Quantitative & design methods in business research
Lecture 1:
Multivariate data analysis comprises all statistical methods that simultaneously analyze
multiple measurements on each individual or object under investigation. à Why: measure
more precise, Explain and predictions AND hypothesis testing
Nonmetric or qualitative data = Nominal and ordinal scale, Metric or quantitative = interval or
ratio scale.
- Nominal scale: size of number is not related to the amount of the characteristic being
measured.
- Ordinal scale: larger numbers indicate more of the characteristics à Grades
- Interval scale: e.g., Likert-scale: equal distance but you cannot multiple (2 is not double of 1)
- Ratio scale: e.g., turnover: equal distance, you can multiple and there is a natural 0 point.
Measurement error = distorts observed relationships and makes multivariate techniques less
powerful.
- Reliability: the observer variables degree of precision and thus the lack of random
measurement error
- Validity: the degree to which a measure accurately represents what it is supposed to.
• Type I error or α= rejecting H0, even if you shouldn’t: you conclude that there is a
difference/ effect but there is no difference or effect
• Type II error or β= not rejecting H0, even if you should have you conclude that there is
no difference/ effect, but there actually is a difference or effect.
• Power (1- β) is the probability of rejecting the null hypothesis when false
• Effect size: the actual magnitude of the effect of interest
• Alpha = 0.05. as alpha is set smaller, power decreases
• Sample size: as sample size increases, power increases.
• Statistical power: 0.80, more stringent significant levels require larger samples to
achieve the desired power level. Power can be increased by choosing a less stringent
alpha level. Any increase in power is most likely achieved by increased sample size
Dependence techniques: a variable is identified as a dependent variable to be predicted or
explained by an independent variable
Interdependence techniques: they involve the simultaneous analysis of all variables in a set,
without distinction between dependent and independent variables. (Factor analysis)
Guidelines for multivariate analysis:
1. Establish practical significance as well as statistical significance.
2. Sample size affects results (the higher the sample size the more reliable the results)
3. Know your data: understand meaning and investigate limitations
4. Strive for model parsimony= simple models with great explanatory predictive power.
,5. Look at your errors: avoid and if not possible explain
6.Validate your results.
A Structural approach to multivariate model building
1. Define the problem, objectives, and right techniques to use
2. Analysis plan: execute a step-by-step analysis
3. Check assumptions for the technique
4. Estimate the model (computer) + assess the model fit (e.g., R2)
5. Interpret coefficients
6. Validate the multivariate models
Frequency distribution à one variable is considered at a time. A frequency distribution for a
variable produces a table of frequency counts, percentages, and cum. Percentages for all the
values associated with that variable à measures of location and of variability
Missing dataà information not available for a subject about who other information is available.
Use observations with complete data only à delete cases or estimate missing values.
Rule of thumb: missing data under 10% for an individual case or observation can be generally
ignored, expect when the missing data occurs is a specific order.
Notes factor analysis
Exploratory factor analysis (EFA) and Principal component analysis (PCA) are two quite similar
techniques:
- Both are used for data reduction and summarization
- Both are independent techniques (no differentiation between dependent and
independent techniques).
Differences:
- PCA considers the total variance of variables, and its main goals is to express the dataset
by a fewer number of variables (data reduction).
- EFA considers only the common variance of variables, and its main goals is to identify
underlying dimensions that explain the correlations among the variables.
When to use it:
- Consumer research: understanding consumer ratings of a restaurant based on
underlying dimensions e.g., food quality and service quality
- Image compressions: capture the main characteristics of pictures.
Steps of EFA & PCA
1. Select variable:
Use could use past research to see what others did, you could use theory or judgement of the
researcher. Usually, the data is collected in a matrix, where each row represents an
observation, and each column contains a variable.
,2. Data screening & testing assumptions:
- All your variables should be interval or ratio scaled. If you have ordinal scaled variables
than use the study of Baglin (2014).
- Sample size needs to be 4 or 5 times as many observations as variables.
- No missing data. Rows in the data matrix containing a missing value should be deleted
(list-wise deletion)
- No outliers as the techniques are sensible to abnormal observations. An outlier is an
observation that is distant compared to the others. They can be detected by descriptive
statistics or scatter plots. Valid observations = observations per variable, then there are
no missing observations.
- Range = minimum value – the maximum value. So, when looking at the range you can
detect a potential outlier. To get a more detailed picture look at the minimum and
maximum values. For example, if you have a very small minimum value there could be
an outlier.
3. Factorability. At least some sizeable correlation among the variables.
- It can be assessed by considering the correlation matrix. There need to be some
correlation above 0.3. Correlation between the variable and itself it always 1. To check
the factorability there needs to be a number above 0.3.
- The diagonal of the anti-image matrix measures the sampling adequacy (MSA) and
should be above 0.5. It tells us how well a variable can be explained by the other. 0
means that the variable cannot be explained by another variable and a 1 means that it
can be perfectly explained.
- Kaiser-Meyer-Olkin (KMO) criterion should be above 0.5.
- Bartlett’s test of sphericity (should be significant). Test whether the variables are
correlated. The null hypothesis needs to be rejected (sig or p value < 0.05), then it is
suitable to use it for a PCA or EFA analysis.
4. Extraction method
- Principle component analysis
- Exploratory factor analysis
o principle component analysis (PCA) is a data reduction technique. We want to
find the smaller set of factors – so called components- that explain as much as
possible of the total variance in the original dataset. Before the analysis
standardize the variables. To start draw a random line of the y-as or differently.
Rotation of the line, like a coordination system. Points on the line are not
maximally spread, therefore not as much variance as possible is captured. The
first principal components do not capture all the variance, so a second
component is necessary to capture the remaining variance. Table total variance
explained: % of variance (total score: number of variables). Communalities tell
us how much of the variance of a variable is explained by the extracted
principal components (number of variables).
, EXTRACT A LOWER NUMBER: The two principal components can explain 44% of the
variance (right row, table of communalities). Lambda is seen in the component
matrix (1 and 2).
o Picture compression: each picture can be seen as a combination or red, green,
and blue. Put rows in the columns and columns in the rows, which will tell you
the strength of the color. From 350 to 40 components gives more or less the
same quality.
- Principle axis factoring: In contrast to PCA, principal axis factoring, which is an approach
to EFA, does not consider the total variance of the dataset, but wants to explain as best
as possible the correlations among the observed variables. In general, PAF is used to
identify several common factors or dimensions underlying the observed variables that
can explain the correlations structure among the observed variables.
EFA model à two variables that explain the variance in the variables.
The measurement error measures the variance that cannot be explained by the
common factors. In EFA we cannot extract as many factors as variables we can only
extract the number of variables – one factor. When looking at extraction sums in the
table, it explains how much variance in the dataset can be explained by the common
, factors (% of variance). The table of communalities tells us how much of the variance in
a variable can be explained by al the extracted common factors. In EFA the
communalities are first estimated (all below 1) and then you have the communalities
after extraction. You need communalities above 0.5 otherwise more than 50% of the
variance of the variable it is explained by measurement errors.
Extraction of only 2 common factors à Communalities second column (right) tell us that
19% of the variance can be explained by the common factors.
There is also a factor matrix representing the correlation between the common factors
and the variables. You want a clearer picture to make sure that variables that are on the
dimension load high on one factor and low on the second factor.
5. Determine number of factors
In practice, it is not desirable to have a full set of factors since we usually want to reduce the
number of dimensions. Therefor we do not consider all the factors. Several methods could be
used: prior knowledge, Eigenvalues, Percentage of variance, Scree plot.
- Based on prior knowledge: the researcher knows how many factors to expect and thus
can specify the number of factors to be extracted beforehand.
- Determination based Eigenvalues: Only factors with Eigenvalues greater than 1.0 are
extracted. Note: an eigenvalue represents the amount of variance associated with the
factor. Hence, only factors with a variance greater than 1.0 are included. Factors with a
variance less than 1.0 are no better than single variance, since due to standardization,
each variable has a variance of 1. If the number of variables is less than 20, this
approach will result in a conservative number of factors. Often used for EFA or Principal
factor analysis.
- Percentage of Variance: The number of extracted factors is determined so that the
cumulative percentage of variance explained by the factors reaches a satisfactory level.
It is recommended that the factors extracted should account for at least 60% of the
variance.
- Determination based on scree plots: a scree plot is a plot of the Eigenvalues against the
number of factors. Experimental evidence indicated that the point at which the scree
begins denotes the true number of factors i.e., left of the elbow. Generally, the number
of factors determined by the scree plot will be one or a few more than that determined
by the Eigenvalue creation. (y-axis Eigenvalue, x-axis number of factors) à look for an
‘elbow’ and go one to the left. However, it does not always give a unique solution.
6. Factor rotation
Although the initial or unrotated factor matrix indicate the relationship between the factors, it
seldom results in factors that can be interpreted, because they are highly correlated with many
variables. To overcome this problem, you can rotate your variables. You can determine the x
and y values of the variables. It would be desired to have a variable with a weights high for one
of the two factors (for example 0.8 on factor 2). In rotation the factors, we would like each
factor to have non-zero, or significant, loadings or coefficient for only some of the variables.
