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)
1
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)
2
: 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
3
> 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$gpaimpr~method, data=gpa,center=mean) Levene's Test for Homogeneity of Variance (center = mean)
Df F value Pr(>F)
group 2 1.94 0.15
57
> #running Levene's test for assumption of HOV (INTERACTION EFFECT)
> leveneTest(gpa$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
54
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
4
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
5
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)
1
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)
2
: 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
3
> 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
57
> #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
54
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
4
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
5