## Thursday, February 15, 2018

### Analysis of Covariance (and) Multivariate Analysis of Covariance (SPSS)

As if MANOVA and ANOVA were not difficult enough to understand, today we will be discussing their most complex forms, MANCOVA and ANCOVA. The “C” in the acronym stands for “covariance”. Which of course generates the inquiry, “What is covariance?”

Covariance is defined as: “a measure of the joint variability of two random variables.” *

In regular terms, this translates to: a variable within the model which will reduce error margin and increase specificity. The covariant, within the context of a model, essentially creates a means for weighing the model's dependent variable(s).

Here are few examples of variable groups which contain a covariant. The covariant in each set is proceeded by a (CV).

Set 1 = { Student Name, Score in AP English, Score in AP English Placement Exam, GPA, Student’s Parents’ Income (CV) }

Set 2 = { Patient’s Name, Weight, Hours Exercised per Week, Gender, Weight at the End of X Weeks, Caloric Intake (CV) }

Set 3 = { Tennant’s Name, Earnings, Late Payment Notices, Number of Children (CV) }

A covariant is a factor which exists outside of experimental parameters, which may assist in the analyzation of the data.

Just like their ANOVA and MANOVA equivalents, ANCOVA and MANCOVA can be multifactor. For our examples, we will perform two-factor analysis.

The dependent and covariate variables in both ANCOVA and MANCOVA models must be continuous. The independent variables must be categorical.

Two-Way ANCOVA (Analysis of Covariance) Example:

Below is the data set which we will be utilizing:

To begin, select “Analyze”, followed by “General Linear Model”, then select “Univariate”.

This series of selections should cause the following screen to appear:

After selecting the independent variables “School” and “Study_Time” as our “Fixed Factor(s)”, we will select the variable “Satisfaction” as our “Dependent Variable”. The “Covariate(s)” that we will be selecting is “Attended_Tutoring”.

We cannot create a “Post Hoc” test while utilizing the ANCOVA model. Therefore, we will proceed with creating our analysis output by clicking “OK”.

This generates the following report:

If we were to write our statistical conclusion based on this output in APA format, the conclusion would resemble:

There was no significant effect of school selection on life satisfaction after controlling for the effect of tutoring attendance, F(1, 23) = .048, p = .829.

There was a significant effect of study time on life satisfaction after controlling for the effect of tutoring attendance, F(2, 23) = 4.197, p = .028.

There was not a significant interaction between the variables school and study time, while controlling for the effect of tutoring attendance, F(2, 23) = .523, p = .600.

The covariate, tutoring attendance, was not significantly related to life satisfaction, F(1, 23) = 3.445, p = .076.

Two-Way MANCOVA (Multivariate Analysis of Covariance) Example:

The sample data set for this exercise can be found below:

We will begin model creation by selecting “Analyze”, then “General Linear Model”, followed by the “Multivariate” option.

This should populate the following screen:

After selecting the independent variables “IndepFactor” and “IndepFactorB” as our “Fixed Factor(s)”, we will select the variables “ContVarA” and “ContVarB” as our “Dependent Variables”. Our “Covariate(s)” will be the variable “CoVar”.

Again, as was the case with the ANCOVA model, we will not be able to run a post hoc test following the model’s creation.

After clicking “OK”, the following output should populate:

From this table, we may state the following conclusions:

Using Pillai’s Trace, there was not a significant effect of “IndepFactor” on “ContVarA” and “ContVarB”, after controlling for the effect of “CoVar”.

Using Pillai’s Trace, there was not a significant effect of the “IndepFactorB” on “ContVarA” and “ContVarB”, after controlling for the effect of “CoVar”.

Using Pillai’s Trace, there was no interaction present between “IndepFactor” and “IndepFactorB”¸ after controlling for the effect of “CoVar”.

Using Pillai’s Trace, there was not a significant effect of the covariant, “CoVar”, on “ContVarA” and “ContVarB”.

* - https://en.wikipedia.org/wiki/Covariance