If the X or Y populations from which data to be analyzed by analysis of covariance (ANCOVA) were sampled violate one or more of the ANCOVA assumptions, the results of the analysis may be incorrect or misleading. For example, if the assumption of independence is violated, then analysis of covariance is not appropriate. If the assumption of normality is violated, or outliers are present, then the analysis of covariance may not be the most informative analysis available, and this could mean the difference between finding a significant difference between the treatment (group) means or not. A transformation may result in a better fit.
If there are other explanatory variables that should be included in the analysis, then the one-way analysis of covariance may not provide the best model for the data. A different linear model, or a nonlinear model may provide a better fit.
Often, the impact of an assumption violation on the ANCOVA result depends on the extent of the violation (such as the how inconstant the residual variance is, or how skewed the Y population distribution is). Some small violations may have little practical effect on the analysis, while other violations may render the ANCOVA result uselessly incorrect or uninterpretable.
Potential assumption violations include:
- Implicit independent variables:Explanatory variables missing from the model
- Lack of independence in Y: lack of independence in the Y variable
- Outliers: apparent nonnormality by a few data points
- Nonnormality: nonnormality of the Y variable or residuals
- X is affected by treatments: there is a relationship between treatment group and the value of X
- Variance of Y is not constant
- Nonparallel treatment regression lines
- Correct model is not linear in X
- There is no linear relation between X and Y
- X variable is random, not fixed
- Patterns in plot of data: detecting violation assumptions graphically
- Special problems with few data points
Apparent lack of independence in the residuals may be caused by the existence of an implicit variable in the data, a continuous X variable or a grouping variable that was not explicitly used in the ANCOVA model. In this case, the best model may be more complicated than the one-way ANCOVA model. The best model may not include the original X variable. If there is a linear trend in the plot of the ANCOVA residuals against the fitted values, then an implicit X variable may be the cause. A plot of the residuals against the prospective new X variable should reveal whether there is a systematic variation; if there is, you may consider adding the new X variable to the ANCOVA model.
If an implicit X variable is not included in the fitted model, the fitted estimates for the individual and common slopes may be biased, and not very meaningful, and the fitted Y values may not be accurate.
Another possible cause of apparent dependence between the Y observations is the presence of another implicit block or treatment effect. (Such an effect can be considered another type of implicit X variable, albeit a discrete one.) If such a variable is suspected, a different model may provide a better fit.
If multiple values of Y are collected at the same values of X for a single group regression line, this can act as another type of blocking, with the unique values of X acting as blocks. These multiple Y measurements may be less variable than the overall variation in Y, and, given their common value of X, they are not truly independent of each other. If there are many replicated X values, and if the variation between Y at replicated values is much smaller than the overall residual variance, then the variance of the estimate of the slope may be too small. This may make a test comparing slopes anticonservative (more likely than the stated significance level to reject the null hypothesis, even when it is true). In this case, an alternative method is to replace each such replicated X value (within the same regression line) by a single data point with the average Y value, and then perform the ANCOVA analysis with the new data set. A possible drawback to this method is that by reducing the number of data points, the degrees of freedom associated with the residual error is reduced, thus potentially reducing the power of the test.
Whether the Y values are independent of each other is generally determined by the structure of the experiment from which they arise. Y values collected over time may be serially correlated (here time is the implicit factor). If the data are in a particular order, consider the possibility of dependence. (If the row order of the data reflect the order in which the data were collected, an index plot of the data [data value plotted against row number] can reveal patterns in the plot that could suggest possible time effects.) For serially correlated Y values, the estimates of the slope and intercept will be unbiased, but the estimates of their variances will not be reliable. If you are unsure whether your Y values are independent, you may wish to consult a statistician or someone who is knowledgeable about the data collection scheme you are using.
Values may not be identically distributed because of the presence of outliers. Outliers are anomalous values in the data. Outliers may have a strong influence over the fitted slope and intercept, giving a poor fit to the bulk of the data points. Outliers tend to increase the estimate of residual variance, lowering the chance of rejecting the null hypothesis. They may be due to recording errors, which may be correctable, or they may be due to the Y values not all being sampled from the same population. Apparent outliers may also be due to the Y values being from the same, but nonnormal, population. Outliers may show up clearly in a X-Y scatterplot of the data for one of the regression lines, as points that do not lie near the general linear trend of the data for that regression line. A point may be an unusual value in either X or Y without necessarily being an outlier in the scatterplot.
Once the analysis of covariance model has been fitted, the boxplot and normal probability plot (normal Q-Q plot) for residuals may suggest the presence of outliers in the data. After the fit, outliers are usually detected by examining the residuals.
The method of least squares used in fitting the analysis of covariance model involves minimizing the sum of the squared vertical distances between each data point and the fitted line. Because of this, fitted lines can be highly sensitive to outliers. (In other words, least squares fitting is not resistant to outliers, and thus, neither is a fitted slope estimate.) Outliers may affect the estimates for the individual slopes and intercepts for the regression lines, and could lead to an incorrect conclusion about whether the slopes are equal, or whether the intercepts are equal.
If you find outliers in your data that are not due to correctable errors, you may wish to consult a statistician as to how to proceed.
The Y values for an analysis of variance will not necessarily come from the same normal population, although they should all have the same variance. For this reason, it may be difficult to assess nonnormality of Y.
After the ANCOVA is performed, the residuals can be examined for signs of nonnormality. The residuals should all come from the same normal distribution, with mean 0 and variance the same as the variance of the Y.
It may be the case that the residuals are indeed from the same population, but not from a normal one. Signs of nonnormality are skewness (lack of symmetry) or light-tailedness or heavy-tailedness. The boxplot, histogram, and normal probability plot (normal Q-Q plot), along with the normality test, can provide information on the normality of the population distribution for the residuals. However, if there are only a small number of data points, nonnormality can be hard to detect. If there are a great many data points, the normality test may detect statistically significant but trivial departures from normality that will have no real effect on the analysis of covariance.
If the residuals come from a normal distribution, normal probability plots should approximate straight lines, and boxplots should be symmetric (median and mean together, in the middle of the box) with no outliers. If the number of data points is not too small, the ANCOVA should not be much affected by small departures from normality.
Just as the blocks and treatments should not interact in a one-way blocked ANOVA, the concomitant (X) variable in the analysis of covariance should not be affected by the treatments (levels of the grouping variable). If both X and Y depend on the group (treatment), then the analysis of covariance can be misleading.
A relationship between X and treatment can be detected informally by examining the X-Y scatterplot of the data before performing the analysis of covariance.
If the variance of the Y is not constant, then the the error variance will not be constant. The most common form of such heteroscedasticity in Y is that the variance of Y may increase as the mean of Y increases, for data with positive X and Y.
Heteroscedasticity of Y is usually detected informally by examining the X-Y scatterplot of the data before performing the ANCOVA. If both nonlinearity and unequal variances are present, employing a transformation of Y might have the effect of simultaneously improving the linearity and promoting equality of the variances.
The analysis of covariance requires that the treatment regression lines have the same slope. The test for equality of treatment (group) means is the test for equality of intercepts, and assumes that slopes are equal.
Inequality of slopes can be ascertained informally by examining the X-Y scatterplot of the data before performing the analysis of covariance. Otherwise, the test of equality of slopes provides a formal test of whether the assumption of parallel treatment regression lines has been violated.
If the linear model assumed by analysis of covariance is not the correct one for the data, then the slope estimates and the fitted values from the ANCOVA will be biased, and not very meaningful. Over a restricted range of X or Y, nonlinear models may be well approximated by linear models (this is in fact the basis of linear interpolation), but for accurate prediction a model appropriate to the data should be selected. An examination of the X-Y scatterplot may reveal whether the linear model is appropriate. If there is a great deal of variation in Y, it may be difficult to decide what the appropriate model is; in this case, the linear model may do as well as any other, and has the virtue of simplicity.
The purpose of using the X variable in the analysis of covariance is to use the information about X to reduce the variation in Y and thus increase the chance of detecting differences between the treatments. (This is similar to the purpose of using a blocking variable in an analysis of variance, except that a blocking variable is discrete instead of continuous.) Choosing an X variable that has no linear relation to Y is pointless: no reduction in variance will be achieved, and the power of the test will be reduced. However, the effect will not generally be serious unless the number of data points is small.
If there is no linear relation between X and Y, then the analysis of covariance offers no improvement over the one-way analysis of variance in detecting differences between the group means.
The lack of a linear relation between X and Y can be detected informally by examining the X-Y scatterplot of the data before performing the ANCOVA. Otherwise, the test of whether all the slopes are equal to 0 provides a formal test of whether there is a linear relation between X and Y. Since this test assumes that all the slopes are equal, it makes little sense if the test for equality of slopes indicates that the slopes are significantly different.
The analysis of covariance model assumes that the observed X variables are fixed, not random. If the X values are are not under the control of the experimenter (i.e., are observed but not set), they may not be fixed. If they have the same variance, the estimates of the slope and intercept may be biased.
If the assumption of parallel straight lines for the different treatment groups is correct, the plot of the observed Y values against X, using different symbols for each group, should suggest parallel linear bands across the graph with no obvious departures from linearity. Outliers may appear as anomalous points in the graph, often in the upper righthand or lower lefthand corner of the graph. (A point may be an outlier in either X or Y without necessarily being far from the general trend of the data.)
If there is no linear relation between X and Y, then the plot of Y vs X for each group will have 0 slope: The bands will all be parallel to the X axis.
If there is no relationship between X and treatment, then a plot of Y vs X for each individual treatment should look like the plot of Y vs X for all treatments combined, except for random variation. In particular, the range of X for each treatment group should be similar.
A plot of the X-Y data that uses a different symbol for each treatment group can help you detect differences in the distribution of Y along the X scale for different groups. If most of the X values for one treatment tend to be larger than the X values for another treatment, for example, then you should investigate the possibility that the value of X depends on the treatment group.
If the ANCOVA model is not correct, the shape of the general trend of the X-Y plot might suggest parallel nonlinear curves. In this case, the shape of the curves might suggest a function to use (e.g., a polynomial, exponential, or logistic function) in a different model. Alternatively, the plot might suggest a reasonable transformation to apply. For example, if the X-Y plot arcs from lower left to upper right so that data points either very low or very high in X lie below the straight line suggested by the data, while the data points with middling X values lie on or above that straight line, taking square roots or logarithms of the X values may promote linearity.
If the plot suggests that the different regression curves are neither parallel nor linear, then the analysis of covariance is not likely to be informative.
If the assumption of equal variances for the Y is correct, the plot of the observed Y values against X for each group should suggest a band across the graph with roughly equal vertical width for all values of X. (That is, the shape of the graph should suggest a tilted cigar and not a wedge or a megaphone.)
A fan pattern like the profile of a megaphone, with a noticeable flare either to the right or to the left as shown in the picture suggests that the variance in the values increases in the direction the fan pattern widens (usually as the sample mean increases), and this in turn suggests that a transformation of the Y values might be useful.
If the number of data points is small, it may be difficult to detect assumption violations. With small samples, violation assumptions such as nonnormality or heteroscedasticity of variances are difficult to detect even when they are present. With a small number of data points, analysis of covariance offers less protection against violation of assumptions. With few data points, it may be hard to determine how well the fitted ANCOVA model matches the data, or whether a different model would be more appropriate.
Even if none of the test assumptions are violated, an analysis of covariance on a small number of data points may not have sufficient power to detect a significant difference between the slope and 0, even if the slope is non-zero. The power depends on the residual error, the observed variation in X, the selected significance (alpha-) level of the test, and the number of data points. Power decreases as the residual variance increases, decreases as the significance level is decreased (i.e., as the test is made more stringent), increases as the variation in observed X increases, and increases as the number of data points increases. If a statistical significance test with a small number of data values produces a surprisingly non-significant P value, then lack of power may be the reason. The best time to avoid such problems is in the design stage of an experiment, when appropriate minimum sample sizes can be determined, perhaps in consultation with a statistician, before data collection begins.