AM 680.01 Recitation 2/10/11

Inferential Statistics Infer properties of the population from what is observed in the sample An inference is a generalization, as inferences go beyond the data Assumption: the sample is representative of the population Random or probability sample

Strategies A point estimate looks at one value and is an unbiased estimate of the population mean Not accurate! An interval estimate or confidence interval can estimate with a specified degree of confidence that the population mean is within a certain range of values

Sampling distribution Probability of obtaining any of the possible values in a statistic Enable researcher to infer with a determined level of confidence, the population parameters from the sample statistics

Central Limit Theory Used with continuous data A statement that the shape of the sampling distribution of the mean will approximate a normal curve if the sample is sufficiently large If random samples of a fixed n are drawn from any population, as n increases the distribution of the sample mean approaches a normal distribution Approximation of a normal distribution is n ≥ 30 Sample sizes less than 30 use t distributions to identify curve

Confidence Interval A range of values or scores from sample data that probably include the true value (population parameter) Necessary to tell the amount of error present or the accuracy of the estimate

Hypothesis Testing A statistical decision on whether the finding in a study reflect chance or real effects, at a given level of probability The larger the sample size (n), the more likely we are to reject H0

Steps in Hypothesis Testing State alternative hypothesis (HA) – claims results are real or significant (independent variable influenced dependent variable) State null hypothesis (H0) –claims that any difference in data was due to chance (independent variable had no effect upon dependent variable)

Steps in Hypothesis Testing 3. Set decision level (α) common level in medicine = 0.05 results are not significant if p > 0.05 4. Calculate probability of H0 being true the smaller decision level α, less likely we are to reject H0

Steps in Hypothesis Testing 5. Decision rule: p (H0 is true) ≤ α; reject H0 (Accept HA) p (H0 is true) > α; retain H0 Important: Cannot PROVE H0, can ONLY reject or retain

Important Note: There is always a risk of making an error when inferences are made To test a null hypothesis requires both: A test of significance, and A selected probability level that indicates how much risk you are willing to take that the decision you make is wrong

Inferential Error: 2 Types Type I- we reject the H0 when it is really true and claim HA is supported when it is actually false Smaller α, less chance of making a Type I error Type II- we retain the H0 when it is really false and conclude the alternate hypothesis was not supported (β error) Larger n, less chance of making Type II error

Directional vs. Non-directional Hypotheses Tests of significance are almost always two- tailed which allows for the possibility that a difference may occur in either direction A one-tailed test assumes that if a difference occurs, it will favor one direction More likely to reject H0 if use a one tail test (directional HA) than a two-tail test (non- directional HA)

Degrees of Freedom (df) Represents the number of scores which are free to vary when calculating the statistic Dependent upon the number of participants and number of groups Example for correlation coefficient (r), df = N (number of participants) – 2 Each test of significance has its own formula for determining degrees of freedom

Steps include: Determine if significance test will be one- tailed or two-tailed; Select a probability level; Compute a test of significance Consult the appropriate tables to determine significance of your results Determined by the intersection of probability level and df

Types of Tests of Significance Parametric test are preferred because they are more powerful, but they require assumptions: Normal population distribution Variables must be interval or ratio Randomized sample Variances of population comparison groups are equal

Types of Tests of Significance Nonparametric tests are less powerful, but they make no assumptions about the shape of the distribution Used for nominal or ordinal data sets Takes a larger sample to reach same level of significance as a parametric test

Common Parametric Tests Interval or Ratio Two groups: t- test for independent groups t- test for dependent groups Three or more groups: Analysis of variance (ANOVA) for independent groups (F ratio) ANOVA for dependent groups (F)

t- tests t-test is used to determine if 2 means are significantly different by comparing the actual mean observed with the difference expected by chance t-value is compared to t-table values; if value is ≥ than table value, the null hypothesis is rejected

ANOVA Computes an F-ratio Total variance is divided into 2 sources: Variance between groups (numerator) Variance within groups (denominator)

Common Nonparametric Test Nominal Chi-square (X2)- df = k (# groups or categories) -1 measure of difference between observed frequency with expected frequency