# Relationship between type 1 error and sample size

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

Statement c ("The probability of a type I or type II error occurring would be reduced by increasing the sample size") is actually false. As the. Figure salonjardin.infocal depiction of the relation between Type I and Type II errors, to estimate accurately, but increasing the sample size always increases power. No, the researcher must decide which type I error use for his test without reference to the sample size. If he enlarges his type I, enlarges the sample size or .

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.

## What are type I and type II errors?

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. 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.

### hypothesis testing - Can a small sample size cause type 1 error? - Cross Validated

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.

### Type I and II Errors

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.

## Type I and II Errors

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.

Type I Errors, Type II Errors, and the Power of the Test

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.

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.

In similar fashion, the investigator starts by presuming the null hypothesis, or no association between the predictor and outcome variables in the population. Based on the data collected in his sample, the investigator uses statistical tests to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis that there is an association in the population.

The standard for these tests is shown as the level of statistical significance. The defendant did not commit crime Null hypothesis: No association between Tamiflu and psychotic manifestations Guilt: As a practical example of the interplay of ideas, I will share a story. Long ago I was asked to recommend a sample size to confirm an environmental cleanup. This was during the pre-cleanup phase before we had any data.

My plan called for analyzing the or so samples that would be obtained during cleanup to establish that enough soil had been removed at each location to assess the post-cleanup mean and variance of the contaminant concentration. Then to simplify greatlyI said we would use a textbook formula--based on specified power and test size--to determine the number of independent confirmation samples that would be used to prove the cleanup was successful.

What made this memorable was that after the cleanup was done, the formula said to use only 3 samples. Suddenly my recommendation did not look very credible!

The reason for needing only 3 samples is that the cleanup was aggressive and worked well. It reduced average contaminant concentrations to about give or take ppm, consistently below the target of ppm. In the end this approach worked because we had obtained the previous samples albeit of lower analytical quality: