False positive and False negative

Type I error

A type I error, also known as an error of the first kind, occurs when the null hypothesis (H₀) is true, but is rejected. It is asserting something that is absent, a false hit. A type I error may be compared with a so called false positive (a result that indicates that a given condition is present when it actually is not present) in tests where a single condition is tested for. Type I errors are philosophically a focus of skepticism and Occam's razor. A Type I error occurs when we believe a falsehood.^[1] In terms of folk tales, an investigator may be "crying wolf" without a wolf in sight (raising a false alarm) (H₀: no wolf).

The rate of the type I error is called the size of the test and denoted by the Greek letter  (alpha). It usually equals the significance level of a test. In the case of a simple null hypothesis  is the probability of a type I error. If the null hypothesis is composite,  is the maximum (supremum) of the possible probabilities of a type I error.

False positive error

A false positive error, commonly called a "false alarm" is a result that indicates a given condition has been fulfilled, when it actually has not been fulfilled. In the case of "crying wolf" - the condition tested for was "is there a wolf near the herd?", the actual result was that there had not been a wolf near the herd. The shepherd wrongly indicated there was one, by calling "Wolf, wolf!".

A false positive error is a Type I error where the test is checking a single condition, and results in an affirmative or negative decision usually designated as "true or false".

Type II error

A type II error, also known as an error of the second kind, occurs when the null hypothesis is false, but it is erroneously accepted as true. It is missing to see what is present, a miss. A type II error may be compared with a so-called false negative (where an actual 'hit' was disregarded by the test and seen as a 'miss') in a test checking for a single condition with a definitive result of true or false. A Type II error is committed when we fail to believe a truth.^[1] In terms of folk tales, an investigator may fail to see the wolf ("failing to raise an alarm"; see Aesop's story of The Boy Who Cried Wolf). Again, H₀: no wolf.

The rate of the type II error is denoted by the Greek letter  (beta) and related to the power of a test (which equals ).

What we actually call type I or type II error depends directly on the null hypothesis. Negation of the null hypothesis causes type I and type II errors to switch roles.

The goal of the test is to determine if the null hypothesis can be rejected. A statistical test can either reject (prove false) or fail to reject (fail to prove false) a null hypothesis, but never prove it true (i.e., failing to reject a null hypothesis does not prove it true).

False negative error

A false negative error is where a test result indicates that a condition failed, while it actually was successful. A common example is a guilty prisoner freed from jail. The condition: "Is the prisoner guilty?" actually had a positive result (yes, he is guilty). But the test failed to realize this, and wrongly decided the prisoner was not guilty.

A false negative error is a type II error occurring in test steps where a single condition is checked for and the result can either be positive or negative.

Example

As it is conjectured that adding fluoride to toothpaste protects against cavities, the null hypothesis of no effect is tested. When the null hypothesis is true (i.e., there is indeed no effect), but the data give rise to rejection of this hypothesis, falsely suggesting that adding fluoride is effective against cavities, a type I error has occurred.

A type II error occurs when the null hypothesis is false (i.e., adding fluoride is actually effective against cavities), but the data are such that the null hypothesis cannot be rejected, failing to prove the existing effect.

In colloquial usage type I error can be thought of as "convicting an innocent person" and type II error "letting a guilty person go free".

Tabularised relations between truth/falseness of the null hypothesis and outcomes of the test:

	Null hypothesis (H0) is true	Null hypothesis (H0) is false
Reject null hypothesis	Type I error False positive	Correct outcome True positive
Fail to reject null hypothesis	Correct outcome True negative	Type II error False negative

Understanding Type I and Type II errors

From the Bayesian point of view, a type I error is one that looks at information that should not substantially change one's prior estimate of probability, but does. A type II error is one that looks at information which should change one's estimate, but does not. (Though the null hypothesis is not quite the same thing as one's prior estimate, it is, rather, one's pro forma prior estimate.)

Hypothesis testing is the art of testing whether a variation between two sample distributions can be explained by chance or not. In many practical applications type I errors are more delicate than type II errors. In these cases, care is usually focused on minimizing the occurrence of this statistical error. Suppose, the probability for a type I error is 1% , then there is a 1% chance that the observed variation is not true. This is called the level of significance, denoted with the Greek letter  (alpha). While 1% might be an acceptable level of significance for one application, a different application can require a very different level. For example, the standard goal of six sigma is to achieve precision to 4.5 standard deviations above or below the mean. This means that only 3.4 parts per million are allowed to be deficient in a normally distributed process

How to Read a Scientific Research Paper

How to Read a Scientific Research Paper-- a four-step guide for students
adapted from Here

Reading research papers ("primary articles") is partly a matter of experience and skill, and partly learning the specific vocabulary of a field. First of all, DON'T PANIC! If you approach it step by step, even an impossible-looking paper can be understood.

1. Skimming. Skim the paper quickly, noting basics like headings, figures and the like. This takes just a few minutes. You're not trying to understand it yet, but just to get an overview.

2. Vocabulary. Go through the paper word by word and line by line, underlining or highlighting every word and phrase you don't understand. Don't worry if there are a lot of underlinings; you're still not trying to make sense of the article.

Now you have several things you might do with these vocabulary and concept questions, depending upon the kind of question each is. You can

1. Look up simple words and phrases. Often the question is simply vocabulary--what's a lateral malleolus, or a christa, or the semilunar valve. A medical or biological dictionary is a good place to look for definitions. A textbook of physiology or anatomy may be a good source, because it give more complete explanations. Your ordinary shelf dictionary is not a good source, because the definitions may not be precise enough or may not reflect the way in which scientists use a word (for example "efficiency" has a common definition, but the physical definition is much more restricted.)
2. Get an understanding from the context in which it is used. Often words that are used to describe the procedures used in an experiment can be understood from the context, and may be very specific to the paper you are reading. Examples are the "lithium-free control group" in a rat experiment or the "carotene extraction procedure" in a biochemical experiment. Of course, you should be careful when deciding that you understand a word from its context, because it might not mean what you think.
3. Flag this phrase as belonging to one of the major concepts of the paper--it's bigger than a vocabulary question. For example, a paper about diet and cancer might refer to "risk reduction," which you would need to understand in context and in some depth.

3. Comprehension, section by section. Try to deal with all the words and phrases, although a few technical terms in the Methods section might remain. Now go back and read the whole paper, section by section, for comprehension.

In the Introduction, note how the context is set. What larger question is this a part of? The author should summarize and comment on previous research, and you should distinguish between previous research and the actual current study. What is the hypothesis of the paper and the ways this will be tested?

In the Methods, try to get a clear picture of what was done at each step. What was actually measured? It is a good idea to make an outline and/or sketch of the procedures and instruments. Keep notes of your questions; some of them may be simply technical, but others may point to more fundamental considerations that you will use for reflection and criticism below.

In Results look carefully at the figures and tables, as they are the heart of most papers. A scientist will often read the figures and tables before deciding whether it is worthwhile to read the rest of the article! What does it mean to "understand" a figure? You understand a figure when you can redraw it and explain it in plain English words.

The Discussion contains the conclusions that the author would like to draw from the data. In some papers, this section has a lot of interpretation and is very important. In any case, this is usually where the author reflects on the work and its meaning in relation to other findings and to the field in general.

4. Reflection and criticism. After you understand the article and can summarize it, then you can return to broader questions and draw your own conclusions. It is very useful to keep track of your questions as you go along, returning to see whether they have been answered. Often, the simple questions may contain the seeds of very deep thoughts about the work--for example, "Why did the authors use a questionnaire at the end of the month to find out about premenstrual tension? Wouldn't subjects forget or have trouble recalling?"

Here are some questions that may be useful in analyzing various kinds of research papers:

              Introduction:

What is the overall purpose of the research?

How does the research fit into the context of its field? Is it, for example, attempting to settle a controversy? show the validity of a new technique? open up a new field of inquiry?

Do you agree with the author's rationale for studying the question in this way?

Methods:

Were the measurements appropriate for the questions the researcher was approaching?

Often, researchers need to use "indicators" because they cannot measure something directly--for example, using babies' birthweight to indicate nutritional status. Were the measures in this research clearly related to the variables in which the researchers (or you) were interested?

If human subjects were studied, do they fairly represent the populations under study?

Results

What is the one major finding?

Were enough of the data presented so that you feel you can judge for yourself how the experiment turned out?

Did you see patterns or trends in the data that the author did not mention? Were there problems that were not addressed?

Discussion

Do you agree with the conclusions drawn from the data?

Are these conclusions over-generalized or appropriately careful?

Are there other factors that could have influenced, or accounted for, the results?

What further experiments would you think of, to continue the research or to answer remaining questions?

数学公式的英语读法 (ZT)

数学公式的英语读法 (ZT)

1.Logic

∃there exist

∀for all

p⇒q p implies q / if p, then q

p⇔q p if and only if q /p is equivalent to q / p and q are equivalent

2.Sets

x∈A x belongs to A / x is an element (or a member) of A

x∉A x does not belong to A / x is not an element (or a member) of A

A⊂B A is contained in B / A is a subset of B

A⊃B A contains B / B is a subset of A

A∩B A cap B / A meet B / A intersection B

A∪B A cup B / A join B / A union B

A\B A minus B / the diference between A and B

A×B A cross B / the cartesian product of A and B

3. Real numbers

x+1 x plus one

x-1 x minus one

x±1 x plus or minus one

xy xy / x multiplied by y

(x-y)(x+y) x minus y, x plus y

= the equals sign

x=5 x equals 5 / x is equal to 5

x≠5 x (is) not equal to 5

x≡y x is equivalent to (or identical with) y

x>y x is greater than y

x≥y x is greater than or equal to y

x<y x is less than y

x≤y x is less than or equal to y

0<x<1 zero is less than x is less than 1

0≤x≤1 zero is less than or equal to x is less than or equal to 1

|x| mod x / modulus x

x² x squared / x (raised) to the power 2

x³ x cubed

x⁴ x to the fourth / x to the power 4

xⁿ x to the nth / x to the power n

x ⁽⁻ⁿ⁾ x to the (power) minus n

x的平方根(square) root x / the square root of x

x的三次根cube root (of) x

x的四次根fourth root (of) x

x的n次根nth root (of) x

(x+y)² x plus y all squared

n! n factorial

x^x hat

x¯ x bar

x˜ x tilde

x_i xi / x subscript i / x suffix i / x sub i

∑(i=1~n) a_i the sum from i equals one to n a_i / the sum as i runs from 1 to n of the a_i

4. Linear algebra

‖x‖the norm (or modulus) of x

OA^→OA / vector OA

OA¯ OA / the length of the segment OA

A^T A transpose / the transpose of A

A⁻¹ A inverse / the inverse of A

5. Functions

f(x) fx / f of x / the function f of x

f:S→T a function f from S to T

x→y x maps to y / x is sent (or mapped) to y

f’(x) f prime x / f dash x / the (first) derivative of f with respect to x

f”(x) f double-prime x / f double-dash x / the second derivative of f with respect to x

f”’(x) triple-prime x / f triple-dash x / the third derivative of f with respect to x

f (4) (x) f four x / the fourth derivative of f with respect to x

∂f/∂x₁ the partial (derivative) of f with respect to x1

∂²f/∂x₁² the second partial (derivative) of f with respect to x1

∫₀^∞ the integral from zero to infinity

lim_x→0the limit as x approaches zero

lim_x→0+the limit as x approaches zero from above

lim_x→0−the limit as x approaches zero from below

log_ey log y to the base e / log to the base e of y / natural log (of) y

lny log y to the base e / log to the base e of y / natural log (of) y

SLAM Paper Repository

SLAM Paper Repository
http://www.cas.kth.se/SLAM/slam-papers.html

Homepage of Tim Bailey
http://www-personal.acfr.usyd.edu.au/tbailey/