Midterm 2 Flashcards
What kind of statistical test only has one mean?
- z-test
- One-sample t-test
What kind of statistical test has 2 means and one factor?
Independent samples t-test
What kind of statistical test has more than 2 means and one factor?
One-way ANOVA
What kind of statistical test has more than 2 means and 2 factors?
Two-way ANOVA
What are the null hypotheses we need to find with a 2-way ANOVA?
Main effect of Factor A
Main effect of Factor B
Interaction between Factor A and B
What would a 3x4 set-up mean for a 2-way ANOVA?
3 levels in Factor A
4 levels in Factor B
What are factorial designs?
- Factorial designs are those in which factors are completely crossed
- They contain all possible combinations of the levels of factors
Ex: when each factor has 3 levels, it is called a 3 × 3 factorial design, resulting in 9 treatment combinations
What does it mean for a design to be fully crossed?
Every level of factor A is combined with every level of factor B
What’s a balanced design?
When sample sizes are equal in each condition
What do Factors represent?
The independent variables
What’s represented in the cells of the 2 × 2 factorial design of a 2-way ANOVA?
Means of all subjects within each cell are displayed
How many effects comprise a two-way factorial experiment?
2 main effects
An interaction effect
What are main effects?
- The effect of one factor when the other factor is ignored (by averaging the means over all levels of the other factor)
- Consists of the differences among marginal means for a factor
What’s the interaction effect?
- The extent which the effect of one factor depends on the level of the other factor
- An interaction is present when the effects of one factor on the DV change at different levels of the other factor
- The presence of an interaction indicates that the main effects along do not fully describe the outcome of a factorial experiment
- Sometimes called a crossover effect
- Considers pattern of results for all cell means
How could you visualize a main effect for Factor A on a plot?
There’s a main effect if there is a difference in average of the 2 dots (coming from both levels) closest to each other -> on both sides of slope
How could you visualize a main effect for Factor B on a plot?
There’s a main effect if the average of both lines (slopes) are different
How could you visualize an interaction effect
between Factor A and Factor B on a plot?
There’s no interaction if lines are parallel
There’s an interaction if they aren’t parallel and indicate that they’ll eventually cross over
The 2-way ANOVA statistically examines the effects of what?
- 2 factors of interest on the DV
(Main effects) - Interaction between the different levels of these 2 factors
(Interaction effect)
What are assumptions of the 2-way ANOVA?
- The population distribution of the DV is normal within each group
- The variance of the population distributions are equal for each group (homogeneity of variance assumption)
- Independence of observations
State the hypotheses for main effects
- Main effect of Factor A
H0A : μA1 = μA2 = ··· = μAa (equal row marginal means)
H1A : Not all μAg are the same - Main effect of Factor B
H0B : μB1 = μB2 = · · · = μBb (equal column marginal means)
H1B : Not all μBj are the same
State the hypotheses for interaction effect
- Hypotheses for interaction effect
H0: A×B : All μAgBj are the same OR The interaction between Factor A and Factor B is equal to zero
H1: A×B : Not all μAgBj are the same OR The interaction between Factor A and Factor B is NOT equal to zero
How is total sums of squares (or total variation) partitioned in 2-way ANOVA?
- It’s divided into 2 parts:
SST = SSM +SSR
SSM = Model (Between-group) variation
SSR = Residual (Within-group) variation
How is sums of squares of the model (SSM) partitioned in 2-way ANOVA?
SSA: Variation between means for Factor A
SSB: Variation between means for Factor B
SSA×B: Variation between cell means
What’s the formula for the F ratio for Factor A in 2-way ANOVA?
FA = MSA / MSR
What’s the formula for the F ratio for Factor B in 2-way ANOVA?
FB = MSB / MSR
What’s the formula for the F ratio for the interaction of Factor A and Factor B in 2-way ANOVA?
FA×B = MSA×B / MSR
When should you reject the null hypothesis in a 2-way ANOVA?
If each observed F value is greater than or equal to its critical value
What does “a” stand for in 2-way ANOVA?
Number of levels for Factor A
What does “b” stand for in 2-way ANOVA?
Number of levels for Factor B
What are treatment sums in 3-way ANOVA?
Sum of raw scores in each treatment group
What’s the grand sum (T) in 3-way ANOVA?
The sum of all the scores in the experiment
How many total effects do we have to compute/find with a 3-way ANOVA?
- 7 effects
- 3 main effects (A, B, C)
- 3 simple (two-way) interactions (AxB, AxC, BxC)
- 1 three-way interaction (AxBxC)
Describe the within-subjects/repeated- measures design
- An experimental design in which the DV is measured several times within the same subject
- Subjects are crossed with at least one experimental factor
- The simplest design of this kind may be a before and after-treatment design (2 conditions)
What is a one-way repeated-measures design comprised of?
- Levels of Factor A (only has 1 factor)
- Subjects (participants)
Describe the one-way repeated measures design
n subjects are measured on the DV under k conditions (or levels) of a single IV or factor
What are possible research questions with the one-way repeated measures design?
- Are there differences in the mean scores of the DV across groups/conditions?
- Within-subject effect of the independent variable (each subject is measured at each time point) -> Variation due to the model
- Are there differences across subjects?
- The variability of subjects (between-subject effect)
- Treat each participant as a different level in an experimental design
What’s the null hypothesis for the between-subject effect in the one-way repeated measures design?
H0: Vs = 0
- this effect represents the variance between subjects
What’s the null hypothesis for the within-subject effect of treatment (IV) in the one-way repeated measures design?
H0: μ1 = μ2 · · · = μk
What are we interested in in a One-way repeated measures ANOVA?
- Usually we are NOT interested in the effect of ‘subjects’ or subject-level variability
- If this effect is significant, it would simply tell us that subjects differ on the dependant variable which has nothing to do with our treatment (IV) so it’s irrelevant
- What we are really interested in is whether the IV has an effect on the subjects, regardless of whether differences existed naturally among the subjects
What’s referred to as SS error in one-way repeated measures ANOVA?
SSaxb
SS within is composed of what 2 types of SS in one-way ANOVA?
SS(S) and SS(AxS)
What are the assumptions in one-way repeated measures ANOVA?
- Normality
- Homogeneity of variance
- Homogeneity of covariance
Describe the normality assumption in one-way repeated measures ANOVA
The distribution of observations on the dependent variable is normal within each level of the factor
Describe the homogeneity of variance assumption in one-way repeated measures ANOVA
The population variance observations is equal at each level of the factor
Describe the homogeneity of covariance assumption in one-way repeated measures ANOVA
The population covariance between any pair of repeated measurements is equal (homogenous covariance)
What 2 assumptions in the one-way repeated measures ANOVA are considered compound symmetry?
- Homogeneity of variance
- Homogeneity of covariance
Describe the assumption of compound symmetry in one-way repeated measures ANOVA
We assume that the variations within experimental conditions is fairly similar and that no 2 conditions are any more dependent than any other two
Describe the assumption of sphericity in one-way repeated measures ANOVA
- Given that hypotheses about treatment effects are tested on differences between scores, the assumption of compound symmetry can be replaced by the assumption of sphericity (or circularity)
- Sphericity means that the variance of differences of a pair of observations is the same across all pairs
- In the assumption of sphericity, we assume that the relationship between pairs of experimental conditions is similar
- This assumption is tested in practice, and it is a necessary condition for validity of the F test in repeated measures ANOVA
What happens when there’s a violation of sphericity in one-way repeated-measures ANOVA and how do we deal with it?
- Use of tests for violations of sphericity, such as Mauchly’s W (1940) - Mauchly’s test
- When Compound Symmetry is violated, the omnibus Ftests in one-way repeated measures ANOVA tend to be inflated, leading to more false rejections of H0
- Violations of CS require adjustments to the F test
- We can use a conservative critical value based on the possible violation of sphericity (conservative Ftest)
- The inflation of the F statistic that occurs when sphericity is violated can be adjusted by evaluating the observed F value against a greater critical value, obtained by reducing the degrees of freedom
- Some of the most popular approaches involve:
1. measuring the degree of violation of sphericity
2. using the critical value equal to the value of the F distribution that corresponds to εdf (the adjustment is made for both df numerator and df denominator)
What’s the formula for finding the conservative critical value?
DF(B) = epsilon x (k-1)
DF(BS) = epsilon x (k-1)(n-1)
What’s the function of epsilon?
It measures the extent to which sphericity was violated
What’s the criteria for determining a violation of sphericity in one-way repeated-measures ANOVA?
- When sphericity holds, epsilon = 1 (i.e., no correction is needed).
- When sphericity is violated, 0 < epsilon < 1
- This reduces both DF(B) and DF(BS), and gives a larger critical value for F
- The further the epsilon value is from 1, the worse the violation
How do we decide the value of epsilon?
- Epsilon ≥ 1/(k-1)
- JASP provides two estimates of epsilon: Greenhouse- Geisser & Huynh-Feldt estimates
What’s the difference between the Greenhouse-Geisser & Huynh-Feldt estimates?
Greenhouse-Geisser is smaller (more conservative)
What’s the effect size for one-way repeated-measures ANOVA?
Partial Omega squared (ω2) that excludes the variability due to differences between subjects (MSS)
What’s the formula for within-participant variation in one-way repeated-measures ANOVA?
SSW =SSA+SSAxS
What’s the F-ratio for one-way between subjects repeated measures ANOVA?
F = MSA / MSsxa
When should you reject the null hypothesis from an F value?
If the observed F value is greater than or equal to its critical value, reject the corresponding null hypothesis
If there’s a significant result p is < or > than .05?
p < .05
If there isn’t a significant result p is < or > than .05?
p > .05
Describe sphericity in one-way repeated-measures ANOVA
Variance of difference scores is equal for all pair-wise comparisons
What do the null and alternative hypotheses indicate in Mauchly’s test
- The null hypothesis in Mauchly’s test is that the assumption of sphericity is met
- Rejecting the null hypothesis indicates that the assumption of sphericity is violated
When sphericity is violated, using the F ratio with unadjusted degrees of freedom leads to what?
- An increase in Type I error rates (false positives)
- When sphericity is violated, the type 1 error rate is no longer .05 but it is greater
What are the 3 possible ways to calculate epsilon?
- Greenhouse-Geisser approach
- Huynh-Feldt approach
- Minimum possible value epsilon can attain which is ε = 1/(a − 1)
What’s the preferred and most generally used adjustment for violation of sphericity
- Huynh-feldt
- Because it tends to have the highest power
- The adjustment we usually use when reporting the results in APA format
What values change when we use the adjustments for violation of sphericity?
- df values
- MS values (since they’re calculated with df)
What values don’t change when we use the adjustments for violation of sphericity?
- SS values
- Observed F ratios -> because if you multiply both sets of degrees of freedom by epsilon, those 2 adjustments cancel each other out so you end up with the same observed F ratio
What test results should always be reported first in an APA summary for a one-way repeated-measures ANOVA?
- (when applicable) Mauchly’s test should always be reported first in APA summary
Why is epsilon denoted as 0 < epsilon < 1?
Epsilon can’t be zero because the formula always includes a minimum of a=2 (2 levels since repeated measures needs a minimum of 2 levels) so the epsilon formula can’t give 0
What measure of effect size do we use for 2-way ANOVA?
Omega-squared (ω2)
In One-way ANOVA, SSR stands for what?
It’s the sum of the squared difference between a group mean and group observations, across all k groups
In Two-way ANOVA, SSR stands for what?
It’s the sum of squared differences between
a cell mean and cell observations, across all (a × b) cells
Describe Mauchly’s Test of Sphericity
- H0 in this test is “variances of differences between conditions are equal”
- If p < .05, the assumption of sphericity (and CS) is violated
- Available in JASP
How are the Greenhouse-Geisser and Huynh-Feldt values obtained?
- Greenhouse-Geisser was obtained with (a-1) x epsilon
- Huynh-Feldt was obtained with 2 x epsilon
