background-image: url("../pic/slide-front-page.jpg") class: center,middle exclude: FALSE # 统计学原理(Statistic) <!--- chakra: libs/remark-latest.min.js ---> ### 胡华平 ### 西北农林科技大学 ### 经济管理学院数量经济教研室 ### huhuaping01@hotmail.com ### 2025-04-02 <div> <style type="text/css">.xaringan-extra-logo { width: 110px; height: 70px; z-index: 0; background-image: url(../pic/logo/nwafu-logo-circle-wb.png); background-size: contain; background-repeat: no-repeat; position: absolute; top:0.2em;left:1em; } </style> <script>(function () { let tries = 0 function addLogo () { if (typeof slideshow === 'undefined') { tries += 1 if (tries < 10) { setTimeout(addLogo, 100) } } else { document.querySelectorAll('.remark-slide-content:not(.title-slide):not(.inverse):not(.hide_logo)') .forEach(function (slide) { const logo = document.createElement('div') logo.classList = 'xaringan-extra-logo' logo.href = null slide.appendChild(logo) }) } } document.addEventListener('DOMContentLoaded', addLogo) })()</script> </div> --- class: center, middle, duke-orange,hide_logo name: chapter02 exclude: FALSE # 第二章 数据收集、整理和清洗 .pull-left[ ### [2.1 数据目标](#target) ### [2.2 数据收集](#collection) ### [2.3 资料整理和数据清洗](#tidy) ### [2.4 数据的数据库化](#database) ] .pull-right[ ### [2.5 数据质量](#quality) ### .emp[[2.6 抽样设计](#sampling)] ### [2.7 抽样分布和抽样误差](#error) ### [2.8 问卷设计技术](#question) ] --- layout: false class: center, middle, duke-softblue,hide_logo name: sampling # 2.6 抽样设计 ### [抽样的要素](#element) ### [抽样的逻辑](#logic) ### [概率/非概率抽样](#prob) ### [非概率抽样](#non-prob) ### [抽样方案和实施](#handle) ### [抽样误差](#bias) --- layout: true <div class="my-header-h2"></div> <div class="watermark1"></div> <div class="watermark2"></div> <div class="watermark3"></div> <div class="my-footer"><span>huhuaping@ &emsp;&emsp; <a href="#chapter02"> 第02章 数据收集、整理和清洗 </a> &emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp; <a href="#sampling"> 2.6 抽样设计 </a> </span></div> --- ## 什么是抽样 假设我们希望通过自己调查来获得一手数据，就需要回答一系列问题： 1. 抽样的基本原理是什么？ 2. 抽样的基本要素有哪些？ 3. 抽样的逻辑是什么？ 4. 什么条件该要采用概率抽样方法？怎么样做概率抽样设计？ 5. 什么条件下要采用非概率抽样方法？又如何做非概率抽样？ 6. 一个抽样方案应该包括哪些内容？ 7. 怎么样去实施抽样工作？ ??? 我们从抽样的基本原理入手， 理解抽样的原理，对理解抽样的方法及其适用的情境，非常重要。 讲枯燥的原理是同学们反对的，我也觉得很无聊，我们还是用例子吧。 还记得我们的课题吗？北京大学本科生入学机会的地区不平等研究。 这个题我们已经做到哪一步啦？我们已经把研究问题操作化了，记得不？ 地区经济发展水平与进入北京大学的机会。 相关的概念我们也操作化了，对吗？ 地区，我们可以根据经济发展水平，把省级单位作为地区区分的基本单位， 依据经济发展水平，区分为高、中、低三个类型。 有同学可能会问："老师，怎么划分高、中、低呢？"。 在方法上可能比较简单，如果有数据，有31个省、市、自治区的数据， 我们就可以看各省级单位的人均GDP， 人均GDP，是可以比较的参数，对吧？我们就用这个参数， 用各省级单位人均GDP进行排序， 由高到低，由低到高都行，但要有相同的规则。 根据变量值分布的相对集中程度吧，我们可以把 31个省级单位分为人均GDP的 高、中、低三个组。 当然还有比较复杂的方法，比如说用多个经济指标进行聚类， 根据多个经济指标对教育的影响计算权重，再分类等等。 这里呢，我们采用最简单的方法，也可能是最有效的，同学们能够做的方法。 再看城乡的区分。 城乡的区分比较简单，我们已经讨论过了，由于学生的 受教育机会主要是受户籍特征的影响， 所以，按照农业户籍和非农业户籍进行划分就好了。 这里我提醒同学们，如果想了解中国的户籍制度，以及户籍制度对人们社会经济生活所带来- 的影响， 建议大家找一些文献进行阅读，做社会研究， 不了解中国社会，是很难做研究的。接着看入学机会， 用某地区当年被北京大学录取的学生数， 除以该地区当年高中毕业生的总数，再乘以100%。 大家还记得这里采用的是能力不平等，对吗？ 是否平等，是我们的因变量， 我们要比较的正是6类地区的入学机会，据此判断入学机会 是否与经济发展水平有关。 从这儿的操作化来看，我们并没有用到超出我们知识水平和能力范围的内容，对吧？ 为下一步操作的方便，有一些问题，还需要 进一步地推敲，比如经济发展水平，虽然采用了人均DGP， 虽然已经做了很强的假设，大家还记得我们假设 不同地区的经济发展水平是同质的， 可我们还是不放心，我们知道一个省内不同地区的经济发展水平是有差异的， 因此会犹豫，是只考虑省级之间的差异呢？ 还是降低层级，比如说，也考虑地市级之间的差异呢？ 如果考虑地市级之间的差异，那么在一个省级单位内， 是把所有的地市级都纳入呢，还是在抽取了省级单位以后， 再把抽到的省级单位的所有地区纳入？ 还是说在抽取省级单位之前，就考虑地市级之间的差距，用地市级单位 进行排队，直接抽取地市级单位呢？ 不管是采用省级单位，还是采用地市级单位， 另一个需要考虑的问题就是，在划分时，要不要把毕业生数量作为一个影响因素？ 再看毕业生，要把31个省级单位的全部高中毕业生都纳入搜集数据的范围吗？ 还是只搜集其中的一部分就够了？ 如果只搜集其中的一部分，我们怎么选高中呢？这一些 都是在调查实施之前要推敲的问题， 之所以要推敲，是因为关系到研究的误差。 在进一步推敲之前，我们先看一些事实，北京大学的本科生， 不是以县级单位为基本单位录取的， 也不是以地市级为基本单位录取的，而是以省级为基本单位录取的，不仅如此， 初等教育的资源配置也是以省级为基本单位的，因此呢，第一个问题似乎就解决了。 那么是否调查全部的高中毕业生呢？还记得我们曾经假设地区内部 具有高度的同质性？事实上同质性是人类社会普遍存在的现象， 对我们的调查而言，同质性意味着变量属性的相似 或者相等，意味着我们无需调查全部的高中毕业生。 我们举一个例子，比如说我们买了100支同样的铅笔， 结账的时候，到底是用乘法，还是用加法？ 可能同学们马上会说："老师，这个问题是不是有点2啊？" 不，对抽样而言，这的确是一个问题， 看起来是生活中一个不需要想的问题，在这儿，却真的要想清楚， 到底是使用乘法，还是使用加法？ 大家都会说："当然使用乘法了！" 的确，如果使用加法，就意味着要对全部的毕业生进行调查； 使用乘法呢，就意味着可以对同类学生中的一位进行调查， 再乘以同类学生的数量。 可是为什么要使用乘法？ 是因为尽管两者的结果是一样的， 使用乘法要快得多。快， 在调查上就意味着节省，省时间、省金钱、 省人力、省物力。这是人类社会行为的动力， 某个变量属性在人群中的分布，有异质性，也有同质性， 前面我们假设同质性，在实践中是很冒险的做法，我们知道有些地区的中学 考上北京大学的人每年都有，另一些地区的中学呢？ 多少年都没有一位，也就是说，在一个省级单位内部， 地区之间就考上北京大学的毕业生而言，是 异质性的，如此，就不能用一位来代表全部的学生，而 需要用不同数量的毕业生来代表不同类型的毕业生，抽样问题也就出现了。 在群体中，抽选代表性群体的做法， 在调查中，就是"抽样"。 我们知道在一个群体中，既有同质性的，也有异质性的， 抽取一个代表不了，那么就地区经济发展水平而言，虽然已经区分了高、中、低三类， 每一类中，其实也存在着很大的异质性。 接下来的问题就是在每一类中抽选多少，才可以代表这一类内部的异质性呢？ 这就是抽样、抽样设计、抽样方法要探讨的问题了。 有同学可能会说："既然存在异质性，干脆把全部高中毕业生都纳入不就得了？" 问题是任何一项研究都会面对资源的约束， 假设这是博士研究生自己的项目，不是教育部委托的，你们认为他有资源调查全部的毕业生吗？ 一方面，资源有限，迫使研究者采用抽样的方法，而不是普查的方法， 从研究对象那里去获得数据； 另一方面呢，在研究对象中，事实上存在着同质性，也迫使普查不仅浪费资源， 也毫无必要。还有，即使你有资源，甚至愿意做普查，也不是想普查就能普查的， 有时候一些研究对象也不可及呀，你不忙人家还忙着呢！ 在这种情况下，采用同质性代表的方法甚至比普查还要靠谱。 当然抽样也有局限性， 如果研究对象的分布非常广泛，对调查而言， 就是很麻烦的事，比如CFPS， 在抽样设计中就受到了研究对象分布太广的影响， 理论上，CFPS要代表的是中国大陆的所有家庭， 可是在新疆西藏的牧区，调查对象的分布实在太广了，在那儿调查一户人家所花费的资源可能是在城镇地区的多少倍，我实在没有这些资源，就只好放弃。还有， 如果研究对象的数量非常稀少，比如说铊中毒， 大家可能比较熟悉，假设我们希望研究铊中毒人群的分布， 由于对象的数量实在太少，采用抽样的方法几乎就不可能 获得研究对象，这时候，采用抽样方法就不管用了。 我做吸毒者研究，做性工作者研究的时候，虽然 研究对象的数量比铊中毒病患的数量要大得多，依然无法采用抽样方法。 还有一种特殊情形，假设是教育部委托博士生在做我们的题， 有足够的时间，也有足够的资源，中国大陆高中毕业生的数量， 考上北京大学的数量，也很容易获得，这个时候，就不需要节省了。 如果我们坚持采用抽样方法，坚持节省，就有可能因此而产生误差， 误差我们一再提到过，不过呢，现在还不是讲误差的时候，大家先记住这个概念。 讲了半天，到底什么是抽样呢？下边，我们对这一节的内容做一个小结，小结中 你就知道答案了。抽样就是用代表来代表同类的做法，注意，这不是科学的定义，是操作定义， 这里并没有在抽样两个字的前面加上任何的定语。 如果讲概率抽样，这个定义就不准确了。概率 抽样我们接下来会讨论，这里大家只需要记住 抽样就是用代表来代表同类。之所以要抽样，是为了节省资源，进而节省成本，也是为了效率，抽样不是无聊的行为。 --- name:element ## 抽样的要素（总体） **总体**：是研究问题指涉对象的集合体，也就是研究问题涉及的全部对象。 - CFPS的总体是中国所有的家庭户 - CGSS的总体是中国所有的个体 - 入学机会的地区不平等研究的总体，就是某年所有的高中毕业生。 -- .pull-left[ 问题是： - 什么叫中国所有家庭户，中国所有个体？ - 什么叫所有，台湾算不算？香港和澳门算不算？ - 住在中国的还是有中国户籍的？住在中国的外国人算不算？ - 长期出国却依然有着中国户籍的人算不算？ ] -- .pull-right[ - 什么叫家庭户？没有生活在一起，户口在一起算不算？生活在一起， - 户口不在一起的，算不算？怎么才算是某个地方的家庭户？ - 户口在甲地，却很少在甲地居住，算不算甲地的家庭户？ - 什么叫所有高中毕业生？没有参加高考的算不算？因非主观原因有参加高考的算不算？ ] --- ## 抽样的要素（研究总体） **研究总体**，是指可操作的研究对象，或称为**可及总体**。 CFPS把总体定义为中国的家庭户，是指有中国户籍的家庭户，指住在一起的，不管户籍是不是在一起的家庭户。 -- <div class="puzzle"> <p><strong>提问</strong>：</p> <ul> <li>CFPS中，家庭户指居住在二十五个省级单位内的家庭户吗？</li> <li>户籍不在本地的算不算？</li> <li>住又是什么意思呢？</li> <li>住多长算是住？</li> <li>一个人打工住在本地算是一户吗？</li> </ul> </div> --- ## 抽样的要素（抽样框和抽样单位） **抽样框**：又叫抽样总体、框总体，是从研究总体中获得的用于抽取样本的研究对象的集合。 - CFPS的**总体**是中国所有的家庭户； - CFPS的**研究总体**是二十五个省、市、自治区的常住户； - CFPS的 **抽样框**是二十五个省、市、自治区在一个地方连续居住六个月或以上常住户。 - 从覆盖面和覆盖的对象数量出发，**总体** `\(\ge\)` **研究总体** `\(\ge\)` **抽样框**。 --- ## 抽样的要素（抽样单位和样本） **抽样单位**：是抽样指涉的基本单位，或包括基本单位的单位集合体。 - CFPS在抽到家庭户之前还要抽样本区县，样本村居。每一次抽样面对的基本单位 就是抽样单位。 **样本**：是从抽样框中运用抽样策略和抽样方法获取的样本单位的集合。 - CFPS的样本是，从25个省市自治区抽取的160个区县样本，从160区县样本中抽取的640村居样本，从640村居样本中抽取的16000个家庭户样本。 --- name:logic ## 抽样的逻辑（样本代表性） **抽样的基本逻辑1**：选择一定数量的样本，来拟合总体中个体变异性的分布，进而代表总体。 **抽样的基本逻辑2**：用尽量少的样本，在可接受的误差范围内，来代表总体的研究特征。 柯西的思想：用代表性的样本就可以估计总体的研究特征。 > 柯西曾去美国国会作证，他反对在美国实施人口普查，认为每十年一次的人口普查，耗费太多的资源，实在没有必要。 如果个体在总体的分布是随机的，根据随机性原则抽取的样本就能代表总体，就是**代表性样本**。 在研究实践中，样本与总体之间总是有差异的。即时在随机条件下，尽管每个抽样单位被抽中的概率是相等的。由**样本特征**与**总体特征**之间，总是有差距的。 --- ## 抽样的逻辑（抽样误差） **抽样误差**：是样本研究特征与总体研究特征之间的差异。误差的大小一般取决于样本的代表性。样本对总体的代表性越好，误差就越小，否则误差就会越大。 依据误差的来源环节，可分为： 1. **随机误差**：误差就是由抽样环节造成的误差。随机误差是希望**尽量**避免的误差。 2. **系统误差**：误差具有规律性，主要是由抽样设计造成的。系统误差是我们**最应当**避免的。因为一旦出现的系统误差，几乎就没有补救的余地， - 假设希望知道性别与成就之间的关系。严格按照抽样方案完成的抽样，抽到的样本却都是男性的，没有女性。 --- ## 抽样的逻辑（抽样误差） 依据抽样活动涉及的对象，可以把误差来源分为： 1. **覆盖性误差**：是抽样活动没有正确的覆盖需要覆盖的总体，要么对总体覆盖过度，要么覆盖不住，过度和不足都会导致误差。 - 假设界定的**总体**为参加高考的高中毕业生 - 如果在抽样中把自愿或者是因为其他原因没有参加高考的毕业生都纳入到了抽样的范围，这就是覆盖过度。 - 如果我们把复读并参加了高考的学生排除在了抽样的范围， 这就是覆盖不足。 2. **选择性偏差**：在设计与执行中，因偏好或者抽样活动而导致某个特定类型的样本的分布出现问题。 - 某一类人群过多或者过少或者缺失。 - 某个人群不在抽样框，被选机会就没了。 --- ## 抽样的逻辑（样本分布） **抽样的逻辑3**：利用重复多次抽样，提高抽样代表性，减小抽样误差。 **抽样分布**：又称统计量分布，指样本估计值的分布。抽样分布可以用来测量抽样方法的稳定性。 **总体分布**：是指总体特征值的分布。总体分布并不总是可得的，即使可得，也不满足经济性原则。 --- name:prob ## 抽样方式（总览） **概率抽样**：就是运用等概率原则进行抽样的总称。**等概率原则**，是指总体中每一个研究对象被抽中的概率是相等的。包括：简单随机抽样、系统抽样、整群抽样、与规模成比例的概率抽样、分层抽样以及隐含的分层抽样、多阶段混合抽样。 **非概率抽样**：抽取样本时不是依据随机原则，而是根据研究目的对数据的要求，采用某种方式从总体中抽出部分单位对其实施调查。包括：方便抽样、判断抽样、自愿样本、滚雪球抽样、配额抽样。 从抽样方式的具体运用，又可分为： - **直接抽样**：一次抽样或独立抽样。简单随机抽样、系统抽样和整群抽样都是直接抽样 - **半截抽样**：通常不可以独立地用，要结合前直接抽样来使用。规模成比例的概率抽样、分层抽样以及隐含的分层抽样、多阶段混合抽样都属于半截抽样。 --- ## 概率抽样1：简单随机抽样（步骤方法） **简单随机抽样（simple random sampling, SRS）**：从总体N个单位中随机地抽取n个单位作为样本，每个单位入抽样本的概率是相等的。它是最基本的抽样方法，也是其它抽样方法的基础。 .pull-left[ **实施方法**： - 第一步，制备抽样框 - 第二步，对要素进行编码 - 第三步，根据抽样的要求抽取样本。 - 直接抽选法 - 抽签法 - 随机数码表法（或kish table） - 软件抽取法 ] --- ## 概率抽样1：简单随机抽样（优缺点） **优点**： - 简单、直观，在抽样框完整时，可直接从中抽取样本 - 用样本统计量对目标量进行估计比较方便 **缺点**： - 当N很大时，不易构造抽样框 - 抽出的单位很分散，给实施调查增加困难 - 没有利用其它辅助信息以提高估计的效率 - 使用随机数表抽样的效率往往比较低下，即使用到，也会使用随机数表的一些变体如kish table --- ## 概率抽样1：简单随机抽样（kish table） > Kish, L. (1949). A Procedure for Objective Respondent Selection Within the Household,Journal of the American Statistical Association, 380-387. - 第一步，制备末端抽样框，将样本家户所有符合要素资格的成员，按照规则顺序编号，依据性别也好，年龄也好，逆序也好，顺序也好，怎么排都行，要求是不重，不漏。 - 第二步，拿出事先准备好的kish表，根据指引，抽取样本。抽样的约定是不管家里有几个要素，只抽取其中的一个要素作为样本。 --- ## 概率抽样1：简单随机抽样（kish table） <img src="../pic/chpt02-kish-table.png" width="761" style="display: block; margin: auto;" /> ??? 我们把家庭人口数量常见的状态都纳入了考量，这就是我们在表左列看到的情况，家庭要素从一到五有不同的抽选方案。比如，家里有四个要素，如果选择a表作为抽选方案，则抽选编号为一的作为样本，同样，如果选中b1表作为抽样方案，也选择编号为一的 作为样本。如果选择e2表作为抽样方案，则选择编号为四的作为样本。 如果选择f表作为抽样方案，同样，也选择编号为四的作为样本。 有的同学可能会问了，老师，到底选择哪个表作为抽样方案呢？ 是怎么确定的？很简单，操作指南中，就已经说明了使用方法。 有的就是随机选择起始表号，按照规则继续和循环，有的呢，直接指定了从哪个表号开始，按照什么规则继续。 --- ## 概率抽样1：简单随机抽样（软件实现） 利用**统计软件**能快速实现简单随机抽样: - SPSS - excel - R 简单随机抽样的两点忠告： - 简单随机抽样是不得已的办法，不是最先选用的办法 - 只有在总体的信息所知甚少的情况下，才用它。 --- exclude: true ## 代码：班级学生案例 --- ### 示例：简单随机抽样（步骤） **任务**：从教学班上随机抽取8人。 - 第一步，确认当前的班级是样本班级，制作抽样框。 - 第二步，对班级的83位同学从1到83实行顺序编码。编码顺序可以按学号、按座位等，只要是有规则，并且保证每一位同学只有一个唯一的编号就行。 - 第三步，选择一个随机数表，大家可以找到很多的随机数表（教材附录）。在查阅随机数表之前，说出第一个样本的行列位置作为起点。 - 第四步，在随机数表上找到上面的起点，然后取一组随机数的固定位置，按照事先制定的规则，依次选中随机数字中的一位。 --- ### 示例：简单随机抽样（数据集） 全体学生名单（按班级和学号排序）：共83人。 <div class="datatables html-widget html-fill-item" id="htmlwidget-e0c14b2e85d3541275a3" style="width:100%;height:auto;"></div> <script type="application/json" 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>序号<\/th>\n <th>学号<\/th>\n <th>姓名<\/th>\n <th>班级<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"序号","targets":1},{"name":"学号","targets":2},{"name":"姓名","targets":3},{"name":"班级","targets":4}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ### 示例：简单随机抽样（抽样结果） 不放回-简单随机抽样：从1-83中产生8个随机数。 .pull-left[ ``` r n1 <- 100 # 定义总体大小 ns1 <- 8 set.seed(202403) spl_srs <- base::sample(1:n1, size = ns1, replace = FALSE) spl_srs # show the result ``` ``` [1] 43 7 9 72 56 69 59 27 ``` ``` r base::sort(spl_srs) # show the sorted result ``` ``` [1] 7 9 27 43 56 59 69 72 ``` ] .pull-right[ <div class="datatables html-widget html-fill-item" id="htmlwidget-669814e13f2012e6ded0" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-669814e13f2012e6ded0">{"x":{"filter":"none","vertical":false,"caption":"<caption>简单随机抽样结果<\/caption>","data":[["1","2","3","4","5","6","7","8"],[7,9,27,43,56,59,69,72],["2018014558","2018014581","2018014672","2018014754","2018014808","2018014823","2018014652","2018014703"],["徐锐谦","王祎霖","杨子凤","李飞","苏诚璐","呼恬","王伟泽","张晓玉"],["农管1801","农管1801","农管1801","农管1802","农管1802","农管1802","营销1801","营销1801"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>序号<\/th>\n <th>学号<\/th>\n <th>姓名<\/th>\n <th>班级<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"序号","targets":1},{"name":"学号","targets":2},{"name":"姓名","targets":3},{"name":"班级","targets":4}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> ] --- ### 示例：简单随机抽样（kish table）-随机数表 下面是一张kish随机数表： <div class="datatables html-widget html-fill-item" id="htmlwidget-cc9d38c0e94d9720ff01" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-cc9d38c0e94d9720ff01">{"x":{"filter":"none","vertical":false,"caption":"<caption>一份kish 随机数表<\/caption>","data":[["1","2","3","4","5","6"],["1","2","3","4","5",">=6"],[1,1,1,1,1,1],[1,1,1,1,2,2],[1,1,1,2,2,2],[1,1,2,2,3,3],[1,2,2,3,4,4],[1,2,3,3,3,5],[1,2,3,4,5,5],[1,2,3,4,5,6]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>adults<\/th>\n <th>A<\/th>\n <th>B1<\/th>\n <th>B2<\/th>\n <th>C<\/th>\n <th>D<\/th>\n <th>E1<\/th>\n <th>E2<\/th>\n <th>F<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"adults","targets":1},{"name":"A","targets":2},{"name":"B1","targets":3},{"name":"B2","targets":4},{"name":"C","targets":5},{"name":"D","targets":6},{"name":"E1","targets":7},{"name":"E2","targets":8},{"name":"F","targets":9}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script> --- ### 示例：简单随机抽样（kish table）-家庭编号 假设需要调查共30户家庭，并对每户的成年人进行了编号： <div class="datatables html-widget html-fill-item" id="htmlwidget-840352ab994dd9154f92" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-840352ab994dd9154f92">{"x":{"filter":"none","vertical":false,"data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30"],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[7,7,3,6,3,2,2,6,3,5,4,6,6,1,2,3,8,5,3,3,1,4,1,1,5,3,8,2,7,2]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>id<\/th>\n <th>adults.num<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"id","targets":1},{"name":"adults.num","targets":2}],"pageLength":6,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ### 示例：简单随机抽样（kish table）-家庭人数汇总 按家庭成年人总数做分类，结合kish表可以得到： <div class="datatables html-widget html-fill-item" 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>id<\/th>\n <th>adults.num<\/th>\n <th>adults<\/th>\n <th>A<\/th>\n <th>B1<\/th>\n <th>B2<\/th>\n <th>C<\/th>\n <th>D<\/th>\n <th>E1<\/th>\n <th>E2<\/th>\n <th>F<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"id","targets":1},{"name":"adults.num","targets":2},{"name":"adults","targets":3},{"name":"A","targets":4},{"name":"B1","targets":5},{"name":"B2","targets":6},{"name":"C","targets":7},{"name":"D","targets":8},{"name":"E1","targets":9},{"name":"E2","targets":10},{"name":"F","targets":11}],"pageLength":6,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ### 示例：简单随机抽样（kish table）-家庭随机表 进一步地，每户都可以在8张表（A-F）中做出随机选择： <div class="datatables html-widget html-fill-item" 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<script type="application/json" data-for="htmlwidget-0239a7e79d4d0ede5fce">{"x":{"filter":"none","vertical":false,"caption":"<caption>Kish随机抽样结果<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30"],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[7,7,3,6,3,2,2,6,3,5,4,6,6,1,2,3,8,5,3,3,1,4,1,1,5,3,8,2,7,2],[">=6",">=6","3",">=6","3","2","2",">=6","3","5","4",">=6",">=6","1","2","3",">=6","5","3","3","1","4","1","1","5","3",">=6","2",">=6","2"],["A","E2","B1","E1","B1","B1","D","F","A","C","C","E2","E1","E2","E2","E1","B2","B1","B1","D","F","A","E1","F","B2","B2","A","D","C","D"],[1,5,1,5,1,1,2,6,1,3,2,5,5,1,2,3,2,2,1,2,1,1,1,1,2,1,1,2,3,2]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>id<\/th>\n <th>adults.num<\/th>\n <th>adults<\/th>\n <th>table<\/th>\n <th>select<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"id","targets":1},{"name":"adults.num","targets":2},{"name":"adults","targets":3},{"name":"table","targets":4},{"name":"select","targets":5}],"pageLength":6,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> ] --- ## 概率抽样2：系统抽样（应用情景） **系统抽样(systematic sampling)**：将总体中的所有单位按一定顺序排列（变量要素），在规定的范围内随机地抽取一个单位作为初始单位，然后按事先规定好的规则确定其它样本单位。**系统抽样**也称为等距抽样。 .pull-left[ **应用情景**： - 总体要素与抽样对象一致 - 总体通常规模也不大 - 变量异质性没有大到需要分层处理的程度 - 要素的特征在排列中没有**周期性**变化 ] --- ## 概率抽样2：系统抽样（实施方法） 系统抽样的具体**实施方法**是： 1. 把抽样框的要素按照规则进行**编码**。 3. 用要素总体数 `\(N\)`除以样本数 `\(n\)`，得到**抽样距** `\(k = N/n\)`（如果不是整数则取数值后一个整数）。 4. 在 `\([1, k]\)`之间选择任何一个**随机起点** `\(R\)`，依照抽样距开展或者顺序抽样或者循环抽样。 5. 得到最终抽样编码标号 `\(R, R+k, R+2k, \cdots, R+(n-1)\times k\)` --- ## 概率抽样2：系统抽样（优缺点） **优点**： - 操作简便，可提高估计的精度 **缺点**： - 系统抽样的框不能太大。太大了就很费事，仅就要素编号就比较费事， - 要素的排列特征不能呈现周期性变化。 --- ## 概率抽样2：系统抽样（注意事项） 如果总体是随机排序的，那么系统抽样样本（systematic sampling）很可能表现得很像一个简单随机样本（simple random sampling），从而是对总体的良好抽样实现。 > 索伦森的一份报告里写道，美国肯尼迪总统过去常常在白宫以系统抽样方式阅读公众写给他的信件。值得注意的是，肯尼迪很清楚，他读的那些信（系统抽样样本），虽然代表了写给白宫的信，但根本不能代表公众的意见。 如果总体排序具有周期性，那么某些系统抽样样本不一定是一份对总体的良好抽样实现。 > 假设一个班级有50个人，男生25位，女生25位，在排列时，每位男生的后面或者前面都是女生。这样男生跟女生之间的排列就是周期性的排列。万一要素的排列的周期，与抽样距吻合了。抽到的就只有一类样本（只有男生，或只有女生）。从而引起选择性偏差。 --- exclude: true ## 代码：班级学生系统抽样 --- ### 示例：系统抽样（规则） **任务**：用**系统抽样**方法从83名学生中随机抽取8人： - 第一步，把班内所有的的学生名单按照按照学号进行排列。 - 第二步，把排列好的学号，从1开始顺序编号。 - 第三步，确定**抽样距** `\(k = \frac{N}{n}=\frac{83}{8}=10.38 \simeq 11\)`。 - 第四步，把抽样编号排列成一个循环圈，在 `\([1,11]\)`上选择一个随机起点编号 `\(R\)`，以 `\(R\)`为起点每隔11抽取一个单位，直到抽取总数为8个单位。 -- </br> > 在排列要素的时候，我们不仅可以排列成循环圈，也可以排列为直线。 > 数到最大编号后不够测量距了怎么办？接着继续回头数，以此类推。 > 为了保证随机结果可复现，需要设定随机种子`set.seed()` --- ### 示例：系统抽样（样本1） .pull-left[ ``` r # 系统抽样1 ns <- 8 k_acr <- n/ns k <- ceiling(k_acr) set.seed(8200) (R1 <- sample(1:k, size =1)) 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概率抽样3：分层抽样（实施步骤） **分层抽样(stratified sampling)**：将抽样单位按某种特征或某种规则划分为不同的层，然后从不同的层中独立、随机地抽取样本。 **实施步骤**： 1. 把研究总体按照研究特征变量进行分层。 2. 在每一层采用合适的方法来抽样 - 简单随机抽样或者等距抽样、整群抽样 - 等比例或者不等比例的抽样，甚至pps抽样都行。 3. 把每个层的样本合起来加总，计算得到对总体进行推断的样本容量。 --- ## 概率抽样3：分层抽样（应用情景） 决定是否采用分层抽样，需要： - 对研究总体同质性程度有一定了解， 知道总体的同质性、异质性如何。 - 了解了总体的异质性的程度是不是大到了必须分层的程度。总体在研究变量上的同质性越高，对分层的要求就越低。 - 分层抽样通常不会独立使用，通常用来构造子抽样框、子总体，它不是独立抽样的方法，也不是末端抽样的方法。 - 对研究变量了解越充分，采用合适的分层方式，就越有利于降低抽样误差。 --- ## 概率抽样3：分层抽样（示例） **任务**：一项研究拟讨论教育模式对高校学生能力的影响，研究者打算采用分层抽样方法抽取n名学生。 - 从学校到院系，简单起见可以先分文和理两个大类的院系。 - 从院系到班，可以采用任何简单抽样的方法。 - 从班抽到学生呢，就可以采用整群抽样的办法。 - 把文和理两类样本加起来，就是一所学校的样本。 - 如果文理之间学生的数量相差的太大，也可以考虑按学生数量的比例分配样本。 --- ## 概率抽样3：分层抽样（分层依据） 分层依据是分层抽样中关键的环节： - 分层依据的变量通常与研究目标有关，与研究变量有关系。 - **分层**并不就是**分等级**，大多数情况下是**分类别**（提问）。 - 研究目的越复杂，分层变量越多，要区分的层数也就越多。 - 实践中一般希望尽可能地选取**主要的分层变量**，因为分层越多，看起来越精准。 - 在抽样实践中，有些分层明显，有一些分层则不太明显，可能实际上还携带着层变量的分层，称之为**内隐分层** 或者叫**隐含分层**。 --- exclude: false ## 概率抽样3：分层抽样（分层依据） 学生教育模式对学生能力的影响研究案例。 有的院系一个年级有多个班，如经济管理学院，有的学院只有一个班，如农学院。 如果有多个班的学院用平均能力对班进行排序，再抽取班级样本，则抽到的班样本不仅携带了院系信息，也携带了能力信息。 不仅按文理院系在分层，也在按照能力进行分层，只是按能力分层被隐含在了按文理院系分层之中。 > 大学里的院系，院系之间是平行的，不是层级关系。同一个院系的不同年级之间的分层，实际上是垂直的序列关系，但也叫分层。 --- ## 概率抽样3：分层抽样（优缺点） - 保证样本的结构与总体的结构比较相近，从而提高估计的精度 - 组织实施调查方便 - 既可以对总体参数进行估计，也可以对各层的目标量进行估计 --- ## 概率抽样3：分层抽样（样本分配） 在各层次中**样本量的分配**有两种基本的方法， - 等比例分层抽样：各层的样本量与要素的规模成比例。 - 不等比例分层抽样：依据经验或者既有的研究结论减少或增加特定群体的样本量比例。 </br> <img src="../pic/chpt02-stratified-sampling.png" width="1621" style="display: block; margin: auto;" /> --- exclude: false ## 概率抽样3：分层抽样（CFPS分层和样本分配） **CFPS的分层和分配示例**： -- - 第一个层：区分大省（5个）和小省（20个）。 -- .pull-left[ - 第二个层1：大省（5个） - 子层1：4个省（辽宁、甘肃、河南、广东），各省为一个抽样框，但遵循相同的抽样策略。 - 子层2：1个省（上海），为一个独立的抽样框 ] .pull-right[ - 第二个层2：小省（20个） - 20个省级行政区，按照人均社会经济指标降序排列 - 每一个省级行政区内，地级市按照人均GDP指标降序排列 - 地级市内，分为区、县级市和县三个层。层内按人均GDP降序排列。 ] --- exclude: false ## 概率抽样3：分层抽样（CFPS分层和样本分配） **CFPS的分层和分配示例**： - 分配样本数。 - 抽取样本县、区的初级抽样单位（PSU），分配各层样本数。 - 实现既有发达的，也有不发达的，既有城市，也有县，人多的地区有样本，人少的地区也有样本。 --- exclude:true ## 代码：普查区域分层抽样 --- ### 示例：分层抽样-数据集 <div 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>id<\/th>\n <th>county<\/th>\n <th>state<\/th>\n <th>region<\/th>\n <th>acres92<\/th>\n <th>acres87<\/th>\n <th>acres82<\/th>\n <th>farms92<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"id","targets":1},{"name":"county","targets":2},{"name":"state","targets":3},{"name":"region","targets":4},{"name":"acres92","targets":5},{"name":"acres87","targets":6},{"name":"acres82","targets":7},{"name":"farms92","targets":8}],"pageLength":6,"scrollX":true,"scrollY":true,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ### 示例：分层抽样-地区规模 ``` r # 地区分类 unique(agpop$region) ``` ``` [1] "W" "S" "NE" "NC" ``` ``` r # 地区数量分布 sort(table(agpop$region)) ``` ``` NE W NC S 220 422 1054 1382 ``` 其中： - NE：东北部 - W：西部 - NC：中北部 - S：南部 --- ### 示例：分层抽样-1按地区成比例（分配公式） 假定最终的抽样数量为 `\(n=300\)`，则按地区规模的分配数量计算公式为： $$ `\begin{aligned} n_h=n \times \left(\frac{N_h}{N}\right) \end{aligned}` $$ .footnote[ 上述数量分配方法被称为成比例分配（Proportional Allocation Methods）。 ] --- ### 示例：分层抽样-1按地区成比例（分配计算） .pull-left[ ``` r n <- 300 al_prop <- as.data.frame( table(agpop$region) #数量统计 ) %>% rename_all(., ~c("region", "popsize")) %>% arrange(popsize) %>% # 按地区计算占比及分配数量 mutate( * prop = popsize/sum(popsize), * propalloc = round(n*prop) ) ``` ] .pull-right[ <div class="datatables html-widget html-fill-item" id="htmlwidget-5778c3a41e60e8a9afc5" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-5778c3a41e60e8a9afc5">{"x":{"filter":"none","vertical":false,"caption":"<caption>按地区计算占比及分配数量<\/caption>","data":[["1","2","3","4","5"],["NE","W","NC","S","合计"],[220,422,1054,1382,3078],[0.07147498375568551,0.1371020142949967,0.3424301494476933,0.4489928525016244,1],[21,41,103,135,300]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>region<\/th>\n <th>popsize<\/th>\n <th>prop<\/th>\n <th>propalloc<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n }"},{"targets":4,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 0, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"region","targets":1},{"name":"popsize","targets":2},{"name":"prop","targets":3},{"name":"propalloc","targets":4}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render"],"jsHooks":[]}</script> ] - 每一个分层的数量占比`prop`为 `\(\pi_{h}=N_h/N\)`； - 给定抽样样本数为 `\(n\)`，则每一个分层的数量分配`propalloc`为 `\(n_h = n\times \pi_{h} = n\times N_h/N\)`（这里进行了取整计算）。 --- ### 示例：分层抽样-1按地区成比例（抽编号） .pull-left[ ``` r set.seed(5324) spl_prop <- agpop %>% left_join(., select(al_prop, region, propalloc), by = "region") %>% group_by(region) %>% nest() %>% mutate( id_selected = map( data, * ~sample_n( tbl = data.frame( id_selected=.$id, propalloc = .$propalloc), * size = unique(.$propalloc) * ) ) ) %>% select(-data) %>% unnest(id_selected) %>% arrange(region, id_selected) ``` ] .pull-right[ <div class="datatables html-widget html-fill-item" id="htmlwidget-d92864b14a1e645e4dc7" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-d92864b14a1e645e4dc7">{"x":{"filter":"none","vertical":false,"caption":"<caption>按地区成比例抽取编号(n=300 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>id<\/th>\n <th>county<\/th>\n <th>state<\/th>\n <th>region<\/th>\n <th>acres92<\/th>\n <th>acres87<\/th>\n <th>acres82<\/th>\n <th>farms92<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"id","targets":1},{"name":"county","targets":2},{"name":"state","targets":3},{"name":"region","targets":4},{"name":"acres92","targets":5},{"name":"acres87","targets":6},{"name":"acres82","targets":7},{"name":"farms92","targets":8}],"pageLength":6,"scrollX":true,"scrollY":true,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ### 示例：分层抽样-2考虑地区差异（分配公式） 假定在总体抽样框中，各个地区之间存在较大差异。并假定我们已经获知了这种差异信息（通过预调研或其他先验知识）。 给定我们通过**方差值** `\(S^2_h\)`来表达地区之间差异信息。那么可以利用下述公式，计算得到基于方差信息的数量分配： $$ `\begin{aligned} n_h=n \times \left(\frac{N_h S_h}{\sum_{l=1}^H N_l S_l}\right) \end{aligned}` $$ .footnote[ 上述数量分配方法被称为**纽曼分配**（Neyman Allocation Methods）。 ] --- ### 示例：分层抽样-2考虑地区差异（分配数量） .pull-left[ ``` r strat_var<- tibble( region = c("NE" , "W", "NC", "S"), # 给定方差信息 * var =c(0.8, 2.0, 1.1, 1.0) ) al_var <- al_prop %>% left_join(., strat_var, by = "region") %>% mutate( * varalloc = n*(popsize*sqrt(var))/sum(popsize*sqrt(var)) ) %>% mutate(varalloc = round(varalloc)) ``` ] .pull-right[ <div class="datatables html-widget html-fill-item" id="htmlwidget-e7ad2e53f5c1654f8a5c" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-e7ad2e53f5c1654f8a5c">{"x":{"filter":"none","vertical":false,"caption":"<caption>（考虑方差）按地区计算占比及分配数量<\/caption>","data":[["1","2","3","4","5"],["NE","W","NC","S","合计"],[220,422,1054,1382,3078],[0.07147498375568551,0.1371020142949967,0.3424301494476933,0.4489928525016244,1],[0.8,2,1.1,1,4.9],[18,55,101,126,300]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>region<\/th>\n <th>popsize<\/th>\n <th>prop<\/th>\n <th>var<\/th>\n <th>varalloc<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n }"},{"targets":4,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 1, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"region","targets":1},{"name":"popsize","targets":2},{"name":"prop","targets":3},{"name":"var","targets":4},{"name":"varalloc","targets":5}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render"],"jsHooks":[]}</script> ] 有了上述分配数量结果，剩下的工作就是（此处略）： - 按地区分层，根据各层的分配数量随机抽取编号 - 然后根据编号获得最后的抽样样本 --- ### 示例：分层抽样-3考虑调研成本（分配公式） 假定在总体抽样框中，各个地区之间存在较大差异，同时考虑到各个地区调研成本（同时假定我们已经获知调研成本差异信息）。 那么可以利用下述公式，计算得到基于方差信息 `\(S^2_h\)`和成本信息 `\(c_h\)`的数量分配： $$ `\begin{aligned} n_h=n \times \left(\frac{\frac{N_h S_h}{\sqrt{c_h}}}{\sum_{l=1}^H \frac{N_l S_l}{\sqrt{c_l}}}\right) \end{aligned}` $$ .footnote[ 上述数量分配方法被称为**最优分配**（Optimal Allocation Methods）。 ] --- ### 示例：分层抽样-3考虑调研成本（分配数量） .pull-left[ ``` r strat_cost<- tibble( region = c("NE" , "W", "NC", "S"), # 给定相对成本信息 * relcost =c(1.0, 1.8, 1.4, 1.0) ) al_cost <- al_var %>% left_join(., strat_cost, by = "region") %>% mutate( costalloc = n*(popsize*sqrt(var/relcost))/ sum(popsize*sqrt(var/relcost)) ) %>% mutate(costalloc = round(costalloc)) ``` ] .pull-right[ <div class="datatables html-widget html-fill-item" id="htmlwidget-7c0c00fe6ddb3330698d" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-7c0c00fe6ddb3330698d">{"x":{"filter":"none","vertical":false,"caption":"<caption>（考虑成本）按地区计算占比及分配数量<\/caption>","data":[["1","2","3","4","5"],["NE","W","NC","S","合计"],[220,422,1054,1382,3078],[0.07147498375568551,0.1371020142949967,0.3424301494476933,0.4489928525016244,1],[0.8,2,1.1,1,4.9],[1,1.8,1.4,1,5.2],[20,45,95,140,300]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>region<\/th>\n <th>popsize<\/th>\n <th>prop<\/th>\n <th>var<\/th>\n <th>relcost<\/th>\n <th>costalloc<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n }"},{"targets":4,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 1, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"region","targets":1},{"name":"popsize","targets":2},{"name":"prop","targets":3},{"name":"var","targets":4},{"name":"relcost","targets":5},{"name":"costalloc","targets":6}],"scrollX":true,"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render"],"jsHooks":[]}</script> ] 有了上述分配数量结果，剩下的工作就是（此处略）： - 按地区分层，根据各层的分配数量随机抽取编号 - 然后根据编号获得最后的抽样样本 --- ### 示例：分层抽样-三种方案下的分配数量比较 下面我们把前述三种分层抽样数量分配方案进行综合比较： <div class="datatables html-widget html-fill-item" id="htmlwidget-082b4f265522b85d48ff" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-082b4f265522b85d48ff">{"x":{"filter":"none","vertical":false,"caption":"<caption>三种数量分配方案的综合比较<\/caption>","data":[["1","2","3","4","5"],["NE","W","NC","S","合计"],[220,422,1054,1382,3078],[0.07147498375568551,0.1371020142949967,0.3424301494476933,0.4489928525016244,1],[0.8,2,1.1,1,4.9],[1,1.8,1.4,1,5.2],[21,41,103,135,300],[18,55,101,126,300],[20,45,95,140,300]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>region<\/th>\n <th>popsize<\/th>\n <th>prop<\/th>\n <th>var<\/th>\n <th>relcost<\/th>\n <th>propalloc<\/th>\n <th>varalloc<\/th>\n <th>costalloc<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n }"},{"targets":4,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 1, 3, \",\", \".\", null);\n }"},{"targets":5,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 1, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"region","targets":1},{"name":"popsize","targets":2},{"name":"prop","targets":3},{"name":"var","targets":4},{"name":"relcost","targets":5},{"name":"propalloc","targets":6},{"name":"varalloc","targets":7},{"name":"costalloc","targets":8}],"scrollX":true,"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render","options.columnDefs.2.render"],"jsHooks":[]}</script> > **提问**：数量区别大么？有什么要注意的？三种方案不同，其本质是什么？（更具代表性的样本） .footnote[ 列`propalloc`表示按地区规模成比例的分配数量；列`varalloc`表示考虑地区方差的分配数量；列`costalloc`表示考虑调研成本的分配数量。 ] --- ## 概率抽样4：整群抽样（应用情景） **整群抽样(cluster sampling)**：将总体中若干个单位合并为组(群)，抽样时直接抽取群，然后对中选群中的所有单位全部实施调查。 **应用情景**： - 是群内具有异质性，不过异质性还没有还没有大到需要专门处理的程度。 - 群之间具有差异性，但也没有大到需要专门处理的程度。 - 通常不作为独立抽样的方法使用，而是用于多阶段、多层次抽样的末端。 --- ## 概率抽样4：整群抽样（步骤方法） .pull-left[ **实施步骤**： 1. 确定抽样框 1. 根据变量或辅助变量把总体分成若干子群 3. 确定样本容量和样本子群数 4. 依照简单随机抽样方法随机抽取子群 ] .pull-right[ <img src="../pic/chpt02-cluster-sampling.png" width="809" style="display: block; margin: auto;" /> ] --- ## 概率抽样4：整群抽样（示例） **任务**：为了分析教学法在16个班级上的效果，请按照**整群抽样法**，抽取90个学生。 -- **注意**： - 采用整群抽样法，是假设了班与班之间特征的差异不大。每个班级同学的学习成绩有一个分布，在班与班之间，具有相似性。 - 整群抽样法实施中，分群过程非常重要。分群的基本原则是： - 在选择研究变量或者辅助变量时，让它在群间具有相似性，在群内具有异质性。 - 如果群内同质群间非常异质，那就不适合用整群抽样了。 - 相似的可以用做分群标准的辅助变量，比如说行政区划、组织、行业、班级、年龄 、性别等等之类。 --- ## 概率抽样4：整群抽样（优缺点） **优点**： - 抽样时只需群的抽样框，可简化工作量 - 调查的地点相对集中，节省调查费用，方便调查的实施 **缺点**： - 估计的精度较差 - 在分群中有一点需要注意，群的规模不宜过大，否则就有可能出现内部同质性。影响抽样的效率，操作起来也很麻烦。 --- ### 单阶段和两阶段整群抽样（等概率原则） 在单阶段整群抽样中，初级抽样单位（PSU）内的所有次要抽样单位（SSU）都会被选中。 但是，在某些情况下，一个群组中的次要抽样单位可能非常相似，以至于测量一个群组中的所有次要抽样单位会浪费资源；或者，相对于采样初级抽样单位的成本，测量次要抽样单位可能是昂贵的。 因此，在两阶段群组抽样中的每个阶段，都以相同的概率对初级抽样单位和次要抽样单位进行抽样（与分层抽样类似）。 --- ### 单阶段和两阶段整群抽样（比较1-总体） .pull-left[ **单阶段抽样** <img src="../pic/chpt02-clusting-compare01-stage-psu.png" width="1819" style="display: block; margin: auto;" /> ] .pull-right[ **两阶段抽样** <img src="../pic/chpt02-clusting-compare01-stage-psu.png" width="1819" style="display: block; margin: auto;" /> ] --- ### 单阶段和两阶段整群抽样（比较2-psu） .pull-left[ **单阶段抽样** <img src="../pic/chpt02-clusting-compare02-stage-psu-srs.png" width="1813" style="display: block; margin: auto;" /> ] .pull-right[ **两阶段抽样** <img src="../pic/chpt02-clusting-compare02-stage-psu-srs.png" width="1813" style="display: block; margin: auto;" /> ] --- ### 单阶段和两阶段整群抽样（比较3-ssu） .pull-left[ **单阶段抽样** <img src="../pic/chpt02-clusting-compare03-stage-ssu-all.png" width="1816" style="display: block; margin: auto;" /> ] .pull-right[ **两阶段抽样** <img src="../pic/chpt02-clusting-compare03-stage-ssu-srs.png" width="1827" style="display: block; margin: auto;" /> ] --- exclude: true ## 代码：学积分案例 --- ### 示例：单阶段整群抽样-psu平衡（学分积案例） 一个学生想估算一下他宿舍的平均绩点(GPA)。他没有得到宿舍所有学生的名单并进行简单随机抽样（SRS），而是注意到学生宿舍由100间套房组成，每间套房有4名学生；他计划从这些套房中随机选择5个，然后得到这5个套房中每个人的GPA。 <div class="datatables html-widget html-fill-item" id="htmlwidget-6e1bea629f97ee985f78" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-6e1bea629f97ee985f78">{"x":{"filter":"none","vertical":false,"caption":"<caption>单阶段整群抽样(样本数=20)<\/caption>","data":[["1","2","3","4"],[1,2,3,4],[3.08,2.6,3.44,3.04],[2.36,3.04,3.28,2.68],[2,2.56,2.52,1.88],[3,2.88,3.44,3.64],[2.68,1.92,3.28,3.2]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>student<\/th>\n <th>suite1<\/th>\n <th>suite2<\/th>\n <th>suite3<\/th>\n <th>suite4<\/th>\n <th>suite5<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"student","targets":1},{"name":"suite1","targets":2},{"name":"suite2","targets":3},{"name":"suite3","targets":4},{"name":"suite4","targets":5},{"name":"suite5","targets":6}],"pageLength":4,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> - 初级抽样单位PSU：套房；次要抽样单位SSU：学生 - 总体群组数（初级抽样单位数） `\(N =100\)`；总体群组中的单位数（次级抽样单位） `\(M=4\)`。样本中的群组数（初级抽样单位数） `\(n = 5\)`。 --- exclude: true ## 代码：数学成绩案例 --- ### 示例：两阶段整群抽样-psu非平衡（数学案例） 在教育调查中，一旦选择了学校或教室，对个别学生的测量就相对容易获得（例如对代数课程进行学生笔试），单阶段整群样本通常是最实用的。 本案例研究感兴趣的响应测度需要研究团队的大量资源投入（学生接受评估专家的个人采访），此时两阶段整群抽样可能会更经济划算。 - 在第一阶段的抽样中，从 `\(N = 75\)`所学校的总体中采用简单随机抽样（SRS）选择 `\(n = 10\)`所学校。 - 第二阶段，从每个抽样学校中采用简单随机抽样（SRS）选出 `\(m_i = 20\)`名学生，并派遣专家对学生进行数学能力采访评估。 --- ### 示例：两阶段整群抽样-psu非平衡（抽样信息） <div class="datatables html-widget html-fill-item" id="htmlwidget-982903bfb43a7bc4dae7" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-982903bfb43a7bc4dae7">{"x":{"filter":"none","vertical":false,"caption":"<caption>第一阶段抽样信息<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10"],[9,17,18,22,35,43,46,55,62,75],[163,180,114,367,109,219,318,259,311,263],[20,20,20,20,20,20,20,20,20,20]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>schoolid<\/th>\n <th>Mi<\/th>\n <th>mi<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"schoolid","targets":1},{"name":"Mi","targets":2},{"name":"mi","targets":3}],"pageLength":5,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> - 初级抽样单位PSU：学校；次要抽样单位SSU：学生 - 总体群组数（初级抽样单位数） `\(N =75\)`；总体群组中的单位数（次级抽样单位） `\(M_i\)`非平衡。样本中的群组数（初级抽样单位数） `\(n = 10\)`。 --- ### 示例：两阶段整群抽样-psu非平衡（抽样数据集） <div class="datatables 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>schoolid<\/th>\n <th>gender<\/th>\n <th>math<\/th>\n <th>mathlevel<\/th>\n <th>Mi<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"schoolid","targets":1},{"name":"gender","targets":2},{"name":"math","targets":3},{"name":"mathlevel","targets":4},{"name":"Mi","targets":5}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ## 概率抽样5：成比例抽样（原理） **成比例的概率抽样(Probability Proportionate to Size Sampling)**：又称按规模大小成比例的概率抽样或PPS抽样。 **原理**： - 如果总体的要素之间在研究变量上有异质性，不同规模要素群体之间异质性的分布不是随机的。在这样的条件下，就要考虑把规模因素纳入抽样的考量了。 - PPS抽样理论上运用了等概率原理，希望让每一个抽样单位被抽中的概率 与抽样单位的规模成比例。 --- ## 概率抽样5：成比例抽样（实施方案） **实施方案**： - **PPS抽样**是一种按概率的比例抽样，在多阶段抽样中，尤其是二阶段抽样中，初级抽样单位被抽中的机率取决于其初级抽样单位的规模大小 - 初级抽样单位规模越大，被抽中机会就越大，初级抽样单位规模越小，被抽中机率就越小。 - PPS抽样也可以运用软件工具执行 - `Stata`编程语言下`ADO`模块的`gsample`或者`samplepps`。 - `R`编程语言下包函数`sampling::cluster()`或者`sampling::mstage()`。 --- exclude: false ## 概率抽样5：成比例抽样（示例） 海淀区西北旺乡有100个社区，4万户。假设抽样要求是，要抽取10个社区，每个样本社区抽取20户，一共抽200户。假设针对海淀区西北旺乡A、B两个社区抽样，其中A社区有2000户，B社区只有500户。 -- .pull-left[ **在社区层次来看**： - A社区的总户数占西北旺乡的总户数的比例为 `\(2000/40000=0.05\)`。 - B社区被抽中的概率为 `\(500/40000=0.0125\)`。 ] .pull-right[ **从社区家户来看**： - A社区家户的被选概率就是 `\(20/2000=0.01\)`； - B社区家户的被选概率为 `\(20/500=0.04\)`。 ] -- **从整个西北旺乡来看**： - A社区家庭户在西北旺乡的被选概率就是 `\(0.05 \ast 0.01 = 0.0005\)`。 - B社区家庭户的被选概率是 `\(0.0125 \ast 0.04 = 0.0005\)`。 --- ## 概率抽样5：成比例抽样（规模度量） 概率抽样基本的条件是具有抽样大小规模的辅助变量，又叫**规模度量**。 规模度量如何选择： - 主要是是代表规模的，比如说社区的家庭户数。 - 规模量度的变量可以有多个，最常用的方法是依据研究变量相关程度来挑选。 - 选择规模度量的影响因素还有获取资料的难易程度、可靠程度等。 - 在两阶段/多阶段抽样中，每一阶段使用的规模度量一定要相同。 > 西北旺乡案例中：第一阶段是社区抽样，被选概率计算还是采用了家庭户数。 第二阶段是家庭户抽样，备选概率的计算也采用了家庭户数。 --- ## 概率抽样5：成比例抽样（特点） 成比例抽样PPS的特点： - 第一，PPS抽样常常会考虑抽样面对的现实，一般是进行多阶段抽样，不是抽一回。 - 第二，有些信息，抽样时并不知道，常常要步步为营，充分利用已经知道的信息。 - 第三，每一个阶段的抽样概率不一定相等。 - 第四，总的原则是总体要素的被选概率一定要相等。 --- exclude: true ## 代码：（班级案例） ``` [1] 2 5 8 11 14 ``` --- ### 示例：成比例抽样（案例介绍） 一所大学有 `\(N=15\)`个班级（初级抽样单位PSU）进行了《统计学原理》课程学习。 下页表是班级信息的总体情况。其中第 `\(i\)`班有 `\(M_i\)`名学生，总共有647名学生上统计学原理课程。 考虑到不同班级人数规模是不同的，如果根据次级抽样单位随机抽样（例如采用简单随机抽样或者系统抽样）来确定初级抽样单位，那么就不是等概率抽样（为什么？从哪里能看出来？）。 我们的目标是按照成比例抽样原理： - **有放回**随机抽取5个班级 - **无放回**随机抽取5个班级 --- ### 示例：成比例抽样（数据集） <div class="datatables html-widget html-fill-item" id="htmlwidget-a9d552f93c82a68bd189" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-a9d552f93c82a68bd189">{"x":{"filter":"none","vertical":false,"caption":"<caption>（总体）班级数据信息<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16"],["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","合计"],[44,33,26,22,76,63,20,44,54,34,46,24,46,100,15,647],[0.06800618238021638,0.05100463678516229,0.0401854714064915,0.03400309119010819,0.1174652241112828,0.0973724884080371,0.03091190108191654,0.06800618238021638,0.08346213292117466,0.05255023183925812,0.07109737248840804,0.03709428129829984,0.07109737248840804,0.1545595054095827,0.0231839258114374,1],["1 - 44","45 - 77","78 - 103","104 - 125","126 - 201","202 - 264","265 - 284","285 - 328","329 - 382","383 - 416","417 - 462","463 - 486","487 - 532","533 - 632","633 - 647","-"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>class<\/th>\n <th>class_size<\/th>\n <th>prop<\/th>\n <th>cum_bin<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 6, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"prop","targets":3},{"name":"cum_bin","targets":4}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script> --- ### 示例：成比例抽样-有放回1（系统抽样法） **任务**：用**系统抽样**方法进行成比例抽样： - 对初级抽样单位（班级编号）排序（列`class`） - 根据每个初级抽样单位包含的次级抽样单位数量（班级人数）制订累加编号区间（列`cum_bin`）。 - 对次级抽样单位进行系统随机抽样：抽样矩 `\(k=\frac{\sum^N_{i=1}{M_i}}{n}=\frac{647}{5} = 129.4 \simeq 130\)`。在 `\([1, 130]\)`区间获得随机抽样起点 `\(R = 53\)`；最后得到系统抽样编号 `\(\{R, R+k, \cdots, R+(n-1)\times k\} = \{53, 183, 313, 443, 573\}\)` - 最后，根据次级抽样单位（学生编号）的系统抽样结果 `\(\{53, 183, 313, 443, 573\}\)`，确定对应的初级抽样单位（班级编号）的抽样编号 `\(\{2, 5, 8, 11, 14\}\)`。 --- ### 示例：成比例抽样-有放回1（系统抽样数据） <div class="datatables html-widget html-fill-item" id="htmlwidget-7d2c43d3431345acd752" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-7d2c43d3431345acd752">{"x":{"filter":"none","vertical":false,"caption":"<caption>成比例抽样结果（系统抽样法）<\/caption>","data":[["1","2","3","4","5","6"],["2","5","8","11","14","合计"],[33,76,44,46,100,299],[0.05100463678516229,0.1174652241112828,0.06800618238021638,0.07109737248840804,0.1545595054095827,0.4621329211746522],["45 - 77","126 - 201","285 - 328","417 - 462","533 - 632","-"],["53","183","313","443","573","-"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>class<\/th>\n <th>class_size<\/th>\n <th>prop<\/th>\n <th>cum_bin<\/th>\n <th>index_ssu<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 6, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"prop","targets":3},{"name":"cum_bin","targets":4},{"name":"index_ssu","targets":5}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script> - 根据次级抽样单位（学生编号）的系统抽样结果 `\(\{53, 183, 313, 443, 573\}\)`，确定对应的初级抽样单位（班级编号）的抽样编号 `\(\{2, 5, 8, 11, 14\}\)`。 - **思考**：同一个班级能否被抽中两次？ --- ### 示例：成比例抽样-有放回2（依概率） 在`R`编程语言基础包中，命令函数`base::sample(x, size, replace = TRUE, prob = NULL)`直接可以设定成比例抽样。 .pull-left[ ``` r # 总体初级抽样单位数量N N <- nrow(classes) # 设定随机种子 *set.seed(78065) # 按照班级规模，依概率成比例、有放回地 # 随机抽取5个班级 pps_units <- sample( 1:N, size = 5, * replace=TRUE, * prob=classes$class_size ) # 抽中的样本编号 pps_units ``` ``` [1] 5 14 6 14 6 ``` ] .pull-right[ <div class="datatables html-widget html-fill-item" id="htmlwidget-7173b0cce2bad2618bbf" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-7173b0cce2bad2618bbf">{"x":{"filter":"none","vertical":false,"caption":"<caption>依班级规模概率pps抽样结果<\/caption>","data":[["1","2","3","4","5","6"],["5","14","6","14","6","合计"],[76,100,63,100,63,402],[0.1174652241112828,0.1545595054095827,0.0973724884080371,0.1545595054095827,0.0973724884080371,0.6213292117465224]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>class<\/th>\n <th>class_size<\/th>\n <th>prop<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 6, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"prop","targets":3}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script> ] --- exclude: true ## 代码：（班级案例）Lahiri拒绝法 ``` r N <- nrow(classes) Mi <- classes$class_size n <- 5 # 初始化变量 num_rand1 <- NULL num_rand2 <- NULL Mi_sel <- NULL cond_true <- NULL *set.seed(12442) # 随机种子 i <- 1 *repeat { # 循环选定 num_rand1[i] <- sample(1:N, size =1, replace = TRUE) Mi_sel[i] <- Mi[num_rand1[i]] num_rand2[i] <- sample(1:max(Mi), size =1, replace = TRUE) cond_true[i] <- Mi_sel[i] > num_rand2[i] * if (sum(cond_true)==n) break # 判断是否达到目标样本量 i <- i+1 } # 最终选定的PSU编号 (id_choose <- num_rand1[which(cond_true==TRUE)]) ``` ``` [1] 7 11 7 6 1 ``` --- ### 示例：成比例抽样-有放回3（Lahiri拒绝法的步骤） `Lahiri(1951)`提出了一种基于拒绝法的PPS抽样方法： 1. 在1和N之间抽一个随机数，这表示你正在考虑哪个PSU。 2. 在1和 `\(max\{M_i\}\)`之间随机抽取一个数字。如果该随机数小于等于 `\(M_i\)`，则将该PSU纳入样本；否则返回步骤1。 3. 重复上述操作，直到获得所需的样本量。 > **思考**：为什么这种方法具有PPS抽样的性质？ --- ### 示例：成比例抽样-有放回3（Lahiri拒绝法的R代码） ``` r N <- nrow(classes) Mi <- classes$class_size n <- 5 # 初始化变量 num_rand1 <- NULL num_rand2 <- NULL Mi_sel <- NULL cond_true <- NULL *set.seed(12442) # 随机种子 i <- 1 *repeat { # 循环选定 num_rand1[i] <- sample(1:N, size =1, replace = TRUE) Mi_sel[i] <- Mi[num_rand1[i]] num_rand2[i] <- sample(1:max(Mi), size =1, replace = TRUE) cond_true[i] <- Mi_sel[i] > num_rand2[i] * if (sum(cond_true)==n) break # 判断是否达到目标样本量 i <- i+1 } # 最终选定的PSU编号 (id_choose <- num_rand1[which(cond_true==TRUE)]) ``` ``` [1] 7 11 7 6 1 ``` --- ### 示例：成比例抽样-有放回3（Lahiri拒绝法的过程） <div class="datatables html-widget html-fill-item" id="htmlwidget-f3943ae525b3435aa5e9" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-f3943ae525b3435aa5e9">{"x":{"filter":"none","vertical":false,"caption":"<caption>Lahiri拒绝法pps抽样过程<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10"],["第1次","第2次","第3次","第4次","第5次","第6次","第7次","第8次","第9次","第10次"],[7,7,11,7,7,7,6,4,4,1],[20,20,46,20,20,20,63,22,22,44],[63,2,22,96,80,4,56,97,94,29],[false,true,true,false,false,true,true,false,false,true],["不成功，重来","成功选定","成功选定","不成功，重来","不成功，重来","成功选定","成功选定","不成功，重来","不成功，重来","成功选定"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>step<\/th>\n <th>rand1<\/th>\n <th>Mi<\/th>\n <th>rand2<\/th>\n <th>choose<\/th>\n <th>decision<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"step","targets":1},{"name":"rand1","targets":2},{"name":"Mi","targets":3},{"name":"rand2","targets":4},{"name":"choose","targets":5},{"name":"decision","targets":6}],"pageLength":10,"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script> --- ### 示例：成比例抽样-有放回3（Lahiri法的抽样结果） 按照前述循环操作步骤，我们得到了初级抽样单位（班级）的编号为 `\(\{7, 11, 7, 6, 1\}\)`。因此最终`Lahiri`拒绝法的抽样结果如下： <div class="datatables html-widget html-fill-item" id="htmlwidget-f4ab880a05dbd4fbd45f" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-f4ab880a05dbd4fbd45f">{"x":{"filter":"none","vertical":false,"caption":"<caption>Lahiri拒绝法的抽样结果<\/caption>","data":[["1","2","3","4","5","6"],["7","11","7","6","1","合计"],[20,46,20,63,44,193],[0.03091190108191654,0.07109737248840804,0.03091190108191654,0.0973724884080371,0.06800618238021638,0.2982998454404946]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>class<\/th>\n <th>class_size<\/th>\n <th>prop<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 6, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"prop","targets":3}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script> --- exclude: true ## 代码：（班级案例）无放回pps-systemetic ``` r library("sampling") # R包 N <- nrow(classes) Mi <- classes$class_size n <- 5 set.seed(330582) # 随机种子 spl_pps_wor <- sampling::cluster( data = classes, clustername=c("class"), * size = n, method = "systematic", # 系统抽样法 * pik = classes$class_size, # 依概率 description= TRUE) ``` ``` Number of selected clusters: 5 Number of units in the population and number of selected units: 15 5 ``` ``` r # 抽到的PSU（班级）编号 (pps_wor_units <- spl_pps_wor$ID_unit) ``` ``` [1] 1 5 8 11 14 ``` --- ### 示例：成比例抽样-无放回1（系统抽样法R代码） 在`R`编程语言中，使用包函数`sampling::cluster()`，可以实现**系统抽样法**下**无放回**成比例抽样。 ``` r library("sampling") # R包 N <- nrow(classes) Mi <- classes$class_size n <- 5 set.seed(330582) # 随机种子 spl_pps_wor <- sampling::cluster( data = classes, clustername=c("class"), * size = n, method = "systematic", # 系统抽样法 * pik = classes$class_size, # 依概率 description= TRUE) ``` ``` Number of selected clusters: 5 Number of units in the population and number of selected units: 15 5 ``` ``` r # 抽到的PSU（班级）编号 (pps_wor_units <- spl_pps_wor$ID_unit) ``` ``` [1] 1 5 8 11 14 ``` --- ### 示例：成比例抽样-无放回1（系统抽样数据） 我们得到了初级抽样单位（班级）的编号为 `\(\{1, 5, 8, 11, 14\}\)`。因此最终**系统抽样法**下**无放回**成比例抽样结果如下： <div class="datatables html-widget html-fill-item" id="htmlwidget-a62b9bfbdd99f140aaa9" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-a62b9bfbdd99f140aaa9">{"x":{"filter":"none","vertical":false,"caption":"<caption>Lahiri拒绝法的抽样结果<\/caption>","data":[["1","2","3","4","5","6"],["1","5","8","11","14","合计"],[44,76,44,46,100,310],[0.3400309119010819,0.5873261205564142,0.3400309119010819,0.3554868624420402,0.7727975270479135,2.395672333848532]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>class<\/th>\n <th>class_size<\/th>\n <th>prop<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 6, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"prop","targets":3}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script> --- exclude: true ## 代码：（班级案例）成比例抽样-两阶段无放回 --- ### 示例：成比例抽样-无放回2（两阶段-数据集） <div class="datatables html-widget html-fill-item" id="htmlwidget-c0eee2f5b83e86b1d2e5" style="width:100%;height:auto;"></div> <script type="application/json" 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>class<\/th>\n <th>class_size<\/th>\n <th>student_id<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"student_id","targets":3}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ### 示例：成比例抽样-无放回2（两阶段-R代码） ``` r library("sampling") # R包 N <- nrow(classes) n <- 5 # psu 班级 mi <- 4 # ssu 学生 *numberselect <- list(n, rep(mi, n)) # 两阶段样本数设定 *prob <-list(classes$class_size/647, # 阶段1概率 * 4/classes_long$class_size) # 阶段2概率 set.seed(75745) # 随机种子 *spl_pps_2stage <- sampling::mstage( # mstage命令函数 data = classes_long, # 长数据集 stage = list("cluster", "stratified"), # 两阶段抽样方法 varnames=list("class","student_id"), # 两阶段抽样单位 size = numberselect, # 两阶段抽样数量 method = list("systematic","srswor"), # 两阶段抽样设定 pik = prob # 两阶段概率设定 ) # 抽到的PSU（班级）编号 dt_st1 <- getdata(classes_long, spl_pps_2stage)[[1]] # 阶段1结果 dt_st2 <- getdata(classes_long, spl_pps_2stage)[[2]] # 阶段2结果 ``` --- ### 示例：成比例抽样-无放回2（两阶段：结果1/2） <div class="datatables html-widget html-fill-item" id="htmlwidget-252e6b19eb91f17b127a" style="width:100%;height:auto;"></div> <script type="application/json" 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","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"student_id","targets":3},{"name":"id_unit","targets":4},{"name":"prob_2_stage","targets":5},{"name":"prob","targets":6}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render"],"jsHooks":[]}</script> --- ## 概率抽样6：多阶段抽样（复习） - 如果总体规模不大，要素在研究变量上的异质性分布具有**随机性**，则我们可以采用**简单随机抽样**、**系统抽样**。 -- - 如果不同群之间的异质性不大，群内的异质性对总体具有代表性，就可以采用**整群抽样**。 -- - 如果总体规模比较大，总体要素的**异质性也比较大**，且与不同特征群体的规模**无关**，研究变量在要素中呈现出某种非随机的分布，则需要采用**分层抽样**。 -- - 如果总体规模比较大，总体要素的**异质性也比较大**，且与不同特征群体的规模**有关**，那么至少要采用两个阶段的抽样，并且采用与群体规模**成比例的概率抽样PPS**。 -- - 如果遇到搜集数据的范围非常大，要素的异质性分布也很复杂，那么采用上述任何一种方法都不足以解决抽样问题，而应该采用**多阶段抽样**。 --- ## 概率抽样6：多阶段抽样（实施方案） **多阶段抽样(multi-stage sampling)**：先抽取子群，但并不是调查群内的所有单位，而是再进行一步抽样，从选中的群中抽取出若干个单位进行调查。在多阶段抽样的每个阶段，采用的**抽样方法**也不一定相同。 **实施方法**： - 先抽大单位（可用分层抽样） - 再在大单位中抽小单位（可用成比例抽样） - 小单位中再抽更小的单位（可用简单随机抽样） --- ## 概率抽样6：多阶段抽样（CGSS抽样方案） `CGSS`调查中基本要素是家庭中年满18岁或以上的个体。 假设研究者希望一次直接抽到个体，就需要编制一份有18岁或以上中国常住人口的抽样框。一个差不多有10亿人口的列表，这是不可能的 `CGSS 2010`年的抽样方案： - 第一阶段，采用了**分层抽样**（覆盖全国区、县级市、县）。 - 第二阶段，抽到了村居，采用**PPS抽样**。 - 第三阶段，抽到了家户，采用了**简单随机抽样**。 - 末端抽样，抽到个体，采用了**Kish表抽样**。 --- ## 概率抽样6：多阶段抽样（CGSS抽样单位） - **初级抽样单位（PSU）**：初级阶段样本框的抽样单位。 - CGSS的PSU就有160个区县 - **次级抽样单位（SSU）**：次级阶段样本框的抽样单位。 - 对于上海，每个PSU只抽两个村居，也就是32乘2等于64，总的SSU的数量与其他大省一致。 - **末端抽样单位（USU）**：最末端阶段样本框的抽样单位。 - CGSS的末端抽样单位是在样本户中抽到个人，是由调查员去抽取的 --- ## 概率抽样6：多阶段抽样（优缺点） - 具有整群抽样优点，保证样本相对集中，节约调查费用 - 需要包含所有低阶段抽样单位的抽样框；同时由于实行了再抽样，使调查单位在更广泛的范围内展开 - 在大规模的抽样调查中，是经常被采用的方法 --- exclude: true ## 代码：（班级案例）复杂多阶段抽样 ``` r classes_complex <- classes_long %>% mutate( * strat = ifelse( class_size > 70, 1, # 班级分层1 ifelse(class_size > 40, 2, 3) # 班级分层2和3 ) ) %>% * arrange(strat) # 按班级分层变量排序 ``` --- ### 示例：班级案例多阶段抽样（介绍） 继续考虑之前的**班级案例**长数据集`classes_long`（包含647个学生信息）。这里我们进一步增加班级类别的分层信息`strat`：小规模班级（3=人数少于等于40人）、中等规模班级（ `\(2=(40,70]\)`人）、大规模班级（1=70人以上），得到新的数据集`classes_complex`。 ``` r classes_complex <- classes_long %>% mutate( * strat = ifelse( class_size > 70, 1, # 班级分层1 ifelse(class_size > 40, 2, 3) # 班级分层2和3 ) ) %>% * arrange(strat) # 按班级分层变量排序 ``` 三阶段抽样设计目标： - 对3个初级抽样单位（规模）采用无放回分层抽样抽取2个次级抽样单位（班级） - 对6个次级抽样单位（班级）分别无放回随机抽取3个末端抽样单位（学生） --- ### 示例：班级案例多阶段抽样（数据集） <div class="datatables html-widget html-fill-item" id="htmlwidget-17b2d092b822a0ad32e6" style="width:100%;height:auto;"></div> <script type="application/json" 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>class<\/th>\n <th>class_size<\/th>\n <th>student_id<\/th>\n <th>strat<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"student_id","targets":3},{"name":"strat","targets":4}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ### 示例：班级案例多阶段抽样（数据概要） <div class="datatables html-widget html-fill-item" id="htmlwidget-644fb35ad09bd1d44602" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-644fb35ad09bd1d44602">{"x":{"filter":"none","vertical":false,"caption":"<caption>数据集摘要<\/caption>","data":[["1","2","3","4"],["1","2","3","合计"],["5;14","1;6;8;9;11;13","2;3;4;7;10;12;15","-"],[2,6,7,15],[176,297,174,647]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>strat<\/th>\n <th>id_class<\/th>\n <th>num_class<\/th>\n <th>num_students<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"strat","targets":1},{"name":"id_class","targets":2},{"name":"num_class","targets":3},{"name":"num_students","targets":4}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> .footnote[ 第1阶段分层抽样（`strat`变量），不需要操作，全部抽中`method = ""`。 ] --- ### 示例：班级案例多阶段抽样（R代码设计） ``` r # 设定阶段抽样数量 number_select <- list( num1 = unname(table(classes_complex$strat)), # 176 297 174 num2 = rep(2, 3), # 2 2 2 num3 = rep(3, 6) # 3 3 3 3 3 3 ) set.seed(75745) # 随机种子 *spl_ms <- sampling::mstage( # 进行三阶段抽样 data = classes_complex, stage = list("stratified", "cluster", "stratified"), # 阶段方法 varnames = list("strat", "class", "student_id"), # 分层变量 size = number_select, # 阶段抽样数量 * method=list("", "srswor", "srswor") # 具体技术 ) # 获得三个阶段抽样数据及抽样概率 dt_sample1 <- getdata(classes_complex, spl_ms)[[1]] #第1阶段样本 dt_sample2 <- getdata(classes_complex, spl_ms)[[2]] #第2阶段样本 dt_sample3 <- getdata(classes_complex, spl_ms)[[3]] #第3阶段样本 ``` --- ### 示例：班级案例多阶段抽样（第1阶段-样本结果） <div class="datatables html-widget html-fill-item" id="htmlwidget-bdd978acd17d311f9700" style="width:100%;height:auto;"></div> <script type="application/json" 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>class<\/th>\n <th>class_size<\/th>\n <th>student_id<\/th>\n <th>strat<\/th>\n <th>id_unit<\/th>\n <th>prob_1_stage<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"targets":6,"render":"function(data, type, row, meta) {\n return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"class","targets":1},{"name":"class_size","targets":2},{"name":"student_id","targets":3},{"name":"strat","targets":4},{"name":"id_unit","targets":5},{"name":"prob_1_stage","targets":6}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script> .footnote[ 第1阶段分层抽样（`strat`变量），全部抽中，概率为1（见列`prob_1_stage`） ] --- ### 示例：班级案例多阶段抽样（第2阶段-样本结果） <div class="datatables 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","targets":0},{"name":"strat","targets":1},{"name":"class","targets":2},{"name":"class_size","targets":3},{"name":"num_student","targets":4},{"name":"prob_3_stage","targets":5}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script> .footnote[ 第3阶段分层抽样（`strat` & `class`& `student_id`变量），抽样概率见列`prob_3_stage` ] --- name:non-prob ## 非概率抽样 **非概率抽样**：抽取样本时不是依据随机原则，而是根据研究目的对数据的要求，采用某种方式从总体中抽出部分单位对其实施调查。 .pull-left[ 1. **方便抽样**（convenience sample） 2. **判断抽样**（purposive sample） 3. **自愿样本**（voluntary response sample） 4. **滚雪球抽样**（snowball sample） 5. **配额抽样**（quota samples） ] .pull-right[ <img src="../pic/chpt02-non-probability-sampling-2.png" width="1333" style="display: block; margin: auto;" /> ] --- ## 非概率抽样1：方便抽样 **方便抽样**（Convenience samples）：调查过程中由调查员依据方便的原则，自行确定入抽样本的单位。 - 调查员在街头、公园、商店等公共场所进行拦截调查 - 厂家在出售产品柜台前对路过顾客进行的调查 **优点**：容易实施，调查的成本低 **缺点**： - 样本单位的确定带有随意性 - 样本无法代表有明确定义的总体 - 调查结果不宜推断总体 --- ## 非概率抽样2：判断抽样 **判断抽样**（Judgement sampels）：研究人员根据经验、判断和对研究对象的了解，有目的选择一些单位作为样本。具体方式有： - 重点抽样 - 典型抽样 - 代表抽样 **缺点**： - 判断抽样是主观的，样本选择的好坏取决于调研者的判断、经验、专业程度和创造性 - 样本是人为确定的，没有依据随机的原则，调查结果不能用于推断总体 **优点**：抽样成本比较低，容易操作 --- ### 示例1：从人口普查表中判断抽样（1/2） 1926年11月，在1921年意大利人口普查的摘要出版之后，包含每个家庭信息的原始数据表将被销毁，以便在存储设施中为新材料腾出空间。 统计学家基尼（Corrado Gini，以提出基尼不平等系数而闻名）和伽伐尼（Luigi Galvani）希望得到一份抽样后的人口普查的信息表，以便"获得整个国家的代表性样本，考虑到其主要的人口、社会、经济和地理特征"，以便将来的研究需要。他们认为，大约15%的记录样本足以产生可靠的结果，但又不至于太大而无法存储。 因为数据已经被制成表格，所以记录是按地区排序的。基尼和伽伐尼决定对地区进行抽样，而不是对个人记录进行抽样，这样他们就可以保留在地区内进行地方比较的能力。他们为214个地区中的每个地区提供了11个特征的表格。 --- ### 示例1：从人口普查表中判断抽样（2/2） 基尼和伽伐尼知道他们可以进行SRS或系统样本。但他们认为，SRS的抽样误差对于他们的目的来说太大了，他们认为通过自己选择抽样单位可以获得更好的代表性。 他们有目的的选择过程是相当繁重和耗时的——比采取任何随机选择的样本都要困难得多。事实上，他们构建了一个有目的的地区平衡样本，该样本被设计成与人口在七个"控制"特征（出生率、死亡率和结婚率、从事农业的10岁以上男性的百分比、城市人口百分比、平均收入、高度)。 判断样本很好地估计了1921年人口普查中控制变量的人口值。然而，Neyman(1934)证明了样本中的其他统计数据与总体值存在很大差异。他从数学上证明，使用随机选择，而不是有目的的选择，可以避免这个问题。 --- ### 示例2：飓风灾害援助分配研究中的判断抽样 Rivera(2016)选取美国新泽西州六个市作为判断样本，调查飓风"桑迪"过后居民申请和获得联邦救灾援助的经历。其中三个城市在该州的北部，另外三个在南部。随机数字拨号，包括固定电话和移动电话号码，被用来选择每个城市的居民样本。这些城市被特意选择来代表飓风造成的不同类型和程度的破坏。 在本研究中，有目的的选择六个城市（判断抽样）满足了研究需要，并允许研究者以六个单位的样本来研究援助分配中的公平问题。 此外，由于在每个城市内选择了一份概率样本（简单随机抽样），因此数据可用于研究飓风破坏程度不同的城市之间的对比。数据实际上包含六个独立的概率样本，每个样本代表其所在的城市。 --- ## 非概率抽样3：自愿样本 **自愿样本**：被调查者自愿参加，成为样本中的一分子，向调查人员提供有关信息 - 参与报刊上和互联网上刊登的调查问卷活动 - 向某类节目拨打热线电话等 **特点**： - 自愿样本与抽样的随机性无关 - 样本是有偏的 - 不能依据样本的信息推断总体 --- ## 非概率抽样4：滚雪球抽样 **滚雪球抽样**：先选择一组调查单位，对其实施调查，再请他们提供另外一些属于研究总体的调查对象；调查人员根据所提供的线索，进行此后的调查。 持续这一过程，就会形成滚雪球效应。 **特点**： - 适合于对稀少群体和特定群体研究 - 容易找到那些属于特定群体的被调查者 - 调查的成本也比较低 --- ## 非概率抽样5：配额抽样 **配额抽样**（Quota samples）：先将总体中的所有单位按一定的标志(变量)分为若干类，然后在每个类中采用方便抽样或判断抽样的方式选取样本单位。 **特点**： - 操作简单，可以保证总体中不同类别的单位都能包括在所抽的样本之中，使得样本的结构和总体的结构类似 - 抽取具体样本单位时，不是依据随机原则，属于非概率抽样 --- ### 示例：配额抽样-在线调查社群 **在线调查社群**（Online panel surveys），是由调查机构为进行调查而招募的一群人。当需要对某个话题进行社会调查时，调查机构要求从符合该调查标准的社群成员中抽取样本进行调查。在线调查社群中往往有一个较大的潜在受访者群体，可以节省每一个调查选择和联系新样本的费用。 - 一些组织使用**概率抽样**来招募社群成员。他们从抽样框架中随机选择地址(有时是电话号码)，并邀请被抽样地址的人员加入社群。尽管如此，社群成员也可能拒绝回应个别调查，造成额外的无回应。 - **非概率社群**，有时被称为选择加入社群，由在调查机构网站上注册的志愿者组成。然后，在进行调查时，机构会在每个感兴趣的配额类别中选择具有所需样本量的成员子样本。此时，使用这类样本的调查获得的都是配额样本。 --- name:handle ## 抽样方案（抽样方法选择） 为了保证抽样过程的严谨，也需要一个文本，用来指导抽样活动，这就是**抽样方案**。 在抽样方案中，抽样方法的选择是核心内容。一般情况下，抽样方法的取舍取决于三个基本因素：要素的同质性、总体的规模、变量的多少。 - 第一，如果总体规模很大，异质性很强，研究变量很多，通常会采用多阶段、分层的PPS抽样。 - 第二，如果总体规模很大，异质性也很强，研究变量很少，通常采用多阶段抽样，末端通常采用整群抽样或者配额抽样。 - 第三，如果总体规模也很大，同质性也很强，这个时候，变量的多少没有太大关系一般的情况下会采用非概率抽样，比如说末端采用就近抽样、判别抽样。 - 第四，如果总体规模很小，异质性很强，变量多少都没关系，通常会采用滚雪球、RDS抽样或者是知情人抽样。 --- ## 抽样方案（文本内容1） 抽样方案一般需要说明，采用什么方法，采用哪些步骤，获得用于收集数据的样本。一份抽样方案在内容上至少要有以下的内容， - 第一，总体。不仅要把总体界定清楚，还要明确地界定研究总体、框总体，或者叫抽样总体。如果采用多阶段抽样、分层抽样的，还要说明每一个阶段、每一层的抽样框。 - 第二，研究对象。包括调查对象或者研究对象，就是收集数据的对象、受访者。比如说CGSS调查对象是家庭中的个人，CFPS调查对象是家庭中的所有成员。 - 第三，样本量。尤其是末端抽样单位的数量要做明确的说明。 - 第四，抽样方法。如果采用复杂设计，例如多阶段混合抽样，那么每一个阶段的抽样方法都要做说明。 --- ## 抽样方案（文本内容2） </br> - 第五，如果采用多阶段混合抽样，或者多阶段抽样，每一个阶段的抽样单位、抽样框、抽样方法、样本量的配置以及末端抽样的方法也都需要写清楚。否则读者就无法知道每一个阶段的权重。 - 第六，如果不是采用大量熟悉的抽样框，自己在制备抽样框，还需要说明抽样框的制备方法。大型调查中的抽样框制备也是一项复杂的工程。 - 第七，还包括估计量的计算方法。比如说，权重到底怎么算，怎么配权重，如果是多阶段抽样，等概率又怎么保证。 ??? 我们举两个例子吧， 两个大型调查的例子。大家已经很熟悉了，我们把两个调查的抽样方案的目录都找过来了。 第一个例子是CFPS抽样方案的目录。我们可以看第一，调查对象与 目标样本量，说明调查对象是谁以及抽多少样本。 第二，抽样设计总原则，介绍分层、分阶段抽样的理由与原则。 第三，各阶段的抽样，分第一阶段抽样、第二阶段抽样、第三阶段抽样，依次到末端抽样。 第四呢，还有一个再抽样。CFPS有六个总抽样框、 五个大省各自一个抽样框，二十个小省 是一个抽样框。当把五个大省与其他二十个小省样本结合在一起 在国家层面作推论时，就需要对五个大省的样本再次抽样。 不然，五个大省的样本与二十个小省的样本量比例就不对， 不是等概率的，需要通过再次抽样， 让二十五个省市自治区作为一个总体的备选概率相等。 第五，说明权数。接下来一共还有五个附录， 说明具体的技术性细节。这一份抽样方案虽然只有十七页，却交代了每一个阶段到底怎么操作， 并证明所有操作作为一个整体如何满足了等概率原则。 我们再看看CGSS的抽样方案，虽然写法上有一些不同， 但主要内容是一样的。第一，调查背景，交代调查的重要性和必要性。 第二，调查的目标总体。第三，抽样设计的原则。第四，抽样设计中的 几个问题，涉及到分层、各阶段的抽样单位、样本量的确定与分配。 第五，具体设计，介绍了分层的方法、各层抽样的方法。第六呢， 最终的样本构成。第七，样本权重的确定。第八，估计量的计算。 第九，方差的计算以及附录， 把PSU都列出来了。对总体的说明，对研究对象的说明， 对样本量的说明，对抽样方法的说明，对多阶段的说明，对抽样框制备的说明，都在这， 一个都不少。对于初学者来讲，尽管不需要做的这么复杂，却也需要清除明白。 为此呢，我给大家一些提示。 你们要知道，做抽样方案的人，总希望有一套完美的方案。 我做抽样方案的时候，甚至希望把每一个异质性因素都纳入考量。 不过，任何抽样方案都会受到资源有限的约束，也会受到可及性不足的约束。 因此最终的抽样方案，总是在资源、可及性与完美性之间的一个取舍， 是一个妥协。可以有理论上比较完美的设计方案，却常常不具备操作性。 任何可以操作的方案呢，却总是有瑕疵的方案。这样，抽样方案只能尽力， 尽量做到把研究对象界定清楚，把抽样每一个环节的对象与边界界定清楚， 比如说CFPS的家庭户，CGSS的个人。 尽量地界定总体、研究总体、抽样框。 避免造成覆盖性误差。说到底，抽样其实是一个遗憾的因素， 最后总要做取舍。做完取舍就会遗憾，这个没有想到，那个没有想到。之所以要做抽样方案，- 就是希望通过文本的 形式，梳理抽样的各个环节，尽量使得没有想到的事少一点，影响小一点。 下边，对这一节的内容做一个小结。 抽样方案是抽样工作必备的文件。 抽样方案的内容需要对涉及抽样 各个环节及其工作做出说明。 这一节的内容就到这里。 --- ## 抽样实施（工作安排） 即使有很好的抽样方案，如果不落到实处还是没有样本。抽样的实施一般来讲，根据抽样方案按照研究设计做就行了。听起来很简单，不过千万别大意。获得样本真的是一个非常艰难的过程。 - 第一，正确地理解方案，制定每一个环节的实施方案。抽样方案只是指引，指南，索引，在实践中在操作中还需要实施方案。 - 第二，组织资源。比如人力，社会关系，设备，后勤保障等等。稍稍大一点的调查就得请人，请学生，请朋友，怎么计酬，怎么支付这就是后勤问题。后勤对社会调查与研究也非常重要。 - 第三，培训抽样人员，督导人员，后勤人员。把实施中可能遇到的问题讲透彻，把合作与分工讲透彻，让每一个人明确的知道自己到底要干什么。 - 第四，逐步实施。一般来讲，前三步工作做完以后就一步一步地实施，先从制作抽样框开始，再抽样， 最后再做质量检验和误差估计。 ??? 举一个例子，末端抽样框的例子，CFPS的例子。我在组织CFPS的时候根据中国国情做了一套末端抽样方法，《地图地址抽样框制备手册》。这是我们综合了已有的末端抽样框制作方法创造的一个方法。比如说，我们根据测试阶段遇到的情况，列出了在制做末端抽样框的时候，如果遇到了一宅多户和一户多宅怎么处理？ 边界怎么界定？无法确定是否是住宅时怎么处理？以及大社区如何拆分等等。 在抽样实施中，每一个抽样都有自己的特异性，不过呢？也有一些常见的错误提出来希望大家能够避免。 第一类错误，理解类的错误。 对设计理解不太准确。比如说，CFPS的家户，到底什么是家户？CFPS对家户有很严格的定义，一般情况下，操作人员认真看定义可能还会有不明白的地方。如果按照不明白的理解去做，就会带来误差。更严重的是根本不看定义，按照自己的日常生活中的理解，就一定会带来误差。为了避免误差操作人员一定要认真阅读操作说明，不明白的一定要问，问明白问清楚再操作。 第二类，操作类的误差。如果理解上没有问题，操作中的误差常常有两类。第一，马虎。比如，遗漏了一宅多户中的户；一户多人中的人。第二，作弊。比如由于执行的困难，故意作弊或者臆想。调查点如果很远不想去，就自己随意的想象了。 在制作CFPS末端抽样框的阶段，我在甘肃省核查的时候去了一个村子。这个村子明明有73户人家，结果绘图员在图上只标明了56户。我问还有十多户上哪去了？我到村子里拿着图一步一步地看，发现绘图员把距村子大概有半米路远的一个小聚居给弄丢了，正好是十多户，这就叫作弊。因为执行的困难而作弊。 --- ## 抽样实施（经验建议） 在抽样的实施中，多问自己几个问题： - 第一，总体到底有多大？到底多大范围的调查？ - 第二，研究总体在哪里？有哪些会影响到对调查对象的识别？ - 第三，有没有可用的抽样框？比如说，有没有可能让执行人提供一个抽样框？ 如果没有怎么制备抽样框？ - 第四，选择什么样的抽样方法可以减少误差？ - 第五，执行的难点到底是什么？怎么样去组织资源能够使得花最少的钱最有效地办事？ - 第六，最重要的一条经验就是多沟通。与相关各方尽可能就抽样设计，抽样实施的目标达成一致。 --- name:bias ## 抽样误差（误差类型） 调研设计总误差（Total Survey Error）包括两个方面： 第一，随机抽样性误差（Sampling Error），是随机抽样造成的误差。 第二，非随机抽样误差（Nonsampling Error），主要是流程 --- ## 随机抽样误差 随机抽样误差主要考察**主要变量的抽样误差**，是由变量特征带来的： - 每一个变量 都有自己的抽样误差 - **主要变量的抽样误差**一般是指的**均值**的误差，用**均值的标准误** `\(\sigma_{\bar{x}}\)`来代表误差。 - **主要变量的抽样误差**也能用相对误差来表示，比如说**均值的变异系数** `\(V_{\bar{x}}\)`。 --- ## 非随机抽样误差 非随机抽样误差主要有： - 第一，流程性误差（processing Error），是抽样活动造成的误差。 - 第二，覆盖性误差（Coverage Error），与抽样设计和抽样活动有关。 - 第三，应答性误差（Nonresponse Error），指访问阶段产生的误差。 - 第四，测量性误差（Measurement Error），指测量、测量工具产生的误差。 --- ## 非随机抽样误差：流程误差1/2 抽样方法搜集数据，误差来源可能会出现在多个阶段： - 第一，在**发起阶段**，由研究者带来的误差。理论假设不好，概念界定不清，对样本要求不明确。 - 第二，在**设计阶段**，由设计者带来的误差。比如测量工具选的不对，实施策略选的不对，抽样设计也有问题。 - 第三，在**抽样阶段**，由抽样员带来的误差。如果末端抽样框的界定不明确，抽样过程监管也不明确，就有可能产生随机性误差。 - 第四，在**访问阶段**，由访问员带来的误差。如果访员作弊、作假，轻易地接受拒访，诱导性提问，不规范的提问，也会造成随机误差、应答误差，甚至系统误差。 --- ## 非随机抽样误差：流程误差2/2 - 第五，在**访问阶段**，由受访者带来的误差。 - 如果受访者拒绝访问，或者没有能力作答，作假、作弊、随意作答、回忆误差，也会造成随机误差、应答误差。 - 第六，在**数据清理阶段**，由数据管理者带来的误差。 - 如果数据的管理者编制的数据录入程序有问题，编码有问题，清理程序有问题，管理程序也有问题，也就有可能会前功尽弃，既可能产生随机误差，也可能产生系统误差。 - 第七，在**数据分析阶段**，由分析者带来的误差。 - 如果分析者分析工具选择不当，模型建构不当，对数据有误读，也会造成研究误差。 --- ## 非随机抽样误差：覆盖性误差 **覆盖性误差**，主要指因抽样方制作不当带来的误差。它属于抽样设计和抽样活动有关一类误差。 - 如果抽样方与研究总体不一致，就会产生误差。 - 假定CGSS使用电话号码作为抽样框，就会出现覆盖性误差。 - 覆盖不足产生的误差：比如有些人没有电话，就会被抽样方忽略，太穷的、太富的都有可能没有电话，或者呢，有电话，却不在电话簿的列表中。 - 覆盖过度产生的误差：比如很多人有多部电话，这些人就有可能被过度代表 - 任何抽样方法都不可避免地会带来误差。 - 忽略样本特征而随意选择抽样方法，就会直接带来误差。 - 即使让抽样方正确地反映了研究总体，抽样活动不可避免地也会带来误差。 > 入学机会的地区不平等研究案例。假设我们对高中毕业生采用家户抽样方法，能抽到毕业生，但是却比直接抽取高中毕业生的学校所产生的误差要大得多。 --- ## 非随机抽样误差：应答性误差 访问阶段的误差也会涉及到抽样误差，尤其是**应答性误差**。它属于访问阶段活动有关一类误差。 - **样本无应答**：又叫**单人无应答**，是指如果受访者对整个访问无论是问卷，还是访谈，都不回答的情形。简言之，就是无法从样本得到任何应答，比如说受访人拒访， 或者根本联系不上。 - **选项无应答**：又叫访题无应答，指受访者接受了访问，可能对某些访题不提供应答。 看起来这样的误差属于纯粹的访问误差，实际上不一定，也可以被认为是抽样误差的一种，比如，某些访题涉及到**稀有应答**，在抽样设计中，就需要予以考虑。 --- ## 非随机抽样误差：测量误差 **测量误差**是对感兴趣目标变量（一般是理论变量或可观测变量）的特征进行实际测度中（测量变量）产生的偏差。 > 我们会简单认为单位$i$上的兴趣特征$Y_i$是一个没有误差的固定量。然而，当存在测量误差时，$Y_i$并不是单位$i$的真正感兴趣的特征。相反，存在一些潜在的"真实"值$\mu_i$, `\(Y_i\)`是从调查中获得的$\mu_i$的测量值。 --- exclude: true ## 误差计算 本学期稍后学习！ （千呼万唤、望穿秋水的《概率论与数理统计》） ??? 我们了解了误差的来源， 也在头脑中有了警醒，在抽样和访问中会努力减少误差来源和误差， 我们也知道了在社会调查与研究中，研究误差指的是具体变量， 由样本值推论到总体值时可能的差距，那么误差怎么计算呢？ 在讨论误差计算之前，我重复一遍， 所有误差来源产生的误差，最后都会反映在样本与总体之间的差距上。 问题是，什么样的差距呢？ 为此，在统计上专门有术语，统计量，用来表达这些差距， 包括偏差、均方误差，还有比如说样本均值，样本方差，样本标准物， 标准差，总体均值，总体方差 等等。注意，这些统计量指的是具体变量的统计量， 要比较的，也是具体变量的统计量。 还有不要忘记了，如果我们把抽样方法也当做是工具， 那么误差的来源只有两类，一类呢，是工具的误差， 一类是既有工具，也有人为因素的误差。 如果我们按照某一抽样方案反复抽样， 用样本估计值的数学期望，与待估参数进行比较， 它们之间的离差就可能是抽样方案造成的误差， 我们称之为偏差，bias，用离差表示。 还记得样本估计量？在抽样的逻辑中讲过，在用同一个抽样方案反复抽样中， 如果把每一次的偏差记下来，就构成了一个分布，样本估计量 的分布，这里算的不是样本分布，是统计量的分布， 就是偏差。 可是如果有各种方案，就会有多种估计值，其中还会有 人为因素的影响，这个时候，就既有偏差又有误差，我们统称之为误差， error，用均方误差来表示。 至于为什么要用离差，为什么要用均方误差，社会科学各学科的统计课程会告诉大家。 在误差中，由抽样活动导致的样本随机性 所造成的样本统计量与总体统计量之间的差异，被叫做 抽样误差。在抽样中呢，抽样误差是一个一般性的概念，包含着不同的统计量， 用于刻画变量变异性的分布。 如果误差不是由样本的随机性带来的，而是由其它的因素带来的，比如说 抽样框误差，测量误差，访问误差等等。 这些误差呢，被统称为非抽样误差。 对抽样而言，我们关注的重点是抽样误差。 我们知道统计量是用来刻画变量变异性分布的， 那么各种统计量的含义究竟指什么呢？ 了解各种统计量的含义之前我们先介绍与统计量密切有关的两个概念，第一，参数值。 参数值，专门用来刻画总体某个变量变异性分布的状态， 比如总体均值，总体方差，总体比例，总体比率， 最常用到的是总体均值，总体方差 和总体比例。比例与比率还不一样。 比例，是指总量为一，p与q的之间的关系， 比率呢，又叫占比，指相对份额。第二，估计量。 估计量专门用来刻画样本某个变量变异性分布的状态， 通常也称统计量，统计值，估计量，在不同的场合有不同的说法。 常见的估计量比如说，样本均值，样本方差，样本比例。 这样，大家就要有一个概念，只要是讲参数值， 指的就是对总体的刻画，如果讲估计量呢，指的就是对样本的刻画。 在讨论中我们一再强调刻画的是 变量变异性的分布，那么在一般意义上如何概括变异性的分布呢？ 最常见的刻画， 是对集中趋势和离散趋势的刻画，还记得在抽样的逻辑中，讨论收入 分布时候的情形？随着样本量的增加，样本估计量的平均值越来越 向总体平均值收敛，对了，刻画变量变异性集中趋势的 就是均值，分布越集中，同质性也就越强，反之，异质性也就越强。均值代表的是 要素的同质性程度或者异质性程度，这里给出的是总体均值的计算方法。 这个公式呢，应该容易理解，x拔代表均值，n代表总体要素数， xi呢，代表一个具体的要素，把总体要素的变异值加总， 除以n，就是总体均值。 样本均值的计算方法一样。为了避免小样本条件下样本量对样本均值的影响， 在除法的分母部分，通常还要用样本量减去一， 可是仅仅知道了均值，并不能全面了解变量变异性分布的状态， 我们还需要了解其异质性，也就是离散趋势， 进而检验集中趋势是不是真实的，以及有多集中，这就需要用方差来刻画。 当知道了均值以后，每一个要素的变量值与均值之间都有一个关系值， 在样本中呢，观察值与平均值之间的差，又叫离差。 这个关系值，要么相等， 要么大于或者小于，如果用要素变量值减去变量均值， 可能得到的结果有三类，零，正数， 负数。如果把这些结果加总，关系值 的正负属性就会搅乱真正的关系属性。为了避免这个问题， 我们对每个关系值进行平方，就得到了每个要素与均值之间关系 值的平方，如果把这些关系值加总再除以总体要素n， 是不是就得到了一个要素值与均值之间差距的平方值？ 这就是总体方差，表达要素在某个变量上与总体之间距离的程度， 当然方差越大，离散程度也就越大，方差越小呢，离散程度也就越小。同样， 我们也可以计算样本方差，把方差开方，就是每个要素与总体均值之间的距离，无论正负， 无论正负，同理呢，也用于样本， 当然在刻画样品估计量分布的时候，还有一些统计量， 比如说，四分位差，极值等等。不过知道均值、方差和标准差是最重要的。 对抽样而言，重要的是样本估计量方差， 又称之为估计量方差，统计量方差。在抽样的逻辑中， 我们提过，那么这个方差到底是什么意思呢？ 举一个例子，假设有一个总体，由三个人组成，他们的年龄分别是两岁，四岁和六岁， 现在假设从中有放回的随机抽取样本， 里边包含了两个个体，也就是样本容量等于二。假设一共抽了九次， 就得到了九组样本，这九组样本分别是二二，二四， 二六，四二，四四，四六，六二，六四和六六。 其实这是简化至极的例子，不太好。为什么呢？一般抽样不会这么抽，三个抽两个， 同时又是一个极好的例子，很容易告诉我们方差从哪里来。 如果我们按照实践来， 譬如假设有一百八十个人抽三十个人，实践上是相符了，但是 对于方差的说明却复杂了。我们用这个例子是希望让大家知道方差从哪里来， 就好了。我们用图表来看，这边是样本， 我一共抽了九组，这边呢，是样本均值，大家看均值的差异有多大， 这是呢总体均值。是四，如果选两个四组样本， 看看估计量方差，假设第一个选这四组， 那么它们的方差就是0.6667，如果选这四组呢？ 这一组的方差就是2.6667，这里大家就知道内部 的差异有多大了。用这个例子我们希望说明，均值与方差， 对理解内部差异性，异质性，是非常重要的估计量。 误差计算，还涉及到其它估计量， 对初学者而言呢，理解均值、方差、标准差也就足够了。 误差计算就讲到这里，谢谢大家。 --- layout:false background-image: url("../pic/thank-you-gif-funny-gentle.gif") class: inverse,center # 本节结束