7118 Lecture Notes - Lecture 4: Repeated Measures Design, Analysis Of Variance, Sphericity

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rosesnake902

20 Jun 2018

School

University of Canberra

Department

University of Canberra

Course

7118

Professor

ALL

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LECTURE 4B

One way between groups anova

- Anova is based on the structural model, which is that each individual value can be defined as

One way repeated measures anova

Between groups factorial anova

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Mixed factorial ANOVA

- Effect of iv a

oType of treatment

- Effect of iv b

oTime

- As with btw groups factorial anova

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oInterested in main effect of A, main effect of B and the interacton btw them

oCalculate MS treatment for each of the 3 effects, and divide them by MS error to work

out the Fs and ps telling us whether each of these is significant or not

- What is new

oIn btw factorial anova we divided all of these by the same MS error – now there is an

error for the main effect of the btw-groups variable, and a different MS error for the

main effect of the within groups variable and the interaction

- F values and p values

oLooking for 3 effects; variable A, variable B and interaction

oAll or none, or some could be significant

oIf using SPSS you will see 3 f values and sig values just as for btw groups factorial but the

within subjects and interaction values are in a separate table from the btw subjects

values as these use different error terms

oCan calculate by hand

- Interactions

oTypically, if the interaction btw the 2 variables is significant, you interpret the findings in

terms of that interactions ie you follow up the interaction rather than following up the

main effects

oBest thing is to look at a graph of the 2 iVs and Dv to examine the interaction

oThen decide which way around you want to look at the data

Effect of a on b

Effect of b on a

oBUT makes more sense to look at effect of within subject variable at each level of the

btw subjects variable separately

- Follow up analyses: interaction

oWhat you’re doing when following up an interaction is looking at the effect of one iv on

the dv at each level of the other iv separately

o‘split file’ by on eiV; usually for the btw subjects iv

oThen conduct t test or one way anova (if 3 levels) examining the effect of the other IV on

the DV

oThis way you’ll end up with separate t tests or anovas

- Follow up analyses; main effect

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Document Summary

Anova is based on the structural model, which is that each individual value can be defined as. Effect of iv a: type of treatment. Effect of b on a: but makes more sense to look at effect of within subject variable at each level of the btw subjects variable separately. Follow up analyses; no effects: just say no effect. Mixed anova; advantages: generalizability of results, interaction effects, economy. Less recruitment, time, money: very useful, common design. Rarely applies in mixed design: homogeneity of variance, homogeneity of covariance. Pattern of covariances btw pairs of repeated measures scores is the same across the groups of the btw groups variable. Consulting with perfume company; want 6 new types released at once. Conduct quick study giving people a) choice of 6 types of perfume and b) choice between 3 types. You measure their willingness to pay for their choice on a scale from 0-10.

Related Questions

QUESTION 1What is the purpose of post hoc tests? Under what circumstances are post hoc tests not needed or appropriate?

a. Under what circumstances is ANOVA a more appropriate statistical technique to use than a t test for independent samples? Please provide an example(s) to support your answer. (2 marks)

b. (2 marks)

c. In testing the difference between group means, how are between group and within group variability related to each other? Provide an example to support your answer. (2 marks)

QUESTION 2

a. If a two-factor ANOVA results in a significant main effect for factor A and a significant AxB interaction, explain why you should be cautious in interpreting the effect of factor A. (1 mark)

b. For the following research situations what would be your main effects and interaction effect. If there was a significant interaction effect what would that mean? (please be as clear as possible in your explanation):

A researcher hypothesizes that the effect of alcohol consumption on one’s motor skills depends on the person’s weight (underweight, normal, overweight), but that this may differ for men versus women. (2 marks)
A teacher is interested in seeing how attending class and reading the assigned chapters is related to her students’ performance on tests. She assesses each student in terms of how often they attend class (rarely, sometimes, regularly) and how often they read the assigned chapters (rarely, sometimes, regularly). (2 marks)

QUESTION 3

Ethel is interested in two methods of note-taking strategies and the effect of these strategies on the overall GPAs of college freshmen. She believes that men would benefit most from Method 1, while women would benefit most from Method 2. After obtaining 30 men and 30 women volunteers in freshmen orientation, she randomly assigns 10 men and 10 women to Method 1, 10 men and 10 women to Method 2, and 10 men and 10 women to a control condition. During the first month of the spring semester individuals in the two note-taking method groups receive daily instruction on the particular note-taking method to which they were assigned. The control group receives no note-taking instruction. Fall and spring GPAs for all participants are recorded and a score that is the difference between the spring and fall semester GPAs is calculated.

Below are the results from running a two-way ANOVA. Please answer the following:

What is the dependent variable? (0.5 marks)
What are the independent variables (factors)? (0.5 marks)
What is the research question(s) being addressed? (1 mark)
What does the descriptive statistics (graphical) tell us about the effects under study?

(1 mark)
What assumptions are being tested in the R output? Are these assumptions met? Please

provide statistical evidence. (2 marks)
What effects are being tested in the R output? (1 mark)
What does output from the ANOVA table tell us? (1 mark)
What type of follow-up analyses was conducted and why? Please interpret. (2 marks)

: Men
: Method1

      median
coef.var

0.300

0.469

: Women
: Method1

      median
coef.var

0.150

0.496

: Men
: Method2

      median
coef.var

0.275

0.027

: Women
: Method2

      median
coef.var

0.600

0.629

: Men
: Control

      median
coef.var

0.175

mean 0.335

mean 0.170

mean 0.305

mean 0.640

mean 0.165

mean 0.105

skew.2SE
   0.408

SE.mean CI.mean.0.95

var 0.052

var 0.033

var 0.037

var 0.032

var 0.022

var 0.021

normtest.W
     0.936

std.dev
  0.229

0.682
    skewness

1.076
    skewness

0.630
    skewness

0.072

SE.mean CI.mean.0.95

normtest.p 0.830 -----------------------------------------------------------------------------

skew.2SE
   0.342

kurtosis
  -0.645

kurt.2SE
  -0.242

normtest.W
     0.964

0.058

SE.mean CI.mean.0.95

std.dev
  0.183

normtest.p 0.096 -----------------------------------------------------------------------------

skew.2SE
   0.361

kurtosis
  -1.363

kurt.2SE
  -0.511

normtest.W
     0.869

0.061

SE.mean CI.mean.0.95

std.dev
  0.192

normtest.p 0.857 -----------------------------------------------------------------------------

0.278
    skewness

0.904
    skewness

skew.2SE
  -0.124

kurtosis
  -1.096

kurt.2SE
  -0.411

normtest.W
     0.979

-0.171

: Women
: Control

      median
coef.var

0.100

1.392
    skewness

0.560

0.046

kurtosis
  -0.759

0.105

kurt.2SE
  -0.284

   std.dev
     0.146

normtest.p
     0.512

skew.2SE
   0.019

kurtosis
  -1.443

kurt.2SE
  -0.541

normtest.W
     0.967

skew.2SE
   0.458

kurtosis
  -0.770

kurt.2SE
  -0.289

normtest.W
     0.906

0.056

SE.mean CI.mean.0.95

std.dev
  0.178

normtest.p 0.254 -----------------------------------------------------------------------------

                                                                      std.dev
                                                                        0.149

normtest.p 0.958 -----------------------------------------------------------------------------

0.047

SE.mean CI.mean.0.95

0.164

0.131

0.137

0.127

0.107

> leveneTest(gpa$gpaimpr~gpa$gender, data=gpa,center=mean)

Levene's Test for Homogeneity of Variance (center = mean) Df F value Pr(>F)

group 1 7.09 0.01* 58

---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > leveneTest(gpa$gpaimpr~method, data=gpa,center=mean) Levene's Test for Homogeneity of Variance (center = mean)

      Df F value Pr(>F)
group  2    1.94   0.15

> #running Levene's test for assumption of HOV (INTERACTION EFFECT)
> leveneTest(gpa$gpaimpr, interaction(gpa$gender,gpa$method),center=mean) Levene's Test for Homogeneity of Variance (center = mean)

      Df F value Pr(>F)
group  5    0.57   0.72

Df Sum Sq Mean Sq F value Pr(>F) gender 1 0.020 0.020 0.61 0.43753

method 2 1.174 0.587 17.81 1.1e-06 *** gender:method 2 0.695 0.348 10.54 0.00014 *** Residuals 54 1.780 0.033
---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

gender        0.0055
method        0.3200
gender:method 0.1894

0.011
0.397
0.281

eta.sq eta.sq.part

#selecting only Method1, run ANOVA, then follow-up tests

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = gpaimpr ~ gender, data = gpaMethod1)
$`gender`
Women-Men -0.165 -0.3594851  0.02948506    0.0915581

diff        lwr        upr     p adj

#selecting only Method2, run ANOVA, then follow-up tests

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = gpaimpr ~ gender, data = gpaMethod2)
$`gender`
Women-Men 0.335 0.161153 0.508847   0.000754

diff      lwr      upr    p adj

aquamarineox434

CLASS NOTES ARE LOCATED WHERE IT SAYS CLASS NOTES!

During the early 1700s, a sailing ship is wrecked on an uninhabited island in the Caribbean Sea. The passengers and crew survive but lose all their possessions. There is no source of fresh water on the island, but coconut palm trees are fairly abundant, and the survivors are able to drink milk harvested from coconuts. There are no mammals living on the island, but many birds roost there, and these become a source of meat. There are a number of mango trees that provide fruit. A long, tough form of sawgrass grows in a number of different places on the island. This grass can be woven into knee-length sleeveless shirts, which are the basic type of clothing for people of both genders. The people on this island have developed a simple system of direct market exchange. Currently, two mangos trade for five coconuts, six coconuts trade for a pound of meat, and three pounds of meat trade for four shirts.

1. Use a formula that was stated in class, and in my notes, to calculate the total number of distinct relative prices in the islandâs economy. Note that when this formula is used to count relative prices, any given pair of goods is considered to have only one relative price. For example, if we count the price of a pound of meat in mangos as one relative price, then we donât count the price of mangos in pounds of meat as a second relative price.

(Class Notes: In a barter economy with markets, each good should have a market price, per unit, in terms of each of the other goods: baskets of fish per bushel of apples, bags of coconuts per bushel of apples, etc. A price of this type â units of one good per unit of the other good â is called a relative price (alternatively, a barter price). If there are different goods traded in an economy, then there turn out to be relative prices for a merchant to keep track of, in each market he visits. Each of the n goods has unit prices: one for each of the other goods. We divide by two to avoid counting reciprocal duplicates: if it costs 2 bags of coconuts to buy a bushel of apples, then it costs 1/2 bushel of apples to buy a bag of coconuts, and so on. )

2. Use the price information you are given to calculate all the distinct relative prices. Remember that the prices we are interested in are unit prices. A unit price is the price of a single unit of one good in units of another good. For each pair of goods, use the order that produces the highest unit price. In the case of coconuts and shirts, for example, if the price of a shirt in coconuts is higher than the price of a coconut in shirts, then report the first price but not the second one. Note: In calculating unit prices, donât worry about issues of divisibility. Assume that any good can be divided in any way, so that fractions or decimals can be components of unit prices.

3. For each relative price you just calculated, calculate the relative price if the order of the two items is switched. (In the scenario above, for example, you would now calculate the price of a coconut in shirts.) What very simple formula can you use to do this? Explain briefly.

4. Now suppose the people on this island start using mangos as money. The relative prices on the island do not change, and the absolute prices are consistent with them. a. Use a formula stated in class, and in my notes, to calculate the total number of absolute prices in the islandâs economy. b. Report these absolute prices.

Class Notes: ( In a monetary economy, in contrast, each good has a single absolute price (alternatively, a money price), which is its unit price in terms of money: dollars per bushel of apples, etc. If we assume that one of the n goods is used as money, then a merchant only needs to keep track of absolute prices. As long as there are more than two different goods ( ), it turns out that , and the difference between these two numbers gets very big as n gets large. Even if there are only 10 different goods, then there are 45 relative prices and only 9 absolute prices. And if there are 100 different goods, which is a more realistic number, then there are 99 absolute prices but almost 5000 relative prices.)

5. Comment briefly on the potential advantages and disadvantages of using mangos as money, using our list of the desirable properties of money as a guide.

6. Suppose you are a merchant on this island, where mangos are used as money in the market (see Item 4 above). A competing merchant who dislikes using money in transactions is willing to trade coconuts for shirts (or vice-versa) directly, at a rate of five coconuts per shirt. Explain, carefully and completely, how you could use this merchantâs willingness to trade these two goods, at this rate, in order to make a profit of one unit of money for yourself by (1) using money to buy units of a good in the market, (2) trading those units of the good to this merchant, in exchange for units of another good, and (3) selling those units in the market for money. Identify the items traded and the quantities, and verify the amount of the profit. Finally, explain what you would need to do in order to make a larger profit (double, for example).

7118 Lecture Notes - Lecture 4: Repeated Measures Design, Analysis Of Variance, Sphericity

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