[Wiley Series in Probability and Statistics] The Analysis of Covariance and Alternatives (Statistical Methods for Experiments, Quasi-Experiments, and Single-Case Studies) || Nonlinear

P1: TIX/OSW P2: ABCJWBS074-c12 JWBS074-Huitema July 30, 2011 8:31 Printer Name: Yet to Come

CHAPTER 12

Nonlinear ANCOVA

12.1 INTRODUCTION

The relationship between the covariate and the dependent variable scores is notalways linear. Because an assumption underlying the ANCOVA model is that thewithin-group relationship between X and Y is linear, researchers should be aware ofthe problem of nonlinearity. If ANCOVA is employed when the data are nonlinear, thepower of the F-test is decreased and the adjusted means may be poor representationsof the treatment effects.

Two reasons for nonlinear relationships between X and Y are inherent nonlinearityof characteristics and scaling error. It is quite possible that the basic characteristicsbeing measured are not linearly related. For example, the relationship between extro-version (X) and industrial sales performance (Y) could be predicted to be nonlinear.Those salespeople with very low extroversion scores may have poor sales perfor-mance because they have difficulty interacting with clients. Those with very highextroversion scores may be viewed as overly social and not serious about their work.Hence, very low or very high extroversion scores may be associated with low salesperformance, whereas intermediate extroversion scores may be associated with highsales performance.

Another example of expected nonlinearity might be found between certain mea-sures of motivation (X) and performance (Y). Psychologists working in the area ofmotivation sometimes hypothesize that there is an optimal level of motivation orarousal for an individual working on a specific task. At very low or very high levelsof arousal, performance is lower than at the optimal level of arousal. In both exam-ples, the relationship between X and Y scores is expected to be nonlinear becausethe relationship between the basic characteristic underlying the observed (measured)scores is expected to be nonlinear. This distinction between the measured and under-lying or basic scores is important. It is quite possible that the relationship betweenobserved X and Y scores is nonlinear when the relationship between the basic X and Ycharacteristics is linear. When this occurs, the problems of scaling error are involved.

The Analysis of Covariance and Alternatives: Statistical Methods for Experiments, Quasi-Experiments,and Single-Case Studies, Second Edition. Bradley E. Huitema.© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

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286 NONLINEAR ANCOVA

There are several types of scaling errors that can produce nonlinearity, but probablythe most frequently encountered type results in either “ceiling” or “floor” effects. Ineither case the problem is that the instrumentation or scale used in the measurement ofeither the X or the Y variable (or both) may not be adequate to reflect real differencesin the characteristics being measured. For example, if most of the subjects employedin a study obtain nearly the highest possible score on a measure, there are likely to beunmeasured differences among those who get the same high score. The measurementprocedure simply does not have sufficient “ceiling” to reflect differences amongthe subjects on the characteristics being measured. Suppose most subjects get ascore of 50 on a 50-point pretest that is employed as a covariate; the test is muchtoo easy for the subjects included in the experiment. If the scores on this measureare plotted against scores on a posttest that is of the appropriate difficulty level,nonlinearity will be observable. Here the inherent relationship between the X and Ycharacteristics is linear, but the obtained relationship between the observed measuresis not linear. Hence, one reason for nonlinearity in the XY relationship is scaling erroror inappropriate measurement. Regardless of the reason for nonlinearity, the linearANCOVA model is inappropriate if the degree of nonlinearity is severe.

12.2 DEALING WITH NONLINEARITY

A routine aspect of any data analysis is to plot the data. This preliminary step involvesplotting the Y scores against the X scores for each group. Severe nonlinearity willgenerally be obvious in both the trend observed in the scatter plot and in the shapeof the marginal distributions. More sensitive approaches for identifying nonlinearityinclude visual inspection of the residuals of the ANCOVA model and fitting variousalternative models to the data. Once it has been decided that nonlinearity is problem-atic, the next step is to either (1) seek a transformation of the original X and/or Yscores that will result in a linear relationship for the transformed data or to (2) fit anappropriate polynomial ANCOVA model to the original data.

Data Transformations

If the relationship between X and Y is nonlinear but monotonic (i.e., Y increases whenX increases but the function is not linear), a transformation of X should be attempted.Logarithmic, square root, and reciprocal transformations are most commonly usedbecause they usually yield the desired linearity. Advanced treatments of regressionanalysis should be consulted for details on these and other types of transformation(e.g., Cohen et al., 2003).

Once a transformation has been selected, ANCOVA is carried out in the usual wayon the transformed data. For example, if there is reason to believe that the relationshipbetween loge X and Y is linear, ANCOVA is carried out using loge X as the covariate.It must be pointed out in the interpretation of the analysis, however, that loge X ratherthan X was the covariate.


DEALING WITH NONLINEARITY 287

A method of determining whether a transformation has improved the fit of themodel to the data is to plot the scores and compute ANCOVA for both untransformedand transformed data. A comparison of the plots and ANCOVAs will reveal the effectof the transformation.

Polynomial ANCOVA Models

If the relationship between X and Y is not monotonic, a simple transformation willnot result in linearity. In the nonlinear-monotonic situation, the values of Y increaseas value of X increases. In the nonlinear-nonmonotonic situation, Y increases as Xincreases only up to a point, and then Y decreases as X increases. If we transform X tologe X for the nonmonotonic situation, the loge X values increase as X increases andnonlinearity is still present when loge X and Y are plotted. The simplest alternativein this case is to fit a second-degree polynomial (quadratic) ANCOVA model. Thismodel is written as

Yij = μ + α j + β1(Xij − X ..

) + β2

(X2

ij −(

X2..))

+ εij,

where

Yij is the dependent variable score of ith individual in jth group;μ is thepopulation mean on Y;αj is the effect of treatment j;β1 is the linear effect regression coefficient;Xij is the covariate score for ith individual in jth group;X .. is the mean of all observations on covariate;β2 is the curvature effect coefficient;X2

ij is the squared covariate score for ith individual in jth group;(X2..

)is the mean of squared observations on covariate (i.e.,

∑ Ni=1 X2

ij/N ); and

εij is the error component associated with ith individual in jth group.

This model differs from the linear model in that it contains the curvature effectterm β2(X2

ij − (X2..)). If the dependent variable scores are a quadratic rather than alinear function of the covariate, this model will provide a better fit and will generallyyield greater power with respect to tests on adjusted means.

The quadratic ANCOVA is computed by using X and X2 as if they were twocovariates in a multiple covariate analysis. The main ANCOVA test, the homogeneityof regression test, the computation of adjusted means, and multiple comparison testsare all carried out as with an ordinary two-covariate ANCOVA. If the relationship be-tween X and Y is more complex than a quadratic function, a higher degree polynomialmay be useful. The third-degree polynomial (cubic) ANCOVA model is written as

Yij = μ + α j + β1(Xij − X ..

) + β2

(X2

ij −(

X2..))

+ β3

(X3

ij −(

X3..))

+ εij.



This model will provide a good fit if the relationship between the covariateand the dependent variable is a cubic function. Cubic ANCOVA is carried out byemploying X, X2, and X3 as covariates in a multiple covariance analysis. Higherdegree polynomials can be employed for more complex functions, but it is veryunusual to encounter such situations.

Higher degree polynomial models virtually always fit sample data better than dosimpler polynomial models, but this does not mean that the more complex modelsare preferable to the simpler ones. Care must be taken not to employ a more complexmodel than is required; there are essentially two reasons to keep the model as simpleas possible. First, a degree of freedom is lost from the ANCOVA error mean square(i.e., MSResw ) for each additional term in the ANCOVA model. If the number ofsubjects is not large, the loss of degrees of freedom can easily offset the sum-of-squares advantage of a better fit afforded by the more complex model. Even thoughthe sum-of-squares residual is smaller with more complex models, the mean-squareerror can be considerably larger with complex models. The consequences of thelarger error term are less precise estimates of the adjusted means, and, correspond-ingly, less precise tests on the difference between adjusted means. This problem isillustrated in Section 12.3. The second reason for not employing a more complexmodel than is required is the law of parsimony. If a linear model fits the data almostas well as a quadratic model, the simpler model should usually be chosen becausethe interpretation and generalization of results is more straightforward.

Two additional points on the use of polynomial regression models are relevantto the polynomial ANCOVA described here. First, it is not necessary that the co-variate be a fixed variable. This point was made earlier in the discussion of as-sumptions for ANCOVA but is reiterated here for nonlinear ANCOVA because, asCramer and Appelbaum (1978) observed, it is sometimes mistakenly believed thatpolynomial regression is appropriate only with X fixed. Second, the parameters ofthe polynomial regression are sometimes difficult to estimate with certain multipleregression computer programs because these programs will not, with certain datasets, yield the inverse of the required matrix. This problem develops because X,X2, X3, and so on are all highly correlated. These computational difficulties cangenerally be reduced by transforming the raw X scores to deviation scores (i.e.,centered scores) before the regression analysis is carried out. That is, in quadraticANCOVA, for example, (X − X ) and (X − X )2 rather than X and X2 should beused as the covariates. Additional details on this problem in the context of con-ventional regression analysis can be found in Bradley and Srivastava (1979) andBudescu (1980).

12.3 COMPUTATION AND EXAMPLE OF FITTINGPOLYNOMIAL MODELS

The computation and rationale for quadratic ANCOVA are essentially the same asfor multiple ANCOVA. Consider the following data:


COMPUTATION AND EXAMPLE OF FITTING POLYNOMIAL MODELS 289

(1) (2)

Experimental Group Control Group

X Y X Y

13 18 11 137 14 2 1

17 7 19 214 14 15 9

3 8 8 1012 19 11 15

Previous research or theoretical considerations may suggest that the relationshipbetween X and Y is best described as a quadratic function. A scatter plot of thesedata appears to support a quadratic model. Hence, the experimenter has a reasonablebasis for deciding to employ the quadratic ANCOVA model. The computation ofthe complete quadratic ANCOVA through the general linear regression procedure isbased on the following variables:

(1) (2) (3) (4) (5) (6)

D X X2 DX DX2 Y

1 13 169 13 169 181 7 49 7 49 141 17 289 17 289 71 14 196 14 196 141 3 9 3 9 81 12 144 12 144 190 11 121 0 0 130 2 4 0 0 10 19 361 0 0 20 15 225 0 0 90 8 64 0 0 100 11 121 0 0 15

We now proceed as if we were performing a multiple covariance analysis usingX and X2 as the covariates. As before, the main test on adjusted treatment effects isbased on the coefficients of multiple determination R2

y X and R2y D,X .

The term R2y X represents the proportion of the total variability explained by the

quadratic regression (i.e., the regression of Y on X and X2), whereas R2y D,X represents

the proportion of the total variability explained by the quadratic regression andthe treatments. Hence, the difference between the two coefficients represents theproportion of the variability accounted for by the treatments that is independent ofthat accounted for by quadratic regression. The proportion of unexplained variabilityis, of course, 1 − R2

y D,X .



Column 1 in this example is the only dummy variable (because there are only J − 1dummy variables), columns 2 and 3 are the covariate columns, columns 4 and 5 arethe interaction columns (not used in the main analysis), and column 6 contains thedependent variable scores. The regression analyses yield the following:

R2y D,X = R2

y123 = 0.918903 and

R2y X = R2

y23 = 0.799091.

Difference or unique contribution of dummy variable beyond quadratic regres-sion = 0.119812.

Total sum of squares = 361.67. The general form of the quadratic ANCOVAsummary is as follows:

Source SS df MS F

Adjusted treatment(R2

y D,X − R2y X

)SST J − 1 SSAT/(J − 1) MSAT/MSResw

Quadratic residualw

(1 − R2

y D,X

)SST N − J − 2 SSResw /(N − J − 2)

Quadratic residualt

(1 − R2

y X

)SST N − 1 − 2

The quadratic ANCOVA summary for the example data is as follows:

Source SS df MS F

Adjusted treatment (0.119812)361.67 = 43.33 1 43.33 11.82 (p = .009)

Quadratic residualw (1 − 0.918903)361.67 = 29.33 8 3.67

Quadratic residualt (1 − 0.799091)361.67 = 72.66 9

Adjusted means and multiple comparison procedures are also dealt with as theyare under the multiple ANCOVA model. The adjusted means for the example dataare obtained through the regression equation associated with R2

y123. The intercept and

regression weights are

b0 = −5.847359

b1 = 3.83111

b2 = 3.66943

b3 = −0.17533

The group 1 dummy score, the grand mean covariate score, and the grand meanof the squared covariate scores are 1, 11, and 146, respectively. Hence, Y1 adj =−5.847359 + 3.83111(1) + 3.66943(11) − 0.17533(146) = 12.75. The group 2dummy score, the grand mean covariate score, and the grand mean of the squaredcovariate scores are 0, 11, and 146, respectively. Hence, Y2 adj = −5.847359 +3.83111(0) + 3.66943(11) − 0.175333(146) = 8.92.



Just as the test of the homogeneity of regression planes is an important adjunct tothe main F test in multiple ANCOVA, the test of the homogeneity of the quadraticregressions for the separate groups should be carried out in quadratic ANCOVA. Thistest is computed in the same manner as the test of the homogeneity of regressionplanes.

The form of the summary is as follows:

Source SS df MS F

Heterogeneity of quadraticregression

(R2

y D,X,DX − R2y D,X

)SST 2(J − 1) MShet

MShet

MSResi

Quadratic residuali

(1 − R2

y D,X,DX

)SST N − (J − 3) MSResi

Quadratic residualw

(1 − R2

y D,X

)SST N − J − 2

A more general form, appropriate for testing the homogeneity of any degree(denoted as C) polynomial regression, is as follows:

Source SS df MS F

Heterogeneity of polynomialregression

(R2

y D,X,DX − R2y D,X

)SST C(J − 1) MShet

MShet

MSResi

Polynomial residuali

(1 − R2

y D,X,DX

)SST N − J(C + 1) MSResi

Polynomial residualw

(1 − R2

y D,X

)SST N − J − C

For the example data, the necessary quantities are

R2y D,X,DX = R2

y12345 = 0.944817 and

R2y D,X = R2

y123 = 0.918903.

Difference or heterogeneity of regression = 0.025914.Total sum of squares = 361.67.

The summary is as follows:

Source SS df MS F

Heterogeneity of polynomialregression

(0.025914)361.67 = 9.37 2 4.68 1.41 (p = .32)

Polynomial residuali (1 – 0.944817)361.67 = 19.96 6 3.33

Polynomial residualw (1 – 0.918903)361.67 = 29.33 8

The obtained F-value is clearly not significant; we conclude that there is littleevidence to argue that the population quadratic regressions for the experimental



and control groups are different. The quadratic ANCOVA model is accepted as areasonable representation of the data.

Comparison of Quadratic ANCOVA with Other Models

It was mentioned earlier that the complexity of the model employed should besufficient to adequately describe the data but that it should not be more complex thanis required. The results of applying four different models to the data of the exampleproblem are tabulated as follows:

Model Obtained F Degrees of Freedom p-value

ANOVA 2.62 1,10 .137Linear ANCOVA 2.38 1,9 .157Quadratic ANCOVA 11.82 1,8 .009Cubic ANCOVA 9.96 1,7 .016

The F of the simplest model, ANOVA, when compared with the linear ANCOVAF, illustrates the fact that ANOVA can be more powerful than ANCOVA when thecorrelation between the covariate and the dependent variable is low. The F of themost complex of the four models, cubic ANCOVA, when compared with the quadraticF, illustrates the fact that more complex models do not necessarily lead to greaterprecision. The greatest precision is obtained with the model that is neither too simplenor more complex than is necessary for an adequate fit.

Minitab Input and Output

Input for estimating the linear ANCOVA model:

MTB > ancova Y=d;SUBC> covariate X;SUBC> means d;SUBC> residuals c7.

Output for linear ANCOVA:

ANCOVA: Y versus dFactor Levels Valuesd 2 0, 1

Analysis of Covariance for YSource DF Adj SS MS F PCovariates 1 3.41 3.41 0.11 0.749d 1 75.00 75.00 2.38 0.157Error 9 283.25 31.47Total 11 361.67



S = 5.61004 R-Sq = 21.68% R-Sq(adj) = 4.28%

Covariate Coef SE Coef T PX 0.1067 0.324 0.3293 0.749

Adjusted Meansd N Y0 6 8.3331 6 13.333

MTB > Plot 'ANCOVA Residuals'*'X';SUBC> Symbol 'd'.

Scatterplot of ANCOVA Residuals vs X

Scatterplot of ANCOVA residuals vs X

AN

CO

VA

res

idu

als

8

6

4

2

–2

–6

–80 5 10 15 20

X

10d

0

–4

It is obvious from inspecting the plot of the residuals of the linear ANCOVA modelshown above that this model is inappropriate. A quadratic model appears to be a goodcontender so it is estimated next.

Input to compute quadratic ANCOVA. The variable d is a (1, 0) dummy variableindicating group membership, c2 = the covariate X, and c3 = X2.

MTB > ancova Y=d;SUBC> covariates c2 c3;SUBC> means d;SUBC> residuals c8.





S = 1.91472 R-Sq = 91.89% R-Sq(adj) = 88.85%

Covariate Coef SE Coef T PX 3.6694 0.4421 8.299 0.000X*X -0.1753 0.0211 -8.322 0.000


MTB > Plot 'Quad ANCOVA Residuals'*'X';SUBC> Symbol 'd'.

Scatterplot of Quad ANCOVA Residuals vs X

Scatterplot of quad ANCOVA residuals vs X

Qu

ad A

NC

OVA

res

idu

als

2

1

0

–1

–2

–30 5 10 15 20

X

10d


SUMMARY 295

Note that the residuals of the quadratic ANCOVA model indicate no additionalforms of nonlinearity or other departures from assumptions. This is confirmed byestimating the cubic ANCOVA model. Note in the output that the p-value on thecubic coefficient is .77.

Input for estimating the cubic ANCOVA model:

MTB > Let c9 = X*X*XMTB > ancova Y=d;SUBC> covariates c2 c3 c9;SUBC> means d;SUBC> residuals c10.

Output for cubic ANCOVA model:



S = 2.03354 R-Sq = 92.00% R-Sq(adj) = 87.42%

Covariate Coef SE Coef T PX 3.2073 1.5909 2.0161 0.084X*X -0.1231 0.1733 -0.7104 0.500X*X*X -0.0016 0.0054 -0.3040 0.770


12.4 SUMMARY

The assumption of the conventional ANCOVA model that the covariate and the de-pendent variable are linearly related will not always be met. Severe nonlinearitygenerally can be easily identified by inspecting the XY scatter plot within groups.If the relationship is nonlinear but monotonic, it is likely that a simple transforma-tion (generally of the X variable) can be found that will yield a linear relationship



between transformed X and Y . Analysis of covariance is then applied by using thetransformed variable as the covariate. If the relationship is not monotonic, the simpletransformation approach will not be satisfactory, and the more complex approach ofemploying some polynomial of X should be attempted. Generally, a quadratic or cubicANCOVA model will fit the data. Complex polynomial models should be employedonly if simpler ones are obviously inadequate. Simpler models are preferred becauseresults based on complex models are (1) more difficult to interpret and generalizeand (2) less stable. When polynomial ANCOVA models are clearly called for, thecomputation involves a straightforward extension of multiple ANCOVA.

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[Wiley Series in Probability and Statistics] The Analysis of Covariance and Alternatives (Statistical Methods for Experiments, Quasi-Experiments, and Single-Case Studies) || Nonlinear