Recap of confidence intervals If the 95%CI does not include the hypothesized μ, we conclude that our sample is statistically different from the assumed population If the 95%CI we calculated includes the hypothesized μ, we conclude that our sample is, is not statistically different from the assumed population

Hypothesis Testing Set-up State the null hypothesis (H 0 ) In statistics, we always start by assuming that the null hypothesis is true (“no effect” or “no difference”) Only if there is convincing evidence do we reject the null hypothesis IQ example In words: “There is no difference in average IQ between group 1 and group 2.” In symbols: μ 1 = μ 2 or μ 1 – μ 2 = 0 Note: The hypothesis is always written in terms of the population parameter, not the sample statistic.

Hypothesis Testing Set-up State the alternate hypothesis (H A, H a, or H 1 ) The alternative hypothesis states that there is a difference Can go in either direction (two-sided or two-tailed) IQ example In words: “There is a difference in average IQ between group 1 and group 2.” In symbols: μ 1 ≠ μ 2 or μ 1 – μ 2 ≠ 0 Note: In the medical literature, specific hypotheses are rarely stated explicitly.

How hypothesis testing is done Define the null and alternate hypotheses Collect relevant data from a sample Calculate the test statistics specific to the null hypothesis Compare the value of the test statistics to that from a known probability distribution Interpret the resultant p-value

What is alpha (α)? The type I error rate The probability threshold beyond which the null hypothesis would be rejected The probability threshold where we allow for the rejection of H 0 when H 0 is true Conventionally set to 5% 2.5%

How is the p-value derived? Look up in table Each test statistic is associated with a p-value

What is the p-value? Under the null hypothesis (H 0 ), the p-value is the probability of obtaining a test statistic at least as extreme as the one observed by chance alone.

What the p-value looks like 0 Theoretical distribution of test statistic Probability Value of test statistic Test statistic Sum of the yellow areas = p-value Area under the curve which represents the probability of obtaining a test statistic at least as extreme as the one observed by chance

Alpha and p-value If p < α then we reject the null hypothesis in favor of the alternate hypothesis. 2.5% Test statistic p-value

Alpha and p-value If p > α then we do not reject the null hypothesis. 2.5% Test statistic p-value

Alpha and p-value: Example CharacteristicCasesControlsp-value Percent female Mean age Mean number of days spent camping Mean daily honey consumption (oz.) Table 1: Baseline characteristics of a sample from a study examining bear attacks in a population of campers We reject H 0 and conclude that there is a statistically significant difference in the sex distribution between cases and controls. We do not reject H 0 and conclude that there is no statistically significant difference in mean age between cases & controls. We do not reject H 0 and conclude that there is no statistically significant difference in the mean of days spent camping between cases & controls. We reject H 0 and conclude that there is a statistically significant difference in mean honey consumption between cases & controls.

Type I and II error Type I error (α) occurs when H 0 is rejected when it shouldn’t be When there truly is no effect or association, but one was observed by chance Type II error (β) occurs when H 0 is not rejected when it should be When there truly is an effect or association, but there was not one detected Is a function of statistical power (1-β)

Power, sample size, alpha, and beta For a given level of α, increasing n (the sample size) will… Increase the power of the study to detect a difference or association Decrease type II error rate (β) Studies with small samples are more likely to be underpowered Large p-values, even if there appears to be an association or difference Wide confidence intervals

Types of error and study conclusions H 0 trueH A true Reject H 0 Type I error (α)Proper decision Do not reject H 0 Proper decisionType II error (β) Unknown reality or truth about population Decision based on study results Analogous to the American justice system… InnocentGuilty Found guiltyType I error (α)Proper decision Found innocentProper decisionType II error (β) Unknown reality or truth about defendant Jury’s decision

Different types of data Age and race are different types of variables State the null hypothesis for the distribution of race. a.The proportion of Whites is the same in cases and controls. b.The proportion of Whites is the different comparing cases to controls. c.The proportion of Whites is lower in the cases than the controls. d.The proportion of Whites is higher in the cases than the controls.