生物学统计手册：SAS与R软件的应用

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"《生物统计学手册_3rd版》由John H. McDonald撰写，是针对实验数据分析中常用统计测试的详细解释和实例集合。这本书虽然以生物学为例，但其统计方法广泛适用于多个领域。手册主要使用SAS软件进行演示，并提供了一些在线计算器或电子表格的链接。为了增加其普适性，特别是对于无法访问SAS的学者和研究者，书中还增加了R语言的示例，R是一个免费且广受欢迎的统计软件包。" 《生物统计学手册》第三版是一部全面介绍生物统计学概念和方法的著作。内容涵盖了数据分析的基础知识、概率理论、假设检验、混淆变量的影响，以及针对名义变量的各种测试。 1. **基础知识**：这部分介绍了数据分析的基本步骤，帮助读者理解如何逐步分析生物学数据。同时，它探讨了生物变量的类型，如定量和定性变量的区别。 2. **概率**：解释了概率的基本概念，这是理解统计学和假设检验的基础。 3. **基本的假设检验概念**：书中详细阐述了假设检验的核心思想，包括显著性水平、p值和拒绝域。 4. **混淆变量**：混淆变量可能会导致统计推断错误，本节讨论如何识别和控制这些变量，以获得更准确的结果。 5. **名义变量的测试**：包括精确拟合性检验、权检验（G-检验）、卡方拟合性检验、独立性检验（卡方检验和G-检验）以及Fisher精确独立性检验。这些测试用于评估分类数据之间的关系。 6. **小样本问题**：针对卡方和G-检验中的小样本问题，提供了处理策略。 7. **重复G-检验**和**Cochran-Mantel-Haenszel检验**：这些方法用于处理多次独立性测试的情况，特别是在有等级结构的数据中。 8. **描述性统计**：涵盖了集中趋势的统计量（如均值、中位数、众数）和离散度的统计量（如方差、标准差）。此外，还讲解了标准误差，它是估计总体参数精度的重要工具。 9. **其他主题**：除了以上内容，书中还涉及置信区间、t检验、ANOVA、回归分析等高级统计方法，以及如何在R和SAS中实现这些分析。这本书不仅适合生物学专业的学生和研究人员，也适用于其他领域需要进行统计分析的学者。通过提供丰富的实例和代码，使得读者能够理解和应用统计学原理解决实际问题。

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agreement to most, which is pretty important information. The other options are to treat it

as a ranked variable or a measurement variable.

Treating a Likert item as a measurement variable lets you summarize the data using a

mean and standard deviation, and analyze the data using the familiar parametric tests

such as anova and regression. One argument against treating a Likert item as a

measurement variable is that the data have a small number of values that are unlikely to

be normally distributed, but the statistical tests used on measurement variables are not

very sensitive to deviations from normality, and simulations have shown that tests for

measurement variables work well even with small numbers of values (Fagerland et al.

2011).

A bigger issue is that the answers on a Likert item are just crude subdivisions of some

underlying measure of feeling, and the difference between “Strongly Disagree” and

“Disagree” may not be the same size as the difference between “Disagree” and “Neither

Agree nor Disagree”; in other words, the responses are not a true “interval” variable. As

an analogy, imagine you asked a bunch of college students how much TV they watch in a

typical week, and you give them the choices of 0=None, 1=A Little, 2=A Moderate

Amount, 3=A Lot, and 4=Too Much. If the people who said “A Little” watch one or two

hours a week, the people who said “A Moderate Amount” watch three to nine hours a

week, and the people who said “A Lot” watch 10 to 20 hours a week, then the difference

between “None” and “A Little” is a lot smaller than the difference between “A Moderate

Amount” and “A Lot.” That would make your 0-4 point scale not be an interval variable.

If your data actually were in hours, then the difference between 0 hours and 1 hour is the

same size as the difference between 19 hours and 20 hours; “hours” would be an interval

variable.

Personally, I don’t see how treating values of a Likert item as a measurement variable

will cause any statistical problems. It is, in essence, a data transformation: applying a

mathematical function to one variable to come up with a new variable. In chemistry, pH is

the base-10 log of the reciprocal of the hydrogen activity, so the difference in hydrogen

activity between a ph 5 and ph 6 solution is much bigger than the difference between ph 8

and ph 9. But I don’t think anyone would object to treating pH as a measurement variable.

Converting 25-44 on some underlying “agreeicity index” to “2” and converting 45-54 to

“3” doesn’t seem much different from converting hydrogen activity to pH, or

micropascals of sound to decibels, or squaring a person’s height to calculate body mass

index.

The impression I get, from briefly glancing at the literature, is that many of the people

who use Likert items in their research treat them as measurement variables, while most

statisticians think this is outrageously incorrect. I think treating them as measurement

variables has several advantages, but you should carefully consider the practice in your

particular field; it’s always better if you’re speaking the same statistical language as your

peers. Because there is disagreement, you should include the number of people giving

each response in your publications; this will provide all the information that other

researchers need to analyze your data using the technique they prefer.

All of the above applies to statistics done on a single Likert item. The usual practice is

to add together a bunch of Likert items into a Likert scale; a political scientist might add

the scores on Likert questions about abortion, gun control, taxes, the environment, etc. and

come up with a 100-point liberal vs. conservative scale. Once a number of Likert items are

added together to make a Likert scale, there seems to be less objection to treating the sum

as a measurement variable; even some statisticians are okay with that.

Independent and dependent variables

Another way to classify variables is as independent or dependent variables. An

independent variable (also known as a predictor, explanatory, or exposure variable) is a

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variable that you think may cause a change in a dependent variable (also known as an

outcome or response variable). For example, if you grow isopods with 10 different

mannose concentrations in their food and measure their growth rate, the mannose

concentration is an independent variable and the growth rate is a dependent variable,

because you think that different mannose concentrations may cause different growth

rates. Any of the three variable types (measurement, nominal or ranked) can be either

independent or dependent. For example, if you want to know whether sex affects body

temperature in mice, sex would be an independent variable and temperature would be a

dependent variable. If you wanted to know whether the incubation temperature of eggs

affects sex in turtles, temperature would be the independent variable and sex would be

the dependent variable.

As you’ll see in the descriptions of particular statistical tests, sometimes it is important

to decide which is the independent and which is the dependent variable; it will determine

whether you should analyze your data with a two-sample t–test or simple logistic

regression, for example. Other times you don’t need to decide whether a variable is

independent or dependent. For example, if you measure the nitrogen content of soil and

the density of dandelion plants, you might think that nitrogen content is an independent

variable and dandelion density is a dependent variable; you’d be thinking that nitrogen

content might affect where dandelion plants live. But maybe dandelions use a lot of

nitrogen from the soil, so it’s dandelion density that should be the independent variable.

Or maybe some third variable that you didn’t measure, such as moisture content, affects

both nitrogen content and dandelion density. For your initial experiment, which you

would analyze using correlation, you wouldn’t need to classify nitrogen content or

dandelion density as independent or dependent. If you found an association between the

two variables, you would probably want to follow up with experiments in which you

manipulated nitrogen content (making it an independent variable) and observed

dandelion density (making it a dependent variable), and other experiments in which you

manipulated dandelion density (making it an independent variable) and observed the

change in nitrogen content (making it the dependent variable).

References

Fagerland, M. W., L. Sandvik, and P. Mowinckel. 2011. Parametric methods outperformed

non-parametric methods in comparisons of discrete numerical variables. BMC

Medical Research Methodology 11: 44.

MacCallum, R. C., S. B. Zhang, K. J. Preacher, and D. D. Rucker. 2002. On the practice of

dichotomization of quantitative variables. Psychological Methods 7: 19-40.

Zar, J.H. 1999. Biostatistical analysis. 4th edition. Prentice Hall, Upper Saddle River, NJ.

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Probability

Although estimating probabilities is a fundamental part of statistics, you will rarely

have to do the calculations yourself. It’s worth knowing a couple of simple rules about

adding and multiplying probabilities.

Introduction

The basic idea of a statistical test is to identify a null hypothesis, collect some data,

then estimate the probability of getting the observed data if the null hypothesis were true.

If the probability of getting a result like the observed one is low under the null hypothesis,

you conclude that the null hypothesis is probably not true. It is therefore useful to know a

little about probability.

One way to think about probability is as the proportion of individuals in a population

that have a particular characteristic. The probability of sampling a particular kind of

individual is equal to the proportion of that kind of individual in the population. For

example, in fall 2013 there were 22,166 students at the University of Delaware, and 3,679

of them were graduate students. If you sampled a single student at random, the

probability that they would be a grad student would be 3,679 / 22,166, or 0.166. In other

words, 16.6% of students were grad students, so if you’d picked one student at random,

the probability that they were a grad student would have been 16.6%.

When dealing with probabilities in biology, you are often working with theoretical

expectations, not population samples. For example, in a genetic cross of two individual

Drosophila melanogaster that are heterozygous at the vestigial locus, Mendel’s theory

predicts that the probability of an offspring individual being a recessive homozygote

(having teeny-tiny wings) is one-fourth, or 0.25. This is equivalent to saying that one-

fourth of a population of offspring will have tiny wings.

Multiplying probabilities

You could take a semester-long course on mathematical probability, but most

biologists just need to know a few basic principles. You calculate the probability that an

individual has one value of a nominal variable and another value of a second nominal

variable by multiplying the probabilities of each value together. For example, if the

probability that a Drosophila in a cross has vestigial wings is one-fourth, and the

probability that it has legs where its antennae should be is three-fourths, the probability

that it has vestigial wings and leg-antennae is one-fourth times three-fourths, or 0.25 × 0.75,

or 0.1875. This estimate assumes that the two values are independent, meaning that the

probability of one value is not affected by the other value. In this case, independence

would require that the two genetic loci were on different chromosomes, among other

things.

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Basic concepts of hypothesis

testing

One of the main goals of statistical hypothesis testing is to estimate the P value, which

is the probability of obtaining the observed results, or something more extreme, if the null

hypothesis were true. If the observed results are unlikely under the null hypothesis, your

reject the null hypothesis. Alternatives to this “frequentist” approach to statistics include

Bayesian statistics and estimation of effect sizes and confidence intervals.

Introduction

There are different ways of doing statistics. The technique used by the vast majority of

biologists, and the technique that most of this handbook describes, is sometimes called

“frequentist” or “classical” statistics. It involves testing a null hypothesis by comparing

the data you observe in your experiment with the predictions of a null hypothesis. You

estimate what the probability would be of obtaining the observed results, or something

more extreme, if the null hypothesis were true. If this estimated probability (the P value) is

small enough (below the significance value), then you conclude that it is unlikely that the

null hypothesis is true; you reject the null hypothesis and accept an alternative hypothesis.

Many statisticians harshly criticize frequentist statistics, but their criticisms haven’t

had much effect on the way most biologists do statistics. Here I will outline some of the

key concepts used in frequentist statistics, then briefly describe some of the alternatives.

Null hypothesis

The null hypothesis is a statement that you want to test. In general, the null hypothesis

is that things are the same as each other, or the same as a theoretical expectation. For

example, if you measure the size of the feet of male and female chickens, the null

hypothesis could be that the average foot size in male chickens is the same as the average

foot size in female chickens. If you count the number of male and female chickens born to

a set of hens, the null hypothesis could be that the ratio of males to females is equal to a

theoretical expectation of a 1:1 ratio.

The alternative hypothesis is that things are different from each other, or different

from a theoretical expectation. For example, one alternative hypothesis would be that

male chickens have a different average foot size than female chickens; another would be

that the sex ratio is different from 1:1.

Usually, the null hypothesis is boring and the alternative hypothesis is interesting. For

example, let’s say you feed chocolate to a bunch of chickens, then look at the sex ratio in

their offspring. If you get more females than males, it would be a tremendously exciting

discovery: it would be a fundamental discovery about the mechanism of sex

determination, female chickens are more valuable than male chickens in egg-laying

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生物学统计手册：SAS与R软件的应用

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