**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____ __** **

- Enter this data into SPSS and carry out the appropriate analysis. Print out the relevant SPSS output as the answer to question one.

- 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:____ __

- What do the ‘Kaiser-Meyer-Olkin Measure of Sampling Adequacy’ and ‘Bartlett’s test of sphericity’ tell us about our data?

- Using the anti-image correlation matrix, describe how good the measures of sampling adequacy are for the individual variables.

- By looking at the ‘Total Variance Explained’ table, how many factors have eigenvalues over 1? How much variation do they explain?

- By using the SPSS output and the answers from the above questions write a paragraph discussing your findings. In your discussion highlight the following:

- Could some items be dropped as a result of poor measures of sampling adequacy?
- Could the Scree plot be used to support a different number of factors?
- 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 | .882^{a} |
.778 | .734 | 4.54799 |

a. Predictors: (Constant), Caffeine, DrinkingFrequency, Smoking, UnitsConsumed |

ANOVA^{a} |
||||||

Model | Sum of Squares | Df | Mean Square | F | Sig. | |

1 | Regression | 1449.519 | 4 | 362.380 | 17.520 | .000^{b} |

Residual | 413.683 | 20 | 20.684 | |||

Total | 1863.202 | 24 | ||||

a. Dependent Variable: HEI | ||||||

b. Predictors: (Constant), Caffeine, DrinkingFrequency, Smoking, UnitsConsumed |

Coefficients^{a} |
||||||||

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 Matrix^{a} |
|||||

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.