Medical Statistics
A practical guide
Analysis of Variance and Covariance: A Gentle Introduction | By: Michael Bramley, Senior Biostatistician, Kendle and Richann Watson, Biostatistician, Kendle
Michael Bramley Richann Watson
The challenge in any clinical trial is always how to document the effects of a given medication, treatment, placebo, etc. Once documented, data must be summarized in such a way as to convey the true results of a study. Two of the most common methods for demonstrating research results are the analysis of variance and the analysis of covariance. An overview of these basic statistical analysis tools follows.
| Analysis of Variance
Analysis of Variance (ANOVA) is a
statistical method to examine and test for differences in a dependent variable’s
FIGURE 1: Scatter Plot of Heart Rate for Three Cardiac Medications
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(response) mean by an independent classification variable (factor). The response represents a variable of interest and the factor represents some criterion that is used to divide this response into groups. A single classification may be called a treatment group. Consider an experiment where the effects of three cardiac medications are evaluated on the heart rates of 15 subjects, with each subject receiving only one medication. In the ANOVA setting, heart rate is the response and cardiac medication is the factor. Figure 1 displays the subjects’ heart rates after treatment, with a dashed horizontal line indicating the response mean for each treatment. The vertical distance from a subject’s response to their mean helps to gauge the extent of the variability. Figure 1 reveals there may be differences
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Heart Rate
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50 Treatment 1 Treatment 2 Treatment 3
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A PRACTICAL GUIDE | Analysis of Variance and Covariance: A Gentle Introduction
between the cardiac medications, but it is uncertain if these differences are real. Answering these questions would require multiple t-tests, but then how are the results interpreted?
| Interpretation of ANOVA Results:
The prime focus of an ANOVA is the ratio of the mean squares. If the ratio of the two mean squares exceeds a prespecified critical value, it can be concluded that not all of the treatment means are equal. However, this does not indicate which treatment means differ, nor what the nature of those differences are. For reporting purposes, the mean value
of the response for a specific treatment equals the response’s overall mean plus the effect for that treatment. Moreover, this value remains the same for all subjects receiving this treatment. In the example, ANOVA reveals that the heart rate means of 62, 75, and 70 (bpm) for treatments 1, 2, and 3, respectively, are not significantly different at the 0.05 level, as the p-value for the ratio is 0.124.
| Why ANOVA?
ANOVA is the tool of choice when an investigator wants to test the response means of two or more treatments for differences. ANOVA organizes the analysis and makes interpretation of the results easier, especially as the number of treatment groups increases. This method also can be used to investigate differences between each of these treatments; for example, to compare each treatment against a standard. ANOVA works by partitioning the response’s total variance into two components: that explained by the factor and that explained by random error, referred to as the mean square treatment and mean square error, respectively.
ANOVA is the tool of choice when an investigator wants to test the response means of two or more treatments for differences.
| Assumptions:
Three assumptions are made about experimental data when using ANOVA: independence, equal variance and normal distribution. • Independence assumes that any one subject’s data is independent from any other subject’s data. This means that each subject is randomly assigned to one and only one treatment, and they are represented in the measurements similarly. • Equal variance and normal distribution assume that measurement errors are normally distributed with a mean of zero and a constant variance, regardless of the factor. These latter two assumptions depend on how the experiment is carried out and may be tested once the data are collected.
FIGURE 2: Reduced Variability with Covariates in ANCOVA
90 Treatment 2 Treatment 3 80 Post-Treatment Heart Rate
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50 40 60 80 Pre-Treatment Heart Rate 100 120
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MEDICAL STATISTICS: A PRACTICAL GUIDE | Analysis of Variance and Covariance: A Gentle Introduction
| Analysis of Covariance:
Analysis of Covariance (ANCOVA) is the natural extension to ANOVA. The idea is to augment ANOVA by including a quantitative variable (covariate) related to the response. The intent is to reduce the variability of the measurement error with the result being an increase in the precision of the analysis. Suppose that in the example each subject’s heart rate was measured prior
is a linear relationship between the covariate and the factor, it is not essential. Another relationship could be employed, e.g. quadratic. However, linearity does lead to a simpler analysis and is often a reasonable approximation. |
for treatments 1, 2, and 3, respectively. In this case, the treatments are significantly different at the 0.05 level as the p-value of the ratio is <0.001.
Summary
ANOVA and ANCOVA are two important methods of analysis that can be extended to meet many experimental needs and provide summary results and insight regarding research data. Each method makes specific assumptions about the data and
| Interpretation of ANCOVA Results:
In ANOVA it is assumed, all subjects receiving the same treatment have the same mean response. This is not so in ANCOVA, as a subject’s mean response
to receiving medication and that this value will be used as a covariate in an ANCOVA. Figure 2 displays the pretreatment versus post-treatment heart rates for each subject, with three treatment regression lines. By including a covariate, the scatter around the treatment regression lines is greatly reduced, relative to Figure 1.This makes the study more powerful for comparing treatment effects.
depends on the treatment and the covariate. Thus, the expected response for a treatment in ANCOVA is reported as a regression line with an intercept of the treatment mean plus an estimate of the slope of the line associated with the covariate. Given that a treatment difference exists, the experimenter can estimate the difference between treatments with the vertical distance between the regression lines for any value of the covariate. This is where the importance of equality of slopes comes in: if the lines are not parallel, then this distance is not constant and ANCOVA is an inappropriate method of analysis. In the example, ANCOVA reveals that the heart rate regression lines have a common slope of 0.3 for the covariate and intercepts of 37.2, 50.8, and 44.8
thus produces different types of summaries. Deciding which method to use is dictated by the study design and type of data available. NOTE: Results may vary. Please consult a statistician for your individual needs.
THE AUTHORS Michael Bramley, Senior Biostatistician, Kendle Richann Watson, Biostatistician, Kendle Bramley and Watson have approximately two decades of biostatistics and SAS programming between them.
| Assumptions:
ANCOVA adds to the assumptions of ANOVA by assuming equality of slopes in the different treatment regression lines. Equality of slopes is important, as it allows an investigator to test for differences between treatments with a single value. This assumption should be checked both visually and statistically. While the above example assumes there
Editor’s Note: This is the third entry in a series of nine articles on basic statistics and clinical research. Up next: Testing for Equivalence and Non-inferiority
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