# Relationship between type 2 error and sample size

### What are type I and type II errors? - Minitab Express

At any given sample size, you can set type I error or type II. Increasing one decreases the other. Higher sample size will (in general) decrease. What are the Differences Between Type I and Type II Errors? rejecting the true null hypothesis and believe a relationship exists when it actually doesn't. It turns out that you committed a type 2 error because your sample size was too small. When you do a hypothesis test, two types of errors are possible: type I and type II. You can do this by ensuring your sample size is large enough to detect a practical difference when one truly exists. To understand the interrelationship between type I and type II error, and to Alternative hypothesis (H 1): μ 1≠ μ 2.

Definition[ edit ] In statisticsa null hypothesis is a statement that one seeks to nullify with evidence to the contrary.

Most commonly it is a statement that the phenomenon being studied produces no effect or makes no difference. An example of a null hypothesis is the statement "This diet has no effect on people's weight. Usually a type I error leads one to conclude that a supposed effect or relationship exists when in fact it doesn't.

Examples of type I errors include a test that shows a patient to have a disease when in fact the patient does not have the disease, a fire alarm going on indicating a fire when in fact there is no fire, or an experiment indicating that a medical treatment should cure a disease when in fact it does not. Examples of type II errors would be a blood test failing to detect the disease it was designed to detect, in a patient who really has the disease; a fire breaking out and the fire alarm does not ring; or a clinical trial of a medical treatment failing to show that the treatment works when really it does.

Thus a type I error is a false positive, and a type II error is a false negative. When comparing two means, concluding the means were different when in reality they were not different would be a Type I error; concluding the means were not different when in reality they were different would be a Type II error. Various extensions have been suggested as " Type III errors ", though none have wide use. In practice, the difference between a false positive and false negative is usually not obvious, since all statistical hypothesis tests have a probability of making type I and type II errors.

These error rates are traded off against each other: For a given test, the only way to reduce both error rates is to increase the sample size, and this may not be feasible. A test statistic is robust if the Type I error rate is controlled. These terms are also used in a more general way by social scientists and others to refer to flaws in reasoning. Many scientists, even those who do not usually read books on philosophy, are acquainted with the basic principles of his views on science.

Popper makes the very important point that empirical scientists those who stress on observations only as the starting point of research put the cart in front of the horse when they claim that science proceeds from observation to theory, since there is no such thing as a pure observation which does not depend on theory.

The first step in the scientific process is not observation but the generation of a hypothesis which may then be tested critically by observations and experiments. It is logically impossible to verify the truth of a general law by repeated observations, but, at least in principle, it is possible to falsify such a law by a single observation. Repeated observations of white swans did not prove that all swans are white, but the observation of a single black swan sufficed to falsify that general statement Popper, It should be simple, specific and stated in advance Hulley et al.

Hypothesis should be simple A simple hypothesis contains one predictor and one outcome variable, e. Here the single predictor variable is positive family history of schizophrenia and the outcome variable is schizophrenia.

A complex hypothesis contains more than one predictor variable or more than one outcome variable, e. Here there are 2 predictor variables, i. Complex hypothesis like this cannot be easily tested with a single statistical test and should always be separated into 2 or more simple hypotheses. Hypothesis should be specific A specific hypothesis leaves no ambiguity about the subjects and variables, or about how the test of statistical significance will be applied.

This is a long-winded sentence, but it explicitly states the nature of predictor and outcome variables, how they will be measured and the research hypothesis.

Often these details may be included in the study proposal and may not be stated in the research hypothesis. However, they should be clear in the mind of the investigator while conceptualizing the study.

### Type I and II Errors

Hypothesis should be stated in advance The hypothesis must be stated in writing during the proposal state. The habit of post hoc hypothesis testing common among researchers is nothing but using third-degree methods on the data data dredgingto yield at least something significant. This leads to overrating the occasional chance associations in the study.

The null hypothesis is the formal basis for testing statistical significance. By starting with the proposition that there is no association, statistical tests can estimate the probability that an observed association could be due to chance. The proposition that there is an association — that patients with attempted suicides will report different tranquilizer habits from those of the controls — is called the alternative hypothesis.

The alternative hypothesis cannot be tested directly; it is accepted by exclusion if the test of statistical significance rejects the null hypothesis.

One- and two-tailed alternative hypotheses A one-tailed or one-sided hypothesis specifies the direction of the association between the predictor and outcome variables. The prediction that patients of attempted suicides will have a higher rate of use of tranquilizers than control patients is a one-tailed hypothesis. A two-tailed hypothesis states only that an association exists; it does not specify the direction. The prediction that patients with attempted suicides will have a different rate of tranquilizer use — either higher or lower than control patients — is a two-tailed hypothesis.

The word tails refers to the tail ends of the statistical distribution such as the familiar bell-shaped normal curve that is used to test a hypothesis. One tail represents a positive effect or association; the other, a negative effect. A one-tailed hypothesis has the statistical advantage of permitting a smaller sample size as compared to that permissible by a two-tailed hypothesis. Unfortunately, one-tailed hypotheses are not always appropriate; in fact, some investigators believe that they should never be used.

However, they are appropriate when only one direction for the association is important or biologically meaningful. An example is the one-sided hypothesis that a drug has a greater frequency of side effects than a placebo; the possibility that the drug has fewer side effects than the placebo is not worth testing.

Whatever strategy is used, it should be stated in advance; otherwise, it would lack statistical rigor.

## Type I and type II errors

Data dredging after it has been collected and post hoc deciding to change over to one-tailed hypothesis testing to reduce the sample size and P value are indicative of lack of scientific integrity.

Because the investigator cannot study all people who are at risk, he must test the hypothesis in a sample of that target population. No matter how many data a researcher collects, he can never absolutely prove or disprove his hypothesis.

There will always be a need to draw inferences about phenomena in the population from events observed in the sample Hulley et al. The absolute truth whether the defendant committed the crime cannot be determined. Instead, the judge begins by presuming innocence — the defendant did not commit the crime.

The judge must decide whether there is sufficient evidence to reject the presumed innocence of the defendant; the standard is known as beyond a reasonable doubt. A judge can err, however, by convicting a defendant who is innocent, or by failing to convict one who is actually guilty.