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ANOVA's differ in 3 main ways:

1. Number of IV's
2. Number of DV's 3. Whether the IV(s) is a betweensubjects variable or a withinsubjects variable 
Betweensubjects variable

There are different people in the different groups that make up the IV. i.e. independent samples

Withinsubjects variable

The same people are tested in all the conditions that make up the IV.

Independent Ttest is _____sbjects and the Dependent Ttest is _____subjects?

Between, Within

One way ANOVA means what?

There is only one IV. (But can have different levels)
Ex: Time out of the house after COG, BxMod, and Rogerian for Agoraphobics (IV is type of therapy); Incomes of doctors, attorneys and CPA's (IV is occupation) 
What does ANOVA do?

It divides the variation inobserved scores into variation due to the treatment (or IV) and variation due to the sampling error (includes individual difference)

Variation in scores of people receiving the same treatment is called?

Withingroup variability (ssquared sub "within"). It is viewed as the "noise" obscuring the treatment effect. i.e. error in measuring the effect of treatment

The group means reflecting a combination of treatment effects and error variation is called what?

Betweengroup variability (ssquared sub "between").

Equation for Total Variability:

Total Variabillty = withingroup variability + betweengroup variability;
ssquared sub "total" = ssquared sub "within" + ssquared sub "between" 
Concluding from the ANVOA we reject the Null Hyp that the group means are all equal when?

When betweengroup variability is much larger than withingroup variability.

Sampling distribution used for ANOVA in Hyp testing:

FTest:
F(observed) = s(squared)"between" / s(squared)"within" When both reflect error only, F=1 
FTesting:
When s(squared)"between" is much larger than s(squared)"within" = ? 
A treatment effect (i.e. group means were not all equal). Because a treatment effect adds to s(squared)"between" and not s(squared)"within".

2 types of Degrees of Freedom with the Fdistribution:

1. Reflects the number of groups that make up the IV.
2. Reflects the number of participants per group. 
Degrees of Freedom notation:
df"between": df"within": 
df"between" = df"B" = # of groups  1 = k 1
df"within": df"W" = k(n1) when each group consists of n participants  or  df"W" = (nsub1  1) + (nsub2  1) + (nsub3  1); when n's are not equal df"W": 
Degrees of Freedom tell you what?

Which row and column to use in the F table when looking up the critical value of F.
If Fobs >= Fcrit, reject Null Hyp 
What is a Mean Square (MS)?

When a sum of squares is divided by its respective degrees of freedom.

What is Fobserved?

MSbetween/MSwithin, which is a ratio of: a measure of difference between group means divided by a measure of variability in each group (for all groups)

What is "p"?

The probability...???

ANOVA assumptions (2):

1. Data are normally distributed in each group.
2. Homogeneity of variance: variances of scores in each population are equal. 
When have unequal sample sizes in the groups, what tests should you use? (2 types in the handout)

1. Use a test to that will test the homogeneity of variance assumption
2. Use a test that does not require this assumption, i.e., Welch's Test or the BrownForsythe procedure. Easiest way is to have equal sized groups. 
When N is ____, you can get a significant F even if the impact/effect is small.

Large

What is the statistic that tells you how much of the variance in the DV can be accounted for by the IV?

Omegasquared
Ex: if Omegasquared value is .44, this would mean that the type of treatment (IV) accounted for 44% of the variability in the DV. 
What do Multiple (Post Hoc) Comparisons do?

They compare all possible pairs of means.

Commonly used Multiple (Post Hoc) Comparisons: (3)

Scheffe's Test
Tukey's HSD Test (When n's in all groups are equal this is the best test.) Bonferroni Test All keep error for all the comparisons at alpha. 
In ANOVA tests with 2 or more IV's, what does "completely crossed" mean?

It means that each level of of one factor (IV) occurs with each level of the other factor (IV).

When would you want to use ANOVA tests that are completely crossed?

When you suspect that the two groups (e.g., men and women) would respond differently to the treatments.

What is the purpose of 2way ANOVA?

To determinethe effects of each factor (IV) and combinations of specific levels of 2 factors on the scores (DV).

Main effects are the effect of what?

One factor alone
Ex. a main effect of type of treatment; a main effect of sex 
Interactions are the effects of what?

A combination of certain levels of the first factor with certain levels of the second factor.
Ex. sex by treatment (Sex X Treatment) 
The main effect of treatment concerns what?

The difference between the means of the three (or more) groups.

If the main effect of treatment is significant, you conclude what?

The 3 (or more) treatment groups' means are not all equal.

The main effect of sex concerns what?

The difference between the two groups men and women.

If the main effect of sex is significant, you conclude what?

The mean number of what was being tested (DV) was not all equal.

The interaction effect concerns what?

The 6 (or more) cells in the table. Ex. each cell represents a combination of one level of sex and one level of treatment.

Significant interactions do what?

Qualify the main effect(s)

A 2way ANOVA tests 3 hypotheses and these concern what? (Three things)

1. The main effect of the first factor
2. The main effect of the second factor 3. The interaction effect 
A 2way ANOVA is better than running 2 1way ANOVAs for two reasons:

1. You get more information. 2 1way ANOVAs would not test the interaction. Information about the effects of certain combinations of the 2 IVs would not be obtained.
2. The error variance (withingroup variability) is reduced in the 2way ANOVA. Variation due to both IV's is not included in the error. (With 1way ANOVA, withingroup variability is calculated from the group mean; in 2way it is calculated from the cell mean. 
When graphing interactions, the DV goes on the __Axis and the IV levels go on the __Axis.

Yaxis, Xaxis

Parallel lines = ________
Not Parallel = ________ 
No interaction
Interaction 
Generally, what does an interaction mean?

The effect of one IV is different at different levels of the other IV.

Two kinds of of situations can occur where the lines are not parallel and there is an interaction:

1. Lines crossing
Ex: the treatments have opposite effects on men and women. 2. Lines not parallel but do not cross Ex. The difference between the effects of the two treatments is greater for women than for men. 
Situations/interactions  where the lines cross  are called what?

Disordinal interactions

Situations/interactions  where the lines are not parallel but do not cross  are called what?

Ordinal interactions

In a 1way ANOVA, the variability is divided how?

Total Variability: Var bet groups + Var within groups (error)

In a 2way ANOVA, the variability is divided how?

Total Variability: Var due to Factor A + Var due to Factor B + Var due to the interaction + Var within groups (error)

1way ANOVA hyp testing structure?

One Null, On Alt Hyp
Ex. For 3 groups Ho = mu1 = mu2 = mu3 
2way ANOVA hyp testing structure?

Three Null, 3 Alt Hyp
One null hyp and one alt hyp for each of the two main effects and for the interaction. See handout 
Disordinal Interaction Example: Factor Analysis of of Variance, Handout p.6

Disordinal Interaction Example: Factor Analysis of of Variance, Handout p.6

Withingroup variability  or error  is based on what?

The differences between each score and the CELL mean. (If you did two 1way ANOVA's the error would be based on the difference between each score and the GROUP mean. )

Are scores generally further from the group mean or cell mean? Why?

Group; because the means of the two cells making up the group are not equal.

What is the purpose of a 2way ANOVA?

To determine the effects of each factor (IV) and combinations of specific levels of 2 factors on the scores (DV).

What is a Main Effect?

The effect of one factor alone.

What are interactions?

The effects of a combination of certain levels of the first factor with certain levels of the second factor.

In a 2way ANOVA where there are three types of treatments and 2 gender types, list the types of:
1. Main effects? 2. Interactions? 
Main effects:
1a. Type of treatment 1b. Sex 2. Interaction effect: Sex X Treatment (Sex by Treatment) 
With three types of therapy and male and female groups, the main effect of treatment concerns what? (DV: number of hours men and women could stay out of the house)

The difference between the means of the three treatment groups. If it is significant, you conclude that the 3 treatment groups' means are not equal and reject the Null Hyp.

With three types of therapy and male and female groups, the main effect of sex concerns what? (DV: number of hours men and women could stay out of the house)

The differences between the means of the two groups men and women. If it is significant, you conclude that the mean number of hours men and women could stay out of the house after therapy was not equal.

With three types of therapy and male and female groups, the interaction effect concerns what? (DV: number of hours men and women could stay out of the house)

The 6 cells in the table. Each cell represents a combination of one level of sex and one level of treatment. Significant interactions qualify the main effects.

A 2way ANOVA tests how many hypotheses and concern what?

Three:
1. The main effect of the first factor 2. The main effect of the second factor 3. The interaction effect 
What is the first of two reasons why running a 2way ANOVA is better than running two 1way ANOVAs?

1. You get more information. The two 1way ANOVAs would test the effects of treatment and the effects of sex, but would not test the interaction. Thus information about the effects of certain combinations of the 2 IVs would not be obtained. Important info would be missed. For ex. wouldn't know the best therapy to use for men and the best for women.

What is the second of two reasons why running a 2way ANOVA is better than running two 1way ANOVAs?

2. The error variance (within group variability) is reduced in the 2way ANOVA. Variation due to to both IVs is not included in the error. If the IVs were tested in separate ANOVAs, variation due to one of them would be included in error.
Another way to look at it: with 1way, within group error is calculated from the group mean; in 2way, within group error is calculated from the cell mean; 
What are the steps to examining interactions?

When examining interactions it is helpful to:
1. Graph them. 1a. The DV goes on the Yaxis and the levels of the IV go on the Xaxis. 2. Draw separate lines connecting the dots for the levels of the factors (e.g., men and women) 3. If the lines cross or are not parallel, there is an interaction, i.e., can see that there is a difference between men and women in the effects of the treatment. 
What does an interaction mean?

The effects of one IV (e.g., type of treatment) is different at different levels of the other IV (e.g., sex)

What are the 3 types of interactions and what are they called?

1. No interaction
2. Lines not parallel but do not cross: "ordinal interaction" 3. Lines cross: "disordinal interaction" 
Ordinal Interaction looks like what and tells you what?

Line are not parallel, but do not cross.
There is a greater difference between levels of Factor A at one level of Factor B than at the other level of Factor B. Ex. The effects of the two treatments is greater for women than for men. 
Disordinal Interaction looks like what and tells you what?

Lines cross.
The treatments have opposite effects on the different levels of Factor A and Factor B. Ex. Cog is better than Behavior Mod for men, but Behavior Mod is better than Cog for women. 
In a 1way ANOVA the variability is divided how?

Tot Var = Var between groups + Var within groups (i.e., error)

In a 2way ANOVA the variability is divided how?

Tot Var = Var due to Factor A + Var due to Factor B + Var due to the interaction +Var within groups (i.e., error)

How many Null Hyp's and Alt Hyp's are there in a 2way ANOVA?

3 and 3.
One Null Hyp and one Alt Hyp for each of the two main effects and for the interaction. 
How to write the Null/Alt Hyps for a 2way ANOVA:

Main Effects:
For Factor A with 3 levels: Ho: Murow1 = Murow2 = Murow3; H1: Mui is not equal to Muj for some i and j For Factor B with 2 levels: Mucol1 = Mucol2; Mu1 is not equal to Mu2 Interaction: Ho: interaction effect = 0 (i.e., no interaction exists) H1: interaction effect is not equal to 0 (i.e., interaction exists) 
The interaction effect will be significant when?

a  b is not equal to c  d
cell structure: ab cd Ex. 65,35 25,45 6535 not equal to 2545 30 not equal to 20 
Withingroup variability  or error  is based on what?

The differences between each score and the cell mean.

How is withingroup variability reduced?

If you did two 1way ANOVAS, instead of a 2way ANOVA, the error would be based on the differences between each score and the group mean.
These differences would be larger than the differences for the 2way because the scores are generally farther from the group mean than from the cell mean. This is because the means of the two cells making up the group are not equal. 
Example of how withingroup variability reduced:

Scores:
55 60 65 (compared to a cell mean of 65 versus a a group mean of 50) 70 75 Max difference: 7565=10 7550=25 
With Factorial ANOVA, a participant may only appear in ___ cell?

One

How do we calculate Strength of Effects:

Betasquared
In a 2way ANOVA, we calculate betasquared for: each effect i.e., for each of the main effects and for the interaction effect. We get the info to calculate beta from the ANOVA Summary Table. Calculate Betasquared only for significant effects. 
Strength of effects example:
Main effect of marital status (assume sig): .22 Child Status (assume not sig) Marital Status x Child Status interaction (assume sig): .33 
Strongest effect is due to the interaction of Marital Status and Child Status. It accounts for 33% of the variability in traditionalism scores.
Marital Status alone (main effect) accounts for 22% of the variability in traditionalism scores. 
In a 2way ANOVA, we make comparisons to see what?

Which groups' means differ for a particular factor
Or, which cell means differ for the interaction. 
What does Tukey's HSD Test compare?

It compares all possible pairs of means, either marginal (group) means for one factor or all cell means.

What does repeated measures ANOVA mean?

More than one measurement is made on the same person on the same variable. Repeated measures ANOVA can have three or more measurements per person.

Examples of repeated measures ANOVA:

1. A group of people are all tested under each of 4 dosages of a drug
2. Or, matched individuals could be tested, where one of four matched people gets one dosage. 
Repeated measures ANOVA is usually one of the 2 following situations: (1 of 2)

1. The same participant is observed under all treatment conditions. (e.g., commonly done in studies of the effects of different dosages of a drug)

What is the advantage of using withinsubjects design in this situation?

Because people differ in their iinitial level on the DV, and this individual difference variability can be removed. (i.e., reaction time example)

Repeated measures ANOVA is usually one of the 2 following situations: (2 of 2)

2. Participants are matched and then assigned to randomly to treatments. Participants are equated on an individual difference variable that is known to be related to the DV. (e.g., if you wanted to compare the effectiveness of two therapies and you knew the outcome was related to verbal ability, you would match the participants on verbal ability in pairs, and then randomly assign one of each pair to each group.

What is a Mixed ANOVA?

When Repeated measures (withinsubjects) ANOVA is combined with a between subjects variable(s). This is a design with one or more betweensubject factors and one or more withinsubject factors. Also called a SplitPlot ANOVA

In a Mixed Design, there is usually also a ____ group?

Control (i.e., each participant is in pre and post, but only in treatment or control)

The pretestposttest with control group experimental design is a common _____ ANOVA; and, the summary table would include:

Mixed
Source: Betweensubjects: Treatment/Control  F for treatment (difference between 2 treatments, over 2 times) Within subjects: Time (pre/post)  F for prepost difference, over both treatments Treatment x Time  F for the interaction of treatment and time 