什么是协变量

http://hi.baidu.com/flykite083/blog/item/920f56636e803b640c33facb.html

http://www.cnblogs.com/ysjxw/archive/2008/05/28/1209033.html

I've found a good passage talking about covariate:

Covariate:

In design of experiments, a covariate is an independent variable not manipulated
by the experimenter but still affecting the response. See Variables (in design
of experiments) for an explanatory example.

协变量:在实验的设计中,协变量是一个独立变量(解释变量),不为实验者所操纵,但仍影响响应。

Variables (in design of experiments):

Many statistical methods rest on a statistical model which states a relationship

Y = f(X₁,..,X_N)

between a dependent variable (Y) and independent variable(s) X₁,...,X_N. In designed experiments, the dependent variable is often named "response", independent variables manipulated by the experimenter "factors", independent variables not manipulated by the experimenter (but still affecting the response) "covariates". The values which a factor can take on are named "levels" of this factor. Tested combinations of levels of all factors are called "treatments".

许多统计方法基于一个统计模型

Y = f(X₁,..,X_N)

描述依存变量（被解释变量）Y 和独立变量（解释变量） X₁,...,X_N 之间的关系。在设计的实验中，依存变量（被解释变量）经常被称为“反应”，被实验者所操纵的独立变量（解释变量）被称为“影响因素”，

不被实验者所操纵（但仍然影响“反应”）的独立变量（解释变量）被称为“协变量”，一个影响因素所取的值称为该因素的“水平”。用于检验的所有影响因素的水平的不同组合被称为“方案”。

Consider an example of a clinical trial of drugs. The question addressed by the trial is how combinations of two drugs affect the survival rate. Two drugs have been tested: "Drug 1" and "Drug 2". Drug 1 has been administered in three ways - "None", "Orally (pills)", "Injection"; drug 2 has been administered in a single way, but in 4 different doses "None", "Low", "Moderate", "High". All the 12 (3x4) possible combinations of administration of the two drugs have been tested. The following model is used to interpret outcomes of the trial:

Y = T_ij + B X + E ;    i=1,...,3;    j=1,...,4;

where Y is the survival rate, indices i and j correspond to the methods of administration of the drug 1 and drug 2 respectively, X is age, and E is random variation in survival rate. Coefficients T_ij, which characterize the effect of the two drugs on the survival rate, are of primary interest.

考虑一个药品临床试验的例子，试验提出的问题是，两种药品的不同组合如何影响救活率。试验的两种药品分别是 “药品1”、“药品2”。药品1的使用方式有3种：“不用”、“口服”、“注射”；药品2的使用方式只有1种，但有4种不同的剂量：“不用”、“低剂量”、“中剂量”、“高剂量”。2种药品在试验中的方案有12种（3*4）可能的组合。试验的结果用模型表示为：

Y = T_ij + B X + E ;    i=1,...,3;    j=1,...,4;

 

The response here is the survival rate. We have two factors - "drug 1" and "drug 2". The first factor has three levels ("none", "orally", "injection"), the second factor - four levels ("none", "low dose", "moderate", "high"). (Note: levels are not necessarily doses, as for factor 2; for factor 1, for example, levels are related to methods of administration). We have here 12 treatments being tested - all the possible combinations of the three levels of factor 1 and the four levels of factor 4 (specified above).

系数T_ij 是相应治疗方式的成效。举个例子，T₂₃ 是药物1口服和药物2中等剂量治疗方案的治疗效果。年龄在这里作为协变量而存在，它虽然不被实验者所控制，但确实会对存活率产生影响。

Coefficients T_ij are "effects" of the corresponding treatments. For example, T₂₃ is the effect of the treatment by drug 1 orally plus drug 2 in moderate dose. Age here is a covariate - it is not manipulated by the experimenter but still may have effect on the survival rate.