PRP4014: Statistics - Research Methods - Assessment Answer

January 07, 2019
Author : Sara Lanning

Solution Code: 1ICG

Question: Statistics Case Study

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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

  1. The data analysis provided on the following pages relates to subjective (Self-report) and objective (Skin Conductance Response) emotional responses to social stimuli. High emotional reactivity is measured by higher scores on each of the variables. Data was collected from two groups of participants: Psychopaths and Non-Psychopaths.

 

Outcome SR Self-reported emotional response
Predictors SCR Skin Conductance Response
Group 0:  Non-Psychopath

1:  Psychopath

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Solution: Statistics Assignment

  1. 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).

    1. Calculate the Omega-squared effect size for the differences in impulsivity scores across conditions from the information below. (6 marks)

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

    1. Calculate the Pearson’s r effect size for the statistically significant contrast result using the information below (4 marks)

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

    1. Explain why a trend analysis would be informative for this design and, given the data, state what would you expect it to show?       (5 marks)

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

decreasing trend for mean also the confidence interval

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.

    1. State the name and design of the statistical test performed below (3 marks)

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

Yiji + eij

The null hypothesis is given by

H0: µ1= µ23=…….µK

Against the alternative hypothesis

H1: µ1? µ2?µ3?…….µK

    1. Briefly summarize the main findings of the data below. (7 marks)

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:

  • The mean for different  levels of impulsivity is following a decreasing trend

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.

  • Using the given  ANOVA table the value of calculated F is 26.874

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.

  • From contrast coefficients, we can say that the contrasts coefficients are orthogonal because they have a zero sum of the products of their coefficients (2x0 + -1x1 + -1x-1 = 0).

  • Ho: The means for different level of impulsivity are same.

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.

  1. The data analysis provided on the following pages relates to subjective (Self-report) and objective (Skin Conductance Response) emotional responses to social stimuli. High emotional reactivity is measured by higher scores on each of the variables. Data was collected from two groups of participants: Psychopaths and Non-Psychopaths.

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:

  1. State the percentage of variability of self-report responses explained by Skin Conductance in each of the two models in Regression 1 (2 marks)

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:

  • The percentage of variability of self- report responses for the two models ( Non Psychopath & psychopath) for regression 1 are 21% & 17 % respectively.

  • The percentage of variability of self- report responses for the two models ( Non Psychopath & psychopath) for regression 2 are 41% & 53 % respectively.

  • Testing of Hypothesis

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.

  • For Regression 1

The linear model equation for non-psychopath is Y=19.334+0.1077X

And for psychopath is 17.7742-00.118X

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