R与SAS在生物统计学中的应用指南

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"《生物统计学手册》第三版是一本由John H. McDonald撰写的关于生物统计学的权威著作，提供了使用SAS和R进行统计分析的指导。书中包括了从基本概念到高级统计测试的详细解释，适用于Windows、Mac和Linux平台的免费统计编程语言R的配套指南也由Salvatore Mangiafico编写并提供，帮助读者理解和应用各种统计方法。" 《生物统计学手册》第三版涵盖了生物统计学的基本原理和各种统计测试，旨在帮助研究人员和学生深入理解生物数据的分析方法。作者首先介绍了数据分析的步骤，并讨论了生物变量的类型。接着，书中探讨了概率论，这是理解统计学的基础，包括基本的假设检验概念。在介绍这些概念时，书中特别强调了混淆变量的影响，这对于正确解读统计数据至关重要。书中的一个重要部分是针对名义变量的测试，如精确的拟合优度检验、功效分析、卡方检验、G检验以及Fisher's精确独立性检验。这些检验用于评估分类数据之间的关联或独立性，尤其在处理小样本时需特别注意方法的选择。书中还讨论了如何处理卡方和G检验中的小数字问题，以及如何进行重复的G检验和Cochran-Mantel-Haenszel检验，以适应不同类型的独立性测试。此外，书中还包括了描述性统计的内容，如中心趋势的统计量和离散度的度量，以及标准误差的概念，这些都是分析数据集基本信息的重要工具。书中还涉及了假设检验，如t检验和ANOVA，以及非参数检验，如Mann-Whitney U检验和Kruskal-Wallis检验，这些对于比较不同组间的差异非常有用。对于那些需要进行大量高级统计分析的用户，作者推荐学习SAS或类似程序，但同时也指出R语言由于其自由、跨平台和日益增长的用户群体，是值得考虑的替代选择。Salvatore Mangiafico编写的R语言伴侣指南为《生物统计学手册》中的许多统计测试提供了实例程序，有助于读者快速上手R语言。《生物统计学手册》第三版是生物科学领域进行统计分析的重要参考书，它不仅提供了理论知识，还提供了实用的计算指导，是研究者和学生提升统计分析技能的理想资源。

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analyzed the data using a test designed for a measurement variable, those two sleepy

isopods would cause the average time for males to be much greater than for females, and

the difference might look statistically significant. When converted to ranks and analyzed

using a non-parametric test, the last and next-to-last isopods would have much less

influence on the overall result, and you would be less likely to get a misleadingly

“significant” result if there really isn’t a difference between males and females.

Some variables are impossible to measure objectively with instruments, so people are

asked to give a subjective rating. For example, pain is often measured by asking a person

to put a mark on a 10-cm scale, where 0 cm is “no pain” and 10 cm is “worst possible

pain.” This is not a ranked variable; it is a measurement variable, even though the

“measuring” is done by the person’s brain. For the purpose of statistics, the important

thing is that it is measured on an “interval scale”; ideally, the difference between pain

rated 2 and 3 is the same as the difference between pain rated 7 and 8. Pain would be a

ranked variable if the pains at different times were compared with each other; for

example, if someone kept a pain diary and then at the end of the week said “Tuesday was

the worst pain, Thursday was second worst, Wednesday was third, etc....” These rankings

are not an interval scale; the difference between Tuesday and Thursday may be much

bigger, or much smaller, than the difference between Thursday and Wednesday.

Just like with measurement variables, if there are a very small number of possible

values for a ranked variable, it would be better to treat it as a nominal variable. For

example, if you make a honeybee sting people on one arm and a yellowjacket sting people

on the other arm, then ask them “Was the honeybee sting the most painful or the second

most painful?”, you are asking them for the rank of each sting. But you should treat the

data as a nominal variable, one which has three values (“honeybee is worse” or

“yellowjacket is worse” or “subject is so mad at your stupid, painful experiment that they

refuse to answer”).

Categorizing

It is possible to convert a measurement variable to a nominal variable, dividing

individuals up into a two or more classes based on ranges of the variable. For example, if

you are studying the relationship between levels of HDL (the “good cholesterol”) and

blood pressure, you could measure the HDL level, then divide people into two groups,

“low HDL” (less than 40 mg/dl) and “normal HDL” (40 or more mg/dl) and compare the

mean blood pressures of the two groups, using a nice simple two-sample t–test.

Converting measurement variables to nominal variables (“dichotomizing” if you split

into two groups, “categorizing” in general) is common in epidemiology, psychology, and

some other fields. However, there are several problems with categorizing measurement

variables (MacCallum et al. 2002). One problem is that you’d be discarding a lot of

information; in our blood pressure example, you’d be lumping together everyone with

HDL from 0 to 39 mg/dl into one group. This reduces your statistical power, decreasing

your chances of finding a relationship between the two variables if there really is one.

Another problem is that it would be easy to consciously or subconsciously choose the

dividing line (“cutpoint”) between low and normal HDL that gave an “interesting” result.

For example, if you did the experiment thinking that low HDL caused high blood

pressure, and a couple of people with HDL between 40 and 45 happened to have high

blood pressure, you might put the dividing line between low and normal at 45 mg/dl.

This would be cheating, because it would increase the chance of getting a “significant”

difference if there really isn’t one.

To illustrate the problem with categorizing, let’s say you wanted to know whether tall

basketball players weigh more than short players. Here’s data for the 2012-2013 men’s

basketball team at Morgan State University:

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Height and weight of the Morgan State University men’s basketball players.

If you keep both variables as measurement variables and analyze using linear regression,

you get a P value of 0.0007; the relationship is highly significant. Tall basketball players

really are heavier, as is obvious from the graph. However, if you divide the heights into

two categories, “short” (77 inches or less) and “tall” (more than 77 inches) and compare

the mean weights of the two groups using a two-sample t–test, the P value is 0.043, which

is barely significant at the usual P<0.05 level. And if you also divide the weights into two

categories, “light” (210 pounds and less) and “heavy” (greater than 210 pounds), you get 6

who are short and light, 2 who are short and heavy, 2 who are tall and light, and 4 who

are tall and heavy. The proportion of short people who are heavy is not significantly

different from the proportion of tall people who are heavy, when analyzed using Fisher’s

exact test (P=0.28). So by categorizing both measurement variables, you have made an

obvious, highly significant relationship between height and weight become completely

non-significant. This is not a good thing. I think it’s better for most biological experiments

if you don’t categorize.

Likert items

Social scientists like to use Likert items: they’ll present a statement like “It’s important

for all biologists to learn statistics” and ask people to choose 1=Strongly Disagree,

2=Disagree, 3=Neither Agree nor Disagree, 4=Agree, or 5=Strongly Agree. Sometimes

they use seven values instead of five, by adding “Very Strongly Disagree” and “Very

Strongly Agree”; and sometimes people are asked to rate their strength of agreement on a

9 or 11-point scale. Similar questions may have answers such as 1=Never, 2=Rarely,

3=Sometimes, 4=Often, 5=Always.

Strictly speaking, a Likert scale is the result of adding together the scores on several

Likert items. Often, however, a single Likert item is called a Likert scale.

There is a lot of controversy about how to analyze a Likert item. One option is to treat

it as a nominal variable with five (or seven, or however many) items. The data would then

be summarized by the proportion of people giving each answer, and analyzed using chi-

square or G–tests. However, this ignores the fact that the values go in order from least

Height

(inches)

Weight

(pounds)

180

185

170

190

220

200

190

225

215

225

210

208

220

270

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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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R与SAS在生物统计学中的应用指南

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