# Type 1 And Type 2 Errors In Statistics Pdf

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The statistical education of scientists emphasizes a flawed approach to data analysis that should have been discarded long ago.

Sign in. If the p-value falls in the confidence interval, we fail to reject the null hypothesis and if it is out of the interval then we reject it. But recently I realized that in the experimental design, the power of the hypothesis test is crucial to understand to choose the appropriate sample size.

Hypothesis testing is an important activity of empirical research and evidence-based medicine. A well worked up hypothesis is half the answer to the research question. For this, both knowledge of the subject derived from extensive review of the literature and working knowledge of basic statistical concepts are desirable.

## Introduction to Type I and Type II errors

In statistical hypothesis testing , a type I error is the rejection of a true null hypothesis also known as a "false positive" finding or conclusion; example: "an innocent person is convicted" , while a type II error is the non-rejection of a false null hypothesis also known as a "false negative" finding or conclusion; example: "a guilty person is not convicted".

By selecting a low threshold cut-off value and modifying the alpha p level, the quality of the hypothesis test can be increased. Intuitively, type I errors can be thought of as errors of commission , i. For instance, consider a study where researchers compare a drug with a placebo.

If the patients who are given the drug get better than the patients given the placebo by chance, it may appear that the drug is effective, but in fact the conclusion is incorrect. In reverse, type II errors as errors of omission. In the example above, if the patients who got the drug did not get better at a higher rate than the ones who got the placebo, but this was a random fluke, that would be a type II error.

The consequence of a type II error depends on the size and direction of the missed determination and the circumstances. An expensive cure for one in a million patients may be inconsequential even if true.

In statistical test theory , the notion of a statistical error is an integral part of hypothesis testing. The test goes about choosing about two competing propositions called null hypothesis , denoted by H 0 and alternative hypothesis , denoted by H 1.

This is conceptually similar to the judgement in a court trial. The null hypothesis corresponds to the position of defendant: just as he is presumed to be innocent until proven guilty, so is the null hypothesis presumed to be true until the data provide convincing evidence against it.

The alternative hypothesis corresponds to the position against the defendant. Specifically, the null hypothesis also involves the absence of a difference or the absence of an association. Thus, the null hypothesis can never be that there is a difference or an association. If the result of the test corresponds with reality, then a correct decision has been made. However, if the result of the test does not correspond with reality, then an error has occurred.

There are two situations in which the decision is wrong. The null hypothesis may be true, whereas we reject H 0. On the other hand, the alternative hypothesis H 1 may be true, whereas we do not reject H 0.

Two types of error are distinguished: Type I error and type II error. The first kind of error is the rejection of a true null hypothesis as the result of a test procedure. This kind of error is called a type I error false positive and is sometimes called an error of the first kind.

In terms of the courtroom example, a type I error corresponds to convicting an innocent defendant. The second kind of error is the failure to reject a false null hypothesis as the result of a test procedure. This sort of error is called a type II error false negative and is also referred to as an error of the second kind. In terms of the courtroom example, a type II error corresponds to acquitting a criminal. The crossover error rate CER is the point at which Type I errors and Type II errors are equal and represents the best way of measuring a biometrics' effectiveness.

See more information in: False positive and false negative. In terms of false positives and false negatives, a positive result corresponds to rejecting the null hypothesis, while a negative result corresponds to failing to reject the null hypothesis; "false" means the conclusion drawn is incorrect. Thus, a type I error is equivalent to a false positive, and a type II error is equivalent to a false negative.

A perfect test would have zero false positives and zero false negatives. However, statistical methods are probabilistic, and it cannot be known for certain whether statistical conclusions are correct. Whenever there is uncertainty, there is the possibility of making an error. Considering this nature of statistics science, all statistical hypothesis tests have a probability of making type I and type II errors. These two types of error rates are traded off against each other: for any given sample set, the effort to reduce one type of error generally results in increasing the other type of error.

The same idea can be expressed in terms of the rate of correct results and therefore used to minimize error rates and improve the quality of hypothesis test.

To reduce the probability of committing a Type I error, making the alpha p value more stringent is quite simple and efficient. To decrease the probability of committing a Type II error, which is closely associated with analyses' power, either increasing the test's sample size or relaxing the alpha level could increase the analyses' power.

Varying different threshold cut-off value could also be used to make the test either more specific or more sensitive, which in turn elevates the test quality.

For example, imagine a medical test, in which experimenter might measure the concentration of a certain protein in the blood sample. Experimenter could adjust the threshold black vertical line in the figure and people would be diagnosed as having diseases if any number is detected above this certain threshold.

According to the image, changing the threshold would result in changes in false positives and false negatives, corresponding to movement on the curve. Since in a real experiment, it is impossible to avoid all the type I and type II error, it is thus important to consider the amount of risk one is willing to take to falsely reject H 0 or accept H 0. For example, if we say, the p-value of a test statistic result is 0. The speed limit of a freeway in the United States is kilometers per hour.

A device is set to measure the speed of passing vehicles. Suppose that the device will conduct three measurements of the speed a passing vehicle, recording as a random sample X 1 , X 2 , X 3. That is to say, the test statistic. In this experiment, the null hypothesis H 0 and the alternative hypothesis H 1 should be. According to change-of-units rule for the normal distribution. Referring to Z-table , we can get.

Here, the critical region. That is to say, if the recorded speed of a vehicle is greater than critical value The type II error corresponds to the case that the true speed of a vehicle is over kilometers per hour but the driver is not fined.

If the true speed is closer to The tradeoffs between type I error and type II error should also be considered. However, if that is the case, more drivers whose true speed is over kilometers per hour, like , would be more likely to avoid the fine. In , Jerzy Neyman — and Egon Pearson — , both eminent statisticians, discussed the problems associated with "deciding whether or not a particular sample may be judged as likely to have been randomly drawn from a certain population": [12] and, as Florence Nightingale David remarked, "it is necessary to remember the adjective 'random' [in the term 'random sample'] should apply to the method of drawing the sample and not to the sample itself".

In , they observed that these "problems are rarely presented in such a form that we can discriminate with certainty between the true and false hypothesis".

They also noted that, in deciding whether to fail to reject, or reject a particular hypothesis amongst a "set of alternative hypotheses", H 1 , H In all of the papers co-written by Neyman and Pearson the expression H 0 always signifies "the hypothesis to be tested". In the same paper they call these two sources of error, errors of type I and errors of type II respectively. It is standard practice for statisticians to conduct tests in order to determine whether or not a " speculative hypothesis " concerning the observed phenomena of the world or its inhabitants can be supported.

The results of such testing determine whether a particular set of results agrees reasonably or does not agree with the speculated hypothesis. This is why the hypothesis under test is often called the null hypothesis most likely, coined by Fisher , p. When the null hypothesis is nullified, it is possible to conclude that data support the " alternative hypothesis " which is the original speculated one. British statistician Sir Ronald Aylmer Fisher — stressed that the "null hypothesis":.

Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis. In the practice of medicine, the differences between the applications of screening and testing are considerable. Screening involves relatively cheap tests that are given to large populations, none of whom manifest any clinical indication of disease e. Testing involves far more expensive, often invasive, procedures that are given only to those who manifest some clinical indication of disease, and are most often applied to confirm a suspected diagnosis.

For example, most states in the USA require newborns to be screened for phenylketonuria and hypothyroidism , among other congenital disorders. Type I error false positive : The true fact is that the newborns do not have phenylketonuria and hypothyroidism but we consider they have the disorders according to the data. Type II error false negative : The true fact is that the newborns have phenylketonuria and hypothyroidism but we consider they do not have the disorders according to the data.

Although they display a high rate of false positives, the screening tests are considered valuable because they greatly increase the likelihood of detecting these disorders at a far earlier stage. The simple blood tests used to screen possible blood donors for HIV and hepatitis have a significant rate of false positives; however, physicians use much more expensive and far more precise tests to determine whether a person is actually infected with either of these viruses.

Perhaps the most widely discussed false positives in medical screening come from the breast cancer screening procedure mammography. One consequence of the high false positive rate in the US is that, in any year period, half of the American women screened receive a false positive mammogram. They also cause women unneeded anxiety. The lowest rates are generally in Northern Europe where mammography films are read twice and a high threshold for additional testing is set the high threshold decreases the power of the test.

The ideal population screening test would be cheap, easy to administer, and produce zero false-negatives, if possible. Such tests usually produce more false-positives, which can subsequently be sorted out by more sophisticated and expensive testing.

False negatives and false positives are significant issues in medical testing. False positives can also produce serious and counter-intuitive problems when the condition being searched for is rare, as in screening. If a test has a false positive rate of one in ten thousand, but only one in a million samples or people is a true positive, most of the positives detected by that test will be false.

The probability that an observed positive result is a false positive may be calculated using Bayes' theorem. False negatives produce serious and counter-intuitive problems, especially when the condition being searched for is common.

This sometimes leads to inappropriate or inadequate treatment of both the patient and their disease. A common example is relying on cardiac stress tests to detect coronary atherosclerosis, even though cardiac stress tests are known to only detect limitations of coronary artery blood flow due to advanced stenosis.

Biometric matching, such as for fingerprint recognition , facial recognition or iris recognition , is susceptible to type I and type II errors. If the system is designed to rarely match suspects then the probability of type II errors can be called the " false alarm rate". On the other hand, if the system is used for validation and acceptance is the norm then the FAR is a measure of system security, while the FRR measures user inconvenience level.

Main articles: explosive detection and metal detector. False positives are routinely found every day in airport security screening , which are ultimately visual inspection systems. The installed security alarms are intended to prevent weapons being brought onto aircraft; yet they are often set to such high sensitivity that they alarm many times a day for minor items, such as keys, belt buckles, loose change, mobile phones, and tacks in shoes.

The ratio of false positives identifying an innocent traveler as a terrorist to true positives detecting a would-be terrorist is, therefore, very high; and because almost every alarm is a false positive, the positive predictive value of these screening tests is very low. The relative cost of false results determines the likelihood that test creators allow these events to occur.

## Curbing type I and type II errors

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth or falseness of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table:. Each of the errors occurs with a particular probability. They are rarely zero. Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test. Suppose the null hypothesis, H 0 , is: Frank's rock climbing equipment is safe.

Two drugs are to be compared in a clinical trial for use in treatment of disease X. Drug A is cheaper than Drug B. Efficacy is measured using a continuous variable, Y, and. Type I error —occurs if the two drugs are truly equally effective, but we conclude that Drug B is better. The consequence is financial loss.

## Outcomes and the Type I and Type II Errors

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth or falseness of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table:. Each of the errors occurs with a particular probability. They are rarely zero.

#### Error probabilities and power

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