Factor Analysis and Principle Component Analysis Case

1 Factor Analysis and Principle Component Analysis Case In this study, I examined factors that predict language development among first-grade students. I believed that some of the variables may be correlated and conducted factor analysis with SPSS. This involved examining 7 variables; reversal (syllable reversal), delete onset, a judgment of tone, learn the word (morpheme learning), odd word (pick up the odd word in four choices), circle word (select the correct "root" for the word), and fill in word (fill in a word in a sentence). The analysis uses DEL618m5.sav dataset that contained 333 cases.

Factor Analysis Results Table 1: Correlation Matrix reversal Correlation reversal

delete

judge

learn

odd

fill in

circle

onset

tone

morpheme

word

word

word

1.000

.275

.348

.186

.160

.137

.262

delete onset

.275

1.000

.318

.096

.058

.088

.209

judge tone

.348

.318

1.000

.096

.113

.182

.264

learn

.186

.096

.096

1.000

.280

.334

.392

odd word

.160

.058

.113

.280

1.000

.208

.319

fill in word

.137

.088

.182

.334

.208

1.000

.610

circle word

.262

.209

.264

.392

.319

.610

1.000

morpheme

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2 The results show a correlation matrix with several correlations above .3, indicating appropriateness to continue with factor analysis (Pallant, 2013).

Table 2: KMO and Bartlett's Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy.

Bartlett's Test of Sphericity

.713

Approx. Chi-Square

369.767

Df

21

Sig.

.000

The Kaiser-Meyer-Olkin Measure of Sampling Adequacy is .713 which is within an acceptable level of .6 or above. Bartlett's Test of Sphericity is significant (p < .001).

Table 3: Communalities Initial

Extraction

Reversal

.188

.315

delete onset

.145

.258

judge tone

.200

.385

learn morpheme

.202

.251

odd word

.135

.149

fill in word

.386

.476

circle word

.471

.723

Extraction Method: Principal Axis Factoring.

The results indicate communalities were low with odd word having the lowest variance (14.9%) in the analysis (Statistics Solutions, 2013).

Table 4: Total Variance Explained Extraction Sums of Squared

Rotation Sums of Squared

Loadings

Loadings

% of

% of

% of

Varianc Cumulativ

Varianc Cumulativ

Varianc Cumulativ

Initial Eigenvalues Facto r

Total

e

e%

Total

e

e%

Total

e

e%

3

1

2.469

35.275

35.275 1.921

27.446

27.446 1.562

22.317

22.317

2

1.283

18.330

53.605

9.091

36.537

14.219

36.537

3

.858

12.260

65.865

4

.721

10.296

76.161

5

.705

10.077

86.239

6

.602

8.593

94.832

7

.362

5.168

100.000

.636

.995

Extraction Method: Principal Axis Factoring.

Using the principal axis factoring method of extraction and applying Kaiser’s eigenvalue greater than 1, I extracted 2 factors. The rest of the factors did not meet the criteria. The first factor has an eigenvalue of 2.469 and accounts for 35.275% of the variance. The second factor has an eigenvalue of 1.283 and accounts for 18.330% of the variance.

Figure 1: Screen plot for exploratory factor analysis (Maxwel, 2014).

4 The screen plot shows the point of inflection at factor 3 indicating a two-dimensional solution.

Table 5: Rotated Factor Matrixa Factor 1

2

Reversal

.533

delete onset

.502

judge tone

.604

learn morpheme

.489

odd word

.369

fill in word

.684

circle word

.813

Extraction Method: Principal Axis Factoring.

Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 3 iterations.

Table 5 shows the latent variable representing each factor. Factor 1 includes learn morpheme, odd word, fill in word, and circle word variables. Factor 2 includes reversal, delete onset, and judge tone variables. The two factors are extracted explaining 36.54% of the variance.

Compare statistical tests Secondly, I compared different statistical tests based on purpose and data assumptions.

Statistical test

Descriptive statistics

Purpose

Assumptions

Describing basic attributes of data using

Normality

summarized coefficients (Statistics Solutions, 2013).

Homogeneity Linearity

5

Independence T-test

Used to show if there is a significant difference between groups or the differences are due to chances.

Normality Equality of variance Random sampling Sample size adequacy

ANOVA

Testing for significant differences between three or more means from unrelated groups.

Normality No multicollinearity Homogeneity of variance

ANCOVA

Testing the main and interaction effects for the categorical variables with dependent variable being continuous. It

Normality Homogeneity

enables effect controlling of selected

Random independence

continues variables that co-vary with

of samples

dependent variable Chi-Square

Correlation

To test association strength among two

Random sampling

categorical variables.

Categorical data

Determine the strength of association

Linearity

among two continuous variables (strength of linear relationship) (Baguley, 2012).

Simple Linear

To determine how changes in a predictor

Regression

variable forecast change levels in outcome variable using a mathematical

Outliers absence Level of measurement

Linearity

6 model.

Homoscedasticity Normality Absence perfect multicollinearity

Multiple Linear

Tests mathematical model on how

Regression

changes among two or more predictor random variables (independent)

linearity Homoscedasticity

determine the level of change in the

Normality

dependent variable

Absence perfect multicollinearity

MANOVA/

Hypothesis testing on the effects of one

MANCOVA

or more predictor variables on two or more dependent variables through p-

Repeated Measures

Linearity Homoscedasticity

value generation

Normality

For controlling factors causing

Independent

variability among subjects

observations

(Statistics Solutions, 2013).

Normality Sphericity

Factor Analysis

A process for data reduction by establishing which linear component exists within the data and how a

Linearity No multicollinearity

particular variable might contribute to

Existence of correlation

that component

among variables and factors.

7

Reference Baguley, T. (2012). A guide to advanced statistics for the behavioral sciences. Macmillan International Higher Education.

Maxwel, A. E. (2014). Factor Analysis. John Wiley & Sons, Inc. Pallant, J. (2013). SPSS survival manual: a step by step guide to data analysis using IBM SPSS. McGraw-Hill.

Statistics Solutions. (2013). Confirmatory Factor Analysis [WWW Document]. Retrieved from http://www.statisticssolutions.com/academic-solutions/resources/directory-of-statisticalanalyses/confirmatory-factor-analysis/