Analysis of Covariance

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    Analysis of covariance (ANCOVA) is a statistical technique that blends analysis of variance and linear regression analysis. It is a more sophisticated method of testing the significance of differences among group means because it adjusts scores on the dependent variable to remove the effect of confounding variables. ANCOVA is based on inclusion of additional variables (known as covariates) into the model that may be influencing scores on the dependent variable. (Covariance simply means the degree to which two variables vary together – the dependent variable covaries with other variables.) This lets the researcher account for inter-group variation associated not with the "treatment" itself, but from extraneous factors on the dependent variable, the covariate(s). ANCOVA can control one or more covariates at the same time.

    The purpose of ANCOVA, then, is the following: to increase the precision of comparison between groups by reducing within-group error variance; and, to “adjust” comparisons between groups for imbalances by eliminating confounding variables.

    In order to accurately identify possible covariates, one needs sufficient background knowledge of theory and research in the topic area. Ideally, there should only be a small number of covariates. Covariates need to be chosen carefully and should have the following qualities:

    Continuous (at interval or ratio level, such as anxiety scores) or dichotomous (such as male/ female); reliable measurement; correlate significantly with the dependent variable; linear relationship with dependent variable; not highly correlated with one another (should not overlap in influence); and relationship with dependent variable the same for each of the groups (homogeneity of regression slopes).

    Each covariate should contribute uniquely to the variance. The covariate must be measured before the intervention is performed. Correct analysis requires that the covariate not be influenced by the treatment – it therefore must be measured prior to treatment.

    The independent variable is a categorical (nominal-level) variable.

    ANCOVA tests whether certain factors have an effect on the outcome variable after removing the covariate effects. It is capable of removing the obscuring effects of pre-existing individual differences among subjects. It allows the researcher to compensate for systematic biases among the samples. The inclusion of covariates can also increase statistical power because it accounts for some of the variability.

    Assumptions: same as ANOVA (normal distribution, homogeneity of variance, random sampling); relationship of the dependent variable to the independent variable(s) must be linear; dependent variables must be independent; regression lines must be parallel; normal distribution with means of zero; and homoscedasticity. The model assumes that the data in the two groups are well described by straight lines that have the same slope.

    An example of ANCOVA is a pretest-posttest randomized experimental design, in which pretest scores are statistically controlled. In this case, the dependent variable is the posttest scores, the independent variable is the experimental/ comparison group status, and the covariate is the pretest scores.

    With ANCOVA, the F-ratio statistic is used to determine the statistical significance (p .05) of differences among group means. Partial Eta Squared is used to determine effect size:

    Small .01
    Medium .06
    Large .138

    There are both one-way and two-way analyses of covariance.

    When a researcher reports the results from analysis of covariance (ANCOVA), he or she needs to include the following information: verification of parametric assumptions; verification that covariate(s) measured before treatment; verification of reliability of the covariate(s); verification that covariates are not too strongly correlated with one another; verification of linearity; verification of homogeneity of regression slopes; dependent variable scores; independent variable, levels; covariate(s); statistical data: significance, F-ratio scores, probability, means, and effect size (partial eta squared). An example is below:

    Presenting the results from one-way ANCOVA (Pallant, 2007, p. 303)

    A one-way between-groups analysis of covariance was conducted to compare the effectiveness of two different interventions designed to reduce participants’ fear of statistics. The independent variable was the type of intervention (math skills, confidence building), and the dependent variable consisted of scores on the Fear of Statistics Test administered after the intervention was completed. Participants’ scores on the pre-intervention administration of the Fear of Statistics Test were used as the covariate in this analysis.

    Preliminary checks were conducted to ensure that there was no violation of the assumptions of normality, linearity, homogenity of variances, homogeneity of regression slopes, and reliable measurement of the covariate. After adjusting for pre-intervention scores, there was no significant difference between the two intervention groups on post-intervention scores on the Fear of Statistics Test , F (1, 27) = .76, p = .39, partial eta squared = .03. There was a strong relationship between the pre-intervention and post-intervention scores on the Fear of Statistics Test, as indicated by a partial eta squared value of .75.

    Presenting the results from two-way ANCOVA (Pallant, 2007, p. 310)

    A 2 by 2 between-groups analysis of covariance was conducted to assess the effectiveness of two programs in reducing fear of statistics for male and female participants. The independent variables were the type of proram (math skills, confidence building) and gender. The dependent variable was scores on the Fear of Statistics Test (FOST), administered following completion of the intervention programs (Time 2). Scores on the FOST administered prior to the commencement of the programs (Time 1) were used as a covariate to control for individual differences.

    Preliminary checks were conducted to ensure that there was no violation of the assumptions of normality, linearity, homogeneity of variances, homogeneity of regression slopes,and reliable measurement of the covariate. After adjusting for FOST scores at Time 1, there was a significant interaction effect. F (1, 25) = 31.7, p < .0005, with a large effect size (partial eta squared = .56). Neither of the main effects were statistically significant, program: F (1, 25) = 1.43, p = .24; gender: F (1, 25) = 1.27, p = .27. These results suggesst that males and females respond differently to the two types of interventions. Males showed a more substantial decrease in fear of statistics after participation in the maths skills program. Females, on the other hand, appeared to benefit more from the confidence building program.

    In presenting the above results, the researcher should also provide a table of means for each of the groups.


    Pallant, J. (2007). SPSS survival manual. New York: McGraw-Hill Education.

    Polit, D. F., & Beck, C. T. (2008). Nursing research: Generating and assessing evidence for nursing practice (8th ed.). Philadelphia: Wolters Kluwer Health.
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  3. by   zuzi
    Vicky is one of the article that warm my stoned a beuaty. I belived that never ever again I will read or hear somenthing related to science, looooool. Thanks for your article for me...means a lot. And by the way is a good presentation of the covariance.