = ∑∑ yijk
j =1 k =1
b
n
y. j . = ∑∑ yijk
SAS 运行结果
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Fisher's Exact Test ----------------------------------------Table Probability (P) 0.0805 Pr <= P 0.6413 Sample Size = 35
SAS程序
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data a; input a $ b $ count @@; cards; A1 B1 8 A1 B2 3 A1 B3 0 A2 B1 13 A2 B2 8 A2 B3 3 run; proc freq data=a; weight count; table a*b/chisq; run;