The document outlines a conceptual framework for selecting appropriate multivariate statistical methods based on the types of variables involved and the research questions posed. It categorizes measurement levels into nominal, ordinal, interval, and ratio, detailing which statistical tests are suitable for different combinations of independent and dependent variables. Various tests and their applications, such as correlation, t-tests, ANOVA, chi-square, and regression analyses, are explained to assist in determining relationships between variables.

1. Conceptual Framework of
Multivariate Statistics©
Anish K.R.,Ph D
Rajagiri College of Social Sciences
Moisés Próspero, Ph.D. ©
College of Social Work
University of Utah

2. Which Statistic to Use?
• The type of statistic will depend on
– The research question
– The level of measurement for the independent
variable (IV)
– The level of measurement for the dependent
variable (DV)
• Therefore, your measuring instrument will
determine what statistic to use.

3. 4 Levels of Measurement
• Nominal measures: Categories without rank-
order, such as sex (female, male).
• Ordinal measures: Rank-ordered categories,
such as Likert-type scales (1=never 7=always).
• Interval measures: Continuous scales with
equal distance between intervals, such as
Celsius temperature (-30…0…+30)
• Ratio measures: Continuous scales with equal
distance between intervals and a true zero,
such as age (0, 1, 2, 3, 4, 5…).

4. Levels of Measurement
• Two Types
– Numerical/Continuous
– Categorical

5. Relationship Between 2 Variables
• Research question is, “Is there an association
between age & income?”
• If 2 nominal/categorical measures, use Chi-
square
– Ex: age (old & young) & income (poor & rich)
• If 2 numerical/ratio measures, use Correlation
– Ex: age (0-65) & income (0-$100,000)

6. Relationship Between 2 Variables
• If 1 nominal IV with 2 categories & 1 ratio
DV, use Independent t-test
– Ex: age (old & young) & income (0-$100,000)
• If 1 nominal IV with 3 categories & 1 ratio
DV, use ANOVA
– Ex: age (elderly, adult & adolescent) & income (0-
$100,000)
• Same research question, 4 different statistics!!

7. Part I
• Use the following statistics when the
Dependent Variable (DV) is continuous
(ordinal, interval, ratio)
• Correlation
• T-test
• ANOVA/ANCOVA
• MANOVA/MANCOVA
• Multiple Regression

8. Correlation
• 2 Continuous Variables
• Ex: age & income
• Age   income

9. Independent T-test
• DV: 1 Continuous
• IV: 1 Categorical (2 attributes/levels)
• Ex:
• DV-Income
• IV-gender (m/f)
• Income  gender

10. Dependent T-test
• DV: 1 Continuous
• IV: 1 Categorical/ Time 1 & Time 2
(repeated measures)
• Ex:
• DV-Reported violent behaviors on a scale 0-10
• IV-time (pretest/posttest)
• Pre/Posttest  violence

11. Analysis of Variance
• Analysis of Variance (ANOVA)
• Analysis of Covariance (ANCOVA)
• Multivariate Analysis of Variance
(MANOVA)
• Multivariate Analysis of Covariance
(MANCOVA)

12. One-Way ANOVA
• DV: 1 Continuous
• IV: 1 Categorical (3+ attributes/levels)
• Ex:
• DV-Income
• IV-Ethnicity (H/AA/W/A)
• Income  ethnicity

13. Two-Way ANOVA
• DV: 1 Continuous
• IV: 2 Categorical (2+ attributes/levels)
(# of IV’s = # of Ways)
• Ex:
• DV-Income
• IV-Ethnicity (H/AA/W/A) & Gender (m/f)
• Income  ethnicity & gender

14. ANCOVA
• DV: 1 Continuous
• IV: 1+ Categorical (# of IV’s = # of Ways)
• COV: 1+ Continuous Covariate
• Ex:
• DV-Stress level
• IV-Exercise (Y/N)
• COV-age & income (continuous)
• Stress level  exercise (control for age & income)

15. MANOVA
• DV: 2+ Continuous (related construct)
• IV: 1+ Categorical (# of IV’s = # of Ways)
• Ex:
• DV-Income, years of education, &
occupational prestige (SES construct)
• IV-Ethnicity (H/AA/W/A) & gender (m/f)
• SES construct  ethnicity & gender

16. MANCOVA
• DV: 2+ Continuous (related construct)
• IV: 1+ Categorical (# of IV’s = # of Ways)
• COV: 1+ Continuous Covariate
• Ex:
• DV-Income, years of education, & occupational
prestige (SES construct)
• IV-Ethnicity (H/AA/W/A) & gender (m/f)
• COV-age, # of homes (continuous)
• SES construct  ethnicity & gender (control for age
& # of homes)

17. Multiple Regression
• DV: 1 Continuous
• IV: 1+ Continuous
• IV: 1+ Categorical (2 attributes/levels)
• Ex:
• DV-Income
• IV-age & years of education (continuous variables)
• IV-minority status & gender (categorical variables: do
dummy variables)
• Income  age, years of education, minority status &
gender

18. Part II
• Use the following statistics when the
Dependent Variable (DV) is categorical or
dichotomous (nominal)
• Chi-square
• Log-linear
• Discriminant Function Analysis
• Logistic Regression

19. Chi-Square
• 2 Nominal Variables
• Ex:
• Sex (Female/Male) &
Adoption disruption (Y/N)
• Sex   Adoption Disruption

20. Log-linear
• DV: 1 Categorical
• IV: 2+ Categorical
• Ex:
• DV-Sex (Female/Male)
• IV-Adoption disruption (Y/N)
• IV-Ethnicity (H/AA/W/A)
• Sex   Adoption disruption & Ethnicity

21. Discriminant Function Analysis
• DV: 1 Categorical
• IV: 1+ Continuous
• Ex:
• DV-child abuse (y/n)
• IV-alcohol use frequency, # of children,
income, # of divorces (continuous variables)
• Child abuse (y/n)  alcohol use frequency, #
of children, income, # of divorces

22. Logistic Regression
• DV: 1 Categorical
• IV: 1+ Continuous
• IV: 1+ Categorical (2 attributes/levels)
• Ex:
• DV-child abuse (yes/no)
• IV-alcohol use frequency, # of children, income, # of
divorces (continuous variables)
• IV-minority (y/n) & parent work (y/n) (cat. vars)
• Child abuse (y/n)  alcohol use frequency, # of
children, income, # of divorces, minority & parent
work

23. Multiple v. Logistic Regression
Multiple Regression
• DV: 1 Continuous***
• IV: 1+ Continuous
• IV: 1+ Categorical
(2 attributes/levels)
Logistic Regression
• DV: 1 Categorical***
• IV: 1+ Continuous
• IV: 1+ Categorical
(2 attributes/levels)

24. Correlation v. Chi-Square
Correlation
• 2 Continuous
Variables***
• Ex:
• Age & Income
• Age   Income
Chi-Square
• 2 Categorical
Variables***
• Ex:
• Sex (Female/Male) &
Adoption disruption
(Y/N)
• Sex   Adoption
Disruption

25. Factor Analysis
• Correlated items put together into factors
• Ex:
• Several Likert scales with personal
characteristic items are grouped into related
personality factors
• Personal Characteristics  Personality factors:
neuroticism & extroversion

26. Structural Equation Modeling
• Constructs related to each other
• Indicator Variables: Measured Variables
• Latent Variables: Constructs
• 1 - Incarceration
• 2 – Antisocial
• 3 – Socioeconomic status

27. Structural Equation Modeling
IV  LV
 LV
IV  LV
Fights
Delinquency  Antisocial
Drinking alcohol
 Incarceration
Single-parent household
Dropout  SES
Low-income community