1 概念解释

1.1 中央极限定理 Central Limit Theorem

  • （总体容量N, 样本容量n, n>30）所有可能取到的样本的均值的分布是趋于正态分布的，所以我们可以用正态分布来描述这些均值。
  • 这些样本均值是原总体的统计量，他们服从\(N(\mu, \sigma^2/n)\)的正态分布
    ## Z 统计
  • z统计如果用来衡量观察值偏离总体均值（用样本均值代替）的程度，用他们差除标准差表示。
  • z统计如果用来衡量样本均值\(\mu\)偏离抽样均值\(\overline X\)（亦即总体均值）的程度，
    用他们差除标准误（即样本均值的标准差\(=\frac{\sigma}{\sqrt n}\)）表示。 \[t\_stat =\frac{\overline x - \mu}{s_s}\]
  • T 因子 T-multiplier 为标准正态分布下，2.5% 和 97.5% 的分为点
  • 因为抽样均值服从正态分布，所以\(样本均值 \pm T因子 \times 标准误\) 可以得到抽样均值的95%置信区间分为点，亦即总体均值的分为点
  • 若要分析总体均值\(\mu\)是否为某个值，则将这个指代入上式计算t_stat，得到这个抽样均值偏离总体均值的程度。
  • 再用t_stat计算\(p_value = 2*(1-pt(abs(t_stat),df))\)

1.2 列联表 contingency table

require(graphics)

## example from Venables and Ripley (2002, pp. 190-2.)
ldose <- rep(0:5, 2)
numdead <- c(1, 4, 9, 13, 18, 20, 0, 2, 6, 10, 12, 16)
sex <- factor(rep(c("M", "F"), c(6, 6)))
SF <- cbind(numdead, numalive = 20-numdead)
budworm.lg <- glm(SF ~ sex*ldose, family = binomial)
summary(budworm.lg)
## 
## Call:
## glm(formula = SF ~ sex * ldose, family = binomial)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -1.39849  -0.32094  -0.07592   0.38220   1.10375  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -2.9935     0.5527  -5.416 6.09e-08 ***
## sexM          0.1750     0.7783   0.225    0.822    
## ldose         0.9060     0.1671   5.422 5.89e-08 ***
## sexM:ldose    0.3529     0.2700   1.307    0.191    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 124.8756  on 11  degrees of freedom
## Residual deviance:   4.9937  on  8  degrees of freedom
## AIC: 43.104
## 
## Number of Fisher Scoring iterations: 4
plot(c(1,32), c(0,1), type = "n", xlab = "dose",
     ylab = "prob", log = "x")
text(2^ldose, numdead/20, as.character(sex))
ld <- seq(0, 5, 0.1)
lines(2^ld, predict(budworm.lg, data.frame(ldose = ld,
   sex = factor(rep("M", length(ld)), levels = levels(sex))),
   type = "response"))
lines(2^ld, predict(budworm.lg, data.frame(ldose = ld,
   sex = factor(rep("F", length(ld)), levels = levels(sex))),
   type = "response"))

“by a factor of” is used commonly to mean the same as “multiplied by” or “divided by.” If x is INCREASED by a factor of 4, it becomes 4x. If x is DECREASED by a factor of 4, it becomes x/4. The key word is the direction of change i.e. (increased / decreased) by a factor of.

2 英语中倍数的表达方法和译法

英语表示倍数增减或倍数对比的句型多种多样，其中有 一些（如下文中的句型⑵、⑸、⑹、⑻、⑿ 等）很容易译错，其主要原因在于：英汉两语在表述或对比倍数方面存在着语言与思维差异。现将常用的英语倍数句型及其正确译法归纳如下：

2.1 倍数增加

（一）A is n times as great (long, much, …) as B. ⑴

A is n times greater (longer, more, …) than B. ⑵

A is n times the size (length, amount, …) of B. ⑶

以上三句都应译为：A的大小（长度、数量、……）是B的n倍，或A比B大（长、多、……）n-1倍。

eg: This book is three times as long as that one. / three times longer than that one. / three times the length of that one.

这本书的篇幅是那本书的3倍（即比那本书长两倍）。

注：当相比的对象B很明显时，than (as, of) B 常被省去。

（二）increase to n times ⑷

increase n times / n-fold ⑸

increase by n times ⑹

increase by a factor of n ⑺

以上四句均应译为：增加到n倍，或增加n-1倍。

eg: The production of integrated circuits has been increased to three times as compared with last year.

集成电路的产量比去年增加了两倍。

eg: The output of chemical fertilizer has been raised five times as against 1986.

化肥产量比1986年增加了4倍。

eg: That can increase metabolic rates by two or three times.

那可使代谢率提高到原来的2倍或3倍（即提高1倍或2倍）。

eg: The drain voltage has been increased by a factor of four.

漏电压增加了3倍（即增加到原来的4倍）。

注：在这类句型中increase常被raise、grow、go/step up、multiply 等词所替代。

（三）There is a n-fold increase / growth… ⑻

应译为：增加n-1倍，或增至n倍。

这个句型还有其它一些形式：

eg: A record high increase in value of four times was reported.

据报道，价值破记录地增长了3倍。

（四）double（增加1倍）、treble（增加2倍）、quadruple（增加3倍）⑼

eg: The efficiency of the machines has been more than trebled or quadrupled.

这些机器的效率已提高了2倍或3倍多。

（五）此外，英语中还有一种用again而不用倍数词来比较倍数的方法，如：

A is as much (large, long, …) again as B. = A is twice as much (large, long, …) as B. ⑽

应译为：A比B多（大、长、……）1倍。

A is half as much (large, long, …) again as B. = A is one and a half times as much (large, long, …) as B. ⑾

应译为：A比B多（大、长、……）一半，即A是B的一倍半。

2.2 倍数减少

（一）A is n times as small (light, slow, …) as B. ⑿

A is n times smaller (lighter, slower, …) than B. ⒀

以上两句均应译为：A的大小（重量、速度、……）是B的1/n，或A比B小（轻、慢、……）(n-1)/n。

eg: The hydrogen atom is nearly 16 times as light as the oxygen atom.

氢原子的重量约为氧原子的1/16（即比氧原子约轻15/16）。

eg: This sort of membrane is twice thinner than ordinary paper.

这种薄膜比普通纸张要薄一半（即是普通纸厚度的1/2）。

注：当相比的对象B很明显时，than/as B 常被省去。

（二）decrease n times / n-fold ⒁

decrease by n times ⒂

decrease by a factor of n ⒃

以上三句均译为：减少到1/n，或减少(n-1)/n。decrease常被reduce、shorten、go/slow down 等词替代。

eg: Switching time of the new-type transistor is shortened 3 times.

新型晶体管的开关时间缩短到1/3（即缩短了2/3）。

eg: When the voltage is stepped up by ten times, the strength of the current is stepped down by ten times.

电压升高9倍，电流强度便降低9/10（即降低90%）。

eg: The equipment reduced the error probability by a factor of 5.

该设备误差概率降低了4/5。

（三）There is a n-fold decrease / reduction… ⒄

应译为：减至1/n，或减少(n-1)/n。

这个句型还有其它一些形式，

eg: A rapid decrease by a factor of 7 was observed.

发现迅速减少到1/7。

eg: The principal advantage of the products is a two-fold（两倍） reduction in weight.

这些产品的主要优点是重量减轻了1/2。

2.3 总结

从以上倍数增减句型及其译法中不难看出：与汉语不同的是，英语在表述或比较倍数时，无论使用什么句型（除了不含倍数词的again句型外）都包括基础倍数在内，因此都不是净增或净减n倍，而是净增或净减n-1倍。所以句型 ⑶、⑸ 表示的倍数增量=句型 ⑷；句型 ⒀ 表示的倍数比差=句型 ⑿，且decrease (by) 3 times应译为“减少2/3”，而不是“减少3/4”。

  1. ordered list
  2. item 2
    1. sub-item 1
      • sub-sub-item 1
  1. A list whose numbering continues after
  2. an interruption