Showing posts with label analysis of variance. Show all posts
Showing posts with label analysis of variance. Show all posts

Tuesday, January 5, 2021

ANCOVA in Counseling & Behavioral Research



ANCOVA is a procedure like ANOVA except researchers can study the effects of one or more independent variables on a dependent variable after adjusting for other variables, called covariates, which were not a primary focus of the study. The letter C in ANCOVA stands for covariate. There can be several covariates in a study. In testing for differences among groups experiencing different leadership styles, we could study the effects on employee satisfaction after adjusting for a covariate of years of employment. A key word in ANCOVA studies is adjusting. Analysts adjust the scores based on information about the covariate before testing for significant differences.

Basic features of an ANCOVA:

Independent or grouping Variable = 1 or more

Dependent or criterion Variable = 1

Covariates = 1 or more

An test indicates significance overall and for specific effects or relationships.

A commonly reported measure of effect size is eta squared.

value reveals the probability of a significant relationship-- one that is not due to chance factors.

Read more about ANCOVA in the following books.

Applied Statistics Concepts for Counselors on AMAZON or GOOGLE

Creating Surveys on AMAZON    or   GOOGLE  Worldwide


Analysis of Covariance

Geoffrey W. Sutton, Ph.D.

            The analysis of covariance is a research strategy that is based upon a two or more groups design that yields at least interval data and could be analyzed using ANOVA. We use the term ANCOVA as an acronym. The letter C in the acronym represents a covariate. We usually refer to the covariate as a CV. A covariate is a variable that is significantly correlated with the dependent variable.

 In experimental research, the covariate helps reduce error variance and makes the F-test more sensitive to any main or interaction effects. The correlation between the covariate and the DV allows for the removal of the effects of a CV on a DV and represents a known source of systematic bias. 

In nonexperimental research, researchers can use the covariate to statistically remove the influence of a variable to help equate groups that could not be formed by random assignment or to better understand another relationships of interest. 

A third purpose is to examine group differences by controlling for the influence of a DV when there are several DVs in an analysis. This latter use is known as multivariate analysis of variance or, MANCOVA.

 You can use more than one CV in a research design or ANCOVA procedure. However, if there is a correlation greater than r =  .80 between two CVs, you should use only one of the CVs because they appear to be measuring a lot of the same variance.

  As with all statistical procedures, there are several assumptions to meet. The first three are basic assumptions for ANOVA and the next three are additional assumptions for ANCOVA.

1. All data are from random samples and independent of other data.

2. The scores on the DV are normally distributed in the population.

3. The distributions of scores on the DV have equal variances.

            Additional assumptions for ANCOVA

4. There is a linear relationship between the CV and the DV.

5. The slope for the regression line (for the CV) is the same in each group.

6. The CV has high reliability and was measured without error.

Research questions and hypotheses

 We generally use a key phrase to identify the CV in a research study. That key phrase can be controlling for or adjusted for. Here are some examples.

1. What is the difference between memory scores for people with right and left hemisphere stroke when adjusted for age?

2. What is the effect of a marriage enrichment program when controlling for years of marriage?

The research hypothesis states there is an effect or a difference when adjusting for the CV and the null hypothesis assumes the usual no difference, or no effect result, when adjusting for the CV.

Following is a hypothesis based on question number one.

H1: When adjusting for age, there is a significant difference between the means on verbal memory between patients who experience right and left hemisphere strokes.

H01: When adjusting for age, there is no significant difference between the means on verbal memory in the population between patients who experienced right and left hemisphere strokes (p < .05).

The research method with a CV

 We would follow usual procedures for delivering the IV (independent variable) or measuring a QIV (quasi-independent variable) along with the DVs. We would consider what variables might affect the DVs and collect data to measure those CVs. After all the data have been entered into our database, we would obtain the descriptive statistics. Next, make any adjustments to the data and calculate correlations between the measured variables. Those variables that were not the primary focus of the experiment or study will be entered as CVs in the ANOVA procedure if they are highly correlated with the DVs. We will perform the usual post hoc analyses, if applicable.


When interpreting the results of an ANCOVA, we will refer to the adjusted means. SPSS reports the results of the analysis. In the Test of Between Subjects Effects table, SPSS reports the CV along with an F-test. If the CV made a significant contribution to the analysis, the p-value for the CV will be less than .05 (or your preferred level of significance). The output will also include adjusted and unadjusted means. When reporting the results, you should report both sets of means. In a small study, the means can be reported in a paragraph. In a larger study, the means should be placed in a table.

Example of ANCOVA reporting for a fictitious study.

IV = communication skills training vs. a no skills training control (2 groups)

DV = some measure of communication on a continuous scale

CV = years of employment-- a continuous scale

Workplace communication skills training for employees significantly improved positive statements when adjusted for years of employment, F(2,38) = 4.56, p = .03, eta2 = .42, Observed Power = .67.


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  ResearchGate   Geoffrey W Sutton 

ANOVA in Counseling & Psychology Research

There are several types of ANOVA procedures. The term ANOVA refers to Analysis of Variance. Variance is a statistical term we will review later. Variance refers to differences, so the ANOVA procedures examine differences in scores among groups of people who complete a survey, a test, or produce a scorable response.

For example, an ANOVA can be used to assess the effects of three temperatures on math. The Independent Variable is temperature varies three ways (75, 85, 95 degrees F). The dependent variable is math. The dependent measure of math is a math test.

When there is only one independent variable (IV), the ANOVA is called a one-way ANOVA. If there are two IVs the ANOVA is a two-way ANOVA, and so forth. It is rare to go beyond a four-way because the interpretation of interactions is complicated.

The ANOVA procedure is usually reported with an F value. The larger the F value, the more likely it is that the differences the researchers found are not due to chance.

There may be several independent variables in a project. The effect of each variable is tested with an F test. When there are two or more variables, researchers also test for possible interaction effects, which results in additional F tests for each interaction. Interactions refer to the possibility that two or more variables combine to produce a change in the dependent variable. As with t tests, researchers include a probability (p) value with each F test. A common effect size associated with F tests is partial eta squared. An ANOVA is used when there are one or more independent variables but only one dependent variable.

Independent or grouping Variable = 1 or more IV

Dependent or criterion Variable = 1 DV

Dependent Measure = DM = a test score or quantitative measure of the DV

IV Groups or levels = 2 or more. An IV may have many levels (e.g., temperature, dosages) or groups (e.g., therapy groups, learning groups, work groups).

Overall tests are used to determine significant effects or differences among the groups or levels of the IVs.

An test indicates significance overall and for specific effects or relationships.

A commonly reported measure of effect size is eta squared. In psychology, researchers have been encouraged to focus on effect sizes rather than p values when analyzing and reporting research results.

value reveals the probability of a significant relationship-- one that is not due to chance factors. The level of significance is set by the researchers. A common level is p is .05. F values yielding a probability below 0.05 are commonly considered significant in psychology and education.

The concept or idea is that a difference between means yields a large F value that would only occur 95 times out of a hundred due to chance. When researchers analyze the differences, they are analyzing variance hence, ANOVA.

If the overall F test is significant, then researchers may compare group means two at a time to determine possible significant differences between pairs of groups. There are many tests of pairs. For example t tests, Tukey HSD, Bonferroni, Neuman-Keuls. These tests are called post hoc tests because they are used only if the overall F test is significant.

Read more about ANOVA and data analyses in the following books.

Applied Statistics Concepts for Counselors on AMAZON   or   GOOGLE

Creating Surveys on AMAZON    or   GOOGLE  Worldwide

Illustration of a three group design project. Volunteers in a math class are randomly assigned to three levels of room temperature (Fahrenheit) to determine if room temperature has an effect on the score. The random assignment should mean that individual characteristics of the participants would not be a factor in the score.

(The temperature is in the 90s today and expected to reach the 100s this week!)




75 degrees

Math score

85 degrees

Math score

95 degrees

Math Score

If the researchers worried that randomization did not truly even out the math skills of the participant's then they could give a math  pretest.

The IV is a research design term. In statistics, the IV is often represented by the letter X and the three groups would be X1, X2, and X3.

The DV if often referred to using the letter Y in statistics. In this study, there would be three Y values Y1, Y2 and Y3.

If the overall F value indicated a significant difference for the variation in scores, then the researchers could compare each pair. There would be three comparisons.

Y1 and Y2
Y2 and Y3
Y1 and Y3

What is compared?

In ANOVA, the differences among the mean scores are analyzed.
In the post hoc tests, the differences between the pairs of means are analyzed.

Serious stats help

If you need help with detailed analysis, one of the best statistics books for details is Discovering Statistics by Andy Field. I've used an earlier edition as a course textbook. It is of course not needed for those who desire only a conceptual understanding. See Discovering Statistics on AMAZON for more information.

Other notes
The letter F in the F value comes from the surname of the British statistician, Sir Ronald Fisher (1890-1962) born in my hometown, London, England.

Daniel Fahrenheit of temperature fame was born in Poland to a German family.

Links to Connections


Checkout My Website


See my Books






JOIN me on


   FACEBOOK   Geoff W. Sutton  


   TWITTER  @Geoff.W.Sutton




Read many published articles:


  Academia   Geoff W Sutton   


  ResearchGate   Geoffrey W Sutton 

Sunday, October 1, 2017

Take a brief Counseling Test Quiz 101

Can you answer these questions that every counselor ought to know?

Choose the BEST available answer.

I'll post the answers below.

1. If the correlation between a test of intelligence and a test of achievement is usually between .88 and .92, how well can you use the intelligence test results to predict achievement test results?

A. Very well
B. Moderately well
C. Not well at all
D. None of the above

2. A personality test score was high on a scale of Extraversion. The validity of the Extraversion scale was reported as .52 to .57 compared to similar tests. How much confidence should the person have that their score is "valid?"

A. A high degree
B. A moderate degree
C. A low degree
D. None of the above

3. School counselors administered a questionnaire to 1,000 students. They calculated results for answers about four school improvements rated on a scale of 1 to 5. Most of the scores were in the range of 18 to 20. The counselors reported a mean rating of 4.6 for each of the 4 items. Based on these data, what should they have reported?

A. The mean is fine-- an average is all that is needed.
B. They should report the Mean and Standard Deviation.
C. They should report the reliability with the mean.
D. They should report the median and range.

4. An agency director asks a counselor to determine if there was evidence of improvement in well-being for clients in one of three treatment groups. Assuming a normal distribution of the data, which of the following statistical procedures could provide the best answer?

A. An independent samples t test
B. A one-way analysis of variance
C. A two-way analysis of variance
D. A chi-square test


1. A. Other things being equal, the correlation between the two tests is strong thus, most of the time the intelligence test score will be a good predictor of the achievement test score. See Chapter 12 in Applied Statistics: Concepts for Counselors.

2. C. We do not know much about the validation of the Extraversion scale ; however, we know the validity values in the .50s are low so the best answer, given the limited data, is C. Validity coefficients range from 0.0 to 1.0. Important note: Validity is a product of the interpretation of data based on scores. Although it is common to refer to a test's validity, tests really do not have validity. Instead, there is a history of validity statistics and interpretations associated with validity. See chapter 20 in Applied Statistics: Concepts for Counselors.

3. D. The data appeared skewed given that 4 items on a 5-point scale would yield a maximum of 20. So, based on the limited data, the median would be the most typical value. When reporting the mean, counselors ought to report the standard deviation, but in this case, the median appears to be the best value. See Chapters 7-10 of Applied Statistics: Concepts for Counselors.

4. A one-way analysis of variance can be used to analyze data from two or more groups. If the overall value is statistically significant, t tests or other post hoc tests can be used to compare pairs of means. See Chapters 15-17 of Applied Statistics: Concepts for Counselors.

APPLIED STATISTICS: CONCEPTS FOR COUNSELORS is available as an eBook or paperback from AMAZON.

Book website

"If you need to review basic statistics and don’t know where to begin, this book is perfect! It makes difficult concepts easy to understand. I would recommend it for my undergraduate students who haven’t had Statistics in a while and need a refresher, or for graduate students facing their first graduate level research class!"
...Heather L. Kelly, Psy.D., Professor of Psychology, Evangel University
Springfield, Missouri, USA

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