Stats Test 3 (Ch. 12-14) Class Notes

One- Way Within- Subjects ANOVA

-A statistical procedure used to test hypotheses for one factor with two or more levels concerning the variance among group means. This test is used when the same participants are observed at each level of a factor and the variance in any one population is unknown -The term "one-way" means that you are testing one factor -The term "within-subjects" means that the same participants are being observed in each group

Tukey Test

-All possible pair-wise comparisons, controlling for alpha error -Find critical mean difference Dt = qt(sqrt(MSw/n)) *look up qt in Tukey Table

Why NOT Do a Bunch of t-Tests?

-Alpha inflation: probably of Type I error increases for each pair-wise comparison (t- test) computed. -If you did 6 t-tests and use alpha = .05, the probability of committing a Type I error would be .25 -ANOVA controls for the alpha inflation, you conduct a single test and it tells you if at least one mean significantly differs from the others

Three Sources of Variation in a One-Way Within-Subjects Design

-Between groups -Within groups -Between persons

Effect Size in ANOVA

-Determined by the variability components 1. η2 = SSB / (SSB + SSW) Proportion of total variability in DV Attributed to the IV AND 2. Omega squared (don't need to know)

Main Effect

-Effect of an F test, effect of one variable (not both) each one has a main effect which can be significant or not -What is assessed in a simple ANOVA - is there an effect of the IV? -When more than one variable is tested - main effect is the impact of each variable independent of the others

Interaction

-Effect of one variable changes based on the effect of another variable -Combined effect of IVs considered simultaneously -An interaction effect occurs when the effect of an independent variable differs depending on the level of a 2nd independent variable

ANOVA Hypotheses

-H0: μ1 = μ2 = μ3 (null hypothesis when using the ANOVA is that the populations being compared all have the same mean) -H1: not all means equal; or at least one mean significantly differs from the others

Accepting or Rejecting Null Hypothesis Based on F Test

-If there is no difference in the means, the between-group variance will be equal to the within-group variance, F will be close to 1—DO NOT reject null hypothesis -When the means differ significantly, the between-group variance will be larger than the within-group variance, F will be significantly greater than 1—reject null hypothesis

Post Hoc Tests

-If you reject H0, you only know there are differences but you do not know where those differences are -Need to do follow-up (post hoc) tests to determine which means significantly differ -Two types: Tukey test and Bonferroni test

F Ratio

-In the F test, two different estimates of the population variance are made. -Between- group variance: finding the variance of the means -Within- group variance: computing the variance using all the data and is not affected by differences in the means

What Graphically Indicates Interaction?

-Non-parallel lines = interaction in a line graph -Uneven bars = interaction in a line graph

Bonferroni Description

-Post hoc tests are computed following a significant one- way ANOVA to determine which pair or pairs of group means significantly differ The Bonferroni procedure adjusts the alpha level, or probability of committing a Type I error, for each test -The alpha level for each test is called testwise alpha -Testwise alpha: the alpha level, or probability of committing a Type I error, for each test or pairwise comparison made on the same data

Bonferroni Test

-Technique to adjust p value to correct for alpha error inflation from multiple t-tests -Alpha Bonferroni = alpha (type I error) / C *C = number of comparisons (how many t- test

Between-Group Variance

-Variation among the means is another way to estimate the variance in the populations from which the samples are chosen -If the null hypothesis is true, all populations are identical and have the same mean variance and shape

Within- Group Variance

-With ANOVA, you estimate the population variance -You assume that all populations have the same variance -You pool the estimated population variances, this is called the within-groups estimate of the population variance

Independent Variable

-e.g., Drug vs. Placebo -Variable considered to be a cause, such as what group a person is part of, for an analysis of variance -2 of them in a two-way analysis of variance

Rules for the Power of the Within- Subjects Design

1. As SSBP increases, power increases 2. As SSE decreases, power increases 3. As MSE decreases, power increases

Assumptions for F Test

1. Normal or approximately normal distribution 2. The samples must be independent of each other 3. Homogeneity of variance - all SDs should be similar

One-Way Within-Subjects ANOVA

1. Normality - assume that data in the population or populations being sampled from are normally distributed 2. Independence Within Groups - assume that participants are independently observed within groups, but not between groups 3. Homogeneity of Variance - assume that the variance in each population is equal to each other 4. Homogeneity of Covariance - assume that participant scores in each group are related because the same participants are observed across or between groups

Dependent Variable - Outcome Variable

A variable that represents the effect of the experimental procedure

Two-Way Analysis of Variance

Analysis of variance for a two-way factorial research design

One-Way Analysis of Variance - Simple ANOVA

Analysis of variance in which there is only one variable

Degrees of Freedom- Between Groups

Degrees of freedom between groups, dfBG , are the number of levels of the factor, or number of groups (k), minus 1 dfBG = k-1

Degrees of Freedom- Between Persons

Degrees of freedom between persons, dfBP , are the number of participants per group (n), minus 1 dfBP = n-1

Degrees of Freedom- Error

Degrees of freedom error, E , are equal to the df between groups (k - 1) times the df between persons (n - 1) dfE = (k-1) (n-1)

Cell

Each grouping in a factorial design

Bonferroni Procedure with the One- Way Within-Subjects ANOVA

Step 1: Calculate the testwise alpha and find the critical values Step 2: Compute a related samples t test for each pairwise comparison and make a decision to retain or reject the null hypothesis for each pairwise comparison

ANOVA

The analysis of variance (ANOVA) procedure is a statistical procedure that tests the variation among the means of more than two groups

Cell Mean

The mean of the scores in each grouping combination