week 9

Power

(1-B) the chance of making *a correct decision of rejecting the null hypothesis when the null is false* if Ho is in fact false, power is the probability that we will reject it as sample size increases, power increases

the null is not rejected

(we fail to reject the null) then we do not have significant evidence in favor of the alternative hypothesis

One-tailed hypothesis

*DIRECTIONAL* specifies the direction (greater than or less than) in which the null hypothesis is incorrect Ho: p≤ 0.2 vs Ha: p>.2

Two-tailed hypothesis

*NON-DIRECTIONAL* does not specify the direction in which the null hypothesis is incorrect 𝐻o:𝑝=0.2 𝑣𝑠. 𝐻a:𝑝≠0.2

Hypothesis testing does not lead to proving a null hypothesis. It only indicates whether the null hypothesis is ...

*supported or not supported by the sample data* - therefore when we fail to reject the null hypothesis we don't say it is true, but say that we don't have any evidence that it is false - therefore, it is improper to say that we accept the null hypothesis. it makes the wrong implication

Null Hypothesis

- Statement of agreement with the conditions presumed to be true in the population of interest - The hypothesis to be tested - It is set up to be discredited - It is the complement of the conclusion that the researcher is seeking to reach. - Also referred to as the hypothesis of no difference

Alternative Hypothesis

- The statement of what the researcher hopes or expects to conclude as a result of the test. - Usually the research hypothesis and the alternative hypothesis are the same. - By rejecting the 𝐻o, you support the 𝐻a.

Decision Rule

All possible values that the test statistic can obtain are split into two groups: the rejection region and the nonrejection region Rejection region - these are test statistic values that are less likely to occur if H0 is true. Nonrejection region - these are test statistic values that are more likely to occur if H0 is true.

Data

Determine they type (or nature) of data that are being evaluated Determine the scale of measurement (nominal, ordinal, interval, ratio) These decide what statistical test should be used to test the hypothesis

Assumptions

Statistical procedures have certain assumptions that must be met ex: - normality of the health outcome - equality of the population variances - independence of the samples These assumptions were also important when constructing confidence intervals

Choosing the type of null hypothesis

The two-tailed H0 encompasses both directions. It should be used whenever you are interested in either way in which the null hypothesis can be incorrect. Use one-tailed when you are only interested in one direction. Two-tailed hypothesis more general and so it is used more frequently.

null vs alternative hypothesis

To summarize: The null hypothesis is always a statement of equality (when testing a hypothesis of superiority) - An indication of equality (either =, ≤, or ≥ ) appear in the null hypothesis The alternative hypothesis states what you want to find evidence to support. Note: In this class we are only covering superiority hypotheses. Non-inferiority hypotheses and equivalence hypotheses are other types of hypotheses that are not covered in this course.

Drawing Conclusions

Two possible statistical decisions: - *Reject* the null hypothesis, or - *Fail to reject* the null hypothesis The hypothesis that is actually being tested is the null hypothesis, so we conclude with respect to 𝐻o. If reject 𝐻o, conclude there is a difference between groups (or an association). If fail to reject 𝐻o, conclude there is no evidence of a difference (or an association).

Hypotheses

Two statistical hypothesis - these should be explicitly stated 1. Null hypothesis (Ho) 2. Alternative hypothesis (Ha, H1)

Types of Hypotheses

Two tailed (non-directional) One tailed (directional)

Distribution of Test Statistic

We must determine and specify the probability distribution of the test statistic

Calculation of Test Statistic

We use the appropriate formula to calculate the test statistic using the sample data Then we compare the computed value of the statistic to the specified rejection and non-rejection regions

What is a hypothesis in statistics?

a statement or claim about one or more populations

Statistical significance

ability to state that the observed difference (or association) in outcome is not due to chance alone necessary for clinical significance but says nothing about the magnitude of the effect

Hypothesis Testing

aka test of significance a standard procedure for testing a claim about a property of a population

A ________ states that there is a difference (an association) in the outcomes of interest

alternative hypothesis

p-value = probability of..

calculating a test statistic at least as extreme as the calculated test statistic, provided the null hypothesis is true

Two general areas of statistical inference

estimation hypothesis testing

statistical hypothesis

hypothesis that are stated in such a way that they may be evaluated by appropriate statistical techniques

If the p-value of the calculated test statistic is _______ to the alpha set by the researcher (usually 0.05) then we can conclude..

less than or equal to the groups are different (or there is an association: statistical significance)

alpha is also called

level of significance

A _______ states that there is no difference (no association) in the outcomes of interest

null hypothesis

e.g. the probability of concluding there is a difference between the groups when a difference actually exists

power

The decision rule tells us to:

reject the H0 if the value of the computed test statistic is one of the values in the rejection region. OR not reject the H0 if the value of the computed test statistic is one of the values in the nonrejection region

General Formula for a test statistic

statistic - hypothesized parameter / SE (statistic)

Single Population Mean You can use the ____ if the sample is from a normally distributed population where the population standard deviation or variance in unknown

t test

p-values

tells us the probability associated with obtaining the computed test statistic or more extreme, *given that the null hypothesis is true* therefore the smaller the p-value the better justification for doubting the truth of Ho

Probability of a Type II Error and Beta

the error committed when we fail to reject a false null hypothesis (false negative) Beta is not directly controlled by the investigator, but is influenced by the sample size, alpha, the hypothesized value of the parameter, and the true value of the parameter (more in the power)

Type I error and Alpha

the error committed when we reject a true null hypothesis since alpha is small, it is unlikely that the test will manifest a value in the rejection region when the null hypothesis is true

Research hypothesis

the question or idea that motivates the research

Clinical significance

the smallest important difference between two treatments the clinical importance of an observed difference in outcome to decide, a clinician should take into account the side effects, long-term complications, and other costs of the two treatments

Critical value

the value of the test statistic that begins the rejection region values of the test statistic that separate the rejection region and the nonrejection region

If the null is rejected,

then we conclude that we have significant evidence in favor of the alternative hypothesis

Purpose of hypothesis testing

to aid the researcher in using a sample from the population to make inferences about that population

the probability of committing a type ___ error is alpha

type I

ex: the probability of concluding that two groups are different when they are actually equal

type I error

Rejecting a Null Hypothesis When we reject a null hypothesis, it is possible that we have committed a type ___ error

type I error Although we never know if we have made an error, in this situation we know that the probability of making this error is alpha. We set alpha as a small number

the probability of committing a type ___ error is beta

type II

ex: the probability of concluding no evidence of a difference between two groups, when in fact they are different (i.e. we should have rejected the null, but did not)

type II error

Failing to Reject a Null Hypothesis If we fail to reject a null hypothesis, it is possible that we have committed a type ___ error

type II error As before, we will not know whether we made an error, but we know the probability of type II error committed (=B)

Single Population Mean You can use the ______ if the sample is from a normally distributed population where the population standard deviation or variance in known

z test

10 Steps for Hypothesis Testing

1. Data 2. Assumptions 3. Hypothesis 4. Test statistic 5. Distribution of test statistics 6. Decision Rule 7. Calculation of test statistic 8. Statistical decision 9. Conclusion 10. p-value

Rules of Thumb for Writing the Hypothesis

1. What you wish to conclude should be in the alternative hypothesis (HA) 2. The null hypothesis (H0) should contain an indication of equality 3. H0 is the hypothesis being tested 4. The H0 and HA are complementary, so the two hypotheses exhaust all possible outcomes regarding the value of the hypothesized parameter.

Test Statistic

A statistic that is computed from the sample data A specific value computed from a sample (thus making it a statistic) that is used in deciding between H0 and HA The decision to reject the null hypothesis (or not to reject it) depends on the magnitude of the test statistic. A statistic, in general, has many possible values because its value depends on the sample that is drawn.

How large is Beta?

Most of the time it is larger than alpha, but we don't directly control Beta as we do alpha

Deciding Rejection Region

The decision on the values that should be included in the rejection region is based on: 1. level of significance (alpha) 2. whether the alternative hypothesis is one-tailed or two-tailed

the level of significance (alpha)

The level of significance is the probability of rejecting a true null hypothesis. (false positive error) If we rejected a true null hypothesis, this would be an error, so it seems that we want to make this probability small. The usual values of are 0.01, 0.05, or 0.10 to make the probability of rejecting a true null hypothesis small.

Statistical Decision

This consists of either: - rejecting the null hypothesis (if computed test statistic falls into rejection region) - failing to reject the null hypothesis (if computed test statistic falls into the non-rejection region)