Chapter 1 Introduction

1Chapter 1 Introduction We are going to talk about: z Review of the basic statistics

z What is survival analysis?

z What is survival data?

z Why use survival analysis?

z Approaches to survival analysis

z What you need to know?

1. Review of basic statistics 1）变量分类： ) 状态变量：年龄、性别、婚姻状况 --- 静态和动态变量 ) 时间变量：年、月、日、小时、分钟、秒 ) 空间变量：公里、米

变量类型： ¾ Qualitative and Quantitative Data 9 Measurement includes two somewhat different things – assignment by category and assignment by amount. 9 When we assign to categories that do not imply amounts, we often refer to the data as qualitative or categorical data (discrete). They differ by attribute not amount. 9 The interval and ratio levels of measurement are concerned with amounts, which are quantitative data (continuous).

The kind of measurement selected will fall into one of four general categories known as levels of measurement or scales: nominal, ordinal, interval, or ratio.

There are two kinds of ordinal data: individual ordinal and grouped ordinal data.

2）统计分布 9 统计分布：指某一类单位数占总体单位数的比例，如年龄分布、性别分布 9 标准分布：具有特定规律的统计分布 2

9 离散变量的概率分布：二项分布、超几何分布、普阿松分布、几何分布、负二项分布等 9 连续变量的概率分布：正态分布、指数分布、伽马分布、对数正态分布、威布尔分布等

3）回归分析 Regression analysis is "a statistical methodology that utilizes the relation between two or more variables so that one variable [the dependent, outcome, or response variable] can be predicted from the other, or others [the independent, explanatory, or predictor variables]."

We show the multiple linear regression function as follows: 011221,1......

(1,2,......,)iiipipiYXXXinββββε−−=+++++

= Where:

011 12,12i

,,......,are parameters,,......, are know constants are independent N(0, )1,2,......,

piiipXXXinβββεσ−−

= z 字母的名称和代表的内容是设定的（Yi是因变量，Xi是自变量，它们是已知的；β0 和 β1 是回归系数——截距和斜率，它们是未知的；εi 是误差项）。

z 对字母之间的关系和某些规定性是假定的（线性关系；Yi是随机变量，Xi是常量；β0 和 β1 是常量；εi是变量，而且 E(ε

i) = 0，

Var(εi) = σ2 ——未知的，Cov(εi, εj) = 0）。

由于在等式的左边和右边各有一个随机变量Yi 和 εi 而β0 + β1Xi 是

常量，所以对εi 的假设和对Yi 的假设是有确定的对应关系的。比如：

z 由于 E(εi) = 0，所以 E（Y

i/Xi）= E(β0 + β1Xi )+E(εi)= β0 + β1Xi

z 由于Var(εi) = σ2 ，所以Var(Yi)=Var(β0 + β1Xi)+Var(εi) = 0 + σ

2

=σ2 。这样 Var(Yi)= Var(εi) = σ2。

z 由于Cov(εi, εj) = 0，所以 Cov(Yi, Yj) = Cov(β

0 + β1Xi , β0 + β1Xj ) +

Cov(εi, εj) = 0 + 0 = 0。这样 Cov(Yi, Yj) = Cov(εi, εj) = 0

Motivation for Multiple Regression Analysis is to support a causal theory by eliminating potential sources of spuriousness. 3

Example: Surgical Unit: Predicting survival in patients undergoing a particular type of liver operation. A random selection of 108 patients was available for analysis. From each patient record, the following information was extracted from the preoperation evaluation:

X1: blood clotting （粘稠） score X2: Prognostic （预后）index X3: enzyme （酶）function test score X4: liver function test score X5: age, in years X6: indicator variable for gender (0=male, 1=female) X7 and X8: indicator variables for history of alcohol use:

Alcohol useX7 X8 None Moderate Severe 0 1 0 0 0 1

The response variable Y is survival time, which was ascertained in a follow-up study. We use only first 54 cases of the 108 patients, and first four explanatory variables. (See data: Surgi-Example).

4) Data for regression analysis Observational data: they are the data from uncontrolled studies, that the explanatory or predictor variables of interest are not controlled. For example, company officials wished to study the relation between age of employee(X) and number of days of illness last year (Y). The needed data for use in regression analysis were obtained from personnel records. Such data are observational data since the explanatory variable, age, is not controlled.

Experimental data: the explanatory variables should be controlled. For example, an insurance company that wishes to study the relation between productivity of its analysts in processing claims and length of training. Nine analysts are to be used in the study. Three of them will be selected at random and trained for two weeks, three for three weeks and three for five weeks. The productivity of the analysis during the next 10 weeks will then be observed. The data will be experimental data because control is exercised over the explanatory variable, length of training. The length of