Stats
Eqn for df factor
# of the levels of the factor - 1 (b - 1)
What questions does a two-way factorial design ask?
(1) Are the means of levels of the first factor different - is there a main effect of Factor 1? (2) Are the means of the levels of the second factor different - is there a main effect of Factor 2? (3) Does the effect of one factor on scores on the dependent variable depend on the level of the other factor - is there a factor 1 x factor 2 interaction?
What does the differences in eta squared and omega squared depend on?
*Sample size = w smaller sample size, eta-squared effect size will be much greater than omega-squared effect size comparatively *Error Variance
What is the advantage of factorial designs?
- Able to test 2 or more IVs, as opposed to 1 IV in one-way ANOVA and t-tests. -can test the GENERALISEABILITY of results -
Which is the factor represented by lines on a graph in a 2x 2 design?
- Least important theoretically or with fewer levels
What are assumptions of ANOVA?
- Treatment populations normally distributed - treatment populations have the same variance (homogeneity of variance) - independent samples - no two measures drawn from the same participant cf repeated-measures ANOVA - independent random sampling for participants - each sample: at least 2 observations and equal n (in each group) - DV measured using a continuous scale (interval or ratio) - math operations don't make sense for other kinds of scales.
within-groups variance
- error variance (due to random chance or unmeasured influences) - distribution of individual DV scores around the group mean
Null Hypothesis in 1-way ANOVA ( 2 or 3 levels)
- no differences between treatment means - no differences between any treatment means
When you are following up a main effect with more than two levels, what are the 2 types of tests that can be conducted?
-+- Protected t-test (pairwise comparison) -+- Linear contrasts
between-groups variance
-systematic variance due to membership in different groups / treatments - distribution of group means around the grand mean
problems with significance testing as a way of determining the importance of findings:
1. firstly and most importantly is the fact that an acceptance criterion results in a binary outcome - significant or non-significant. This unashamedly reductionist approach results in the loss of a lot of important information, namely, how IMPORTANT the result is. It is still a matter of livid debate, but it can be argued successfully that equating significant results with important results is overly simplistic - I will provide some examples shortly. 3. Another problem is an obstinate statistical anomaly - the fact that increases in sample size affect the magnitude of p - eventually, a non-significant p WILL slip under the acceptance criterion
df error
= total # of observations - # of treatments
What is a cell mean?
A cell is the mean result for a crossed level
What is a main effect?
A significant main effect of example type tells use that the marginal means of levels of one factor are different. e.g. There is a difference between 3 marginal means, but not sure exactly where, follow-up comparisons would be necessary.
What question does a one-way design ask?
Are the mean DV scores of EACH LEVEL of the factor different from the grand mean (from each other)?
What is a marginal mean?
Average of one level of the IV collapsed across the levels of the other IV.
What are advantages in factorial designs?
Can have two or more IVs Fewer participants required to test 2 IVs (can cross them, rather than have 2 sets of participants for 2 IVs)
All ____ means in a factorial design must be represented on the graph by points
Cell means!
What are the two main approaches to estimating effect sizes in ANOVA?
Eta-squared - n2 And omega-squared - w2
Compare eta squared, omega-squared and partial eta-squared.
INFLATED, sometimes massively, but common - proportion of residual variance after other variables/effects are controlled/accounted for by our effect Eta-squared - slightly too large, but common, easily interpretable, proportion of sample variance in DV accounted for by effect Omega-squared - more conservative, but uncommon, proportion of population variance in DV accounted for
What is the generaliseability of results?
Is the difference described by a main effect the same across levels of the other factor.
What questions does a 2-way factorial design ask?
Is there a main effect of Factor 1? Is there a main effect of Factor 2? Does the effect of one factor on the DV depend on the level of the other factor? (Interaction)
When the F ratio is calculated for each omnibus test in a 2-way ANOVA, identify which MS terms are used in the numerator and denominator in each case.
MS treatment / MS error.
Calculation for F statistic?
MStreat/ MS error
What is MS treatment?
MStreatment = SStreatment/dftreatment
What are omnibus tests?
Main effects and interactions
Eqn for df total
N - 1
In most beween-subjects factorial designs there are n observations per cell (the same). The number of cells multiplied by n gives you ...
N, the total number of observations
Are follow up test required for a main effect of an IV with two levels?
No - easily able to interpret direction
What is the difference between eta-squared and omega-squared ?
Omega-squared: more conservative, less biased, proportion of variance in the population's DV that is accounted for by the effect versus Eqa-squared: describes the proportion of variance in the SAMPLE's DV scores that is accoun for by the effect considered a biased estimate, most commonly reported (because effect size measure easily interpretable)
When should simple comparisons be conducted?
Only significant simple effects should be followed up with simple comparisons.
What is eta squared
Proportion of total variance accounted for by the effect
What are some issues with follow-up comparisons?
REDUNDANCY INCREASES FAMILY-WISE ERROR RATE (solutions use Bonferroni Adjustment for critical t OR conduct contracts a priori --- do fewer contrasts)
What is the formula for eta-squared?
SS effect / SS total
What is the formula for omega-squared?
SSeffect - (df effect) x MS error / SS total + MS error
What is SS error?
SSerror = SStotal - SStreatment
What is MS error?
SSerror/dferror
Are simple effects or main effects used to interpret an interaction?
Simple effects
What does the additivity of simple effects mean
Simple effects repartition the main effect and interaction variance
What is a simple effect?
The effect of one factor at ONE LEVEL of the other factor.
What is an interaction?
The effect of one factor is conditional upon the levels of the other factor.
In a 2x2 design, which factor goes on the x-axis?
The factor with the most levels OR the factor which is most theoretically important.
What is the grand mean?
The mean of all observations - not as important for inferential purposes but forms part of the structural model of ANOVA.
What is the procedure for conducting simple comparisons?
The procedure is identical to that used to follow up main effects.
What is SStreatment in a one-way ANOVA?
This means we: 1. Subtract the grand mean, or the mean of all of the individual data points, from each group mean 2. Square these numbers 3. Multiply them by the number of subjects from that particular group 4. Sum them Note: n = number of subjects per group Hint: The number of numbers that you sum should equal the number of groups
When is a t-test used?
To compare two means, e.g. two levels of one IV
What does an F value larger or smaller than 1 mean?
When the F ratio is > 1, the treatment effect (variability between groups) is bigger than the "error" variability (variability within groups). Or more specifically: The sum of the squared differences between the group means and the grand mean x the number of people in each group, divided by the number of groups minus 1, is bigger than the sum of the squared differences between the observations and the group means, weighted by the number of observations in each group minus 1 x the # of groups
When do you need to follow-up a main effect?
When the IV has more than 2 levels
Why would you use a one-way ANOVA instead of a t-test?
When the IV has more than two levels, a one-way ANOVA is preferred. Multiple comparisons increase the risk of a Type 1 Error, so by 3 comparisons, the risk of a type 1 error is almost 15% (this is unacceptably high)
error variance
can't be explained (random or due to unmeasured influences)
alternative hypothesis in 1-way ANOVA (2 or 3 level)
difference between treatment means at least one difference between treatment means
What is Cohen's differentiation of effect sizes?
differentiating effect sizes (Cohen, 1973): 0.2 = small 0.5 = medium 0.8 = large
What is variance?
dispersion or spread of scores around a point of central tendency - the mean
Explain what it means to say that interactions and main effects in a factorial design are independent.
interactions and main effects can occur in any combination, they are independent
Eqn for df interaction
product of each df factor e.g. (b -1 ) x (a - 1)
What is partial eta square?
residual variance = variance left over to be explained (not accounted for by any other IV in the model)
Xij = u. + tj + eij
structural model of 1-way ANOVA Xij, any DV score is a combination of: u. the grand mean tj - the effect of the j-th treatment (uj - u.) eij = error for i person in j-th treatment
treatment variance
systematic differences due to our IV (e.g., experimental manipulation)
Explain the difference between treatment variance and error variance
systematic differences due to our IV (e.g., experimental manipulation) vs variance that can't be explained (random or due to unmeasured influences)
A significant interaction must always be followed up with ...
tests of the simple effects: the effects of one factor at each level of the other factor
what is the magnitude of experimental effect,
the effect size gives you another way of assessing the reliability of the result in terms of variance can compare size of effects within a factorial design: how much variance explained by factor 1, factor 2, their interaction, etc.
what is a significant interaction?
the interaction effect will be significant if the simple effects of one factor differ at the various levels of the other factor
what has been proposed as an accompaniment (if not an outright replacement) to significance testing ?
the magnitude of experimental effect, or effect size,
In a 1-way ANOVA with multiple levels, what does a significant effect mean?
there is a difference among the group means. A follow-up test tells us exactly where that difference lies.
Lines crossing indicate that...
there is a disordinal interaction
lines not parallel but not crossing indicate that..
there is an ordinal interaction
Parallel lines indicate that ...
there is no interaction