PRP4014: Statistics - Research Methods - Assessment Answer

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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.

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

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

1. 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)

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. 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 Yij=µi + eij The null hypothesis is given by H0: µ1= µ2=µ3=…….µK Against the alternative hypothesis H1: µ1? µ2?µ3?…….µK

1. 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.

4. 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