Analysis
process
Factor Problem Objectives: data summarization / data reduction
analysis formulation Variables: ratio/interval, sample size 4-5 N per variable
Exploratory factor analysis Confirmatory factor analysis
Find an underlying structure A priori ideas of underlying structure
Assumptions that factors cause correlations Relationships between variables and factors
between variables. Errors uncorrelated before conducting the analysis. Errors could
correlate
Analyze correlation matrix Analyze variance-covariance matrix
Purpose: generation of hypothesis Purpose: testing of hypothesis
Constructing Useful matrix based on:
correlation - KMO measure of sampling adequacy. Does the sample represent the population? KMO
matrix above .50
- Bartlett’s test of sphericity: test H0 that variables are uncorrelated in the population.
Rejection is needed because you want correlation. Sig <0.05
Selecting Principal components analysis Common factor analysis
extraction Total variance (1,000) Common variance
method Unities Communalities
Primary concern: minimum number of Primary concern: identify the underlying
factors that will account for maximum dimensions and their common variance
variance (data reduction). Factors are (data summarization). Known as principal
principal components axis factoring.
Extraction result: factor matrix. Factor loading: correlation between variable and factor.
Minimum: around 0.5. Significant: above 0.5. Desirable: above 0.7.
Determining - A priori determination
number of - Eigenvalues > 1 (also called latent root criterion)
factors - Scree plot
- Percentage of variance (in total >0.60)
Rotating Each factor should have significant loadings for only some variables, each variable with only a
factors few, most ideal only 1. Rotation for interpretation reasons
Orthogonal rotation (Varimax) Oblique rotation (Oblimin)
Axes maintained at right angles Axes NOT maintained at right angles
Assumes factors are not correlated Assumes factors are correlated
Given objective of data reduction or Meaningfulness of corelated constructs for a
subsequent use in other analysis contexts specific context of the study
Use rotated factor matrix Use pattern matrix
If a correlation >.30, use oblique (see factor correlation matrix). However, theory is most
important!
Interpreting Factor can be interpreted in terms of the variables that load high on it
factors
Using factors - Reliability: Cronbach’s alpha: above 0.7
in other Validity:
analyses - Factor scores: composite measure created for each observation on each factor extracted in
the factor analysis
- Surrogate variables: selection of a single variable with the highest factor loading to
represent a factor in the data reduction stage
- Summated scores: method of combining several variables that measure the same concept
into a single variable to increase reliability of the measurement.
Which reduction method to select?
, Need for simplicity surrogate variables
Replication in other studies summated scales
esire for orthogonality of the measures factor scores
Determining Residuals: comparing differences between observed correlations (as given in input correlation
model fit matrix) and reproduced correlations (as estimated from the factor matrix).
SPSS Cross loading: variable who has two or more factor loadings exceeding the threshold value
Orthogonal rotation rotated factor matrix
Oblique rotation pattern matrix
Remove variables when:
- Factor loadings: <.30
- Cross loader: if highest correlation and second highest correlation <│.20│
- Communalities after extraction >.20 (see table communalities)
AN(C)OVA Identify Independent variable: at least one is categorical (different categories), levels are independent
independent (except repeated-measure ANOVA) so called factors.
Categorical and Dependent variable: must be metrically scaled, like Likert scale.
IV + Metric dependent ANCOVA: independent variables contain both categorical (still factors) and metric variables.
DV variables Covariates are the metric independent variables, to include statistical control variables.
Decompose F= between groups/ within groups (the larger, the more different means)
ANOCAVA the total
+ Metric IV variation
Measure the Assumptions:
effects - Normality (no problem if N of each group >30)
- Independence of errors
Test the - Independent scores
significance - Sample size
- Homogeneity of variance
Interpret the Effect size: >0.01 small >0.06 medium >0.14 high
results Test of main effect hypotheses:
SPSS - H0: the group averages of the diverse groups are equal
- H1: the group averages of the diverse groups are unequal (desirable)
Test of interaction effect
- H0: no interaction effect occurs
- H1: an interaction effect occurs
Interaction effect ordinal/disordinal:
- No interaction: lines are parallel
- Ordinal: Lines don’t cross, direction of change is always the same
- Disordinal with non-crossover: direction of change differs, order is the same
- Disordinal with crossover: the order is even different
Homogeneity of variance Levene’s test.
- H0= equal variances, so homogeneity (desired, so non significance)
- H1= unequal variances, so heterogeneity. when group have equal sizes, it’s not harmful.
In case of unequal sizes, use Welch statistic.
Post hoc analysis:
- Games Howell: in case of heterogeneity
- Hochberg: in case of homogeneity and unequal group sizes
- Tukey: in case of homogeneity and equal group sizes
Group sizes equal or unequal biggest N / smallest N
- Outcome <1.5 than equal group sizes
- Outcome >1.5 than unequal group sizes
process
Factor Problem Objectives: data summarization / data reduction
analysis formulation Variables: ratio/interval, sample size 4-5 N per variable
Exploratory factor analysis Confirmatory factor analysis
Find an underlying structure A priori ideas of underlying structure
Assumptions that factors cause correlations Relationships between variables and factors
between variables. Errors uncorrelated before conducting the analysis. Errors could
correlate
Analyze correlation matrix Analyze variance-covariance matrix
Purpose: generation of hypothesis Purpose: testing of hypothesis
Constructing Useful matrix based on:
correlation - KMO measure of sampling adequacy. Does the sample represent the population? KMO
matrix above .50
- Bartlett’s test of sphericity: test H0 that variables are uncorrelated in the population.
Rejection is needed because you want correlation. Sig <0.05
Selecting Principal components analysis Common factor analysis
extraction Total variance (1,000) Common variance
method Unities Communalities
Primary concern: minimum number of Primary concern: identify the underlying
factors that will account for maximum dimensions and their common variance
variance (data reduction). Factors are (data summarization). Known as principal
principal components axis factoring.
Extraction result: factor matrix. Factor loading: correlation between variable and factor.
Minimum: around 0.5. Significant: above 0.5. Desirable: above 0.7.
Determining - A priori determination
number of - Eigenvalues > 1 (also called latent root criterion)
factors - Scree plot
- Percentage of variance (in total >0.60)
Rotating Each factor should have significant loadings for only some variables, each variable with only a
factors few, most ideal only 1. Rotation for interpretation reasons
Orthogonal rotation (Varimax) Oblique rotation (Oblimin)
Axes maintained at right angles Axes NOT maintained at right angles
Assumes factors are not correlated Assumes factors are correlated
Given objective of data reduction or Meaningfulness of corelated constructs for a
subsequent use in other analysis contexts specific context of the study
Use rotated factor matrix Use pattern matrix
If a correlation >.30, use oblique (see factor correlation matrix). However, theory is most
important!
Interpreting Factor can be interpreted in terms of the variables that load high on it
factors
Using factors - Reliability: Cronbach’s alpha: above 0.7
in other Validity:
analyses - Factor scores: composite measure created for each observation on each factor extracted in
the factor analysis
- Surrogate variables: selection of a single variable with the highest factor loading to
represent a factor in the data reduction stage
- Summated scores: method of combining several variables that measure the same concept
into a single variable to increase reliability of the measurement.
Which reduction method to select?
, Need for simplicity surrogate variables
Replication in other studies summated scales
esire for orthogonality of the measures factor scores
Determining Residuals: comparing differences between observed correlations (as given in input correlation
model fit matrix) and reproduced correlations (as estimated from the factor matrix).
SPSS Cross loading: variable who has two or more factor loadings exceeding the threshold value
Orthogonal rotation rotated factor matrix
Oblique rotation pattern matrix
Remove variables when:
- Factor loadings: <.30
- Cross loader: if highest correlation and second highest correlation <│.20│
- Communalities after extraction >.20 (see table communalities)
AN(C)OVA Identify Independent variable: at least one is categorical (different categories), levels are independent
independent (except repeated-measure ANOVA) so called factors.
Categorical and Dependent variable: must be metrically scaled, like Likert scale.
IV + Metric dependent ANCOVA: independent variables contain both categorical (still factors) and metric variables.
DV variables Covariates are the metric independent variables, to include statistical control variables.
Decompose F= between groups/ within groups (the larger, the more different means)
ANOCAVA the total
+ Metric IV variation
Measure the Assumptions:
effects - Normality (no problem if N of each group >30)
- Independence of errors
Test the - Independent scores
significance - Sample size
- Homogeneity of variance
Interpret the Effect size: >0.01 small >0.06 medium >0.14 high
results Test of main effect hypotheses:
SPSS - H0: the group averages of the diverse groups are equal
- H1: the group averages of the diverse groups are unequal (desirable)
Test of interaction effect
- H0: no interaction effect occurs
- H1: an interaction effect occurs
Interaction effect ordinal/disordinal:
- No interaction: lines are parallel
- Ordinal: Lines don’t cross, direction of change is always the same
- Disordinal with non-crossover: direction of change differs, order is the same
- Disordinal with crossover: the order is even different
Homogeneity of variance Levene’s test.
- H0= equal variances, so homogeneity (desired, so non significance)
- H1= unequal variances, so heterogeneity. when group have equal sizes, it’s not harmful.
In case of unequal sizes, use Welch statistic.
Post hoc analysis:
- Games Howell: in case of heterogeneity
- Hochberg: in case of homogeneity and unequal group sizes
- Tukey: in case of homogeneity and equal group sizes
Group sizes equal or unequal biggest N / smallest N
- Outcome <1.5 than equal group sizes
- Outcome >1.5 than unequal group sizes
