种群的空间分布型及抽样
李典谟
中科院动物研究所
Email:lidm@
2005年10月
(一）空间分布 型
• 1. 意义
种群生态特性：空间是聚集 分布还是 随机分布， 解
决抽样方法，提供理论依据。
• 2.分类
随机分布：泊松(Poisson)分布
聚集分布：负二项分布(negative binomial
distribution)
奈曼分布(Neyman)
泊松二项分布
The simplest view of spatial patterning can be obtained
by adopting an individual orientation, and asking the question,
Given the location of one individual, what is the probability
that another individual is nearby? There are three
possibilities:
1. This probability is increased—aggregated pattern
2. This probability is reduced—uniform pattern
3. This probability is unaffected—random pattern
Random Aggregated Uniform
Three possible types of spatial patterning of
individual animals or plant in a population.
• 3.频次分布理论公式
（1）泊松（普阿松）分布
例：蝗蝻的田间分布
0
2
0 5 0
1 0
1
2
0
0 1
1 2
(1)普阿松分布（Poisson 分布）
例：对公共汽车客流进行调查，统计某天上午10∶30—11∶47
左右每隔20秒钟来到的乘客批数，共得到230个记录。
来到批数i 0 1 2 3 4 总共
频数ni 100 81 34 9 6 230
频率
普阿松分布的意义
已经发现许多随机现象服从普阿松分布
（1）社会生活，服务行业
如：电话交换台中来到的呼叫数
公共汽车站来到的 乘客数
（2）物理学
放射性分裂落到某区域的质点数
（3）昆虫个体的空间分布
普阿松分布的特点
以交换台电话呼叫数为例
（1）平衡性
在[t0,t0+t]中来到的呼叫数只与时间间隔长度t
有关，而与时间起点T0无关
（2）独立增量性（无后效性）
在[t0,t0+t]内来到k个呼叫这一事件与时刻T0前
发生的事件独立
（3）普通性
在充分小的时间间隔中，最多只来到一个呼叫
例：蝗蝻分布型调查，共取样例：蝗蝻分布型调查，共取样408408个个
虫数 x 频率 f f*x
0 225 0
1 130 130
2 40 80
3 10 30
4 3 12
408 252
计算方法计算方法
• 另样的理论数
n*p0=408*=
• 有一头虫的样本的理论数
n*p1=
观察值与理论值比较
虫数 x 观察值 (o) 理论值(c)
0 225
1 130 135..9
2 40
3 10
4 3
自由度自由度=n-2=3=n-2=3，失去两个自由度，失去两个自由度
（（11）用来限制实际样本数）用来限制实际样本数NN
(2) (2) 用来估计用来估计
意味不是一个小概率事件（p>),没有
理由否定假设
要求各组内的预计数都不少于5，当某组的Y少
于5时，须把它和相邻的一组或几组合并直到Y
大于5，然后再用上式计算 x2值。
检验的理论与方法
1 公式
O为实际观测值，E为理论推算值。
其基本原理是应用理论推算值与实际观测值
之间的偏离程度来决定其 值的大小。
是理论分布总体的频数
是观察分布总体的频数
两个样本来自不同的总体
2 分布的特点
df=1
df=3
df=5
（1） 分布于区间[1， ），偏斜度随自由度
降低而增大，当自由度df=1时，曲线以纵轴
为渐近线。
（2）随自由度df增大， 分布趋左右对称，当
df>30时， 分布接近正态。
3 检验的基本步骤
（1）建立检验假设，确定检验水平。
（2）计算检验统计量
（3）确定概率P值，计算自由度df＝k-1
由 和自由度查统计表 的临界值
（4）判断结果
临界值检验假设的关系
值 P 假设 判断
< > 不拒绝 差异无显著性
拒绝 差异有显著性
例：假定某地婴儿出生的男女比例为1：1。
研究者抽取了一个含10，000名婴儿的样品，男
孩5100，女孩4900，问他是否证实了假设或否定了
假设。
某地婴儿出生性比为1：1
> 拒绝 婴儿性比不为1：1
注：在自由度df＝1时，需进行连续性矫正，其矫
正的 为：
适合性检验
比较观测数与理论数是否符合的假设检验叫适
合性检验。例如在遗传学上，常用 检验来测定所
得的结果是否符合孟德尔分离规律，自由组合定律
等。
例 有一鲤鱼遗传试验，以荷包红鲤（红色）与湘
江野鲤（青灰色）杂交，其 代获得如表5-2所列
得体色分离尾数，问这一资料的实际观察值是否
符合孟德尔的青：红=3：1一对等为基因的遗传规
律？
表 鲤鱼遗传试验 F2观察结果
体 色 青 灰 色 红 色 总数
F2观测尾数 1503 99 1602
（1） 鲤鱼体色 分离符合3：1比率。
（2）取显著水平
（3）计算
青灰色理论数
红色理论数
（4）差 值表。df=1时，
故否定 ，接受
即鲤鱼体色 分离不符合3：1比率。
（2）负二项分布
• 正二项分布是( p+q)n 的展开式的各项,其中n为个体总数，
p,q为分成对比两类期望的比例。[Student (1907).]
展开上述式子，于是一个样本单位有r个个体的概率为
可以估算出p,k。矩法
• 由此可以推出
（二）分布型指数
上述蝗蝻例子中
说明上述蝗蝻属Poisson分布。
2. David&Moore (1954)方法
Index of Dispersion Test. We define an index of dispersion
I to be
For the theoretical Poisson distribution, the variance equals
the mean, so the expected value of I is always in a Poisson
world. The simplest test statistic for the index of dispersion is
a chi-squared one:
where I = Index of dispersion (as defined in equation )
n = Number of quadrats counted
= value of chi-squared with (n-1) degrees of
freedom.
0 4
1 8
2 2
3 5
4 2
5 3
6 1
虫数 频率
25
例：取了25个样，调查蚯蚓的田间分布。
由于 observed chi-squared
所以，我们接受原假设：蚯蚓田间分布符合Poisson
分布。
3. Waters(1959)
• 提出 负二项分布中的K
k’的特性：当种群密度因为随机死亡而减小时，k’保
持不变，表示种群空间分布的内在特点，而与密度无
关
4. Tayloz (1961,1965,1978)方法
密度越高，种群分布越均匀，（聚集度越低）
5. 平均拥挤度指标Lloyd,M.(1967)
例：
a
1
b
0
c
2
d
3
X1=1; x2=0
X3=2; x4=3
n=4
• A： 一头“独居” 1*(1-1)
• B: 没有邻居
• C: 有两头，各以对方为邻居；2*(2-1)=2
• D: 每个有两个邻居， 3*（3-1）=6，总共“邻居”数为：
0+0+2+6=8
平均每个个体有个邻居
Lloyd定义
• 聚集度指标：
• Iwao 发现
The idealized index should have three properties.
(Elliott 1977)
1. It should change in a smooth manner as moves
from maximum uniformity to randomness to
maximum aggregation.
2. It should not be affected by sample size (n),
population density ( ), or by variation in the size
and shape of the sampling quadrat.
3. It should be statistically tractable , so that a
confidence belt can be specified and comparixons
between samples can be tested for significance.
Morisita’s Index of Dispersion
(1)
Morisita’s index of dispersion
样本大小
sum of the quadnat counts =
Morisita (1962) 证明 随机分布的假设下：
Standardized Morisita Index
Uniform index = (2)
Clumped index = (3)
= Value of chi-squared from table with (n-1)
degrees of freedom that has % of the
area to the right.
When
When
When
When
取值以到+
带着95%置信区间
随机分布
聚集分布
均匀分布
In a simulation study Myers (1978) found
the standardized Morisita index to be one of
the best measures of dispersion because it was
independent of population density and sample
size.
例：E. Sinclair 在26个10公顷的样点调查大象的数量，
其中一个样点有20头，令一样点30头，还有一样
点10头，其他23点为零。
（1）计算 Morisita‘s index
（2）以公式(2)(3)中计算临界点。
当自由度= n-1=25,
Uniform index
Clumped index
（3）计算 Standardized Morisita index:
由于
（4）因为 于是我们得到结论：
在置信水平95%下，在我们取样区大象是聚集分布
的。
（三）Sample and Experimental Design
Sampling and experimental design are statistical jargon
for the three most obvious questions that can occur to a field
ecologist: Where should I take my samples, how should I
collect the data in space and time, and how many samples
should I try to take?
抽样理论及在生态学中的应用
• 1908年以“Student”笔名将“t-检验”发表于
《biometrika>上,文章中说:“任何实验可以作为是许多可能在
相同条件下作出的实验的总体中的一个个体.一系列的实验则
是以从这个总体中所抽得的一个样品”
1.总体与抽样
设一块棉田有N株棉株,每株上某种害虫数分别为
X1,X2…..XN,
• 从总体N中,随机抽取n株(n<N)样本,每株虫
数分别为X1,X2,……,Xn.
目的:通过样本对总体做出推断
抽样误差估计及t分布
• 1908年,“Student”发表了t分布
• 例: 棉田中随机调查50株棉株,以估计该棉田
中害虫的数量.
Sample Size for Continuous Variable
理论抽样数模型
例: 洪泽湖蝗区
虫数 样本数
(f)
fx
0 17 0
1 53 53
2 18 36
3 10 30
4 2 8
100 127
如果，我们引入变异系数（coefficient of variation）
这儿， =标准差
=观察平均数
那么，绝对误差
可写成相对误差 ，（以百分比形式）
（方程1）
两个平均数的比较
例如，我们要比较两个池塘中同一种鱼的重量是否有差异，
典型的方法是个抽取一定数量的样本用t检验来检验两样本
平均数是否有差异。但是，如何在抽样前回答应该取多少
样？
Snedecor and Cochran (1967,113)提出了如下的近似公
式：
一般
这儿 =从两个种群中的每一个抽取的样本大小；
=水平为 的标准正态离差值
（ ）
=水平为 的Ⅱ型错误概率下的标准正态离差值
（见下表）
=测量的方差。（已知，或推测）。
=你希望以 概率能检测出的两平均
值的最小差异。
Type Ⅱ error Power Two-tailed
决策
Power越大，决策结果越可靠
不拒绝H0 拒绝H0
H0 是真 决策正确（概率＝1－α） I型错误（概率＝α）
H0 是假 II型错误（P＝β） 决策正确（P＝1－β）＝power
例. 如果上例中我们希望检测出的平均数差异是：
（从以前的研究中知道）
如果， 则
条。
2. SAMPLE SIZE FOR DISCRETE VARIABLES
Counts of the numbers of plants in a quadrat or the numbers of eggs in a
nest differ from continuous variables in their statistical properties. The
frequency distribution of counts will often be described by either the
binomial distribution, the Poisson distribution or the negative binomial
distribution (Elliott 1977). The sampling properties of these distributions
differ, so we require a different approach to estimating sample sizes needed
for counts.
(1) Proportions and Percentages
Proportions like the sex ratio or fraction of juveniles in a population are
described statistically by the binomial distribution. All the organisms are
classified into two classes, and the distribution has only two parameters:
Proportion of types in the population
Proportion of types in the population
If sample size is above 20, we can use the normal approximation
to the confidence interval:
Where Observed proportion
Value of Student’s t-distribution for n-1
degrees of freedom
Standard error of
Thus the desired margin of error is
Solving for n, the sample size required is
where n=Sample size needed for estimating the proportion p
d=Desired margin of error in our estimate
As a first approximation for we can use We
need to have an approximate value of p to use in this equation. Prior
information, or a guess, should be used; the only rule-of-thumb is that
when in doubt, pick a value of p closer to than you guess. This will
make your answer conservative.
As an example, suppose you wish to estimate the sex ratio of a
deer population. You expect p to be about , and you would like to
estimate p within an error limit of with . From
equation
(2) Counts from a Poisson Distribution
Sample size estimation is very simple for any variable that can be
described by the Poisson distribution, in which the variance equals the
mean. From this it follows that
or
Thus from equation,(1) assuming :
where Sample size required for a Poisson variable
Desired relative error (as percentage)
Coefficient of variation =
For example ,if you are counting eggs in starling nests and know
that these counts fit a Poisson distribution and that the mean is
about , then if you wish to estimate this mean with precision of
(width of confidence interval), you have:
nests
Equation (2) can be simplified for the normal range of relative
errors as follows:
For precision
3. STATISTICAL POWER ANALYSIS
Decision
State of real world Do not reject null hypothesis Reject the null hypothesis
Null hypothesis is Correct decision Type Ⅰerror
actually true (probability =1- ) (probability = )
Null hypothesis is Type Ⅱerror Correct decision
actually false (probability = ) (probability =(1- )=power)
Most ecologists worry about , the probability of a Type Ⅰerror,
but there is abundant evidence now that we should worry just as
much or more about ,the probability of a Type Ⅱerror (Peterman
1990; Fairweather 1991).
Power analysis can carried out before you begin your study (a
priori, or prospective power analysis) or after you have finished
(retrospective power analysis). Here we discuss a priori power analysis as
it is used for the planning of experiments. Thomas (1997)discussed
retrospective power analysis.
The key point you should remember is that there are four variables
affecting any statistical inference:
sample size
Probability of a Probability of a
Type Ⅰerror Type Ⅱerror
Magnitude of the
effect = effect size
These four variables are interconnected, and once any three of them are
fixed, the fourth is automatically determined. Looked at from another
perspective, given any three of these, you can determine the fourth.
Figure An example of how power calculations can be visualized.
In this simple example, a t-test to be carried out to determine if the
plant nitrogen level has changed from the base level of % (the null hypothese )to
the improved level of % (the alternative hypothese). Given n=100, s
SUMMARY
The most common question in ecological research is, how large a
sample should I take? This chapter attempts to give a general answer
to this question by providing a series of equations from which sample
size may be calculated. It is always necessary to know something
about the population you wish to analyze unless you use guesswork
or prior observations. You must also make some explicit decision
about how much error you will allow in your estimates (or how small
a confidence interval you wish to have).
For continuous variables like weight or length, we can assume a
normal distribution and calculate the required sample sizes for means
and for variances quite precisely. For counts, we need to know the
underlying statistical distribution—binomial, Poisson, or negative
binomial—before we can specify sample sizes needed.
Power analysis explores the relationships between the
four interconnected variables (probability of Type
Ⅰerror), (probability of Type Ⅱerror), effect size, and
sample size. Fixing three of these automatically fixes the
fourth, and ecologists should explore these relationships
before they begin their experiments. Significant effect sizes
should be specified on ecological grounds before a study is
begun.
Sampling Designs: Random, Adaptive and Systematic
Sampling
(1)Simple Random Sampling
(2)Stratilied Random Sampling
(3)Adaptive Sampling
(4)Systematic Sampling
Simple random sampling is the easiest and most
common sampling design. Each possible sample unit must
have an equal chance of being selected to obtain a random
sample. All the formulas of statistics are based on random
sampling, and probability theory is the foundation of
statistics. Thus you should always sample randomly when
you have a choice.
In some cases the statistical population is finite in size,
and the idea of a finite population correction must be added
into formulas for variances and standard errors. These
formulas are reviewed for measurements, ratios, and
proportion.
Often a statistical population can be subdivided into homogeneous
subpopulations, and random sampling can be applied to each
subpopulation separately. This is stratified random sampling, and
represents the single most powerful sampling design that ecologists can
adopt in the field with relative ease. Stratified sampling is almost
always more precise than simple random sampling, and every ecologist
should use it whenever possible.
Sample size allocation in stratified sampling can be determined
using proportional or optimal allocation. To use optimal allocation,
you need rough estimates of the variances in each of the strata and the
cost of sampling each strata. Optimal allocation is more precise than
proportional allocation, and is to be preferred. Some simple rules are
presented to allow you to estimate the optimal number of strata you
should define in setting up a program of stratified random sampling.
If organisms are rare and patchily distributed, you
should consider using adaptive cluster sampling to
estimate abundance. When a randomly placed quadrat
contains a rare species, adaptive sampling adds
quadrats in the vicinity of the original quadrat to
sample the potential cluster. This additional nonrandom
sampling requires special formulas to estimate
abundance without bias.
Systematic sampling is easier to apply in the field
than random sampling, but may produce biased estimates
of means and confidence limits if there are periodicities in
the data. In field ecology this is usually not the case, and
systematic samples seem to be the equivalent of random
samples in many field situations. If a gradient exists in the
ecological community, systematic sampling will be better
than random sampling for describing it.
Systematic Sampling
• What is the likelihood that problems like periodic
variation will occur in actual field data?
Milne(1959) attempted to answer this question
by looking at systematic samples taken on
biological populations that had been completely
enumerated. He analyzed data from 50
populations and found that, in practice, there
was no error introduced by that a centric
systematic sample is a simple random sample,
and using all the appropriate formulas from
random sampling theory.
Step 1. Calculate the average abundance of each of the networks:
()
where =Average abundance of the i-th network
=Abundance of the organism in each of the k quadrats
in the i-th network
=Number of quadrats in the i-th netwrok
Step 2. From these values we obtain an estimator of the mean abundance as
follows:
()
where Unbiased estimate of mean abundance from adaptive cluster sampling
Number of initial sampling units selected via random sampling
If the initial sample is selected with replacement, the variance of this mean is
given by:
()
where Estimated variance of mean abundance for sampling with
replacement and all other terms are as defined above.
If the initial sample is selected without replacement, the variance of the mean
is given by:
()
where N = Total number of possible sample quadrats in the sampling
universe
The example shown in Figure . in the initial random sample of n =
10 quadrats, from equation().
plants per quadrat
Since we were sampling without replacement, we use equation ()
to estimate the variance of this mean:
We can obtain confidence limits from these estimates in the usual way:
For this example with n = 10, for 95% confidence limits , and the
confidence limits become:
or from to plants per quadrat.
When should one consider using adaptive sampling? Much depends on
the abundance and the spatial pattern of the animals or the plants being
studied. In general the more clustered the population and the rarer the
organism, the more efficient it will be to use adaptive cluster sampling
.Thompson(1992) shows, in Figure ,that adaptive sampling is about 12%
more efficient than simple random sampling for n = 10 quadrats and nearly
50% more efficient when n = 30 quadrats. In any particular situation it may
well pay to conduct a pilot experiment with simple random sampling and
adaptive cluster sampling to determine the size of the resulting variances.
序贯抽样法的基本原理
• 特点:在抽样时不预先指定子样容量,而是要求给出
一组停止采样的规则.
检验的步骤
拒绝
则接受
拒绝
拒绝
令
例：东亚飞蝗蝗蝻的序贯抽样，田间分布属负二项分布，其
公共k值为，规定：每平方丈（1丈=10/3米）平均虫
口在1头以下为轻度发生；2-3头为中等发生；5头以上为
严重发生。于是有以下假设检验：
（1）发生程度在轻度与中等之间
H0:平均虫口密度D≦1头/平方丈为轻度发生
H1:D≧2头/平方丈为中等发生
（2）发生在中等与严重之间
H0 ： D≦3头/平方丈为中等
H1: D≧5头/平方丈为严重
常数 发生程度 发生程度
轻度 中等 中等 严重
Ho H1 H1 H2
