# Analysis of covariance

Last updated

Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s). [1]

## Contents

The ANCOVA model assumes a linear relationship between the response (DV) and covariate (CV):

${\displaystyle y_{ij}=\mu +\tau _{i}+\mathrm {B} (x_{ij}-{\overline {x}})+\epsilon _{ij}.}$

In this equation, the DV, ${\displaystyle y_{ij}}$ is the jth observation under the ith categorical group; the CV, ${\displaystyle x_{ij}}$ is the jth observation of the covariate under the ith group. Variables in the model that are derived from the observed data are ${\displaystyle \mu }$ (the grand mean) and ${\displaystyle {\overline {x}}}$ (the global mean for covariate ${\displaystyle x}$). The variables to be fitted are ${\displaystyle \tau _{i}}$ (the effect of the ith level of the IV), ${\displaystyle B}$ (the slope of the line) and ${\displaystyle \epsilon _{ij}}$ (the associated unobserved error term for the jth observation in the ith group).

Under this specification, the categorical treatment effects sum to zero ${\displaystyle \left(\sum _{i}^{a}\tau _{i}=0\right).}$ The standard assumptions of the linear regression model are also assumed to hold, as discussed below. [2]

## Uses

### Increase power

ANCOVA can be used to increase statistical power (the probability a significant difference is found between groups when one exists) by reducing the within-group error variance. [3] In order to understand this, it is necessary to understand the test used to evaluate differences between groups, the F-test. The F-test is computed by dividing the explained variance between groups (e.g., medical recovery differences) by the unexplained variance within the groups. Thus,

${\displaystyle F={\frac {MS_{between}}{MS_{within}}}}$

If this value is larger than a critical value, we conclude that there is a significant difference between groups. Unexplained variance includes error variance (e.g., individual differences), as well as the influence of other factors. Therefore, the influence of CVs is grouped in the denominator. When we control for the effect of CVs on the DV, we remove it from the denominator making F larger, thereby increasing our power to find a significant effect if one exists at all.

Another use of ANCOVA is to adjust for preexisting differences in nonequivalent (intact) groups. This controversial application aims at correcting for initial group differences (prior to group assignment) that exists on DV among several intact groups. In this situation, participants cannot be made equal through random assignment, so CVs are used to adjust scores and make participants more similar than without the CV. However, even with the use of covariates, there are no statistical techniques that can equate unequal groups. Furthermore, the CV may be so intimately related to the IV that removing the variance on the DV associated with the CV would remove considerable variance on the DV, rendering the results meaningless. [4]

## Assumptions

There are several key assumptions that underlie the use of ANCOVA and affect interpretation of the results. [2] The standard linear regression assumptions hold; further we assume that the slope of the covariate is equal across all treatment groups (homogeneity of regression slopes).

### Assumption 1: linearity of regression

The regression relationship between the dependent variable and concomitant variables must be linear.

### Assumption 2: homogeneity of error variances

The error is a random variable with conditional zero mean and equal variances for different treatment classes and observations.

### Assumption 3: independence of error terms

The errors are uncorrelated. That is, the error covariance matrix is diagonal.

### Assumption 4: normality of error terms

The residuals (error terms) should be normally distributed ${\displaystyle \epsilon _{ij}}$ ~ ${\displaystyle N(0,\sigma ^{2})}$.

### Assumption 5: homogeneity of regression slopes

The slopes of the different regression lines should be equivalent, i.e., regression lines should be parallel among groups.

The fifth issue, concerning the homogeneity of different treatment regression slopes is particularly important in evaluating the appropriateness of ANCOVA model. Also note that we only need the error terms to be normally distributed. In fact both the independent variable and the concomitant variables will not be normally distributed in most cases.

## Conducting an ANCOVA

### Test multicollinearity

If a CV is highly related to another CV (at a correlation of 0.5 or more), then it will not adjust the DV over and above the other CV. One or the other should be removed since they are statistically redundant.

### Test the homogeneity of variance assumption

Tested by Levene's test of equality of error variances. This is most important after adjustments have been made, but if you have it before adjustment you are likely to have it afterwards.

### Test the homogeneity of regression slopes assumption

To see if the CV significantly interacts with the IV, run an ANCOVA model including both the IV and the CVxIV interaction term. If the CVxIV interaction is significant, ANCOVA should not be performed. Instead, Green & Salkind [5] suggest assessing group differences on the DV at particular levels of the CV. Also consider using a moderated regression analysis, treating the CV and its interaction as another IV. Alternatively, one could use mediation analyses to determine if the CV accounts for the IV's effect on the DV.

### Run ANCOVA analysis

If the CV×IV interaction is not significant, rerun the ANCOVA without the CV×IV interaction term. In this analysis, you need to use the adjusted means and adjusted MSerror. The adjusted means (also referred to as least squares means, LS means, estimated marginal means, or EMM) refer to the group means after controlling for the influence of the CV on the DV.

### Follow-up analyses

If there was a significant main effect, it means that there is a significant difference between the levels of one IV, ignoring all other factors. [6] To find exactly which levels are significantly different from one another, one can use the same follow-up tests as for the ANOVA. If there are two or more IVs, there may be a significant interaction, which means that the effect of one IV on the DV changes depending on the level of another factor. One can investigate the simple main effects using the same methods as in a factorial ANOVA.

## Power considerations

While the inclusion of a covariate into an ANOVA generally increases statistical power by accounting for some of the variance in the dependent variable and thus increasing the ratio of variance explained by the independent variables, adding a covariate into ANOVA also reduces the degrees of freedom. Accordingly, adding a covariate which accounts for very little variance in the dependent variable might actually reduce power.

• MANCOVA (Multivariate analysis of covariance)

## Related Research Articles

Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.

An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact "F-tests" mainly arise when the models have been fitted to the data using least squares. The name was coined by George W. Snedecor, in honour of Sir Ronald A. Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s.

In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable. Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations. Interactions are often considered in the context of regression analyses or factorial experiments.

In statistics, a vector of random variables is heteroscedastic if the variability of the random disturbance is different across elements of the vector. Here, variability could be quantified by the variance or any other measure of statistical dispersion. Thus heteroscedasticity is the absence of homoscedasticity. A typical example is the set of observations of income in different cities.

The general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as

In statistics and in particular in regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables of a set of objects. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object. The design matrix is used in certain statistical models, e.g., the general linear model. It can contain indicator variables that indicate group membership in an ANOVA, or it can contain values of continuous variables.

Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification.

Multilevel models are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models, although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.

Omnibus tests are a kind of statistical test. They test whether the explained variance in a set of data is significantly greater than the unexplained variance, overall. One example is the F-test in the analysis of variance. There can be legitimate significant effects within a model even if the omnibus test is not significant. For instance, in a model with two independent variables, if only one variable exerts a significant effect on the dependent variable and the other does not, then the omnibus test may be non-significant. This fact does not affect the conclusions that may be drawn from the one significant variable. In order to test effects within an omnibus test, researchers often use contrasts.

The sample mean and the sample covariance are statistics computed from a sample of data on one or more random variables.

In statistics, the intraclass correlation, or the intraclass correlation coefficient (ICC), is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly units in the same group resemble each other. While it is viewed as a type of correlation, unlike most other correlation measures it operates on data structured as groups, rather than data structured as paired observations.

In statistics, one-way analysis of variance is a technique that can be used to compare whether two sample's means are significantly different or not. This technique can be used only for numerical response data, the "Y", usually one variable, and numerical or (usually) categorical input data, the "X", always one variable, hence "one-way".

Multivariate analysis of covariance (MANCOVA) is an extension of analysis of covariance (ANCOVA) methods to cover cases where there is more than one dependent variable and where the control of concomitant continuous independent variables – covariates – is required. The most prominent benefit of the MANCOVA design over the simple MANOVA is the 'factoring out' of noise or error that has been introduced by the covariant. A commonly used multivariate version of the ANOVA F-statistic is Wilks' Lambda (Λ), which represents the ratio between the error variance and the effect variance.

In statistics, a mixed-design analysis of variance model, also known as a split-plot ANOVA, is used to test for differences between two or more independent groups whilst subjecting participants to repeated measures. Thus, in a mixed-design ANOVA model, one factor is a between-subjects variable and the other is a within-subjects variable. Thus, overall, the model is a type of mixed-effects model.

In statistics and regression analysis, moderation occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the moderator variable or simply the moderator. The effect of a moderating variable is characterized statistically as an interaction; that is, a categorical or quantitative variable that affects the direction and/or strength of the relation between dependent and independent variables. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables, or the value of the slope of the dependent variable on the independent variable. In analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation.

In statistics, Tukey's test of additivity, named for John Tukey, is an approach used in two-way ANOVA to assess whether the factor variables are additively related to the expected value of the response variable. It can be applied when there are no replicated values in the data set, a situation in which it is impossible to directly estimate a fully general non-additive regression structure and still have information left to estimate the error variance. The test statistic proposed by Tukey has one degree of freedom under the null hypothesis, hence this is often called "Tukey's one-degree-of-freedom test."

Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.

In statistics, one purpose for the analysis of variance (ANOVA) is to analyze differences in means between groups. The test statistic, F, assumes independence of observations, homogeneous variances, and population normality. ANOVA on ranks is a statistic designed for situations when the normality assumption has been violated.

In statistics, the two-way analysis of variance (ANOVA) is an extension of the one-way ANOVA that examines the influence of two different categorical independent variables on one continuous dependent variable. The two-way ANOVA not only aims at assessing the main effect of each independent variable but also if there is any interaction between them.

In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

## References

1. Keppel, G. (1991). Design and analysis: A researcher's handbook (3rd ed.). Englewood Cliffs: Prentice-Hall, Inc.
2. Montgomery, Douglas C. "Design and analysis of experiments" (8th Ed.). John Wiley & Sons, 2012.
3. Tabachnick, B. G.; Fidell, L. S. (2007). Using Multivariate Statistics (5th ed.). Boston: Pearson Education.
4. Miller, G. A.; Chapman, J. P. (2001). "Misunderstanding Analysis of Covariance". Journal of Abnormal Psychology. 110 (1): 40–48. doi:10.1037/0021-843X.110.1.40. PMID   11261398.
5. Green, S. B., & Salkind, N. J. (2011). Using SPSS for Windows and Macintosh: Analyzing and Understanding Data (6th ed.). Upper Saddle River, NJ: Prentice Hall.
6. Howell, D. C. (2009) Statistical methods for psychology (7th ed.). Belmont: Cengage Wadsworth.