Solution Code: 1ICG
This assignment is related to ” Statistics Case Study” and experts at My Assignment Services AU successfully delivered HD quality work within the given deadline.
Case Scenario/ Task
You have studied the impact of cognitive fatigue on behavioural impulsivity. You measured impulsivity using a decision-making task where a higher score represents less impulsivity. 60 participants were allocated to one of three conditions: a control condition and two conditions inducing cognitive fatigue (Mild or High). After the treatment, participants completed the decision-making task. The output for the One-way independent ANOVA is below (assumptions were met).
Descriptives | ||||||||
Impulsivity | ||||||||
N | Mean | Std. Deviation | Std. Error | 95% Confidence Interval for Mean | Minimum | Maximum | ||
Lower Bound | Upper Bound | |||||||
Control | 20 | 100.80 | 8.817 | 1.972 | 96.67 | 104.93 | 79 | 114 |
Mild | 20 | 85.05 | 11.009 | 2.462 | 79.90 | 90.20 | 65 | 100 |
High | 20 | 81.10 | 6.601 | 1.476 | 78.01 | 84.19 | 64 | 96 |
Total | 60 | 88.98 | 12.319 | 1.590 | 85.80 | 92.17 | 64 | 114 |
ANOVA | |||||
Impulsivity | |||||
Sum of Squares | df | Mean Square | F | Sig. | |
Between Groups | 4345.033 | 2 | 2172.517 | 26.874 | .000 |
Within Groups | 4607.950 | 57 | 80.841 | ||
Total | 8952.983 | 59 |
Contrast Coefficients | ||||
Contrast | Fatigue Condition | |||
Control | Mild | High | ||
1 | -2 | 1 | 1 | |
2 | 0 | -1 | 1 |
Contrast Tests | ||||||||
Contrast | Value of Contrast | Std. Error | t | df | Sig. (2-tailed) | |||
Impulsivity | Assume equal variances | 1 | -35.45 | 4.925 | -7.198 | 57 | .000 | |
2 | -3.95 | 2.843 | -1.389 | 57 | .170 | |||
Does not assume equal variances | 1 | -35.45 | 4.877 | -7.268 | 37.957 | .000 | ||
2 | -3.95 | 2.870 | -1.376 | 31.096 | .179 |
Outcome | SR | Self-reported emotional response |
Predictors | SCR | Skin Conductance Response |
Group | 0: Non-Psychopath
1: Psychopath |
These assignments are solved by our professional Statistics Assignment Help Experts at My Assignment Services AU and the solution are high quality of work as well as 100% plagiarism free. The assignment solution was delivered within 2-3 Days.
Our Assignment Writing Experts are efficient to provide a fresh solution to this question. We are serving more than 10000+ Students in Australia, UK & US by helping them to score HD in their academics. Our Experts are well trained to follow all marking rubrics & referencing style.
Solution 3
Omega squared, ?2
Omega squared is an estimate of the dependent variance accounted for by the independent variable in the population for a fixed effects model. The between-subjects, fixed effects, form of the ?2 formula is :-
?2 = (SSeffect - (dfeffect)(MSerror)) / MSerror + SStotal
The values from the ANOVA table that are used in the calculation of ?2 are shown in Table 1.
Table 1. Values used in computing Omega squared. | ||||||
Effect | SSeffect | dfeffect | MSerror | SStotal | SSeffect-(dfeffect)(MSerror)
MSerror + Sstotal |
?2 |
Impulsivity | 4345.03 | 2 | 80.841 | 8952.983 | 4183.351 / 9033.824 | 0.463076434 |
Omega-squared effect size for the differences in impulsivity scores across conditions is 0.46
Contrast Tests | |||||||||
Contrast | Value of Contrast | Std. Error | t | df | Sig. (2-tailed) | rcontrast | |||
Impulsivity | Assume equal variances | 1 | -35.45 | 4.925 | -7.198 | 57 | 0 | -0.69004 | |
2 | -3.95 | 2.843 | -1.389 | 57 | 0.17 | -0.18094 | |||
Does not assume equal variances | 1 | -35.45 | 4.877 | -7.268 | 37.957 | 0 | -0.76281 | ||
2 | -3.95 | 2.87 | -1.376 | 31.096 | 0.179 | -0.23957 |
The pearson’s r effect size are given in the lst column of the above table.
Formula for rcontrast = t/?t2+dfwithin
The trend analysis for mean and standard deviation is done in the following chart
Impulsivity | Mean | Standard Deviation |
Control | 100.8 | 8.817 |
Mild | 85.05 | 11.009 |
High | 81.1 | 6.601 |
From the above chart it can be easily shown that there is a decreasing trend for mean also the confidence interval or the lower and upper bounds are decreasing
But, for the standard deviation there is no specific trend.
The line of best fit for mean is y = -9.85x + 108.6
& standard deviation is y = -1.108x + 11.02
And hence the trend analysis is informative for mean and not for standard deviation.
One way ANOVA F – Test
One-factor analysis of variance or one-way analysis of variance is a special case of ANOVA for one factor of variable of interest and a generalization of the two sample t-test.
A One–Way ANOVA is a statistical technique by which we can test if three or more means are equal. It tests if the value of a single variable differs significantly among three or more levels of a factor.
We can say we have a framework for one way ANOVA when we have a single factor three or more levels & multiple observations at each level.
In this kind of layout, we can calculate the mean of the observations within each level of our factor.
The linear mathematical model for one-way classified data can be written as
Yij=µi + eij
The null hypothesis is given by
H0: µ1= µ2=µ3=…….µK
Against the alternative hypothesis
H1: µ1? µ2?µ3?…….µK
Descriptives | ||||||||
Impulsivity | ||||||||
N | Mean | Std. Deviation | Std. Error | 95% Confidence Interval for Mean | Minimum | Maximum | ||
Lower Bound | Upper Bound | |||||||
Control | 20 | 100.80 | 8.817 | 1.972 | 96.67 | 104.93 | 79 | 114 |
Mild | 20 | 85.05 | 11.009 | 2.462 | 79.90 | 90.20 | 65 | 100 |
High | 20 | 81.10 | 6.601 | 1.476 | 78.01 | 84.19 | 64 | 96 |
Total | 60 | 88.98 | 12.319 | 1.590 | 85.80 | 92.17 | 64 | 114 |
ANOVA | |||||
Impulsivity | |||||
Sum of Squares | df | Mean Square | F | Sig. | |
Between Groups | 4345.033 | 2 | 2172.517 | 26.874 | .000 |
Within Groups | 4607.950 | 57 | 80.841 | ||
Total | 8952.983 | 59 |
Contrast Coefficients | ||||
Contrast | Fatigue Condition | |||
Control | Mild | High | ||
1 | -2 | 1 | 1 | |
2 | 0 | -1 | 1 |
Contrast Tests | ||||||||
Contrast | Value of Contrast | Std. Error | t | df | Sig. (2-tailed) | |||
Impulsivity | Assume equal variances | 1 | -35.45 | 4.925 | -7.198 | 57 | .000 | |
2 | -3.95 | 2.843 | -1.389 | 57 | .170 | |||
Does not assume equal variances | 1 | -35.45 | 4.877 | -7.268 | 37.957 | .000 | ||
2 | -3.95 | 2.870 | -1.376 | 31.096 | .179 | |||
We can draw below mentioned interpretations from the given data:
Considering part (c) there is a decreasing trend for mean also the confidence interval or the lower and upper bounds are decreasing
But, for the standard deviation there is no specific trend.
And the tabulated value of F (2, 57) at 5% level of significance is 2.99
Since the calculated F (i.e.26.8874) is greater than the tabulated value of F (i.e. 2.99) therefore we reject the null hypothesis at 5 % level of significance. So the effects of different levels of impulsivity are different and we need to test which level of impulsivity has significant effect.
Hence there is a need to perform pair-wise comparison.
H1: The means for different level of impulsivity are different.
From the contrast tests, the calculated value of t when variances are assumed to be equal and when variances are assumed to be different:
Contrast | t | df | Tabulated t0.05(df) | |
Assume equal variances | 1 | -7.198 | 57 | 1.96 |
2 | -1.389 | 57 | 1.96 | |
Does not assume equal variances | 1 | -7.268 | 37.957 | 1.96 |
2 | -1.376 | 31.096 | 1.96 |
From the table above we can see that the calculated value of t is less than the tabulated value of t for all the contrast and assumptions.
Hence we can accept the null hypothesis Ho: The means for different level of impulsivity are same.
Outcome | SR | Self-reported emotional response |
Predictors | SCR | Skin Conductance Response |
Group | 0: Non-Psychopath
1: Psychopath |
The output on the following pages is taken from a series of analyses attempting to understand the relationships between these variables:
From the model summary the adjusted R2 for non-psychopath & psychopath are 0.207 & 0.170.
Therefore by the definition of adjusted R2 the percentage of variability of self- report responses for the two models is:
Group | Adjusted R2 | Percentage of variability |
Non-Psychopath | 0.207 | 21% |
Psychopath | 0.17 | 117% |
(b) From the data presented in Regression 1, state the basic linear model equation and predict the Self-report emotional response for BOTH a Psychopath and Non-Psychopath with an SCR score of 9.
(6 marks)
The equation for simple linear regression model is Y=a+bX
From the given regression analysis the unstandardized coefficients are
Group | Unstandardized Coefficients | |
SCR | Constant | |
Non-psychopath | 0.1077 | 19.334 |
Psychopath | -0.118 | 17.742 |
The linear model equation for non-psychopath is Y=19.334+0.1077X
And for psychopath is 17.7742-00.118X
Self-report emotional response for both psychopath and non-psychopath is given by:
Group | Self-report emotional response |
Non-psychopath | 20.3033 |
Psychopath | 16.7122 |
(c) Explain why the best predictors chosen by a Backwards regression does not have to be the same as that suggested by the Forced entry regression. (4 marks)
In backward regression method all p variables are included in the model then the least important variable is deleted if its contribution to the sum of squares is not significant. At the second stage the next least important variable whose contribution is not significant is deleted this process goes on till a variable cannot be deleted from the model.
It is considered to be an economical procedure as it attempts to examine only the best regression equation containing a certain number of variable.
In forced entry method all the variables are forced into the model simultaneously,
This method relies on good theoretical reasons for including the chosen variable but here the experimenter makes no decision about the order in which variables are entered.
Hence the above two methods of selection of independent variables provides us with different predictors.
(d)Sketch two plots you would find if the regression assumptions hold for these analyses. (3 marks)
Xi | Fitted values of Yi for Non-Psychopath | Fitted values of Yi for Psychopath |
1 | 19.4417 | 17.6562 |
2 | 19.5494 | 17.5382 |
3 | 19.6571 | 17.4202 |
4 | 19.7648 | 17.3022 |
5 | 19.8725 | 17.1842 |
6 | 19.9802 | 17.0662 |
7 | 20.0879 | 16.9482 |
8 | 20.1956 | 16.8302 |
(e) Summarise the key findings of the data presented in Regression analyses 1 and 2. Sketch an appropriate scatterplot with lines of best fit which illustrate these key findings. (10 marks)
Regression 1
Variables Entered/Removeda | ||||
Group | Model | Variables Entered | Variables Removed | Method |
Non-Psychopath | 1 | SCRb | . | Enter |
Psychopath | 1 | SCRb | . | Enter |
a. Dependent Variable: SR | ||||
b. All requested variables entered. |
Model Summary | |||||
Group | Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |
Non-Psychopath | 1 | .484a | .234 | .207 | 5.868 |
Psychopath | 1 | .446a | .199 | .170 | 5.683 |
a. Predictors: (Constant), SCR |
ANOVAa | |||||||
Group | Model | Sum of Squares | df | Mean Square | F | Sig. | |
Non-Psychopath | 1 | Regression | 294.450 | 1 | 294.450 | 8.550 | .007b |
Residual | 964.250 | 28 | 34.437 | ||||
Total | 1258.700 | 29 | |||||
Psychopath | 1 | Regression | 224.249 | 1 | 224.249 | 6.942 | .014b |
Residual | 904.551 | 28 | 32.305 | ||||
Total | 1128.800 | 29 | |||||
a. Dependent Variable: SR | |||||||
b. Predictors: (Constant), SCR |
(Cont.)
Coefficientsa | |||||||
Group | Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||
B | Std. Error | Beta | |||||
Non-Psychopath | 1 | (Constant) | 19.334 | 1.950 | 9.913 | .000 | |
SCR | .107 | .036 | .484 | 2.924 | .007 | ||
Psychopath | 1 | (Constant) | 17.742 | 2.012 | 8.817 | .000 | |
SCR | -.118 | .045 | -.446 | -2.635 | .014 | ||
a. Dependent Variable: SR |
Regression 2
Variables Entered/Removeda | |||
Model | Variables Entered | Variables Removed | Method |
1 | Group, SCRb | . | Enter |
2 | SCR_BY_Groupb | . | Enter |
a. Dependent Variable: SR | |||
b. All requested variables entered. |
Model Summary | ||||
Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |
1 | .657a | .432 | .412 | 6.448 |
2 | .743b | .552 | .528 | 5.776 |
a. Predictors: (Constant), Group, SCR | ||||
b. Predictors: (Constant), Group, SCR, SCR_BY_Group |
(Cont.)
ANOVAa | ||||||
Model | Sum of Squares | df | Mean Square | F | Sig. | |
1 | Regression | 1799.630 | 2 | 899.815 | 21.641 | .000b |
Residual | 2370.020 | 57 | 41.579 | |||
Total | 4169.650 | 59 | ||||
2 | Regression | 2300.849 | 3 | 766.950 | 22.982 | .000c |
Residual | 1868.801 | 56 | 33.371 | |||
Total | 4169.650 | 59 | ||||
a. Dependent Variable: SR | ||||||
b. Predictors: (Constant), Group, SCR | ||||||
c. Predictors: (Constant), Group, SCR, SCR_BY_Group |
Coefficientsa | ||||||
Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||
B | Std. Error | Beta | ||||
1 | (Constant) | 23.189 | 1.833 | 12.649 | .000 | |
SCR | .020 | .031 | .065 | .648 | .519 | |
Group | -10.775 | 1.676 | -.646 | -6.429 | .000 | |
2 | (Constant) | 19.334 | 1.920 | 10.070 | .000 | |
SCR | .107 | .036 | .341 | 2.970 | .004 | |
Group | -1.592 | 2.805 | -.095 | -.567 | .573 | |
SCR_BY_Group | -.224 | .058 | -.682 | -3.875 | .000 | |
a. Dependent Variable: SR |
The key findings for the two regression analysis are:
H0: ?=0
H1: At least one of the regression coefficients is not zero.
For regression 1
The calculated value of F for model 1 and model 2 are 8.5550 & 6.942 respectively.
And the tabulated value of F (1, 28) is 4.20
Since the calculated value of is greater than the tabulated value
Therefore, we may reject H0
Hence at least one of the regression coefficients is not zero.
For regression 2
The calculated value of F for model 1 and model 2 are 21.641 & 22.982 respectively.
And the tabulated value of F (2, 57) is 3.162 and F (3, 56) is 2.776
Since the calculated value of is greater than the tabulated value
Therefore, we may reject H0
Hence at least one of the regression coefficients is not zero.
The linear model equation for non-psychopath is Y=19.334+0.1077X
And for psychopath is 17.7742-00.118X
Find Solution for Statistics case study assignment by dropping us a mail at help@myassignmentservices.com.au along with the question’s URL. Get in Contact with our experts at My Assignment Services AU and get the solution as per your specification & University requirement.
Trending now
The Student Corner
Subscribe to get updates, offers and assignment tips right in your inbox.
Popular Solutions
Popular Solutions
Request Callback
Doing your Assignment with our resources is simple, take Expert assistance to ensure HD Grades. Here you Go....