Bartlett Test of sphericity

Bartlett Test of sphericity

  • Multiple Regression

Maladaptive coping behaviours and Diet Quality 

THE DATA:

A new health psychologist wanted to explore alcohol consumption along with other maladaptive coping behaviours and the effect they had on diet quality.  An advert was placed in a local hospital for volunteers and a group of 25 people volunteered to take part in this pilot study.  The respondents reported on their Healthy Eating Index (HEI), drinking frequency (in units per week), smoking frequency (in cigarettes per day) and caffeine consumption (in cups of coffee per day).  You have been asked to carry out analysis and provide an interpretation for the health psychologist.

THE RATIONALE:

Previous research (Breslow et al, 2006) has found a relationship between alcohol drinking patterns and diet quality. 

THE PREDICTION:

Null: Level of alcohol consumption and other maladaptive coping behaviours will not significantly predict HEI scores.

Alternative: Alcohol consumption levels and other maladaptive coping behaviours will significantly predict HEI scores.

 Questions  

  1. Enter this data into SPSS and carry out the appropriate analysis. Print out the relevant SPSS output as the answer to question one.
  1. Write up the analysis as a results section which might appear in a health psychology journal, including a brief interpretation of the results. 

2) Exploratory Factor Analysis

PCA and the Dysfunctional Attitude Scale 

THE INVESTIGATION AND RATIONALE: 

The Dysfunctional Attitude Scale (DAS; Weissman, 1979; Beck, Brown, Steer and Weissman, 1991) was developed to measure beliefs reflecting relatively stable schema that might ‘interact with a congruent stressor to produce clinical symptomatology’ (Beck et al, 1991, p 478).  Power, Katz, McGuffin, Duggan, Lam and Beck (1994) described the development of a short (24 item), three factor version of the DAS measuring achievement, self control and dependency.   The majority of the achievement items originally loaded onto the ‘success-perfectionism’ dimension described by Beck et al (1991), while most of the items in the dependency factor came from the ‘imperatives’ list from Beck et al.  The self-control factor mainly comprised items from Beck et al’s factors of ‘vulnerability’ and ‘need for approval’.  The three 8 item scales showed reasonable reliability (alphas of .85, .74, .68 respectively) though only equivocal information on validity was provided. 

ANALYSIS 1

Load the data file ‘das441ad.sav’ (on BREO session 6) into SPSS.

Perform a PCA on items das1 to das24.  In your analysis:

From Descriptives: select defaults plus univariate, anti-image, and KMO

From Extraction: select defaults plus scree plot

From Rotation: select Varimax

From Options: select defaults plus ‘sorted by size’.

SPECIFIC QUESTIONS ABOUT THE INVESTIGATION AND ANALYSIS: 

  1. What do the ‘Kaiser-Meyer-Olkin Measure of Sampling Adequacy’ and ‘Bartlett’s test of sphericity’ tell us about our data?
  1. Using the anti-image correlation matrix, describe how good the measures of sampling adequacy are for the individual variables.
  1. By looking at the ‘Total Variance Explained’ table, how many factors have eigenvalues over 1?  How much variation do they explain?
  1. By using the SPSS output and the answers from the above questions write a paragraph discussing your findings.  In your discussion highlight the following:
  1. Could some items be dropped as a result of poor measures of sampling adequacy?
  2. Could the Scree plot be used to support a different number of factors?
  3. Do the item-component correlations in the rotated component matrix suggest that 5 factors is a good solution? 

Solution 

Q 1) 

Q1.

Model Summary
Model R R Square Adjusted R Square Std. Error of the Estimate
1 .882a .778 .734 4.54799
a. Predictors: (Constant), Caffeine, DrinkingFrequency, Smoking, UnitsConsumed
ANOVAa
Model Sum of Squares Df Mean Square F Sig.
1 Regression 1449.519 4 362.380 17.520 .000b
Residual 413.683 20 20.684
Total 1863.202 24
a. Dependent Variable: HEI
b. Predictors: (Constant), Caffeine, DrinkingFrequency, Smoking, UnitsConsumed
Coefficientsa
Model Unstandardized Coefficients Standardized Coefficients t Sig. 95.0% Confidence Interval for B
B Std. Error Beta Lower Bound Upper Bound
1 (Constant) 86.338 3.433 25.148 .000 79.176 93.499
DrinkingFrequency -2.428 .654 -.442 -3.713 .001 -3.792 -1.064
UnitsConsumed -3.651 .819 -.537 -4.457 .000 -5.360 -1.942
Smoking -.111 .090 -.145 -1.233 .232 -.300 .077
Caffeine -.237 .517 -.051 -.459 .651 -1.317 .842
a. Dependent Variable: HEI

Q2.

From this study, it is observed that drinking frequency affects the healthy eating index (HEI) significantly (p-value = 0.01) and the HEI deceases by 2.428 units with an increase in 1 unit of drinking frequency per week. It is also observed that the HEI decreases by 0.111 units with an increase in 1 cigarette smoking frequency per day and the HEI decreases by 0.237 units with an increase in 1 cup caffeine consumption per day but the effect of smoking consumption (p-value = 0.232) and the caffeine consumption (p-value = 0.651) on Healthy Eating Index are not statistically significant. The multiple linear regression model to predict Healthy Eating Index based on other variables is statistically significant and the variables explain 77.8% of the variability in the HEI as the adjusted R-squared statistic for this model is 0.778 

Q 2) 

Q1.

KMO and Bartlett’s Test
Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .895
Bartlett’s Test of Sphericity  

Approx. Chi-Square

 

4537.830

df 276
Sig. .000

The Kaiser-Meyer-Olkin Measure of Sampling Adequacy for the data is 0.895, which indicates the sampling is adequate to perform Principal Component Analysis (PCA).

The p-value of the Bartlett’s Test of Sphericity is very low, which means that we reject the null hypothesis that the correlation matrix is an identity matrix and hence, we conclude from this test that there is a scope for reduction in the number of dimensions in the data.

Q2.

The anti-image correlation matrix contains the negative partial covariances and correlations.

The diagonals of this matrix are used as a measure of sampling adequacy (MSA).

The diagonals of the anti-image correlation matrix are:

Variable MSA
A1 0.911
A2 0.888
A3 0.921
A4 0.953
A5 0.926
A6 0.897
A7 0.926
A8 0.948
A9 0.874
A10 0.886
A11 0.901
A12 0.914
A13 0.909
A14 0.659
A15 0.662
A16 0.940
A17 0.853
A18 0.847
A19 0.882
A20 0.877
A21 0.596
A22 0.935
A23 0.904
A24 0.706

Hence, the measures of sampling adequacy are very good for most of the individual variables as the MSA for the individual variables are greater than 0.8

The measure of sampling adequacy for the variables 14, 15, 21 and 24 as the MSA for these variables are 0.659, 0.662, 0.596 and 0.706 respectively.

Q3.

Total Variance Explained
Component Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
1 7.480 31.167 31.167 7.480 31.167 31.167 5.089 21.204 21.204
2 2.408 10.032 41.198 2.408 10.032 41.198 3.213 13.389 34.593
3 1.942 8.094 49.292 1.942 8.094 49.292 3.184 13.266 47.859
4 1.424 5.934 55.226 1.424 5.934 55.226 1.581 6.588 54.446
5 1.038 4.325 59.551 1.038 4.325 59.551 1.225 5.105 59.551
6 .951 3.961 63.512
7 .917 3.821 67.334
8 .828 3.450 70.784
9 .719 2.996 73.779
10 .668 2.783 76.562
11 .634 2.640 79.202
12 .591 2.463 81.665
13 .554 2.308 83.973
14 .511 2.131 86.104
15 .426 1.775 87.879
16 .405 1.688 89.567
17 .388 1.616 91.183
18 .365 1.519 92.702
19 .345 1.438 94.140
20 .330 1.374 95.514
21 .303 1.264 96.778
22 .298 1.242 98.020
23 .252 1.049 99.069
24 .224 .931 100.000
Extraction Method: Principal Component Analysis.

By looking at the ’Total Variance Explained’ table, 5 factors have eigenvalues over 1 and they explain a variation of 59.551% of the variation.

Q4.

(a)

The measures of sampling adequacy are good for most of the variables. So, the items could not be dropped as a measure of poor sampling adequacy.

(b)

The scree plot indicates that 5 factors have eigen value greater than 1. So, the Scree plot cannot be used to support a different number of factors.

(c)

Rotated Component Matrixa
Component
1 2 3 4 5
a3 .803 .160 .068 -.094 .082
a6 .784 .070 .068 -.165 .102
a5 .781 .059 .195 -.137 .172
a4 .773 .137 .184 -.075 .129
a7 .749 .235 .213 .007 -.139
a2 .732 .255 .058 .072 -.044
a1 .651 .201 .201 .072 -.044
a8 .638 .241 .254 .002 -.110
a16 .418 .229 .338 .177 -.397
a9 .164 .807 .095 -.032 -.097
a10 .208 .780 .098 .002 -.043
a12 .228 .697 .131 -.178 .018
a11 .192 .695 .044 -.059 .049
a13 .122 .680 .067 -.185 -.028
a19 .228 .081 .806 .014 -.025
a18 .140 .121 .803 -.022 .161
a17 .092 .180 .725 .027 .116
a20 .145 -.080 .690 -.073 .140
a22 .418 .104 .503 -.042 -.151
a23 .189 .243 .449 -.061 -.369
a15 -.099 -.153 -.028 .818 .225
a14 -.079 -.202 -.057 .814 .085
a24 .111 .052 .172 .086 .687
a21 .061 -.030 .090 .240 .454
Extraction Method: Principal Component Analysis.

Rotation Method: Varimax with Kaiser Normalization.

a. Rotation converged in 6 iterations.

The item-component correlations in the rotated component matrix suggests that 5 factors is a good solution.

(Q4.) Findings:

The Kaiser-Meyer-Olkin Measure of Sampling Adequacy (0.895) indicates that the sampling is adequate for the Principal Component Analysis and Bartlett’s test of sphericity indicates that there is a scope for dimensional reduction in the data. The measures of sampling adequacy are good for the individual variables and the items could not be dropped as a measure of poor sampling adequacy. The Principal Component Analysis and the Scree plot indicate 5 factors have eigen values over 1 and these factors explain 59.551% of the variation. The item-component correlations in the rotated component matrix and the Scree plot indicate that 5 factors is a good solution for the factor analysis.